Speculation: E6 and 26-dim. string theory [Archive]

Tony Smith

May9-04, 11:01 AM

<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>Alejandro Rivero wrote ...\n\n> Perhaps your question in this forum could be, if string theory has\n> something to say about the group [E6] ...\n\nIn that context, my question would be:\nString theory is fundamentally 26-dimensional.\n\n[Moderator\'s note: First of all, this sentence is not\na question. Second, it is not true. "String theory" is\na shorthand for "superstring theory" which is at most\n10-dimensional - and its extension "M-theory" is\n11-dimensional. 26 is the spacetime dimension of the\nbosonic string - the old theory suffering from the tachyon,\nthe absence of fermions, and so on. LM]\n\nAs Pierre Ramond noted in hep-th/0112261\n"... The traceless Jordan matrices [ J3(O)o ] ...\n(3x3) traceless octonionic hermitian matrices,\neach labelled by 26 real parameters ... span the 26 representation\nof [ the 52-dimensional exceptional Lie algebra F4 ].\n\n[Moderator\'s note: This seems as a pure numerology. The dimension\n26 of the bosonic string coincides with the dimension\nof the fundamental representation of F_4 but because there does\nnot seem to be any priviliged action of F_4 on the 26 coordinates,\nit seems as a clear coincidence. It is not so nontrivial\nto get the number 26 in two ways. Similar comments would\nprobably apply to the rest of the text, too. LM]\n\nOne can supplement the F4 transformations by an additional 26 parameters\n.... leading to a group with 78 parameters. These extra transformations\nare non-compact, and close on the F4 transformations,\nleading to the exceptional group E6(-26).\nThe subscript in parenthesis denotes the number of non-compact minus\nthe number of compact generators. ...".\n\nAs Soji Kaneyuki noted in Graded Lie Algebras, Related Geometric Structures,\nand Pseudo-hermitian Symmetric Spaces, a section of the book\nAnalysis and Geometry on Complex Homogeneous Domains, by Jacques Faraut,\nSoji Kaneyuki, Adam Koranyi, Qi-keng Lu, and Guy Roos (Birkhauser 2000),\nE6 can be seen as a Graded Lie Algebra with 5 grades:\ng = E6 = g(-2) + g(-1) + g(0) + g(1) + g(2)\n*such that\ng(0) = so(8) + R + R has 28 + 2 dimensions\ndimR g(-1) = dimR g(1) = 16 = 8 + 8\ndimR g(-2) = dimR g(2) = 8\n(dimR refers to real, as opposed to complex, dimension)\n\nCould the 8 + 8 + 8 + 2 = 26 dimensions of the\ng(-2), g(-1), and g(0) (other than the 28 so(8) dimensions) parts\nof the graded structure of E6 correspond to\nthe 26 dimensions of traceless part J3(O)o of the\n27-dimensional Jordan algebra J3(O) of 3x3 hermitian octonionic matrices\nin a way that could give a physical interpretation to the 26 dimensions ?\n\nSome of my thoughts (set out in CERN CDS preprint EXT-2004-031)\nalong those lines are:\n\n1 - the 8 + 8 + 8 dimensions of the g(1) and g(2) parts could\ncorrespond to the complexification of\nthe 8 + 8 + 8 dimensions of the g(-1) and g(-2) parts;\n\n2 - the "real part" 8-dim g(-2) might represent an 8-dimensional spacetime;\n\n3 - the "real part" 8+8-dim g(-1) might be useful as a representation\nspace of the 8 first-generation fermion particles and\nthe 8 first-generation antiparticles,\nif the 8+8-dim "real part" were to be "discretized" by orbifolding it; and\n\n4 - the so(8) might act effectively as a gauge group,\nreminiscent of the so(8) in 11-dim supergravity,\nbut acting in an effective 8-dim "spacetime" and\nacting on fermions determined by (3) instead\nof by 1-1 fermion-boson supersymmetry.\n\nNote that these thoughts are consistent with considering the\n"odd" graded parts g(-1) and g(1) of E6 as fermionic,\nand its "even" parts as bosonic (spacetime or gauge bosons),\nand that another way of looking at the above graded structure is\nthat E6 can be constructed from:\n\nthe 28-dim adjoint representation of so(8) plus\na complexification of the 8-dim vector representation of so(8) plus\na complexification of the two 8-dim half-spinor representations of so(8)\nplus 2 more dimensions, related to the complexifications.\n\nIn other words, this might be\nan alternative (instead of postulating 1-1 fermion-boson supersymmetry)\nway to bring fermions, as spinors, into string theory.\n\nFor visualisation,\nit might be helpful to recall these two symmetric spaces:\n\nE6 / so(10) x U(1) with dimension 78 - 45 - 1 = 32 and\nso(10) / so(8) x U(1) with dimenson 45 - 28 - 1 = 16.\n\nIf it would be better to start a new thread called something\nlike "E6 and 26-dim string theory", then that is OK with me.\n\nTony Smith\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>Alejandro Rivero wrote ...

