# Open-closed string correspondence

Recently, after many years of searching for the right source, I came across the excellent lecture by Aspinwall, "Some Applications of Commutative Algebra to String Theory", in Eisenbud's Festschrift. He presents a coherent exposition explains branes, TQFTs and all the stuff that you, perhaps, like me, always wanted to know, but were afraid to ask.

In particular, he mentions the boundary conditions of open strings as carrying the structure of triangulated category, such as the familiar derived categories of Lagrangians and coherent sheaves. On the other hand, I found out recently that there is a classical example of triangulated category, that of stable homotopy types (say, like in this excellent exposition by Drozd).

Now, a few years ago, Lipschitz and Sarkar have constructed the stable homotopy type of Khovanov homology, and used a modification of it to, in turn, construct a of refinement of Rasmussen's s-invariant.

Lipshitz and Sarkar, as mentioned above, have already constructed two homotopy types that compute two different version of Khovanov homology, the original and a Lee-Rasmussen deformation. I expect there to be a whole array of such objects, corresponding to different deformations. The morphisms between them I expect to be the Frobenius extensions, as per Khovanov, and each such extension will lead, with any luck, to a triangle.

Finally, the question:

Are there papers in physics that cover triangulated structures on collections of TQFT's, i.e., the correspondence between closed string TQFT's and D-branes?

EDIT: I added a link to the correct paper of Lipshitz and Sarkar (the original link was to a precursor paper where they constructed the Steenrod square on the Khovanov homology).

EDIT 2: I tried to add a little more clarity. Specifically, I am looking for an open string TQFT that assigns a closed string TQFT to a boundary.

EDIT 3 I was going to radically rewrite the question, but I don't know if it's allowed, so here's an edit instead:

Consider a stable category, associated to a given commutative Frobenius algebra $A$. This has a triangulated structure. I would imagine that the subcategory $B$ of commutative Frobenius algebras over $A$ inherits a triangulated structure as well. Does a category like $B$ appear as a category of boundary conditions for an open string TQFT anywhere in the physical literature?