Re: Supergravity Cosmological Billards and the BIG group

Urs Schreiber

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Thu, 15 Apr 2004, Urs Schreiber wrote:\n\n> The most natural idea is to try to quantize the sigma model\n> on E10/K(E10), or one of its consistent truncations. In\n> principle this is straightforward, since quantizing 1+0 dim\n> theories just involves quantum mechanics. But of course there\n> are some thorny details. In any case, it hasn\'t been done yet.\n\n\nActually I now see that steps in this direction have been\ndone at least in\n\nJ. Brown, O. Ganor, C. Helfgott,\nM-Theory on E10: Billiards, Branes, and Imaginary Roots,\nhep-th/0401053 .\n\nIn section 7 the authors describe how to iteratively\nconstruct the E10 Laplacian Delta (i.e. the "wave operator"\non the group G10). And they also discuss how certain\nsolutions of the conjectured "Wheeler-deWitt equation\nof M-theory"\n\nDelta |psi> = 0\n\nshould describe states containing brane/anti-brane pairs\nas a pure quantum effect of this framework.\n\nFurthermore they discuss a related "quantum test" of the\n\'Delta |psi> = 0\'-conjecture related to the masses of the\nbranes.\n\nOne big open question is: What about fermions?\n\nSo far all these constructions seem to be purely bosonic,\neven though on p.50 of the above paper the possibility\nis mentioned that the fermions might be part of the E10\nvariables (in bosonized form).\n\nBut apparently one rather expects that a sort of\nsupersymmetrization of the E10 1+0d sigma model is\nnecessary.\n\nI still think that it should be interesting to study the\ndeRahm operators d and del on E10. We know that every 1+0d\nsigma model admits a straightforward N=2 susy extension\nby replacing Delta by {d,del}. It seems to me that with\nthe technology presented in hep-th/0401053 it should be\neasy to construct\n\nd = d_h + d_0 + d_1 + d_2 + ...\n\nand similarly for del in complete analogy to what is\ndone in section 7.2 of the above paper for\n\nDelta = Delta_h + Delta_0 + Delta_1 + ... .\n\nHopefully I\'ll find the time to look into that\nquestion. One should check if the fermions (differential\nforms) introduced this way match the dynamics of\nmodes of the 11d sugra fermions, in analoggy to the bosonic\nsector.\n\nOne important point at least seems to be immediate for\nthis kind of supersymmetrization of the E10 sigma model:\n\nAt the end of section 7.4 on p.53 of the above paper it\nis mentioned that in the bosnic theory there is a problem\nwith zero-point energies which should better vanish. Of\ncourse these do vanish automatically for the deRahm\nmodel for harmonic forms of the respective d_i.\n\n\nI see that Lubos is acknowledged by the above authors\nfor helpful discussion. Maybe he can comment on\nthe above issues.~\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Thu, 15 Apr 2004, Urs Schreiber wrote:

> The most natural idea is to try to quantize the [itex]\sigma[/itex] model
> on [itex]E10/K(E10),[/itex] or one of its consistent truncations. In
> principle this is straightforward, since quantizing 1+0 dim
> theories just involves quantum mechanics. But of course there
> are some thorny details. In any case, it hasn't been done yet.

Actually I now see that steps in this direction have been
done at least in

J. Brown, O. Ganor, C. Helfgott,
M-Theory on E10: Billiards, Branes, and Imaginary Roots,
hep-th/0401053 .

In section 7 the authors describe how to iteratively
construct the E10 Laplacian [itex]\Delta (i.e.[/itex] the "wave operator"
on the group G10). And they also discuss how certain
solutions of the conjectured "Wheeler-deWitt equation
of M-theory"

[tex]\Delta |\psi> =[/tex]

should describe states containing brane/anti-brane pairs
as a pure quantum effect of this framework.

Furthermore they discuss a related "quantum test" of the
[itex]'\Delta |\psi> =[/itex] 0'-conjecture related to the masses of the
branes.

One big open question is: What about fermions?

So far all these constructions seem to be purely bosonic,
even though on p.50 of the above paper the possibility
is mentioned that the fermions might be part of the E10
variables (in bosonized form).

But apparently one rather expects that a sort of
supersymmetrization of the E10 [itex]1+0d \sigma[/itex] model is
necessary.

I still think that it should be interesting to study the
deRahm operators d and del on E10. We know that every [itex]1+0d[/itex]
[itex]\sigma[/itex] model admits a straightforward N=2 susy extension
by replacing [itex]\Delta[/itex] by {d,del}. It seems to me that with
the technology presented in hep-th/0401053 it should be
easy to construct

[itex]d = d_h + d_0 + d_1 + d_2 + .[/itex]..

and similarly for del in complete analogy to what is
done in section 7.2 of the above paper for

[itex]\Delta = \Delta_h + \Delta_0 + \Delta_1 + .[/itex].. .

Hopefully I'll find the time to look into that
question. One should check if the fermions (differential
forms) introduced this way match the dynamics of
modes of the 11d sugra fermions, in analoggy to the bosonic
sector.

One important point at least seems to be immediate for
this kind of supersymmetrization of the E10 [itex]\sigma[/itex] model:

At the end of section 7.4 on p.53 of the above paper it
is mentioned that in the bosnic theory there is a problem
with zero-point energies which should better vanish. Of
course these do vanish automatically for the deRahm
model for harmonic forms of the respective [itex]d_i[/itex].

I see that Lubos is acknowledged by the above authors
for helpful discussion. Maybe he can comment on
the above issues.~

lumidek

Dear Urs,

I've been intrigued by Ori Ganor's proposals from the beginning. Several years ago, he conjectured that the whole M-theory is encoded in the Laplace equation defined on the [itex]E_{10}[/itex] group manifold. Partition sums of gravities, worldvolume theories and perhaps all other things should be encoded in this allegedly unique function that satisfies this Laplace equation on this huge manifold, see

http://arxiv.org/abs/hep-th/9903110
http://arxiv.org/abs/hep-th/9910236

This would have the advantage of making all U-dualities etc. manifest. I think that the new paper by Ori et al. continues in this direction, and they would also like to find a one-to-one map between the basis vectors of the full M-theoretical Hilbert space on one side and the roots of [itex]E_{10}[/itex] on the other side, or something along these lines.

We should perhaps ask Ori to write something interesting about it on SPS.

Another purpose of this posting is to test LaTeX posted from physicsforums.com. Click the link below to see the math version of this post, including the nonsensical equation

[tex]
E=mc^2 + \frac{\sqrt{1-x^2}}{\log(3)} + \int {\mathrm{d}} x\, \bar f(x) g(x) + \dots
[/tex]

All the best
Lubos

Urs Schreiber

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Fri, 16 Apr 2004, lumidek wrote:\n\n> I\'ve been intrigued by Ori Ganor\'s proposals from the beginning.\n\nGreat to hear! :-)\n\nBTW, do you know if the approach by Ganor et al. started independently\nof the cosmological Billiard approach by Damour Henneaux and Nicolai\nand later merged, or what is the historical development?\n\n> Several years ago, he conjectured that the whole M-theory is encoded in\n> the Laplace equation defined on the E_{10} group manifold.\n\nYes. That would be remarkable, if true. It is entertainingly\nreminiscent of this famous saying, apparently due to Feynman,\nthat most of physics is about "Laplace phi = 0" in some\ngeneralized sense.\n\n> We should perhaps ask Ori to write something interesting about it on\n> SPS.\n\nAre you going to contact him or shall I?\n\nBy the way, while talking to Hermann Nicolai about these things\nin Golm I learned about the interesting fact that E10 can be\nregarded as sitting inside the vertex operator algebra of the\nstring. Of course that\'s long and well known (e.g. hep-th/9411188)\nbut I didn\'t fully appreciate before how already the symmetry\nalgebra of the single string knows about "M-theory" in this sense.\n\nRight now I am visiting Ioannis Giannakis at Rockefeller University\nand he also emphasizes this fact. As you know, I am here in order to\ndiscuss deformations of superconformal worldsheet theories\n(as in hep-th/0401175 and references given there) which in particular\ninclude the symmetry transformations of the background (gauge\ntransformations, dualities, etc.). Ten years ago Evans, Giannakis and\nNanopoulos in\n\nEvans, Giannakis & Nonopoulos,\nAn infinte dimensional symmetry algebra in string theory\nhep-th/9401075\n\nhave already discussed the immense symmetry algebra that is\nfound by deforming the worldsheet CFT by similarity transformations\nof the form\n\nA -> exp(-W) A exp(W)\n\nfor W a general integrated vertex. I am not completely sure\nwhat currently the state of the art is with respect to our\nunderstanding of this algebra, though. I am being told that\nphysicists are currently better prepared to discuss E10\nthan mathematicians are.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Fri, 16 Apr 2004, lumidek wrote:

> I've been intrigued by Ori Ganor's proposals from the beginning.

Great to hear! :-)

BTW, do you know if the approach by Ganor et al. started independently
of the cosmological Billiard approach by Damour Henneaux and Nicolai
and later merged, or what is the historical development?

> Several years ago, he conjectured that the whole M-theory is encoded in
> the Laplace equation defined on the [itex]E_{10}[/itex] group manifold.

Yes. That would be remarkable, if true. It is entertainingly
reminiscent of this famous saying, apparently due to Feynman,
that most of physics is about "Laplace [itex]\phi =[/itex] " in some
generalized sense.

> We should perhaps ask Ori to write something interesting about it on
> SPS.

Are you going to contact him or shall I?

By the way, while talking to Hermann Nicolai about these things
in Golm I learned about the interesting fact that E10 can be
regarded as sitting inside the vertex operator algebra of the
string. Of course that's long and well known (e.g. http://www.arxiv.org/abs/hep-th/9411188)
but I didn't fully appreciate before how already the symmetry
algebra of the single string knows about "M-theory" in this sense.

Right now I am visiting Ioannis Giannakis at Rockefeller University
and he also emphasizes this fact. As you know, I am here in order to
discuss deformations of superconformal worldsheet theories
(as in http://www.arxiv.org/abs/hep-th/0401175 and references given there) which in particular
include the symmetry transformations of the background (gauge
transformations, dualities, etc.). Ten years ago Evans, Giannakis and
Nanopoulos in

Evans, Giannakis & Nonopoulos,
An infinte dimensional symmetry algebra in string theory
http://www.arxiv.org/abs/hep-th/9401075

have already discussed the immense symmetry algebra that is
found by deforming the worldsheet CFT by similarity transformations
of the form

[tex]A -> \exp(-W) A \exp(W)[/tex]

for W a general integrated vertex. I am not completely sure
what currently the state of the art is with respect to our
understanding of this algebra, though. I am being told that
physicists are currently better prepared to discuss E10
than mathematicians are.

Lubos Motl

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Hi Urs,\n\nthanks for your response! Ori et al. is writing another paper, so he is\npretty busy - and at least one of his collaborator is a student that is\nfinishing his/her PhD thesis, but I was told that eventually they will\nwrite something about it on SPS.\n\n> BTW, do you know if the approach by Ganor et al. started independently\n> of the cosmological Billiard approach by Damour Henneaux and Nicolai\n> and later merged, or what is the historical development?\n\nMy guess would be that it was independent, but they must know the answer\nmuch better.\n\n> Yes. That would be remarkable, if true.\n\nDefinitely. If true. ;-)\n\n> It is entertainingly reminiscent of this famous saying, apparently due\n> to Feynman, that most of physics is about "Laplace phi = 0" in some\n> generalized sense.\n\nFeynman also advertised the equation U=0 for the theory of everything\nwhere U is Feynman\'s universal U function. Was the Laplacian comment more\nserious?\n\n> Are you going to contact him or shall I?\n\nWe have exchanged a couple of mails, but your mail might encourage him to\nwrite something even more than one mail from me. ;-)\n\n> By the way, while talking to Hermann Nicolai about these things\n> in Golm I learned about the interesting fact that E10 can be\n> regarded as sitting inside the vertex operator algebra of the\n> string. Of course that\'s long and well known (e.g. hep-th/9411188)\n\nThis might be interesting, but let me ask a simple question. Are you/they\nsaying something beyond the simple claim that the E_{10} current algebra\ncan be represented by a compactified bosonic CFT?\n\n> have already discussed the immense symmetry algebra that is\n> found by deforming the worldsheet CFT by similarity transformations\n> of the form\n>\n> A -> exp(-W) A exp(W)\n>\n> for W a general integrated vertex.\n\nBy a general integrated vertex, do you mean an integral of a (1,1) primary\nfield? Or the integral of an arbitrary field? Don\'t you just get the\nalgebra of all operators in the CFT? Although I used to think about\nexactly these ideas in the past, today I don\'t quite know how big object\nshould I imagine when you talk about these intriguing concepts.\n\nMoreover, it reminds me of some of our discussions about Thiemann\'s stuff.\nDo you realize that exp(W).exp(-W) is not equal to identity if W is a\ngeneral enough operator in a quantum field theory? For example,\nexp(i.k.X(z)) and exp(-i.k.X(z)) have huge short-distance singularities in\ntheir OPEs, and so on. I hope you don\'t want to ignore the subtleties of\nQFT and build something along these lines of Thiemann\'s papers - such an\napproach contradicts all properties of quantum field theory that\ndistninguish it from simple quantum mechanics.\n\n> I am not completely sure what currently the state of the art is with\n> respect to our understanding of this algebra, though. I am being told\n> that physicists are currently better prepared to discuss E10 than\n> mathematicians are.\n\nYou are obviously a fan of the idea that the deep underlying principle\nbehind M-theory is some huge symmetry/group. A favorite idea of many great\npeople many decades ago, as well as of Thomas Larsson and others today.\nWell, let me admit that my belief that the things are that simple has been\nreduced significantly (all symmetries in string theory tend to be local\nsymmetries, and local symmetries describe a redundancy of your description\nwhich is not quite a physical and measurable concept, and can differ\nbetween different descriptions), and moreover the E_{10} group (or\nhyperbolic algebra) just does not seem mysterious enough to me today. But\nsuch exceptional groups must play *some* role.\n\nDo you think that the existence of the E_9 and E_{10} symmetries of\nM-theory on 9-dimensional and 10-dimensional tori is equally well\nestablished as in the case of E_8 and lower groups?\n\nAll the best\nLubos\n__________________________________________________________ ____________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)\n^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hi Urs,

thanks for your response! Ori et al. is writing another paper, so he is
pretty busy - and at least one of his collaborator is a student that is
finishing [itex]his/her[/itex] PhD thesis, but I was told that eventually they will
write something about it on SPS.

> BTW, do you know if the approach by Ganor et al. started independently
> of the cosmological Billiard approach by Damour Henneaux and Nicolai
> and later merged, or what is the historical development?

My guess would be that it was independent, but they must know the answer
much better.

> Yes. That would be remarkable, if true.

Definitely. If true. ;-)

> It is entertainingly reminiscent of this famous saying, apparently due
> to Feynman, that most of physics is about "Laplace [itex]\phi =[/itex] " in some
> generalized sense.

Feynman also advertised the equation U=0 for the theory of everything
where U is Feynman's universal U function. Was the Laplacian comment more
serious?

> Are you going to contact him or shall I?

We have exchanged a couple of mails, but your mail might encourage him to
write something even more than one mail from me. ;-)

> By the way, while talking to Hermann Nicolai about these things
> in Golm I learned about the interesting fact that E10 can be
> regarded as sitting inside the vertex operator algebra of the
> string. Of course that's long and well known (e.g. http://www.arxiv.org/abs/hep-th/9411188)

This might be interesting, but let me ask a simple question. Are [itex]you/they[/itex]
saying something beyond the simple claim that the [itex]E_{10}[/itex] current algebra
can be represented by a compactified bosonic CFT?

> have already discussed the immense symmetry algebra that is
> found by deforming the worldsheet CFT by similarity transformations
> of the form
>
> [itex]A -> \exp(-W) A \exp(W)[/itex]
>
> for W a general integrated vertex.

By a general integrated vertex, do you mean an integral of a (1,1) primary
field? Or the integral of an arbitrary field? Don't you just get the
algebra of all operators in the CFT? Although I used to think about
exactly these ideas in the past, today I don't quite know how big object
should I imagine when you talk about these intriguing concepts.

Moreover, it reminds me of some of our discussions about Thiemann's stuff.
Do you realize that [itex]\exp(W)[/itex].[itex]\exp(-W)[/itex] is not equal to identity if W is a
general enough operator in a quantum field theory? For example,
[itex]\exp(i.k.X(z))[/itex] and [itex]\exp(-i.k.X(z))[/itex] have huge short-distance singularities in
their OPEs, and so on. I hope you don't want to ignore the subtleties of
QFT and build something along these lines of Thiemann's papers - such an
approach contradicts all properties of quantum field theory that
distninguish it from simple quantum mechanics.

> I am not completely sure what currently the state of the art is with
> respect to our understanding of this algebra, though. I am being told
> that physicists are currently better prepared to discuss E10 than
> mathematicians are.

You are obviously a fan of the idea that the deep underlying principle
behind M-theory is some huge symmetry/group. A favorite idea of many great
people many decades ago, as well as of Thomas Larsson and others today.
Well, let me admit that my belief that the things are that simple has been
reduced significantly (all symmetries in string theory tend to be local
symmetries, and local symmetries describe a redundancy of your description
which is not quite a physical and measurable concept, and can differ
between different descriptions), and moreover the [itex]E_{10}[/itex] group (or
hyperbolic algebra) just does not seem mysterious enough to me today. But
such exceptional groups must play *some* role.

Do you think that the existence of the [itex]E_9[/itex] and [itex]E_{10}[/itex] symmetries of
M-theory on 9-dimensional and 10-dimensional tori is equally well
established as in the case of [itex]E_8[/itex] and lower groups?

