View Full Version : Nonabelian 2-forms (was: on gerbes and boundary states)
Urs Schreiber
Jun18-04, 11:17 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>I wrote in news:2jdjt1F10cbmhU1-100000@uni-berlin.de...\n\n> What I don\'t understand yet is how for instance B-fields arise in string\n> theory which take values in a _non_-abelian group. Hm, how does that work?\n> Of course B really mixes with the (non-abelian) gauge field A in that only\n> the combination B-(dA)/T is physically meaningful.But the non-abelian\n> property of A is related to the fact that this comes from the open string\n> sector, where stings carry Chan-Paton factors, while B comes from the\nclosed\n> string sector without such factors. Hm. Apparently I am missing an\n> elementary insight here.\n\nI was kindly contacted by private email and pointed to the following two\npapers:\n\nAmitabha Lahiri:\nLocal symmetries of the non-Abelian two-form\nhep-th/0109220\n\nAmitabha Lahiri:\nParallel transport on non-Abelian flux tubes,\nhep-th/0312112\n\nThese papers come from a field theoretic viewpoint, not from strings, and\ndiscuss some aspects of the general problem of dealing with non-Abelian\n2-forms. In particular, these papers point out some nice \'pedestrian\' ways\nto understand various transformation laws for 2-form connections.\n\nAs for these general properties, I am fully aware that I should also go back\nand read extensive discussion of this issue on s.p.r., for instance in this\nthread:\n\nhttp://groups.google.de/groups?hl=de&lr=&ie=UTF-8&threadm=acmkg0%247o5%241%40glue.ucr.edu&rnum=1&prev=/groups%3Fhl%3Dde%26lr%3D%26ie%3DUTF-8%26selm%3Dacmkg0%25247o5%25241%2540glue.ucr.edu\n \nThere I found the very useful reference\n\nChristiaan Hofman:\nNonabelian 2-forms,\nhep-th/0207017 .\n\nFirst of all, this reference clarifies where the non-Abelian B-field appears\nin string theory.It\'s pretty obvious now that I think about it: The B field\ncouples to the NS 5-brane (by definition, essentially). So just pick a stack\nof such NS5-branes to obtain a matrix-valued B-field.\n\nEr, or something along these lines. What is it that plays the role of the\nChan-Paton factors here?\n\nIn the above paper it says:\n\n"It has long been speculated that stacks of these 5-branes at low energy\nare described by nonabelian 2-forms. [...] A strong hint for the appearance\nof nonabelian 2-forms in these systems comes from string duality. According\nto this, the dimensional reduction of the (2,0) LST should give rise to N=4\nsuper Yang-Mills. In the abelian case it can easily be seen how the\n(self-dual) 2-form reduces to the abelian 1-form gauge field. The question\nis how the nonabelian 1-form in the Yang-Mills lifts to a 2-form gauge\nfield."\n\nSo apparently the question how the nonabelian B field arises in string\ntheory is almost but not completely understood (at least as of 2002).\n\nThis paper by Hofman is very valuable to me, since it nicely emphasizes the\nloop space point of view.\n\nIn fact, it is a nice exercise to check that the deifnition of the exterior\nderivative that he gives in equation (5) of hep-th/0207017 follows from\nusing the simple form\n\nd = E^{\\dagger I} \\partial_I\n\nwhich I mentioned in a previous message\n(http://groups.google.de/groups?selm=2jdjt1F10cbmhU1-100000%40uni-berlin.de)\n..\n\nHofman also managed to make me understand why the 2-form connection B needs\nto come along with a 1-form connection A: We need some connection A to pull\nback the algebra-valued forms, that distribute over the loop as we compute\npath-ordered integrals, back to the (arbitrary) base point of the loop,\nbecause only there can we locally and safely compute the non-abelian\nproduct.\n\nI still need to figure out the following: As I have mentioned before the\nabelian 2-form covariant exterior derivative is equal to\n\nd_B = d + exp(\\int B) K-> exp(-\\int B) .\n\nIt seems natural to conjecture that we get the non-abelian covariant\nderivative (equation (15) in Hofman\'s paper) by simply replacing exp(\\int B)\nby its path-ordered version, equipped with the above mentioned pull-back by\nsome A. I am pretty sure that this must be true, but I cannot quite see it\nyet.\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>I wrote in news:2jdjt1F10cbmhU1-100000@uni-berlin.de...
> What I don't understand yet is how for instance B-fields arise in string
> theory which take values in a _non_-abelian group. Hm, how does that work?
> Of course B really mixes with the (non-abelian) gauge field A in that only
> the combination B-(dA)/T is physically meaningful.But the non-abelian
> property of A is related to the fact that this comes from the open string
> sector, where stings carry Chan-Paton factors, while B comes from the
closed
> string sector without such factors. Hm. Apparently I am missing an
> elementary insight here.
I was kindly contacted by private email and pointed to the following two
papers:
Amitabha Lahiri:
Local symmetries of the non-Abelian two-form
http://www.arxiv.org/abs/hep-th/0109220
Amitabha Lahiri:
Parallel transport on non-Abelian flux tubes,
http://www.arxiv.org/abs/hep-th/0312112
These papers come from a field theoretic viewpoint, not from strings, and
discuss some aspects of the general problem of dealing with non-Abelian
2-forms. In particular, these papers point out some nice 'pedestrian' ways
to understand various transformation laws for 2-form connections.
As for these general properties, I am fully aware that I should also go back
and read extensive discussion of this issue on s.p.r., for instance in this
thread:
http://groups.google.de/groups?hl=de&lr=&ie=UTF-8&threadm=acmkg0%247o5%241%40glue.ucr.edu&rnum=1&prev=/groups%3Fhl%3Dde%26lr%3D%26ie%3DUTF-8%26selm%3Dacmkg0%25247o5%25241%2540glue.ucr.edu
There I found the very useful reference
Christiaan Hofman:
Nonabelian 2-forms,
http://www.arxiv.org/abs/hep-th/0207017 .
First of all, this reference clarifies where the non-Abelian B-field appears
in string theory.It's pretty obvious now that I think about it: The B field
couples to the NS 5-brane (by definition, essentially). So just pick a stack
of such NS5-branes to obtain a matrix-valued B-field.
Er, or something along these lines. What is it that plays the role of the
Chan-Paton factors here?
In the above paper it says:
"It has long been speculated that stacks of these 5-branes at low energy
are described by nonabelian 2-forms. [...] A strong hint for the appearance
of nonabelian 2-forms in these systems comes from string duality. According
to this, the dimensional reduction of the (2,0) LST should give rise to N=4
super Yang-Mills. In the abelian case it can easily be seen how the
(self-dual) 2-form reduces to the abelian 1-form gauge field. The question
is how the nonabelian 1-form in the Yang-Mills lifts to a 2-form gauge
field."
So apparently the question how the nonabelian B field arises in string
theory is almost but not completely understood (at least as of 2002).
This paper by Hofman is very valuable to me, since it nicely emphasizes the
loop space point of view.
In fact, it is a nice exercise to check that the deifnition of the exterior
derivative that he gives in equation (5) of http://www.arxiv.org/abs/hep-th/0207017 follows from
using the simple form
d = E^{\dagger I} \partial_I
which I mentioned in a previous message
(http://groups.google.de/groups?selm=2jdjt1F10cbmhU1-100000%40uni-berlin.de)
..
Hofman also managed to make me understand why the 2-form connection B needs
to come along with a 1-form connection A: We need some connection A to pull
back the algebra-valued forms, that distribute over the loop as we compute
path-ordered integrals, back to the (arbitrary) base point of the loop,
because only there can we locally and safely compute the non-abelian
product.
I still need to figure out the following: As I have mentioned before the
abelian 2-form covariant exterior derivative is equal to
d_B = d + \exp(\int B) K-> \exp(-\int B) .
It seems natural to conjecture that we get the non-abelian covariant
derivative (equation (15) in Hofman's paper) by simply replacing \exp(\int B)
by its path-ordered version, equipped with the above mentioned pull-back by
some A. I am pretty sure that this must be true, but I cannot quite see it
yet.
Urs Schreiber
Jun18-04, 04:37 PM
<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>"Urs Schreiber" <Urs.Schreiber@uni-essen.de> schrieb im Newsbeitrag\nnews:2jgisnF11d328U1-100000@uni-berlin.de...\n\n> Christiaan Hofman:\n> Nonabelian 2-forms,\n> hep-th/0207017 .\n\nThis text is a revelation.\n\nI am thinking about how the construction on p. 6-7 could translate into a\nboundary state describing nonabelian B field plus gauge connection A on,\nsay, a D9. We know from Hashimoto, Maeda, Nakatsu&Oonmishi\n(http://golem.ph.utexas.edu/string/archives/000381.html) that in the abelian\ncase and without the B field the supersymmetric boundary state is something\nlike\n\n\\exp( \\int (A \\cdot X^\\prime + \\psi \\cdot (dA) \\cdot \\psi ) |\\alpha>\n(1)\n\nwhere |\\alpha> is the bare boundary state of just the flat brane without\nanything turned on, and \\psi are worldsheet fermions.\n\nNow the crucial insight of Hofman\'s paper above seems to be that which is\nassociated with his equation (13), namely that in the case of non-abelian\nhigher gauge theory the path-ordered integral has to include pull-back\nholonomies of A in order to make sense, so that in particular the 2-form\ngauge covariant exterior derivative (15) on loop space contains not just the\nterm\n\n\\psi \\cdot B \\cdot X^\\prime\n\n(where \'\\cdot\' is the loop space index contraction which includes\nintegration over the loop parameter)\n\nbut that what Hofman writes in (14) as\n\n\\oint_A (B).\n\nBut somehow it looks unnatural to include these A-holonomies by hand as in\n(12). That would give rise to a weirdly complicated expression for any\nboundary state, as far as I can see.\n\nBut Hashimoto\'s formula (1) above seems to suggest that the solution could\ncome about more elgantly. There the field strenght F = dA appears and we\nknow that with a B-field we should really have the invariant combination B +\ndA (when the string tension is set to unity, for convenience).\n\nSo let\'s substitute dA \\to dA + B in (1) (and go to the nonabelian case,\nintroducing an appropriate path ordering). Now apply the worldsheet super\nViraoro constraints on that boundary state. As we pull the polar combination\nd = G + i \\bar G throught this boundary state we pick up a couple of terms,\nin particular one term linear in the worldsheet fermions, which should hence\ngive the gauge connection (a 1-form, the fermions play the roles of 1-forms\non loop space) and this term is precisely of the form\n\n\\oint_A (dA + B)\n\nas required by Hofman. So the loop space exterior derivate d picks up the\ncorrect 2-form gauge connection term. The A and the B field automatically\nand naturally conspire so as to give the otherwise rather contrived looking\nconnection term \\oint_A (dA + B).\n\n(Or so I think, actually I am too tired for true thinking.)\n\n(Of course also other terms appear, but that\'s fine in general, since we\nknow that the string constraints deform not just by picking up a generalized\ncovariant but also other terms.)\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>"Urs Schreiber" <Urs.Schreiber@uni-essen.de> schrieb im Newsbeitrag
news:2jgisnF11d328U1-100000@uni-berlin.de...
> Christiaan Hofman:
> Nonabelian 2-forms,
> http://www.arxiv.org/abs/hep-th/0207017 .
This text is a revelation.
I am thinking about how the construction on p. 6-7 could translate into a
boundary state describing nonabelian B field plus gauge connection A on,
say, a D9. We know from Hashimoto, Maeda, Nakatsu&Oonmishi
(http://golem.ph.utexas.edu/string/archives/000381.html) that in the abelian
case and without the B field the supersymmetric boundary state is something
like
\exp( \int (A \cdot X^\prime + \psi \cdot (dA) \cdot \psi ) |\alpha>[/itex]
(1)
where |\alpha> is the bare boundary state of just the flat brane without
anything turned on, and \psi are worldsheet fermions.
Now the crucial insight of Hofman's paper above seems to be that which is
associated with his equation (13), namely that in the case of non-abelian
higher gauge theory the path-ordered integral has to include pull-back
holonomies of A in order to make sense, so that in particular the 2-form
gauge covariant exterior derivative (15) on loop space contains not just the
term
\psi \cdot B \cdot X^\prime
(where '\cdot' is the loop space index contraction which includes
integration over the loop parameter)
but that what Hofman writes in (14) as
\oint_A (B).
But somehow it looks unnatural to include these A-holonomies by hand as in
(12). That would give rise to a weirdly complicated expression for any
boundary state, as far as I can see.
