The n-Category Café

QFT of Charged n-particle: Chan-Paton Bundles

Posted by Urs Schreiber

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 $$\array{ && \mathrm{conf}\times \mathrm{par} \\ & \multiscripts{^{\mathrm{ev}}}{\swarrow}{} && \searrow \\ \mathrm{tar} &&&& \mathrm{par} } \,.$$

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 $\mathrm{tar} = \Sigma(\href{http://golem.ph.utexas.edu/category/2006/11/chernsimons_lie3algebra_inside.html}{\mathrm{INN}(\mathrm{String}_k(G))})$ and coupled to a 2-gerbe $\mathrm{tra} : \mathrm{tar} \to \href{http://golem.ph.utexas.edu/category/2006/11/the_1dimensional_3vector_space.html}{1d3\mathrm{Vect}}$, 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 $\mathrm{tar} = \Sigma(\href{http://golem.ph.utexas.edu/string/archives/000547.html}{\mathrm{String}_k(G)})$ and coupled to a trivial 1-gerbe $\mathrm{tra} : \mathrm{tar} \to \href{http://golem.ph.utexas.edu/category/2006/12/my_personal_spy_has_just.html#c006474}{\mathrm{Bim} \hookrightarrow 2\mathrm{Vect}}$ in

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

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 $\mathrm{par} = \bullet$ 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 $\array{ && \mathrm{conf}\times \mathrm{par} \\ & \multiscripts{^{\mathrm{ev}}}{\swarrow}{} && \searrow \\ \mathrm{tar} &&&& \mathrm{par} }$ 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. $$\mathrm{tra} : \left( \array{ & \nearrow \searrow \\ x &\Downarrow& y \\ & \searrow \nearrow } \right) \mapsto \left( \array{ & \nearrow \searrow^\mathrm{Id} \\ \mathbb{C} &\;\;\Downarrow^\mathrm{Id}& \mathbb{C} \\ & \searrow \nearrow_{\mathrm{Id}} } \right)$$ (“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 $$q(\mathrm{tra}) : (a \to b) \mapsto ( C(X) \stackrel{C(P X)}{\to} C(X) ) \,.$$

Here $C(X)$ is the algebra of (complex) functions on $X$ and $C(P X)$ 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 $P X$ by pointwise multiplication at either of the two endpoints of the path.

So much for now.