The n-Category Café

Chas and Sullivan famously noticed, that the homology groups of the free loop space $L X$ of a compact oriented smooth manifold $X$ are equipped with an interesting paring operation

$$H_\bullet(L X) \otimes H_\bullet(L X) \to H_{\bullet - dim X}(L X)$$

that generalizes the Goldmann bracket on $H_0(L X) \simeq F \pi_1(X)$. Since this operation is induced from concatenating loops, they called it the string product . Its study has come to be known as string topology , now a branch of differential topology.

It was soon realized that there indeed ought to be a relation to string physics: there ought to be a 2-dimensional quantum field theory associated with $X$, as follows:

for $\Sigma$ a 2-dimensional surface with incoming and outgoing boundary components $\partial_{in} \Sigma \stackrel{in}{\to} \Sigma \stackrel{out}{\leftarrow} \partial_{out} \Sigma$, the “space of states” of the theory ought to be given by the homology groups $H_\bullet(X^{\partial_{in} \Sigma})$ and $H_\bullet(X^{\partial_{out} \Sigma})$, and the path integral as a pull-push transform along $\Sigma$ ought to be given by push-forward and dual fiber integration

$$(X^{out})_* \circ (X^{in})^! : H_\bullet(X^{\partial_{in} \Sigma}) \to H_{\bullet- dim X}(X^{\partial_{out} \Sigma})$$

induced by the mapping space span

$$\array{ && X^{\Sigma} \\ & {}^{\mathllap{X^{in}}}\swarrow && \searrow^{\mathrlap{X^{out}}} \\ X^{\partial_{in} \Sigma} &&&& X^{\partial_{out} \Sigma} } \,.$$

For $\Sigma \simeq S^1 \vee S^1$ the 3-holed sphere with two incoming and one outgoing circle, this would describe an operation on string states induced by the merging of two closed strings to a single one

$$H_\bullet(X^{S^1 \coprod S^1}) \simeq H_\bullet(L X) \times H_\bullet(L X) \to H_\bullet(L X)$$

and this ought to be Chas-Sullivan string product operation.

That this is indeed the case was finally demonstrated by Veronique Godin. (See string topology for all references.) While the idea is rather simple, the concrete realization, especially when taking open strings into account, is fairly technical (see for instance this MO discussion).

But there should be more to it: one expects that these operations on homology groups are just a shadow of a refined construction on chain complexes (for instance singular chains): while Godin’s construction gives an HQFT – a quantum field theory that depends only on the homology of the moduli spaces of the relevant cobordisms – one expect that this is the homology of a genuine extended TQFT (which in this dimension is widely but somewhat unfortunately known under the term “TCFT”). Remarks on how that might be obtained have been made in print by Costello and Lurie.

In the context of our discussion of $\sigma$-models, we would want to refine this even one further step and ask: is the string-topology TCFT of a manifold (given that it exists) formally a $\sigma$-model with target space that manifold, and using some suitable background gauge field?

Given that the string topology TCFT itself has not been fully identified yet, we cannot expect a complete answer to this at the moment, but I will try to discuss a crucial ingredient that is available.

Notably we can first ignore the dynamics of the system, just consider the kinematics and ask the simple question: which quantum $\sigma$-models on $X$ have (∞,n)-vector spaces of states whose decategorification are graded homology groups $H_\bullet(X^{\partial \Sigma})$ of mapping spaces of $X$?

The answer to this question is more more transparent after we formulate the question in more generality: as observed by Cohen and Godin in their A Polarized View of String Topology , we may assume without restriction that the homology groups here are with respect to the generalized homology with coefficients in any commutative ∞-ring $K$ (as long as this is an “$\infty$-field” and as long as $X$ is $K$-oriented):

$$H_\bullet(X^{\partial_{in} \Sigma}) := H_\bullet(X^{\partial_{in}} \Sigma, K) \,.$$

Most every statement about ordinary commutative rings has its analog for commutative ∞-rings, and so we can just follow our nose:

An (∞,1)-vector space over $K$ is an $K$-module spectrum and we have an (∞,1)-category $K$Mod of such $\infty$-vector spaces. Notice that these are a categorification of the ordinary notion of vector space only in the “$r$“-direction of the lattice of $(r,n)$-categories. A genuine (∞,n)-vector space over $K$ – as appears in the description of general $n$-dimensional $\sigma$-models – is instead an object of $(\cdots ((K Mod) Mod) \cdots ) Mod$. Here we should be fine with just $(\infty,1)$-vector spaces.

This means that an (∞,1)-vector bundle with flat ∞-connection over some manifold $X$ is equivalently encoded by an (∞,1)-functor

$$\alpha : \Pi X \to K Mod$$

out of the fundamental ∞-groupoid of $X$. This assigns to each point of $X$ a $K$-module – the fiber of the (∞,1)-vector bundle thus encoded – to each path in $X$ an equivalence between the fibers over its endpoints, and so on: this is the higher parallel transport of a flat $\infty$-connection. We can also think of this as an ∞-representation of $\Pi X$ on $K$-modules, also called a representation up to homotopy . For instance if $X$ is the classifying space $X = B G$ of a discrete ∞-group, then flat (∞,1)-vector bundles on $X$ are precisely ∞-representations of $G$.

