Seiberg–Witten Differential via Primitive Forms

  • Si LiEmail author
  • Dan Xie
  • Shing-Tung Yau


Three-fold quasi-homogeneous isolated rational singularity is argued to define a four dimensional \({\mathcal{N}=2}\) SCFT. The Seiberg–Witten geometry is built on the mini-versal deformation of the singularity. We argue in this paper that the corresponding Seiberg–Witten differential is given by the Gelfand–Leray form of K. Saito’s primitive form. Our result also extends the Seiberg–Witten solution to include irrelevant deformations.


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The work of S.T Yau is supported by NSF Grant DMS-1159412, NSF Grant PHY-0937443, and NSF Grant DMS-0804454. The work of DX is supported by Center for Mathematical Sciences and Applications at Harvard University, and in part by the Fundamental Laws Initiative of the Center for the Fundamental Laws of Nature, Harvard University. SL is partially supported by Grant 20151080445 of Independent Research Program at Tsinghua University, Grant 11801300 of NSFC, and Grant Z180003 of Beijing Natural Science Foundation. Part of this work was done when SL was visiting the Center for Mathematical Sciences and Applications at Harvard University and Max Planck Institute for Mathematics in January 2018. SL thanks each for their hospitality and provision of an excellent working enviroment.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA
  2. 2.Center of Mathematical Sciences and ApplicationsHarvard UniversityCambridgeUSA
  3. 3.Jefferson Physical LaboratoryHarvard UniversityCambridgeUSA
  4. 4.Department of Mathematical SciencesTsinghua UniversityBeijingChina
  5. 5.Yau Mathematical Sciences CenterTsinghua UniversityBeijingChina

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