But Hashimoto's formula (1) above seems to suggest that the solution could
come about more elgantly. There the field strenght F = dA appears and we
know that with a B-field we should really have the invariant combination B +
dA (when the string tension is set to unity, for convenience).
So let's substitute dA \to dA + B in (1) (and go to the nonabelian case,
introducing an appropriate path ordering). Now apply the worldsheet super
Viraoro constraints on that boundary state. As we pull the polar combination
d = G + i \bar G throught this boundary state we pick up a couple of terms,
in particular one term linear in the worldsheet fermions, which should hence
give the gauge connection (a 1-form, the fermions play the roles of 1-forms
on loop space) and this term is precisely of the form
[itex]\oint_A (dA + B)
as required by Hofman. So the loop space exterior derivate d picks up the
correct 2-form gauge connection term. The A and the B field automatically
and naturally conspire so as to give the otherwise rather contrived looking
connection term \oint_A (dA + B).
(Or so I think, actually I am too tired for true thinking.)
(Of course also other terms appear, but that's fine in general, since we
know that the string constraints deform not just by picking up a generalized
covariant but also other terms.)
Charlie Stromeyer Jr.
Jun21-04, 02:14 PM
<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>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:\n\n> There I found the very useful reference\n>\n> Christiaan Hofman:\n> Nonabelian 2-forms,\n> hep-th/0207017 .\n>\n> First of all, this reference clarifies where the non-Abelian B-field appears\n> in string theory.It\'s pretty obvious now that I think about it: The B field\n> couples to the NS 5-brane (by definition, essentially). So just pick a stack\n> of such NS5-branes to obtain a matrix-valued B-field.\n>\n> Er, or something along these lines. What is it that plays the role of the\n> Chan-Paton factors here?\n\nThis is not made clear in the paper. Instead, the author hopes that\nhis approach might be relevant for D-branes physics.\n\nThe algebraic structures involved with the B-field change depending\nupon whether there is the presence of NS 5-brane charges in which case\nthe B-field is non-torsion or whether the B-field is flat (torsion),\nthe effects of the KK-monopole electric charges associated with the\nB-field etc., and whether the B-field is e.g. on a gerbe or a quantum\ngerbe (at least in the paper I referred you to about quantum gerbes\n[hep-th/0206101] there is a natural contravariant connection).\n\n\n--------------------------------------------------------------------------------\n\n"These results [of Srinivasa Ramanujan] must be true because, if they\nwere not true, no one would have had the imagination to invent them."\n- G.H. Hardy\n\n"The most beautiful thing we can experience is the Mysterious." - A.\nEinstein\n\n"Superstring/M-theory is the language in which God wrote the world."\n- L. Motl\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>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:
> There I found the very useful reference
>
> Christiaan Hofman:
> Nonabelian 2-forms,
> http://www.arxiv.org/abs/hep-th/0207017 .
>
> First of all, this reference clarifies where the non-Abelian B-field appears
> in string theory.It's pretty obvious now that I think about it: The B field
> couples to the NS 5-brane (by definition, essentially). So just pick a stack
> of such NS5-branes to obtain a matrix-valued B-field.
>
> Er, or something along these lines. What is it that plays the role of the
> Chan-Paton factors here?
This is not made clear in the paper. Instead, the author hopes that
his approach might be relevant for D-branes physics.
The algebraic structures involved with the B-field change depending
upon whether there is the presence of NS 5-brane charges in which case
the B-field is non-torsion or whether the B-field is flat (torsion),
the effects of the KK-monopole electric charges associated with the
B-field etc., and whether the B-field is e.g. on a gerbe or a quantum
gerbe (at least in the paper I referred you to about quantum gerbes
[http://www.arxiv.org/abs/hep-th/0206101] there is a natural contravariant connection).
--------------------------------------------------------------------------------
"These results [of Srinivasa Ramanujan] must be true because, if they
were not true, no one would have had the imagination to invent them."
- G.H. Hardy
"The most beautiful thing we can experience is the Mysterious." - A.
Einstein
"Superstring/M-theory is the language in which God wrote the world."
- L. Motl
--------------------------------------------------------------------------------
Urs Schreiber
Jun21-04, 02:30 PM
<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 Mon, 21 Jun 2004, Charlie Stromeyer Jr. wrote:\n\n> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:\n>\n> > There I found the very useful reference\n> >\n> > Christiaan Hofman:\n> > Nonabelian 2-forms,\n> > hep-th/0207017 .\n> >\n> > First of all, this reference clarifies where the non-Abelian B-field appears\n> > in string theory.It\'s pretty obvious now that I think about it: The B field\n> > couples to the NS 5-brane (by definition, essentially). So just pick a stack\n> > of such NS5-branes to obtain a matrix-valued B-field.\n> >\n> > Er, or something along these lines. What is it that plays the role of the\n> > Chan-Paton factors here?\n>\n> This is not made clear in the paper. Instead, the author hopes that\n> his approach might be relevant for D-branes physics.\n\nI find it hard to tell what in this paper is derived fact and what is a\nproposal for a construction. For instance the paper seems to suggest that\nall non-commutative forms on loop space should be built using that \\oint_A\ninstead of \\oint. I am not sure if that\'s the right approach, for\ninstance to string theory, where the states of the superstring are\nprecisely such p-forms on loop space. Also, the author does not show that\nthe 2-form gauge connection he uses is the only sensible construction that one\ncould call by such a name, or does he?\n\nI am asking in particular because I made some SCFT deformation analysis\nwhich in a rather nice way gives something that looks like a 2-form gauge\nconnection on loop space induced by the B-field, and which also is almost\nexactly what Christiaan Hofman discusses - except for a small difference.\nI have the details at\n\nhttp://golem.ph.utexas.edu/string/archives/000385.html .\n\n> The algebraic structures involved with the B-field change depending\n> upon whether there is the presence of NS 5-brane charges in which case\n> the B-field is non-torsion or whether the B-field is flat (torsion),\n> the effects of the KK-monopole electric charges associated with the\n> B-field etc., and whether the B-field is e.g. on a gerbe or a quantum\n> gerbe (at least in the paper I referred you to about quantum gerbes\n> [hep-th/0206101] there is a natural contravariant connection).\n\nThis reference seems to be interesting. I\'ll have a look at it. (From a\nfirst glance I am not sure if the authors are concerned with non-abelian\ngauge groups on branes, non-commutative spacetime due to branes or both!?)\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 Mon, 21 Jun 2004, Charlie Stromeyer Jr. wrote:
> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:
>
> > There I found the very useful reference
> >
> > Christiaan Hofman:
> > Nonabelian 2-forms,
> > http://www.arxiv.org/abs/hep-th/0207017 .
> >
> > First of all, this reference clarifies where the non-Abelian B-field appears
> > in string theory.It's pretty obvious now that I think about it: The B field
> > couples to the NS 5-brane (by definition, essentially). So just pick a stack
> > of such NS5-branes to obtain a matrix-valued B-field.
> >
> > Er, or something along these lines. What is it that plays the role of the
> > Chan-Paton factors here?
>
> This is not made clear in the paper. Instead, the author hopes that
> his approach might be relevant for D-branes physics.
I find it hard to tell what in this paper is derived fact and what is a
proposal for a construction. For instance the paper seems to suggest that
all non-commutative forms on loop space should be built using that \oint_A
instead of \oint. I am not sure if that's the right approach, for
instance to string theory, where the states of the superstring are
precisely such p-forms on loop space. Also, the author does not show that
the 2-form gauge connection he uses is the only sensible construction that one
could call by such a name, or does he?
I am asking in particular because I made some SCFT deformation analysis
which in a rather nice way gives something that looks like a 2-form gauge
connection on loop space induced by the B-field, and which also is almost
exactly what Christiaan Hofman discusses - except for a small difference.
I have the details at
http://golem.ph.utexas.edu/string/archives/000385.html .
> The algebraic structures involved with the B-field change depending
> upon whether there is the presence of NS 5-brane charges in which case
> the B-field is non-torsion or whether the B-field is flat (torsion),
> the effects of the KK-monopole electric charges associated with the
> B-field etc., and whether the B-field is e.g. on a gerbe or a quantum
> gerbe (at least in the paper I referred you to about quantum gerbes
> [http://www.arxiv.org/abs/hep-th/0206101] there is a natural contravariant connection).
This reference seems to be interesting. I'll have a look at it. (From a
first glance I am not sure if the authors are concerned with non-abelian
gauge groups on branes, non-commutative spacetime due to branes or both!?)
Charlie Stromeyer Jr.
Jun23-04, 02:50 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>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:\n\n> Also, the author does not show that\n> the 2-form gauge connection he uses is the only sensible construction that one\n> could call by such a name, or does he?\n\nHofman is only considering the local case. I was not able to see how\nonly the approach of his paper could be used for constructing a K-R\nfield for spacetimes that are not flat. However, Hofman says in his\npaper that a category theory approach would be needed for a more\nfundamental consideration, and so it is interesting that his 1 and 2\nforms seem very similar to those of John Baez in his category theory\npaper [hep-th/0206130] since they are both using a crossed module.\n\nI could explain in more detail some of what I think about this if you\nwant me to, however, I may be getting the sense that you might be\nabout to discover something interesting if you keep pursuing this path\nof inquiry and I would not want to attempt to steal any of your\nthunder by trying to beat you to the finish line !-)\n\nInstead, I suggest a hint that you look at another paper by Y. Zunger\nin which he does the boundary state formalism in the noncommutative\ncase and applies his theory to studying an infinite number of D0\nbranes [hep-th/0210175].\n\n\n-----------------------------------------------------------------------------\n\n"These results [of Srinivasa Ramanujan] must be true because, if they\nwere not true, no one would have had the imagination to invent them."\n- G.H. Hardy\n\n"The most beautiful thing we can experience is the Mysterious."\n- Albert Einstein\n\n"Superstring/M-theory is the language in which God wrote the world."\n- Lubos Motl\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>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:
> Also, the author does not show that
> the 2-form gauge connection he uses is the only sensible construction that one
> could call by such a name, or does he?
Hofman is only considering the local case. I was not able to see how
only the approach of his paper could be used for constructing a K-R
field for spacetimes that are not flat. However, Hofman says in his
paper that a category theory approach would be needed for a more
fundamental consideration, and so it is interesting that his 1 and 2
forms seem very similar to those of John Baez in his category theory
paper [http://www.arxiv.org/abs/hep-th/0206130] since they are both using a crossed module.
I could explain in more detail some of what I think about this if you
want me to, however, I may be getting the sense that you might be
about to discover something interesting if you keep pursuing this path
of inquiry and I would not want to attempt to steal any of your
thunder by trying to beat you to the finish line !-)
Instead, I suggest a hint that you look at another paper by Y. Zunger
in which he does the boundary state formalism in the noncommutative
case and applies his theory to studying an infinite number of D0
branes [http://www.arxiv.org/abs/hep-th/0210175].
-----------------------------------------------------------------------------
"These results [of Srinivasa Ramanujan] must be true because, if they
were not true, no one would have had the imagination to invent them."
- G.H. Hardy
"The most beautiful thing we can experience is the Mysterious."
- Albert Einstein
"Superstring/M-theory is the language in which God wrote the world."
