Quantum curves and conformal field theory

Abstract

To a given algebraic curve we assign an infinite family of quantum curves (Schrödinger equations), which are in one-to-one correspondence with, and have the structure of, Virasoro singular vectors. For a spectral curve of a matrix model we build such quantum curves out of an appropriate representation of the Virasoro algebra, encoded in the structure of the $\alpha /\beta$-deformed matrix integral and its loop equation. We generalize this construction to a large class of algebraic curves by means of a refined topological recursion. We also specialize this construction to various specific matrix models with polynomial and logarithmic potentials, and among other results, show that various ingredients familiar in the study of conformal field theory (Ward identities, correlation functions and a representation of Virasoro operators acting thereon, BPZ equations) arise upon specialization of our formalism to the multi-Penner matrix model.

90 pages, published version

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Cited by in corpus (12)

• A slow review of the AGT correspondence
• CFT approach to constraint operators for ($\beta$-deformed) hermitian one-matrix models
• BPS counting for knots and combinatorics on words
• Super-quantum curves from super-eigenvalue models
• Integrable differential systems of topological type and reconstruction by the topological recursion
• From CFT to Ramond super-quantum curves
• Quantum curves from refined topological recursion: the genus 0 case
• Loop equations from differential systems
• Quantum curves as quantum distributions
• Deformation and quantisation condition of the $\mathcal{Q}$-top recursion
• Refined topological recursion revisited -- properties and conjectures
• Irregular Conformal States and Spectral Curve: Irregular Matrix Model Approach