Spectral theory and mirror symmetry

Abstract

Recent developments in string theory have revealed a surprising connection between spectral theory and local mirror symmetry: it has been found that the quantization of mirror curves to toric Calabi–Yau threefolds leads to trace class operators, whose spectral properties are conjecturally encoded in the enumerative geometry of the Calabi–Yau. This leads to a new, infinite family of solvable spectral problems: the Fredholm determinants of these operators can be found explicitly in terms of Gromov–Witten invariants and their refinements; their spectrum is encoded in exact quantization conditions, and turns out to be determined by the vanishing of a quantum theta function. Conversely, the spectral theory of these operators provides a non-perturbative definition of topological string theory on toric Calabi–Yau threefolds. In particular, their integral kernels lead to matrix integral representations of the topological string partition function, which explain some numbertheoretic properties of the periods. In this paper we give a pedagogical overview of these developments with a focus on their mathematical implications.