The n-Category Café

Parallel transport in a vector bundle is a functor

which sends paths in base space $X$ to morphisms of vector spaces.

Propagation in quantum mechanics is a functor

which sends 1-dimensional Riemannian cobordisms to morphisms of Hilbert spaces #.

Notice that these two concepts are closely related. The standard example is the single non-relativistic charged particle propagating on $X$ in the presence of an electromagnetic field.

The electromagnetic field itself is the functor

namely a line bundle $E$ with connection.

The propagation functor of our particle

is obtained by performing the path integral over $\mathrm{tra}_\mathrm{EM}$:

where the right hand side denotes the operator defined by the integral kernel $K(x,y)$ which is obtained by the path integral over all paths from $x$ to $y$, using the Wiener measure $D\gamma$.

This operator acts on the Hilbert space

(or simply $L^2(X)$ if E is trivial), the space of square integrable sections of the line bundle $E$.

Notice how we can think of the space of sections of the bundle $E \to X$ as something like

So the Hilbert space assigned to the point in the quantum theory is indeed the “sum” of the vector spaces assigned to the many points of target space by the action! (See below for the relevance of this statement).

Notice how the parallel transport functor plays the role of the action functional (except for the kinetic contribution, which should really be thought of as being part of the measure of the path integral), whose integration yields the propagation functor.

Probably the right way to formalize this passage from parallel transport functors to propagation functors is to use a theory of spans as indicated in

E. Lupercio & B. Uribe
Topological Quantum Field Theory, String and Orbifolds
hep-th/0605255.

This is one of the things I want to better understand here. Especially the categorified version of the above (which plays a central role in Freed’s observation).

Before moving on, we should take the opportunity and identify some terminology which will be needed later on.

Notice how in our little theory of the electrically charged non-relativistic particle we were really describing a 1-dimensional quantum field theory.

The set of fields on a given cobordism $\bullet \stackrel{t}{\to} \bullet$ is the set of (suitably well behaved) maps from the interval $[0,t]$ to $X$, namely the set of parameterized paths in $X$ with parameter range $[0,t]$. Following Freed, we denote this space by

Accordingly, the space of fields over the point $\bullet$ may be identified with $X$ itself

(Notice that the electromagnetic field encoded in $\mathrm{tra}_\mathrm{EM}$ is a background structure in the present case, and not part of the field content of our quantum theory. It would become so only after “second quantization” - whatever that really means.)

All of the above is supposed to make you want to say the following:

Parallel transport in an $n$-vector bundle is an $n$-functor

Propagation in $n$-dimensional quantum field theory is an $n$-functor

Here $n\mathrm{Vect}$ and $n\mathrm{Hilb}$ are to be thought of as $(n+1)$-categories of some notion of $n$-vector spaces and $n$-Hilbert spaces, respectively. $P_n(X)$ is some notion of $n$-paths in $X$, and $n\mathrm{Cob}$ is a placeholder for $n$-cobordisms with possibly extra structure on them, regarded as an $n$-category (instead of as a 1-category).

For instance, you can think of a line bundle gerbe with connection as a “rank-1” 2-vector bundle with parallel transport, where to each point $x \in X$ we associate a $\mathrm{Vect}$-vector space ${}_{A_x}\mathrm{Mod}$, to each path $x \stackrel{\gamma}{\to} y$ a $\mathrm{Vect}$-linear map ${}_{A_x}\mathrm{Mod} \stackrel{N_\gamma}{\to} {}_{A_y}\mathrm{Mod}$, and so on.

Notice that, for closed paths, this $n$-functor associates

• 2-vector spaces to 0-paths
• 1-vector spaces to 1-paths
• 0-vector spaces to 2-paths

Here we would like to understand in which sense an analogous reasoning lets us cook up from the data of an ordinary $n$-dimensional QFT something like a propagation $n$-functor

Crucial observations in this direction have been made long ago in

Daniel S. Freed
Quantum Groups from Path Integrals
q-alg/9501025

and

Daniel S. Freed
Higher Algebraic Structures and Quantization
hep-th/9212115 .

A nice overview of the key concepts is provided in section 5.4 of

Bruce Bartlett
Categorical Aspects of Topological Quantum Field Theories
math.QA/0512103.

Dan Freed takes the 2-dimensional Wess-Zumino(-Witten) model and in particular 3-dimensional Chern-Simons theory apart, to show that from the action functionals defining them, usually thought of as producing numbers from field configurations, one actually obtains a structure of the following sort:

An action denoted $\exp(iS(\cdot))$ which is a list of morphisms

which maps a field $\phi$ defined over a closed $d$-dimensional space to an $(n-d)$-Hilbert space $\exp(iS(\phi))$.

In particular, if we agree to address elements of the ground field $K$ itself (usually $K=\mathbb{C}$ here) as a 0-Hilbert space, then the action takes closed $n$-dimensional spaces to numbers. This is the familiar notion of action, as used by ordinary physicists.

But now, we want the action also to assign data to lower-dimensional spaces. Very roughly the general idea for that is always the following:

Suppose we have an “action” defined on “field configurations” of $n$-dimensional spaces. If our $n$-dimensional space $X$ has an $(n-1)$-dimensional boundary $\partial X$, then we may restrict our action to those fields on $X$ which have a specified restriction on $\partial X$.

But this way the action becomes an element in the ($n$-Hilbert) space of maps on fields on $(n-1)$-dimensional spaces. This space we identitfy as the value of our original action to $(n-1)$-dimensional spaces.

This is roughly the idea. Dan Freed spells it out in great detail for 2-dimensional WZW theory as well as for Dijkgraaf-Witten theory. I have, however, not seen a truly general statement of this idea. There should be some.

From the action, we want to produce a quantum field theory propagator. (Recall the toy example of the charged particle at the beginning.) This is now obtained by some sort of “path integral” involving the above generalized notion of action.

Accordingly, the quantum field theoretic structure we can distill here involves assignments of

• integrals of numbers to $n$-dimensional spaces (known, then, as the partition function)
• “integrals of Hilbert spaces” to $(n-1)$-dimensional spaces
• in general, “integrals of $r$-Hilbert spaces” to $(n-r)$-dimensional spaces

Personally, I feel that a really good general understanding of the mechanism at work here has not yet been written down. Among other things, we expect all of the above structure to assemble into some $n$-functor. Maybe we can discuss this in the comment section.

What has however nicely been worked out are a couple of concrete examples. As I said, most notably the Dijkgraaf-Witten model. This I shall talk about in a followup entry.