All the best
Lubos
__{____________________________________________________________________ ________}
E-mail: lumo@matfyz.cz fax: [itex]+1-617/496-0110[/itex] Web: http://lumo.matfyz.cz/
eFax: [itex]+1-801/454-1858[/itex] work: [itex]+1-617/496-8199[/itex] home: [itex]+1-617/868-4487[/itex] (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^

Urs Schreiber

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Sat, 17 Apr 2004, Lubos Motl wrote:\n\n> > Yes. That would be remarkable, if true.\n>\n> Definitely. If true. ;-)\n\nHeh. Well, so it should be checked! I guess the task is to proceed as\nin section 7 of hep-th/0401053 for higher orders\nand try to identify the degrees of\nfreedom and their dynamical relations (e.g. as in section 7.4)\ndescribed by the various orders Delta_i of the Laplace operator.\n\nAt least for the pure supergravity degrees of freedom it\nhas been checked in\n\nDamour, Henneaux, Nicolai\nE10 and a "samll tension expansion" of M Theory\nhep-th/0207267\n\nthat the E10 sigma-model reproduces these up to 30th order in\nsome appropriate expansion. So I guess there can be no doubt that\nthe E10 sigma-model does describe full 11d sugra plus lots of\nother stuff. Wouldn\'t it be pretty weird if this other stuff is\n_not_ that what we would hope it to be?\n\n> Feynman also advertised the equation U=0 for the theory of everything\n> where U is Feynman\'s universal U function. Was the Laplacian comment more\n> serious?\n\nI don\'t recall where I read it (must have been in the Feynman lectures,\nsomewhere) but, yes, in the context that I read it it was supposed to\nbe more serious than writing\n\n0 = U = (F-ma)^2 + ... .\n\nPersonally I think the more refined version of the statement that I am\nthinking about is the point of view expressed in\n\nFroehlich, Grandjean, Recknagel,\nSupersymmetric Quantum Theory, non-commutative geometry, and gravitation\nhep-th/9706132\n\nwhere it is emphasized that any system of susy QM can be viewed as\ndescribing a particular (generalized) geometry. From this point of\nview it wouldn\'t be surprising if a TOE is solved by "harmonic forms"\nin some generalized sense (like e.g. harmonic functions on the group of\nE10 in the case which we are discussing).\n\nBut I am digressing...\n\n> This might be interesting, but let me ask a simple question. Are you/they\n> saying something beyond the simple claim that the E_{10} current algebra\n> can be represented by a compactified bosonic CFT?\n\nI have barely begun learning about E10, so please bear with me.\n\nFirst of all I am confused why you refer to the "E10 current algebra".\nFrom hep-th/9411188 I seemed to learn that the whole point is that\nE10 is not a current algebra, because of its indefinite Cartan\nmatrix. Only a Kac-Moody algebra with (semi)definite Cartan\nmatrix is a current algebra.\n\nBut what I was referring to was the fact that, according to\nequation (0.11) of the above mentioned paper E10 sits inside the\nLie algebra of physical states of a completly compactified bosonic\nstring. Is that a "simple" claim?\n\nIf you feel that I am lacking some basic knowledge about E10,\nplease go ahead and educate me! :-)\n\n> > have already discussed the immense symmetry algebra that is\n> > found by deforming the worldsheet CFT by similarity transformations\n> > of the form\n> >\n> > A -> exp(-W) A exp(W)\n> >\n> > for W a general integrated vertex.\n>\n> By a general integrated vertex, do you mean an integral of a (1,1) primary\n> field? Or the integral of an arbitrary field? Don\'t you just get the\n> algebra of all operators in the CFT? Although I used to think about\n> exactly these ideas in the past, today I don\'t quite know how big object\n> should I imagine when you talk about these intriguing concepts.\n>\n> Moreover, it reminds me of some of our discussions about Thiemann\'s stuff.\n> Do you realize that exp(W).exp(-W) is not equal to identity if W is a\n> general enough operator in a quantum field theory? For example,\n> exp(i.k.X(z)) and exp(-i.k.X(z)) have huge short-distance singularities in\n> their OPEs, and so on. I hope you don\'t want to ignore the subtleties of\n> QFT and build something along these lines of Thiemann\'s papers - such an\n> approach contradicts all properties of quantum field theory that\n> distninguish it from simple quantum mechanics.\n\nOk, good point. The exponents used in hep-th/9401075 are supposed\nto be normal orderd, but the full exponentials are not. So more explicitly\nI was referring to\n\nA -> exp(-:W:) A exp(:W:) .\n\n(cf. eq. (2.5) of hep-th/9401075).\n\nSo this way exp(-:W:) is indeed the inverse of exp(:W:), at least\nif one of them is well defined in the first place. The point of\nall this is that given any stress-energy tensor T of the\nworlsheet CFT you get a new worldsheet CFT by setting A=T in\nthe above formula. The new CFT can be interpreted as coming from\nthe old one by a generalized symmetry operation on the\nbackground.\n\nBut, yes the W are supposed to be integrals over weight 1\nfields, such that the sigma-reparameterization generator\nT- bar T is left unaffected by the operation, as it must be.\n\nThe algebra of the :W: is huge and should have some close relation\nto that discussed on pp3 of hep-th/9411188.\n\n> > I am not completely sure what currently the state of the art is with\n> > respect to our understanding of this algebra, though. I am being told\n> > that physicists are currently better prepared to discuss E10 than\n> > mathematicians are.\n>\n> You are obviously a fan of the idea that the deep underlying principle\n> behind M-theory is some huge symmetry/group. A favorite idea of many great\n> people many decades ago, as well as of Thomas Larsson and others today.\n> Well, let me admit that my belief that the things are that simple has been\n> reduced significantly (all symmetries in string theory tend to be local\n> symmetries, and local symmetries describe a redundancy of your description\n> which is not quite a physical and measurable concept, and can differ\n> between different descriptions), and moreover the E_{10} group (or\n> hyperbolic algebra) just does not seem mysterious enough to me today. But\n> such exceptional groups must play *some* role.\n\nI fully agree with what you are saying here.\n\n> Do you think that the existence of the E_9 and E_{10} symmetries of\n> M-theory on 9-dimensional and 10-dimensional tori is equally well\n> established as in the case of E_8 and lower groups?\n\nI don\'t know. I still know too little about this stuff to make an\neducated guess. What I find impressive though is that E10 definitely\nknows all about 11d sugra (at least of the bosonic sector, that is) while\nstill having way more variables. This alone seems to be very strong\nevidence for the fact that E10 must say something about M theory,\nI\'d say.\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Sat, 17 Apr 2004, Lubos Motl wrote:

> > Yes. That would be remarkable, if true.
>
> Definitely. If true. ;-)

Heh. Well, so it should be checked! I guess the task is to proceed as
in section 7 of http://www.arxiv.org/abs/hep-th/0401053 for higher orders
and try to identify the degrees of
freedom and their dynamical relations (e.g. as in section 7.4)
described by the various orders [itex]\Delta_i[/itex] of the Laplace operator.

At least for the pure supergravity degrees of freedom it
has been checked in

Damour, Henneaux, Nicolai
E10 and a "samll tension expansion" of M Theory
http://www.arxiv.org/abs/hep-th/0207267

that the E10 [itex]\sigma-model[/itex] reproduces these up to 30th order in
some appropriate expansion. So I guess there can be no doubt that
the E10 [itex]\sigma-model[/itex] does describe full 11d sugra plus lots of
other stuff. Wouldn't it be pretty weird if this other stuff is
_not_ that what we would hope it to be?

> Feynman also advertised the equation U=0 for the theory of everything
> where U is Feynman's universal U function. Was the Laplacian comment more
> serious?

I don't recall where I read it (must have been in the Feynman lectures,
somewhere) but, yes, in the context that I read it it was supposed to
be more serious than writing

[tex]= U = (F-ma)^2 + .[/itex].. .

Personally I think the more refined version of the statement that I am
thinking about is the point of view expressed in

Froehlich, Grandjean, Recknagel,
Supersymmetric Quantum Theory, non-commutative geometry, and gravitation
http://www.arxiv.org/abs/hep-th/9706132

where it is emphasized that any system of susy QM can be viewed as
describing a particular (generalized) geometry. From this point of
view it wouldn't be surprising if a TOE is solved by "harmonic forms"
in some generalized sense (like e.g. harmonic functions on the group of
E10 in the case which we are discussing).

But I am digressing...

> This might be interesting, but let me ask a simple question. Are [itex]you/they[/itex]
> saying something beyond the simple claim that the [itex]E_{10}[/itex] current algebra
> can be represented by a compactified bosonic CFT?

I have barely begun learning about E10, so please bear with me.

First of all I am confused why you refer to the "E10 current algebra".
From http://www.arxiv.org/abs/hep-th/9411188 I seemed to learn that the whole point is that
E10 is not a current algebra, because of its indefinite Cartan
matrix. Only a Kac-Moody algebra with (semi)definite Cartan
matrix is a current algebra.

But what I was referring to was the fact that, according to
equation (0.11) of the above mentioned paper E10 sits inside the
Lie algebra of physical states of a completly compactified bosonic
string. Is that a "simple" claim?

If you feel that I am lacking some basic knowledge about E10,
please go ahead and educate me! :-)

> > have already discussed the immense symmetry algebra that is
> > found by deforming the worldsheet CFT by similarity transformations
> > of the form
> >
> > [itex]A -> \exp(-W) A \exp(W)[/itex]
> >
> > for W a general integrated vertex.
>
> By a general integrated vertex, do you mean an integral of a (1,1) primary
> field? Or the integral of an arbitrary field? Don't you just get the
> algebra of all operators in the CFT? Although I used to think about
> exactly these ideas in the past, today I don't quite know how big object
> should I imagine when you talk about these intriguing concepts.
>
> Moreover, it reminds me of some of our discussions about Thiemann's stuff.
> Do you realize that [itex]\exp(W)[/itex].[itex]\exp(-W)[/itex] is not equal to identity if W is a
> general enough operator in a quantum field theory? For example,
> [itex]\exp(i.k.X(z))[/itex] and [itex]\exp(-i.k.X(z))[/itex] have huge short-distance singularities in
> their OPEs, and so on. I hope you don't want to ignore the subtleties of
> QFT and build something along these lines of Thiemann's papers - such an
> approach contradicts all properties of quantum field theory that
> distninguish it from simple quantum mechanics.

Ok, good point. The exponents used in http://www.arxiv.org/abs/hep-th/9401075 are supposed
to be normal orderd, but the full exponentials are not. So more explicitly
I was referring to

[itex]A -> \exp(-:W:) A \exp(:W:) .[/tex]

(cf. eq. (2.5) of http://www.arxiv.org/abs/hep-th/9401075).

So this way [itex]\exp(-:W:)[/itex] is indeed the inverse of [itex]\exp(:W:), at[/itex] least
if one of them is well defined in the first place. The point of
all this is that given any stress-energy tensor T of the
worlsheet CFT you get a new worldsheet CFT by setting A=T in
the above formula. The new CFT can be interpreted as coming from
the old one by a generalized symmetry operation on the
background.

But, yes the W are supposed to be integrals over weight 1
fields, such that the [itex]\sigma-reparameterization[/itex] generator
T- bar T is left unaffected by the operation, as it must be.

The algebra of the :W: is huge and should have some close relation
to that discussed on pp3 of http://www.arxiv.org/abs/hep-th/9411188.

> > I am not completely sure what currently the state of the art is with
> > respect to our understanding of this algebra, though. I am being told
> > that physicists are currently better prepared to discuss E10 than
> > mathematicians are.
>
> You are obviously a fan of the idea that the deep underlying principle
> behind M-theory is some huge symmetry/group. A favorite idea of many great
> people many decades ago, as well as of Thomas Larsson and others today.
> Well, let me admit that my belief that the things are that simple has been
> reduced significantly (all symmetries in string theory tend to be local
> symmetries, and local symmetries describe a redundancy of your description
> which is not quite a physical and measurable concept, and can differ
> between different descriptions), and moreover the [itex]E_{10}[/itex] group (or
> hyperbolic algebra) just does not seem mysterious enough to me today. But
> such exceptional groups must play *some* role.

I fully agree with what you are saying here.

> Do you think that the existence of the [itex]E_9[/itex] and [itex]E_{10}[/itex] symmetries of
> M-theory on 9-dimensional and 10-dimensional tori is equally well
> established as in the case of [itex]E_8[/itex] and lower groups?

I don't know. I still know too little about this stuff to make an
educated guess. What I find impressive though is that E10 definitely
knows all about 11d sugra (at least of the bosonic sector, that is) while
still having way more variables. This alone seems to be very strong
evidence for the fact that E10 must say something about M theory,
I'd say.

Lubos Motl

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Sat, 17 Apr 2004, Urs Schreiber wrote:\n\n> Heh. Well, so it should be checked!\n\nYes, it should be tried.\n\n> that the E10 sigma-model reproduces these up to 30th order in\n> some appropriate expansion.\n\nI would probably believe this particular technical claim even with a small\nnumber of orders than 30. ;-) I guess that you only talk about the bosonic\nsector of SUGRA?\n\n> So I guess there can be no doubt that\n> the E10 sigma-model does describe full 11d sugra plus lots of\n> other stuff. Wouldn\'t it be pretty weird if this other stuff is\n> _not_ that what we would hope it to be?\n\nIf you ask me, my answer would be No, it would not be too weird.\nConstructing the full covariant formulation of M-theory is a big task, and\nit would seem more weird to me if someone suddenly found it using an\nobscure and apparently a slightly ill-defined sigma-model, especially\nbecause I don\'t expect M-theory to be *just* another string theory. :-)\nNevertheless, I am gonna look at these things because what you say sounds\nhighly nontrivial.\n\n> First of all I am confused why you refer to the "E10 current algebra".\n> From hep-th/9411188 I seemed to learn that the whole point is that\n> E10 is not a current algebra, because of its indefinite Cartan\n> matrix. Only a Kac-Moody algebra with (semi)definite Cartan\n> matrix is a current algebra.\n\nOK, so I wanted to say a more general word for the algebra that does not\nhave any such constraints of (semi)definiteness. Does it make a big\ndifference?\n\n> But what I was referring to was the fact that, according to\n> equation (0.11) of the above mentioned paper E10 sits inside the\n> Lie algebra of physical states of a completly compactified bosonic\n> string. Is that a "simple" claim?\n\nI may misunderstand something, but what you say sounds as a simple claim.\nIf you read the paragraph below the equation (0.11) that you mentioned,\nyou will see that the nontrivial statement is not that g(A) is included in\ng_\\Lambda, but rather the fact that the inclusion is "proper", i.e. that\nthere are some missing states.\n\nI thought that just like the bosons compactified on an even self-dual\nlattice \\Gamma_8 gives you an E_8 current algebra, the compactification on\nanother even self-dual lattice \\Gamma_{9+1} gives you the E_{10}\nhyperbolic Kac-Moody algebra. The 10-dimensional Cartan subalgebra\n(indefinite one, in this case) has vertex operators \\partial{X^i}, while\nthe roots are represented by \\exp(ik_iX_i(z)) where the momenta k_i belong\nto the lattice, and they can be, in the E_{10} case, both time-like as\nwell as spacelike. Is there some huge difference that I missed?\n\n> ...So this way exp(-:W:) is indeed the inverse of exp(:W:), at least\n> if one of them is well defined in the first place.\n\nNo, the colons don\'t change anything about my claim. Let me repeat my\nstatements with the colons - which incidentally don\'t change the operators\nin my example at all:\n\n> > Do you realize that exp(:W:).exp(-:W:) is not equal to identity if W is a\n> > general enough operator in a quantum field theory? For example,\n> > exp(:i.k.X(z):) and exp(-:i.k.X(z):) have huge short-distance\n> > singularities in their OPEs, and so on.\n\nIt is just not true that in quantum field theory you can write the inverse\noperator to \\exp(:W:) as \\exp(-:W:) for a nontrivial operator W. The\nsubtlety is hidden in the OPE of W with itself. The exceptions where the\nnaive formula works are the operators W that are constructed as the\nintegrals of the (1,1) primary fields. The primary fields only have the\nsimple singularities with the stress energy tensor, and they lead to\n"nonsingular" expressions after integration. Among these operators whose\nexponentials are invertible, you will find the total momentum, the\ngenerators of the Lorentz group (and its individual contributions, if\nproperly defined), and some others. But it is not correct to assume that\nthe exponential of a generic operator can be inverted in this classical\nway.\n\n> The point of all this is that given any stress-energy tensor T of the\n> worlsheet CFT you get a new worldsheet CFT by setting A=T in the above\n> formula. The new CFT can be interpreted as coming from the old one by\n> a generalized symmetry operation on the background.\n\nThat\'s probably interesting, but I did not get it. ;-/\n\n> The algebra of the :W: is huge and should have some close relation\n> to that discussed on pp3 of hep-th/9411188.\n\nDo you say that it is huge in a general case? The (1,1) primary fields are\ncertainly rare. For example, the Ising model CFT contains 3 of them.\n\n> I fully agree with what you are saying here.\n\nNice to hear.\n\n> I don\'t know. I still know too little about this stuff to make an\n> educated guess. What I find impressive though is that E10 definitely\n> knows all about 11d sugra (at least of the bosonic sector, that is) ...\n\nThe absence of fermions is another reason why I feel that this E_{10} is\nunlikely to be far-reaching and even more unlikely to contain the whole\nM-theory. The inclusion of the fermions would try to lead us to\nsuperalgebras such as OSp(1|32) or something similar, which is very\ndifferent from E_{10}.\n\nCheers\nLubos\n_____________________________________________ _________________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)\n^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Sat, 17 Apr 2004, Urs Schreiber wrote:

> Heh. Well, so it should be checked!

Yes, it should be tried.

> that the E10 [itex]\sigma-model[/itex] reproduces these up to 30th order in
> some appropriate expansion.

I would probably believe this particular technical claim even with a small
number of orders than 30. ;-) I guess that you only talk about the bosonic
sector of SUGRA?

> So I guess there can be no doubt that
> the E10 [itex]\sigma-model[/itex] does describe full 11d sugra plus lots of
> other stuff. Wouldn't it be pretty weird if this other stuff is
> _not_ that what we would hope it to be?

If you ask me, my answer would be No, it would not be too weird.
Constructing the full covariant formulation of M-theory is a big task, and
it would seem more weird to me if someone suddenly found it using an
obscure and apparently a slightly ill-defined [itex]\sigma-model,[/itex] especially
because I don't expect M-theory to be *just* another string theory. :-)
Nevertheless, I am gonna look at these things because what you say sounds
highly nontrivial.

> First of all I am confused why you refer to the "E10 current algebra".
> From http://www.arxiv.org/abs/hep-th/9411188 I seemed to learn that the whole point is that
> E10 is not a current algebra, because of its indefinite Cartan
> matrix. Only a Kac-Moody algebra with (semi)definite Cartan
> matrix is a current algebra.

OK, so I wanted to say a more general word for the algebra that does not
have any such constraints of (semi)definiteness. Does it make a big
difference?

> But what I was referring to was the fact that, according to
> equation (0.11) of the above mentioned paper E10 sits inside the
> Lie algebra of physical states of a completly compactified bosonic
> string. Is that a "simple" claim?

I may misunderstand something, but what you say sounds as a simple claim.
If you read the paragraph below the equation (0.11) that you mentioned,
you will see that the nontrivial statement is not that g(A) is included in
[itex]g_\Lambda,[/itex] but rather the fact that the inclusion is "proper", i.e. that
there are some missing states.

I thought that just like the bosons compactified on an even self-dual
lattice [itex]\Gamma_8[/itex] gives you an [itex]E_8[/itex] current algebra, the compactification on
another even self-dual lattice [itex]\Gamma_{9+1}[/itex] gives you the [itex]E_{10}[/itex]
hyperbolic Kac-Moody algebra. The 10-dimensional Cartan subalgebra
(indefinite one, in this case) has vertex operators [itex]\partial{X^i},[/itex] while
the roots are represented by [itex]\exp(ik_iX_i(z))[/itex] where the momenta [itex]k_i[/itex] belong
to the lattice, and they can be, in the [itex]E_{10}[/itex] case, both time-like as
well as spacelike. Is there some huge difference that I missed?

> ...So this way [itex]\exp(-:W:)[/itex] is indeed the inverse of [itex]\exp(:W:), at[/itex] least
> if one of them is well defined in the first place.

No, the colons don't change anything about my claim. Let me repeat my
statements with the colons - which incidentally don't change the operators
in my example at all:

> > Do you realize that [itex]\exp(:W:)[/itex].[itex]\exp(-:W:)[/itex] is not equal to identity if W is a
> > general enough operator in a quantum field theory? For example,
> > [itex]\exp(:i.k.X(z):)[/itex] and [itex]\exp(-:i.k.X(z):)[/itex] have huge short-distance
> > singularities in their OPEs, and so on.

