Mathematical Research Letters

Volume 28 (2021)

Number 1

$4d$ $N=2$ SCFT and singularity theory Part IV: Isolated rational Gorenstein non-complete intersection singularities with at least one-dimensional deformation and nontrivial $T^2$

Pages: 1 – 23

DOI: https://dx.doi.org/10.4310/MRL.2021.v28.n1.a1

Authors

Bingyi Chen (Department of Mathematical Sciences, Tsinghua University, Beijing, China)

Stephen S.-T. Yau (Department of Mathematical Sciences, Tsinghua University, Beijing, China; and Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Huairou, China)

Shing-Tung Yau (Department of Mathematics, Center of Mathematical Sciences and Applications, Jefferson Physical Laboratory, Harvard University, Cambridge, Massachusetts, U.S.A.)

Huaiqing Zuo (Department of Mathematical Sciences, Tsinghua University, Beijing, China)

Abstract

We study the miniversal deformations of minimally elliptic two-dimensional singularities of multiplicities of $5$, $6$ and $7$. By restricting the miniversal deformations on the line transverse to the discriminant locus, we construct many new three-dimensional isolated rational Gorenstein singularities with one-dimensional equisingular deformation and nontrivial $T^2$. In fact the three-dimensional isolated rational Gorenstein singularities constructed from minimally elliptic singularities of multiplicity $5$ has four-dimensional family of deformation, of which one-dimensional family is equisingular in the sense of Hilbert polynomial. On the other hand, the three-dimensional isolated rational Gorenstein singularities constructed from minimally elliptic singularities of multiplicity $6$ and $7$ respectively has nontrivial $T^2$ and has one-dimensional equisingular family of deformation. These singularities define many new interesting four dimensional $N = 2$ superconformal field theories.

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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 Stephen S.-T. Yau is supported by NSFC grant 11961141005, Tsinghua University start-up fund, and Tsinghua University Education Foundation fund 042202008. The work of Huaiqing Zuo is supported by NSFC (grant nos. 11771231, 11961141005).

Received 17 March 2019

Accepted 14 October 2019

Published 24 May 2022