The String Coffee Table

So here is the first revision of the idea outlined last time:

1) 1-QM

Start by formulating ordinary quantum mechanics (QM) in an nicely algebraic way. There is the worldline of a particle, described by some geometric category $P_1$ whose morphisms are 1-dimensional ‘spaces’ of some sort and whose composition is just gluing of such spaces.

To each object in $P_1$ we want to associate a Hilbert space $H$ of states. Being a vector space, let me regard $H$ as a (left, say) $\mathbb{C}$-module.

But $H$ has more structure. We also want the algebra of (bounded) observables being represented on it by bounded operators. We think of these as being some ring $R$ of ($\mathbb{C}$-valued) functions over configuration space. $H$ is hence supposed to be a (right, say) $R$-module, too.

So over each object of $P_1$ we want a $\mathbb{C}$-$R$-bimodule. (Of course this is just a right $R$-module, but for the following it helps to think of it this way.)

Now, to a morphism in $P_1$ we want to associate a linear operator, hence a morphism of left $\mathbb{C}$-modules. So we can say quantum mechanics is a functor

where the target category is supposed to be that of $\mathbb{C}$-$R$ bimodules with morphisms respecting only the left $\mathbb{C}$ action.

This is straightforward to categorify and doing so yields all kinds of known structures that are being used elsewhere or have been proposed elsewhere.

$\,$

2) 2-QM

Let $P_2$ be some geometric 2-category modelling some sort of surface elements and horizontal as well as vertical gluing of them.

The target 2-category should have

- objects being $C_2$-$R_2$ bimodules

where both $C_2$ and $R_2$ are “2-rings” but where $R_2$ is like a 2-ring of 2-functions with values in $C_2$.

That’s easy. A 2-ring should be an abelian monoidal category. Let’s fix any such category and call it $C_2$. If $C_2$ is sufficiently nice (see this entry on bimodules), we can apply Ostrik’s theorem and find that a (left) $C_2$-module is nothing but a category $C_A$ of right $A$-modules internal to $C_2$, where $A$ is some algebra internal to $C_2$.

That’s cool, because we also have the category ${}_A C_A$ of $A$-$A$-bimodules internal to $C_2$. This beast is also a 2-ring and it acts from the right on $C_A$. Hence $C_A$ is actually a

$C_2$ - ${}_A C_A$-(2-)bimodule

- morphisms in the target 2-category should be functors $C_A \to C_A$ which respect the left action by $C_2$ on $C_A$. By Ostrik’s theorem, this are nothing but (where ‘nothing but’ means up to the above mentioned conjecture) $A$-$A$-bimodules internal to $C_2$

- 2-morphisms are natural trafos between these, which are internal bimodule homomorphisms (again, up to the above conjecture, see the above mentioned notes for details).

If we call this target 2-category $T$ we have that 2-QM is a 2-functor

This is nice, for several reasons.

First, this concept actually appears (up to technicalities) in the work by Stolz and Teichner as a proposal for a formulation of 2D CFT. It is in principle known how it relates to path-integral prescriptions of field theory.

Second, I claim to have been able to prove large parts of my conjecture that “locally trivializing” such a 2-functor yields the structure of CFT formalism as it appears in the work by Fuchs-Runkel-Schweigert. See here for the latest version of this work.

Third, this does incorporate the approach proposed in HDAII, I think. This requires a little explanation:

One very nice categorical way to get internal modules is to use quivers. Let $Q$ be any category and $A(Q)$ its category algebra. An $A(Q)$-module in $C_2$ is nothing but a functor

An $A(Q)$-$A(Q')$ bimodule is a functor

We can identify $C_2$ with the category $C^{(1)}$ of functors from the category $(1)$ containing a single morphism to $C_2$. Hence we may regard $C_A$ as the category of functors

But if we regard $Q$ as a categorified version of a spacetime, this are nothing but the 2-functions considered in HDAII.

A module morphism is just a natural transformation between the functors $Q \to C_2$ . The module tensor product over $A$ is the functor $\mathrm{AntiHom}(-.-)$ which sends functors

$Q \to C_2$ and $Q^\mathrm{op} \to C_2$

to the space of antinatural transformations between them. This also serves as the categorified inner product on $C_A$, and hence really gives $C_A$ the structure of a 2-Hilbert space.

What is kind of remarkable is how the bimodule structure as well as the inner product actually become more natural (more functorial) after categorification.