### (String) Physics from (Higher) Algebra, II

#### Posted by Urs Schreiber

Motivated by some private correspondence on the content of the previous entry, (String) Physics from (Higher) Algebra, I would like to improve some of the statements made there.

1) First of all, the phrasing in the previous entry suggested that the 2-category of bimodules internal to $\mathrm{BiMod}({R}_{2})$ is equivalent to the 2-category of left ${R}_{2}$-modules when Ostrik’s theorem applies. Actually, what is pretty obvious is only that the former sits inside the latter in a certain sense. That’s good enough for the general argument I was interested in. Still, I suspect that the two really are equivalent, but that’s not so obvious. I have tried to cleanly describe the situation and the corresponding conjecture in these notes:

$\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}$Module Categories and internal Bimodules

If anyone feels like helping to prove or disprove the conjecture stated at the very end of these notes, please let me know.

2) Second, the formulations in the previous entry didn’t make it clear which (2-)ring action the morphism in the various (2-)categories were supposed to respect. I give a new, somewhat more accurate re-formulation of the general idea below. Further refinements are obviously still necessary.

Finally, I should mention that Jeffrey Morton has a new preprint concerned with categorifying quantum mechanics:

Jeffrey Morton
**Categorified Algebra and Quantum Mechanics**

math.QA/0601458 .

Superficially this looks very different from what I am talking about here. But I think it might be related. However, I have not fully digested that paper yet.

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) $\u2102$-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 ($\u2102$-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 $\u2102$-$R$-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 $\u2102$-modules. So we can say quantum mechanics is a functor

where the target category is supposed to be that of $\u2102$-$R$ bimodules with morphisms respecting only the left $\u2102$ 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.

$\phantom{\rule{thinmathspace}{0ex}}$

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 $A(Q)$-$A(Q\prime )$ 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.

## Re: (String) Physics from (Higher) Algebra, II

I’d be interested to hear how you think your ideas are related to those in Morton’s paper.