This is a ver very good answer. I am wondering if you could show the same for the $\mathcal{N}=4$ twist though. There things are harder since the global rotation group is $SU(2)_L \times SU(2)_R \times SU(4)_R$ which can be written as $SU(2)_L \times SU(2)_R \times SU(2)_A \times SU(2)_B \times U(1)$. Now, I do not see how exactly the spinor $(2,1,\bar{4}) \oplus (1,2,4)$ representation changes (that is how the supercharges change under the twist).

Hi, thanks. I know that book. My confusion is in what sense the "cobordism" is equal to the SW invariants in 4d. These are polynomial invariants. On the other hand homologies are vector spaces. How are polynomial invariants categorification of a vector space, I thought that that should be a 2 category!

@ChrisGerig Firstly homology should be categorified to a 2-category if I am correct. Secondly physically what is this cobordism in 4d? Any references?

@AHusain I actually cannot find the chat you mentioned. If you see this message please let me know. Thanks a lot!

Hi, I just saw this comment. Sure, I am up for a chat although I have not used it before.

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.

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 $\mathbb{R} \times M_4 \times CY_3$.

Thanks for the answer. I feel a bit confused by the notion of quantum product though. Its it possible to define it (or give reference for it)? Does it correspond to some product used by physicists?