> Perhaps your question in this forum could be, if string theory has
> something to say about the group [E6] ...

In that context, my question would be:
String theory is fundamentally 26-dimensional.

[Moderator's note: First of all, this sentence is not
a question. Second, it is not true. "String theory" is
a shorthand for "superstring theory" which is at most
10-dimensional - and its extension "M-theory" is
11-dimensional. 26 is the spacetime dimension of the
bosonic string - the old theory suffering from the tachyon,
the absence of fermions, and so on. LM]

As Pierre Ramond noted in http://www.arxiv.org/abs/hep-th/0112261
"... The traceless Jordan matrices [ J3(O)o ] ...
(3x3) traceless octonionic hermitian matrices,
each labelled by 26 real parameters ... span the 26 representation
of [ the 52-dimensional exceptional Lie algebra F4 ].

[Moderator's note: This seems as a pure numerology. The dimension
26 of the bosonic string coincides with the dimension
of the fundamental representation of F_4 but because there does
not seem to be any priviliged action of F_4 on the 26 coordinates,
it seems as a clear coincidence. It is not so nontrivial
to get the number 26 in two ways. Similar comments would
probably apply to the rest of the text, too. LM]

One can supplement the F4 transformations by an additional 26 parameters
.... leading to a group with 78 parameters. These extra transformations
are non-compact, and close on the F4 transformations,
leading to the exceptional group E6(-26).
The subscript in parenthesis denotes the number of non-compact minus
the number of compact generators. ...".

As Soji Kaneyuki noted in Graded Lie Algebras, Related Geometric Structures,
and Pseudo-hermitian Symmetric Spaces, a section of the book
Analysis and Geometry on Complex Homogeneous Domains, by Jacques Faraut,
Soji Kaneyuki, Adam Koranyi, Qi-keng Lu, and Guy Roos (Birkhauser 2000),
E6 can be seen as a Graded Lie Algebra with 5 grades:
g = E6 = g(-2) + g(-1) + g(0) + g(1) + g(2)
*such that
g(0) = so(8) + R + R has 28 + 2 dimensions
dimR g(-1) = dimR g(1) = 16 = 8 + 8
dimR g(-2) = dimR g(2) = 8
(dimR refers to real, as opposed to complex, dimension)

Could the 8 + 8 + 8 + 2 = 26 dimensions of the
g(-2), g(-1), and g(0) (other than the 28 so(8) dimensions) parts
of the graded structure of E6 correspond to
the 26 dimensions of traceless part J3(O)o of the
27-dimensional Jordan algebra J3(O) of 3x3 hermitian octonionic matrices
in a way that could give a physical interpretation to the 26 dimensions ?

Some of my thoughts (set out in CERN CDS preprint EXT-2004-031)
along those lines are:

1 - the 8 + 8 + 8 dimensions of the g(1) and g(2) parts could
correspond to the complexification of
the 8 + 8 + 8 dimensions of the g(-1) and g(-2) parts;

2 - the "real part" 8-dim g(-2) might represent an 8-dimensional spacetime;

3 - the "real part" 8+8-dim g(-1) might be useful as a representation
space of the 8 first-generation fermion particles and
the 8 first-generation antiparticles,
if the 8+8-dim "real part" were to be "discretized" by orbifolding it; and

4 - the so(8) might act effectively as a gauge group,
reminiscent of the so(8) in 11-dim supergravity,
but acting in an effective 8-dim "spacetime" and
acting on fermions determined by (3) instead
of by 1-1 fermion-boson supersymmetry.

Note that these thoughts are consistent with considering the
"odd" graded parts g(-1) and g(1) of E6 as fermionic,
and its "even" parts as bosonic (spacetime or gauge bosons),
and that another way of looking at the above graded structure is
that E6 can be constructed from:

the 28-dim adjoint representation of so(8) plus
a complexification of the 8-dim vector representation of so(8) plus
a complexification of the two 8-dim half-spinor representations of so(8)
plus 2 more dimensions, related to the complexifications.

