The n-Category Café

Abelian higher gauge theory of Yang-Mills type is the study of of a certain function on the space of all line $n$-bundles with connection on a given Riemannian manifold $X$.

A line $n$-bundle with connection is modeled by any of the following concepts: an abelian (n-1) (bundle) gerbe with connection, a Cheeger-Simons differential character of degree $(n+1)$, a Deligne cocycle of degree $(n+1)$. Whatever model you use, such a gadget $\nabla$ comes with its curvature $(n+1)$-form $$F_\nabla \,.$$ The higher Yang-Mills like function(al) on the space of all such line $n$-bundles with connection is the assignment $$e^{-S} : \nabla \mapsto \exp\left( - \int_X F_\nabla \wedge \star F_\nabla \right) \in \mathbb{R} \,,$$ where “$\star$” is the Hodge star operator with respect to the chosen Riemannian structure on $X$.

In a more general setup we are looking at higher Yang-Mills theory with electric charges. As recalled in Charges and twisted $n$-bundles, II, “electric charge” for an “electric field” given by a line $n$-bundle with connection on a $dim(X)$-dimensional base space is itself a line $(d-n-1)$-bundle with connection, called $\hat j_E$.

There is a product of differential cocycles which on their curvature $(n+1)$-forms reproduces the ordinary wedge product of differential forms. If the $n$-bundle $\nabla$ has locally the connection form $A$ and if $\hat j_E$ has curvature $(d-n)$-form $j_e$, then the product line $d$-bundle with connection $$\nabla \cdot \hat j_E$$ has locally the connection $d$-form $$A \wedge j_E \,.$$ The action functional of (higher) abelian Yang-Mills theory in the presence of electric charges is $$e^{-S} : (\nabla, \hat j_E) \mapsto \exp\left( - \int_X F_\nabla \wedge \star F_\nabla \right) \exp\left( 2 \pi i \int_X \hat j_E \cdot \nabla \right) \in \mathbb{C} \,.$$

Here the integral in the term on the right denotes push-forward in differential cohomology, which reduces to the ordinary integral over the $d$-form $j_E \wedge A$ in the case that $A$ is a globally defined connection $n$-form on the line $n$-bundle $\nabla$.

So far, $e^{-S}$ is clearly an ordinary function (on the space of pairs consisting of one $n$-bundle and one $d-n-1$-bundle on $X$.) No anomaly so far.

Before coming to that, we need to quickly see what $e^{-S}$ really is: we want it in the end to be a smooth function on the space of all gauge field configurations. So far that is the space of all pairs consisting of a line $n$-bundle and a line $(n+1)$-bundle on $X$. To make that configuration space a smooth space we use the reasoning from Space and Quantity and give it the structure of a sheaf (might be an $n$-stack unless we do something about it, but that’s not of relevance for the moment) on smooth test domains $U$: $$conf : U \mapsto \{ pairs (\nabla, \hat j_E) on U \times X\} \,.$$ The action functional $e^{-S}$ we have now extends to a function $$e^{-S} : conf \to \mathbb{C}$$ from the smooth space $conf$ to the complex numbers by thinking of $\int_X$, which had been push-forward to the point along $X \to pt$ with the coresponding fiber integration obtained by push-forward to $U$ along $X \times U \to U$.

Here the complex numbers are equipped with their canonical structure of a smooth space: $$\mathbb{C} : U \mapsto \{smooth complex-valued functions on U \} \,.$$ So $e^{-S}$ is now a function of “smooth spaces”, which are really sheaves on smooth test domains, but apart from that it is still a function. (Freed does not mention sheaves this way, but that’s what he means to say.)

This changes as we further generalize the theory to also include magnetic charges:

as also recalled in Charges and twisted $n$-bundles, II, magnetic charge for a gauge field given by a line $n$-bundle with connection $\nabla$ is itself a line $(n+1)$-bundle with connection, $\hat j_B$. But now something changes: if $\hat j_B$ is nontrivial, then $\nabla$ is no longer an ordinary line $n$-bundle, but becomes a “twisted” line $n$-bundle, a “section” of $\hat j_B$.

As a result of that, one can show that the second term in our action functional $$e^{-S} : (\hat j_E, 0 \stackrel{\nabla}{\to} \hat j_B)_{X \times U} \mapsto \exp\left( - \int_X F_\nabla \wedge \star F_\nabla \right) \exp\left( 2 \pi i \int_X \hat j_E \cdot \nabla \right)$$ is no longer a complex function, hence a line 0-bundle – but a twisted line 0-bundle: a section of a line 1-bundle. That line 1-bundle turns out to be given, over each local probe $U$, by the expression $$ anom := \exp\left( 2 \pi i \int_X \hat j_E \cdot \hat j_B \right) : U \mapsto \{ line 1-bundles with connection over $U$ \} \,. $$

This line bundle (over each test domain $U$) is our anomaly. It is the obstruction to interpreting our expression for $e^{-S}$ in the naive fashion as a complex function on configuration space: in general it will now only be a section of the anomaly line bundle over configuration space:

$$\array{ anom \\ \downarrow & \uparrow^{e^{-S}} \\ conf } \,.$$

I wanted to say more, but it took me longer than I thought to get this far, and now boarding for my plane over the pond will start any minute. I’ll stop here for the moment and try to continue later.