- Lubos Motl
-----------------------------------------------------------------------------
Urs Schreiber
Jun23-04, 09:26 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>"Charlie Stromeyer Jr." <cstromey@hotmail.com> schrieb im Newsbeitrag\nnews:61773ed7.0406221501.3e9c3996-100000@posting.google.com...\n> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:\n\n> Hofman is only considering the local case.\n\nTrue.\n\n> However, Hofman says in his\n> paper that a category theory approach would be needed for a more\n> fundamental consideration, and so it is interesting that his 1 and 2\n> forms seem very similar to those of John Baez in his category theory\n> paper [hep-th/0206130] since they are both using a crossed module.\n\nOn p. 6 of hep-th/0206130 it says that only _trivial_ 2-bundles are\nconsidered. I think this amounts to the same as studying things locally. Of\ncourse in the end one should apply all these overlap conditions to get the\nglobal picutre. But as far as I understand that part is not the problem.\n\n> I was not able to see how\n> only the approach of his paper could be used for constructing a K-R\n> field for spacetimes that are not flat.\n\nWhat do you mean by K-R field?\n\n> However, Hofman says in his\n> paper that a category theory approach would be needed for a more\n> fundamental consideration, and so it is interesting that his 1 and 2\n> forms seem very similar to those of John Baez in his category theory\n> paper [hep-th/0206130] since they are both using a crossed module.\n\nI think this must all be the same, the only difference being in terms of\nlanguage. I.e. Zunger is using loop-space formalism.\n\n> I could explain in more detail some of what I think about this if you\n> want me to, however, I may be getting the sense that you might be\n> about to discover something interesting if you keep pursuing this path\n> of inquiry and I would not want to attempt to steal any of your\n> thunder by trying to beat you to the finish line !-)\n\nI am always grateful for all input that you can come up with.\n\n> Instead, I suggest a hint that you look at another paper by Y. Zunger\n> in which he does the boundary state formalism in the noncommutative\n> case and applies his theory to studying an infinite number of D0\n> branes [hep-th/0210175].\n\nThis paper\n\nYoantan Zunger,\nConstructing exotic D-branes with infinite matrices in type IIA string\ntheory,\nhep-th/0210175\n\nis very nice. I sort of knew most aspects of what Zunger writes here, but\nthis paper put them in a big picture for me, which I didn\'t have before.\n\nFirst of all it is great to see Zunger discuss boundary states as modules\nover the non-commutative algebras that live on a stack of branes. This is\nsomething I used in http://golem.ph.utexas.edu/string/archives/000385.html\n(that\'s why I say there "Don\'t take a trace, since the matrices must act on\nthe CP factors.") but which at the same time confused me, because it is\n_different_ from what Maeda&Nakatsu&Oonishi do in hep-th/0312260. They do\ntake the trace and their boundary state is therefore just an ordinary "C(M)\nmodule", as far as I can see, even thought it is advertised as describing\nbranes with non-abelian gauge fields turned on.\n\nMaybe I should add a remark to that blog-entry saying that defining the\ndeformed exterior derivative on loop space the way I did there really\nassumes the ground state that all these operators act on (the bare boundary\nstate) to be a \\mathcal{A}-modules, where \\mathcal{A} = M_N(H) is the NxN\nmatrix algebra with values in the Heisenberg algebra H of string\noscillators.\n\nBTW, there is one point in Zunger\'s paper that seems to be a little\nunprecise to me, assuming that I fully understand it, of course: On p. 2\n\\mathcal{A} is the algebra of gauge connections on spacetime, but on the\nbottom of p.3 it is suddenly that of string oscillators and the\nM_N(\\mathcal{A}) of p.4 seems to be more directly related to the \\mathcal{A}\nfrom p.2\n\nThen it is fun to see how Zunger manages to describe a whole lot of brane\nphysics by just using a couple of simple facts about noncommutative algebras\nand K-theory. He identifies two conserved charges of the brane, the K-theory\ncharge K_0(\\mathcal{A}) which sort of measures how many branes are stacked\ntogether and then the H^1(\\mathcal{A}) \'charge\' which counts the number of\nouter derivations of the algebra \\mathcal{A}, which is nothing but the\ndimension of the brane!\n\nThis distinction between inner and outer derivations of the non-commutatiev\nalgebra and how the inner ones describe \'inner\' gauge groups, while the\nouter ones are related to the space-time symmetry is very neat. It goes back\nall the way to Connes\n(http://golem.ph.utexas.edu/string/archives/000298.html#c000695) and was\nused in\n\nAlain Connes, Michael R. Douglas, Albert Schwarz:\nNoncommutative Geometry and Matrix Theory: Compactification on Tori\nhep-th/9711162\n\nin the context of Matrix-Models. But I wasn\'t aware before that one can\ndiscuss brane stability using the set of outer derivations (and hence\nouter-Automorphisms of the algebra).\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>"Charlie Stromeyer Jr." <cstromey@hotmail.com> schrieb im Newsbeitrag
news:61773ed7.0406221501.3e9c3996-100000@posting.google.com...
> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:
> Hofman is only considering the local case.
True.
> However, Hofman says in his
> paper that a category theory approach would be needed for a more
> fundamental consideration, and so it is interesting that his 1 and 2
> forms seem very similar to those of John Baez in his category theory
> paper [http://www.arxiv.org/abs/hep-th/0206130] since they are both using a crossed module.
On p. 6 of http://www.arxiv.org/abs/hep-th/0206130 it says that only _trivial_ 2-bundles are
considered. I think this amounts to the same as studying things locally. Of
course in the end one should apply all these overlap conditions to get the
global picutre. But as far as I understand that part is not the problem.
> I was not able to see how
> only the approach of his paper could be used for constructing a K-R
> field for spacetimes that are not flat.
What do you mean by K-R field?
> However, Hofman says in his
> paper that a category theory approach would be needed for a more
> fundamental consideration, and so it is interesting that his 1 and 2
> forms seem very similar to those of John Baez in his category theory
> paper [http://www.arxiv.org/abs/hep-th/0206130] since they are both using a crossed module.
I think this must all be the same, the only difference being in terms of
language. I.e. Zunger is using loop-space formalism.
> I could explain in more detail some of what I think about this if you
> want me to, however, I may be getting the sense that you might be
> about to discover something interesting if you keep pursuing this path
> of inquiry and I would not want to attempt to steal any of your
> thunder by trying to beat you to the finish line !-)
I am always grateful for all input that you can come up with.
> Instead, I suggest a hint that you look at another paper by Y. Zunger
> in which he does the boundary state formalism in the noncommutative
> case and applies his theory to studying an infinite number of D0
> branes [http://www.arxiv.org/abs/hep-th/0210175].
This paper
Yoantan Zunger,
Constructing exotic D-branes with infinite matrices in type IIA string
theory,
http://www.arxiv.org/abs/hep-th/0210175
is very nice. I sort of knew most aspects of what Zunger writes here, but
this paper put them in a big picture for me, which I didn't have before.
First of all it is great to see Zunger discuss boundary states as modules
over the non-commutative algebras that live on a stack of branes. This is
something I used in http://golem.ph.utexas.edu/string/archives/000385.html
(that's why I say there "Don't take a trace, since the matrices must act on
the CP factors.") but which at the same time confused me, because it is
_different_ from what Maeda&Nakatsu&Oonishi do in http://www.arxiv.org/abs/hep-th/0312260. They do
take the trace and their boundary state is therefore just an ordinary "C(M)
module", as far as I can see, even thought it is advertised as describing
branes with non-abelian gauge fields turned on.
Maybe I should add a remark to that blog-entry saying that defining the
deformed exterior derivative on loop space the way I did there really
assumes the ground state that all these operators act on (the bare boundary
state) to be a \mathcal{A}-modules, where \mathcal{A} = M_N(H) is the NxN
matrix algebra with values in the Heisenberg algebra H of string
oscillators.
BTW, there is one point in Zunger's paper that seems to be a little
unprecise to me, assuming that I fully understand it, of course: On p. 2
\mathcal{A} is the algebra of gauge connections on spacetime, but on the
bottom of p.3 it is suddenly that of string oscillators and the
M_N(\mathcal{A}) of p.4 seems to be more directly related to the \mathcal{A}
from p.2
Then it is fun to see how Zunger manages to describe a whole lot of brane
physics by just using a couple of simple facts about noncommutative algebras
and K-theory. He identifies two conserved charges of the brane, the K-theory
charge K_0(\mathcal{A}) which sort of measures how many branes are stacked
together and then the H^1(\mathcal{A}) 'charge' which counts the number of
outer derivations of the algebra \mathcal{A}, which is nothing but the
dimension of the brane!
This distinction between inner and outer derivations of the non-commutatiev
algebra and how the inner ones describe 'inner' gauge groups, while the
outer ones are related to the space-time symmetry is very neat. It goes back
all the way to Connes
(http://golem.ph.utexas.edu/string/archives/000298.html#c000695) and was
used in
Alain Connes, Michael R. Douglas, Albert Schwarz:
Noncommutative Geometry and Matrix Theory: Compactification on Tori
http://www.arxiv.org/abs/hep-th/9711162
in the context of Matrix-Models. But I wasn't aware before that one can
discuss brane stability using the set of outer derivations (and hence
outer-Automorphisms of the algebra).
Charlie Stromeyer Jr.
Jun23-04, 06:07 PM
<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>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:\n\n> On p. 6 of hep-th/0206130 it says that only _trivial_ 2-bundles are\n> considered. I think this amounts to the same as studying things locally. Of\n> course in the end one should apply all these overlap conditions to get the\n> global picutre. But as far as I understand that part is not the problem.\n\nWhich problem, more specifically, are you thinking of?\n\n> > I was not able to see how\n> > only the approach of his paper could be used for constructing a K-R\n> > field for spacetimes that are not flat.\n>\n> What do you mean by K-R field?\n\nIn the sense of papers [hep-th/0311183 and 0310156].\n\n> I think this must all be the same, the only difference being in terms of\n> language. I.e. Zunger is using loop-space formalism.\n\nYes, they seem the same at first, but since John Baez is continuously\ntrying to envision how to approach complicated physics issues in a way\nthat may be more simple and unifying via category theory, he might be\nable to see some subtleties in this case that we would miss.\n\n[Moderator\'s note: Let\'s wish John Baez good luck in his\nattempt to simplify and unify physics using "abstract nonsense",\nas category theory is known. ;-) Sorry, I could not resist. LM]\n\n> BTW, there is one point in Zunger\'s paper that seems to be a little\n> unprecise to me, assuming that I fully understand it, of course: On p. 2\n> \\mathcal{A} is the algebra of gauge connections on spacetime, but on the\n> bottom of p.3 it is suddenly that of string oscillators and the\n> M_N(\\mathcal{A}) of p.4 seems to be more directly related to the \\mathcal{A}\n> from p.2\n\nYou may first want to look at the Moyal plane case I mentioned earlier\n[hep-th/0206101], but note that where Zunger refers to this paper he\nmentions that the argument will extend to a general A.\n\nIt seems to me that the key result of this paper is for where the\ncommutative and NC cases coincide, i.e. for A = C(R^p+1) x M_N which\nis where the BI action reproduces that of a stack of N p-branes (since\nthe BI action still holds for the case of infinite matrices).\n\nAs I have time for it, I am currently trying to think about\nfundamental math notions of sequence, covergence and limit. In the\nmeantime, you might also want to look at yet another paper by Zunger\n[hep-th/0106030]in which he shows that the group of area-preserving\ndiffeomorphisms of an Riemann surface is SU(infinity) or that the\nPoisson brackets on any Riemann surface form the algebra U(infinity),\nand the two groups differ by a U(1) factor.\n\nHowever, there are some papers about regularization on NCFT which I\nhave not even had time to look at yet but which may be important such\nas [hep-th/0312047, 0401072 and 0310127]. Also, keep in mind the view\nof the Witten paper [hep-th/0007175].\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>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:
> On p. 6 of http://www.arxiv.org/abs/hep-th/0206130 it says that only _trivial_ 2-bundles are
> considered. I think this amounts to the same as studying things locally. Of
> course in the end one should apply all these overlap conditions to get the
> global picutre. But as far as I understand that part is not the problem.
Which problem, more specifically, are you thinking of?
> > I was not able to see how
> > only the approach of his paper could be used for constructing a K-R
> > field for spacetimes that are not flat.
>
> What do you mean by K-R field?
In the sense of papers [http://www.arxiv.org/abs/hep-th/0311183 and 0310156].
> I think this must all be the same, the only difference being in terms of
> language. I.e. Zunger is using loop-space formalism.
Yes, they seem the same at first, but since John Baez is continuously
trying to envision how to approach complicated physics issues in a way
that may be more simple and unifying via category theory, he might be
able to see some subtleties in this case that we would miss.
[Moderator's note: Let's wish John Baez good luck in his
attempt to simplify and unify physics using "abstract nonsense",
as category theory is known. ;-) Sorry, I could not resist. LM]
> BTW, there is one point in Zunger's paper that seems to be a little
> unprecise to me, assuming that I fully understand it, of course: On p. 2
> \mathcal{A} is the algebra of gauge connections on spacetime, but on the
> bottom of p.3 it is suddenly that of string oscillators and the
> M_N(\mathcal{A}) of p.4 seems to be more directly related to the \mathcal{A}
> from p.2
You may first want to look at the Moyal plane case I mentioned earlier
[http://www.arxiv.org/abs/hep-th/0206101], but note that where Zunger refers to this paper he
mentions that the argument will extend to a general A.
It seems to me that the key result of this paper is for where the
commutative and NC cases coincide, i.e. for A = C(R^p+1) x M_N which
is where the BI action reproduces that of a stack of N p-branes (since
the BI action still holds for the case of infinite matrices).
