The n-Category Café

It’s me again, chewing on the other end of the bone.

On my side the bone looks like this:

A “background gauge field” is a parallel transport $n$-functor $$\mathrm{tra}:\mathrm{tar}\to \mathrm{phas}.$$ An $n$-particle is determined by its shape $\mathrm{par}$ and its configurations $$\mathrm{par}\stackrel{\gamma \in \mathrm{conf}}{\to }\mathrm{tar}.$$ Whatever quantization is, we know that in the Schrödinger picture it is something that reads in the above data and spits out another $n$-functor: $$q(\mathrm{tra}):\mathrm{par}\to \mathrm{phas}.$$

There is a natural canonical way this could arise from the above data, namely as the pull-push of $\mathrm{tra}$ through the correspondence $$\begin{array}{ccc}& & \mathrm{conf}\times \mathrm{par}\\ & {}^{\mathrm{ev}}\swarrow & & \searrow \\ \mathrm{tar}& & & & \mathrm{par}\end{array}.$$

After collecting some evidence I decided that this must be right and gave a discussion of this definition in

The Globular Extended QFT of the Charged n-Particle: Definition .

My motivation and goal is to understand Chern-Simons theory as the quantization of the 3-particle (“membrane”) $\mathrm{par}={\Pi }_2(S^2)$ propagating on $\mathrm{tar}=\Sigma (\mathrm{INN}({\mathrm{String}}_k(G)))$ and coupled to a 2-gerbe $\mathrm{tra}:\mathrm{tar}\to 1d3\mathrm{Vect}$, and to understand the FFRS state sum model this way.

As a preparation for that, I discussed something like a decategorified Chern-Simons theory, describing a 2-particle (“string”) $\mathrm{par}={\Pi }_1(S^1)$, propagating on $\mathrm{tar}=\Sigma ({\mathrm{String}}_k(G))$ and coupled to a trivial 1-gerbe $\mathrm{tra}:\mathrm{tar}\to \mathrm{Bim}\hookrightarrow 2\mathrm{Vect}$ in

Globular Extended QFT of the Charged n-Particle: String on $BG$.

While this example is nice for understanding things related to Chern-Simons, we would want to see more familiar examples of “ordinary” $n$-particles that zip around in ordinary spacetime.

For the ordinary 1-particle $\mathrm{par}=•$ propagating on some space $\mathrm{tar}=P_1(X)$ and coupled to an ordinary vector bundle $\mathrm{tra}:\mathrm{tra}\to \mathrm{Vect}$ the above pull-push quantization reproduces the familiar result.

As we move from 1-particles to 2-particles (“strings”), we want to see our arrow-theoretic formalism reproduce standard stringy phenomena. In particular, the formalism should know that the ends of an open 2-particle $$\mathrm{par}=(a\to b)$$ couple to a Chan-Paton vector bundle on a D-brane.

See Brodzki, Mathai, Rosenberg & Szabo: D-Branes, RR-Fields and Duality for details on what this means.

In my second talk at Fields I indicated how a coupling of the open 2-particle $\mathrm{par}=(a\to b)$ to a nontrivial line 2-bundle transport $\mathrm{tra}:P_2(X)\to 1d2\mathrm{Vect}$ (also known as a line bundle gerbe with connection and curving) yields, by the above pull-push quantization, a coupling of the endpoints $a$ and $b$ to a gerbe module. This is nothing but a Chan-Paton bundle twisted by a gerbe – and is what, in parts of the mathematical string literature, is taken as the definition of a D-brane.

Among other things, this gives a nice definition of a gerbe module simply as a morphism $$E:1\to \mathrm{tra},$$ where $\mathrm{tra}$ is the 2-anafunctor respresenting the gerbe, and $1$ is the tensor unit in the 2-category of all these.

But before even getting into the discussion of these twisted Chan-Paton bundles, it is interesting and instructive to consider in more detail the simple situation of an open 2-particle $\mathrm{par}=(a\to b)$ coupled to the trivial line-2-bundle $1:P_2(X)\to 2\mathrm{Vect}$, forgetting about all twists and turns.

From looking at the standard string theory textbooks, we expect this to have (2-)states that involve a combination of sections on path space of $X$ and a morphism between two vector bundles on $X$.

Here I want to talk about how this comes about in the arrow-theoretic framework of pull-push quantization of the charged $n$-particle.

So the exercise is this:

Exercise: Determine the pull-push through $\begin{array}{ccc}& & \mathrm{conf}\times \mathrm{par}\\ & {}^{\mathrm{ev}}\swarrow & & \searrow \\ \mathrm{tar}& & & & \mathrm{par}\end{array}$ for the charged 2-particle $$\left(\mathrm{par}\stackrel{\gamma \in \mathrm{conf}}{\to }\mathrm{tar}\stackrel{\mathrm{tra}}{\to }\mathrm{phas}\right)$$ that is defined as follows. Target space is $$\mathrm{tar}=P_2(X),$$ the 2-groupoid of 2-paths in a manifold $X$ (“spacetime”). The parallel transport is $$\mathrm{tra}:P_2(X)\to \mathrm{Bim}\hookrightarrow 2\mathrm{Vect},$$ the trivial line-2bundle on $X$, i.e. $$\mathrm{tra}:\left(\begin{array}{cc}& \nearrow \searrow \\ x& \Downarrow & y\\ & \searrow \nearrow \end{array}\right)\mapsto \left(\begin{array}{cc}& \nearrow {\searrow }^{\mathrm{Id}}\\ ℂ& {\Downarrow }^{\mathrm{Id}}& ℂ\\ & \searrow {\nearrow }_{\mathrm{Id}}\end{array}\right)$$ (“trivial background field”, like a vanishing electromagnetic field). The “shape and inner structure” of the 2-particle (“string”) is $$\mathrm{par}=(a\to b).$$ The space of its configurations $$\mathrm{conf}\subset [(a\to b),P_2(X)]$$ is the discrete 2-category $$\mathrm{conf}=\mathrm{Disc}(\mathrm{Obj}([(a\to b),P_2(X)])),$$ which regards all paths in $X$ as distinct (gauge inequivalent) configurations of our 2-particle.

Since I have already spent so much time introducing this example, I will close this entry simply by stating the result and postponing the discussion of the derivation.

Result:

One finds that the 2-space of states of the above 2-particle, or rather that part of it which is “local” over the endpoints of our string, consists of triples $$e=(E_1,E_2,f),$$ where $E_1$ and $E_2$ are vector bundles on $X$, and where $f$ is a map that sends each path $x\stackrel{\gamma }{\to }y$ in $X$ to a morphism $$f(\gamma ):E_1(x)\to E_2(y)$$ between the fibers of these bundles over the endpoints of this path.

Accordingly, the quantization produces a 2-functor $$q(\mathrm{tra}):(a\to b)\to \mathrm{Bim}\hookrightarrow 2\mathrm{Vect}$$ which is such that its states $$\psi :1\to \mathrm{q}(\mathrm{tra})$$ are equivalent to these $e$.

This is the case when $$q(\mathrm{tra}):(a\to b)\mapsto (C(X)\stackrel{C(PX)}{\to }C(X)).$$

Here $C(X)$ is the algebra of (complex) functions on $X$ and $C(PX)$ is the algebra of functions on the free path space of $X$, which is regarded as a $C(X)$-bimodule. A function on $X$ acts on a function on $PX$ by pointwise multiplication at either of the two endpoints of the path.

So much for now.