Higher Gauge Theory in Edinburgh – Part II

Continuing from the previous post, we’ll take a detour in a different direction. The physics-oriented talks were by Martin Wolf, Sam Palmer, Thomas Strobl, and Patricia Ritter. Since my background in this subject isn’t particularly physics-y, I’ll do my best to summarize the ones that had obvious connections to other topics, but may be getting things wrong or unbalanced here…

Dirac Sigma Models

Thomas Strobl’s talk, “New Methods in Gauge Theory” (based on a whole series of papers linked to from the conference webpage), started with a discussion of of generalizing Sigma Models. Strobl’s talk was a bit high-level physics for me to do it justice, but I came away with the impression of a fairly large program that has several points of contact with more mathematical notions I’ll discuss later.

In particular, Sigma models are physical theories in which a field configuration on spacetime $\Sigma$ is a map $X : \Sigma \rightarrow M$ into some target manifold, or rather $(M,g)$ , since we need a metric to integrate and find differentials. Given this, we can define the crucial physics ingredient, an action functional
$S[X] = \int_{\Sigma} g_{ij} dX^i \wedge (\star d X^j)$
where the $dX^i$ are the differentials of the map into $M$ .

In string theory, $\Sigma$ is the world-sheet of a string and $M$ is ordinary spacetime. This generalizes the simpler example of a moving particle, where $\Sigma = \mathbb{R}$ is just its worldline. In that case, minimizing the action functional above says that the particle moves along geodesics.

The big generalization introduced is termed a “Dirac Sigma Model” or DSM (the paper that introduces them is this one).

In building up to these DSM, a different generalization notes that if there is a group action $G \rhd M$ that describes “rigid” symmetries of the theory (for Minkowski space we might pick the Poincare group, or perhaps the Lorentz group if we want to fix an origin point), then the action functional on the space $Maps(\Sigma,M)$ is invariant in the direction of any of the symmetries. One can use this to reduce $(M,g)$ , by “gauging out” the symmetries to get a quotient $(N,h)$ , and get a corresponding $S_{gauged}$ to integrate over $N$ .

To generalize this, note that there’s an action groupoid associated with $G \rhd M$ , and replace this with some other (Poisson) groupoid instead. That is, one thinks of the real target for a gauge theory not as $M$ , but the action groupoid $M \/\!\!\/ G$ , and then just considers replacing this with some generic groupoid that doesn’t necessarily arise from a group of rigid symmetries on some underlying $M$ . (In this regard, see the second post in this series, about Urs Schreiber’s talk, and stacks as classifying spaces for gauge theories).

The point here seems to be that one wants to get a nice generalization of this situation – in particular, to be able to go backward from $N$ to $M$ , to deal with the possibility that the quotient $N$ may be geometrically badly-behaved. Or rather, given $(N,h)$ , to find some $(M,g)$ of which it is a reduction, but which is better behaved. That means needing to be able to treat a Sigma model with symmetry information attached.

There’s also an infinitesimal version of this: locally, invariance means the Lie derivative of the action in the direction of any of the generators of the Lie algebra of $G$ – so called Killing vectors – is zero. So this equation can generalize to a case where there are vectors where the Lie derivative is zero – a so-called “generalized Killing equation”. They may not generate isometries, but can be treated similarly. What they do give, if you integrate these vectors, is a foliation of $M$ . The space of leaves is the quotient $N$ mentioned above.

The most generic situation Thomas discussed is when one has a Dirac structure on $M$ – this is a certain kind of subbundle $D \subset TM \oplus T^*M$ of the tangent-plus-cotangent bundle over $M$ .

Supersymmetric Field Theories

Another couple of physics-y talks related higher gauge theory to some particular physics models, namely $N=(2,0)$ and $N=(1,0)$ supersymmetric field theories.

The first, by Martin Wolf, was called “Self-Dual Higher Gauge Theory”, and was rooted in generalizing some ideas about twistor geometry – here are some lecture notes by the same author, about how twistor geometry relates to ordinary gauge theory.