As I have time for it, I am currently trying to think about
fundamental math notions of sequence, covergence and limit. In the
meantime, you might also want to look at yet another paper by Zunger
[http://www.arxiv.org/abs/hep-th/0106030]in which he shows that the group of area-preserving
diffeomorphisms of an Riemann surface is SU(infinity) or that the
Poisson brackets on any Riemann surface form the algebra U(infinity),
and the two groups differ by a U(1) factor.
However, there are some papers about regularization on NCFT which I
have not even had time to look at yet but which may be important such
as [http://www.arxiv.org/abs/hep-th/0312047, 0401072 and 0310127]. Also, keep in mind the view
of the Witten paper [http://www.arxiv.org/abs/hep-th/0007175].
Thomas Larsson
Jun24-04, 07:52 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>"Charlie Stromeyer Jr." <cstromey@hotmail.com> wrote in message news:<61773ed7.0406231029.270033f4-100000@posting.google.com>...\n\n> As I have time for it, I am currently trying to think about\n> fundamental math notions of sequence, covergence and limit. In the\n> meantime, you might also want to look at yet another paper by Zunger\n> [hep-th/0106030]in which he shows that the group of area-preserving\n> diffeomorphisms of an Riemann surface is SU(infinity) or that the\n> Poisson brackets on any Riemann surface form the algebra U(infinity),\n> and the two groups differ by a U(1) factor.\n\n???\n\nI thought that the group of area-preserving diffeomorphisms of a\nRiemann surface was the the group of area-preserving diffeomorphisms of\na Riemann surface = the group of canonical transformations since\na Riemann surface is two-dimensional. The Poisson brackets probably\nform the algebra of Poisson brackets. It differs from the algebra of\nHamiltonian vector fields by a central extension - the constant function\n(which must be your U(1) factor) gets mapped to the zero vector field.\n\nYou probably mean that these groups/algebras can be embedded in a\nsuitable version of SU(infinity) / U(infinity). That would be the same\nas saying that they have unitary representations.\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>"Charlie Stromeyer Jr." <cstromey@hotmail.com> wrote in message news:<61773ed7.0406231029.270033f4-100000@posting.google.com>...
> As I have time for it, I am currently trying to think about
> fundamental math notions of sequence, covergence and limit. In the
> meantime, you might also want to look at yet another paper by Zunger
> [http://www.arxiv.org/abs/hep-th/0106030]in which he shows that the group of area-preserving
> diffeomorphisms of an Riemann surface is SU(infinity) or that the
> Poisson brackets on any Riemann surface form the algebra U(infinity),
> and the two groups differ by a U(1) factor.
???
I thought that the group of area-preserving diffeomorphisms of a
Riemann surface was the the group of area-preserving diffeomorphisms of
a Riemann surface = the group of canonical transformations since
a Riemann surface is two-dimensional. The Poisson brackets probably
form the algebra of Poisson brackets. It differs from the algebra of
Hamiltonian vector fields by a central extension - the constant function
(which must be your U(1) factor) gets mapped to the zero vector field.
You probably mean that these groups/algebras can be embedded in a
suitable version of SU(infinity) / U(infinity). That would be the same
as saying that they have unitary representations.
Urs Schreiber
Jun24-04, 08:35 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>"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag\nnews:24a23f36.0406240442.70c2f9b1-100000@posting.google.com...\n> "Charlie Stromeyer Jr." <cstromey@hotmail.com> wrote in message\nnews:<61773ed7.0406231029.270033f4-100000@posting.google.com>...\n>\n> > As I have time for it, I am currently trying to think about\n> > fundamental math notions of sequence, covergence and limit. In the\n> > meantime, you might also want to look at yet another paper by Zunger\n> > [hep-th/0106030]in which he shows that the group of area-preserving\n> > diffeomorphisms of an Riemann surface is SU(infinity) or that the\n> > Poisson brackets on any Riemann surface form the algebra U(infinity),\n> > and the two groups differ by a U(1) factor.\n>\n> ???\n>\n> I thought that the group of area-preserving diffeomorphisms of a\n> Riemann surface was the the group of area-preserving diffeomorphisms of\n> a Riemann surface = the group of canonical transformations since\n> a Riemann surface is two-dimensional.\n\nSure, and the claim is that this is isomorphic to U(N\\to \\infty) when this\nlimit is suitably defined.\n\n> You probably mean that these groups/algebras can be embedded in a\n> suitable version of SU(infinity) / U(infinity). That would be the same\n> as saying that they have unitary representations.\n\nZunger claims to show that they are not just embedded, but in fact equal.\nHis point is that there is only one "simple peudocompact algebra", in his\nterminology, and that both SU(\\infty) as well as Poisson/U(1) are simple\npseudocompact.\n\nAs far as I know for simple topologies, such as the sphere and the torus,\nthis has been known for a long time, Zunger\'s point being that it\ngeneralizes to arbitrary topologies.\n\n>From the physical point of view, it is an almost obvious consequence of the\nmatrix regularization technique. The induced metric of a surface embedded\ninto a space with metric g_{ij} = \\delta_{ij} is\n\n{X^i,X^j}{X_i,X_j},\n\nwhere {.,.} are the Poisson brackets in the given coordinates on the\nsurface. This expression is approximated by the commutator of large\nhermitian matrices (the same step as in quantization) to become\n\n[X^i,X^j][X_i,X_j],\n\nwith X^i now hermitian matrices.\n\nIntegrating the induced metric (not its square root!) over the surface\ntransaltes into taking the trace\n\n\\int_M {X^i,X^j}{X_i,X_j} \\approx Tr [X^i,X^j][X_i,X_j] .\n\nObviously U(N\\to \\infty) is the symmetry group.\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>"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag
news:24a23f36.0406240442.70c2f9b1-100000@posting.google.com...
> "Charlie Stromeyer Jr." <cstromey@hotmail.com> wrote in message
news:<61773ed7.0406231029.270033f4-100000@posting.google.com>...
>
> > As I have time for it, I am currently trying to think about
> > fundamental math notions of sequence, covergence and limit. In the
> > meantime, you might also want to look at yet another paper by Zunger
> > [http://www.arxiv.org/abs/hep-th/0106030]in which he shows that the group of area-preserving
> > diffeomorphisms of an Riemann surface is SU(infinity) or that the
> > Poisson brackets on any Riemann surface form the algebra U(infinity),
> > and the two groups differ by a U(1) factor.
>
> ???
>
> I thought that the group of area-preserving diffeomorphisms of a
> Riemann surface was the the group of area-preserving diffeomorphisms of
> a Riemann surface = the group of canonical transformations since
> a Riemann surface is two-dimensional.
Sure, and the claim is that this is isomorphic to U(N\to \infty) when this
limit is suitably defined.
> You probably mean that these groups/algebras can be embedded in a
> suitable version of SU(infinity) / U(infinity). That would be the same
> as saying that they have unitary representations.
Zunger claims to show that they are not just embedded, but in fact equal.
His point is that there is only one "simple peudocompact algebra", in his
terminology, and that both SU(\infty) as well as Poisson/U(1) are simple
pseudocompact.
As far as I know for simple topologies, such as the sphere and the torus,
this has been known for a long time, Zunger's point being that it
generalizes to arbitrary topologies.
>From the physical point of view, it is an almost obvious consequence of the
matrix regularization technique. The induced metric of a surface embedded
into a space with metric g_{ij} = \delta_{ij} is
{X^i,X^j}{X_i,X_j},
where {.,.} are the Poisson brackets in the given coordinates on the
surface. This expression is approximated by the commutator of large
hermitian matrices (the same step as in quantization) to become
[X^i,X^j][X_i,X_j],
with X^i now hermitian matrices.
Integrating the induced metric (not its square root!) over the surface
transaltes into taking the trace
\int_M {X^i,X^j}{X_i,X_j} \approx Tr [X^i,X^j][X_i,X_j] .
Obviously U(N\to \infty) is the symmetry group.
Charlie Stromeyer Jr.
Jun24-04, 08:54 PM
<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>"Charlie Stromeyer Jr." <cstromey@hotmail.com> wrote in message news:\n\n> [Moderator\'s note: Let\'s wish John Baez good luck in his\n> attempt to simplify and unify physics using "abstract nonsense",\n> as category theory is known. ;-) Sorry, I could not resist. LM]\n\nThis comment brings to mind a question which is implicitly involved\nwith my ongoing thinking about Bousso\'s question, higher n-gerbes etc.\nfor M-theory. At the bottom of page 4 of their description of the Clay\nMath Institute Millenium Prize Problem of Quantum YM theory, Arthur\nJaffe and Edward Witten write:\n\n"Finally, QFT is the jumping off point for a quest that may prove\ncentral in twenty-first century physics - the effort to unify gravity\nand quantum mechanics, perhaps in string theory".\n\nHowever, I suspect that new and non-trivial insights about the famous\nproblems with 4d gauge theory may be more likely to come by first\nstarting from a "jumping off point" or basis that is something like\nM-theory. My question is whether there may be any compelling reasons\nto believe otherwise?\n\nIf it should make more sense to begin first from a potential\n(non-perturbative) quantum gravity framework, then I feel certain that\nat least Lubos will fully share my opinion of alternative techniques\nfrom approaches that are not superstring/M-theory:\n\nNa, krivak; zda se mi, ze se neda dobre brousit !-)\n\n[Moderator\'s note: Haha, I am afraid that others won\'t\nunderstand this comment in Czech, but I appreciated it. LM]\n\nFurthermore, there may be reasons for our shared view which are not\nentirely silly or Californian, despite the fact that I was born at\nStanford !-)\n\nMore seriously, as I have time for it, I will at least try to answer\nany specific enough questions about the math within Y. Zunger\'s\npapers. In the meantime, keep these three points in mind:\n\n1) His paper about Riemann surfaces [hep-th/0210175] only deals with\ntrivial backgrounds.\n\n2) Some of the assumptions in his papers regarding convergence and\nlimits might no longer hold in a more general mathematical context.\nThis is what I am currently investigating, but I cannot really say\nmore about this now because I still need to review various ideas from\ngeneral topology, algebraic geometry, the theory of buildings and even\ncategory theory.\n\n(Note that, for example, since Stephen Hawking is a logical positivist\nhe won\'t mind if we try to use bizarre math as long as the physics\nresults are valid in the end. Although I have perhaps a more aesthetic\nview of what theoretical physics and pure math should be like, I do\nagree with Hawking\'s logical positivist view for many practical\napplications, e.g. about 100 years ago a man invented aspirin to help\ntreat his father\'s headaches but it was only about 10 years ago that\nscientists really started to figure out how aspirin works.)\n\n3) Although the math within Zunger\'s papers still seems valid to me,\nstring theorists such as Urs may need to consider whether the physical\nassumptions still seem valid, e.g. I have never looked at any papers\nabout regularization on NCFT such as the three papers on this topic\nthat I referred to earlier.\n\n\n-----------------------------------------------------------------------\n\n"These results [of Srinivasa Ramanujan] must be true because, if they\nwere not true, no one would have had the imagination to invent them."\n- G.H. Hardy\n\n"The most beautiful thing we can experience is the Mysterious."\n- Albert Einstein\n\n"Superstring/M-theory is the language in which God wrote the world."\n- Lubos Motl\n\n[Moderator\'s note: So that\'s a slightly overhyped place for\nthis signature, but thanks anyway. ;-) LM]\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>"Charlie Stromeyer Jr." <cstromey@hotmail.com> wrote in message news:
> [Moderator's note: Let's wish John Baez good luck in his
> attempt to simplify and unify physics using "abstract nonsense",
> as category theory is known. ;-) Sorry, I could not resist. LM]
This comment brings to mind a question which is implicitly involved
with my ongoing thinking about Bousso's question, higher n-gerbes etc.
for M-theory. At the bottom of page 4 of their description of the Clay
Math Institute Millenium Prize Problem of Quantum YM theory, Arthur
Jaffe and Edward Witten write:
"Finally, QFT is the jumping off point for a quest that may prove
central in twenty-first century physics - the effort to unify gravity
and quantum mechanics, perhaps in string theory".
However, I suspect that new and non-trivial insights about the famous
problems with 4d gauge theory may be more likely to come by first
starting from a "jumping off point" or basis that is something like
M-theory. My question is whether there may be any compelling reasons
to believe otherwise?