In other words, this might be
an alternative (instead of postulating 1-1 fermion-boson supersymmetry)
way to bring fermions, as spinors, into string theory.

For visualisation,
it might be helpful to recall these two symmetric spaces:

E6 / so(10) x U(1)[/itex] with dimension 78 - 45 - 1 = 32 and
so(10) / so(8) x U(1) with dimenson [itex]45 - 28 - 1 = 16.

If it would be better to start a new thread called something
like "E6 and 26-dim string theory", then that is OK with me.

Tony Smith

kneemo

May15-04, 07:43 AM

The Moderator may want to read hep-th/0212085 before commenting on the 26-dimensional bosonic theory. There the authors (Chattaraputi et al) recover fermions from the 26-dimensional theory via a carefully engineered toroidal compactification.

A Matrix theory such as Tony Smith's would then be a nice formulation of (bosonic) M-theory (as Susskind refers to the 27-dimensional theory), from where we work down dimensionally--using a more dynamical version of the Chattaraputi et al procedure--to recover fermions.

Lubos Motl

May15-04, 08:05 AM

<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, 15 May 2004, kneemo wrote:\n\n> The Moderator may want to read hep-th/0212085 before commenting on the\n> 26-dimensional bosonic theory. There the authors (Chattaraputi et al)\n> recover fermions from the 26-dimensional theory via a carefully\n> engineered toroidal compactification.\n\nI was always intrigued by the idea that one could get the superstring from\na configuration of the bosonic string - because this would settle the\nquestion whether bosonic string theory is part of the "real" string\ntheory, or just a pedagogical semi-consistent toy model. The answer would\nbe the former if you could find a configuration - a compactification, or a\n"9-brane" or a "10-brane" within the bosonic theory - that would describe\nthe superstring or M-theoretical spacetime.\n\nHowever this particular program, initiated already in 1986 by Casher,\nEnglert, Nicolai, and Taormina, just does not seem physical to me. Yes,\nI\'ve read these articles. They are "truncating" the degrees of freedom of\nthe original bosonic string, in order to get the superstring.\n\n"Truncation" is something done by hand, a human intervention, an\nartificial change of dynamics, unlike Sen\'s tachyon condensation which is\na physical process. It seems to me that their results only say that the\nbosonic string has a larger number of degrees of freedom - in some way of\ncounting - and one can manually "erase" some of them to get the "smaller"\nsuperstring. This is a very different statement from finding an\nequivalence or a physical relation between these two theories.\n\nIt\'s like if we related, before the discovery of T-duality, type IIA and\ntype IIB string theories by switching the chirality of the right-moving\nfermions. Such an operation is a game with symbols, not a physical\nmechanism, I think.\n\n> A Matrix theory such as Tony Smith\'s would then be a nice formulation\n> of (bosonic) M-theory (as Susskind refers to the 27-dimensional\n> theory), from where we work down dimensionally--using a more dynamical\n> version of the Chattaraputi et al procedure--to recover fermions.\n\nBosonic Matrix theory (for example, one written by Soo-Jong Rey) is a\ntypical example of a theory where nothing works at quantum level. A\nBFSS-like matrix model with no fermions and 25 bosons has no moduli space,\nand therefore no spacetime interpretation. All objects tend to shrink to a\npoint. A more interesting result is obtained when we compactify on circle:\nbosonic matrix string theory does not work either, but at least the lowest\ndimension twist field has dimension (3/2,3/2), just like the superstring.\n\nHorowitz and Susskind\'s bosonic M-theory does not seem to offer a\nnon-trivial consistency check of the type that we are used to from\ndualities between the supersymmetric theories. If they were thinking\nnaturally, they would start with identifying the bosonic string with\nbosonic M-theory on a circle. But there are no stable, conserved D0-branes\nin bosonic string theory, those that would have to follow from the\nKaluza-Klein mechanism.\n\nSo they must change the shape of the 27th dimension and choose a line\ninterval instead because the U(1) isometry - and the D0-branes - then\ndisappear. Horava-Witten domain walls must carry an E_8 gauge symmetry to\ncancel the gravitational anomalies; however, the Horowitz-Susskind domain\nwall carries no anomalies to be canceled. In fact, they are wrong that\nthis means that the gauge group is nothing. In reality it means that the\ngauge group can be *anything* because there are no anomalies anywhere. No\nanomalies, no BPS objects to be compared - no constraints. The only thing\nthat they compare are the dimensions of spacetime (and membranes), but\nthey check as many numbers (such as 3 indices of a C-field) as many pieces\nof input they use to construct their theory, and therefore I believe it is\ncorrect to say that the evidence in favor of bosonic M-theory as the\nstrong coupling limit of the bosonic string equals zero.\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, 15 May 2004, kneemo wrote:

