# Relation between Donaldson invariants and GW invariants

What is known about the relation of Donaldson invariants on a complex surface $\Sigma$ and GW invariants (or equivalent) of local Calabi-Yau 3folds such as the canonical bundle of $\Sigma$? (if any of course)

I would like to understand any such relation either from a physics or a maths point of view.

Are there any known relations, results, formulae, papers? Any kind of references would be very much appreciated.

• At least physically, Donaldson is doing a 4d field theory with source $\Sigma$ while GW is doing a 2d field theory with target $\Sigma$. – AHusain Mar 19 '17 at 8:54
• Thanks, that is quite incorrect. $\Sigma$ for Donaldson has to be a 4 manifold while for GW (at least in the classical sense) has to be a 3fold. The connection should somehow arise from M-theory construction as $\mathbb{R} \times M_4 \times CY_3$. – Gorbz Mar 19 '17 at 12:45
• No, need to be M-theory, just leave it as topological string. But I'm drawing distinction between string/M theories and ordinary field theories because of world/brane-sheet formalism. Also, yes forgot to make the target a 3-fold. – AHusain Mar 19 '17 at 22:25
• Target does not necessarily need to be 3-fold. For M5 brane 6d (2,0) theory though I think it does. This is where upon proper constructions one can relate Nekrasov part. function on $\mathbb{R}^4$ and GW with target the 3fold with time direction remaining. I cannot find any reference for this story though. – Gorbz Mar 20 '17 at 9:29
• Let's form a chat to discuss separately. See on chat.stackexchange.com with title this question. – AHusain Mar 22 '17 at 23:01