It is just not true that in quantum field theory you can write the inverse
operator to [itex]\exp(:W:)[/itex] as [itex]\exp(-:W:)[/itex] for a nontrivial operator W. The
subtlety is hidden in the OPE of W with itself. The exceptions where the
naive formula works are the operators W that are constructed as the
integrals of the (1,1) primary fields. The primary fields only have the
simple singularities with the stress energy tensor, and they lead to
"nonsingular" expressions after integration. Among these operators whose
exponentials are invertible, you will find the total momentum, the
generators of the Lorentz group (and its individual contributions, if
properly defined), and some others. But it is not correct to assume that
the exponential of a generic operator can be inverted in this classical
way.

> The point of all this is that given any stress-energy tensor T of the
> worlsheet CFT you get a new worldsheet CFT by setting A=T in the above
> formula. The new CFT can be interpreted as coming from the old one by
> a generalized symmetry operation on the background.

That's probably interesting, but I did not get it[itex]. ;-/[/itex]

> The algebra of the :W: is huge and should have some close relation
> to that discussed on pp3 of http://www.arxiv.org/abs/hep-th/9411188.

Do you say that it is huge in a general case? The (1,1) primary fields are
certainly rare. For example, the Ising model CFT contains 3 of them.

> I fully agree with what you are saying here.

Nice to hear.

> I don't know. I still know too little about this stuff to make an
> educated guess. What I find impressive though is that E10 definitely
> knows all about 11d sugra (at least of the bosonic sector, that is) ...

The absence of fermions is another reason why I feel that this [itex]E_{10}[/itex] is
unlikely to be far-reaching and even more unlikely to contain the whole
M-theory. The inclusion of the fermions would try to lead us to
superalgebras such as OSp(1|32) or something similar, which is very
different from [itex]E_{10}[/itex].

Cheers
Lubos
__{____________________________________________________________________ ________}
E-mail: lumo@matfyz.cz fax: [itex]+1-617/496-0110[/itex] Web: http://lumo.matfyz.cz/
eFax: [itex]+1-801/454-1858[/itex] work: [itex]+1-617/496-8199[/itex] home: [itex]+1-617/868-4487[/itex] (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^

Thomas Larsson

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Lubos Motl wrote (although Google was unable to retrieve it, so I\nresponded to the next message instead):\n\n> You are obviously a fan of the idea that the deep underlying principle\n> behind M-theory is some huge symmetry/group. A favorite idea of many great\n> people many decades ago, as well as of Thomas Larsson and others today.\n> Well, let me admit that my belief that the things are that simple has been\n> reduced significantly (all symmetries in string theory tend to be local\n> symmetries, and local symmetries describe a redundancy of your description\n> which is not quite a physical and measurable concept, and can differ\n> between different descriptions), and moreover the E_{10} group (or\n> hyperbolic algebra) just does not seem mysterious enough to me today. But\n> such exceptional groups must play *some* role.\n\nHmrph. It is hardly some random symmetry such as E_10 that I suggest. The\ndiffeomorphism group is *the* correct symmetry of GR (ask Rovelli!).\n\n[Moderator\'s note: I hope that Rovelli is not necessary to answer basic\nquestions about classical general relativity. LM]\n\nAs Urs Schreiber has pointed out, LQG is not really canonical\nquantization, which generically would give rise to lowest-energy reps of\nthe symmetry group - this is true even for gauge symmetries such as\nworldsheet conformal symmetry. Thus, combining the symmetry principles\nunderlying GR and QM *and nothing else* - experimentally confirmed physics\n- leads to LW reps of the diff group, and non-trivial such reps are\nanomalous. That\'s just the way it is.\n\n[Moderator\'s note: Sorry, I did not realize that you started to promote\nanother group as the theory of everything, beyond your original mb(3|8)\nfrom http://arxiv.org/abs/hep-th/0208185 . I guess that E_{10} is closer\nto the truth. LM]\n\nIt seems that the basic idea in West\'s approach is that M-theory and\n11D SUGRA should for some reason be described by non-linear realization\nof the Kac-Moody algebra E11. At least, this is what West writes in\nhttp://www.arxiv.org/abs/hep-th/0104081, which seems to be the seminal\npaper. A striking observation is that NO SUCH REALIZATION EXISTS! (if\nthe underlying space is finite-dimensional or mildly infinite-\ndimensional). This follows immedately from the well-known fact that\nnon-affine Kac-Moody algebras like E11 are of expontential growth.\n\nFor the last 20 years at least, Victor Kac has emphasized that the lack\nof a natural realization, linear or nonlinear, is a major problem for\nnon-affine Kac-Moody algebras; without a realization, these are somewhat\nuseless. Kac states this very explicitly in\nhttp://www.arxiv.org/abs/q-alg/9709008:\n\n"It is a well kept secret that the theory of Kac-Moody algebras has\nbeen a disaster."\n\n[Moderator\'s note: this disaster has not influenced physics too much,\nand Kac-Moody algebras are very important and well functioning tool\nto study conformal field theories and perhaps other realizations.\nThe users of physicsforums.com should probably click at "View the post\nin the original ASCII form" because otherwise they might have problems\nto see the equations and pictures in this post correctly.]\n\nThus, if West and others would find a nice realization of E_11 it would\nbe a major mathematical breakthrough, irrespective of its relevance to\nM-theory (and the relevance of M-theory to physics). However, I have\nglanced at quite a few of these E_11 papers, and I have never seen any\nmentioning of exponential growth. This suggests that string theorists\ndo not realize that this is a major difficultly, which has stalled\nVictor and other mathematicians for decades.\n\nNevertheless, West\'s original paper contains some beautiful math. First\nrecall that nonlinear realizations are the same thing as gradings.\nE.g., the conformal realization of so(n+1,1) is a 3-grading:\n\nso(n+1,1) = g_-1 + g_0 + g_1,\n\nwhere g_-1 = translations, g_0 = rotations and dilatations, and\ng_1 = conformal boosts. The grading is specified by defining the\ngrading operator, which here is the dilatation - the degree is the\ndilatation eiginvalue. I had a nice discussion with John Baez and Tony\nSmith about 16 months ago. The spr thread "Structures preserved by e8"\nshould be available from some archive.\n\nWests starts from a 7-grading of E8, eqs (4.1-11)\n\nE8 = g_-3 + g_-2 + g_-1 + g_0 + g_1 + g_2 + g_3.\n\nIf you formulate E8 in Cartan-Weil basis, H^i, E^\\alpha, a grading\noperator is of the form Z = alpha_i H^i. There is thus a 1-1 correspondence\nbetween gradings and roots alpha. The relevant root here is\n\n*\n|\no-o-o-o-o-o-o\n\nwhich gives\n\nE8 = V* + A^6 V + A^3 V + (sl(8)+C) + A^3 V* + A^6 V* + V.\n\n248 = 8 + 28 + 56 + 64 + 56 + 28 + 8\n\nHere V is the 8-dim vector rep of sl(8) and V* its dual and A^n V is\nthe n:th anti-symmetric power. Note that g_0 = sl(8)+C and you get the\nDynkin diagram of sl(8) by removing the starred root - this is a general\nphenomenon. This grading corresponds to a realization on the dual of\ng_- = g_-3 + g_-2 + g_-1, according to the following table:\n\ng_-1 x_abc R^abc = d/dx_abc + ...\n\ng_-2 y_abcdef R^abcdef = d/dy_abcdef + .....\n\ng_-3 z^a S_a = d/dz^a\n\nAside: other E8 gradings have also occurred in the literature. The one\ncorresponding to\n\no\n|\no-o-o-o-o-o-*\n\n(g_0 = E7) was discussed in a 2000 CMP paper by Gunaydin-Koepsell-Nicolai\nand\n\no\n|\n*-o-o-o-o-o-o\n\n(g_0 = so(14)) is implicit in GSW, (6.A.1-3).\n\nHowever, E8 cannot be right for M-theory, because the vector rep of g_0\n= sl(8) is 8-dimensional and describes translations in 8D rather than\n11D. Therefore, West seems to propose that the E11 grading corresponding to\n\n*\n|\no-o-o-o-o-o-o-o-o-o\n\nshould be relevant to M-theory. Alas, this grading extends to infinity\nin both directions,\n\nE11 = ... + g_-4 + g_-3 + g_-2 + g_-1 + g_0 + g_1 + g_2 + g_3 + ... .\n\nMoreover, dim g_k ~ exp(ck) for some constant c, i.e. E11 is of\nexponential growth. This means that E11 is HUGE, much bigger than any\nlocal symmetry. Local symmetries, e.g. diff and Yang-Mills in n dimensions,\ngrow polynomially, dim g_k ~ k^(n-1), i.e. there are finitely many\nsymmetry generators per spacetime point.\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Lubos Motl wrote (although Google was unable to retrieve it, so I
responded to the next message instead):

> You are obviously a fan of the idea that the deep underlying principle
> behind M-theory is some huge symmetry/group. A favorite idea of many great
> people many decades ago, as well as of Thomas Larsson and others today.
> Well, let me admit that my belief that the things are that simple has been
> reduced significantly (all symmetries in string theory tend to be local
> symmetries, and local symmetries describe a redundancy of your description
> which is not quite a physical and measurable concept, and can differ
> between different descriptions), and moreover the [itex]E_{10}[/itex] group (or
> hyperbolic algebra) just does not seem mysterious enough to me today. But
> such exceptional groups must play *some* role.

Hmrph. It is hardly some random symmetry such as [itex]E_{10}[/itex] that I suggest. The
diffeomorphism group is *the* correct symmetry of GR (ask Rovelli!).

[Moderator's note: I hope that Rovelli is not necessary to answer basic
questions about classical general relativity. LM]

As Urs Schreiber has pointed out, LQG is not really canonical
quantization, which generically would give rise to lowest-energy reps of
the symmetry group - this is true even for gauge symmetries such as
worldsheet conformal symmetry. Thus, combining the symmetry principles
underlying GR and [itex]QM *and[/itex] nothing [itex]else* -[/itex] experimentally confirmed physics
- leads to LW reps of the diff group, and non-trivial such reps are
anomalous. That's just the way it is.

[Moderator's note: Sorry, I did not realize that you started to promote
another group as the theory of everything, beyond your original mb(3|8)
from http://arxiv.org/abs/http://www.arxi...hep-th/0208185 . I guess that [itex]E_{10}[/itex] is closer
to the truth. LM]

It seems that the basic idea in West's approach is that M-theory and
11D SUGRA should for some reason be described by non-linear realization
of the Kac-Moody algebra E11. At least, this is what West writes in
http://www.arxiv.org/abs/http://www....ep-th/0104081, which seems to be the seminal
paper. A striking observation is that NO SUCH REALIZATION EXISTS! (if
the underlying space is finite-dimensional or mildly infinite-
dimensional). This follows immedately from the well-known fact that
non-affine Kac-Moody algebras like E11 are of expontential growth.

For the last 20 years at least, Victor Kac has emphasized that the lack
of a natural realization, linear or nonlinear, is a major problem for
non-affine Kac-Moody algebras; without a realization, these are somewhat
useless. Kac states this very explicitly in
http://www.arxiv.org/abs/q-alg/9709008:

"It is a well kept secret that the theory of Kac-Moody algebras has
been a disaster."

[Moderator's note: this disaster has not influenced physics too much,
and Kac-Moody algebras are very important and well functioning tool
to study conformal field theories and perhaps other realizations.
The users of physicsforums.com should probably click at "View the post
in the original ASCII form" because otherwise they might have problems
to see the equations and pictures in this post correctly.]

Thus, if West and others would find a nice realization of [itex]E_{11} it[/itex] would
be a major mathematical breakthrough, irrespective of its relevance to
M-theory (and the relevance of M-theory to physics). However, I have
glanced at quite a few of these [itex]E_{11}[/itex] papers, and I have never seen any
mentioning of exponential growth. This suggests that string theorists
do not realize that this is a major difficultly, which has stalled
Victor and other mathematicians for decades.

Nevertheless, West's original paper contains some beautiful math. First
recall that nonlinear realizations are the same thing as gradings.
E.g., the conformal realization of [itex]so(n+1,1)[/itex] is a 3-grading:

[tex]so(n+1,1) = g_-1 + g_0 + g_1,[/tex]

where [itex]g_-1 =[/itex] translations, [itex]g_0 =[/itex] rotations and dilatations, and
[itex]g_1 =[/itex] conformal boosts. The grading is specified by defining the
grading operator, which here is the dilatation - the degree is the
dilatation eiginvalue. I had a nice discussion with John Baez and Tony
Smith about 16 months ago. The spr thread "Structures preserved by e8"
should be available from some archive.

Wests starts from a 7-grading of E8, eqs [itex](4.1-11)E8 = g_-3 + g_-2 + g_-1 + g_0 + g_1 + g_2 + g_3[/itex].

If you formulate E8 in Cartan-Weil basis, [itex]H^i, E^\alpha, a[/itex] grading
operator is of the form [itex]Z = \alpha_i H^i[/itex]. There is thus [itex]a 1-1[/itex] correspondence
between gradings and roots [itex]\alpha.[/itex] The relevant root here is

*
|
[itex]o-o-o-o-o-o-o[/itex]

which gives

[tex]E8 = V* + A^6 V + A^3 V + (sl(8)+C) + A^3 V* + A^6 V* + V[/itex].

248 [itex]= 8 +[/itex] 28 + 56 + 64 + 56 + 28 + 8

Here V is the [itex]8-dim[/itex] vector rep of sl(8) and V* its dual and [itex]A^n V[/itex] is
the n:th anti-symmetric power. Note that [itex]g_0 = sl(8)+C[/itex] and you get the
Dynkin diagram of sl(8) by removing the starred root - this is a general
phenomenon. This grading corresponds to a realization on the dual of
[itex]g_- = g_-3 + g_-2 + g_-1,[/itex] according to the following table:

[itex]g_-1 x_{abc} R^{abc} = d/dx_abc + .[/itex]..

[itex]g_-2 y_{abcdef} R^{abcdef} = d/dy_abcdef + .[/itex]....

[itex]g_-3 z^a S_a = d/dz^a[/tex]

Aside: other E8 gradings have also occurred in the literature. The one
corresponding to

o
|
[itex]o-o-o-o-o-o-*(g_0 = E7)[/itex] was discussed in a 2000 CMP paper by Gunaydin-Koepsell-Nicolai
and

o
|
[itex]*-o-o-o-o-o-o(g_0 = so(14))[/itex] is implicit in GSW, [itex](6.A.1-3).[/itex]

However, E8 cannot be right for M-theory, because the vector rep of [itex]g_0= sl(8)[/itex] is 8-dimensional and describes translations in 8D rather than
11D. Therefore, West seems to propose that the E11 grading corresponding to

*
|
[itex]o-o-o-o-o-o-o-o-o-o[/itex]

should be relevant to M-theory. Alas, this grading extends to infinity
in both directions,

E11 = ..[itex]. + g_-4 + g_-3 + g_-2 + g_-1 + g_0 + g_1 + g_2 + g_3 + .[/itex].. .

Moreover, dim [itex]g_k ~ \exp(ck)[/itex] for some constant c, i.e. E11 is of
exponential growth. This means that E11 is HUGE, much bigger than any
local symmetry. Local symmetries, e.g. diff and Yang-Mills in n dimensions,
grow polynomially, dim [itex]g_k ~ k^(n-1), i[/itex].e. there are finitely many
symmetry generators per spacetime point.

Urs Schreiber

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Sun, 18 Apr 2004, Thomas Larsson wrote:\n\n> As Urs Schreiber has pointed out, LQG is not really canonical\n> quantization, which generically would give rise to lowest-energy reps of\n> the symmetry group - this is true even for gauge symmetries such as\n> worldsheet conformal symmetry. Thus, combining the symmetry principles\n> underlying GR and QM *and nothing else* - experimentally confirmed physics\n> - leads to LW reps of the diff group, and non-trivial such reps are\n> anomalous. That\'s just the way it is.\n\nBTW, do we now agree that this fact is not in contradiction to any\nknown fact in string theory? You seemed to claim that the fact that\ngravitational anomalies for chiral fermions occur in\n4k + 2 dimensions is in contradiction with the above somehow. But\nclearly it is not, since what you write above has nothing to do\nwith chiral fermions.\n\n> For the last 20 years at least, Victor Kac has emphasized that the lack\n> of a natural realization, linear or nonlinear, is a major problem for\n> non-affine Kac-Moody algebras; without a realization, these are somewhat\n> useless. Kac states this very explicitly in\n> http://www.arxiv.org/abs/q-alg/9709008:\n>\n> "It is a well kept secret that the theory of Kac-Moody algebras has\n> been a disaster."\n>\n> [Moderator\'s note: this disaster has not influenced physics too much,\n> and Kac-Moody algebras are very important and well functioning tool\n> to study conformal field theories and perhaps other realizations.\n\nLubos, please note that Larsson (and Kac for that matter) referred to\n_non-affine_ Kac-Moody algebras, while it seems to me what you have\nin mind are current algebras, i.e. _affine_ KM algebras.\n\nKM algebras fall into 3 classes:\n\n1) Cartan matrix positive definite --> finite algebra\n\n2) Cartan matrix positive semi-definite --> affine algebra = current\nalgebra in 2d\n\n3) Cartan matrix indefinite --> exponential growth, desaster and all that\n\n(paraphrased from hep-th/9411188)\n\n> Thus, if West and others would find a nice realization of E_11 it would\n> be a major mathematical breakthrough, irrespective of its relevance to\n> M-theory (and the relevance of M-theory to physics). However, I have\n> glanced at quite a few of these E_11 papers, and I have never seen any\n> mentioning of exponential growth. This suggests that string theorists\n> do not realize that this is a major difficultly, which has stalled\n> Victor and other mathematicians for decades.\n\nThis is at least not true for all physicist. In\n\nGebert, Nicolai,\nE_10 for beginners,\nhep-th/9411188\n\nit is emphasized quite strongly that there is exponential growth and\nthat things are pretty problematic concerning E_10. But I am being\ntold that the point is that one can study consistent finite truncations\nwhich can be handled and in terms of which we can do physics.\n\n\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Sun, 18 Apr 2004, Thomas Larsson wrote:

> As Urs Schreiber has pointed out, LQG is not really canonical
> quantization, which generically would give rise to lowest-energy reps of
> the symmetry group - this is true even for gauge symmetries such as
> worldsheet conformal symmetry. Thus, combining the symmetry principles
> underlying GR and [itex]QM *and[/itex] nothing [itex]else* -[/itex] experimentally confirmed physics
> - leads to LW reps of the diff group, and non-trivial such reps are
> anomalous. That's just the way it is.

BTW, do we now agree that this fact is not in contradiction to any
known fact in string theory? You seemed to claim that the fact that
gravitational anomalies for chiral fermions occur in
[itex]4k + 2[/itex] dimensions is in contradiction with the above somehow. But
clearly it is not, since what you write above has nothing to do
with chiral fermions.

