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Spectral Theory and Mirror Symmetry

Marcos Marino

TL;DR

The work surveys a deep correspondence between spectral theory and local mirror symmetry, showing that quantized mirror curves of toric Calabi–Yau threefolds yield trace-class operators whose spectra are encoded by Gromov–Witten and refined invariants. It presents the main conjecture that Fredholm determinants of these operators are computable from topological-string data via a quantum theta function, offering a non-perturbative definition of topological strings. The framework connects semiclassical spectra, exact quantization conditions, and NS/GV free energies, and reveals a rich interplay with random-matrix models and conifold physics. It also discusses non-perturbative completions, resurgence, and number-theoretic aspects, outlining future directions for higher-genus, eigenfunctions, and refined theories.

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

Spectral Theory and Mirror Symmetry

TL;DR

The work surveys a deep correspondence between spectral theory and local mirror symmetry, showing that quantized mirror curves of toric Calabi–Yau threefolds yield trace-class operators whose spectra are encoded by Gromov–Witten and refined invariants. It presents the main conjecture that Fredholm determinants of these operators are computable from topological-string data via a quantum theta function, offering a non-perturbative definition of topological strings. The framework connects semiclassical spectra, exact quantization conditions, and NS/GV free energies, and reveals a rich interplay with random-matrix models and conifold physics. It also discusses non-perturbative completions, resurgence, and number-theoretic aspects, outlining future directions for higher-genus, eigenfunctions, and refined theories.

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

Paper Structure

This paper contains 14 sections, 1 theorem, 141 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. A problem in spectral theory
  3. A quantum curve
  4. A first approach to the spectrum
  5. An exact quantization condition
  6. From topological strings to spectral theory
  7. Mirror symmetry and topological strings
  8. Quantizing curves
  9. Fredholm determinants from topological strings
  10. From spectral theory to topological strings
  11. Random matrices from operators and topological strings
  12. Non-perturbative topological strings
  13. Number-theoretic aspects
  14. Outlook

Key Result

Proposition 3.4

The operator on $L^2(\mathbbm R)$ is positive-definite and of trace class. Let us define normalized Heisenberg operators ${\mathsf{q}}$, ${\mathsf{p}}$, satisfying the normalized commutation relation They are related to ${\mathsf{x}}$, ${\mathsf{y}}$ by the linear canonical transformation, so that $\hbar$ is related to ${\mathsf{b}}$ by In the momentum representation associated to ${\mathsf{p}}

Figures (6)

  • Figure 1: The region $\mathcal{R}(E)$ for $E=15$.
  • Figure 2: The function $\xi(E)-1/4$, as a function of $E$, determining the energy spectrum of the operator (\ref{['op-p2']}) for $\hbar=2 \pi$.
  • Figure 3: The vectors (\ref{['p2-vec']}) defining the local $\mathbbm P^2$ geometry, together with the polyhedron $\Delta_{\mathbbm P^2}$ (in thick lines) and the dual polyhedron (in dashed lines).
  • Figure 4: The Fredholm determinant $\Xi_{\mathbbm P^2} (\kappa, 2 \pi)$ as a function of $\kappa$, showing the first three zeroes on the negative real axis, corresponding to the first three energy levels $-{\rm e}^{E_n}$, $n=0,1,2$.
  • Figure 5: Different points in the moduli space lead to different expansions for the spectral quantities.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Proposition 3.4
  • Conjecture 3.5
  • Example 3.6