The n-Category Café

1) the setup

Because there are enough issues to discuss already in this case, I’ll concentrate here on a trivial background gerbe, however with arbitrary 2-connection. This is, on target space $$P_2(X)$$ a trivial line-2-bundle with connection, given by the smooth parallel transport 2-functor $$\mathrm{tra} : P_2(X) \to \mathrm{Bim}$$ which sends any 2-path $$\array{ & \nearrow\searrow^{\gamma} & \\ x &\Downarrow^\Sigma& y \\ & \searrow\nearrow_{\gamma'} & }$$ simply to $$\array{ & \nearrow\searrow^{\mathbb{C}} & \\ \mathbb{C} &\Downarrow^{e^{i \int_\Sigma B}}& \mathbb{C} \\ & \searrow\nearrow_{\mathbb{C}} & \,, }$$ where $B \in \Omega^2(X)$ is some globally (by simplifying assumption) defined 2-form.

I used to think of the open 2-particle as modeled by the category $$\mathrm{par} = (a \to b)$$ with two objects and a single nontrivial morphism connecting them. Since we are now approaching true string dynamics, there is need to refine this eventually to $$\mathrm{par} = P_1([0,\pi]) \,,$$ i.e. to the category of paths on the interval. But here I won’t explicitly get to the point where this distinction becomes crucial, so I suggest that we keep thinking of the string as looking like $$\mathrm{par} = (a \to b) \,,$$ as there are other issues to worry about.

The configuration space of this string is then that subcategory $$\mathrm{conf} \subset [\mathrm{par},P_2(X)]$$ of the category of cobordisms of $\mathrm{par}$ in $P_2(X)$ which contains only trivial morphisms, $$\mathrm{conf} := \mathrm{Disc}(\mathrm{Obj}([\mathrm{par},P_2(X)])) \,.$$ This expresses the fact that every path in target space is considered to be a distinct configuration of the string.

(This, too, is slightly more subtle if we do it in full detail: we really need to work in a setup where everything is locally smoothly trivializable, which will make us replace target space $P_2(X)$ by the equivalent 2-category of paths in the transition groupoid of a cover of $X$. This implies including jumps between patches as morphisms in $\mathrm{conf}$ (cf. p. 15 of Quantum 2-States: Sections of 2-vector bundles) ).

2) the 2-space of states

A section of our line-bundle is a morphism $$e : 1_* \to \mathrm{tra}_* \,,$$ of 2-functors $$1_*, \mathrm{tra}_* : \mathrm{conf} \to [\mathrm{par},\mathrm{Bim}] \,.$$ Here $1_*$ is the tensor unit in this monoidal 2-category of 2-functors, while $\mathrm{tra}_*$ is our Kalb-Ramond line 2-bundle restricted to configuration space. Since this simply coincides (by our simplifying assumption) with $1_*$, a section is really the same as an endomorphism $$e : 1_* \to 1_* \,,$$ Hence these 2-sections are, in the present setup, mere 2-functions (i.e. 2-wave functions, if you wish. This is nothing but the 2-analog of the statement that a (complex) function is nothing but an endomorphism of the trivial line bundle.)

By the magic of arrow-theoretic quantum theory, these 2-wave functions know automatically about the Chan-Paton degrees of freedom over the endpoints of the string: that’s because such a morphism, $e$, of 2-functors is nothing but, over each configuration $$(x \stackrel{\gamma}{\to} y) \in \mathrm{Mor}(P_2(X))$$ of the string, a 2-cell $$\array{ \mathbb{C} & \stackrel{\mathrm{Id}}{\to} & \mathbb{C} \\ V_x \downarrow \;\; &\Downarrow e(\gamma)& \;\; \downarrow V_y \\ \mathbb{C} & \stackrel{\mathrm{Id}}{\to} & \mathbb{C} }$$ in $\mathrm{Bim}$.

Suppose first that $V_x$ and $V_y$ were the identity morphism everywhere. Then this would be nothing but a complex function on path space. Once measures enter the game we’d get something like $L^2(P X)$ from that.

But $V_x$ and $V_y$ may be any other $\mathbb{C}$-bimodules, hence any vector spaces. Choosing a basis of these, we see that $e(\gamma)$ now is a matrix of complex numbers. The components of such a matrix would be addressed as Chan-Paton components of the wave function of the open string in – for historical reasons and therefore in almost every string theory introducory text.

