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.