If it should make more sense to begin first from a potential
(non-perturbative) quantum gravity framework, then I feel certain that
at least Lubos will fully share my opinion of alternative techniques
from approaches that are not superstring/M-theory:
Na, krivak; zda se mi, ze se neda dobre brousit !-)
[Moderator's note: Haha, I am afraid that others won't
understand this comment in Czech, but I appreciated it. LM]
Furthermore, there may be reasons for our shared view which are not
entirely silly or Californian, despite the fact that I was born at
Stanford !-)
More seriously, as I have time for it, I will at least try to answer
any specific enough questions about the math within Y. Zunger's
papers. In the meantime, keep these three points in mind:
1) His paper about Riemann surfaces [http://www.arxiv.org/abs/hep-th/0210175] only deals with
trivial backgrounds.
2) Some of the assumptions in his papers regarding convergence and
limits might no longer hold in a more general mathematical context.
This is what I am currently investigating, but I cannot really say
more about this now because I still need to review various ideas from
general topology, algebraic geometry, the theory of buildings and even
category theory.
(Note that, for example, since Stephen Hawking is a logical positivist
he won't mind if we try to use bizarre math as long as the physics
results are valid in the end. Although I have perhaps a more aesthetic
view of what theoretical physics and pure math should be like, I do
agree with Hawking's logical positivist view for many practical
applications, e.g. about 100 years ago a man invented aspirin to help
treat his father's headaches but it was only about 10 years ago that
scientists really started to figure out how aspirin works.)
3) Although the math within Zunger's papers still seems valid to me,
string theorists such as Urs may need to consider whether the physical
assumptions still seem valid, e.g. I have never looked at any papers
about regularization on NCFT such as the three papers on this topic
that I referred to earlier.
-----------------------------------------------------------------------
"These results [of Srinivasa Ramanujan] must be true because, if they
were not true, no one would have had the imagination to invent them."
- G.H. Hardy
"The most beautiful thing we can experience is the Mysterious."
- Albert Einstein
"Superstring/M-theory is the language in which God wrote the world."
- Lubos Motl
[Moderator's note: So that's a slightly overhyped place for
this signature, but thanks anyway. ;-) LM]
-----------------------------------------------------------------------
Urs Schreiber
Jun30-04, 07:45 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>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:<2jgisnF11d328U1-100000@uni-berlin.de>...\n\n> Christiaan Hofman:\n> Nonabelian 2-forms,\n> hep-th/0207017 .\n\nLuckily, I have just met Christiaan Hofman here in Paris at Strings04.\nI have asked him if he thinks that the nonabelian 2-form connection on\nloop space which I derive at\n\nhttp://golem.ph.utexas.edu/string/archives/000385.html\n\nfrom SCFT deformation/boundary state formalism, and which differs from\nHofman\'s connection in that it reads\n\n\\int d\\sigma U(0,\\sigma) B(\\sigma) U(\\sigma,0)\n\ninstead of just\n\n\\int d\\sigma U(0,\\sigma) B(\\sigma) ,\n\nlooks reasonable to him. Essentially this seems to be an issue of\nwhether B "takes values" in the fundamental or in the adjoint of A\n(where "take values" is deliberately vague). So if A acts by taking\nadjoints on B the form that I give looks in fact plausible. I think\nthis is the case that arises naturally in string theory.\n\nOne questions that we could not fully clarify is the following:\n\nBy S-duality one would expect also a nonabelian RR 2-form to play a\nrole, like for the D-string. Does anyone konw any literature on\nnonabelian RR 2-forms?\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>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:<2jgisnF11d328U1-100000@uni-berlin.de>...
> Christiaan Hofman:
> Nonabelian 2-forms,
> http://www.arxiv.org/abs/hep-th/0207017 .
Luckily, I have just met Christiaan Hofman here in Paris at Strings04.
I have asked him if he thinks that the nonabelian 2-form connection on
loop space which I derive at
http://golem.ph.utexas.edu/string/archives/000385.html
from SCFT deformation/boundary state formalism, and which differs from
Hofman's connection in that it reads
\int d\sigma U(0,\sigma) B(\sigma) U(\sigma,0)
instead of just
\int d\sigma U(0,\sigma) B(\sigma) ,
looks reasonable to him. Essentially this seems to be an issue of
whether B "takes values" in the fundamental or in the adjoint of A
(where "take values" is deliberately vague). So if A acts by taking
adjoints on B the form that I give looks in fact plausible. I think
this is the case that arises naturally in string theory.
One questions that we could not fully clarify is the following:
By S-duality one would expect also a nonabelian RR 2-form to play a
role, like for the D-string. Does anyone konw any literature on
nonabelian RR 2-forms?
Thomas Larsson
Jul1-04, 12:56 PM
<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>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:<2k03iaF164tv0U1-100000@uni-berlin.de>...\n> "Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag\n> news:24a23f36.0406240442.70c2f9b1-100000@posting.google.com...\n\n> > I thought that the group of area-preserving diffeomorphisms of a\n> > Riemann surface was the the group of area-preserving diffeomorphisms of\n> > a Riemann surface = the group of canonical transformations since\n> > a Riemann surface is two-dimensional.\n>\n> Sure, and the claim is that this is isomorphic to U(N\\to \\infty) when this\n> limit is suitably defined.\n>\n\nNow I feel like a complete fool. I didn\'t actually have a look\nat Zunger\'s paper; the assertion seemed wrong so I unjustly\nassumed that Charlie Stromeyer had misunderstood something. But\nit turned out that I was the confused one. Apologies to\neveryone.\n\nWhat I doubted was that the algebra of Hamiltonian vector fields\ncould be identified with su(infinity) locally. But now I realize\nthat this is an old result which I already knew about; the\nconstruction is explained simply in section 2 of hep-th/9912130.\n\nWhat\'s even more embarassing is that I studied several of\nZunger\'s older references when they first came out, probably\n[4,5,18]. In those days, Bachas and Fairlie hadn\'t yet realized\nthat their "sine algebra" had already been discovered by Moyal\nand others; it is basically deformation quantization.\n\nIn case somebody is interested, which I seriously doubt, the\nMoyal algebra (and in particular the Hamiltonian algebra), has a\ncentral extension, at least on the torus:\n\n[T_m, T_n] = sin(mxn) T_m+n + k.m delta_m+n\n\nwhere m, n are momenta in Z^2, mxn is the cross product, and k\nis a constant vector. This is not related to the Virasoro\nalgebra, though, since it is only linear in m. The Hamiltonian\nalgebra also has a non-central extension, which follows by\nrestriction from the Virasoro algebra in 2D. It must be quintic\nin momenta.\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>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:<2k03iaF164tv0U1-100000@uni-berlin.de>...
> "Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag
> news:24a23f36.0406240442.70c2f9b1-100000@posting.google.com...
> > I thought that the group of area-preserving diffeomorphisms of a
> > Riemann surface was the the group of area-preserving diffeomorphisms of
> > a Riemann surface = the group of canonical transformations since
> > a Riemann surface is two-dimensional.
>
> Sure, and the claim is that this is isomorphic to U(N\to \infty) when this
> limit is suitably defined.
>
Now I feel like a complete fool. I didn't actually have a look
at Zunger's paper; the assertion seemed wrong so I unjustly
assumed that Charlie Stromeyer had misunderstood something. But
it turned out that I was the confused one. Apologies to
everyone.
What I doubted was that the algebra of Hamiltonian vector fields
could be identified with su(infinity) locally. But now I realize
that this is an old result which I already knew about; the
construction is explained simply in section 2 of http://www.arxiv.org/abs/hep-th/9912130.
What's even more embarassing is that I studied several of
Zunger's older references when they first came out, probably
[4,5,18]. In those days, Bachas and Fairlie hadn't yet realized
that their "sine algebra" had already been discovered by Moyal
and others; it is basically deformation quantization.
In case somebody is interested, which I seriously doubt, the
Moyal algebra (and in particular the Hamiltonian algebra), has a
central extension, at least on the torus:
[T_m, T_n] = sin(mxn) T_m+n + k[/itex].[itex]m \delta_m+n
where m, n are momenta in Z^2, mxn is the cross product, and k
is a constant vector. This is not related to the Virasoro
algebra, though, since it is only linear in m. The Hamiltonian
algebra also has a non-central extension, which follows by
restriction from the Virasoro algebra in 2D. It must be quintic
in momenta.
Thomas Larsson
Jul2-04, 04:37 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>Thomas Larsson <thomas_larsson_01@hotmail.com> wrote in message news:<24a23f36.0407010505.4d99dcb4-100000@posting.google.com>...\n\n> [4,5,18]. In those days, Bachas and Fairlie hadn\'t yet realized\n> that their "sine algebra" had already been discovered by Moyal\n\nThat should be Zachos, not Bachas.\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>Thomas Larsson <thomas_larsson_01@hotmail.com> wrote in message news:<24a23f36.0407010505.4d99dcb4-100000@posting.google.com>...
> [4,5,18]. In those days, Bachas and Fairlie hadn't yet realized
> that their "sine algebra" had already been discovered by Moyal
That should be Zachos, not Bachas.
Thomas Larsson
Jul2-04, 12:36 PM
<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>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:<2k03iaF164tv0U1-100000@uni-berlin.de>...\n> "Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag\n> news:24a23f36.0406240442.70c2f9b1-100000@posting.google.com...\n\n> > I thought that the group of area-preserving diffeomorphisms of a\n> > Riemann surface was the the group of area-preserving diffeomorphisms of\n> > a Riemann surface = the group of canonical transformations since\n> > a Riemann surface is two-dimensional.\n>\n> Sure, and the claim is that this is isomorphic to U(N\\to \\infty) when this\n> limit is suitably defined.\n>\n> > You probably mean that these groups/algebras can be embedded in a\n> > suitable version of SU(infinity) / U(infinity). That would be the same\n> > as saying that they have unitary representations.\n>\n> Zunger claims to show that they are not just embedded, but in fact equal.\n> His point is that there is only one "simple peudocompact algebra", in his\n> terminology, and that both SU(\\infty) as well as Poisson/U(1) are simple\n> pseudocompact.\n>\n> As far as I know for simple topologies, such as the sphere and the torus,\n> this has been known for a long time, Zunger\'s point being that it\n> generalizes to arbitrary topologies.\n\n\nNow I realize why I instinctively reacted against an\nequivalence between area-preserving diffeomorphisms and\nSU(infinity) - I can\'t see how the irreps correspond to each\nother. This problem has nothing to do with doing things on\narbitrary symplectic manifolds, but is present already\nlocally.\n\nWe know how the classical irreps of the symplectomorphism\ngroup look like - they act on symplectic tensor fields. These\nare like ordinary tensor fields, except that we use the\nsymplectic metric to lower contravariant indices and to\neliminate any anti-symmetric pair of indices. In fact, there\nis a 1-1 correspondence between irreps of the\nsymplectomorphism group in n dimensions (n even) and Sp(n).\n\nIt is easy to see that this is true locally for the algebra\nof polynomial Hamiltonian vector fields H_n. Namely, g = H_n\nis a graded simple Lie algebra, where the degree is given by\nthe dilatation eigenvalue. In this situation, there is always\na 1-1 correspondence between g and g_0 irreps. The typical\nexample is the algebra of all vector fields, where the\nzeroth component g_0 = gl(n). Typically the irreducible\nmodule is the corresponding tensor field, although things are\na little more complicated if there is an invariant morphism\nlike the exterior derivative. In our case, g = H_n, it is\neasy to see that g_0 = sp(n), and therefore there is a 1-1\ncorrespondence between H_n and sp(n) irreps.\n\nI am pretty sure that this correspondence can be extended\nglobally and to the group level. In particular, when n = 2\nthe irreps of the group of area-preserving diffs correspond\nto Sp(2) = SU(2); a spin j irrep is carried by a totally\nsymmetric tensor field with j indices. Somehow this must\ncorrespond to irreps of SU(infinity). I fail to see how this\ncan be true.\n\nOTOH, I can follow the calculations on the torus and I see how\none formally recovers H_2. Perhaps there is an SU(2) sitting\ninside SU(infinity) which somehow decouples, but I don\'t see\nhow that works.\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>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:<2k03iaF164tv0U1-100000@uni-berlin.de>...
> "Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag
> news:24a23f36.0406240442.70c2f9b1-100000@posting.google.com...
> > I thought that the group of area-preserving diffeomorphisms of a
> > Riemann surface was the the group of area-preserving diffeomorphisms of
> > a Riemann surface = the group of canonical transformations since
> > a Riemann surface is two-dimensional.
>
> Sure, and the claim is that this is isomorphic to U(N\to \infty) when this
> limit is suitably defined.
>
> > You probably mean that these groups/algebras can be embedded in a
> > suitable version of SU(infinity) / U(infinity). That would be the same
> > as saying that they have unitary representations.
