Urs Schreiber
Dec15-04, 11:56 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>This is a followup to Peter Woit\'s recent blog entry\nhttp://www.math.columbia.edu/~woit/blog/archives/000122.html\n\nThere, Peter mentions the following\n\n> Part of this story involves the Montonen-Olive duality of N=4\n> supersymmetric Yang-Mills. This duality interchanges the coupling constant\n> with its inverse, whiile taking the gauge group G to the Langlands dual group\n> (group with dual weight lattice). The symmetry that inverts the coupling constant\n> is actually part of a larger {SL}(2, Z) symmetry.\n>\n> One possible explanation for this {SL}(2,Z) symmetry is the conjectured\n> existence of a six-dimensional superconformal QFT with certain\n> properties. Witten explains more about this in his lectures at Graeme Segal\'s 60th\n> birthday conference in 2002. His article from the proceedings volume,\n> entitled \'Conformal Field Theory in Four and Six Dimensions\' doesn\'t\n> seem to be available online, but his slides are, and they cover much\n> the same material.\n\nThese slides can be found here:\nhttp://www.maths.ox.ac.uk/notices/events/special/tgqfts/photos/witten/\n\nThe abelian case is well understood. The \\${SL}(2,\\mathbb{Z})\\$ symmetry of\nabelian YM follows (at least classically obviously) from realizing it as a\ntoroidal compactification of the theory of an abelian 2-form with self-dual\nfield strength in six dimensions, where the \\${SL}(2,\\mathbb{Z})\\$ is just the\nmodular group of the internal torus.\n\nIt is believed that something analogous holds true for nonabelian\n(super)Yang-Mills (for any A-D-E gauge group), i.e. that its Montone-Olive\nsymmetry comes from a toroidal compactification of some 6-dimensional theory\ninvolving a non-abelian 2-form.\n\nIn this set of slides, Witten calls this nonabelian 6D theory a nonabelian\ngerbe theory. But certainly that is just a name, to be filled with content,\nright?\n\nThe most glaring problem with making this concrete seems to be this:\n\nWhat precisely is the duality condition in the nonabelian case and under\nwhich conditions can it be imposed?\n\nWhen I talked to nonabelian gerbe people about this, one thing they said is\nthat it is not clear that in the nonabelian case the self-duality should\nstill be ordinary Hodge self-duality, but that it might involve in addition\nto the Hodge star an operation on the Lie algebra factor. But I am not quite\nsure what that should be.\n\nIn lack of a better idea, let me assume in the following that we want\nordinary Hodge duality. Now, one sufficient condition fulfilled by an\nordinary bundle to admit a self-dual field strength is that the field\nstrength transforms covariantly.\n\nSo if \\$U = \\{ U_i \\}_{i\\in I}\\$ is a good covering of the base space with\nopen sets and \\$F_{A_i}\\$ is the field strength on \\$U_i\\$, then on double\noverlaps\n\n\\[\nF_i = g_ij F_j g_ij^{-1}\\,,\n\\]\n\nobviously.\n\nSince the covariant transformation respects Hodge self-duality, it is\nconsistent to impose Hodge self-duality in overlapping patches \\$U_i\\$.\n\nIt is not clear at all that this remains true in general for nonabelian\ngerbes!\n\nFor nonabelian gerbes the general transition law for the nonabelian 3-form\nfield strenth \\$H_i\\$ has a covariant part\n\n\\[\nH_i = g_ij(H_j) + ...\n\\]\n\nplus a mess of noncovariant terms\n\n\\[\n... + \\mathbf{d} d_{ij} + [a_{ij},d_{ij}] - A_i(d_{ij}) + ...\n\\]\n\nand in particular involving this term\n\n\\[\n... + (F_{A_i} + {ad}(B_i))(a_{ij}) \\,.\n\\]\n\n(The notation here is taken from equation (55) in hep-th/0409200.)\n\nSuppose we want \\$H\\$ to be Hodge self-dual and hence \\$H_i\\$ to be\nHodge-self-dual on each \\$U_i\\$. This implies that on every double overlap all\nthese additional terms in the above transition law have to be self-dual by\nthemselves!\n\nSo self-duality on \\$H\\$ implies further self-duality conditions on the fields\n\\$A_i\\$, \\$B_i\\$, \\$a_{ij}\\$, \\$d_{ij}\\$ (which are the connection 1-form, it\'s\n2-form cousin and two \'transition forms\' that measure the failure of \\$A_i\\$\nand \\$B_i\\$ to transform as usual.)