Mathematics

A system of Bethe-Ansatz type equations, which specify a unique array of Young diagrams responsible for the leading contribution to the Nekrasov partition function in the $\epsilon_2\rightarrow 0$ limit is derived. All the expectation values of the chiral fields $\langle \text{Tr} \phi^J \rangle $ are simple symmetric functions of the column lengths. An entire function $Q(z)$ whose zeros are determined by the column lengths is introduced. It satisfies a functional equation, resembling Baxter's T-Q equation in 2d integrable models. This functional relation directly leads to a nice generalization of the equation defining Seiberg-Witten curve. A sort of Fourier transform maps the difference equation for $Q(z)$ into a differential equation which can be identified with the equation satisfied by 2d CFT conformal blocks including a degenerate field. Using AGT relation, in the case of the gauge group $U(2)^r$ and generic $\Omega$ background we find a difference-differential equation for Q(z) generalizing Baxter's T-Q relation.