>
> Zunger claims to show that they are not just embedded, but in fact equal.
> His point is that there is only one "simple peudocompact algebra", in his
> terminology, and that both SU(\infty) as well as Poisson/U(1) are simple
> pseudocompact.
>
> As far as I know for simple topologies, such as the sphere and the torus,
> this has been known for a long time, Zunger's point being that it
> generalizes to arbitrary topologies.
Now I realize why I instinctively reacted against an
equivalence between area-preserving diffeomorphisms and
SU(infinity) - I can't see how the irreps correspond to each
other. This problem has nothing to do with doing things on
arbitrary symplectic manifolds, but is present already
locally.
We know how the classical irreps of the symplectomorphism
group look like - they act on symplectic tensor fields. These
are like ordinary tensor fields, except that we use the
symplectic metric to lower contravariant indices and to
eliminate any anti-symmetric pair of indices. In fact, there
is a 1-1 correspondence between irreps of the
symplectomorphism group in n dimensions (n even) and Sp(n).
It is easy to see that this is true locally for the algebra
of polynomial Hamiltonian vector fields H_n. Namely, g = H_n
is a graded simple Lie algebra, where the degree is given by
the dilatation eigenvalue. In this situation, there is always
a 1-1 correspondence between g and g_0 irreps. The typical
example is the algebra of all vector fields, where the
zeroth component g_0 = gl(n). Typically the irreducible
module is the corresponding tensor field, although things are
a little more complicated if there is an invariant morphism
like the exterior derivative. In our case, g = H_n, it is
easy to see that g_0 = sp(n), and therefore there is a 1-1
correspondence between H_n and sp(n) irreps.
I am pretty sure that this correspondence can be extended
globally and to the group level. In particular, when n = 2
the irreps of the group of area-preserving diffs correspond
to Sp(2) = SU(2); a spin j irrep is carried by a totally
symmetric tensor field with j indices. Somehow this must
correspond to irreps of SU(infinity). I fail to see how this
can be true.
OTOH, I can follow the calculations on the torus and I see how
one formally recovers H_2. Perhaps there is an SU(2) sitting
inside SU(infinity) which somehow decouples, but I don't see
how that works.
As I have time for it, I am currently trying to think about
fundamental math notions of sequence, covergence and limit.
Hi Charlie, you may want to discard the sequence concept altogether and use a net (http://planetmath.org/encyclopedia/Net.html).
Thomas Larsson
Jul6-04, 12:14 PM
<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>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:<2jgisnF11d328U1-100000@uni-berlin.de>...\n> There I found the very useful reference\n>\n> Christiaan Hofman:\n> Nonabelian 2-forms,\n> hep-th/0207017 .\n>\n\n\nIn case you are interested in older literature, Jean-Michel\nMaillet and Frank Nijhoff wrote a series of papers around\n1989 about gauge theory on loop space, multi-dimensional\nintegrability, simplex equations, etc, which very much\nresembles the modern gerbe stuff. Here are some references:\n\nM&N, Phys Lett B224 (1989) 389\nM&N, Phys Lett B229 (1989) 71\nM&N, Phys Lett A134 (1989) 221\nM&N, preprint CERN.TH-5595/89 (1989)\nFreidel+M, Phys Lett B296 (1992) 353\nM, Nucl Phys B (Proc Suppl) 18B (1990) 212\n\nAfter that, these papers stopped coming, which was a pity for\nme because I found them very inspiring. I guess that they\nfound it difficult to move beyond the definition phase and\nreally start producing important results. Let us hope that\npeople will succeed better this time.\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>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:<2jgisnF11d328U1-100000@uni-berlin.de>...
> There I found the very useful reference
>
> Christiaan Hofman:
> Nonabelian 2-forms,
> http://www.arxiv.org/abs/hep-th/0207017 .
>
In case you are interested in older literature, Jean-Michel
Maillet and Frank Nijhoff wrote a series of papers around
1989 about gauge theory on loop space, multi-dimensional
integrability, simplex equations, etc, which very much
resembles the modern gerbe stuff. Here are some references:
M&N, Phys Lett B224 (1989) 389
M&N, Phys Lett B229 (1989) 71
M&N, Phys Lett A134 (1989) 221
M&N, preprint CERN.TH-5595/89 (1989)
Freidel+M, Phys Lett B296 (1992) 353
M, Nucl Phys B (Proc Suppl) 18B (1990) 212
After that, these papers stopped coming, which was a pity for
me because I found them very inspiring. I guess that they
found it difficult to move beyond the definition phase and
really start producing important results. Let us hope that
people will succeed better this time.
Urs Schreiber
Jul6-04, 12:49 PM
<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, 6 Jul 2004, Thomas Larsson wrote:\n\n> In case you are interested in older literature,\n\n\n<snip references>\n\nThanks for the references. I\'ll have a look at them as soon as I get home.\nCurrently I am in Karlstad attendig at MathPhys conference on NCG and\nrelated stuff (mostly related stuff, actually)\n\n\nhttp://golem.ph.utexas.edu/string/archives/000393.html .\n\nI have just given a talk on my stuff in which I also mentioned that I\nthink I can write down a constistent boundary superconformal field theory\nwith a non-abelian 2-form background\n\nhttp://www-stud.uni-essen.de/~sb0264/p9.pdf\n\nand in particular read off the nonabelian 2-form connection on the\n1-gerbe. I also think that I can show that this connection is well defined\nas an operator on an appropriate bundle over loop space only if the\nnonabelian background field obeys its equation of motion. If you look at\nthe second and third line of equation (3.12) of the above pdf-file,\nconsider _constant_ gauge connection A and B and\nsimply contract two (dX/d\\sigma), you get a divergence which is\nproportional\n\n\n[A^\\mu, ([A_\\mu,A_\\nu] + G_{\\mu\\nu}) ].\n\nThe vansihing of this is just the A-divergence condition, as it should be.\n\n\nAfter the talk Martin Cederwall asked if I can say anything about an\naction that this equation of motion comes from. He says that the action\ngiven in hep-th/0206130 doesn\'t work, because it doesn\'t have the right\ninvariances.\n\n(I haven\'t checked yet. I took this paper with me to Sweden but\nScandinavian Airlines SAS lost my luggage somewhere in the vicinity of\nCopenhagen. Now I am waiting for them to find it and send it to me... )\n\n\nI have had a long and very informative discussion with Martin Cederwall\nabout the general problem of nonabelian 2-forms and of those in string\ntheory in particular. Apparently the configurations which give strings in\nsuch backgrounds are not at all well understood.\n\nMaybe this even makes my claim that I have a sensible BSCFT for a\nnonabelian 2-form background bold enough to be interesting. ;-)\nIf anyone sees a problem with it, please let me know.\n\nDo the papers that you cited consider connection on loop space? I wouldn\'t\nbe surprised if my equation (3.12) had been written down a long time ago.\n\nMartin Cederwall suggested that one should check if the slicing of the\nworldsheet into spatial slices along which the A field acts, and temporal\ndirections along which the B field acts (horizontal and vertical action\nin the language of 2-groups and crossed modules) is really arbitrary, as\nit should be. For instance for a torus worldhseet it should not matter\nwhich 1-cyle we associate with worldsheet time tau and which one we\nassociate with sigma. This would then really correspond to a "surface\nholonmy", as one would expect.\n\nI think that this is manifestly true for the construction I give. One\nsimply has to write the deformation operator as a super-Pohlmeyer\ninvariant and then apply T-duality.\n\nP.S.\n\nI like Sweden a lot. I have an uncle who lives in Sveg, near Oestersund,\nand when i was a child we very often spend our holidays in Sweden. Now\nwhen I landed on tiny Karlstad airport I stepped out of the plane and\nimmediately had the smell of wood and forest in my nose, which took me\ndirectly back to some good childhood memories.\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, 6 Jul 2004, Thomas Larsson wrote:
> In case you are interested in older literature,
<snip references>
Thanks for the references. I'll have a look at them as soon as I get home.
Currently I am in Karlstad attendig at MathPhys conference on NCG and
related stuff (mostly related stuff, actually)
http://golem.ph.utexas.edu/string/archives/000393.html .
I have just given a talk on my stuff in which I also mentioned that I
think I can write down a constistent boundary superconformal field theory
with a non-abelian 2-form background
http://www-stud.uni-essen.de/~sb0264/p9.pdf
and in particular read off the nonabelian 2-form connection on the
1-gerbe. I also think that I can show that this connection is well defined
as an operator on an appropriate bundle over loop space only if the
nonabelian background field obeys its equation of motion. If you look at
the second and third line of equation (3.12) of the above pdf-file,
consider _constant_ gauge connection A and B and
simply contract two (dX/d\sigma), you get a divergence which is
proportional
[A^\mu, ([A_\mu,A_\nu] + G_{\mu\nu}) ].
The vansihing of this is just the A-divergence condition, as it should be.
After the talk Martin Cederwall asked if I can say anything about an
action that this equation of motion comes from. He says that the action
given in http://www.arxiv.org/abs/hep-th/0206130 doesn't work, because it doesn't have the right
invariances.
(I haven't checked yet. I took this paper with me to Sweden but
Scandinavian Airlines SAS lost my luggage somewhere in the vicinity of
Copenhagen. Now I am waiting for them to find it and send it to me... )
I have had a long and very informative discussion with Martin Cederwall
about the general problem of nonabelian 2-forms and of those in string
theory in particular. Apparently the configurations which give strings in
such backgrounds are not at all well understood.
Maybe this even makes my claim that I have a sensible BSCFT for a
nonabelian 2-form background bold enough to be interesting. ;-)
If anyone sees a problem with it, please let me know.
Do the papers that you cited consider connection on loop space? I wouldn't
be surprised if my equation (3.12) had been written down a long time ago.
Martin Cederwall suggested that one should check if the slicing of the
worldsheet into spatial slices along which the A field acts, and temporal
directions along which the B field acts (horizontal and vertical action
in the language of 2-groups and crossed modules) is really arbitrary, as
it should be. For instance for a torus worldhseet it should not matter
which 1-cyle we associate with worldsheet time \tau and which one we
associate with \sigma. This would then really correspond to a "surface
holonmy", as one would expect.
I think that this is manifestly true for the construction I give. One
simply has to write the deformation operator as a super-Pohlmeyer
invariant and then apply T-duality.
P.S.
I like Sweden a lot. I have an uncle who lives in Sveg, near Oestersund,
and when i was a child we very often spend our holidays in Sweden. Now
when I landed on tiny Karlstad airport I stepped out of the plane and
immediately had the smell of wood and forest in my nose, which took me
directly back to some good childhood memories.
Charlie Stromeyer Jr.
Jul6-04, 01:50 PM
<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@hotmail.com> wrote in message news:\n\n> Charlie Stromeyer Jr. Wrote:\n> >\n> > As I have time for it, I am currently trying to think about\n> > fundamental math notions of sequence, covergence and limit.\n>\n> Hi Charlie, you may want to discard the sequence concept altogether and\n> use a net (http://planetmath.org/encyclopedia/Net.html).\n\nThanks for your suggestion, but I am already thinking about more\ngeneral ideas of net, e.g. see the link within this post I made\nearlier here in s.p.s. [1]. Also, the term "net" was once called a\nMoore-Smith sequence, and even when this concept of net is further\nextended as in the above planetmath.org link there is still involved a\ngeneralized concept of sequence.\n\n\n[1] http://physicsforums.com/showthread.php?t=26370\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@hotmail.com> wrote in message news:
> Charlie Stromeyer Jr. Wrote:
> >
> > As I have time for it, I am currently trying to think about
> > fundamental math notions of sequence, covergence and limit.
>
> Hi Charlie, you may want to discard the sequence concept altogether and
> use a net (http://planetmath.org/encyclopedia/Net.html).
Thanks for your suggestion, but I am already thinking about more
general ideas of net, e.g. see the link within this post I made
earlier here in s.p.s. [1]. Also, the term "net" was once called a
Moore-Smith sequence, and even when this concept of net is further
extended as in the above planetmath.org link there is still involved a
generalized concept of sequence.