\n\nBut these fields don\'t transform covariantly themselves. So the self-duality\ncondition on them involves still more conditions, now on triple overlaps.\nAnd so on. It is a huge mess of ever more complicated conditions that arise\nthis way. (Unless there is some simplifying principle hidden in them, which\nI currently cannot see.)\n\nIt will be hard to find solutions to these conditions. One solution, though,\nis easy to see. Obviously, for \\$H\\$ to be self-dual it is sufficient that\n\n\\[\nd_{ij} = 0\n\\]\n\n(actually this seems to be easy to weaken somewhat)\n\nand\n\n\\[\n{ad}(B_i) + F_{A_i} = 0 \\,.\n\\]\n\nThe big question is: <em>Are there any further restrictions on the cocycle\ndata of a nonabelian gerbe that would allow Hodge-self-dual H?</em> In\nparticular, are there any with \\${ad}(B_i) + F_{A_i} \\neq 0\\$?\n\nThe above choice is curious, since it implies that, while \\$A_i\\$ and \\$B_i\\$\nare nonabelian, \\$H_i\\$ takes value in an abelian subalgebra of the full\nnonabelian Lie algebra.\n\nIt is also the only case so far in which we know (so far) how to associate a\nnonabelian 2-holonomy with the nonabelian gerbe. (A paper on that is due out\nby end of the year. Really, I should not be blogging but be working on\nthat...)\n\nThe existence of that nonabelian 2-holonomy seems to be, apart from the\nself-duality of \\$H\\$, a further important condition on whatever Witten may\nmean by nonabelian gerbe field theory:\n\nWe known that when lifted to M-theory these nonabelian 6-D theories come\nfrom stacks of coinciding M5s with M2s ending in them. The action of these\nM2s should involve the abelian <em>volume holonomy</em> of an abelian\n2-gerbe characterized by the 4-form \\$dC_3\\$, where \\$C_3\\$ is the supergravity\n3-form potential, over the world-volume of the membrane, call that\nsuggestively but by abuse of the integral notation \\$\\exp(i \\int_V C_3)\\$,\ntimes a <em>non</em>abelian surface holonomy of the nonabelian 2-form living\non the M5s over the worldsheet of the boundary of the M2, call that\n\\${Tr}{hol}_{\\partial V}(B)\\$.\n\nDue to global issues (completely analogous to how the coupling of the string\nto an abelian 2-form involves abelian gerbe holonomy) the product\n\n\\[\n\\exp\\left(i \\int_V C_r\\right) {Tr}{hol}_{\\partial V}\\left(B\\right)\n\\]\n\nhas a couple of subtleties. (For the case of 1-dimension lower these, and\ntheir solution, are nicely discussed in the above mentioned paper by\nAschieri& Jurčo).\n\nTherefore, in order to understand nonabelian theories in 6D (and,\nincidentally, the general configuration of the fundamental objects of\nM-theory) it would be very helpful to have a notion of nonabelian surface\nholonomy \\${hol}_{\\partial V}(B)\\$ that makes the above expression\nwell-defined.\n\nI do have a (global!) nonabelian surface holonomy for nonabelian 2-bundles\nand nonabelian gerbes for the case \\${ad}(B_i) + F_{A_i} = 0\\$, i.e. for the\nonly known case in which the existence of a self-dual 3-form field strength\nis known. But I have not yet checked if it makes the above action for the M2\nbrane globally well defined.\n\n\n(For better readable formulas see the original blog entry:\nhttp://golem.ph.utexas.edu/string/archives/000485.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>This is a followup to Peter Woit's recent blog entry
http://www.math.columbia.edu/~woit/blog/archives/000122.html
There, Peter mentions the following
> Part of this story involves the Montonen-Olive duality of N=4
> supersymmetric Yang-Mills. This duality interchanges the coupling constant
> with its inverse, whiile taking the gauge group G to the Langlands dual group
> (group with dual weight lattice). The symmetry that inverts the coupling constant
> is actually part of a larger {SL}(2, Z) symmetry.
>
> One possible explanation for this {SL}(2,Z) symmetry is the conjectured
> existence of a six-dimensional superconformal QFT with certain
> properties. Witten explains more about this in his lectures at Graeme Segal's 60th
> birthday conference in 2002. His article from the proceedings volume,
> entitled 'Conformal Field Theory in Four and Six Dimensions' doesn't
> seem to be available online, but his slides are, and they cover much
> the same material.