The idea of twistor geometry is somewhat analogous to the idea of a Fourier transform, which is ultimately that the same space of fields can be described in two different ways. The Fourier transform goes from looking at functions on a position space, to functions on a frequency space, by way of an integral transform. The Penrose-Ward transform, analogously, transforms a space of fields on Minkowski spacetime, satisfying one set of equations, to a set of fields on “twistor space”, satisfying a different set of equations. The theories represented by those fields are then equivalent (as long as the PW transform is an isomorphism).

The PW transform is described by a “correspondence”, or “double fibration” of spaces – what I would term a “span”, such that both maps are fibrations:

$P \stackrel{\pi_1}{\leftarrow} K \stackrel{\pi_2}{\rightarrow} M$

The general story of such correspondences is that one has some geometric data on $P$ , which we call $Ob_P$ – a set of functions, differential forms, vector bundles, cohomology classes, etc. They are pulled back to $K$ , and then “pushed forward” to $M$ by a direct image functor. In many cases, this is given by an integral along each fibre of the fibration $\pi_2$ , so we have an integral transform. The image of $Ob_P$ we call $Ob_M$ , and it consists of data satisfying, typically, some PDE’s.In the case of the PW transform, $P$ is complex projective 3-space $\mathbb{P}^3/\mathbb{P}^1$ and $Ob_P$ is the set of holomorphic principal $G$ bundles for some group $G$ ; $M$ is (complexified) Minkowski space $\mathbb{C}^4$ and the fields are principal $G$ -bundles with connection. The PDE they satisfy is $F = \star F$ , where $F$ is the curvature of the bundle and $\star$ is the Hodge dual). This means cohomology on twistor space (which classifies the bundles) is related self-dual fields on spacetime. One can also find that a point in $M$ corresponds to a projective line in $P$ , while a point in $P$ corresponds to a null plane in $M$ . (The space $K = \mathbb{C}^4 \times \mathbb{P}^1$ ).

Then the issue to to generalize this to higher gauge theory: rather than principal $G$ -bundles for a group, one is talking about a 2-group $\mathcal{G}$ with connection. Wolf’s talk explained how there is a Penrose-Ward transform between a certain class of higher gauge theories (on the one hand) and an $N=(2,0)$ supersymmetric field theory (on the other hand). Specifically, taking $M = \mathbb{C}^6$ , and $P$ to be (a subspace of) 6D projective space $\mathbb{P}^7 / \mathbb{P}^1$ , there is a similar correspondence between certain holomorphic 2-bundles on $P$ and solutions to some self-dual field equations on $M$ (which can be seen as constraints on the curvature 3-form $F$ for a principal 2-bundle: the self-duality condition is why this only makes sense in 6 dimensions).

This picture generalizes to supermanifolds, where there are fermionic as well as bosonic fields. These turn out to correspond to a certain 6-dimensional $N = (2,0)$ supersymmetric field theory.

Then Sam Palmer gave a talk in which he described a somewhat similar picture for an $N = (1,0)$ supersymmetric theory. However, unlike the $N=(2,0)$ theory, this one gives, not a higher gauge theory, but something that superficially looks similar, but in fact is quite different. It ends up being a theory of a number of fields – form valued in three linked vector spaces

$\mathfrak{g}^* \stackrel{g}{\rightarrow} \mathfrak{h} \stackrel{h}{\rightarrow} \mathfrak{g}$

equipped with a bunch of maps that give the whole setup some structure. There is a collection of seven fields in groups (“multiplets”, in physics jargon) valued in each of these spaces. They satisfy a large number of identities. It somewhat resembles the higher gauge theory that corresponds to the $N=(1,0)$ case, so this situation gets called a “ $(1,0)$ -gauge model”.

There are some special cases of such a setup, including Courant-Dorfman algebras and Lie 2-algebras. The talk gave quite a few examples of solutions to the equations that fall out. The overall conclusion is that, while there are some similarities between $(1,0)$ -gauge models and the way Higher Gauge Theory appears at the level of algebra-valued forms and the equations they must satisfy, there are some significant differences. I won’t try to summarize this in more depth, because (a) I didn’t follow the nitty-gritty technical details very well, and (b) it turns out to be not HGT, but some new theory which is less well understood at summary-level.

Theoretical Atlas