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February 7, 2007

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 nn-functor tra:tarphas. \mathrm{tra} : \mathrm{tar} \to \mathrm{phas} \,. An nn-particle is determined by its shape par\mathrm{par} and its configurations parγconftar. \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 nn-functor: q(tra):parphas. 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 tra\mathrm{tra} through the correspondence conf×par ev tar par. \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”) par=Π 2(S 2)\mathrm{par} = \Pi_2(S^2) propagating on tar=Σ(INN(String k(G)))\mathrm{tar} = \Sigma(\href{}{\mathrm{INN}(\mathrm{String}_k(G))}) and coupled to a 2-gerbe tra:tar1d3Vect\mathrm{tra} : \mathrm{tar} \to \href{}{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”) par=Π 1(S 1)\mathrm{par} = \Pi_1(S^1), propagating on tar=Σ(String k(G))\mathrm{tar} = \Sigma(\href{}{\mathrm{String}_k(G)}) and coupled to a trivial 1-gerbe tra:tarBim2Vect\mathrm{tra} : \mathrm{tar} \to \href{}{\mathrm{Bim} \hookrightarrow 2\mathrm{Vect}} in

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

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

For the ordinary 1-particle par=\mathrm{par} = \bullet propagating on some space tar=P 1(X)\mathrm{tar} = P_1(X) and coupled to an ordinary vector bundle tra:traVect\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 par=(ab) \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 par=(ab)\mathrm{par} = (a\to b) to a nontrivial line 2-bundle transport tra:P 2(X)1d2Vect\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 aa and bb 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:1tra, E : 1 \to \mathrm{tra} \,, where tra\mathrm{tra} is the 2-anafunctor respresenting the gerbe, and 11 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 par=(ab)\mathrm{par} = (a \to b) coupled to the trivial line-2-bundle 1:P 2(X)2Vect1 : 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 XX and a morphism between two vector bundles on XX.

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

So the exercise is this:

Exercise: Determine the pull-push through conf×par ev tar par \array{ && \mathrm{conf}\times \mathrm{par} \\ & \multiscripts{^{\mathrm{ev}}}{\swarrow}{} && \searrow \\ \mathrm{tar} &&&& \mathrm{par} } for the charged 2-particle (parγconftartraphas) \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 tar=P 2(X), \mathrm{tar} = P_2(X) \,, the 2-groupoid of 2-paths in a manifold XX (“spacetime”). The parallel transport is tra:P 2(X)Bim2Vect, \mathrm{tra} : P_2(X) \to \mathrm{Bim} \hookrightarrow 2\mathrm{Vect} \,, the trivial line-2bundle on XX, i.e. tra:( x y )( Id Id Id) \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 par=(ab). \mathrm{par} = (a \to b) \,. The space of its configurations conf[(ab),P 2(X)] \mathrm{conf} \subset [(a \to b), P_2(X)] is the discrete 2-category conf=Disc(Obj([(ab),P 2(X)])), \mathrm{conf} = \mathrm{Disc}(\mathrm{Obj}([(a \to b), P_2(X)])) \,, which regards all paths in XX 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.


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), e = (E_1, E_2, f) \,, where E 1E_1 and E 2E_2 are vector bundles on XX, and where ff is a map that sends each path xγyx \stackrel{\gamma}{\to} y in XX to a morphism f(γ):E 1(x)E 2(y) 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(tra):(ab)Bim2Vect q(\mathrm{tra}) : (a \to b) \to \mathrm{Bim} \hookrightarrow 2\mathrm{Vect} which is such that its states ψ:1q(tra) \psi : 1 \to \mathrm{q}(\mathrm{tra}) are equivalent to these ee.

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

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

So much for now.

Posted at February 7, 2007 5:46 PM UTC

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