> The Moderator may want to read http://www.arxiv.org/abs/hep-th/0212085 before commenting on the
> 26-dimensional bosonic theory. There the authors (Chattaraputi et al)
> recover fermions from the 26-dimensional theory via a carefully
> engineered toroidal compactification.

I was always intrigued by the idea that one could get the superstring from
a configuration of the bosonic string - because this would settle the
question whether bosonic string theory is part of the "real" string
theory, or just a pedagogical semi-consistent toy model. The answer would
be the former if you could find a configuration - a compactification, or a
"9-brane" or a "10-brane" within the bosonic theory - that would describe
the superstring or M-theoretical spacetime.

However this particular program, initiated already in 1986 by Casher,
Englert, Nicolai, and Taormina, just does not seem physical to me. Yes,
I've read these articles. They are "truncating" the degrees of freedom of
the original bosonic string, in order to get the superstring.

"Truncation" is something done by hand, a human intervention, an
artificial change of dynamics, unlike Sen's tachyon condensation which is
a physical process. It seems to me that their results only say that the
bosonic string has a larger number of degrees of freedom - in some way of
counting - and one can manually "erase" some of them to get the "smaller"
superstring. This is a very different statement from finding an
equivalence or a physical relation between these two theories.

It's like if we related, before the discovery of T-duality, type IIA and
type IIB string theories by switching the chirality of the right-moving
fermions. Such an operation is a game with symbols, not a physical
mechanism, I think.

> A Matrix theory such as Tony Smith's would then be a nice formulation
> of (bosonic) M-theory (as Susskind refers to the 27-dimensional
> theory), from where we work down dimensionally--using a more dynamical
> version of the Chattaraputi et al procedure--to recover fermions.

Bosonic Matrix theory (for example, one written by Soo-Jong Rey) is a
typical example of a theory where nothing works at quantum level. A
BFSS-like matrix model with no fermions and 25 bosons has no moduli space,
and therefore no spacetime interpretation. All objects tend to shrink to a
point. A more interesting result is obtained when we compactify on circle:
bosonic matrix string theory does not work either, but at least the lowest
dimension twist field has dimension (3/2,3/2), just like the superstring.

Horowitz and Susskind's bosonic M-theory does not seem to offer a
non-trivial consistency check of the type that we are used to from
dualities between the supersymmetric theories. If they were thinking
naturally, they would start with identifying the bosonic string with
bosonic M-theory on a circle. But there are no stable, conserved D0-branes
in bosonic string theory, those that would have to follow from the
Kaluza-Klein mechanism.

So they must change the shape of the 27th dimension and choose a line
interval instead because the U(1) isometry - and the D0-branes - then
disappear. Horava-Witten domain walls must carry an E_8 gauge symmetry to
cancel the gravitational anomalies; however, the Horowitz-Susskind domain
wall carries no anomalies to be canceled. In fact, they are wrong that
this means that the gauge group is nothing. In reality it means that the
gauge group can be *anything* because there are no anomalies anywhere. No
anomalies, no BPS objects to be compared - no constraints. The only thing
that they compare are the dimensions of spacetime (and membranes), but
they check as many numbers (such as 3 indices of a C-field) as many pieces
of input they use to construct their theory, and therefore I believe it is
correct to say that the evidence in favor of bosonic M-theory as the
strong coupling limit of the bosonic string equals zero.
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^

