Advances in Theoretical and Mathematical Physics

Volume 24 (2020)

Number 7

The first of two special issues in honor of Cumrun Vafa’s 60th birthday

Opers, surface defects, and Yang-Yang functional

Pages: 1789 – 1916



Saebyeok Jeong (C.N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, New York, U.S.A.)

Nikita Nekrasov (C.N. Yang Institute for Theoretical Physics and Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, New York, U.S.A.)


We explore the non-perturbative Dyson–Schwinger equations obeyed by the partition functions of the $\Omega$-deformed $\mathcal{N}=2, d=4$ supersymmetric linear quiver gauge theories in the presence of surface defects. We demonstrate that the partition functions of different types of defects (orbifold or vortex strings) are related by analytic continuation. We introduce Darboux coordinates on a patch of the moduli space of flat $SL(N)$-connections on a sphere with special punctures, which generalize the NRS coordinates defined in the $SL(2)$ case. Finally, we compare the generating function of the Lagrangian variety of opers in these Darboux coordinates with the effective twisted superpotential of the linear quiver theory in the two-dimensional $\Omega$-background, thereby proving the NRS conjecture and its generalization to the $SL(3)$ case.

Published 8 September 2021