These slides can be found here:
http://www.maths.ox.ac.uk/notices/events/special/tgqfts/photos/witten/
The abelian case is well understood. The ${SL}(2,\mathbb{Z})$ symmetry of
abelian YM follows (at least classically obviously) from realizing it as a
toroidal compactification of the theory of an abelian 2-form with self-dual
field strength in six dimensions, where the ${SL}(2,\mathbb{Z})$ is just the
modular group of the internal torus.
It is believed that something analogous holds true for nonabelian
(super)Yang-Mills (for any A-D-E gauge group), i.e. that its Montone-Olive
symmetry comes from a toroidal compactification of some 6-dimensional theory
involving a non-abelian 2-form.
In this set of slides, Witten calls this nonabelian 6D theory a nonabelian
gerbe theory. But certainly that is just a name, to be filled with content,
right?
The most glaring problem with making this concrete seems to be this:
What precisely is the duality condition in the nonabelian case and under
which conditions can it be imposed?
When I talked to nonabelian gerbe people about this, one thing they said is
that it is not clear that in the nonabelian case the self-duality should
still be ordinary Hodge self-duality, but that it might involve in addition
to the Hodge star an operation on the Lie algebra factor. But I am not quite
sure what that should be.
In lack of a better idea, let me assume in the following that we want
ordinary Hodge duality. Now, one sufficient condition fulfilled by an
ordinary bundle to admit a self-dual field strength is that the field
strength transforms covariantly.
So if $U = \{ U_i \}_{i\in I}$ is a good covering of the base space with
open sets and $F_{A_i}$ is the field strength on $U_i$, then on double
overlaps
\[
F_i = g_{ij} F_j g_{ij}^{-1}\,,
\]
obviously.
Since the covariant transformation respects Hodge self-duality, it is
consistent to impose Hodge self-duality in overlapping patches $U_i$.
It is not clear at all that this remains true in general for nonabelian
gerbes!
For nonabelian gerbes the general transition law for the nonabelian 3-form
field strenth $H_i$ has a covariant part
\[
H_i = g_{ij}(H_j) + ...
\]
plus a mess of noncovariant terms
\[
... + \mathbf{d} d_{ij} + [a_{ij},d_{ij}] - A_i(d_{ij}) + ...
\]
and in particular involving this term
\[
... + (F_{A_i} + {ad}(B_i))(a_{ij}) \,.
\]
(The notation here is taken from equation (55) in http://www.arxiv.org/abs/hep-th/0409200.)
Suppose we want $H$ to be Hodge self-dual and hence $H_i$ to be
Hodge-self-dual on each $U_i$. This implies that on every double overlap all
these additional terms in the above transition law have to be self-dual by
themselves!
So self-duality on $H$ implies further self-duality conditions on the fields
$A_i$, $B_i$, $a_{ij}$, $d_{ij}$ (which are the connection 1-form, it's
2-form cousin and two 'transition forms' that measure the failure of $A_i$
and $B_i$ to transform as usual.)
But these fields don't transform covariantly themselves. So the self-duality
condition on them involves still more conditions, now on triple overlaps.
And so on. It is a huge mess of ever more complicated conditions that arise
this way. (Unless there is some simplifying principle hidden in them, which
I currently cannot see.)
It will be hard to find solutions to these conditions. One solution, though,
is easy to see. Obviously, for $H$ to be self-dual it is sufficient that
\[
d_{ij} =
\]
(actually this seems to be easy to weaken somewhat)
and
\[
{ad}(B_i) + F_{A_i} = \,.
\]
The big question is: <em>Are there any further restrictions on the cocycle
data of a nonabelian gerbe that would allow Hodge-self-dual H?</em> In
particular, are there any with ${ad}(B_i) + F_{A_i} \neq 0$?
The above choice is curious, since it implies that, while $A_i$ and $B_i$
are nonabelian, $H_i$ takes value in an abelian subalgebra of the full
nonabelian Lie algebra.
It is also the only case so far in which we know (so far) how to associate a
nonabelian 2-holonomy with the nonabelian gerbe. (A paper on that is due out
by end of the year. Really, I should not be blogging but be working on
that...)