[1] http://physicsforums.com/showthread.php?t=26370
Urs Schreiber
Jul7-04, 08:35 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, 6 Jul 2004, Urs Schreiber wrote:\n\n> action that this equation of motion comes from. He says that the action\n> given in hep-th/0206130 doesn\'t work, because it doesn\'t have the right\n> invariances.\n\n\nI think that the problem is that the A-gauge-covariant field strength G of\nthe B-field\n\nG = dB + A /\\ B - B /\\ A\n\nis not invariant under the B-gauge transformations\n\n\nB -> B + d_A \\Lambda\n\nA -> A + \\Lambda .\n\n\nThat\'s why the action\n\n\nS = \\int Tr ( F/\\*F + G/\\*G )\n\nisn\'t either.\n\n\nBut I also think that instead of trying to guess the correct gauge\ninvarint field strength we can just read it off from the boundary state\ndeformations that I have mentioned. From the abelian case we know that the\ngrade 1 term in the deformed loop space exterior derivative is the\nconnection term, while the grade 3 piece is the B-field strength.\n\nIn the nonabelian case this yields the above G plus a correction term,\nsomething like\n\n\nU_A ( dB + A/\\B - B/\\A + (K->B) U_A B + B U_A (K->B) ) U_A .\n\n(Sigma-integrations along the loop are implicit, U_A is the A-holonomy and\nK->B the contraction of B with the rep Killing vector.)\n\n\nThis is the above field strength G plus a corection term that is only\nvisible on loop space.\n\nInvariance under the above mentioned gauge transformation is easy to see\nwhen one looks at it the right way.\n\n\nWell, at least I think so. Since we have free afternoon today I\'ll go back\nto my hotel room and try to write this up in detail.\n\n\n(In writing this post here I should acknowledge inspiring lunch\nconversation with Jens Fjelstad about this topic.)\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, 6 Jul 2004, Urs Schreiber wrote:
> action that this equation of motion comes from. He says that the action
> given in http://www.arxiv.org/abs/hep-th/0206130 doesn't work, because it doesn't have the right
> invariances.
I think that the problem is that the A-gauge-covariant field strength G of
the B-field
G = dB + A /\ B - B /\ A
is not invariant under the B-gauge transformations
B -> B + d_A \LambdaA -> A + \Lambda .
That's why the action
S = \int Tr ( F/\*F + G/\*G )
isn't either.
But I also think that instead of trying to guess the correct gauge
invarint field strength we can just read it off from the boundary state
deformations that I have mentioned. From the abelian case we know that the
grade 1 term in the deformed loop space exterior derivative is the
connection term, while the grade 3 piece is the B-field strength.
In the nonabelian case this yields the above G plus a correction term,
something like
U_A ( dB + A/\B - B/\A + (K->B) U_A B + B U_A (K->B) ) U_A .(\Sigma-integrations along the loop are implicit, U_A is the A-holonomy and
K->B the contraction of B with the rep Killing vector.)
This is the above field strength G plus a corection term that is only
visible on loop space.
Invariance under the above mentioned gauge transformation is easy to see
when one looks at it the right way.
Well, at least I think so. Since we have free afternoon today I'll go back
to my hotel room and try to write this up in detail.
(In writing this post here I should acknowledge inspiring lunch
conversation with Jens Fjelstad about this topic.)
Urs Schreiber
Jul8-04, 06:30 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 Thu, 1 Jul 2004, Thomas Larsson wrote:\n\n> the algebra of Hamiltonian vector fields\n> could be identified with su(infinity) locally. But now I realize\n> that this is an old result which I already knew about; the\n> construction is explained simply in section 2 of hep-th/9912130.\n\nI have just listened to a Mambrane/Matrix theory review talk by Martin\nCederwall and was reminded of a well known fact which is relevant for this\ndiscussion here, probably: Namely if one looks not at the commutative\nmembrane (surface) but at its Moyal deformation, such that the two\ncoordinates x1 and x2 satisfy\n\n[x1,x2] = i \\theta\n\nfor some real number \\theta, then then regularization of the membrane\nHamiltonian by means of su(N) elements is much less tricky, which\nphysically implies for instance that the finite-N truncation is actually a\nconsistent truncation in the sense that its solutions are also solutions\nfor N to infinity.\n\nThis suggests that a proof somewhat along the lines as\nin Zunger\'s paper could be maybe more transparent if one first turned on\n\\theta \\neq 0 and let \\theta \\to 0 in the end.\n\n\nCederwall mentioned that at least at first sight the \\theta\\neq 0\nnoncommutativity on the membrane breaks the nonlinear realization of\nPoincare invariance in the lightcone quantization and that hence the\nphysical viability of the noncommutative membrane (i.e. with \\theta no\nvanishing) may be unclear.\n\nThis made me wonder if the noncommutative membrane might show up as the\ntarget space of a N=(2,1) string, as described in hep-th/9602049.\nCertainly I should have a closer look at this paper, but naively it\nseems that there should also be a 2-form in the spectrum of these (2,1)\nstrings. Giving this a vev should give a physical justification for the\nnoncommutative membrane, shouldn\'t it?\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 Thu, 1 Jul 2004, Thomas Larsson wrote:
> the algebra of Hamiltonian vector fields
> could be identified with su(infinity) locally. But now I realize
> that this is an old result which I already knew about; the
> construction is explained simply in section 2 of http://www.arxiv.org/abs/hep-th/9912130.
I have just listened to a Mambrane/Matrix theory review talk by Martin
Cederwall and was reminded of a well known fact which is relevant for this
discussion here, probably: Namely if one looks not at the commutative
membrane (surface) but at its Moyal deformation, such that the two
coordinates x1 and x2 satisfy
[x1,x2] = i \theta
for some real number \theta, then then regularization of the membrane
Hamiltonian by means of su(N) elements is much less tricky, which
physically implies for instance that the finite-N truncation is actually a
consistent truncation in the sense that its solutions are also solutions
for N to infinity.
This suggests that a proof somewhat along the lines as
in Zunger's paper could be maybe more transparent if one first turned on
\theta \neq and let \theta \to in the end.
Cederwall mentioned that at least at first sight the \theta\neq
noncommutativity on the membrane breaks the nonlinear realization of
Poincare invariance in the lightcone quantization and that hence the
physical viability of the noncommutative membrane (i.e. with \theta no
vanishing) may be unclear.
This made me wonder if the noncommutative membrane might show up as the
target space of a N=(2,1) string, as described in http://www.arxiv.org/abs/hep-th/9602049.
Certainly I should have a closer look at this paper, but naively it
seems that there should also be a 2-form in the spectrum of these (2,1)
strings. Giving this a vev should give a physical justification for the
noncommutative membrane, shouldn't it?
Urs Schreiber
Jul8-04, 07:15 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 Wed, 7 Jul 2004, Urs Schreiber wrote:\n\n> Well, at least I think so. Since we have free afternoon today I\'ll go back\n> to my hotel room and try to write this up in detail.\n\nOk, what I wrote before is essentially correct, but very intransparent, as\nyou may have noticed. ;-)\n\nThere is a much cleaner way to think about this:\n\n\nSo consider on loop space the nonabelian 2-form gauge covariant exterior\nderivative\n\n\nnabla = d + \\oint_A (B) ,\n\nwhere I use the notation as in Hofmann\'s paper hep-th/0207017, but, as I\nhave emphasized before, with the difference that I have A-holonomies\nU_A(\\sigma_1,\\sigma_2) on both sides of the insertion, so that explicitly\n\n\nnabla =\n\\int_0^{2\\pi}d\\sigma E^{\\dagger \\mu}(\\sigma) (\n\\partial_\\mu(\\sigma)\n+\nU_A(0,\\sigma) B_{\\mu\\nu}X^{\\prime \\nu}(\\sigma) U_A(\\sigma,0)\n)\n\nwhere E^{\\dagger} is the operator of exterior multiplication by the\nrespective differential form on loop space.\n\nNow, the nice thing about the loop space perspective is that here we are\ndealing simply with an ordinary connection. The worldsheet is a line in\nloop space and this line couples to the loop space 1-form \\oint_A (B) in\njust the usual way.\n\nThis means that gauge transformation of this connection will follow the\nusual rules. We can work them out and see what they imply for the target\nspace theory, instead of just trying to guess the latter.\n\nThe gauge transformation on the connection \\oint_A (B) is obtained as\nusual by specifying any group-valued function U on loop space and setting\n\n\n\\oint_A (B)\n\\mapsto\nU \\circ \\oint_A (B) \\circ U^\\dagger + U (dU^\\dagger) .\n\n\nNow first of all let\'s look at _global_ gauge transformations, i.e. those\nfor which U(dU^\\dagger) = 0, which is the case when U does not depend on\nthe embedding coordinates of the string/loop.\n\nGiven any such U, a little calculation demonstrates the following:\n\nGiven _any_ group-valued function V on the loop which coincides with U at\nthe arbitrary basepoint\n\nV = V(\\sigma)\n\nV(0) = U\n\nwe have the identity\n\nU \\circ \\oint_A (B) \\circ U^\\dagger = \\oint_{A\'} (B\')\n\nwhere\n\nA\' = V A V^\\dagger + V (d V^dagger)\n\nB\' = V B V^\\dagger .\n\n\nThis is the first of the gauge invariances of a 2-form connection, namely\nessentially the ordinary gauge transformation of the 1-form connection A\ntogether with the obvious action on B.\n\nSo this demonstrates that _global_ gauge transformations on loop space\nare equivalent to ordinary gauge transformations in target space.\n\nI should emphasize again that this crucially depends on that second U_A\nfactor in my definition of the loop space connection, which is derived\nfrom boundary state deformation theory. Without that factor the above does\nnot seem to have a meaningful analogue.\n\n\nThe next step is the more interesting one. There should be a further gauge\ntransformation associated to the cohomology equivalence classes of the\n2-form B, roughly.\n\nFor instance as stated in Lahiri\'s papers, e.g. hep-th/0109220, one\nexpects there to be a gauge invariance of the form\n\nB \\mapsto B + d_A A\n\nA \\mapsto A .\n\n\nAs far as I can see this is the invariance that is not respected by the\naction which is given in hep-th/0206130 .\n\n\nI am a little confused about the status of the proposal for this second\ngauge transformation. Why is this expected to be the correct form for\nnonabelian 2-forms? Is this a definition, a derived result, or a guess?\n\nThe reason I am asking this question is that I believe to have evidence\nthat this gauge transformation needs some correction terms. In order to\ndemonstrate this I\'ll just consider a local gauge transforation on loop\nspace, where I believe to know the correct form of the 2-form gauge\n_connection_, and simply see what the resulting effect on that connection\nis.\n\n\nSo consider a group-valued 1-form \\Lambda on loop space, and consider to\nfirst order the gauge transformation\n\nU = 1 + i \\oint_A (\\Lambda) + \\cdots ,\n\nin direct analogy with the respective expression for target space\ntheories.\n\nComputing the transformation of the connection \\oint_A (B) with respect to\nthis transformation gives, by the above formula, something like\n\n\n\\oint_A (B)\n\\mapsto\n\\oint_A (B + d_A B)\n+\ni\\oint_A (\\Lambda,B - d_A A) - i \\oint_A (B + d_A A,\\Lambda)\n\n\nInterestingly, the first line is precisely the expected gauge\ntransformation on B. But, due to the action of d on the U_A holonomies and\ndue to the commutation with \\oint (\\Lambda), there appear correction terms\nin the second line. As these involve path-ordered integrals of two\nquantities over the loop, they do not have any analogy on target space at\nall! (I think.)\n\nBy construction, the above is a valid gauge transformation on loop space.\nUnless I am confused this shows that we won\'t easily find any target space\naction in terms of a _local_ field theory of point particles which\nrespects this invariance.\n\nOn the other hand, it is of course possible to compute the gauge curvature\n\n\n(d + \\oint_A (B))^2\n\non loop space, take it\'s Hodge-square, trace over it and integrte it over\nloop space to obtain a YM-like theory whose parameter space is loop space.\nI\'d expect, but haven\'t checked, that this action, which manifestly has\nthe full loop space gauge symmetry, flows to some of the target space actions\nwhich have been written down 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 Wed, 7 Jul 2004, Urs Schreiber wrote:
> Well, at least I think so. Since we have free afternoon today I'll go back
> to my hotel room and try to write this up in detail.
Ok, what I wrote before is essentially correct, but very intransparent, as
you may have noticed. ;-)
There is a much cleaner way to think about this:
So consider on loop space the nonabelian 2-form gauge covariant exterior
derivative
nabla = d + \oint_A (B) ,
where I use the notation as in Hofmann's paper http://www.arxiv.org/abs/hep-th/0207017, but, as I
have emphasized before, with the difference that I have A-holonomies
U_A(\sigma_1,\sigma_2) on both sides of the insertion, so that explicitly
nabla =
\int_0^{2\pi}d\sigma E^{\dagger \mu}(\sigma) (\partial_\mu(\sigma)
+
U_A(0,\sigma) B_{\mu\nu}X^{\prime \nu}(\sigma) U_A(\sigma,0)
)
where E^{\dagger} is the operator of exterior multiplication by the
respective differential form on loop space.
