Duke Mathematical Journal

A Landau–Ginzburg mirror theorem without concavity

Jérémy Guéré

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We provide a mirror symmetry theorem in a range of cases where state-of-the-art techniques that rely on concavity or convexity do not apply. More specifically, we work on a family of FJRW potentials (named for the Fan, Jarvis, Ruan, and Witten quantum singularity theory) which is viewed as the counterpart of a nonconvex Gromov–Witten potential via the physical Landau–Ginzburg/Calabi–Yau (LG/CY) correspondence. The main result provides an explicit formula for Polishchuk and Vaintrob’s virtual cycle in genus zero. In the nonconcave case of the so-called chain invertible polynomials, it yields a compatibility theorem with the FJRW virtual cycle and a proof of mirror symmetry for FJRW theory.

Article information

Duke Math. J., Volume 165, Number 13 (2016), 2461-2527.

Received: 1 December 2013
Revised: 1 October 2015
First available in Project Euclid: 12 May 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
Secondary: 14J33: Mirror symmetry [See also 11G42, 53D37]

mirror symmetry FJRW theory nonconcavity non-concavity virtual cycle quantum product matrix factorization LG model spin curves recursive complex invertible polynomial chain polynomial Givental’s formalism J-function


Guéré, Jérémy. A Landau–Ginzburg mirror theorem without concavity. Duke Math. J. 165 (2016), no. 13, 2461--2527. doi:10.1215/00127094-3477235. https://projecteuclid.org/euclid.dmj/1463080786

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