mitchell porter

May16-04, 12:26 AM

<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 <motl@feynman.harvard.edu> wrote in message news:<Pine.LNX.4.31.0405150744340.18361-100000@feynman.harvard.edu>...\n\n> I was always intrigued by the idea that one could get the superstring from\n> a configuration of the bosonic string - because this would settle the\n> question whether bosonic string theory is part of the "real" string\n> theory, or just a pedagogical semi-consistent toy model. The answer would\n> be the former if you could find a configuration - a compactification, or a\n> "9-brane" or a "10-brane" within the bosonic theory - that would describe\n> the superstring or M-theoretical spacetime.\n\nThere is an obscure paper:\n\nMEMBRANES IN STRING THEORY, TREES, THE WEIL CONJECTURES AND THE RAMANUJAN NUMBERS.\nBy Bernard Grossman (Rockefeller U.),. DOE/ER/40325-34, (Received Oct 1988). 21pp.\nPublished in J.Math.Phys.30:506,1989\n\n.... which I have long thought might be relevant to this question,\nbut I\'ve never understood it. From the introduction:\n\n"... we will use p-adic group theory to understand the\nfactorization of the inverse bosonic string partition\nfunction. We interpret this factorization in terms of\nthe existence of an 11-dimensional membrane."\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 <motl@feynman.harvard.edu> wrote in message news:<Pine.LNX.4.31.0405150744340.18361-100000@feynman.harvard.edu>...

> I was always intrigued by the idea that one could get the superstring from
> a configuration of the bosonic string - because this would settle the
> question whether bosonic string theory is part of the "real" string
> theory, or just a pedagogical semi-consistent toy model. The answer would
> be the former if you could find a configuration - a compactification, or a
> "9-brane" or a "10-brane" within the bosonic theory - that would describe
> the superstring or M-theoretical spacetime.

There is an obscure paper:

MEMBRANES IN STRING THEORY, TREES, THE WEIL CONJECTURES AND THE RAMANUJAN NUMBERS.
By Bernard Grossman (Rockefeller U.),. DOE/ER/40325-34, (Received Oct 1988). 21pp.
Published in J.Math.Phys.30:506,1989

.... which I have long thought might be relevant to this question,
but I've never understood it. From the introduction:

"... we will use p-adic group theory to understand the
factorization of the inverse bosonic string partition
function. We interpret this factorization in terms of
the existence of an 11-dimensional membrane."

Charlie Stromeyer Jr.

May16-04, 12:28 AM

<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>kneemo <kneemo@physicsforums.com> wrote in message news:\n\n> A Matrix theory such as Tony Smith\'s would then be a nice formulation\n> of (bosonic) M-theory (as Susskind refers to the 27-dimensional\n> theory), from where we work down dimensionally--using a more dynamical\n> version of the Chattaraputi et al procedure--to recover fermions.\n\nAccording to this paper by A. Keurentjes and as noted on page 36 of\nthis paper, there does not appear to be a way to make the hypothetical\nBosonic M-theory compatible with a realistic classification of the\nsimple Lie algebras:\n\nhttp://arxiv.org/abs/hep-th/0210178\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>kneemo <kneemo@physicsforums.com> wrote in message news:

> A Matrix theory such as Tony Smith's would then be a nice formulation
> of (bosonic) M-theory (as Susskind refers to the 27-dimensional
> theory), from where we work down dimensionally--using a more dynamical
> version of the Chattaraputi et al procedure--to recover fermions.

According to this paper by A. Keurentjes and as noted on page 36 of
this paper, there does not appear to be a way to make the hypothetical
Bosonic M-theory compatible with a realistic classification of the
simple Lie algebras:

http://arxiv.org/abs/http://www.arxiv.org/abs/hep-th/0210178

Tony Smith

May17-04, 09:34 AM

<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>In reply to kneemo\'s remark in a post to this sps thread\nRe: Speculation: E6 and 26-dim. string theory\n"... A Matrix theory such as Tony Smith\'s would then be\na nice formulation of (bosonic) M-theory (as Susskind refers\nto the 27-dimensional theory), from where we work down dimensionally-\n-using a more dynamical version of the Chattaraputi et al procedure-\n....[ http://xxx.lanl.gov/abs/hep-th/0212085 ]...\n-to recover fermions. ...",\n\nCharlie Stromeyer Jr. (formerly known as Zirkus) said in that thread:\n"... According to this paper by A. Keurentjes ...\n....and as noted on page 36 of this paper,\nhttp://arxiv.org/abs/hep-th/0210178\nthere does not appear to be a way to make the hypothetical\nBosonic M-theory compatible with a realistic classification\nof the simple Lie algebras ...".\n\nAlthough I agree with LM that my model may fairly be\nlabelled speculation, my biased personal view is that\nthere is a possibility that it may be correct,\nand that the Keurentjes paper does not refute it.\n\nWith respect to what is known as bosonic (26-dim) string theory,\nthe Keurentjes paper, entitled "The group theory of oxidation",\nsays, at page 36:\n"... although our methods are easily applicable to\nthe massless sector of the bosonic string,\n(governed by the split form of D24, SO(24, 24)),\nwe find no hints for "bosonic M-theory" ...\n[33 ... G. T. Horowitz and L. Susskind, "Bosonic M theory,"\nJ. Math. Phys. 42 (2001) 3152 ... arXiv:hep-th/0012037 ] ...\n.... Again\nit can be argued that our assumptions are not suitable\nfor dealing with this issue,\nbut\nit is less clear to us why (from a physical perspective)\nthis hypothetical theory should not fit somewhere in our\nframework (it cannot fit anywhere by virtue of the classification\nof simple Lie algebra\'s). ...".\n\nAs to the assumptions of Keurentjes,\nat page 2 of his paper, Keurentjes says:\n"... In this paper we study the well established coset theories.\nThe cosets are of the form G/H, where G is a non-compact Lie\ngroup, and H its maximal compact subgroup. ...\n.... the coset symmetries that occur in theories that are related\nto higher dimensional theories by toroidal compactification,\nand subsequently truncating to the massless sector ...\nIn this paper, we will refer to this procedure as "dimensional\nreduction", even though one can of course reduce over other\nmanifolds than tori. ...".\n\nWith respect to the classification of simple Lie algebras,\nKeurentjes has at page 30 a Table 1: The E8 triangle,\ncontaining as part of its bottom two lines the following:\n"... E8 E7 E6 D5 ...\n... n = 8 n = 7 n = 6 n = 5 ...".\n\nAs to how all this is related to my model:\n\n1 - My model is not based on coset space supergravity oxidation,\nnor does it use "the split form of D24, SO(24,24)",\nbut is based on a Graded Lie Algebra 5-graded structure of E6,\nwhich is\ng = E6 = g(-2) + g(-1) + g(0) + g(1) + g(2)\n*such that\ng(0) = so(8) + R + R has 28 + 2 dimensions\ndimR g(-1) = dimR g(1) = 16 = 8 + 8\ndimR g(-2) = dimR g(2) = 8\n(dimR refers to real, as opposed to complex, dimension).\n\n2 - However, the 16-dim g(-1) and 16-dim g(1) parts\nare, in my model, related to the 32-dim coset space\nE6 / D5 x U(1), and I note Keurentjes lists D5 following E6\nin the penultimate line of his Table 1 (and that they\nare also similarly adjacent in the second column of that table).\nKeurentjes does not discuss this coset in his section B.5\nentitled "E6" (at pages 43-44), perhaps because it may not be\nrelevant to his supergravity oxidation structures that are\nsubject to the assumptions of his paper (unlike my model, which\nis based on a different structure and a different physical\nperspective).\nRoughly, in my model the space E6 / D5 x U(1) is related\nto 16 of the 26 dimensions of string theory,\nwhich 16 dimensions represent fermionic structure.\n\n3 - The orbifolding of 16 of the 26 dimensions in my model is\nnot exactly toroidal compactification, although it\nmay be related thereto.\n\n4 - Since E6 is roughly a complexification of F4, which is\nthe automorphism group of the exceptional 27-dim Jordan algebra J3(O)\nof 3x3 hermitian octonionic matrices (whose traceless part is\n26-dim J3(O)o),\nthe structure of my model is less related to supergravity oxidation\nthan it is to the role of Jordan algebras (and nonassociative\noctonions) in string theory, such as discussed by Zirkus at\nhttp://www.lns.cornell.edu/spr/2003-02/msg0048479.html\n\nIn light of such things, it is my personal opinion\nthat the assumptions of Keurentjes are not applicable to my model,\nand that the Keurentjes paper does not refute it.\n\nTony Smith\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>In reply to kneemo's remark in a post to this sps thread
Re: Speculation: E6 and 26-dim. string theory
"... A Matrix theory such as Tony Smith's would then be
a nice formulation of (bosonic) M-theory (as Susskind refers
to the 27-dimensional theory), from where we work down dimensionally-
-using a more dynamical version of the Chattaraputi et al procedure-
....[ http://xxx.lanl.gov/abs/http://www.arxiv.org/abs/hep-th/0212085 ]...
-to recover fermions. ...",

Charlie Stromeyer Jr. (formerly known as Zirkus) said in that thread:
"... According to this paper by A. Keurentjes ...
....and as noted on page 36 of this paper,
http://arxiv.org/abs/http://www.arxiv.org/abs/hep-th/0210178
there does not appear to be a way to make the hypothetical
Bosonic M-theory compatible with a realistic classification
of the simple Lie algebras ...".