In effect, we find that our 2-wave function is in fact three things in one:

a) a vector bundle over $X$ associated with the left endpoint, $a$, of the string

b) a vector bundle over $X$ associated with the right endpoint, $b$, of the string

c) for each path $x \stackrel{\gamma}{\to} y$ a morphism of the respective fibers of these vector bundles over the endpoints.

(I am again glossing over a subtlety: a priori these vector bundles live over path space. For topological reasons these may just as well be thought of as vector bundles over $X$ itself. This is an issue which requires further discussion. Maybe I’ll do that later. For the moment I just assume, by decree, that all my vector bundles on paths are pulled back via one of the endpoint restriction maps from target space $X$.)

This way, we understand 2-wave functions of the open string as elements of the space of sections $$\mathrm{Hom}_{[\mathrm{conf},[\mathrm{par},\mathrm{Bim}]]}(1_*,\mathrm{tra}_*) \,.$$ By the quantization prescription, we are asked to find the quantum functor $$q(\mathrm{tra}) : \mathrm{par} \to \mathrm{Bim}$$ which assigns the 2-space of states to the parameter space of the string, such that $$\mathrm{Hom}_{[\mathrm{conf},[\mathrm{par},\mathrm{Bim}]]}(1_*,\mathrm{tra}_*) \simeq \mathrm{Hom}_{[\mathrm{par},\mathrm{Bim}]}(1,q(\mathrm{tra}))$$ i.e. such that we identify 2-wave functions with 2-states $$e \stackrel{\sim}{\mapsto} \psi_e \,.$$

I think this yields $$q(\mathrm{tra}) : (a \to b) \mapsto (C(X) \stackrel{C(P X)}{\to} C(X)) \,.$$

Here $C(X)$ denotes the algebra of functions on $X$, $C(P X)$ the algebra of functions on path space. The latter is regarded as a bimodule for the former by multiplication over the respective endpoint.

To check this, it is convenient to first look at cosections $$\psi^\dagger : q(\mathrm{tra}) \to 1$$ which are given by 2-cells in $\mathrm{Bim}$ of the form $$\array{ C(X) &\stackrel{C(P X)}{\to} & C(X) \\ \psi_a \downarrow\;\; &\Downarrow^{\psi(a\to b)}& \;\;\downarrow \psi_b \\ \mathbb{C} &\stackrel{\mathbb{C}}{\to}& \mathbb{C} } \,.$$ Since a $C(X)$-module (at least if finitely generated and projective – I guess I am currently glossing over such technical details) is the same as (the space of sections of) a vector bundle on $X$, we see that $\psi(a \to b)$ is indeed, over each path in $X$, a morphism between the fibers over the two endpoints.

3) evolution over the disk

With the space of states in hand, we want to find the action of “time propagation” on it.

Recall what this problem looks like in quantum mechanics:

There the space of states is just an ordinary vector space $H$, and a state is a morphism $$\array{ \mathbb{C} \\ \psi \downarrow \;\; \\ H } \,.$$ Dirac invented the habit of drawing these arrows in a slightly ideosyncratic (but by now fairly standard ;-) notation as $$|\psi\rangle \,.$$ Evolving the quantum system over a piece of worldline of length $t$, amounts to postcomposing this arrow with the respective operator $$\array{ H \\ U(t) \downarrow \;\; \\ H }$$ otherwise known as $U(t) = e^{i t \Delta}$.

The result of this, $$\array{ \mathbb{C} \\ \psi \downarrow \;\; \\ H \\ U(t) \downarrow \;\; \\ H } \,,$$ is what Dirac suggested to write as $$H | \psi \rangle \,.$$

In order to measure what this result is like, with respect to some reference co-state $$\array{ H \\ \phi^\dagger \downarrow \;\; \\ \mathbb{C} }$$ we would form the pairing $$\array{ \mathbb{C} \\ \psi \downarrow \;\; \\ H \\ U(t) \downarrow \;\; \\ H \\ \phi^\dagger \downarrow \;\; \\ \mathbb{C} } \,,$$ which Dirac would write as $$\langle \phi | U(t) | \psi \rangle \,.$$

This is a number, which we address as the 2-point 1-disk amplitude over a 1-disk (an interval) of modulus $t$.

By categeorification, an analogous procedure produces for us the 2-point 2-disk amplitude of the string.