> For the last 20 years at least, Victor Kac has emphasized that the lack
> of a natural realization, linear or nonlinear, is a major problem for
> non-affine Kac-Moody algebras; without a realization, these are somewhat
> useless. Kac states this very explicitly in
> http://www.arxiv.org/abs/q-alg/9709008:
>
> "It is a well kept secret that the theory of Kac-Moody algebras has
> been a disaster."
>
> [Moderator's note: this disaster has not influenced physics too much,
> and Kac-Moody algebras are very important and well functioning tool
> to study conformal field theories and perhaps other realizations.

Lubos, please note that Larsson (and Kac for that matter) referred to
_non-affine_ Kac-Moody algebras, while it seems to me what you have
in mind are current algebras, i.e. _affine_ KM algebras.

KM algebras fall into 3 classes:

1) Cartan matrix positive definite --> finite algebra

2) Cartan matrix positive semi-definite --> affine algebra = current
algebra in 2d

3) Cartan matrix indefinite --> exponential growth, desaster and all that

(paraphrased from http://www.arxiv.org/abs/hep-th/9411188)

> Thus, if West and others would find a nice realization of [itex]E_{11} it[/itex] would
> be a major mathematical breakthrough, irrespective of its relevance to
> M-theory (and the relevance of M-theory to physics). However, I have
> glanced at quite a few of these [itex]E_{11}[/itex] papers, and I have never seen any
> mentioning of exponential growth. This suggests that string theorists
> do not realize that this is a major difficultly, which has stalled
> Victor and other mathematicians for decades.

This is at least not true for all physicist. In

Gebert, Nicolai,
[itex]E_{10}[/itex] for beginners,
http://www.arxiv.org/abs/hep-th/9411188

it is emphasized quite strongly that there is exponential growth and
that things are pretty problematic concerning [itex]E_{10}[/itex]. But I am being
told that the point is that one can study consistent finite truncations
which can be handled and in terms of which we can do physics.

Urs Schreiber

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Sat, 17 Apr 2004, Lubos Motl wrote:\n\n> > that the E10 sigma-model reproduces these up to 30th order in\n> > some appropriate expansion.\n>\n> I would probably believe this particular technical claim even with a small\n> number of orders than 30. ;-) I guess that you only talk about the bosonic\n> sector of SUGRA?\n\nYes. But I was told that using some appropriate susy extension (I am not\nsure which one, though) you can similarly check that also the fermionic\ndegrees of freedom of sugra are there. Aparently this is not published\nyet, though.\n\n> If you ask me, my answer would be No, it would not be too weird.\n> Constructing the full covariant formulation of M-theory is a big task, and\n> it would seem more weird to me if someone suddenly found it using an\n> obscure and apparently a slightly ill-defined sigma-model, especially\n> because I don\'t expect M-theory to be *just* another string theory. :-)\n\nHm, the sigma model wouldn\'t be "another string theory", somehow, I think.\nIt would rather be a vast extension of what in quantum cosmology is\nsometimes called "miDi superspace", i.e. the configuration space of all\nmodes of all objects in the theory.\n\nI must say that I find the claim that such a 1+0 dim sigma model of all\nof M-theory exists (BTW, this is not a 1+1d sigma model describing string\nmotion in some background!) plausible, because it is quite similar to\nthe M=Matrix proposal, where the claim is, too, that all the degrees of\nfreedom can be parameterized by a 1+0 dimensional quantum mechanics.\n\nIndeed, I have a little private speculation how one could maybe make\na possible relation between the two 1+0d models which we know to\nreduce to 11d sugra in some limit, namely the E_10 model and BFSS.\n\nI have talked about that at the Coffee Table:\n\nhttp://golem.ph.utexas.edu/string/archives/000342.html .\n\nThe rough idea is the following: We know that close to a\nspacelike singularity (super)gravity decouples in the sense that\nnearby points on spatial hyperslices no longer interact because\nthey are outside each other\'s backward light cones and that\nin this limit the dynamics of each of the thus disconnected\nsmall causal patches of space are described in config space by\nthe billiard with walls given by the Weyl chamber of E_10\n(in the case of 11d sugra). This billiard motion is chaotic,\na generalization of the old BKL "mixmaster" idea.\n\nNow the "first law" of quantum chaos is that the universal\nbehaviour of any classically chaotic system (i.e. that\nbehaviour which remains when you slightly coarse-grain the\ndynamics so as to get rid of system-dependent details of the\ndynamics on small scales) is described by Random Matrix Theory,\ni.e. by a theory where you pick a random ensemble of Hamiltonians\nand average over their properties (see the above link for more on\nthis). In particular, the ensemble is in general Gaussian in the\nsense that the probability to find a particular u(N) matrix M\nin the ensemble is proportional to\n\nexp(- tr(M^2)) .\n\nTaking this well known fact in the context of the above\ncosmological Billiard tells us that 11d sugra in the limit\nwhere inteaction are negligible is well approximated by\na canonical matrix ensemble exp(- tr(M^2)).\n\nBut isn\'t that suggestive? Take the BFSS canonical ensemble\nin the limit where the interaction terms are negected. It,\ntoo, looks like exp(- tr P^2).\n\n> Nevertheless, I am gonna look at these things because what you say sounds\n> highly nontrivial.\n\nGreat. I hope you\'ll let us know what you come up with! :-)\n\n> > First of all I am confused why you refer to the "E10 current algebra".\n> > From hep-th/9411188 I seemed to learn that the whole point is that\n> > E10 is not a current algebra, because of its indefinite Cartan\n> > matrix. Only a Kac-Moody algebra with (semi)definite Cartan\n> > matrix is a current algebra.\n>\n> OK, so I wanted to say a more general word for the algebra that does not\n> have any such constraints of (semi)definiteness. Does it make a big\n> difference?\n\nYes, apparently. I am no expert yet on this stuff, but according\nto the introductory article hep-th/9411188 it makes a huge\ndifference. Non-affine KM algebras with indefinite Cartan matrices\nare apparently mind-bogglingly huge beasts which greatly\nexceed the complexity of the affine KM algebras. Hermann Nicolai\nhas emphasized that to me over and over again. In order to\nemphasize this point he had shown me files of piles of pages\nof computer output where order-by-order the affine KM algebras contained\nin E_10, as well as their irreps and the multiplicity with which\nthese appear, were listed. He said that mathematicians don\'t\nunderstand this huge algebra well yet, but that string theorists\nare slowly beginning to make progress in these matters.\n\n> I thought that just like the bosons compactified on an even self-dual\n> lattice \\Gamma_8 gives you an E_8 current algebra, the compactification on\n> another even self-dual lattice \\Gamma_{9+1} gives you the E_{10}\n> hyperbolic Kac-Moody algebra. The 10-dimensional Cartan subalgebra\n> (indefinite one, in this case) has vertex operators \\partial{X^i}, while\n> the roots are represented by \\exp(ik_iX_i(z)) where the momenta k_i belong\n> to the lattice, and they can be, in the E_{10} case, both time-like as\n> well as spacelike. Is there some huge difference that I missed?\n\nHm, no, probably not. It is just important that E_10 is not\na current algebra, as far as I understand.\n\n> > ...So this way exp(-:W:) is indeed the inverse of exp(:W:), at least\n> > if one of them is well defined in the first place.\n>\n> No, the colons don\'t change anything about my claim. Let me repeat my\n\nThe point of the colons was to distinguish\n\n:exp(W):\n\nfrom\n\nexp(:W:) .\n\nDue to OPEs we have\n\n:exp(W): :exp(-W): = possibly lots of correction terms .\n\nBut on the other hand\n\nexp(:W:)exp(-:W:) = exp(:W:-:W:) = 1 .\n\nAh, now I see what your are perhaps thinking: It is important\nthat here W is the same object everywhere, i.e. what I wrote is\n_not_ supposed to be a shorthand for\n\nexp(:W:(z)) exp(-:W:(w))\n\nwith z \\neq w . In fact, W is not supposed to depend on any coordinate\nanymore, because it is an integrated object.\n\n> > The point of all this is that given any stress-energy tensor T of the\n> > worlsheet CFT you get a new worldsheet CFT by setting A=T in the above\n> > formula. The new CFT can be interpreted as coming from the old one by\n> > a generalized symmetry operation on the background.\n>\n> That\'s probably interesting, but I did not get it. ;-/\n\nThere are really to statements here:\n\n1) Take the Virasor algebra generated by L_m satisfying\n[L_m,L_n] = (m-n)L_{m+n} + anomaly .\nNow define\n\nL\'_m := exp(-W) L_m exp(W)\n\nwith the exponentials defined as discussed above so that indeed\nexp(W)exp(-W) = 1 . Then obviously we have defined an algebra\nisomorphism and the new generators still satisfy the Virasoro\nalgebra:\n\n[L\'_m, L\'_n] = (m-n)L_{m+n} + anomaly .\n\n2) The second statement is that one can indeed interpret the\nnew L\'_m as the generators corresponding to a different background\nof the string theory. Since the states annihilated by the L\'_m\nare just exp(-W) times the states annihilated by the original\nL_m and the spectrum is the same and everything, the new\nbackground must be related to the original one by a symmetry\nof the background theory.\n\n> > The algebra of the :W: is huge and should have some close relation\n> > to that discussed on pp3 of hep-th/9411188.\n>\n> Do you say that it is huge in a general case? The (1,1) primary fields are\n> certainly rare. For example, the Ising model CFT contains 3 of them.\n\nI say that it is huge in genral. It is the full algebra of DDF invariants,\ni.e. of polynomials in the A_n^mu where\n\nA_n^mu \\propto \\int dz \\partial X^mu exp(in k \\cdot X)\n\n(for the bosonic string).\n\n> The absence of fermions is another reason why I feel that this E_{10} is\n> unlikely to be far-reaching and even more unlikely to contain the whole\n> M-theory. The inclusion of the fermions would try to lead us to\n> superalgebras such as OSp(1|32) or something similar, which is very\n> different from E_{10}.\n\nAs I said above, there is apparently a susy extension of E_10 which does\ncorrectly reproduce the fermionic degrees of freedom of 11d sugra\n(among other things). As far as I understood the corresponding\npublications are in preparation.\n\nAnother (mayb equivalent) straighforward inclusion of fermions\nis the promotion of the 1+0d E_10 model to the respective\ndeRahm model, as I have mentioned before.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Sat, 17 Apr 2004, Lubos Motl wrote:

> > that the E10 [itex]\sigma-model[/itex] reproduces these up to 30th order in
> > some appropriate expansion.
>
> I would probably believe this particular technical claim even with a small
> number of orders than 30. ;-) I guess that you only talk about the bosonic
> sector of SUGRA?

Yes. But I was told that using some appropriate susy extension (I am not
sure which one, though) you can similarly check that also the fermionic
degrees of freedom of sugra are there. Aparently this is not published
yet, though.

> If you ask me, my answer would be No, it would not be too weird.
> Constructing the full covariant formulation of M-theory is a big task, and
> it would seem more weird to me if someone suddenly found it using an
> obscure and apparently a slightly ill-defined [itex]\sigma-model,[/itex] especially
> because I don't expect M-theory to be *just* another string theory. :-)

Hm, the [itex]\sigma[/itex] model wouldn't be "another string theory", somehow, I think.
It would rather be a vast extension of what in quantum cosmology is
sometimes called "miDi superspace", i.e. the configuration space of all
modes of all objects in the theory.

I must say that I find the claim that such [itex]a 1+0[/itex] dim [itex]\sigma[/itex] model of all
of M-theory exists (BTW, this is not [itex]a 1+1d \sigma[/itex] model describing string
motion in some background!) plausible, because it is quite similar to
the M=Matrix proposal, where the claim is, too, that all the degrees of
freedom can be parameterized by [itex]a 1+0[/itex] dimensional quantum mechanics.

Indeed, I have a little private speculation how one could maybe make
a possible relation between the two [itex]1+0d[/itex] models which we know to
reduce to 11d sugra in some limit, namely the [itex]E_{10}[/itex] model and BFSS.

I have talked about that at the Coffee Table:

http://golem.ph.utexas.edu/string/archives/000342.html .

The rough idea is the following: We know that close to a
spacelike singularity (super)gravity decouples in the sense that
nearby points on spatial hyperslices no longer interact because
they are outside each other's backward light cones and that
in this limit the dynamics of each of the thus disconnected
small causal patches of space are described in config space by
the billiard with walls given by the Weyl chamber of [itex]E_{10}[/itex]
(in the case of 11d sugra). This billiard motion is chaotic,
a generalization of the old BKL "mixmaster" idea.

Now the "first law" of quantum chaos is that the universal
behaviour of any classically chaotic system (i.e. that
behaviour which remains when you slightly coarse-grain the
dynamics so as to get rid of system-dependent details of the
dynamics on small scales) is described by Random Matrix Theory,
i.e. by a theory where you pick a random ensemble of Hamiltonians
and average over their properties (see the above link for more on
this). In particular, the ensemble is in general Gaussian in the
sense that the probability to find a particular u(N) matrix M
in the ensemble is proportional to

[tex]\exp(- tr(M^2)) .[/tex]

Taking this well known fact in the context of the above
cosmological Billiard tells us that 11d sugra in the limit
where inteaction are negligible is well approximated by
a canonical matrix ensemble [itex]\exp(- tr(M^2))[/itex].

But isn't that suggestive? Take the BFSS canonical ensemble
in the limit where the interaction terms are negected. It,
too, looks like [itex]\exp(- tr P^2)[/itex].

> Nevertheless, I am gonna look at these things because what you say sounds
> highly nontrivial.

Great. I hope you'll let us know what you come up with! :-)

> > First of all I am confused why you refer to the "E10 current algebra".
> > From http://www.arxiv.org/abs/hep-th/9411188 I seemed to learn that the whole point is that
> > E10 is not a current algebra, because of its indefinite Cartan
> > matrix. Only a Kac-Moody algebra with (semi)definite Cartan
> > matrix is a current algebra.
>
> OK, so I wanted to say a more general word for the algebra that does not
> have any such constraints of (semi)definiteness. Does it make a big
> difference?

Yes, apparently. I am no expert yet on this stuff, but according
to the introductory article http://www.arxiv.org/abs/hep-th/9411188 it makes a huge
difference. Non-affine KM algebras with indefinite Cartan matrices
are apparently mind-bogglingly huge beasts which greatly
exceed the complexity of the affine KM algebras. Hermann Nicolai
has emphasized that to me over and over again. In order to
emphasize this point he had shown me files of piles of pages
of computer output where order-by-order the affine KM algebras contained
in [itex]E_{10},[/itex] as well as their irreps and the multiplicity with which
these appear, were listed. He said that mathematicians don't
understand this huge algebra well yet, but that string theorists
are slowly beginning to make progress in these matters.

> I thought that just like the bosons compactified on an even self-dual
> lattice [itex]\Gamma_8[/itex] gives you an [itex]E_8[/itex] current algebra, the compactification on
> another even self-dual lattice [itex]\Gamma_{9+1}[/itex] gives you the [itex]E_{10}[/itex]
> hyperbolic Kac-Moody algebra. The 10-dimensional Cartan subalgebra
> (indefinite one, in this case) has vertex operators [itex]\partial{X^i},[/itex] while
> the roots are represented by [itex]\exp(ik_iX_i(z))[/itex] where the momenta [itex]k_i[/itex] belong
> to the lattice, and they can be, in the [itex]E_{10}[/itex] case, both time-like as
> well as spacelike. Is there some huge difference that I missed?

Hm, no, probably not. It is just important that [itex]E_{10}[/itex] is not
a current algebra, as far as I understand.

> > ...So this way [itex]\exp(-:W:)[/itex] is indeed the inverse of [itex]\exp(:W:), at[/itex] least
> > if one of them is well defined in the first place.
>
> No, the colons don't change anything about my claim. Let me repeat my

The point of the colons was to distinguish

[tex]:\exp(W):[/tex]

from

[tex]\exp(:W:) .[/tex]

Due to OPEs we have

[tex]:\exp(W): :\exp(-W): =[/itex] possibly lots of correction terms .

But on the other hand

[itex]\exp(:W:)\exp(-:W:) = \exp(:W:-:W:) = 1 .[/tex]

Ah, now I see what your are perhaps thinking: It is important
that here W is the same object everywhere, i.e. what I wrote is
_not_ supposed to be a shorthand for

[tex]\exp(:W:(z)) \exp(-:W:(w))[/tex]

with [itex]z \neq w .[/itex] In fact, W is not supposed to depend on any coordinate
anymore, because it is an integrated object.

> > The point of all this is that given any stress-energy tensor T of the
> > worlsheet CFT you get a new worldsheet CFT by setting A=T in the above
> > formula. The new CFT can be interpreted as coming from the old one by
> > a generalized symmetry operation on the background.
>
> That's probably interesting, but I did not get it[itex]. ;-/[/itex]

There are really to statements here:

1) Take the Virasor algebra generated by [itex]L_m[/itex] satisfying
[itex][L_m,L_n] = (m-n)L_{m+n} +[/itex] anomaly .
Now define

[tex]L'_m := \exp(-W) L_m \exp(W)[/tex]

with the exponentials defined as discussed above so that indeed
[itex]\exp(W)\exp(-W) = 1 .[/itex] Then obviously we have defined an algebra
isomorphism and the new generators still satisfy the Virasoro
algebra:

[tex][L'_m, L'_n] = (m-n)L_{m+n} +[/itex] anomaly .

2) The second statement is that one can indeed interpret the
new [itex]L'_m[/itex] as the generators corresponding to a different background
of the string theory. Since the states annihilated by the [itex]L'_m[/itex]
are just [itex]\exp(-W)[/itex] times the states annihilated by the original
[itex]L_m[/itex] and the spectrum is the same and everything, the new
background must be related to the original one by a symmetry
of the background theory.

> > The algebra of the :W: is huge and should have some close relation
> > to that discussed on pp3 of http://www.arxiv.org/abs/hep-th/9411188.
>
> Do you say that it is huge in a general case? The (1,1) primary fields are
> certainly rare. For example, the Ising model CFT contains 3 of them.

I say that it is huge in genral. It is the full algebra of DDF invariants,
i.e. of polynomials in the [itex]A_n^\mu[/itex] where

[itex]A_n^\mu \propto \int dz \partial X^\mu \exp(in k \cdot X)[/tex]

(for the bosonic string).

> The absence of fermions is another reason why I feel that this [itex]E_{10}[/itex] is
> unlikely to be far-reaching and even more unlikely to contain the whole
> M-theory. The inclusion of the fermions would try to lead us to
> superalgebras such as OSp(1|32) or something similar, which is very
> different from [itex]E_{10}[/itex].

As I said above, there is apparently a susy extension of [itex]E_{10}[/itex] which does
correctly reproduce the fermionic degrees of freedom of 11d sugra
(among other things). As far as I understood the corresponding
publications are in preparation.

Another (mayb equivalent) straighforward inclusion of fermions
is the promotion of the [itex]1+0d E_{10}[/itex] model to the respective
deRahm model, as I have mentioned before.

Urs Schreiber

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Sun, 18 Apr 2004, Urs Schreiber wrote:\n\n> I say that it is huge in genral. It is the full algebra of DDF invariants,\n> i.e. of polynomials in the A_n^mu where\n>\n> A_n^mu \\propto \\int dz \\partial X^mu exp(in k \\cdot X)\n>\n> (for the bosonic string).\n\nLet me make that more precise:\n\nPick any physical state by applying lots of the A_n^\\mu to the\ntachyon state. By the state/operator correspondence the resulting\nstate comes from an integrated weight 1 vertex (the V(\\psi,z) in\nequation (0.10) of hep-th/9411188).\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Sun, 18 Apr 2004, Urs Schreiber wrote:

> I say that it is huge in genral. It is the full algebra of DDF invariants,
> i.e. of polynomials in the [itex]A_n^\mu[/itex] where
>
> [itex]A_n^\mu \propto \int dz \partial X^\mu \exp(in k \cdot X)[/itex]
>
> (for the bosonic string).

