The String Coffee Table

Actually, I won’t talk about $\mathrm{String}(n)$ here at all, but instead try to make some general facts about 2-reps explicit. I don’t claim that the following is particularly deep. In fact, the considerations are quite elementary. The motivation for spelling this out is that by slightly enriching the following discussion (in particular by replacing vector spaces by Hilbert spaces) this should describe 2-reps of the $\mathrm{String}(n)$-2-group in terms of bimodules of vonNeumann algebras.

Claim: Every faithful representation

of any group $H$ on a vector space $V$ (over some field) gives rise in a canonical way to a 2-representation

of the automorphism 2-group of $H$ on a 2-vector space (a module category over $\mathrm{Vect}$).

This representation factors through the 2-category of bimodules.

The simple idea is this. Last time ($\to$) I tried to indicate how every faithful representation $\rho$ of $H$ on the vector space $V$ gives rise to a 2-representation of $\mathrm{Aut}(H)$ on $\mathrm{Im}(\rho)$ by

- sending an object $g$ of $\mathrm{Aut}(H)$ (an automorphism of $H$) to the autofunctor

- sending a morphism $g \overset{h}{\to} g'$ of $\mathrm{Aut}(H)$ to the natural isomorphism given by these naturality squares:

But we may want a representation on a proper 2-vector space. By this I shall mean, somewhat loosely, any module category over some monoidal category $C$.

Here we can choose $C = \mathrm{Vect}$. There is a close relation between the 2-category of module categories and the 2-category of bimodules of algebras in $\mathrm{Vect}$ ($\to$). So let’s first construct a representation in terms of bimodules from the above one.

This is obvious. We let $A_\rho \subset \mathrm{End}(V)$ be the algebra generated from the operators in $\mathrm{Im}(\rho)$ – nothing but the category algebra of $\mathrm{Im}(\rho)$

(In the more general case where we have representations on infinite dimensional Hilbert spaces we’d take the double commutant of $\mathrm{Im}(\rho)$ and obtain a vonNeumann algebra $A_\rho$.)

The above autofunctors sending $\rho(h)$ to $\rho(g(h))$ extend to automorphisms

of this algebra. By a standard construction, from every morphism $A \overset{\phi}{\to} A'$ of algebras we obtain an $A$-$A'$ bimodule

which, as a vector space, is simply $A'$ itself, where the right action by $A'$ is simply the product in $A'$ and where the left action by $A$ is obtained by first sending $A$ to $A'$ using $\phi$ and then acting from the left on $A'$ by multiplication.

Hence for any two objects $g$ and $g'$ of $\mathrm{Aut}(H)$ we obtain two $A_\rho$-$A_\rho$ bimodules $N_g = (A_\rho,\phi_g)$ and $N_{g'} = (A_\rho,\phi_{g'})$.

Now, one can easily check that for every morphism $g \overset{h}{\to} g'$ in $\mathrm{Aut}(H)$ we get a homomorphism of $A_\rho$-$A_\rho$-bimodules

by multiplying from the left with $\rho(h)$. The condition for this to preserve the left $A_\rho$ action is precisely the commutativity of the above naturality squares. (The right action is preserved trivially.)

One also checks that horizontal and vertical composition of bimodule homomorhisms $N_g \overset{\phi_h}{\to} N_{g'}$ reproduces the horizontal and vertical composition in $\mathrm{Aut}(H)$ and $\mathrm{Aut}(\mathrm{Im}(\rho)))$.

Finally, we can regard homomorphisms of bimodules as 2-morphisms in ${\mathrm{Vect}}\mathrm{Mod}$ by a standard construction ($\to$).

Spelled out, the 2-representation

obtained this way from $\rho : H \to \mathrm{Vect}$ works as follows.

The 2-group is represented on the 2-vector $C = \mathrm{Mod}_{A_\rho}$, which is the category of right modules over the algebra $A_\rho = \langle \rho(h) \;|\; h \in H\rangle$.

Every object $g \in \mathrm{Obj}(\mathrm{Aut}(H))$ is sent to the $\mathrm{Vect}$-linear functor

Every morphism $g \overset{h}{\to} g'$ \in \mathrm{Mor}(\mathrm{Aut}(H)) is sent to the obvious natural isomorphism of these functors. (See the diagram at the end of these notes.)