## October 18, 2005

### Kalkkinen: Nonabelian Gerbes from twisted SYM

#### Posted by urs

As I had briefly mentioned last time Jussi Kalkkinen had worked out a BRST-cohomology-like formulation of the cocycle description of a nonabelian gerbe in hep-th/0510069.

In a followup preprint which appeared today

Jussi Kalkkinen
Non-Geometric Magnetic Flux and Crossed Modules
hep-th/0510135

he now makes the suggestive relation to BRST symmetry of physical theories, namely $N=4$ Super Yang-Mills, more explicit and presents some arguments concerning the relation of nonabelian gerbes to the physics of M5-branes.

The main point is (section 3) that the ‘BRST operator’ for a nonabelian gerbe (in the sense of the previous paper hep-th/0510069) can be related to that appearing in equation (2.24) of

J. M. F. Labastida, C. Lozano,
Mathai-Quillen Formulation of Twisted $N=4$ Supersymmetric Gauge Theories in Four Dimensions
hep-th/9702106.

More precisely, both are claimed to coincide when restricted to a single patch of a good covering (the latter has only been defined there) with all data on double overlaps assumed to be trivial and when, of course, fields are suitably idenitfied.

There is a lot of notation involved (plenty of local data on the gerbe side matched to plenty of data on the SYM side) which I cannot claim to have sorted out for myself, but this seems to be plausible in the light of the fact that generally twisted bundles are described by gerbes.

In the last part of section 3 Jussi Kalkkinen discusses certain mismatches between the gerbe data and that of the SYM. In particular he describes how he imagines extending locally defined twisted SYM to a globally defined theory by using the nonabelian gerbe cocycle data on double and triple overlaps to glue such local theories together.

In the discussion section 7 a heuristic physical interpretation for this procedure is proposed. There the author argues that double overlaps should be thought of as domain walls where two different twisted SYM theories meet. Furthermore, these domain walls are proposed to be thought of as the worldvolumes of membranes moving in the 4-dimensional bulk spacetime. Since boundaries of these membranes are strings Jussi Kalkkinen is lead to propose that triple overlaps should hence be thought of as worldvolumes of strings.

These conclusions look surprising to me, though maybe there is more evidence for them than I have learned of from this paper.

In particular, it is argued that the gerbe cocycle data ${g}_{\mathrm{ijk}}\in {\Omega }^{0}\left({U}_{\mathrm{ijk}},G\right)$ which map points in triple intersections to elements of the gauge group, is hence a map from homology classes of string worldsheets into the group and therefore a surface holonomy for them.

I have to admit that I do not quite follow the reasoning at this point. This seems to be analogous to claiming that the transition functions ${h}_{\mathrm{ij}}\in \Omega \left({U}_{\mathrm{ij}},G\right)$ of an ordinary bundle ‘are’ the holonomy of a curve.

Possibly I am missing something, in which case I’d be glad to be educated. I do understand the motivating example discussed by Jussi Kalkkinen in section 5.1. In a twisted bundle the transition functions satisfy their ordinary cocycle condition only up to a nontrivial function ${a}_{\mathrm{ijk}}$

(1)${h}_{\mathrm{ij}}{h}_{\mathrm{jk}}{h}_{\mathrm{ki}}={a}_{\mathrm{ijk}}\phantom{\rule{thinmathspace}{0ex}}.$

If we have a vanishing 1-form connection on each patch ${U}_{i}$ this equation may be read as the holonomy of a curve going from ${U}_{i}$ to ${U}_{j}$ to ${U}_{k}$ back to ${U}_{i}$. Hence ${a}_{\mathrm{ijk}}$ is just this holonomy, which could be adressed as a ‘magnetic flux through’ or else the surface holonomy of any surface bounded by that curve.

I could imagine that in certain special cases where the physics one is dealing with is purely topological and it is sufficient to compute surface holonomy in single patches (triple overlaps of them, even), it would be appropriate to interpret the ${g}_{\mathrm{ijk}}$ as surface holonomuies. More generally (even in the well-understood abelian case!), global surface holonomy is the product of a ${g}_{\mathrm{ijk}}$ for each vertex together with further group-valued data on edges and faces of a triangulation of the surface in question.

Another important aspect of the paper is the attempt to incorporate S-duality in SYM into the structure of a nonabelian gerbe (section 4). The idea is that two different SYM theories defined on a common overlap may be related by an S-duality transformation and that one might be able to re-interpret this S-duality transformation in terms of the transition data of a gerbe.

These transition rules say that for a gerbe the connection 1-form defined on single patches has the transition rule known from ordinary bundles but up to an additional additive term. If I understood correctly Jussi Kalkkinen argues that it may be possible to choose this addive term in such a way that the curvatures of the 1-forms thus related are mutually Hodge dual (or at least Hodge dual up to an action by some automorphism). Since curvatures of S-dual theories are mutually Hodge dual this might make it possible to include S-duality among the ordinary gauge transformations on common overlaps.

To me, this looks like a very interesting idea. Is it however obvious that the above mentioned additive term can really be chosen in such a way that Hodge-duality of the respective curvatures is (under some conditions?) achievable?