Although I agree with LM that my model may fairly be
labelled speculation, my biased personal view is that
there is a possibility that it may be correct,
and that the Keurentjes paper does not refute it.

With respect to what is known as bosonic (26-dim) string theory,
the Keurentjes paper, entitled "The group theory of oxidation",
says, at page 36:
"... although our methods are easily applicable to
the massless sector of the bosonic string,
(governed by the split form of D24, SO(24, 24)),
we find no hints for "bosonic M-theory" ...
[33 ... G. T. Horowitz and L. Susskind, "Bosonic M theory,"
J. Math. Phys. 42 (2001) 3152 ... arXiv:http://www.arxiv.org/abs/hep-th/0012037 ] ...
.... Again
it can be argued that our assumptions are not suitable
for dealing with this issue,
but
it is less clear to us why (from a physical perspective)
this hypothetical theory should not fit somewhere in our
framework (it cannot fit anywhere by virtue of the classification
of simple Lie algebra's). ...".

As to the assumptions of Keurentjes,
at page 2 of his paper, Keurentjes says:
"... In this paper we study the well established coset theories.
The cosets are of the form G/H, where G is a non-compact Lie
group, and H its maximal compact subgroup. ...
.... the coset symmetries that occur in theories that are related
to higher dimensional theories by toroidal compactification,
and subsequently truncating to the massless sector ...
In this paper, we will refer to this procedure as "dimensional
reduction", even though one can of course reduce over other
manifolds than tori. ...".

With respect to the classification of simple Lie algebras,
Keurentjes has at page 30 a Table 1: The E8 triangle,
containing as part of its bottom two lines the following:
"... E8 E7 E6 D5 ...
... n = 8 n = 7 n = 6 n = 5 ...".

As to how all this is related to my model:

1 - My model is not based on coset space supergravity oxidation,
nor does it use "the split form of D24, SO(24,24)",
but is based on a Graded Lie Algebra 5-graded structure of E6,
which is
g = E6 = g(-2) + g(-1) + g(0) + g(1) + g(2)
*such that
g(0) = so(8) + R + R has 28 + 2 dimensions
dimR g(-1) = dimR g(1) = 16 = 8 + 8
dimR g(-2) = dimR g(2) = 8
(dimR refers to real, as opposed to complex, dimension).

2 - However, the 16-dim g(-1) and 16-dim g(1) parts
are, in my model, related to the 32-dim coset space
E6 / D5 x U(1), and I note Keurentjes lists D5 following E6
in the penultimate line of his Table 1 (and that they
are also similarly adjacent in the second column of that table).
Keurentjes does not discuss this coset in his section B.5
entitled "E6" (at pages 43-44), perhaps because it may not be
relevant to his supergravity oxidation structures that are
subject to the assumptions of his paper (unlike my model, which
is based on a different structure and a different physical
perspective).
Roughly, in my model the space E6 / D5 x U(1) is related
to 16 of the 26 dimensions of string theory,
which 16 dimensions represent fermionic structure.

3 - The orbifolding of 16 of the 26 dimensions in my model is
not exactly toroidal compactification, although it
may be related thereto.

4 - Since E6 is roughly a complexification of F4, which is
the automorphism group of the exceptional 27-dim Jordan algebra J3(O)
of 3x3 hermitian octonionic matrices (whose traceless part is
26-dim J3(O)o),
the structure of my model is less related to supergravity oxidation
than it is to the role of Jordan algebras (and nonassociative
octonions) in string theory, such as discussed by Zirkus at
http://www.lns.cornell.edu/spr/2003-02/msg0048479.html

In light of such things, it is my personal opinion
that the assumptions of Keurentjes are not applicable to my model,
and that the Keurentjes paper does not refute it.

