Donaldson-Thomas invariants are the (virtual) Euler characteristics of moduli spaces of elements of the derived category of coherent sheaves (with some fixed Chern class, satisfying some stability condition, etc.) which bear some relation to "holomorphic Chern Simons theory", whatever that is.
Should I think of these as "A-model" or "B-model" invariants?
On the one hand, DT invariants come from the bounded derived category of coherent sheaves, which is what features in the B-model. On the other, there is the MNOP conjecture which tells me that the DT invariants of a CY 3-fold are “the same'' as the Gromov-Witten invariants of the SAME 3-fold, which are A-model things.
As I understand it, according to Costello, if I take the cyclic A-infinity category built out of d-b-coh and run it through his machinery, I should get the topological string corresponding to the B-model. But (modulo my confusion on the matter) according to Kontsevich and Soibelman, if I take the cyclic A-infinity category built out of d-b-coh and run it through their machinery, I should get DT invariants. So what is going on?
Best Answer
I got a (very) short answer to this question from Nikita Nekrasov who emailed me with "[DT theory is the] B model per se. The GW/DT correspondence is the duality between the A model and the B model (S-duality)".
I had not been aware that the GW/DT correspondence is an instance of S-duality. Recall that S-duality occurs in various contexts and is characterized by taking the coupling constant from one theory $g_{s}$ to the inverse $1/g_s$ in the other. So in the GW/DT correspondence, the GW expansion is when $g_s$ is small and (this point was clarified for me by Ooguri) the DT expansion in the variable $q=exp(-g_s)$ is valid when $1/g_s$ is small.