There is an evident full sub-(∞,1)-category

$$K Line \hookrightarrow K Mod$$

of 1-dimensional $(\infty,1)$-vector spaces: $K$-lines – $K$-modules that are equivalent to $K$ itself regarded as a $K$-module.

An $(\infty,1)$-vector bundle $\nabla : \Pi(X) \to K Mod$ that factors through this inclusion is a $K$-line $\infty$-bundle .

One finds, as for the case of ordinary 1-vector spaces, that

$$K Line \simeq B GL_1(K) \simeq B Aut(K)$$

is the delooping of the automorphism ∞-group of $K$. This means that $K$-line $\infty$-bundles are equivalently $GL_1(K)$-principal ∞-bundles. We can think of the inclusion

$$\rho : B GL_1(K) \stackrel{\simeq}{\to} K Line \hookrightarrow K Mod$$

as being the canonical linear ∞-representation of $GL_1(K)$; and for $g : \Pi X \to B GL_1(K)$ a $GL_1(K)$-principal ∞-bundle of the $(\infty,1)$-vector bundle

$$\Pi X \stackrel{g}{\to} B GL_1(K) \stackrel{\rho}{\to} K Mod$$

as the corresponding associated ∞-bundle.

Therefore it makes sense to consider $\sigma$-models with target space $X$ and background gauge field given by a $K$-line $\infty$-bundle.

For instance for $K = K U$ the K-theory spectrum, there is a canonical morphism $B^2 U(1) \to B GL_1(KU)$ and hence to every circle 2-bundle $\alpha : \Pi X \to B^2 U(1)$ is associated the corresponding $K U$-line $\infty$-bundle

$$\Pi X \stackrel{\alpha}{\to} B^2 U(1) \stackrel{}{\to} B GL_1(K U) \stackrel{\rho}{\to} K U Mod \,.$$

Or for $K = tmf$ the tmf spectrum, there is a canonical morphism $B^3 U(1) \to B GL_1(tmf)$ and hence to every circle 3-bundle $\alpha : \Pi X \to B^3 U(1)$ is associated the corresponding $tmf$-line $\infty$-bundle

$$\Pi X \stackrel{\alpha}{\to} B^3 U(1) \stackrel{}{\to} B GL_1(tmf) \stackrel{\rho}{\to} tmf Mod \,.$$

This was amplified by Ando, Blumberg, Gepner, Hopkins, and Rezk (see the references here), who notice much of the theory of $K$-(co)homology – including notably its Thom spectrum theory and its twisted cohomology – is neatly captured by simple statements about such $A$-line $(\infty,1)$-bundles. For instance the notion of orientation in generalized cohomology simply boils down to the notion of trivialization of such $K$-line $\infty$-bundles:

a vector bundle $E \to X$ is $K$-oriented precisely if the corresponding Thom space-bundle – which is a sphere spectrum-line $\infty$-bundle $V: \Pi(X) \to S Line$ is such that the canonically associated $K$-line bundle is trivializable:

$$(V is K-orientable) \Leftrightarrow ( (\Pi(X) \stackrel{V}{\to} S Line \to K Line) \simeq const_K ) \,.$$

For our discussion here this means that Cohen-Godin’s finding that the string topology HQFT exists for $K$ such that $X$ is $K$-orientable meaks that the $K$-line $\infty$-bundle background field that we are to consider in this context are to be trivializable.

Recall that if we interpret such an $K$-line bundle as a background gauge field for a $\sigma$-model, then for $\Sigma$ any cobordism the corresponding (∞,1)-vector space of states assigned to, say, the incoming boundary $\partial_{in} \Sigma$ is defined to be the $\infty$-vector space of sections of the transgression of this $\infty$-vector bundle to the mapping space. The transgression of a trivial bundle is again the trivial bundle. And the $\infty$-vector space of (co)sections is, in the discrete case, as we had discussed before, the (∞,1)-colimit

$$\Gamma_{\partial_{in} \Sigma} := \lim_\to(\Pi X^{\partial_{in} \Sigma} \to S Line \to K Mod ) \,.$$

This one can compute. By triviality of the bundle, Ando-Blumberg-Gepner-Hopkins-Rezk observe that this is the $K$-homology spectrum of $X^{\partial_{in} \Sigma}$

$$\cdots \simeq (\Sigma^\infty X^{\partial_{in} \Sigma}) \wedge K \in K Mod \,.$$

(Here the main point is that for the bundle not being trivial the result encodes the corresponding twisted cohomology , but for our purposes at the moment we want the oriented/trivializable case.)

This is hence the $\infty$-vector space of states over $\partial_{in} \Sigma$ assigned by a $\sigma$-model with background gauge field a $K$-line $\infty$-bundle over a $K$-oriented target space.

Its decategorification is precisely the tower of homology groups of $X$:

$$\pi_\bullet(\Gamma_{\partial \Sigma}) = H_\bullet(X^{\partial_{in} \Sigma}, K) \,.$$

Which is the decategorified space of states that we set out to find in a $\sigma$-model.

So it looks like there ought to be a chance that we understand “string topology TCFT” as a sigma-model induced from a trivial $(\infty,1)$-vector bundle background gauge field.