Exact eigenfunctions and the open topological string

Abstract

Mirror curves to toric Calabi-Yau threefolds can be quantized and lead to trace class operators on the real line. The eigenvalues of these operators are encoded in the BPS invariants of the underlying threefold, but much less is known about their eigenfunctions. In this paper we first develop methods in spectral theory to compute these eigenfunctions. We also provide a matrix integral representation which allows to study them in a 't Hooft limit, where they are described by standard topological open string amplitudes. Based on these results, we propose a conjecture for the exact eigenfunctions which involves both the WKB wavefunction and the standard topological string wavefunction. This conjecture can be made completely explicit in the maximally supersymmetric, or self-dual case, which we work out in detail for local P1xP1. In this case, our conjectural eigenfunctions turn out to be closely related to Baker-Akhiezer functions on the mirror curve, and they are in full agreement with first-principle calculations in spectral theory.

50 pages, 8 figures; v2: corrections, references, and many clarifications added

References in corpus (12)

• Open string amplitudes and large order behavior in topological string theory
• Remodeling the B-model
• Exact ABJM Partition Function from TBA
• Topological Strings and Quantum Spectral Problems
• Hofstadter's Butterfly in Quantum Geometry
• Operators and higher genus mirror curves
• Large N expansion of convergent matrix integrals, holomorphic anomalies, and background independence
• Quantization condition from exact WKB for difference equations
• TBA-like integral equations from quantized mirror curves
• Topological Strings and (Almost) Modular Forms
• Bethe/Gauge correspondence in odd dimension: modular double, non-perturbative corrections and open topological strings
• Spectral determinants and quantum theta functions

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• Exact relativistic Toda chain eigenfunctions from Separation of Variables and gauge theory
• Bootstrapping Calabi-Yau Quantum Mechanics
• Bethe/Gauge correspondence in odd dimension: modular double, non-perturbative corrections and open topological strings
• Instanton Corrections of 1/6 BPS Wilson Loops in ABJM Theory
• Spectral equations for the modular oscillator
• New results in $\mathcal{N}=2$ theories from non-perturbative string
• 2d QCD and Integrability, Part I: 't Hooft model
• Geometric transition in Non-perturbative Topological string
• Matrix models for topological strings: exact results in the planar limit
• On the spectrum of the local ${\mathbb{P}}^2$ mirror curve
• Quantum curves as quantum distributions
• Painlevé Kernels and Surface Defects at Strong Coupling