The n-Category Café

General context.

As recalled here recently in Fivebrane Structures (blog, arXiv, pdf) and a bit in Charges and twisted $n$-bundles (blog) an important concept in (higher) gauge theory is that of electric-magnetic duality:

Given a line line $n$-bundle $(P,\nabla )$ (aka bundle ($n-1$)-gerbe, aka Deligne $n$-cocycle, aka differential $n$-character) on a $d$-dimensional Riemannian space $(X,g)$, equipped with a connection $\nabla$ with integral curvature $(n+1)$-form $F_{n+1}$ it may happen that the Hodge dual form $${\star }_g{F}_{n+1}\in {\Omega }^{d-n-1}(X)$$ is again closed and integral. If so, it means there is a line $(d-n-2)$-bundle with connection which has this curvature. So somehow the Hodge star in this case extends from a duality operation on deRham classes to a duality operation on differential cohomology classes (not, in general, uniquely so, since there may be several inequivalent $n$-bundles with connection but with the same curvature, in general).

This is important, because this duality operation describes electric-magnetic duality in physics. It plays a big role in various contexts, not the least since it was discovered by Witten and Kapustin that Geometric Langlands (I, II) is apparently just a tiny aspect (I, II, III) of this general mechanism.

But it is hard to find any literature concretely on the extension of the Hodge duality on curvature forms to a relation between the full differential cocycles. The bulk of the physics literature concentrates on things that can be described by globally defined differential forms.

Now recently, Hisham Sati pointed me to

M. Caicedo, I. Martín and A. Restuccia
Gerbes and Duality
(arXiv:hep-th/0205002).

While not addressing exactly the lift of Hodge duality from closed integral to differential cocycles, they do discuss some aspects of electric magnetic duality paying close attention to the full differential cocycle structure of line $n$-bundles with connection.

The duality-symmetric Lagrangians for higher abelian Yang-Mills theory

The bulk of their article is a review of Deligne cohomology, which I’ll take for granted here. The main argument which I want to talk about is that described for $n=1$ and $d=4$ on p.13,14 and then extended to the general case on p. 35,36. (Their main application, to the relation between M2-branes and D2-branes in section 6 shall not concern me here.

We’ll be talking about “path integrals” in gauge theory, which means we’ll not be rigorous. I hope you can deal with that.

As recalled in Charges and twisted $n$-Bundles (I, II, III, IV) the exponentiated action functional for abelian $n$-Yang-Mills theory on the Riemannian space $(X,g)$ in the presence of electric flux is the assignment $$({\hat{F}}_A,{\hat{j}}_E)\mapsto \exp (-\frac{1}{2g}{\int }_X{F}_A\wedge \star F_A)\exp (i{\int }_X\hat{A}\cdot j),$$ where ${\hat{F}}_A$ denotes a line $n$-bundle with connection (locally given by an $n$-form $A$) whose curvature $(n+1)$-form is $F_A$, and where ${\hat{j}}_E$, the electric flux, is itself a $(d-n-1)$-line bundle, which however here we take to be trivial, given by a global curvatrure $(d-n)$-form $j_E$. The dot on the right is the product in differential cohomology.

REMARK on notation: This corresponds to equation (5.92) in CMR. Notice that these authors have the habit of writing ${\int }_X$ when they really mean: choose a good open cover $U{\bigsqcup }_{i\in I}{U}_i$ of $X$, and choose a triangulation $T$ of $X$ subordinate to that cover, i.e. with a lift $s:\{F_k^{(d)}\}\to I$ of each $d$-dimensional face $F_k^{(d)}$ to $U$, a lift of each $(d-1)$-dimensional face to $U{\times }_{X}U$ and so on. Then read expressions like ${\int }_{X}A$ as $$\sum _k{\int }_{F_k^{(d)}}{A}_{s(F_k^{(d)})}.$$ That kind of implicit gymnastics is necessary to make sense of equations (2.23) and (2.25) for instance.

We want to assume further here, for simplicity, that the electric current $j_E$ is Poincaré-dual to an $n$-boundary $\partial \Sigma$ of an $(n+1)$-dimensional submanifold $\Sigma \subset X$. Then the action functional simplifies to $$({\hat{F}}_A,j_E)\mapsto \exp (-\frac{1}{2g}{\int }_X{F}_A\wedge \star F_A)\exp (i{\int }_{\Sigma }{F}_A),$$

The main point now is the following:

the path integral for our abelian $n$-Yang-Mills theory

is over all line $n$-bundles $F_A$. But only the curvature $(n+1)$-form $F_A$ of these $n$-bundles actually appears in the integrand. Therefore we could imagine replacing this by an intregral over all $(n+1)$-forms $F\in {\Omega }^{n+1}(X)$ and including a suitable factor which I’ll denote $${\delta }_{\mathrm{closed},\mathrm{integral}}(F)$$ in the integrand which restricts the integral back to those $(n+1)$-forms which actually appear as curvature $(n+1)$-forms of line $n$-bundles, i.e. those which are closed and integral:

Now CMR argue (that’s the argument between (2.23) and (2.28) generalized to higher $n$) that this factor is itself the path integral over dual line $(d-n-2)$-bundles ${\hat{F}}_{\tilde{A}}$ over the natural pairing of their curvature forms:

where

I think a hidden assumption for their argument to make sense is that $$H^{(d-n-1)}(X,\mathbb{Z})=\mathbb{Z},$$ i.e. that there are no line $(d-n-2)$-line bundles on $X$ whose class is a torsion element. Because they decompose the “path integral” over line $(d-n-2)$-bundles with connection into a sum over the corresponding bundles, which then reduces to a sum over integers, and a sum over the possible connections one can put on these bundles. I suppose with this non-torsion assumption (2.25) surely implies (2.28) if one gets the path integral measure under control. That the remaining integral over line $(d-n-2)$-bundles with connection implies (2.24) appears plausible, but much more subtle. (But maybe I am missing something).

In any case, let’s assume that (3) is right.

Then (2) becomes

We now imagine that we can switch the integration order

and perform the inner integral. The result is supposedly (5.101)

This is now manifestly the kind of path integral that we started with, but now for the dual entities: not for line $n$-bundles with connection, but for line $(d-n-2)$-bundles with connection.

The path integrals being equal means: the physical theories, the original $n$-gauge theory and the dual $(d-n-2)$-gauge theory are equivalent.