The existence of that nonabelian 2-holonomy seems to be, apart from the
self-duality of $H$, a further important condition on whatever Witten may
mean by nonabelian gerbe field theory:
We known that when lifted to M-theory these nonabelian 6-D theories come
from stacks of coinciding M5s with M2s ending in them. The action of these
M2s should involve the abelian <em>volume holonomy</em> of an abelian
2-gerbe characterized by the 4-form $dC_3$, where $C_3$ is the supergravity
3-form potential, over the world-volume of the membrane, call that
suggestively but by abuse of the integral notation $\exp(i \int_V C_3)$,
times a <em>non</em>abelian surface holonomy of the nonabelian 2-form living
on the M5s over the worldsheet of the boundary of the M2, call that
${Tr}{hol}_{\partial V}(B)$.
Due to global issues (completely analogous to how the coupling of the string
to an abelian 2-form involves abelian gerbe holonomy) the product
\[
\exp\left(i \int_V C_r\right) {Tr}{hol}_{\partial V}\left(B\right)
\]
has a couple of subtleties. (For the case of 1-dimension lower these, and
their solution, are nicely discussed in the above mentioned paper by
Aschieri& Jurčo).
Therefore, in order to understand nonabelian theories in 6D (and,
incidentally, the general configuration of the fundamental objects of
M-theory) it would be very helpful to have a notion of nonabelian surface
holonomy ${hol}_{\partial V}(B)$ that makes the above expression
well-defined.
I do have a (global!) nonabelian surface holonomy for nonabelian 2-bundles
and nonabelian gerbes for the case ${ad}(B_i) + F_{A_i} = 0$, i.e. for the
only known case in which the existence of a self-dual 3-form field strength
is known. But I have not yet checked if it makes the above action for the M2
brane globally well defined.
(For better readable formulas see the original blog entry:
http://golem.ph.utexas.edu/string/archives/000485.html)
http://www.math.columbia.edu/~woit/blog/archives/000122.html
There, Peter mentions the following
> Part of this story involves the Montonen-Olive duality of N=4
> supersymmetric Yang-Mills. This duality interchanges the coupling constant
> with its inverse, whiile taking the gauge group G to the Langlands dual group
> (group with dual weight lattice). The symmetry that inverts the coupling constant
> is actually part of a larger {SL}(2, Z) symmetry.
>
> One possible explanation for this {SL}(2,Z) symmetry is the conjectured
> existence of a six-dimensional superconformal QFT with certain
> properties. Witten explains more about this in his lectures at Graeme Segal's 60th
> birthday conference in 2002. His article from the proceedings volume,
> entitled 'Conformal Field Theory in Four and Six Dimensions' doesn't
> seem to be available online, but his slides are, and they cover much
> the same material.
These slides can be found here:
http://www.maths.ox.ac.uk/notices/events/special/tgqfts/photos/witten/
The abelian case is well understood. The ${SL}(2,\mathbb{Z})$ symmetry of
abelian YM follows (at least classically obviously) from realizing it as a
toroidal compactification of the theory of an abelian 2-form with self-dual
field strength in six dimensions, where the ${SL}(2,\mathbb{Z})$ is just the
modular group of the internal torus.
It is believed that something analogous holds true for nonabelian
(super)Yang-Mills (for any A-D-E gauge group), i.e. that its Montone-Olive
symmetry comes from a toroidal compactification of some 6-dimensional theory
involving a non-abelian 2-form.
In this set of slides, Witten calls this nonabelian 6D theory a nonabelian
gerbe theory. But certainly that is just a name, to be filled with content,
right?
The most glaring problem with making this concrete seems to be this:
What precisely is the duality condition in the nonabelian case and under
which conditions can it be imposed?
When I talked to nonabelian gerbe people about this, one thing they said is
that it is not clear that in the nonabelian case the self-duality should
still be ordinary Hodge self-duality, but that it might involve in addition
to the Hodge star an operation on the Lie algebra factor. But I am not quite
sure what that should be.
In lack of a better idea, let me assume in the following that we want
ordinary Hodge duality. Now, one sufficient condition fulfilled by an
ordinary bundle to admit a self-dual field strength is that the field
strength transforms covariantly.
So if $U = \{ U_i \}_{i\in I}$ is a good covering of the base space with
open sets and $F_{A_i}$ is the field strength on $U_i$, then on double
overlaps
\[
F_i = g_{ij} F_j g_{ij}^{-1}\,,
\]
obviously.
