Spectral Theory and Mirror Symmetry
read the original 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 number-theoretic properties of the periods. In this paper we give a pedagogical overview of these developments with a focus on their mathematical implications
This paper has not been read by Pith yet.
Forward citations
Cited by 5 Pith papers
-
The non-perturbative topological string: from resurgence to wall-crossing of DT invariants
An isomorphism is shown between the algebra of alien derivatives acting on the topological string partition function and the Kontsevich-Soibelman Lie algebra, linking resurgence to DT wall-crossing with numerical matc...
-
The non-perturbative topological string: from resurgence to wall-crossing of DT invariants
Links resurgence of the topological string partition function to DT wall-crossing via an isomorphism of alien derivative algebras to the Kontsevich-Soibelman Lie algebra, with Borel singularities matched to specific D...
-
Thou shalt not tunnel: Complex instantons and tunneling suppression in deformed quantum mechanics
Deformed quantum mechanics from Seiberg-Witten curves shows phases with real or complex instantons, leading to tunneling suppression at Toda points and anomalous scaling at critical monopole points.
-
Large Order Enumerative Geometry, Black Holes and Black Rings
Numerical study of high-genus GV invariants reveals 5D indices matching BMPV black-hole entropy below a critical angular momentum and black-ring dominance above, with additional phase transitions and growth laws in PT...
-
Sine-Liouville gravity as a Vertex Model on Planar Graphs
The seven-vertex matrix model realizes sine-Liouville gravity through a shared classical spectral curve with matrix quantum mechanics but distinct branes, with dilute-dense flow analogous to a gravitational massless s...
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.