Exact partition functions for the $\Omega$-deformed $\mathcal N=2^{*}$ $SU(2)$ gauge theory

Understand

We study the low energy effective action of the $\Omega$-deformed $\mathcal N =2^{*}$ $SU(2) $ gauge theory.

• It depends on the deformation parameters $\epsilon_{1},\epsilon_{2}$, the scalar field expectation value $a$, and the hypermultiplet mass $m$.
• We explore the plane $(\frac{m}{\epsilon_{1}}, \frac{\epsilon_{2}}{\epsilon_{1}})$ looking for special features in the multi-instanton contributions to the prepotential, motivated by what happens in the Nekrasov-Shatashvili limit $\epsilon_{2}\to 0$.
• We propose a simple condition on the structure of poles of the $k$-instanton prepotential and show that it is admissible at a finite set of points in the above plane.