S-dual of Hamiltonian $\mathbf{G}$ spaces and relative Langlands duality

Abstract

The S-dual $({\mathbf{G}}^{\vee }\curvearrowright {\mathbf{M}}^{\vee })$ of the pair $(\mathbf{G}\curvearrowright \mathbf{M})$ of a smooth affine algebraic symplectic manifold $\mathbf{M}$ with hamiltonian action of a complex reductive group $\mathbf{G}$ was introduced implicitly in [arXiv:1706.02112] and explicitly in [arXiv:1807.09038] under the cotangent type assumption. The definition was a modification of the definition of Coulomb branches of gauge theories in [arXiv:1601.03586]. It was motivated by the S-duality of boundary conditions of 4-dimensional $\mathcal{N}=4$ super Yang-Mills theory, studied by Gaiotto and Witten [arXiv:0807.3720]. It is also relevant to the relative Langlands duality proposed by Ben-Zvi, Sakellaridis and Venkatesh. In this article, we review the definition and properties of S-dual.

11 pages