Since the covariant transformation respects Hodge self-duality, it is
consistent to impose Hodge self-duality in overlapping patches $U_i$.
It is not clear at all that this remains true in general for nonabelian
gerbes!
For nonabelian gerbes the general transition law for the nonabelian 3-form
field strenth $H_i$ has a covariant part
\[
H_i = g_{ij}(H_j) + ...
\]
plus a mess of noncovariant terms
\[
... + \mathbf{d} d_{ij} + [a_{ij},d_{ij}] - A_i(d_{ij}) + ...
\]
and in particular involving this term
\[
... + (F_{A_i} + {ad}(B_i))(a_{ij}) \,.
\]
(The notation here is taken from equation (55) in http://www.arxiv.org/abs/hep-th/0409200.)
Suppose we want $H$ to be Hodge self-dual and hence $H_i$ to be
Hodge-self-dual on each $U_i$. This implies that on every double overlap all
these additional terms in the above transition law have to be self-dual by
themselves!
So self-duality on $H$ implies further self-duality conditions on the fields
$A_i$, $B_i$, $a_{ij}$, $d_{ij}$ (which are the connection 1-form, it's
2-form cousin and two 'transition forms' that measure the failure of $A_i$
and $B_i$ to transform as usual.)
But these fields don't transform covariantly themselves. So the self-duality
condition on them involves still more conditions, now on triple overlaps.
And so on. It is a huge mess of ever more complicated conditions that arise
this way. (Unless there is some simplifying principle hidden in them, which
I currently cannot see.)
It will be hard to find solutions to these conditions. One solution, though,
is easy to see. Obviously, for $H$ to be self-dual it is sufficient that
\[
d_{ij} =
\]
(actually this seems to be easy to weaken somewhat)
and
\[
{ad}(B_i) + F_{A_i} = \,.
\]
The big question is: <em>Are there any further restrictions on the cocycle
data of a nonabelian gerbe that would allow Hodge-self-dual H?</em> In
particular, are there any with ${ad}(B_i) + F_{A_i} \neq 0$?
The above choice is curious, since it implies that, while $A_i$ and $B_i$
are nonabelian, $H_i$ takes value in an abelian subalgebra of the full
nonabelian Lie algebra.
It is also the only case so far in which we know (so far) how to associate a
nonabelian 2-holonomy with the nonabelian gerbe. (A paper on that is due out
by end of the year. Really, I should not be blogging but be working on
that...)
The existence of that nonabelian 2-holonomy seems to be, apart from the
self-duality of $H$, a further important condition on whatever Witten may
mean by nonabelian gerbe field theory:
We known that when lifted to M-theory these nonabelian 6-D theories come
from stacks of coinciding M5s with M2s ending in them. The action of these
M2s should involve the abelian <em>volume holonomy</em> of an abelian
2-gerbe characterized by the 4-form $dC_3$, where $C_3$ is the supergravity
3-form potential, over the world-volume of the membrane, call that
suggestively but by abuse of the integral notation $\exp(i \int_V C_3)$,
times a <em>non</em>abelian surface holonomy of the nonabelian 2-form living
on the M5s over the worldsheet of the boundary of the M2, call that
${Tr}{hol}_{\partial V}(B)$.
Due to global issues (completely analogous to how the coupling of the string
to an abelian 2-form involves abelian gerbe holonomy) the product
\[
\exp\left(i \int_V C_r\right) {Tr}{hol}_{\partial V}\left(B\right)
\]
has a couple of subtleties. (For the case of 1-dimension lower these, and
their solution, are nicely discussed in the above mentioned paper by
Aschieri& Jurčo).
Therefore, in order to understand nonabelian theories in 6D (and,
incidentally, the general configuration of the fundamental objects of
M-theory) it would be very helpful to have a notion of nonabelian surface
holonomy ${hol}_{\partial V}(B)$ that makes the above expression
well-defined.
I do have a (global!) nonabelian surface holonomy for nonabelian 2-bundles
and nonabelian gerbes for the case ${ad}(B_i) + F_{A_i} = 0$, i.e. for the
only known case in which the existence of a self-dual 3-form field strength
is known. But I have not yet checked if it makes the above action for the M2
brane globally well defined.
(For better readable formulas see the original blog entry:
http://golem.ph.utexas.edu/string/archives/000485.html)