Darboux coordinates, Yang–Yang functional, and gauge theory

The moduli space of flat $SL_2$ connections on a punctured Riemann surface $\Sigma$ with fixed conjugacy classes of the monodromies around the punctures is endowed with a system of holomorphic Darboux coordinates in which the generating function of the variety of $SL_2$-opers is identified with the universal part of the effective twisted superpotential of the corresponding four-dimensional $\mathcal{N}=2$ supersymmetric theory subject to the two-dimensional $\Omega$-deformation. This allows defining the Yang–Yang functionals for the quantum Hitchin system in terms of the classical geometry of the moduli space of local systems for the dual gauge group and relating it to the instanton counting of the four-dimensional gauge theories in the rank-one case.