Let me make that more precise:

Pick any physical state by applying lots of the [itex]A_n^\mu[/itex] to the
tachyon state. By the state/operator correspondence the resulting
state comes from an integrated weight 1 vertex (the [itex]V(\psi,z)[/itex] in
equation (0.10) of http://www.arxiv.org/abs/hep-th/9411188).

Lubos Motl

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Sun, 18 Apr 2004, Urs Schreiber wrote:\n\n> Yes. But I was told that using some appropriate susy extension (I am not\n> sure which one, though) you can similarly check that also the fermionic\n> degrees of freedom of sugra are there. Aparently this is not published\n> yet, though.\n\nI would like to see it first, to avoid building on someone\'s wishful\nthinking. Be sure that I will be among the first believers if such a\ntheory really works, including the fermions. ;-)\n\n> I must say that I find the claim that such a 1+0 dim sigma model of all\n> of M-theory exists (BTW, this is not a 1+1d sigma model describing string\n> motion in some background!) plausible, because it is quite similar to\n> the M=Matrix proposal, where the claim is, too, that all the degrees of\n> freedom can be parameterized by a 1+0 dimensional quantum mechanics.\n\nSure, why not. :-) But there is usually a long path from these big dreams\nto actual results.\n\n> ...dynamics so as to get rid of system-dependent details of the\n> dynamics on small scales) is described by Random Matrix Theory,\n\nDo you want to reproduce M-theory amplitudes approximately, or exactly?\nIs this connection with Random Matrix Theory a trick to get the right\nresults? Or is it evidence that this description cannot work exactly?\n\n> But isn\'t that suggestive? Take the BFSS canonical ensemble\n> in the limit where the interaction terms are negected. It,\n> too, looks like exp(- tr P^2).\n\nThere are also many systems that I don\'t like where you can obtain\n\\exp(-\\tr P^2), too. :-) This exponential is not such a difficult concept\nfor us to be shocked if we see it somewhere.\n\n> Yes, apparently. I am no expert yet on this stuff, but according\n> to the introductory article hep-th/9411188 it makes a huge\n> difference. Non-affine KM algebras with indefinite Cartan matrices\n> are apparently mind-bogglingly huge beasts which greatly...\n\nWell, I understand why E_9 is the affine E_8, and why the hyperbolic\nalgebra E_{10} is even "bigger". There are many generators for a single\nroot, and infinitely many roots organized in a Minkowski-like Cartan\nspace, and so forth. Using the words "mind-boggling" for such things does\nnot look quite appropriate to me. These objects are simply large, which is\nboth a good as well as a bad message. I don\'t see anything amazing about\nan object only because it is "large"; and it makes such an object less\nready for various operations. For example, we don\'t want to build\nYang-Mills theories based on infinite-dimensional non-compact groups, at\nleast so far. :-)\n\nWhat I still think is straightforward is that one can obtain such\nalgebras, with the correct commutation relations, from a compactified\nbosonic CFT, and the maths justifying this fact is as trivial as in the\nE_8 case. Am I wrong? This is about the very definition of E_{10}. The\ncompactified bosonic CFT simply reproduces (and allows you to check) the\ndefining axioms of E_{10} step by step. If there is something nontrivial\nabout these things, you will have to be more specific to explain it.\n\n> The point of the colons was to distinguish\n> :exp(W):\n> from\n> exp(:W:) .\n\nOK, understood. But then one must note that the non-ordered exp(:W:)\ncontains a lot of hard-to-interpret singular terms which make such an\noperator useless and, in fact, also ill-defined if W is a local operator.\n\n> Ah, now I see what your are perhaps thinking: It is important\n> that here W is the same object everywhere, i.e. what I wrote is\n> _not_ supposed to be a shorthand for\n>\n> exp(:W:(z)) exp(-:W:(w))\n>\n> with z \\neq w .\n\nThat\'s exactly one of the problems. If W(z) is a local operator, there is\nno working easy way to define W(z)^2. The only way to do so is to consider\nW(z)W(w) for w approaching z. You will see that W(z)W(w) has a lot of\nsingularities that you must deal with, subtract, normal-order, and so on,\nand these operations will destroy the naive identities that you are\nthinking about.\n\n> In fact, W is not supposed to depend on any coordinate\n> anymore, because it is an integrated object.\n\nI just don\'t see anything sharp about these objects. Of course, you can\ndefine the U(\\infty) group acting on the whole Hilbert space - which is\nthe only thing I can imagine where you are led if you try to define an\nalgebra that contains more than just a "couple" of operators. But every\nquantum mechanical theory has this U(\\infty) "symmetry" group and we use\nit extensively all the time e.g. when we switch from one basis to another\n(it is only a symmetry of the norm, not a symmetry of dynamics). I don\'t\nprecisely understand whether you want to define something more useful,\nconstrained, and original than this universal and well-known U(\\infty),\nand if you want, what it is. Depending on your constraints what the "W"\noperators can be, you can obtain a couple of algebras, but because you\nhave not described any clear constraints so far, it seems that you talk\nabout the U(\\infty) algebra.\n\n> There are really to statements here:\n>\n> 1) Take the Virasor algebra generated by L_m satisfying\n> [L_m,L_n] = (m-n)L_{m+n} + anomaly .\n> Now define\n>\n> L\'_m := exp(-W) L_m exp(W)\n>\n> with the exponentials defined as discussed above so that indeed\n> exp(W)exp(-W) = 1 . Then obviously we have defined an algebra\n> isomorphism and the new generators still satisfy the Virasoro\n> algebra:\n>\n> [L\'_m, L\'_n] = (m-n)L_{m+n} + anomaly .\n\nWell, that\'s a basic lesson of undergraduate linear algebra that\nconjugation does not change the commutation relations. I guess that both\nof us realize that every physicist must know these things, and that we are\nusing them all the time - and it is unlikely to make a revolutionary\ndiscovery just by noting that conjugation preserves commutators.\n\n> 2) The second statement is that one can indeed interpret the\n> new L\'_m as the generators corresponding to a different background\n> of the string theory. Since the states annihilated by the L\'_m\n> are just exp(-W) times the states annihilated by the original\n> L_m and the spectrum is the same and everything, the new\n> background must be related to the original one by a symmetry\n> of the background theory.\n\nI find this statement physically vacuous. You have not constructed any new\ntheory at all; you just renamed the basis vectors of its Hilbert space in\na chaotic and arbitrary fashion. If you want to get a spacetime\ninterpretation of the "transformed" theory, you will have to undo this\nunphysical exp(W) transformation, and return to the original operators\nsuch as X^\\mu(z) which encode the position of the string in spacetime -\ninstead of \\exp(-W) X^\\mu(z) \\exp(W) - and you will finally obtain\n*identical* physical results. You have not even changed any moduli in the\ntheory. It seems that what you did is nothing more than the U(\\infty)\nsymmetry discussed above, and this whole "large" symmetry is not\nphysically useful unless you restrict it to something that preserves some\nother structure of the theory beyond the norm on the Hilbert space.\n\n> I say that it is huge in genral. It is the full algebra of DDF invariants,\n\nOK, once again, it seems that you are talking about the full U(\\infty)\nacting on the Hilbert space which I find useless for physics. It is an\nalgebra that does not commute with the spacetime Hamiltonian, for example.\nIt does not even have any simple commutation relations with these\nspacetime Poincare generators. I think that what you are talking about is\nthe algebra of everything - and everything is more or less isomorphic to\nnothing. It is the sort of group that allows you to "prove" that a\nharmonic oscillator is "equivalent" to the Standard Model: both of them\nhave an infinite-dimensional Hilbert space, and they can be mapped to one\nanother, and every operator in one theory can be identified with an\noperator in the other theory. But I say that such a conclusion is\nunphysical because the structures that physics requires are more than just\nthe norm. You need some sort of dynamics and interpretable operators - for\nexample the Hamiltonian - and the only interesting transformations are\nthose under which the Hamiltonian (...) is invariant, or at least\ntransforms in a controllable way.\n\n> As I said above, there is apparently a susy extension of E_10 which does\n> correctly reproduce the fermionic degrees of freedom of 11d sugra\n> (among other things). As far as I understood the corresponding\n> publications are in preparation.\n\nI thought that there is a theorem that no useful superalgebra extending\nE_{10} ( or at least E_8 ) exists.\n\n> Another (mayb equivalent) straighforward inclusion of fermions\n> is the promotion of the 1+0d E_10 model to the respective\n> deRahm model, as I have mentioned before.\n\nOK, but it is a random guessing, is not it? I sort of feel that you want\nto say that if you construct anything huge enough, it must be M-theory.\nWell, this is an assumption that I certainly don\'t share. But it is\nprobably still more rational than the assumption of Lee Smolin that\nanything *simple* enough - e.g. the 3D Chern-Simons theory - must be\nequivalent to M-theory. ;-)\n\nCheers,\nLubos\n__________________________________________________ ____________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)\n^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Sun, 18 Apr 2004, Urs Schreiber wrote:

> Yes. But I was told that using some appropriate susy extension (I am not
> sure which one, though) you can similarly check that also the fermionic
> degrees of freedom of sugra are there. Aparently this is not published
> yet, though.

I would like to see it first, to avoid building on someone's wishful
thinking. Be sure that I will be among the first believers if such a
theory really works, including the fermions. ;-)

> I must say that I find the claim that such [itex]a 1+0[/itex] dim [itex]\sigma[/itex] model of all
> of M-theory exists (BTW, this is not [itex]a 1+1d \sigma[/itex] model describing string
> motion in some background!) plausible, because it is quite similar to
> the M=Matrix proposal, where the claim is, too, that all the degrees of
> freedom can be parameterized by [itex]a 1+0[/itex] dimensional quantum mechanics.

Sure, why not. :-) But there is usually a long path from these big dreams
to actual results.

> ...dynamics so as to get rid of system-dependent details of the
> dynamics on small scales) is described by Random Matrix Theory,

Do you want to reproduce M-theory amplitudes approximately, or exactly?
Is this connection with Random Matrix Theory a trick to get the right
results? Or is it evidence that this description cannot work exactly?

> But isn't that suggestive? Take the BFSS canonical ensemble
> in the limit where the interaction terms are negected. It,
> too, looks like [itex]\exp(- tr P^2)[/itex].

There are also many systems that I don't like where you can obtain
[itex]\exp(-\tr P^2),[/itex] too. :-) This exponential is not such a difficult concept
for us to be shocked if we see it somewhere.

> Yes, apparently. I am no expert yet on this stuff, but according
> to the introductory article http://www.arxiv.org/abs/hep-th/9411188 it makes a huge
> difference. Non-affine KM algebras with indefinite Cartan matrices
> are apparently mind-bogglingly huge beasts which greatly...

Well, I understand why [itex]E_9[/itex] is the affine [itex]E_8,[/itex] and why the hyperbolic
algebra [itex]E_{10}[/itex] is even "bigger". There are many generators for a single
root, and infinitely many roots organized in a Minkowski-like Cartan
space, and so forth. Using the words "mind-boggling" for such things does
not look quite appropriate to me. These objects are simply large, which is
both a good as well as a bad message. I don't see anything amazing about
an object only because it is "large"; and it makes such an object less
ready for various operations. For example, we don't want to build
Yang-Mills theories based on infinite-dimensional non-compact groups, at
least so far. :-)

What I still think is straightforward is that one can obtain such
algebras, with the correct commutation relations, from a compactified
bosonic CFT, and the maths justifying this fact is as trivial as in the
[itex]E_8[/itex] case. Am I wrong? This is about the very definition of [itex]E_{10}[/itex]. The
compactified bosonic CFT simply reproduces (and allows you to check) the
defining axioms of [itex]E_{10}[/itex] step by step. If there is something nontrivial
about these things, you will have to be more specific to explain it.

> The point of the colons was to distinguish
> [itex]:\exp(W):[/itex]
> from
> [itex]\exp(:W:) .[/itex]

OK, understood. But then one must note that the non-ordered [itex]\exp(:W:)[/itex]
contains a lot of hard-to-interpret singular terms which make such an
operator useless and, in fact, also ill-defined if W is a local operator.

> Ah, now I see what your are perhaps thinking: It is important
> that here W is the same object everywhere, i.e. what I wrote is
> _not_ supposed to be a shorthand for
>
> [itex]\exp(:W:(z)) \exp(-:W:(w))[/itex]
>
> with [itex]z \neq w .[/itex]

That's exactly one of the problems. If W(z) is a local operator, there is
no working easy way to define [itex]W(z)^2[/itex]. The only way to do so is to consider
W(z)W(w) for w approaching z. You will see that W(z)W(w) has a lot of
singularities that you must deal with, subtract, normal-order, and so on,
and these operations will destroy the naive identities that you are
thinking about.

> In fact, W is not supposed to depend on any coordinate
> anymore, because it is an integrated object.

I just don't see anything sharp about these objects. Of course, you can
define the [itex]U(\infty)[/itex] group acting on the whole Hilbert space - which is
the only thing I can imagine where you are led if you try to define an
algebra that contains more than just a "couple" of operators. But every
quantum mechanical theory has this [itex]U(\infty) "[/itex]symmetry" group and we use
it extensively all the time e.g. when we switch from one basis to another
(it is only a symmetry of the norm, not a symmetry of dynamics). I don't
precisely understand whether you want to define something more useful,
constrained, and original than this universal and well-known [itex]U(\infty),[/itex]
and if you want, what it is. Depending on your constraints what the "W"
operators can be, you can obtain a couple of algebras, but because you
have not described any clear constraints so far, it seems that you talk
about the [itex]U(\infty)[/itex] algebra.

> There are really to statements here:
>
> 1) Take the Virasor algebra generated by [itex]L_m[/itex] satisfying
> [itex][L_m,L_n] = (m-n)L_{m+n} +[/itex] anomaly .
> Now define
>
> [itex]L'_m := \exp(-W) L_m \exp(W)[/itex]
>
> with the exponentials defined as discussed above so that indeed
> [itex]\exp(W)\exp(-W) = 1 .[/itex] Then obviously we have defined an algebra
> isomorphism and the new generators still satisfy the Virasoro
> algebra:
>
> [itex][L'_m, L'_n] = (m-n)L_{m+n} +[/itex] anomaly .

Well, that's a basic lesson of undergraduate linear algebra that
conjugation does not change the commutation relations. I guess that both
of us realize that every physicist must know these things, and that we are
using them all the time - and it is unlikely to make a revolutionary
discovery just by noting that conjugation preserves commutators.

> 2) The second statement is that one can indeed interpret the
> new [itex]L'_m[/itex] as the generators corresponding to a different background
> of the string theory. Since the states annihilated by the [itex]L'_m[/itex]
> are just [itex]\exp(-W)[/itex] times the states annihilated by the original
> [itex]L_m[/itex] and the spectrum is the same and everything, the new
> background must be related to the original one by a symmetry
> of the background theory.

I find this statement physically vacuous. You have not constructed any new
theory at all; you just renamed the basis vectors of its Hilbert space in
a chaotic and arbitrary fashion. If you want to get a spacetime
interpretation of the "transformed" theory, you will have to undo this
unphysical [itex]\exp(W)[/itex] transformation, and return to the original operators
such as [itex]X^\mu(z)[/itex] which encode the position of the string in spacetime -
instead of [itex]\exp(-W) X^\mu(z) \exp(W) -[/itex] and you will finally obtain
*identical* physical results. You have not even changed any moduli in the
theory. It seems that what you did is nothing more than the [itex]U(\infty)[/itex]
symmetry discussed above, and this whole "large" symmetry is not
physically useful unless you restrict it to something that preserves some
other structure of the theory beyond the norm on the Hilbert space.

> I say that it is huge in genral. It is the full algebra of DDF invariants,

OK, once again, it seems that you are talking about the full [itex]U(\infty)[/itex]
acting on the Hilbert space which I find useless for physics. It is an
algebra that does not commute with the spacetime Hamiltonian, for example.
It does not even have any simple commutation relations with these
spacetime Poincare generators. I think that what you are talking about is
the algebra of everything - and everything is more or less isomorphic to
nothing. It is the sort of group that allows you to "prove" that a
harmonic oscillator is "equivalent" to the Standard Model: both of them
have an infinite-dimensional Hilbert space, and they can be mapped to one
another, and every operator in one theory can be identified with an
operator in the other theory. But I say that such a conclusion is
unphysical because the structures that physics requires are more than just
the norm. You need some sort of dynamics and interpretable operators - for
example the Hamiltonian - and the only interesting transformations are
those under which the Hamiltonian (...) is invariant, or at least
transforms in a controllable way.

> As I said above, there is apparently a susy extension of [itex]E_{10}[/itex] which does
> correctly reproduce the fermionic degrees of freedom of 11d sugra
> (among other things). As far as I understood the corresponding
> publications are in preparation.

I thought that there is a theorem that no useful superalgebra extending
[itex]E_{10} ( or at[/itex] least [itex]E_8 )[/itex] exists.

> Another (mayb equivalent) straighforward inclusion of fermions
> is the promotion of the [itex]1+0d E_{10}[/itex] model to the respective
> deRahm model, as I have mentioned before.