The incoming 2-state $$\array{ \mathbb{C} & \stackrel{\mathrm{Id}}{\to} & \mathbb{C} \\ \psi_a \downarrow\;\; &\Downarrow^{\psi(a\to b)}& \;\;\downarrow \psi_b \\ C(X) &\stackrel{C(P X)}{\to}& C(X) } \,,$$ which we might allow ourselves to denote $$| \psi \rangle$$ will be postcomposed with a 2-cell $$\array{ C(X) &\stackrel{C(P X)}{\to}& C(X) \\ U(t)_a \downarrow\;\; &\Downarrow^{U(t)_{a\to b}}& \;\;\downarrow U(t)_b \\ C(X) &\stackrel{C(P X)}{\to}& C(X) }$$ that encodes the propagation over a disk of modulus $t$ (now we are almost being bitten by our slightly too crude model for the parameter space of the string – but don’t worry for the moment).

The result $$U(t) | \psi \rangle$$ we pair with an outgoing co-2-section $$\array{ \mathbb{C} & \stackrel{\mathrm{Id}}{\to} & \mathbb{C} \\ \psi_a \downarrow\;\; &\Downarrow^{\psi(a\to b)}& \;\;\downarrow \psi_b \\ C(X) &\stackrel{C(P X)}{\to}& C(X) \\ U(t)_a \downarrow\;\; &\Downarrow^{U(t)_{a\to b}}& \;\;\downarrow U(t)_b \\ C(X) &\stackrel{C(P X)}{\to}& C(X) \\ \phi_a \downarrow\;\; &\Downarrow^{\phi(a\to b)}& \;\;\downarrow \phi_b \\ \mathbb{C} &\stackrel{\mathbb{C}}{\to}& \mathbb{C} }$$ to get $$\langle \phi | U(t) | \psi \rangle \,.$$

Now something happens that Dirac could not have foreseen (on the other hand: what do I know about what a mind like Dirac could have foreseen!): at this point $$\langle \phi | U(t) | \psi \rangle$$ could be the propagation over a disk-like worldhseet, with boundary sitting on some D-brane, just as well as propagation over just a slice of worldsheet – say over a sliced cylinder!

In the former case we’d have to do a second order pairing of sections. This amounts to gluing 2-cells to the left and right boundary of the above diagram that encode the propation of the endpoint of the string along the boundary of the disk, coupling to the non-abelian Chan-Paton vector bundle, as it were.

In the latter case we would have to perform a 2-trace to glue the $b$ to the $a$ boundary.

But first, we need to find the 2-propagator $$\array{ C(X) &\stackrel{C(P X)}{\to}& C(X) \\ U(t)_a \downarrow\;\; &\Downarrow^{U(t)_{a\to b}}& \;\;\downarrow U(t)_b \\ C(X) &\stackrel{C(P X)}{\to}& C(X) } \,.$$

This is, by definition, such that its action on the 2-space of states by postcomposition reproduces the pull-push quantum propagation action on this space.

To get this started, we define a worldsheet $$\mathrm{worldvol} = \left( \array{ a &\stackrel{t}{\to}& a' \\ \downarrow &\Leftarrow& \downarrow \\ b &\stackrel{t}{\to}& b' } \right)$$ of length $t$, with the two obvious injections $$\array{ & \mathrm{worldvol} & \\ \mathrm{par}\nearrow && \nwarrow \mathrm{par} }$$ of our string into it.

Then we turn the crank and let things flow their way.

5) a little bit on the result

I think I claim that as a result of turning the crank, we find that the propagator $$\array{ C(X) &\stackrel{C(P X)}{\to}& C(X) \\ U(t)_a \downarrow\;\; &\Downarrow^{U(t)_{a\to b}}& \;\;\downarrow U(t)_b \\ C(X) &\stackrel{C(P X)}{\to}& C(X) }$$ has vertical morphisms that are again the algebra of functions on path space $C(P X)$ (we need to be a little careful with what kinds of paths we have here, though).

This makes the propgator an endohomomorphism of the bimodule of functions on path space. $$\array{ C(X) &\stackrel{C(P X)}{\to}& C(X) \\ C(P X) \downarrow\;\; &\Downarrow^{U(t)_{a\to b}}& \;\;\downarrow C(P X) \\ C(X) &\stackrel{C(P X)}{\to}& C(X) } \,.$$ I think this morphism is given by the path integral over all surfaces cobounding these in- and outgoing path, weighted by the respective kinetic measure and as well as the coupling term to the $B$-field.

But, having written so much already, I feel exhausted now and will stop here.