The String Coffee Table

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 $g_{\mathrm{ijk}}\in {\Omega }^0(U_{\mathrm{ijk}},G)$ 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 $h_{\mathrm{ij}}\in \Omega (U_{\mathrm{ij}},G)$ 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 $a_{\mathrm{ijk}}$

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 $U_i\times T^2$. The transition functions in this torus bundle act by $\mathrm{SL}(2,\mathbb{Z})$-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.