OK, but it is a random guessing, is not it? I sort of feel that you want
to say that if you construct anything huge enough, it must be M-theory.
Well, this is an assumption that I certainly don't share. But it is
probably still more rational than the assumption of Lee Smolin that
anything *simple* enough - e.g. the 3D Chern-Simons theory - must be
equivalent to M-theory. ;-)

Cheers,
Lubos
__{____________________________________________________________________ ________}
E-mail: lumo@matfyz.cz fax: [itex]+1-617/496-0110[/itex] Web: http://lumo.matfyz.cz/
eFax: [itex]+1-801/454-1858[/itex] work: [itex]+1-617/496-8199[/itex] home: [itex]+1-617/868-4487[/itex] (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^

Urs Schreiber

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Sun, 18 Apr 2004, Urs Schreiber wrote:\n\n> 1) Take the Virasor algebra generated by L_m satisfying\n> [L_m,L_n] = (m-n)L_{m+n} + anomaly .\n> Now define\n>\n> L\'_m := exp(-W) L_m exp(W)\n>\n> with the exponentials defined as discussed above so that indeed\n> exp(W)exp(-W) = 1 . Then obviously we have defined an algebra\n> isomorphism and the new generators still satisfy the Virasoro\n> algebra:\n>\n> [L\'_m, L\'_n] = (m-n)L_{m+n} + anomaly .\n\nIn my discussions with Ioannis Giannakis I have come up\nwith an alternative description in terms of string\nfield theory, which might be instructive:\n\n\nConsider the equation of motion of open bosonic cubic string\nfield theory:\n\nQ \\phi + \\phi \\star \\phi = 0\n\n(setting the constant factor to unity). Here Q is the BRST operator\nfor a string and \\star is the star product which\nmerges two string states. \\phi is the string field.\n\nNow suppose we want to study single strings in non-trivial\nbackgrounds. We could do that by trying to find a consistent\nworldsheet CFT. But the above equation should give us\nan alternative prescription:\n\nPick any solution \\phi to the above string field equations of motion.\nNow perturb this solution by adding an infinitesimal "test field"\n\\psi, namely the field of a single "test string" probing the background\ndescribed by phi:\n\n\\phi \\to \\phi + \\psi .\n\nBy inserting this ansatz into the above equation of motion, using the\nfact that the original \\phi is a solution all by itself and that\n\\psi is infinitesimal, we get the equation\n\nQ \\psi + \\phi \\star \\psi + \\psi \\star \\phi = 0 .\n\nIf I introduce the notation\n\n\\{\\phi, \\cdot\\} = \\psi \\star \\cdot + \\cdot \\star \\phi\n\nthis becomes\n\n(Q \\psi + \\{\\phi, \\cdpt\\}) \\psi = 0\n\nwhich should be the quantum consraint for the single string \\psi\nin the background described by the string field \\phi .\n\nIs this interpretation consistent?\n\nIf it is, it should be possible ot interpret\n\nQ^\\prime := Q + \\{\\phi , \\cdot \\}\n\nas the BRST operator of the string in the background \\phi.\n\nCan that be? Is Q^\\prime nilpotent?\n\nHm, maybe I am confused. Comments are welcome. Anyway, what I\nwanted to say is that if the excitation \\phi is a symmetry of the\ntheory in the sense that turning on \\phi does not really change\nthe physics, then we would expect that\n\nQ^\\prime = \\exp(-W) Q \\exp(W) .\n\nThis are symmetries of the type studied in\n\nIoannis Giannkis,\nString in nontrivial gravitino and RR backgrounds\nhep-th/0205219\n\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Sun, 18 Apr 2004, Urs Schreiber wrote:

> 1) Take the Virasor algebra generated by [itex]L_m[/itex] satisfying
> [itex][L_m,L_n] = (m-n)L_{m+n} +[/itex] anomaly .
> Now define
>
> [itex]L'_m := \exp(-W) L_m \exp(W)[/itex]
>
> with the exponentials defined as discussed above so that indeed
> [itex]\exp(W)\exp(-W) = 1 .[/itex] Then obviously we have defined an algebra
> isomorphism and the new generators still satisfy the Virasoro
> algebra:
>
> [itex][L'_m, L'_n] = (m-n)L_{m+n} +[/itex] anomaly .

In my discussions with Ioannis Giannakis I have come up
with an alternative description in terms of string
field theory, which might be instructive:

Consider the equation of motion of open bosonic cubic string
field theory:

[tex]Q \phi + \phi \star \phi =[/tex]

(setting the constant factor to unity). Here Q is the BRST operator
for a string and [itex]\star[/itex] is the star product which
merges two string states[itex]. \phi[/itex] is the string field.

Now suppose we want to study single strings in non-trivial
backgrounds. We could do that by trying to find a consistent
worldsheet CFT. But the above equation should give us
an alternative prescription:

Pick any solution [itex]\phi[/itex] to the above string field equations of motion.
Now perturb this solution by adding an infinitesimal "test field"
[itex]\psi,[/itex] namely the field of a single "test string" probing the background
described by [itex]\phi:\phi \to \phi + \psi .[/itex]

By inserting this ansatz into the above equation of motion, using the
fact that the original [itex]\phi[/itex] is a solution all by itself and that
[itex]\psi[/itex] is infinitesimal, we get the equation

[tex]Q \psi + \phi \star \psi + \psi \star \phi =[/itex] .

If I introduce the notation

[itex]\{\phi, \cdot\} = \psi \star \cdot + \cdot \star \phi[/tex]

this becomes

[tex](Q \psi + \{\phi, \cdpt\}) \psi =[/tex]

which should be the quantum consraint for the single string [itex]\psi[/itex]
in the background described by the string field [itex]\phi .[/itex]

Is this interpretation consistent?

If it is, it should be possible ot interpret

[tex]Q^\prime := Q + \{\phi , \cdot \}[/tex]

as the BRST operator of the string in the background [itex]\phi[/itex].

Can that be? Is [itex]Q^\prime[/itex] nilpotent?

Hm, maybe I am confused. Comments are welcome. Anyway, what I
wanted to say is that if the excitation [itex]\phi[/itex] is a symmetry of the
theory in the sense that turning on [itex]\phi[/itex] does not really change
the physics, then we would expect that

[tex]Q^\prime = \exp(-W) Q \exp(W) .[/tex]

This are symmetries of the type studied in

Ioannis Giannkis,
String in nontrivial gravitino and RR backgrounds
http://www.arxiv.org/abs/hep-th/0205219

Urs Schreiber

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Sun, 18 Apr 2004, Lubos Motl wrote:\n\n> Sure, why not. :-) But there is usually a long path from these big dreams\n> to actual results.\n\nSure.\n\n> > ...dynamics so as to get rid of system-dependent details of the\n> > dynamics on small scales) is described by Random Matrix Theory,\n>\n> Do you want to reproduce M-theory amplitudes approximately, or exactly?\n> Is this connection with Random Matrix Theory a trick to get the right\n> results? Or is it evidence that this description cannot work exactly?\n\nThis was supposed to be evidence that at least in the discussed limit\nthe E_10 model is indeed equivalent to the BFSS model. Of course\nthe limit is pretty radical and you may argue that it doesn\'t allow\nany conclusion about the equivalence of the two models\naway from this limit.\n\n> OK, understood. But then one must note that the non-ordered exp(:W:)\n> contains a lot of hard-to-interpret singular terms which make such an\n> operator useless and, in fact, also ill-defined if W is a local operator.\n>\n> > Ah, now I see what your are perhaps thinking: It is important\n> > that here W is the same object everywhere, i.e. what I wrote is\n> > _not_ supposed to be a shorthand for\n> >\n> > exp(:W:(z)) exp(-:W:(w))\n> >\n> > with z \\neq w .\n>\n> That\'s exactly one of the problems. If W(z) is a local operator, there is\n> no working easy way to define W(z)^2. The only way to do so is to consider\n> W(z)W(w) for w approaching z. You will see that W(z)W(w) has a lot of\n> singularities that you must deal with, subtract, normal-order, and so on,\n> and these operations will destroy the naive identities that you are\n> thinking about.\n\nWe are still not talking about the same constructions, it seems to me.\nI am not considering things like W(z)^2. As I said, the W are integrated\nobjects W = \\int dz something(z).\n\nFor instance the W which induces T-duality is of the form\n\nW \\propto \\int d\\sigma (exp(i \\sqrt{2}k\\cdot X) - h.c.) .\n\nCompare equation (3.18) of hep-th/9511061 .\n\n> I just don\'t see anything sharp about these objects. Of course, you can\n> define the U(\\infty) group acting on the whole Hilbert space - which is\n> the only thing I can imagine where you are led if you try to define an\n> algebra that contains more than just a "couple" of operators.\n\nThat\'s too large and indeed without physical content. But as I said\nbefore, the restriction on the allowed W is that the similarity\ntransformation preserves T - \\bar T. Only under this condition can the\nresulting new algebra be interpreted as coming from a string in a certain\nbackground.\n\n> > [L\'_m, L\'_n] = (m-n)L_{m+n} + anomaly .\n>\n> Well, that\'s a basic lesson of undergraduate linear algebra that\n\nYes. :-)\n\n> conjugation does not change the commutation relations. I guess that both\n> of us realize that every physicist must know these things, and that we are\n> using them all the time - and it is unlikely to make a revolutionary\n> discovery just by noting that conjugation preserves commutators.\n\nIt is interesting to examine how different admissable W give\nrise to certain symmetries/dualities of the string backgrounds.\nYou can use this also to find deformations which are not symmetries\nbut give new backgrounds, inequivalent to the one one started\nfrom.\n\n> I find this statement physically vacuous. You have not constructed any new\n> theory at all; you just renamed the basis vectors of its Hilbert space in\n\nTrue, but that\'s precisely what a symmetry/duality is supposed to do.\nAs you know a string state with winding number n and momentum number m\nis sent to a state with winding m and momentum n and this is\na symmetry. So T-duality, too, could be described as just a\nrenaming of vectors in a Hilbert space, which it is. But of\ncourse that does not mean that it is not interesting.\n\n> I thought that there is a theorem that no useful superalgebra extending\n> E_{10} ( or at least E_8 ) exists.\n\nAha, don\'t know about that. Do you have a reference?\n\n> > is the promotion of the 1+0d E_10 model to the respective\n> > deRahm model, as I have mentioned before.\n>\n> OK, but it is a random guessing, is not it?\n\nNot fully random, but, yes, it is guessing. :-)\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Sun, 18 Apr 2004, Lubos Motl wrote:

> Sure, why not. :-) But there is usually a long path from these big dreams
> to actual results.

Sure.

> > ...dynamics so as to get rid of system-dependent details of the
> > dynamics on small scales) is described by Random Matrix Theory,
>
> Do you want to reproduce M-theory amplitudes approximately, or exactly?
> Is this connection with Random Matrix Theory a trick to get the right
> results? Or is it evidence that this description cannot work exactly?

This was supposed to be evidence that at least in the discussed limit
the [itex]E_{10}[/itex] model is indeed equivalent to the BFSS model. Of course
the limit is pretty radical and you may argue that it doesn't allow
any conclusion about the equivalence of the two models
away from this limit.

> OK, understood. But then one must note that the non-ordered [itex]\exp(:W:)[/itex]
> contains a lot of hard-to-interpret singular terms which make such an
> operator useless and, in fact, also ill-defined if W is a local operator.
>
> > Ah, now I see what your are perhaps thinking: It is important
> > that here W is the same object everywhere, i.e. what I wrote is
> > _not_ supposed to be a shorthand for
> >
> > [itex]\exp(:W:(z)) \exp(-:W:(w))[/itex]
> >
> > with [itex]z \neq w .[/itex]
>
> That's exactly one of the problems. If W(z) is a local operator, there is
> no working easy way to define [itex]W(z)^2[/itex]. The only way to do so is to consider
> W(z)W(w) for w approaching z. You will see that W(z)W(w) has a lot of
> singularities that you must deal with, subtract, normal-order, and so on,
> and these operations will destroy the naive identities that you are
> thinking about.

We are still not talking about the same constructions, it seems to me.
I am not considering things like [itex]W(z)^2[/itex]. As I said, the W are integrated
objects [itex]W = \int dz[/itex] something(z).

For instance the W which induces T-duality is of the form

[tex]W \propto \int d\sigma (\exp(i \sqrt{2}k\cdot X) - h.c.) .[/tex]

Compare equation (3.18) of http://www.arxiv.org/abs/hep-th/9511061 .

> I just don't see anything sharp about these objects. Of course, you can
> define the [itex]U(\infty)[/itex] group acting on the whole Hilbert space - which is
> the only thing I can imagine where you are led if you try to define an
> algebra that contains more than just a "couple" of operators.

That's too large and indeed without physical content. But as I said
before, the restriction on the allowed W is that the similarity
transformation preserves [itex]T - \bar T[/itex]. Only under this condition can the
resulting new algebra be interpreted as coming from a string in a certain
background.

> > [itex][L'_m, L'_n] = (m-n)L_{m+n} +[/itex] anomaly .
>
> Well, that's a basic lesson of undergraduate linear algebra that

Yes. :-)

> conjugation does not change the commutation relations. I guess that both
> of us realize that every physicist must know these things, and that we are
> using them all the time - and it is unlikely to make a revolutionary
> discovery just by noting that conjugation preserves commutators.

It is interesting to examine how different admissable W give
rise to certain symmetries/dualities of the string backgrounds.
You can use this also to find deformations which are not symmetries
but give new backgrounds, inequivalent to the one one started
from.

> I find this statement physically vacuous. You have not constructed any new
> theory at all; you just renamed the basis vectors of its Hilbert space in

True, but that's precisely what a symmetry/duality is supposed to do.
As you know a string state with winding number n and momentum number m
is sent to a state with winding m and momentum n and this is
a symmetry. So T-duality, too, could be described as just a
renaming of vectors in a Hilbert space, which it is. But of
course that does not mean that it is not interesting.

> I thought that there is a theorem that no useful superalgebra extending
> [itex]E_{10} ( or at[/itex] least [itex]E_8 )[/itex] exists.

Aha, don't know about that. Do you have a reference?

> > is the promotion of the [itex]1+0d E_{10}[/itex] model to the respective
> > deRahm model, as I have mentioned before.
>
> OK, but it is a random guessing, is not it?

Not fully random, but, yes, it is guessing. :-)

Lubos Motl

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Sun, 18 Apr 2004, Urs Schreiber wrote:\n\n> This was supposed to be evidence that at least in the discussed limit\n> the E_10 model is indeed equivalent to the BFSS model.\n\nDoes this proof explain why the BFSS model is the discrete light-cone\nquantization of the covariant E_{10} model, and why "N" of BFSS is the\nlight-like momentum? (If it does not, it must be wrong.)\n\n> U(\\infty) is too large and indeed without physical content. But as I said\n> before, the restriction on the allowed W is that the similarity\n> transformation preserves T - \\bar T.\n\nT-\\bar T ? Do you mean the different between T_{zz} and T_{\\bar z \\bar z} ?\nThat\'s very strange. You cannot just add different components of the\nsame tensor. More precisely, this difference is not Virasoro invariant.\nThe exact conformal symmetry is essential for string theory to work, at\nleast at the loop level, and the condition `` {T-\\bar T} is invariant\nunder the elements of your group\'\' is not Virasoro invariant. So I guess\nthat you now understand that I consider the group preserving {T-\\bar T} to\nbe as (un)natural as the group that preserves L_{2004}+2005 \\tilde\nL_{2006} or anything else. What\'s exactly your reason to think that there\nis anything interesting about a group defined in this way, as opposed to\nany other random combination of words and symbols that respect the\nsyntactic rules of our mathematical formalism?\n\n> > I find this statement physically vacuous. You have not constructed any new\n> > theory at all; you just renamed the basis vectors of its Hilbert space in\n>\n> True, but that\'s precisely what a symmetry/duality is supposed to do.\n\nNo, it\'s not. A duality (and similarly for a self-duality, which is\nessentially the same thing as a symmetry) is a map redefining the\nvariables of a theory, a redefinition that transforms the known\nobservables in a first known theory AB exactly into some known observables\nof another known theory CD (which may be the same thing as AB). For\nexample, if we work with the same theory AB and all the relevant operators\nsuch as the Hamiltonian are *invariant* under the map, then we talk about\nthe symmetry of the theory AB.\n\nOn the other hand, your conjugation by a random operator exp(W) transforms\nthe quantities of a theory AB - for example, the Virasoro worldsheet\ngenerators or the spacetime Lorentz generators - into some other,\ndifferent randomly modified operators in a theory EF whose form has never\nbeen written before. The best way to solve the theory EF is to conjugate\nthem back, and realize that EF is just an awkward way to write the theory\nAB; they are equivalent, and we don\'t earn anything by talking about a\n"new" theory EF. Such a conjugation by exp(W) is not a symmetry of AB,\nbecause the Hamiltonian and other operators are not invariant under it,\nand it is also not a duality because it does not relate two useful\ntheories in a nontrivial way - it relates one theory to the same theory\nwritten in awkward variables.\n\nA duality or a symmetry are very priviliged words, and a random\nredefinition of variables or a conjugation by a random operator certainly\ndon\'t deserve to be called a "duality" or a "symmetry". In CFT and\nperturbative string theory, you should only use the word "symmetry" or\n"duality" for the transformations that preserve the form of all Virasoro\ngenerators, and be sure that the known spacetime symmetries and T-duality\nare the only solutions.\n\n> As you know a string state with winding number n and momentum number m\n> is sent to a state with winding m and momentum n and this is\n> a symmetry. So T-duality, too, could be described as just a\n> renaming of vectors in a Hilbert space, which it is. But of\n> course that does not mean that it is not interesting.\n\nBut there is a huge difference between T-duality, which is a textbook\nexample of a duality, on one side, and your generic conjugation on the\nother side. T-duality is a map, namely X_L\\to X_L, X_R\\to -X_R, that\npreserves all Virasoro generators, for example, and at the self-dual\nradius, it even preserves the periodicities of all fields, and therefore\nis a nontrivial symmetry from the Hilbert space to the same Hilbert space\n(the same superselection sector). T-duality is a symmetry of string theory\nmapping two backgrounds that were thought to be independent (such as the\nUniverse with radii R and 1/R of a circle) into one another. At the\nself-dual radius, it is even a symmetry of the background - in fact, it is\nan element of the SU(2)^2 enhanced gauge symmetry.\n\nOn the other hand, your generic conjugation is a meaningless redefinition\nof fields that preserves neither the Virasoro algebra nor the spacetime or\nother operators. It is an uninteresting change of the conventions\ndescribing the same theory, not a map between two different theories or a\nsymmetry of a single theory. You just can\'t use the word "symmetry" or\n"duality" for such contentless constructions.\n\n> > I thought that there is a theorem that no useful superalgebra extending\n> > E_{10} ( or at least E_8 ) exists.\n>\n> Aha, don\'t know about that. Do you have a reference?\n\nUnfortunately I don\'t. I was asking the people who were telling me about\nthis theorem, but no one has given me any reference either.\n\n> > > is the promotion of the 1+0d E_10 model to the respective\n> > > deRahm model, as I have mentioned before.\n> >\n> > OK, but it is a random guessing, is not it?\n>\n> Not fully random, but, yes, it is guessing. :-)\n\nOK, so if it is not fully random :-), do you have some explanation why you\nsaid deRham model and not something else, for example the topological\nB-model on the E_{10} group manifold? ;-)\n____________________________________________________________________ __________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)\n^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Sun, 18 Apr 2004, Urs Schreiber wrote:

> This was supposed to be evidence that at least in the discussed limit
> the [itex]E_{10}[/itex] model is indeed equivalent to the BFSS model.

Does this proof explain why the BFSS model is the discrete light-cone
quantization of the covariant [itex]E_{10}[/itex] model, and why "N" of BFSS is the
light-like momentum? (If it does not, it must be wrong.)

> [itex]U(\infty)[/itex] is too large and indeed without physical content. But as I said
> before, the restriction on the allowed W is that the similarity
> transformation preserves [itex]T - \bar T[/itex].

[itex]T-\bar T ?[/itex] Do you mean the different between [itex]T_{zz}[/itex] and [itex]T_{\bar z \bar z} ?[/itex]
That's very strange. You cannot just add different components of the
same tensor. More precisely, this difference is not Virasoro invariant.
The exact conformal symmetry is essential for string theory to work, at
least at the loop level, and the condition [itex]`` {T-\bar T}[/itex] is invariant
under the elements of your group'' is not Virasoro invariant. So I guess
that you now understand that I consider the group preserving [itex]{T-\bar T}[/itex] to
be as (un)natural as the group that preserves [itex]L_{2004}+2005 \tildeL_{2006}[/itex] or anything else. What's exactly your reason to think that there
is anything interesting about a group defined in this way, as opposed to
any other random combination of words and symbols that respect the
syntactic rules of our mathematical formalism?

> > I find this statement physically vacuous. You have not constructed any new
> > theory at all; you just renamed the basis vectors of its Hilbert space in
>
> True, but that's precisely what a symmetry/duality is supposed to do.

