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Seiberg–Witten Differential via Primitive Forms

  • Si LiEmail author
  • Dan Xie
  • Shing-Tung Yau
Article
  • 14 Downloads

Abstract

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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Notes

Acknowledgements

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.

References

  1. 1.
    Sierra, G., Townsend, P.K.: An introduction to N = 2 rigid supersymmetry. In: 19th Winter School and Workshop on Theoretical Physics: Supersymmetry and Supergravity Karpacz, Poland, February 14–26, 1983, pp. 396–430 (1983)Google Scholar
  2. 2.
    Gates S.J. Jr: Superspace formulation of new non-linear sigma models. Nucl. Phys. B 238(2), 349–366 (1984)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    de Wit B., Van Proeyen A.: Potentials and symmetries of general gauged n = 2 supergravity-Yang–Mills models. Nucl. Phys. B 245, 89–117 (1984)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Strominger A.: Special geometry. Commun. Math. Phys. 133(1), 163–180 (1990)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Seiberg N., Witten E.: Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD. Nucl.Phys. B 431, 484–550 (1994) arXiv:hep-th/9408099 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Seiberg N., Witten E.: Monopole condensation, and confinement in N = 2 supersymmetric Yang–Mills theory. Nucl. Phys. B 426, 19–52 (1994) arXiv:hep-th/9407087 ADSCrossRefzbMATHGoogle Scholar
  7. 7.
    Shapere, A.D., Vafa, C.: BPS structure of Argyres–Douglas superconformal theories. arXiv:hep-th/9910182
  8. 8.
    Xie, D., Yau, S.-T.: 4d N=2 SCFT and singularity theory Part I: classification. arXiv:1510.01324 [hep-th]
  9. 9.
    Saito K.: Period mapping associated to a primitive form. Publ. Res. Inst. Math. Sci. 19(3), 1231–1264 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dubrovin, B.: Geometry of 2D topological field theories. In: Integrable Systems and Quantum Groups (Montecatini Terme, 1993), vol. 1620 of Lecture Notes in Math., pp. 120–348. Springer, Berlin (1996).  https://doi.org/10.1007/BFb0094793
  11. 11.
    Hertling C.: Frobenius Manifolds and Moduli Spaces for Singularities. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  12. 12.
    Losev A.: Descendants constructed from matter field in Landau–Ginzburg theories coupled to topological gravity. Theor. Math. Phys. 95(2), 595–603 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Losev, A.: Hodge strings and elements of k. Saito’s theory of primitive form. In: Kashiwara, A., Matsuo, A., Saito, K., Satake, I. (eds.) Topological Field Theory, Primitive Forms and Related Topics, pp. 305–335. Springer (1998)Google Scholar
  14. 14.
    Costello, K.J., Li, S.: Quantum BCOV theory on Calabi–Yau manifolds and the higher genus B-model (2012). arXiv preprint arXiv:1201.4501
  15. 15.
    Dijkgraaf R., Verlinde H., Verlinde E.: Topological strings in \({d < 1}\). Nucl. Phys. B 352(1), 59–86 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Li, C., Li, S., Saito, K.: Primitive forms via polyvector fields (2013). arXiv preprint arXiv:1311.1659
  17. 17.
    Li C., Li S., Saito K., Shen Y.: Mirror symmetry for exceptional unimodular singularities. J. Eur. Math. Soc. (JEMS) 19(4), 1189–1229 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Craps B., Roose F., Troost W., Van Proeyen A.: What is special Kahler geometry?. Nucl. Phys. B 503, 565–613 (1997) arXiv:hep-th/9703082 [hep-th]ADSCrossRefzbMATHGoogle Scholar
  19. 19.
    Donagi R., Witten E.: Supersymmetric Yang–Mills theory and integrable systems. Nucl.Phys. B 460, 299–334 (1996) arXiv:hep-th/9510101 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Griffiths P, Harris J: Principles of Algebraic Geometry. Wiley, Hoboken (2014)zbMATHGoogle Scholar
  21. 21.
    Arnold V.I., Varchenko A.N., Gusein-Zade S.M.: Singularities of Differentiable Maps: Volume II Monodromy and Asymptotic Integrals. Springer, Berlin (2012)CrossRefzbMATHGoogle Scholar
  22. 22.
    Gaiotto D.: N=2 dualities. JHEP 08, 034 (2012) arXiv:0904.2715 [hep-th]ADSCrossRefGoogle Scholar
  23. 23.
    Xie D.: General Argyres–Douglas theory. JHEP 01, 100 (2013) arXiv:1204.2270 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Saito M.: On the structure of brieskorn lattice. Ann. Inst. Fourier (Grenoble) 39(1), 27–72 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Douai, A., Sabbah, C.: Gauss–Manin systems, brieskorn lattices and frobenius structures (i)(systèmes de Gauss–Manin, réseaux de brieskorn et structures de frobenius (i)). In: Annales de l’institut Fourier, Vol. 53, pp. 1055–1116 (2003)Google Scholar
  26. 26.
    Douai, A., Sabbah, C.: Gauss–Manin systems, brieskorn lattices and frobenius structures (ii). In: Hertling, C., Marcolli, M. (eds.) Frobenius Manifolds, pp. 1–18. Springer (2004)Google Scholar
  27. 27.
    Cecotti S., Vafa C.: Topological–anti-topological fusion. Nucl. Phys. B 367(2), 359–461 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Del Zotto M., Vafa C., Xie D.: Geometric engineering, mirror symmetry and \({6{\mathrm{d}}_{\left(1,0\right)} \to 4{\mathrm{d}}_{\left({\mathcal{N}}=2\right)}}\). JHEP 11, 123 (2015) arXiv:1504.08348 [hep-th]ADSCrossRefGoogle Scholar
  29. 29.
    Chen B., Xie D., Yau S.-T., Yau S.S.T., Zuo H.: 4D \({{\mathcal{N}} = 2}\) SCFT and singularity theory. Part II: complete intersection. Adv. Theor. Math. Phys. 21, 121–145 (2017) arXiv:1604.07843 [hep-th]MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wang Y., Xie D., Yau S.S.T., Yau S.-T.: \({4d {\mathcal{N}} = 2}\) SCFT from complete intersection singularity. Adv. Theor. Math. Phys. 21, 801–855 (2017) arXiv:1606.06306 [hep-th]MathSciNetCrossRefzbMATHGoogle Scholar

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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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