The n-Category Café

The following is transcribed from the notes I have taken in the lecture. Personal comments are set in italics.

We want to study 3-dimensional topological field theory built from elements

in the fourth integral singular cohomology of the classifying space $B G$ of some group $G$.

As usual in this business, we will assume “for convenience” that $G$ is simple and simply connected. (At the end of the lecture the general case was considered too, but I will not have anything to report on that here.).

(This may be understood as related by transgression to 2-dimensional field theories characterized by elements in $H^3(G,\mathbb{Z})$, namely Wess-Zumino-Witten CFT. This explains why $\tau$ controls many of the objects of interest in the following. In particular, it specifies the level of Kac-Moody central extensions of the loop group on $G$.)

Our TFT will involve 3-dimensional manifolds

In its Lagrangian formulation, we would compute on $M$ the integrals over $M$ of the Chern-Simons 3-forms of flat $G$-bundles with connection on $M$. Hence denote by

the moduli space (moduli stack) of $G$-bundles with connection on $M$.

Every such bundle is classified by a map from $M$ to $B G$. By forgetting the connection we hence get a homotopy class of maps

In fact, not forgetting the connection corresponds to remembering the representative of this homotopy class of maps.

By composing this map with the evaluation map we get from $M \times A_G(M)$ to $B G$

Pulling our chosen element $\tau$ in $H^4(B G,\mathbb{Z})$ back along this map and integrating the result over $M$ yields

an element in the first integral singular cohomology on our “configuration space” $A_G(M)$.

Generally, this first cohomology is isomorphic to homotopy classes of maps to the circle

As mentioned before, we may hence refine $H^1(X,\mathbb{Z})$ by looking at continuous maps

without dividing out by homotopy.

(This is an example of passing from cohomology to differential cohomology.)

What all this means is that for every element $\tau \in H^4(B G , \mathbb{Z})$, we naturally get a map

(from configuration space $A_G(M)$ to the space of phases $U(1)$.

This map defines for us what we mean by the exponentiated action that Chern-Simons theory for fixed $\tau$ assigns to any one field configuration (a $G$ bundle with connection) on $M$.)

If now we were lucky enough to know of a measure $d\mu$ on configuration space $A_G(M)$, we would define Chern-Simons TFT by the assignment

(One of the crucial points is that, in the end, neither $e^{2\pi i S}$ nor $d\mu$ will be well defined seperately, but only their combination, in general. I am afraid, though, that I cannot report much on that more general case.)

Next we need to know what our TFT functor $Z$ assigns to 2-dimensional boundaries of 3-dimensional cobordisms, i.e. to 2-dimensional manifolds.

So let

be the boundary of a 3-manifold. By restricting bundles with connection on $M$ to $N$, we get a map

Analogous to the above construction, this now allows to pull back $\tau$ along

and integrate the result over $N$

Since the integration domain now has one dimension less, the resulting class is one degree higher. Accordingly, instead of yielding a map from $N$ to $U(1)$, it yields a map to $U(1)$-torsors. In other words: $\int_N p^*\tau$ classifies a $U(1)$-bundle (or complex line bundle)

on $N$.

Recall that before we obtained the action

and integrated that over all $\nabla \in A_G(M)$ to obtain the amplitude

Following Freed we now want to do an analogous integration with $\int_N r^*\tau$ to obtain a “2-amplitude” - a Hilbert space.

We hence write

and want this collection of symbols to denote a Hilbert space (the space of states over $N$).

The computation of this Hilbert space was done, of course, by Witten in

E. Witten
Quantum Field Theory and the Jones Polynomial
Commun.Math.Phys.121:351,1989
(spires)

using geometric quantization. This formalism leads to a line bundle (the so-called pre-quantum line bundle)

on the moduli space of flat $G$-connections on the 2-dimensional manifold $N$.

The Hilbert space that we are looking for can be roughly thought of as the “direct integral over all fibers” of this bundle. More precisely, it is the space of square integrable holomorphic sections of $E$:

Our intuition of thinking of this as a sort of integral

can be made precise by defining this in terms of pushforward to a point of the bundle in K-theory (going the wrong way).

It turns out that the Hilbert space obtained this way can be identitfied with the space of conformal blocks over $N$ of the Wess-Zumino-Witten 2-dimensional conformal field theory determined by $\tau$.

In particular, this implies that for

the 2-dimensional torus, the corresponding vector space of states is

This is supposed to make it plausible that to a single circle we want to assign the category $\mathrm{Rep}^\tau(L G)$ of these reps, whose tensor product is fusion of highest weight reps. In any case, we set

This happens to be a modular tenor category. (Every 2-dimensional rational conformal field theory comes from some modular tensor category $C$. For WZW on $G$ it happens to be $C = \mathrm{Rep}^\tau(L G)$.).

It is at this point that the Freed-Hopkins-Teleman theorem enters the game.

Because this theorem tells us that $Z(S^1 \times S^1)$ is nothing but the $\tau$-twisted K-theory of $G$, which is equivariant with respect to the adjoint action of $G$ on itself:

Now, we can regard the K-theory of $G$ as a module over itself. This means that we are apparently saying that our $n$-tiered Chern-Simons theory assigns K-modules to 1-dimensional manifolds.

After meditating about this fact for a while, Michael Hopkins et al. wrote down the table:

For instance for $d=2$, you imagine going from the second to the third column by observing that

- starting with $\mathrm{Vect}_\mathbb{C}$

- restricting to its core (containg only isomorphisms)

- decategorifying and group completing with respect to the $\oplus$-monoidal structure of direct sums of vector spaces

leads to the space

which is the classifying space for K-theory.

This statement is hence to be thought of as a 0-version of the Freed-Hopkins-Teleman result - a statement about passing from $n$-categories to spectra.

We can formulate functoriality of our TFT $Z$ entirely in terms of gadgets present in the third column.

To a 2-dimensional surface

with incoming and outgoing boundary

we associate the span

where $\mathcal{M}_G$ denotes the moduli space of flat $G$-bundles with connection and where the arrows are the above-mentioned push-forwards in K-theory. Recall that for push forward to a point this already appeared in the $n$-categorical path integral above.)

In this “third-column-language” composition of 2-dimensional manifolds corresponds to composition of these spans by pullback along adjacent legs.

That’s what I understood. The remainder of the third lecture now related all this to the Madsen-Tillmann cobordism theories that I briefly mentioned in part I. Unfortunately, I couldn’t follow this at all and hence cannot reasonably report about it here.