S-dual of Hamiltonian G spaces and relative Langlands duality

Algebraic Geometry and Number Theory Seminar

The S-dual (G∨M∨) of the pair (GM) of a smooth affine algebraic symplectic manifold M with hamiltonian action of a complex reductive group 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 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.