Now, the nice thing about the loop space perspective is that here we are
dealing simply with an ordinary connection. The worldsheet is a line in
loop space and this line couples to the loop space 1-form \oint_A (B) in
just the usual way.
This means that gauge transformation of this connection will follow the
usual rules. We can work them out and see what they imply for the target
space theory, instead of just trying to guess the latter.
The gauge transformation on the connection \oint_A (B) is obtained as
usual by specifying any group-valued function U on loop space and setting
\oint_A (B)\mapstoU \circ \oint_A (B) \circ U^\dagger + U (dU^\dagger) .
Now first of all let's look at _global_ gauge transformations, i.e. those
for which U(dU^\dagger) = 0, which is the case when U does not depend on
the embedding coordinates of the string/loop.
Given any such U, a little calculation demonstrates the following:
Given _any_ group-valued function V on the loop which coincides with U at
the arbitrary basepoint
V = V(\sigma)
V(0) = U
we have the identity
U \circ \oint_A (B) \circ U^\dagger = \oint_{A'} (B')
where
A' = V A V^\dagger + V (d V^{dagger})B' = V B V^\dagger .
This is the first of the gauge invariances of a 2-form connection, namely
essentially the ordinary gauge transformation of the 1-form connection A
together with the obvious action on B.
So this demonstrates that _global_ gauge transformations on loop space
are equivalent to ordinary gauge transformations in target space.
I should emphasize again that this crucially depends on that second U_A
factor in my definition of the loop space connection, which is derived
from boundary state deformation theory. Without that factor the above does
not seem to have a meaningful analogue.
The next step is the more interesting one. There should be a further gauge
transformation associated to the cohomology equivalence classes of the
2-form B, roughly.
For instance as stated in Lahiri's papers, e.g. http://www.arxiv.org/abs/hep-th/0109220, one
expects there to be a gauge invariance of the form
B \mapsto B + d_A AA \mapsto A .
As far as I can see this is the invariance that is not respected by the
action which is given in http://www.arxiv.org/abs/hep-th/0206130 .
I am a little confused about the status of the proposal for this second
gauge transformation. Why is this expected to be the correct form for
nonabelian 2-forms? Is this a definition, a derived result, or a guess?
The reason I am asking this question is that I believe to have evidence
that this gauge transformation needs some correction terms. In order to
demonstrate this I'll just consider a local gauge transforation on loop
space, where I believe to know the correct form of the 2-form gauge
_connection_, and simply see what the resulting effect on that connection
is.
So consider a group-valued 1-form \Lambda on loop space, and consider to
first order the gauge transformation
U = 1 + i \oint_A (\Lambda) + \cdots ,
in direct analogy with the respective expression for target space
theories.
Computing the transformation of the connection \oint_A (B) with respect to
this transformation gives, by the above formula, something like
\oint_A (B)\mapsto\oint_A (B + d_A B)[/itex]
+
[itex]i\oint_A (\Lambda,B - d_A A) - i \oint_A (B + d_A A,\Lambda)
Interestingly, the first line is precisely the expected gauge
transformation on B. But, due to the action of d on the U_A holonomies and
due to the commutation with \oint (\Lambda), there appear correction terms
in the second line. As these involve path-ordered integrals of two
quantities over the loop, they do not have any analogy on target space at
all! (I think.)
By construction, the above is a valid gauge transformation on loop space.
Unless I am confused this shows that we won't easily find any target space
action in terms of a _local_ field theory of point particles which
respects this invariance.
On the other hand, it is of course possible to compute the gauge curvature
(d + \oint_A (B))^2
on loop space, take it's Hodge-square, trace over it and integrte it over
loop space to obtain a YM-like theory whose parameter space is loop space.
I'd expect, but haven't checked, that this action, which manifestly has
the full loop space gauge symmetry, flows to some of the target space actions
which have been written down before.
Urs Schreiber
Jul8-04, 07:19 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>\nMy apologies, there were two silly typos in my formulas:\n\nOn Thu, 8 Jul 2004, Urs Schreiber wrote:\n\n> For instance as stated in Lahiri\'s papers, e.g. hep-th/0109220, one\n> expects there to be a gauge invariance of the form\n>\n> B \\mapsto B + d_A A\n>\n> A \\mapsto A .\n\n\nOf course the first line must be\n\nB \\mapsto B + d_A \\Lambda\n\n\n> Computing the transformation of the connection \\oint_A (B) with respect to\n> this transformation gives, by the above formula, something like\n>\n>\n> \\oint_A (B)\n> \\mapsto\n> \\oint_A (B + d_A B)\n> +\n> i\\oint_A (\\Lambda,B - d_A A) - i \\oint_A (B + d_A A,\\Lambda)\n\n\nHere, too, the first line after \\mapsto must read\n\n\\oint_A (B + d_A \\Lambda)\n\n\nSorry.\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>My apologies, there were two silly typos in my formulas:
On Thu, 8 Jul 2004, Urs Schreiber wrote:
> For instance as stated in Lahiri's papers, e.g. http://www.arxiv.org/abs/hep-th/0109220, one
> expects there to be a gauge invariance of the form
>
> B \mapsto B + d_A A
>
> A \mapsto A .
Of course the first line must be
B \mapsto B + d_A \Lambda
> Computing the transformation of the connection \oint_A (B) with respect to
> this transformation gives, by the above formula, something like
>
>
> \oint_A (B)
> \mapsto
> \oint_A (B + d_A B)
> +
> i\oint_A (\Lambda,B - d_A A) - i \oint_A (B + d_A A,\Lambda)
Here, too, the first line after \mapsto must read
\oint_A (B + d_A \Lambda)
Sorry.
Urs Schreiber
Jul12-04, 12:56 PM
<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>I think I am beginning to understand this whole business with nonabelian\n2-form connections much better. Some thinking over the weekend has produced\nthe following ideas:\n\nNobody seems to know precisely how the physical scenario looks like which\ngives us strings in nonabelian 2-form backgrounds, but one thing should be\nclear: Since the closed string cannot carry any Chan-Paton factors it must\nnot couple to the nonabelian 2-form (even though it of course does couple to\nthe abelian one). In any case, let\'s concentrate on cases where the coupling\nis a pure worldsheet boundary effect.\n\nBut that means that the \'surfac holonomy\' assigned to any torus-like\nworldsheet must vanish. For the 1-form connection on loop space induced by\nthe nonabelian 2-form this implies that it must be _flat_ because tori are\nclosed paths in loop space.\n\nNow one can check that the condition for the connection on loop space to be\nflat is\n\nF + B = 0 .\n\nThis is also the condition which makes all the troublesom \'correction\nterms\', which I had mentioned before, disappear, and it ensures that local\ngauge transformations on loop space give the correct target space\ntransformations.\n\nSo this is not only heuristically the correct condition, but also formally\nthe one that makes things consistent.\n\nAfter I had found this, I saw that in\n\nF. Girellu & H. Pfeiffer:\nHigher gauge theory - differential versus integral formulation\nhep-th/0309173\n\nprecisely the same condition F + B = 0 is derived as a conistency condition\n(their (3.25)). The authors of that paper use a completely different\napproach, based on Lie 2-groups.\n\nFor more details see the recent SCT entry\n\nhttp://golem.ph.utexas.edu/string/archives/000396.html .\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>I think I am beginning to understand this whole business with nonabelian
2-form connections much better. Some thinking over the weekend has produced
the following ideas:
Nobody seems to know precisely how the physical scenario looks like which
gives us strings in nonabelian 2-form backgrounds, but one thing should be
clear: Since the closed string cannot carry any Chan-Paton factors it must
not couple to the nonabelian 2-form (even though it of course does couple to
the abelian one). In any case, let's concentrate on cases where the coupling
is a pure worldsheet boundary effect.
But that means that the 'surfac holonomy' assigned to any torus-like
worldsheet must vanish. For the 1-form connection on loop space induced by
the nonabelian 2-form this implies that it must be _flat_ because tori are
closed paths in loop space.
Now one can check that the condition for the connection on loop space to be
flat is
F + B = .
This is also the condition which makes all the troublesom 'correction
terms', which I had mentioned before, disappear, and it ensures that local
gauge transformations on loop space give the correct target space
transformations.
So this is not only heuristically the correct condition, but also formally
the one that makes things consistent.
After I had found this, I saw that in
F. Girellu & H. Pfeiffer:
Higher gauge theory - differential versus integral formulation
http://www.arxiv.org/abs/hep-th/0309173
precisely the same condition F + B = is derived as a conistency condition
(their (3.25)). The authors of that paper use a completely different
approach, based on Lie 2-groups.
For more details see the recent SCT entry
http://golem.ph.utexas.edu/string/archives/000396.html .
Urs Schreiber
Jul15-04, 05:06 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>Yesterday I have finally sent my ideas on nonabelian 2-forms to the arXiv,\nwhere it appears as hep-th/0407122. I would like to thank everybody involved\nin the discussion here as well as for all the private email that I have\nreceived, and in particular thank Charlie Strohmeyer for making me look into\nthe issue of nonabelian 2-forms in the first place. This paper would not\nexist without sci.physics.strings.\n\nBTW, as a spin-off I think I could show that the method used in\n\nKoji Hashimoto:\nGeneralized supersymetric boundary state\nhttp://www.iop.org/EJ/abstract/1126-6708/2000/04/023/\n\nfor computing background equations of motion from divergences in deformed\nboundary states can also be applied to the case of nonablian 2-forms - where\nthe computation is in fact much more transparent, see\n\nhttp://golem.ph.utexas.edu/string/archives/000397.html .\n\n\nNext I want to show how hep-th/0407122 relates to\n\nMaeda, Nakatsu and Oonishi:\nNon-Linear Field Equation from Boundary State Formalism,\nhep-th/0312260\n\nand to super-Pohlmeyer invariants. In this context I was wondering about the\nfollowing:\n\nIt is known that the content of the Yang-Mills equations\n\ndiv_A F_A = 0\n\nis essentially preserved when we take _constant_ gauge connections A with\nvalues in su(N \\to \\infty), where it becomes\n\n[A^m [A_m, A_n]] = 0 .\n\nNow I am wondering if this is compatible with various gauge choices. For\ninstance if we pick a transversal gauge with A_+ = 0, for instance, the\nabove of course reduces to\n\n\\sum_i [A_i [A_i, A_n]] = 0 \\forall n .\n\nDoes this still capture the full information? The reason why I am wondering\nis that _after_ having gone to [A^m [A_m, A_n]] = 0 there is in general no\ngauge trafo, now reading A \\mapsto UAU^\\dagger, which achieves A_+ = 0.\n\nAny ideas?\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>Yesterday I have finally sent my ideas on nonabelian 2-forms to the arXiv,
where it appears as http://www.arxiv.org/abs/hep-th/0407122. I would like to thank everybody involved
in the discussion here as well as for all the private email that I have
received, and in particular thank Charlie Strohmeyer for making me look into
the issue of nonabelian 2-forms in the first place. This paper would not
exist without sci.physics.strings.
BTW, as a spin-off I think I could show that the method used in
Koji Hashimoto:
Generalized supersymetric boundary state
http://www.iop.org/EJ/abstract/1126-6708/2000/04/023/
for computing background equations of motion from divergences in deformed
boundary states can also be applied to the case of nonablian 2-forms - where
the computation is in fact much more transparent, see
http://golem.ph.utexas.edu/string/archives/000397.html .
Next I want to show how http://www.arxiv.org/abs/hep-th/0407122 relates to
Maeda, Nakatsu and Oonishi:
Non-Linear Field Equation from Boundary State Formalism,
http://www.arxiv.org/abs/hep-th/0312260
and to super-Pohlmeyer invariants. In this context I was wondering about the
following:
It is known that the content of the Yang-Mills equations
div_A F_A =
is essentially preserved when we take _constant_ gauge connections A with
values in su(N \to \infty), where it becomes
[A^m [A_m, A_n]] = .
Now I am wondering if this is compatible with various gauge choices. For
instance if we pick a transversal gauge with A_+ = 0, for instance, the
above of course reduces to
\sum_i [A_i [A_i, A_n]] =\forall n .
Does this still capture the full information? The reason why I am wondering
is that _after_ having gone to [A^m [A_m, A_n]] = there is in general no
gauge trafo, now reading A \mapsto UAU^\dagger, which achieves A_+ = .
Any ideas?
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