Physical interpretation of categorical structures related to Dirichlet Branes

Question

In Dirichlet Branes and Mirror Symmetry by Aspinwall et al, section 5.9 discusses various questions that remain open. In particular they say:

"There are many constructions from homological algebra which played little or no role in our physics discussion, such as the derived versions of tensor product, push-forward, and pull-back. Furthermore, while we made essential use of correspondences and derived autoequivalences, this was as monodromies, which are rather complicated operations from a physical point of view.

It might be useful to give more direct physical definitions of these constructions, both to flesh out the picture and to guide physics work in related contexts, such as the theory of Dirichlet branes with less or no supersymmetry."

They then go on give an example where such a physical picture is known (e.g. interpreting the derived correspondence from $\S 4.6.1$ in terms of a topologically twisted version of the defect line), but my question is:

Is anyone aware of further work that has been done in this direction? Has there been, at least partial, progress in creating a sort of categorical-construction-to-physics dictionary? Or, even better (as I feel that this is ultimately the direction this work will have to go), is anyone aware of applications of higher categories to the physics related to mirror symmetry?

(Of course, any references would be appreciated, but any inkling as to what we know in this direction is definitely enough! :) )

This post imported from StackExchange Mathematics at 2015-05-07 17:36 (UTC), posted by SE-user Ralph Mellish

Answer 1

The relation of derived categories to the physics of topological strings has been understood in full beauty via the results on "TCFT" and the absorption of these results into the classification of extended topological field theories via the proof of the cobordism hypothesis. Notably the (derived) tensor product structure on the objects of the derived category is at the very heart of this relation. Required reading on the route to enlightment here is Costello 04 and then Lurie 09, section 4.2.