December 15, 2004

Nonabelian Self-Duality

Posted by urs

This is a followup to Peter Woit’s recent blog entry Langlands Program and Physics.

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 $\mathrm{SL}\left(2,ℤ\right)$ symmetry.

One possible explanation for this $\mathrm{SL}\left(2,ℤ\right)$ 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.

The abelian case is well understood. The $\mathrm{SL}\left(2,ℤ\right)$ 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 $\mathrm{SL}\left(2,ℤ\right)$ 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 $U=\left\{{U}_{i}{\right\}}_{i\in I}$ 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

(1)${F}_{i}={g}_{\mathrm{ij}}{F}_{j}{g}_{\mathrm{ij}}^{-1}\phantom{\rule{thinmathspace}{0ex}},$

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

(2)${H}_{i}={g}_{\mathrm{ij}}\left({H}_{j}\right)+...$

plus a mess of noncovariant terms

(3)$...+d{d}_{\mathrm{ij}}+\left[{a}_{\mathrm{ij}},{d}_{\mathrm{ij}}\right]-{A}_{i}\left({d}_{\mathrm{ij}}\right)+...$

and in particular involving this term

(4)$...+\left({F}_{{A}_{i}}+\mathrm{ad}\left({B}_{i}\right)\right)\left({a}_{\mathrm{ij}}\right)\phantom{\rule{thinmathspace}{0ex}}.$

(The notation here is taken from equation (55) in 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 ${A}_{i}$, ${B}_{i}$, ${a}_{\mathrm{ij}}$, ${d}_{\mathrm{ij}}$ (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

(5)${d}_{\mathrm{ij}}=0$

(actually this seems to be easy to weaken somewhat)

and

(6)$\mathrm{ad}\left({B}_{i}\right)+{F}_{{A}_{i}}=0\phantom{\rule{thinmathspace}{0ex}}.$

The big question is: Are there any further restrictions on the cocycle data of a nonabelian gerbe that would allow Hodge-self-dual H? In particular, are there any with $\mathrm{ad}\left({B}_{i}\right)+{F}_{{A}_{i}}\ne 0$?

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 volume holonomy of an abelian 2-gerbe characterized by the 4-form ${\mathrm{dC}}_{3}$, 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 $\mathrm{exp}\left(i{\int }_{V}{C}_{3}\right)$, times a nonabelian surface holonomy of the nonabelian 2-form living on the M5s over the worldsheet of the boundary of the M2, call that $\mathrm{Tr}{\mathrm{hol}}_{\partial V}\left(B\right)$.

Due to global issues (completely analogous to how the coupling of the string to an abelian 2-form involves abelian gerbe holonomy) the product

(7)$\mathrm{exp}\left(i{\int }_{V}{C}_{r}\right)\mathrm{Tr}{\mathrm{hol}}_{\partial V}\left(B\right)$

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 ${\mathrm{hol}}_{\partial V}\left(B\right)$ 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 $\mathrm{ad}\left({B}_{i}\right)+{F}_{{A}_{i}}=0$, 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.

Posted at December 15, 2004 11:29 AM UTC

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