## October 27, 2006

### Hopkins Lecture on TFT: Chern-Simons

#### Posted by Urs Schreiber

In the third part (following part I and part II) of his 2006 lecture series in Göttingen on topological field theory, Michael Hopkins considered the special case of 3-dimensional topological field theories characterized by classes

(1)$\tau \in H^4(B G, \mathbb{Z}) \,.$

These are known as Chern-Simons field theories at level $\tau$.

The first part of the talk reviewed some basic concepts in suitable language.

Then the seminal theorem in

D. Freed, M. Hopkins, C. Teleman
Twisted K-theory and loop group representations
math.AT/0312155,

which relates the modular tensor category encoding $G$-Chern-Simons theory with the twisted $\mathrm{Ad}G$-equivariant K-theory on $G$ - is used as a key for extracting topological information from Chern-Simons TFT 3-functors and reformulating everything in terms of K-theory.

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

(1)$\tau \in H^4(B G, \mathbb{Z})$

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

(2)$M \,.$

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

(3)$A_G(M)$

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

(4)$A_G(M) \to [M,B G] \,.$

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 $B G$

(5)$\array{ M \times A_G(M) &\to& M \times [M,B G] \\ & \searrow_p & \;\;\downarrow \mathrm{ev} \\ && B G } \,.$

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

(6)$\array{ p^* \tau &\in& H^4(M \times A_G(M),\mathbb{Z}) \\ \;\;\downarrow \int_M \\ \int_M p^* \tau &\in& H^1(A_G(M),\mathbb{Z}) }$

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

(7)$H^1(X, \mathbb{Z}) \simeq [X, S^1] \,.$

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

(8)$X \to S^1$

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

(9)$e^{2\pi i S(\cdot)} \; : \; A_G(M) \to U(1)$

(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

(10)$Z : M \mapsto \int_{A_G(M)} e^{2\pi i S} \; d\mu \,.$

(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

(11)$N := \partial M$

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

(12)$A_G(M) \to A_G(N) \,.$

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

(13)$\array{ N \times A_G(N) &\to& N \times [N,B G] \\ & \searrow_{r} & \;\;\downarrow \mathrm{ev} \\ && B G }$

and integrate the result over $N$

(14)$\array{ r^* \tau &\in& H^4(N \times A_G(M),\mathbb{Z}) \\ \;\;\downarrow \int_N \\ \int_N r^* \tau &\in& H^2(A_G(M),\mathbb{Z}) } \,.$

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)

(15)$\array{ L_\tau \\ \downarrow \\ N }$

on $N$.

Recall that before we obtained the action

(16)$e^{2\pi i S(\nabla)} := \int_M p^* \tau$

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

(17)$Z(M) := \int_{A_G(M)} e^{2\pi i S} \,.$

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

(18)$Z(N) := \int_{A_G(M^2)} L_\tau$

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)

(19)$E \to \mathcal{M}_G(N)$

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 $E$:

(20)$Z(N) := \Gamma^2_\mathrm{hol}(E) \,.$

Our intuition of thinking of this as a sort of integral

(21)$Z(N) = \int_{\mathcal{M}_G(N)} E$

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

(22)$N = T^1 = S^1 \times S^1$

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

(23)$Z(S^1 \times S^1) = \left\lbrace \array{ \text{Grothendieck group of} \\ \text{highest weight reps of} \\ L G\;\text{at level}\; \tau } \right\rbrace \otimes \mathbb{C} \,.$

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

(24)$Z(S^1) := \mathrm{Rep}^\tau(L G) \,.$

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:

(25)$Z(S^1 \times S^1) = K^{\tau}_G(G) \,.$

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:

(26)$\array{ d = 3 & \text{number} & ? \\ d = 2 & \text{Vector space} & \text{K-class} \\ d = 1 & modular tensor category & \text{K-module} \\ d = 0 & ? & \text{K-linear category} } \,.$

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

(27)$\to (\sqcup B \mathrm{GL}_N(\mathbb{C}))_{\text{grp. compl.}} = \mathbb{Z} \times B U \,,$

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

(28)$\Sigma$

with incoming and outgoing boundary

(29)$\partial \Sigma = \partial^\mathrm{in}\Sigma \sqcup \partial^\mathrm{out} \Sigma^{-}$

we associate the span

(30)$\array{ K(\mathcal{M}_G(\Sigma)) &\to& K(\mathcal{M}_G(\partial^\mathrm{out}\Sigma)) \\ \downarrow \\ K(\mathcal{M}_G(\partial^\mathrm{in}\Sigma)) } \,,$

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.

Posted at October 27, 2006 11:32 AM UTC

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### Re: Hopkins Lecture on TFT: Chern-Simons

Is this CFT WZW on the boundary of TFT CS just a boundary condition or an holograhic dual of a TFT bulk?

Posted by: Daniel de França MTd2 on July 12, 2009 1:06 AM | Permalink | Reply to this

### Re: Hopkins Lecture on TFT: Chern-Simons

This is not a random question. I am really confused after I found this article:

(Non-)Abelian Kramers-Wannier duality and topological field theory Authors: Pavol Severa (Submitted on 18 Jun 2002)

Abstract: “We study a connection between duality and topological field theories. First, 2d Kramers-Wannier duality is formulated as a simple 3d topological claim (more or less Poincare duality), and a similar formulation is given for higher-dimensional cases. In this form they lead to simple TFTs with boundary coloured in two colours. The statistical models live on the boundary of these TFTs, as in the CS/WZW or AdS/CFT correspondence. Classical models (Poisson-Lie T-duality) suggest a non-abelian generalization in the 2dcase, with abelian groups replaced by quantum groups. Amazingly, the TFT formulation solves the problem without computation: quantum groups appear in pictures, independently of the classical motivation. Connection with Chern-Simons theory appears at the symplectic level, and also in the pictures of the Drinfeld double: Reshetikhin-Turaev invariants of links in 3-manifolds, computed from the double, are included in these TFTs. All this suggests nice phenomena in higher dimensions.”

Posted by: Daniel de França MTd2 on July 13, 2009 2:34 PM | Permalink | Reply to this

### Re: Hopkins Lecture on TFT: Chern-Simons

And not only that. It seems that urs said something about this on Peter Woit’s blog.

Posted by: Daniel de França MTd2 on July 13, 2009 3:00 PM | Permalink | Reply to this

### Re: Hopkins Lecture on TFT: Chern-Simons

Alright, I think things are clearer now. I have just to study the links in Peter’s blog.

Posted by: Daniel de França MTd2 on July 14, 2009 12:02 AM | Permalink | Reply to this

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