One might apply similar arguments to other sorts of dualities between gauge theories. I once argued that if one interprets Seiberg duality as a gauge transformation in a 2-bundle one obtains an interesting notion of a vector 2-bundle with possibly nice conceptual relations to the derived category description of D-branes. That’s in section 4.4 of hep-th/0509163.

Jussi Kalkkinen proposes an explicit physical picture for the above interpretation of S-duality in section 6. The idea is to consider the worldvolume theory of an M5-brane which is a torus-bundle over a 4-dimensional base. Picking a good covering of the base the total space of the M5-brane is build up from patches of the form ${U}_{i}×{T}^{2}$. The transition functions in this torus bundle act by $\mathrm{SL}\left(2,ℤ\right)$-tgransformations on the ${T}^{2}$-factor. In terms of the dimensionally reduced SYM theory on the base this action is nothing but Olive-Montonen (S-)duality.

While this is interesting, it should also be well-known. I would like to see in more detail what this now implies for the conjecture that S-duality might be interpretable in terms of gauge transformations in a nonabelian gerbe.

Posted at October 18, 2005 10:51 AM UTC

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### Re: Kalkkinen: Nonabelian Gerbes from twisted SYM

Here are some short answers to points raised in Urs’ post.

The evidence for interpreting overlaps as dynamical objects is that there are more degrees of freedom on an overlap than what there is on a single chart: one boson, one fermion, one vector. This is precisely what happens on physical branes in superstring theory as well. The small excitations of a D-brane show up on the worldvolume in terms of a matter multiplet with one scalar for each transverse dimension.
Note that this jump in degrees of freedom is forced on one by the respective structures of the SYM and the gerbe, not a choice.

(Having said this, an overlap is an open subset in spacetime, and not a closed surface as a membrane, so the analogue is not perfect. What I have in mind is in fact a triangulation where double intersections are replaced by sides and triple intersections by edges. This is how the local data shows up in the concrete formula for an Abelian surface holonomy in Brylinski, for instance.)

I agree with Urs that g_ijk does not look like anything you would call traditionally a surface holonomy. The reason why I proposed this terminology was again by pure analogue from what happens in Yang-Mills. There, namely, the magnetic flux of ‘t Hooft’s appears precisely as the centre of a Wilson line (=holonomy along a loop). As argued in the paper, this flux generalises in the context of a gerbe to the class of the gerbe (\lambda_ij, g_ijk). Perhaps a more appropriate name would have been the “non-geometric Wilson surface” … which is cumbersome.

Finally, given two connections whose curvatures are Hodge dual, they are still two connections in the same space of affine connections, which is contractible. The difference is therefore well-defined as well. The only restriction is in fact that the difference is an Inner automorphism but this is precisely what \lambda in the transition rule is for.

Posted by: Jussi Kalkkinen on October 18, 2005 6:34 PM | Permalink | Reply to this

### Re: Kalkkinen: Nonabelian Gerbes from twisted SYM

Dear Jussi Kalkkinen,

The evidence for interpreting overlaps as dynamical objects is that there are more degrees of freedom on an overlap than what there is on a single chart […]

Note that this jump in degrees of freedom is forced on one by the respective structures of the SYM and the gerbe, not a choice.

That’s an interesting observation. Maybe I am not seeing the full picture yet. Would you make an analogous statement for ordinary YM using ordinary bundles? There, too, we have ‘more degrees of freedom’ on overlaps (the transition functions). Would you therefore ‘interpret the overlaps as dynamical objects’ for ordinary YM, too? If not, what is it that makes this seem advisable in the case of gerbes? And isn’t there a general problem in that the overlaps are ‘not physical’ but just a choice one makes for describing the total structure?

Finally, given two connections whose curvatures are Hodge dual, they are still two connections in the same space of affine connections, which is contractible. The difference is therefore well-defined as well. The only restriction is in fact that the difference is an Inner automorphism but this is precisely what $\lambda$ in the transition rule is for.

True. I was wondering about how to glue things together. Suppose that there are no outer automorphisms (e.g. assume the gauge group is ${E}_{8}$). Then on a double overlap ${U}_{\mathrm{ij}}$ we can always arrange $\kappa \left({A}_{i}\right)=\star \kappa \left({A}_{j}\right)$. Now this forces certain choices on triple overlaps on us, doesn’t it?

I haven’t thought about this in detail. But I would expect realizing S-duality in terms of gerbe transition data should involve more structure then noticing that the transition for ${A}_{i}$ includes an extra additive shift. Something, for instance, which works for the relation $\kappa \left({A}_{i}\right)=\star \kappa \left({A}_{j}\right)$ but not for any other old relation like $\kappa \left({A}_{i}\right)=-\kappa \left({A}_{j}\right)$ or something.

Posted by: Urs Schreiber on October 19, 2005 11:19 AM | Permalink | Reply to this

### Re: Kalkkinen: Nonabelian Gerbes from twisted SYM

First of all, the topological class of the gerbe (\lambda_ij, g_ijk) is kept fixed all the time, in the same way as transition functions are usually kept fixed in Yang-Mills. These don’t count as degrees of freedom.