Tony Smith

Lubos Motl

Jul18-04, 02:54 AM

<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 Tue, 13 Jul 2004, kneemo wrote:\n\n> Yes, "truncation" is indeed an artificial process, as was voiced by the\n> authors themselves. That it was done at all, even artificially, is\n> enough to motivate a search for the natural mechanism which\n> accomplishes truncation dynamically. I look at it as a road map of\n> what needs to be achieved by physical means.\n\nDear Mike, I share your excitement from the possibility to obtain the\nsuperstring as a configuration in the bosonic string - a sort of\ncondensation of the bosonic string vacuum into the superstring vacuum. But\nI don\'t quite agree that the first proposal must be immediately a road\nmap. If a proposal does not work in details - and the need to truncate by\nhand shows that this proposal does not quite work - the correct solution\nis often completely different. The idea that the worldsheet degrees of\nfreedom of the superstring must be directly taken from some similar\ndegrees of freedom of the bosonic string - this idea is just a conjecture,\nand most likely it is a wrong and arbitrary conjecture. The correct map,\nif it exists at all, can be must more subtle.\n\n> Yes, it would be beautiful to recover the superstring from a\n> configuration of the bosonic string via a physical mechanism. I am\n> particularly fond of Sen\'s tachyon condensation, not only as a physical\n> process, but for its interplay of D-branes and tachyons. I\'d like to\n> think, physically, something like Sen\'s tachyon condensation would be\n> the natural version of truncation.\n\nThat\'s the goal, but such a mechanism is not known (yet) in the case of\nthe bosonic-superstring relation.\n\n> As for finding the configuration, how have you went about finding a\n> 9-brane or 10-brane in the bosonic theory? I\'m curious as to how you\n> approached the problem in the first place.\n\nThis is my favorite speculation. I want to start with a theory in 26\ndimensions that already has a E8 gauge group - satisfy my requirement in\nany way you know.\n\nThen one can have nonzero integrals of F /\\ F - the instanton number -\nthat would give some types of 21-branes.\n\nThe next nonzero invariant is the integral of F^8 (with the wedge product\nunderstood) because E8 has nontrivial \\pi_{15}. You must arrange a E8\ngauge field in a 16-dimensional space transverse to the 10D spacetime in\nsuch a way that the integral of the 8th wedge power of F is N - and this\nconfiguration could then look like (or condense into) N 10-dimensional\nsuperstring. A superstring spacetime would be a sort of codimension-16\nbrane in the bosonic string theory.\n\nOf course, it is just a speculation based on numerology, and nothing\nguarantees that a true picture like this one exists.\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 Tue, 13 Jul 2004, kneemo wrote:

> Yes, "truncation" is indeed an artificial process, as was voiced by the
> authors themselves. That it was done at all, even artificially, is
> enough to motivate a search for the natural mechanism which
> accomplishes truncation dynamically. I look at it as a road map of
> what needs to be achieved by physical means.

Dear Mike, I share your excitement from the possibility to obtain the
superstring as a configuration in the bosonic string - a sort of
condensation of the bosonic string vacuum into the superstring vacuum. But
I don't quite agree that the first proposal must be immediately a road
map. If a proposal does not work in details - and the need to truncate by
hand shows that this proposal does not quite work - the correct solution
is often completely different. The idea that the worldsheet degrees of
freedom of the superstring must be directly taken from some similar
degrees of freedom of the bosonic string - this idea is just a conjecture,
and most likely it is a wrong and arbitrary conjecture. The correct map,
if it exists at all, can be must more subtle.

> Yes, it would be beautiful to recover the superstring from a
> configuration of the bosonic string via a physical mechanism. I am
> particularly fond of Sen's tachyon condensation, not only as a physical
> process, but for its interplay of D-branes and tachyons. I'd like to
> think, physically, something like Sen's tachyon condensation would be
> the natural version of truncation.

That's the goal, but such a mechanism is not known (yet) in the case of
the bosonic-superstring relation.

> As for finding the configuration, how have you went about finding a
> 9-brane or 10-brane in the bosonic theory? I'm curious as to how you
> approached the problem in the first place.

This is my favorite speculation. I want to start with a theory in 26
dimensions that already has a E8 gauge group - satisfy my requirement in
any way you know.

Then one can have nonzero integrals of F /\ F - the instanton number -
that would give some types of 21-branes.

The next nonzero invariant is the integral of F^8 (with the wedge product
understood) because E8 has nontrivial \pi_{15}. You must arrange a E8
gauge field in a 16-dimensional space transverse to the 10D spacetime in
such a way that the integral of the 8th wedge power of F is N - and this
configuration could then look like (or condense into) N 10-dimensional
superstring. A superstring spacetime would be a sort of codimension-16
brane in the bosonic string theory.

Of course, it is just a speculation based on numerology, and nothing
guarantees that a true picture like this one exists.

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