### Picard and Brauer 2-Groups

#### Posted by Urs Schreiber

Picard and Brauer groups of representation categories of vertex algebras encode symmetries of full CFTs ($\to $, $\to $). In the CFT context the elements of the Picard group are called *simple currents* (e.g. Int. J.Mod.Phys. A 5 (1990) 2903).

It is rarely explicitly admitted that the Picard group is really a *2-group* ($\to $), and so is the Brauer group.

Even better, both fit into a certain 3-group. All this has been well known to a handful of experts, apparently ($\to $) - which of course does not stop me from enjoying rediscovering this from my personal point of view.

Namely, this 3-group is closely related to 2-dimensional field theory, and in particular to CFT. The fact that it is an $n$-group for precisely $n=3$ is closely related to the *holography*-like phenomenon that 2D CFT can be described in terms of 3D TFT.

Before talking about the big 3-group, let me describe how the Picard group and the Brauer group look like as 2-groups. This is nicely described in

Enrico M. Vitale
*A Picard-Brauer exact sequence of categorical groups*

pdf.

The Picard 2-group $\mathrm{Pic}(C)$ of a monoidal category $C$ is the 2-group of invertible objects of $C$.

There is a better way to say this: let $\Sigma (C)$ be the suspension of the monoidal category $C$, which is the (possibly weak) 2-category with a single object $\u2022$, 1-morphisms the objects of $C$ and 2-morphisms the morphsims of $C$. Then

The

Picard 2-groupof $C$ is the automorphism 2-group of the single object $\u2022$ in the suspension of $C$:(1)$$\mathrm{Pic}(C)={\mathrm{Aut}}_{\Sigma (C)}(\u2022)\phantom{\rule{thinmathspace}{0ex}}.$$

What is ordinarily called the Picard group is the decategorification of $\mathrm{Pic}(C)$, namely the group ${\pi}_{0}(\mathrm{Pic}(C))$ of isomorphism classes of objects of $\mathrm{Pic}(C))$.

A special subclass of Picard 2-groups are the Brauer 2-groups.

Consider the bicategory $\mathrm{BiMod}(C)$ of bimodules of algebra objects internal to $C$. Its objects are algebras, its 1-morphisms are bimodules and its 2-morphisms are bimodule homomorphisms.

When $C$ is not just monoidal, but *braided monoidal* (experts will take this as a first hint of a secret 3-categorical structure) we get a monoidal structure - now on the bicategory $\mathrm{BiMod}(C)$ (the periodic table of $n$-categories is at work here).

Taking 2-isomorphism classes in $\mathrm{BiMod}(C)$, thus obtaining a monoidal category called $\mathrm{cl}(\mathrm{BiMod}(C))$, we can again form a Picard 2-group. This is the Brauer 2-group:

The

Brauer 2-group$\mathrm{B}(C)$ is the Picard 2-group of 2-isomorphism classes of bimodules internal to $C$.(2)$$\mathrm{B}(C):=\mathrm{Pic}(\mathrm{cl}(\mathrm{BiMod}(C)))\phantom{\rule{thinmathspace}{0ex}}.$$

The fact that I used one suspension in the definition of the Picard 2-group, and one taking of isomorphism classes in the definition of the Brauer 2-group all indicates that the 2-group nature of Picard and Brauer is still not the full truth.

The need to go to three dimensions is nicely seen in the the way 2D CFT correlators are written in terms of Wilson graphs decorated in a modular tensor category ($\to $).

Take a worldsheet, pick a dual triangulation, label the resulting graph in some tensor category. One quickly finds (see section 2.3 of $\to $ for an illustration) that one ultimately needs to pass some of the graph’s edges “beneath” or “above” each other - to “braid” them. Visually, this requires a third dimension. But this is not just heuristics, there is a precise theorem behind this:

Every

braidedmonoidal category $C$ has a double suspension $\Sigma (\Sigma (C))$, i.e. a 3-category with a single object and a single 1-morphism.

(See for instance page xx here.)

Now realize that $C$ can be thought of as sitting embedded inside its own 2-category of internal bimodules

where each object of $C$ is regarded as a bimodule over the trivial algebra constituted by the tensor unit in $C$.

Hence, by simply generalizing to arbitrary bimodules the way that $C$ sits inside a 3-category, we get the full 3-catgegoy

(recall that we are assuming $C$ to be braided monoidal for this to make sense!).

This is the 3-category John Baez describes in TWF 209.

(And, guess what, I am once again running out of time…)

## Re: Picard and Brauer 2-Groups

I wonder: is this 3-category equivalent to a strict one? The unit group, the Picard group, and the Brauer group of A are the 0th, 1st, and 2nd cohomology groups of the sheaf O^* on Spec A. It’s interesting that the same groups appear in this 3-category upside-down. Behind the cohomology interpretation there is of course a chain complex. From one point of view or another a chain complex is some kind of *strict* infinity-category, right?

D.