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

Hiraku Nakajima
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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.
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哈密顿$\mathbf G$空间的S对偶性和相对朗兰兹对偶性
S-dual $(\mathbf G^\vee\curvearrowright\mathbf M^\vee)$ of the pair$(\mathbf G\curvearrowright\mathbf M)$ of a smooth affine algebraic symplecticmanifold $\mathbf M$ with hamiltonian action of a complex reductive group$\mathbf G$ 在[arXiv:1706.02112]中隐含地提出,并在[arXiv:1807.09038]中根据余切型假设明确地提出。这个定义是对[arXiv:1601.03586]中规理论库仑分支定义的修正。它是由 Gaiotto 和 Witten [arXiv:0807.3720]研究的 4 维 $\mathcal N=4$ 超级杨-米尔斯理论边界条件的 S 对偶性激发的。它也与 Ben-Zvi、Sakellaridis 和 Venkatesh 提出的相对朗兰兹对偶性有关。本文回顾了 S 对偶的定义和性质。
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