André Henriques speaks the about the following question (my paraphrase):

What is a definition of *2-trace* that can be applied to a suitable surface diagram of 2-vector spaces in analogy to how the ordinary trace can be applied to a suitable string diagram of (finite dimensional) vector spaces?

Or rather: given an evident guess for a definition, is it well defined?

This question arises for instance when you have a parallel 2-transport or extended 2d QFT with values in 2-vector spaces

$P_2(X) \to 2 Vect$

and want to compute its higher holonomy.

Since the 2-category $2 Vect$ over some field $k$ is equivalent to that of $k$-algebras, algebra bimodules between these as 1-morphisms, and bimodule homomorphisms between those as 2-morphisms, this question more concretely reads as follows:

suppose you have a surface, say a torus, tringaulated in an oriented way, or otherwise cell-decomposed, with

an algebra $A_v$ assigned to each vertex $v$;

an $A_{v_1}-A_{v_2}$-bimodule $N_e$ assigned to each edge $e$ between vertices $v_1$ and $v_2$;

a bimodule homomorphism $\phi_\Sigma : N_{e_1} \to N_{e_2}$ for every 2-cell $\Sigma$ with incoming boundary $e_1$ and outgoing boundary $e_2$.

Then a *double trace* should send this to a number by producing tensor product of bimodules along vertices and traces of bimodule homomorphisms, at least provided that the bimodules involved are suitably “trace class”.

André gives a comparatively simple example of this for the case of a 2-torus.

The next question is how to formalize what it means for a morphism of 2-vector spaces, hence for a bimodule, to be “trace class”.

To address this, it should help to first see how “trace class” can be defined generally in monoidal categories. This questions was addressed and answered in

- Stephan Stolz, Peter Teichner,
*Traces in monoidal categories* (arXiv:1010.4527)

André says he is generalizing the kind of argument given there now to the 2-dimensional case, terming the concept “double trace class” for the moment. This involves drawing some nice surface diagrams, which I can’t try to indicate here.

After this definition, André turns to the intended application to (extended) 2-dimensional super-conformal field theory. Here we think of the above bimodules as spaces of states of the theory, and the bimodule homomorphisms as correlators. The 2-trace of these on a given surface is going to be the partition function of the theory. Specifically for the heterotic string this is going to be a modular form on the moduli of the given torus.

So I asked how this relates to

- Kate Ponto, Michael Shulman,
*Shadows and traces in bicategories* (arXiv:0910.1306)

André says its similar, but probably a little different. Certainly the “shadow” he uses is not defined globally on the 2-category as in Ponto-Shulman – that’s precisely what the “2-trace class” condition above is about. But to which extent the concepts coincides once this is taken care of seems to be unclear at the moment.

## Re: Strings and Automorphic Forms in Topology

Is there a path from your material to the other term in the conference title –

automorphic forms?Hmm, $n$Lab has a link to automorphic form, e.g. from Langlands program, but no entry yet.

But the conference description specifies

topologicalautomorphic forms, and we certainly don’t have anything on them. Topological automorphic forms by Behrens and Lawson seems to be the standard referenceSo does the topic arise at this conference through something like a parallel for TAF of Stolz-Teichner’s rendition of TMF in terms of supersymmetric Euclidean field theories?