{"title":"S-dual of Hamiltonian $\\mathbf G$ spaces and relative Langlands duality","authors":"Hiraku Nakajima","doi":"arxiv-2409.06303","DOIUrl":null,"url":null,"abstract":"The S-dual $(\\mathbf G^\\vee\\curvearrowright\\mathbf M^\\vee)$ of the pair\n$(\\mathbf G\\curvearrowright\\mathbf M)$ of a smooth affine algebraic symplectic\nmanifold $\\mathbf M$ with hamiltonian action of a complex reductive group\n$\\mathbf G$ was introduced implicitly in [arXiv:1706.02112] and explicitly in\n[arXiv:1807.09038] under the cotangent type assumption. The definition was a\nmodification of the definition of Coulomb branches of gauge theories in\n[arXiv:1601.03586]. It was motivated by the S-duality of boundary conditions of\n4-dimensional $\\mathcal N=4$ super Yang-Mills theory, studied by Gaiotto and\nWitten [arXiv:0807.3720]. It is also relevant to the relative Langlands duality\nproposed by Ben-Zvi, Sakellaridis and Venkatesh. In this article, we review the\ndefinition and properties of S-dual.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"6 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Representation Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.06303","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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.
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。