Virasoro constraints in quantum singularity theory

In this talk, we introduce Virasoro operators in quantum singularity theories for nondegenerate quasi-homogeneous polynomials with nontrivial diagonal symmetries. Using Givental's quantization formula of quadratic Hamiltonians, these operators satisfy the Virasoro relations. Inspired by the famous Virasoro conjecture in Gromov-Witten theory, we conjecture that the genus g generating functions arise in quantum singularity theories are annihilated by these Virasoro operators. We verify the conjecture in various examples and discuss the connections to mirror symmetry of LG models and LG/CY correspondence. This talk is based on work joint with Weiqiang He.