No, it's not. A duality (and similarly for a self-duality, which is
essentially the same thing as a symmetry) is a map redefining the
variables of a theory, a redefinition that transforms the known
observables in a first known theory AB exactly into some known observables
of another known theory CD (which may be the same thing as AB). For
example, if we work with the same theory AB and all the relevant operators
such as the Hamiltonian are *invariant* under the map, then we talk about
the symmetry of the theory AB.

On the other hand, your conjugation by a random operator [itex]\exp(W)[/itex] transforms
the quantities of a theory [itex]AB -[/itex] for example, the Virasoro worldsheet
generators or the spacetime Lorentz generators - into some other,
different randomly modified operators in a theory EF whose form has never
been written before. The best way to solve the theory EF is to conjugate
them back, and realize that EF is just an awkward way to write the theory
AB; they are equivalent, and we don't earn anything by talking about a
"new" theory EF. Such a conjugation by [itex]\exp(W)[/itex] is not a symmetry of AB,
because the Hamiltonian and other operators are not invariant under it,
and it is also not a duality because it does not relate two useful
theories in a nontrivial way [itex]- it[/itex] relates one theory to the same theory
written in awkward variables.

A duality or a symmetry are very priviliged words, and a random
redefinition of variables or a conjugation by a random operator certainly
don't deserve to be called a "duality" or a "symmetry". In CFT and
perturbative string theory, you should only use the word "symmetry" or
"duality" for the transformations that preserve the form of all Virasoro
generators, and be sure that the known spacetime symmetries and T-duality
are the only solutions.

> As you know a string state with winding number n and momentum number m
> is sent to a state with winding m and momentum n and this is
> a symmetry. So T-duality, too, could be described as just a
> renaming of vectors in a Hilbert space, which it is. But of
> course that does not mean that it is not interesting.

But there is a huge difference between T-duality, which is a textbook
example of a duality, on one side, and your generic conjugation on the
other side. T-duality is a map, namely [itex]X_L\to X_L, X_R\to -X_R,[/itex] that
preserves all Virasoro generators, for example, and at the self-dual
radius, it even preserves the periodicities of all fields, and therefore
is a nontrivial symmetry from the Hilbert space to the same Hilbert space
(the same superselection sector). T-duality is a symmetry of string theory
mapping two backgrounds that were thought to be independent (such as the
Universe with radii R and 1/R of a circle) into one another. At the
self-dual radius, it is even a symmetry of the background - in fact, it is
an element of the [itex]SU(2)^2[/itex] enhanced gauge symmetry.

On the other hand, your generic conjugation is a meaningless redefinition
of fields that preserves neither the Virasoro algebra nor the spacetime or
other operators. It is an uninteresting change of the conventions
describing the same theory, not a map between two different theories or a
symmetry of a single theory. You just can't use the word "symmetry" or
"duality" for such contentless constructions.

> > I thought that there is a theorem that no useful superalgebra extending
> > [itex]E_{10} ( or at[/itex] least [itex]E_8 )[/itex] exists.
>
> Aha, don't know about that. Do you have a reference?

Unfortunately I don't. I was asking the people who were telling me about
this theorem, but no one has given me any reference either.

> > > is the promotion of the [itex]1+0d E_{10}[/itex] model to the respective
> > > deRahm model, as I have mentioned before.
> >
> > OK, but it is a random guessing, is not it?
>
> Not fully random, but, yes, it is guessing. :-)

OK, so if it is not fully random :-), do you have some explanation why you
said deRham model and not something else, for example the topological
B-model on the [itex]E_{10}[/itex] group manifold? ;-)
__{____________________________________________________________________ ________}
E-mail: lumo@matfyz.cz fax: [itex]+1-617/496-0110[/itex] Web: http://lumo.matfyz.cz/
eFax: [itex]+1-801/454-1858[/itex] work: [itex]+1-617/496-8199[/itex] home: [itex]+1-617/868-4487[/itex] (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^

Urs Schreiber

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>NNTP-Posting-Host: feynman.harvard.edu\nMime-Version: 1.0\nContent-Type: TEXT/PLAIN; charset=US-ASCII\nX-X-Sender: <sps@feynman.harvard.edu>\nIn-Reply-To: <Pine.LNX.4.31.0404181249060.10181-100000@feynman.harvard.edu>\nXref: solaris.cc.vt.edu sci.physics.strings:38\n\nOn Sun, 18 Apr 2004, Lubos Motl wrote:\n\n> T-\\bar T ? Do you mean the different between T_{zz} and T_{\\bar z \\bar z} ?\n> That\'s very strange. You cannot just add different components of the\n> same tensor.\n\nT - \\bar T (in modes L_m - \\bar L_{-m})\nis always the generator of spatial reparameterizations\nof the string at appropriately fixed worldsheet time. That it\nmust be preserved under a\nchange of background can be seen by either making an ADM-like analysis\nof the worldsheet gravity theory or else by considering\ndeformations\n\nT \\to T + \\delta T\n\n\\bar T \\to \\bar T + \\delta \\bar T\n\nand noting that consistency requires\n\n\\delta T = \\delta \\bar T .\n\n\n> L_{2006} or anything else. What\'s exactly your reason to think that there\n> is anything interesting about a group defined in this way, as opposed to\n\nIt is precisely the group which gives you isomorphisms of the\nleft+right Virasoro algebra. It is possible to identify the group elements\nwhich, for instance, correspond to background T-duality, background\ndiffeo transformations, background gauge transformations. (see below)\n\n> > True, but that\'s precisely what a symmetry/duality is supposed to do.\n>\n> No, it\'s not. A duality (and similarly for a self-duality, which is\n> essentially the same thing as a symmetry) is a map redefining the\n> variables of a theory, a redefinition that transforms the known\n> observables in a first known theory AB exactly into some known observables\n> of another known theory CD (which may be the same thing as AB).\n\nSure. And the conjugation which I mentioned in my last post does\nexactly that for T-duality.\n\n> On the other hand, your generic conjugation is a meaningless redefinition\n> of fields that preserves neither the Virasoro algebra\n\nI don\'t understand why you are now saying that it does not preserve the\nVirasoro algebra. In your last post you said that it is known to\nevery undergraduate that conjugation preserves the algebra.\n\n> other operators. It is an uninteresting change of the conventions\n> describing the same theory, not a map between two different theories or a\n> symmetry of a single theory. You just can\'t use the word "symmetry" or\n> "duality" for such contentless constructions.\n\nLubos, please have a look at\n\nF. Lizzi, R. Szabo,\nDuality symmetries and noncommutative geometry of string spacetimes,\nhep-th/9707202\n\nwhere it is shown in great detail that and how the duality symmetries\nof the string are related to the conjugations in question.\n\n> > > OK, but it is a random guessing, is not it?\n> >\n> > Not fully random, but, yes, it is guessing. :-)\n>\n> OK, so if it is not fully random :-), do you have some explanation why you\n> said deRham model and not something else, for example the topological\n> B-model on the E_{10} group manifold? ;-)\n\nYes, I have a reason for this guess:\n\nI once checked that the supersymmetry generators of N=1 D=3+1 SUGRA\n(or rather appropriate linear combinations of them) are\nthe exterior derivative d and its adjoint del on the configuration space\nof the theory (really the Dolbeault operators \\partial and \\bar \\partial\nso that d = \\partial + \\bar \\partial). The susy algebra of the constraints\nof the theory are hence equivalent to the algebra which is schematically\nof the form\n\n{d,del} = H + ...\n\nwhere H is the Hamiltonian constraint of the theory (and in particular\nsomething like the Laplace-Beltrami operator on config space)\nand the ellipsis indicates sa sum of patial diffeomorphism and\nLorentz rotation constraints.\n\nIt is to be expected that a similar statement applies to the\nconstraints of 11d sugra. For 11d sugra Damour,Henneux&Nicolai have\nshown that the bosonic part of configuration space (which, to emphasize it\nagain, is the space on which the above operators d and del act) is\n(a subspace of) the group manifold of E10/K(E10) and that the\n(bosonic part of) the Hamiltonian constraint H is the Laplace\noperator on this manifold. Hence by the above it is a reasonable\nguess that the supersymmetric extension of this Hamiltonian\nconstraint is the Laplace-Beltrami operator {d,del} on the\nexterior bundle over E10/K(E10) (or rather of finite truncations\nof this object, since we don\'t know how to handle the full thing.)\n\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>NNTP-Posting-Host: feynman.harvard.edu
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Xref: solaris.cc.vt.edu sci.physics.strings:38

On Sun, 18 Apr 2004, Lubos Motl wrote:

> [itex]T-\bar T ?[/itex] Do you mean the different between [itex]T_{zz}[/itex] and [itex]T_{\bar z \bar z} ?[/itex]
> That's very strange. You cannot just add different components of the
> same tensor.

[tex]T - \bar T (in[/itex] modes [itex]L_m - \bar L_{-m})[/itex]
is always the generator of spatial reparameterizations
of the string at appropriately fixed worldsheet time. That it
must be preserved under a
change of background can be seen by either making an ADM-like analysis
of the worldsheet gravity theory or else by considering
deformations

[itex]T \to T + \delta T\bar T \to \bar T + \delta \bar T[/tex]

and noting that consistency requires

[tex]\delta T = \delta \bar T .[/tex]

> [itex]L_{2006}[/itex] or anything else. What's exactly your reason to think that there
> is anything interesting about a group defined in this way, as opposed to

It is precisely the group which gives you isomorphisms of the
left+right Virasoro algebra. It is possible to identify the group elements
which, for instance, correspond to background T-duality, background
diffeo transformations, background gauge transformations. (see below)

> > True, but that's precisely what a symmetry/duality is supposed to do.
>
> No, it's not. A duality (and similarly for a self-duality, which is
> essentially the same thing as a symmetry) is a map redefining the
> variables of a theory, a redefinition that transforms the known
> observables in a first known theory AB exactly into some known observables
> of another known theory CD (which may be the same thing as AB).

Sure. And the conjugation which I mentioned in my last post does
exactly that for T-duality.

> On the other hand, your generic conjugation is a meaningless redefinition
> of fields that preserves neither the Virasoro algebra

I don't understand why you are now saying that it does not preserve the
Virasoro algebra. In your last post you said that it is known to
every undergraduate that conjugation preserves the algebra.

> other operators. It is an uninteresting change of the conventions
> describing the same theory, not a map between two different theories or a
> symmetry of a single theory. You just can't use the word "symmetry" or
> "duality" for such contentless constructions.

Lubos, please have a look at

F. Lizzi, R. Szabo,
Duality symmetries and noncommutative geometry of string spacetimes,
http://www.arxiv.org/abs/hep-th/9707202

where it is shown in great detail that and how the duality symmetries
of the string are related to the conjugations in question.

> > > OK, but it is a random guessing, is not it?
> >
> > Not fully random, but, yes, it is guessing. :-)
>
> OK, so if it is not fully random :-), do you have some explanation why you
> said deRham model and not something else, for example the topological
> B-model on the [itex]E_{10}[/itex] group manifold? ;-)

Yes, I have a reason for this guess:

I once checked that the supersymmetry generators of [itex]N=1 D=3+1[/itex] SUGRA
(or rather appropriate linear combinations of them) are
the exterior derivative d and its adjoint del on the configuration space
of the theory (really the Dolbeault operators [itex]\partial[/itex] and [itex]\bar \partial[/itex]
so that [itex]d = \partial + \bar \partial)[/itex]. The susy algebra of the constraints
of the theory are hence equivalent to the algebra which is schematically
of the form

{d,del} [itex]= H + .[/itex]..

where H is the Hamiltonian constraint of the theory (and in particular
something like the Laplace-Beltrami operator on config space)
and the ellipsis indicates sa sum of patial diffeomorphism and
Lorentz rotation constraints.

It is to be expected that a similar statement applies to the
constraints of 11d sugra. For 11d sugra Damour,Henneux&Nicolai have
shown that the bosonic part of configuration space (which, to emphasize it
again, is the space on which the above operators d and del act) is
(a subspace of) the group manifold of [itex]E10/K(E10)[/itex] and that the
(bosonic part of) the Hamiltonian constraint H is the Laplace
operator on this manifold. Hence by the above it is a reasonable
guess that the supersymmetric extension of this Hamiltonian
constraint is the Laplace-Beltrami operator {d,del} on the
exterior bundle over [itex]E10/K(E10) (or[/itex] rather of finite truncations
of this object, since we don't know how to handle the full thing.)

Urs Schreiber

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>NNTP-Posting-Host: feynman.harvard.edu\nMime-Version: 1.0\nContent-Type: TEXT/PLAIN; charset=US-ASCII\nX-X-Sender: <sps@feynman.harvard.edu>\nIn-Reply-To: <Pine.LNX.4.31.0404181249060.10181-100000@feynman.harvard.edu>\nXref: solaris.cc.vt.edu sci.physics.strings:39\n\nOn Sun, 18 Apr 2004, Lubos Motl wrote:\n\n> On Sun, 18 Apr 2004, Urs Schreiber wrote:\n>\n> > This was supposed to be evidence that at least in the discussed limit\n> > the E_10 model is indeed equivalent to the BFSS model.\n>\n> Does this proof explain why the BFSS model is the discrete light-cone\n> quantization of the covariant E_{10} model, and why "N" of BFSS is the\n> light-like momentum? (If it does not, it must be wrong.)\n\nThanks for asking this question! (And thanks, in fact, for all of this\nrapid and very critical discussion.)\n\nI have thought about precisely this type of question a lot in the last\ncouple of days. The point is: Can we, apart from noting that we have\nensembles of the form exp(-Tr(M^2)) on both sides in the given limit,\ncan we identify the physical meaning of the matrices on both sides\nof the conjectured equivalence?\n\nIn order to answer the question we would of course first of all have\nto figure out what the "meaning" of the matrices on the E_10 side\nof the conjectured equivalence actually is.\n\nWithout any further work, all that we know is that the semiclassical\nlimit of 11d sugra close to a spacelike sinmgularity is, due to\nchaoticity, equal to that of a theory of an ensemble of randomly\nchosen NxN matrices for large N.\n\nBut what is the physical interpretation of N in this context?\n\nIn fact this is a very old question. It has been known for a long\ntime empirically (i.e. using numerics) that Random Matrix Theory (RMT)\ndoes universally describe the semiclassical limit of all chaotic\nquantum systems (i.e. the predictions obtained from the random\nensemble of matrices, for instance concerning the statistic of\nthe spectrum of the system\'s Hamiltonian, precisely coincide with\nthe predictions obtained from the origina theory).\n\nBut I am being told by specialists working on quantum chaos that\nthe reason for this "unreasonable effectiveness" of RMT in\nquantum chaos has so far remained a mystery.\n\nIn fact, there is quite some excitement at my institution about\nthe recent results of one of our groups (S. Mueller, S. Heusler,\nP. Braun, F. Haake et al.) who have solved the long-standing\nproblem of actually explicitly calculating the universal\nsemiclassical spectrum of general chaotic quantum systems and\nchecking equivalence with the resuts of RMT. So this finally\nimproves the comparison with numerical results to a formal\ncalculation. The prediction of RMT are now proven to be exactly\nthose found from a full semiclassical analysis of the true\nsystems.\n\nSo this is reassuring, because it shows that the RMT-conjecture\n("every chaotic quantum system is described by RMT") is correct.\n\nBut unfortunately, at least as far as I can see, this still does\nnot tell us WHY the conjecture is correct, i.e. why this\nensemble of matrices describes single chaotic systems. It still\ndoes not give us a physical interpretation of the large matrices\nand of the parameter N, which is what we would need to answer\nyour question above.\n\nFor precisely this reason I have tried to figure out the answer\nmyself, recently. You can find my proposed solution as well as\nsome background information and discussion at the Coffee Table:\n\nhttp://golem.ph.utexas.edu/string/archives/000342.html#c000927\n\nI don\'t know if this works out as expected, but I think it looks\npromising.\n\nSo my proposed solution is that we have to interpret the\nensemble of systems used in RMT as associated with the ensemble\nof points in a "non-universal cell" of the config space\nof the chaotic system.\n\nMore precisely, the classical chaoticity of the system\nimplies that in the semiclassical limit (which is dominated\nby classical contributions to the path integral) we can\nintroduce a coarse-graining of the confuguration space\ninto cells, such that within each cell matrix elements of\nthe Hamiltonian are correlated, while outside they are not.\nThis reflects precisely the fact that on very small scales\nthe system will have non-universal behaviour while on\nsufficiently large scales it will exhibt the universality of\nergodicity and chaos.\n\nPlease refer to the above links for some more details of\nthis, admittedly simple but apparently original, idea.\n\nLet\'s assume this interpretation is approximately correct and\ntry to understand what it implies for the interpretation of the\nmatrices which would describe the rmt of 11d sugra close to\na spacelike singularity.\n\nNow config space is the mini-superspace, and in particular the\nWeyl-chamber of E10, where every point corresponds to a\nspecific set of values of the scale factors of the\nsugra universe. The matrices would now describe transition\namplitudes between different "cells" of this config space\nand N would be the total number of such cells.\n\nHence N would correspond roughly to a discretzation of the\nscale factors, I\'d think.\n\nThs does not look like it could have any relation to the\ninterpretation of N in the BFSS model, does it?\n\nSo maybe my conjecture is wrong. But maybe we should bea little more\ncareful before making this conclusion, because of various\nreinterpretations which are possible.\n\nFor instance in order for my conjecture to make sense we would\nreally be looking at the canonical ensemble of BFSS theory\nand hence at BFSS at _finite temperature_.\nBut we know that BFSS at finite temperature is nothing but\nthe IKKT model!\n\nSo it seems that we would rather have to identify the matrices\nwith those of the IKKT model.\n\nThere the above interpretation might make much more sense, since\nit is known that we can interpret in the IKKT model the integer\nN as the number of points in a discretization of spacetime.\n(i assume that that\'s what you were referring to when mentioning\nthe relation of IKKT to "quantum foam".) So this would match\nnicely with the interpretation of N as characterizing discrete\nscale factors of the universe.\n\nOf course I am aware that all this is rather vague and very\nspeculative. But I think it is a fun speculation and not\nuninteresting. Maybe it is wrong. Maybe not. In fact, if there\nis any truth to the E10 model then something along the lines\nsketched above must be true.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>NNTP-Posting-Host: feynman.harvard.edu
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In-Reply-To: <Pine.LNX.4.31.0404181249060.10181-1...an.harvard.edu>
Xref: solaris.cc.vt.edu sci.physics.strings:39

On Sun, 18 Apr 2004, Lubos Motl wrote:

> On Sun, 18 Apr 2004, Urs Schreiber wrote:
>
> > This was supposed to be evidence that at least in the discussed limit
> > the [itex]E_{10}[/itex] model is indeed equivalent to the BFSS model.
>
> Does this proof explain why the BFSS model is the discrete light-cone
> quantization of the covariant [itex]E_{10}[/itex] model, and why "N" of BFSS is the
> light-like momentum? (If it does not, it must be wrong.)

Thanks for asking this question! (And thanks, in fact, for all of this
rapid and very critical discussion.)

