Duke Math. J. 165 (13), 2461-2527, (15 September 2016) DOI: 10.1215/00127094-3477235
KEYWORDS: 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, 14N35, 14J33
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.