The String Coffee Table

First, what is quantum mechanics?

Let $P_1$ be the worldline category. This has 0-dimensional manifolds as objects and 1-dimensional manifolds (equipped with extra structure as appropriate) stretching between these as boundaries.

Usually one thinks of quantum mechanics as a functor (the propagator) from $P_1$ to $\mathrm{Hilb}$. But this can be refined:

Let $R$ be the ring of (measurable, continuous, smooth,…, whatever) functions on the configuration space of the system under consideration. More generally, we should be able to generalize to the case where R is the ring of any scheme playing the role of configuration space.

The crucial point of QM is that $R$ is represented by bounded operators on some Hilbert space $H$, which hence plays the role of an $R$-module. Hence we can think of the QM propagator $p$ as a functor

from the worldline category to modules of the ring of functions over configuration space.

Using the monoidal structure of $R-\mathrm{Mod}$ (for commutative $R$) this applies similarly to perturbative quantum field theory (or many-particle QM). Here $P_1$ is replaced by ${\mathrm{Feyn}}_1$, the category of (unlabelled) ‘Feynman diagrams’ (1-dimensional cobordisms with merging/splitting).

Second quantization tells us to look at solutions of equations of motion inside objects in $R-\mathrm{Mod}$ which are acted on by the symmetry group $G$ of the system (e.g. the Poincaré group). We may thus think of $\mathrm{Rep}(G)$ as sitting ‘inside’ $R-\mathrm{Mod}$ (I suppose), such that a Feynman graph is a functor

up to renormalization issues.

This (well-known) point of view is nicely reviewed in section 2.6 of Bruce Bartlett’s thesis.

The point is that Ostrik’s theorem indicates that categorifying this situation yields strings described along the lines of enriched elliptic objects.

The natural categorification of a ring is an abelian monoidal category. Let’s fix such a category and call it $R_2$. As discusssed last time, a (2-)module for $R_2$ is a category $M$ with a functor

satisfying the usual axioms of a left action up to coherent isomorphism. These module categories $M$ live in a 2-category $R_2-\mathrm{Mod}$ of $R_2$-modules.

Hence we suspect that categorified QM is a 2-functor

from some 2-category of 2D-manifolds with boundary (and with additional structure as appropriate) to the 2-category of $R_2$-modules.

But using Ostrik’s result, we find that (under some mild conditions)

- objects in $R_2-\mathrm{Mod}$ are labelled by (Morita classes) of algebra objects $A,B,...$ internal to $R_2$.

and it follows by an easy argument that

- 1-morphisms in $R_2-\mathrm{Mod}$ are labelled by internal $A-B$-bimodules

- 2-morphisms in $R_2-\mathrm{Mod}$ are labelled by bimodule homomorphisms.

In other words, we have that $R_2-\mathrm{Mod}$ is essentially nothing but $\mathrm{Bimod}(R_2)$, the 2-category of bimodules internal to $R_2$. Hence the above 2-functor can be viewed as

which assigns algebras to points, bimodules to string segments and bimodule morphisms to pieces of worldsheet.

That’s the rough structure of 2D CFT as described here. I have indicated before how this should translate to other categorical descriptions of CFT. Hopefully I’ll someday also see the translation to Kevin Costello’s viewpoint.

One fun thing to note is that (finite, hereditary) algebras are encoded by ‘quiver diagrams’ (directed graphs, essentially), while their modules are encoded by functors from the free category of the quiver to $\mathrm{Vect}$. By ‘dimensional deconstruction’ we may think of at least some of these quivers as latticized spacetimes. But from this point of view functors from the quiver to $\mathrm{Vect}$ are like are a latticed baby version of functors

and hence of perturbabtive QFT of point particles. So from this point of view we may say that the 2-functor

assigns quivers to points, quiver representations to string segments and morphisms of quiver (bi-)representations to worldsheet segments

or even that it assigns latticized spacetimes to points, QFT’s to string segments (that’s very loosely speaking) and transformations between these to worldsheet segments.

I had previously made some comments on what this might mean in section 4.4 of my thesis. It’s speculation, but there seems to be some interesting pattern emerging here.