I have thought about precisely this type of question a lot in the last
couple of days. The point is: Can we, apart from noting that we have
ensembles of the form [itex]\exp(-Tr(M^2))[/itex] on both sides in the given limit,
can we identify the physical meaning of the matrices on both sides
of the conjectured equivalence?

In order to answer the question we would of course first of all have
to figure out what the "meaning" of the matrices on the [itex]E_{10}[/itex] side
of the conjectured equivalence actually is.

Without any further work, all that we know is that the semiclassical
limit of 11d sugra close to a spacelike sinmgularity is, due to
chaoticity, equal to that of a theory of an ensemble of randomly
chosen NxN matrices for large N.

But what is the physical interpretation of N in this context?

In fact this is a very old question. It has been known for a long
time empirically (i.e. using numerics) that Random Matrix Theory (RMT)
does universally describe the semiclassical limit of all chaotic
quantum systems (i.e. the predictions obtained from the random
ensemble of matrices, for instance concerning the statistic of
the spectrum of the system's Hamiltonian, precisely coincide with
the predictions obtained from the origina theory).

But I am being told by specialists working on quantum chaos that
the reason for this "unreasonable effectiveness" of RMT in
quantum chaos has so far remained a mystery.

In fact, there is quite some excitement at my institution about
the recent results of one of our groups (S. Mueller, S. Heusler,
P. Braun, F. Haake et al.) who have solved the long-standing
problem of actually explicitly calculating the universal
semiclassical spectrum of general chaotic quantum systems and
checking equivalence with the resuts of RMT. So this finally
improves the comparison with numerical results to a formal
calculation. The prediction of RMT are now proven to be exactly
those found from a full semiclassical analysis of the true
systems.

So this is reassuring, because it shows that the RMT-conjecture
("every chaotic quantum system is described by RMT") is correct.

But unfortunately, at least as far as I can see, this still does
not tell us WHY the conjecture is correct, i.e. why this
ensemble of matrices describes single chaotic systems. It still
does not give us a physical interpretation of the large matrices
and of the parameter N, which is what we would need to answer
your question above.

For precisely this reason I have tried to figure out the answer
myself, recently. You can find my proposed solution as well as
some background information and discussion at the Coffee Table:

http://golem.ph.utexas.edu/string/ar...2.html#c000927

I don't know if this works out as expected, but I think it looks
promising.

So my proposed solution is that we have to interpret the
ensemble of systems used in RMT as associated with the ensemble
of points in a "non-universal cell" of the config space
of the chaotic system.

More precisely, the classical chaoticity of the system
implies that in the semiclassical limit (which is dominated
by classical contributions to the path integral) we can
introduce a coarse-graining of the confuguration space
into cells, such that within each cell matrix elements of
the Hamiltonian are correlated, while outside they are not.
This reflects precisely the fact that on very small scales
the system will have non-universal behaviour while on
sufficiently large scales it will exhibt the universality of
ergodicity and chaos.

Please refer to the above links for some more details of
this, admittedly simple but apparently original, idea.

Let's assume this interpretation is approximately correct and
try to understand what it implies for the interpretation of the
matrices which would describe the rmt of 11d sugra close to
a spacelike singularity.

Now config space is the mini-superspace, and in particular the
Weyl-chamber of E10, where every point corresponds to a
specific set of values of the scale factors of the
sugra universe. The matrices would now describe transition
amplitudes between different "cells" of this config space
and N would be the total number of such cells.

Hence N would correspond roughly to a discretzation of the
scale factors, I'd think.

Ths does not look like it could have any relation to the
interpretation of N in the BFSS model, does it?

So maybe my conjecture is wrong. But maybe we should bea little more
careful before making this conclusion, because of various
reinterpretations which are possible.

For instance in order for my conjecture to make sense we would
really be looking at the canonical ensemble of BFSS theory
and hence at BFSS at _finite temperature_.
But we know that BFSS at finite temperature is nothing but
the IKKT model!

So it seems that we would rather have to identify the matrices
with those of the IKKT model.

There the above interpretation might make much more sense, since
it is known that we can interpret in the IKKT model the integer
N as the number of points in a discretization of spacetime.
(i assume that that's what you were referring to when mentioning
the relation of IKKT to "quantum foam".) So this would match
nicely with the interpretation of N as characterizing discrete
scale factors of the universe.

Of course I am aware that all this is rather vague and very
speculative. But I think it is a fun speculation and not
uninteresting. Maybe it is wrong. Maybe not. In fact, if there
is any truth to the E10 model then something along the lines
sketched above must be true.

Lubos Motl

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>NNTP-Posting-Host: feynman.harvard.edu\nMime-Version: 1.0\nContent-Type: TEXT/PLAIN; charset=US-ASCII\nIn-Reply-To: <Pine.LNX.4.31.0404181329450.10319-100000@feynman.harvard.edu>\nXref: solaris.cc.vt.edu sci.physics.strings:40\n\nOn Sun, 18 Apr 2004, Urs Schreiber wrote:\n\n> T - \\bar T (in modes L_m - \\bar L_{-m}) is always the generator of\n> spatial reparameterizations of the string at appropriately fixed\n> worldsheet time.\n\nRight, that\'s the very correct answer. So anything that you will obtain\nfrom such a special treatment of the "fixed" worldsheet time will be\nnon-invariant under the general conformal transformations of the 2D\nworldsheet which can act nontrivially on the worldsheet time. These\ngeneral conformal symmetries are necessary for decoupling of ghosts, and\nfor calculations of the loop diagram, among many other things. If you take\nthe comment about the loop diagrams seriously, your group of spatial\nreparameterizations does not even exist for loop diagrams. Such a group\nonly looks meaningful if you stare at the cylindrical worldsheet of a\nsingle string, but it has no invariant meaning if you consider the\ninteractions. The whole conformal symmetry (and the Virasoro algebra) must\nbe treated as one group if you want to use its power - and believe me that\nyou need its full power to calculate the scattering amplitudes, especially\nat the loop level.\n\n> It is precisely the group which gives you isomorphisms of the\n> left+right Virasoro algebra. It is possible to identify the group elements\n> which, for instance, correspond to background T-duality, background\n> diffeo transformations, background gauge transformations. (see below)\n\nThese transformations (T-duality etc.; the spacetime diffeomorphisms are\nnot the best example unless they are isometries, because the other\ntransformations *do* change the background i.e. they *do* change the form\nof the worldsheet action and consequently of the Virasoro generators)\npreserve the full left as well as the full right Virasoro group, and\ntherefore it is not surprising that they also preserve its subgroup or\nsubalgebra of the "diagonal spatial" Virasoro generators.\n\nWhat I\'ve been trying to say for some time is that the transformations\nthat preserve the whole Virasoro algebra are interesting (they are\nsymmetries of string theory, such as T-duality), while the remaining\ngeneric transformations that only preserve a part of it (such as the\nspatial diffeomorphisms) are not interesting (and are not physical) in\nstring theory, and they are not symmetries, much like any other\ntransformation or field redefinition that preserves nothing.\n\nThere is a huge difference between the words "field redefinition" and a\n"symmetry" (or "duality"). Symmetries and dualities are much more special\nand rare concepts!\n\n> > No, it\'s not. A duality (and similarly for a self-duality, which is\n> > essentially the same thing as a symmetry) is a map redefining the\n> > variables of a theory, a redefinition that transforms the known\n> > observables in a first known theory AB exactly into some known observables\n> > of another known theory CD (which may be the same thing as AB).\n>\n> Sure. And the conjugation which I mentioned in my last post does\n> exactly that for T-duality.\n\nI don\'t quite understand. Is the conjugation that you talk about now the\nconjugation by a generic exp(W)? T-duality is not a generic conjugation,\nbut a very specific transformation that preserves - unlike the conjugation\nby a generic exp(W) - every single generator of the Virasoro algebra, and\nas far as I know, it was not you who discovered T-duality. ;-)\n\n> > On the other hand, your generic conjugation is a meaningless redefinition\n> > of fields that preserves neither the Virasoro algebra\n>\n> I don\'t understand why you are now saying that it does not preserve the\n> Virasoro algebra. In your last post you said that it is known to\n> every undergraduate that conjugation preserves the algebra.\n\nI guess that you only want to entertain us right now. What I wrote is that\nevery kid knows that every conjugation preserves the form of the\n*commutation relations* between various generators, i.e. the form of the\nidentities of the form [A,B]=C which is transformed to [A\',B\']=C\'. But\nthat\'s not enough for an operation to be called a symmetry. For a\ntransformation to be called "symmetry", the explicit form of every\nVirasoro generator, in terms of the fundamental field, must be unchanged,\ni.e. A=A\',B=B\',C=C\' in the example above, or for example L_{m} must be\nstill equal to\n\n\\frac{1}{2}\\sum_{n} \\alpha_{m+n}^\\mu \\alpha_{-n}_\\mu.\n\nWhile the generic conjugation by \\exp(W) preserves the form of the\n*commutation relations*, it certainly does not preserve the generators\nthemselves (a generic transformation preserves neither L_m nor\n\\alpha^\\mu_m) - even though you seem to be saying exactly this (!), and\ntherefore you cannot call it a "symmetry", and therefore it is also much\nless interesting than the real symmetries such as T-duality.\n\n> F. Lizzi, R. Szabo,\n> Duality symmetries and noncommutative geometry of string spacetimes,\n> hep-th/9707202\n\nThis paper offers a slightly non-standard approach, especially because of\ntheir usage of the Dirac operators, but nothing against it. I am not\narguing against the fact that T-duality is important, interesting, and it\nis a very special example of a transformation i.e. conjugation. ;-) On the\ncontrary, I was opposing your statement that you can extend the well-known\nT-duality to a much larger group, and I claim that there exists nothing\nbeyond T-dualities (plus spacetime gauge symmetries and isometries) that\nis a symmetry of perturbative string theory. T-duality is very special.\nSome generic transformations that only preserve a subgroup of the Virasoro\nalgebra are physically uninteresting.\n\n> Yes, I have a reason for this guess:\n>\n> I once checked that the supersymmetry generators of N=1 D=3+1 SUGRA\n> (or rather appropriate linear combinations of them) are\n> the exterior derivative d and its adjoint del on the configuration space\n> of the theory (really the Dolbeault operators \\partial and \\bar \\partial\n> so that d = \\partial + \\bar \\partial). The susy algebra of the constraints\n> of the theory are hence equivalent to the algebra which is schematically\n> of the form\n>\n> {d,del} = H + ...\n>\n> where H is the Hamiltonian constraint of the theory (and in particular\n> something like the Laplace-Beltrami operator on config space)\n> and the ellipsis indicates sa sum of patial diffeomorphism and\n> Lorentz rotation constraints.\n\nLorentz rotation constraints? The RHS of the SUSY algebra in the flat\nspace should contain the momenta, but no Lorentz generators or other\ndiffeomorphisms. I also don\'t understand what you did with the spinor\nindices of the supercharges. Your schematic formulae look very\nLorentz-non-invariant, also because of the special treatment of the\nHamiltonian, and it just seems difficult at this moment to imagine a\nvariation of these arguments that would be believable. Sorry, I am sure\nthat you will convince me later. ;-)\n\n> It is to be expected that a similar statement applies to the\n> constraints of 11d sugra. For 11d sugra Damour,Henneux&Nicolai have\n> shown that the bosonic part of configuration space (which, to emphasize it\n> again, is the space on which the above operators d and del act) is\n> (a subspace of) the group manifold of E10/K(E10) and that the\n> (bosonic part of) the Hamiltonian constraint H is the Laplace\n> operator on this manifold. Hence by the above it is a reasonable\n> guess that the supersymmetric extension of this Hamiltonian\n> constraint is the Laplace-Beltrami operator {d,del} on the\n> exterior bundle over E10/K(E10) (or rather of finite truncations\n> of this object, since we don\'t know how to handle the full thing.)\n\nOK, I think that some work is needed before others will see what this\nproposal really means and why it is a reasonable guess. Good luck, LM\n___________________________________________________________________ ___________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)\n^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^\n\n\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>NNTP-Posting-Host: feynman.harvard.edu
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Xref: solaris.cc.vt.edu sci.physics.strings:40

On Sun, 18 Apr 2004, Urs Schreiber wrote:

> [itex]T - \bar T (in[/itex] modes [itex]L_m - \bar L_{-m})[/itex] is always the generator of
> spatial reparameterizations of the string at appropriately fixed
> worldsheet time.

Right, that's the very correct answer. So anything that you will obtain
from such a special treatment of the "fixed" worldsheet time will be
non-invariant under the general conformal transformations of the 2D
worldsheet which can act nontrivially on the worldsheet time. These
general conformal symmetries are necessary for decoupling of ghosts, and
for calculations of the loop diagram, among many other things. If you take
the comment about the loop diagrams seriously, your group of spatial
reparameterizations does not even exist for loop diagrams. Such a group
only looks meaningful if you stare at the cylindrical worldsheet of a
single string, but it has no invariant meaning if you consider the
interactions. The whole conformal symmetry (and the Virasoro algebra) must
be treated as one group if you want to use its power - and believe me that
you need its full power to calculate the scattering amplitudes, especially
at the loop level.

> It is precisely the group which gives you isomorphisms of the
> left+right Virasoro algebra. It is possible to identify the group elements
> which, for instance, correspond to background T-duality, background
> diffeo transformations, background gauge transformations. (see below)

These transformations (T-duality etc.; the spacetime diffeomorphisms are
not the best example unless they are isometries, because the other
transformations *do* change the background i.e. they *do* change the form
of the worldsheet action and consequently of the Virasoro generators)
preserve the full left as well as the full right Virasoro group, and
therefore it is not surprising that they also preserve its subgroup or
subalgebra of the "diagonal spatial" Virasoro generators.

What I've been trying to say for some time is that the transformations
that preserve the whole Virasoro algebra are interesting (they are
symmetries of string theory, such as T-duality), while the remaining
generic transformations that only preserve a part of it (such as the
spatial diffeomorphisms) are not interesting (and are not physical) in
string theory, and they are not symmetries, much like any other
transformation or field redefinition that preserves nothing.

There is a huge difference between the words "field redefinition" and a
"symmetry" (or "duality"). Symmetries and dualities are much more special
and rare concepts!

> > No, it's not. A duality (and similarly for a self-duality, which is
> > essentially the same thing as a symmetry) is a map redefining the
> > variables of a theory, a redefinition that transforms the known
> > observables in a first known theory AB exactly into some known observables
> > of another known theory CD (which may be the same thing as AB).
>
> Sure. And the conjugation which I mentioned in my last post does
> exactly that for T-duality.

I don't quite understand. Is the conjugation that you talk about now the
conjugation by a generic [itex]\exp(W)[/itex]? T-duality is not a generic conjugation,
but a very specific transformation that preserves - unlike the conjugation
by a generic [itex]\exp(W) -[/itex] every single generator of the Virasoro algebra, and
as far as I know, it was not you who discovered T-duality. ;-)

> > On the other hand, your generic conjugation is a meaningless redefinition
> > of fields that preserves neither the Virasoro algebra
>
> I don't understand why you are now saying that it does not preserve the
> Virasoro algebra. In your last post you said that it is known to
> every undergraduate that conjugation preserves the algebra.

I guess that you only want to entertain us right now. What I wrote is that
every kid knows that every conjugation preserves the form of the
*commutation relations* between various generators, i.e. the form of the
identities of the form [itex][A,B]=C[/itex] which is transformed to [itex][A',B']=C'[/itex]. But
that's not enough for an operation to be called a symmetry. For a
transformation to be called "symmetry", the explicit form of every
Virasoro generator, in terms of the fundamental field, must be unchanged,
i.e[itex]. A=A',B=B',C=C'[/itex] in the example above, or for example [itex]L_{m}[/itex] must be
still equal to

[itex]\frac{1}{2}\sum_{n} \alpha_{m+n}^\mu \alpha_{-n}_\mu[/itex].

While the generic conjugation by [itex]\exp(W)[/itex] preserves the form of the
*commutation relations*, it certainly does not preserve the generators
themselves (a generic transformation preserves neither [itex]L_m[/itex] nor
[itex]\alpha^\mu_m) -[/itex] even though you seem to be saying exactly this (!), and
therefore you cannot call it a "symmetry", and therefore it is also much
less interesting than the real symmetries such as T-duality.

> F. Lizzi, R. Szabo,
> Duality symmetries and noncommutative geometry of string spacetimes,
> http://www.arxiv.org/abs/hep-th/9707202

This paper offers a slightly non-standard approach, especially because of
their usage of the Dirac operators, but nothing against it. I am not
arguing against the fact that T-duality is important, interesting, and it
is a very special example of a transformation i.e. conjugation. ;-) On the
contrary, I was opposing your statement that you can extend the well-known
T-duality to a much larger group, and I claim that there exists nothing
beyond T-dualities (plus spacetime gauge symmetries and isometries) that
is a symmetry of perturbative string theory. T-duality is very special.
Some generic transformations that only preserve a subgroup of the Virasoro
algebra are physically uninteresting.

> Yes, I have a reason for this guess:
>
> I once checked that the supersymmetry generators of [itex]N=1 D=3+1[/itex] SUGRA
> (or rather appropriate linear combinations of them) are
> the exterior derivative d and its adjoint del on the configuration space
> of the theory (really the Dolbeault operators [itex]\partial[/itex] and [itex]\bar \partial[/itex]
> so that [itex]d = \partial + \bar \partial)[/itex]. The susy algebra of the constraints
> of the theory are hence equivalent to the algebra which is schematically
> of the form
>
> {d,del} [itex]= H + .[/itex]..
>
> where H is the Hamiltonian constraint of the theory (and in particular
> something like the Laplace-Beltrami operator on config space)
> and the ellipsis indicates sa sum of patial diffeomorphism and
> Lorentz rotation constraints.

Lorentz rotation constraints? The RHS of the SUSY algebra in the flat
space should contain the momenta, but no Lorentz generators or other
diffeomorphisms. I also don't understand what you did with the spinor
indices of the supercharges. Your schematic formulae look very
Lorentz-non-invariant, also because of the special treatment of the
Hamiltonian, and it just seems difficult at this moment to imagine a
variation of these arguments that would be believable. Sorry, I am sure
that you will convince me later. ;-)

> It is to be expected that a similar statement applies to the
> constraints of 11d sugra. For 11d sugra Damour,Henneux&Nicolai have
> shown that the bosonic part of configuration space (which, to emphasize it
> again, is the space on which the above operators d and del act) is
> (a subspace of) the group manifold of [itex]E10/K(E10)[/itex] and that the
> (bosonic part of) the Hamiltonian constraint H is the Laplace
> operator on this manifold. Hence by the above it is a reasonable
> guess that the supersymmetric extension of this Hamiltonian
> constraint is the Laplace-Beltrami operator {d,del} on the
> exterior bundle over [itex]E10/K(E10) (or[/itex] rather of finite truncations
> of this object, since we don't know how to handle the full thing.)

OK, I think that some work is needed before others will see what this
proposal really means and why it is a reasonable guess. Good luck, LM
__{____________________________________________________________________ ________}
E-mail: lumo@matfyz.cz fax: [itex]+1-617/496-0110[/itex] Web: http://lumo.matfyz.cz/
eFax: [itex]+1-801/454-1858[/itex] work: [itex]+1-617/496-8199[/itex] home: [itex]+1-617/868-4487[/itex] (call)
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