The new degrees of freedom I listed in the previous post show up on double intersections. On triple intersections the situation is more complicated. In principle the cocycle conditions of a non-Abelian gerbe tell you that there will not be new local fields there; what will happen, however, is that the local fields will have up to three different local values depending on which chart you are working on. [These values are nevertheless constrained as in (39)-(40) in the paper.] The information content of (65) is that, if the rhs commutator vanishes, there are no new degrees of freedom on a triple intersection either.

(The most concervative argument for the presense of string-defects is still the fact that if there are membranes in the above sense, they will have boundaries.)

The last point in your post was about well-definedness of a non-geometric background put together using S-duality.

Whether a background is well-defined depends very much of how it is put together precisely. I looked at one example, where on a triple intersection all three theories were respectively S-dual, and checked that the Cech differential of the constraint vanishes. I think this should be a sufficient condition for the well-definedness of this specific background. It would be very interesting to work out more examples.

Posted by: Jussi Kalkkinen on October 19, 2005 12:20 PM | Permalink | Reply to this

### Re: Kalkkinen: Nonabelian Gerbes from twisted SYM

Thanks again.

BTW, did you think about how ${n}^{3}$-scaling behaviour on M5-branes might arise from the point of view that you develop?

Posted by: Urs Schreiber on October 19, 2005 2:21 PM | Permalink | Reply to this

### Re: Kalkkinen: Nonabelian Gerbes from twisted SYM

Well, yes, but I haven’t got much to say about it. I can’t see how any local QFT could reproduce that scaling behaviour at the moment. I suspect this has to do with the full non-local structure of the properly six-dimensional worldvolume string theory. Any ideas?

Posted by: Jussi Kalkkinen on October 19, 2005 2:50 PM | Permalink | Reply to this

### n-cube again

Any ideas?

I have thought about it and have an educated guess, but not more. In section 4.1.2 of hep-th/0509163 I discuss what currently looks like the best candidate mechanisms to me.

As you know, E. Akhmedov noticed recently (in my words that he did not fully agree with) that given a 2D TFT on $\Sigma$ with correlator $〈\cdots {〉}_{\Sigma }$ and given a 2-form $B$ with values in vertex operators of the TFT, the expression

(1)$F\left(\Sigma \right):={〈\mathrm{exp}\left({\int }_{\Sigma }B\right)〉}_{\Sigma }$

can be regarded as a 2-dimensional generalization of the ordinary holonomy along a line. (The latter can be regarded as the correlator of an exponentiated vertex-valued 1-form in a trivial 1D TFT.)

A large class of 2D TFTs, as described in the famous hep-th/9212154 has correlators that arise as follows:

For the $n$-point correlator $〈{\varphi }_{1}{\varphi }_{2}\dots {\varphi }_{n}{〉}_{\Sigma }$, pick any triangulation of $\Sigma$ with $n$-disjoint (no common edges) triangles labeled by the ${\varphi }_{i}$. Pick an associative semisimple algebra $A$. Let each ${\varphi }_{i}$ be a rank-3 tensor in this algebra (regarded as a vector space). Assign to each triangle of the triangulation which is not labeled by one of the $\varphi$ the structure constants of $A$ (regarded as a rank-3 tensor). Then contract all tensors that sit in triangles which share a common edge. The result is a rank-0 tensor which we identify with the sought-after $n$-point correlator.

The intersting thing is, that the vertex operators here are tensors precisely of rank 3. We could define a 2-form $B$ on $\Sigma$ with values in such tensors

(2)$B\in {\Omega }^{2}\left(\Sigma ,{A}^{\otimes 3}\right)$

and compute

(3)$F\left(\Sigma \right):={〈\mathrm{exp}\left({\int }_{\Sigma }B\right)〉}_{\Sigma }$

using the above prescription, thus obtaining something like a surface holonomy from $B$.

Here $B$, ‘carrying 3 indices’, should naturally account for an ${n}^{3}$ scaling precisely analogous to how an ordinary matrix (rank-2 tensor!)-valued 1-form accounts for the ${n}^{2}$-scaling of ordinary gauge theories.

The above description shows that the numbers 2, 3 appearing here are nothing but the number of corners of the standard simplex in 1 and 2 dimensions, respectively.

I find this a nice idea. However, it is so far defined only locally. Just like A. Gustavsson’s approach, which was inspired by this, it requires globally defined 2-forms.

But I expect that it should be possible to identify a 2-group/crossed module that governs the above setup. It should involve loop groups of some kind. I have made some comments on that here, but have not seriously thought about it since then.

If it were possible to identify the 2-group/crossed module governing the above local surface holonomy, we should be able, using nonabelian gerbes/2-bundles, to obtain the global version of the above construction.

It does however seem that for a 2-group to have any chance to capture this situation, it must be non-strict. This would give rise to something like a ‘lax crossed module’ and would require weaking the standard concept of a gerbe, probably.

Posted by: Urs Schreiber on October 19, 2005 3:34 PM | Permalink | Reply to this

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