Quantum spectral problems and isomonodromic deformations

Abstract

We develop a self-consistent approach to study the spectral properties of a class of quantum mechanical operators by using the knowledge about monodromies of $2\times 2$ linear systems (Riemann-Hilbert correspondence). Our technique applies to a variety of problems, though in this paper we only analyse in detail two examples. First we review the case of the (modified) Mathieu operator, which corresponds to a certain linear system on the sphere and makes contact with the Painlevé ${\mathrm{III}}_3$ equation. Then we extend the analysis to the 2-particle elliptic Calogero-Moser operator, which corresponds to a linear system on the torus. By using the Kiev formula for the isomonodromic tau functions, we obtain the spectrum of such operators in terms of self-dual Nekrasov functions (${\epsilon }_1+{\epsilon }_2=0$). Through blowup relations, we also find Nekrasov-Shatashvili type of quantizations (${\epsilon }_2=0$). In the case of the torus with one regular singularity we obtain certain results which are interesting by themselves. Namely, we derive blowup equations (filling some gaps in the literature) and we relate them to the bilinear form of the isomonodromic deformation equations. In addition, we extract the ${\epsilon }_2\to 0$ limit of the blowup relations from the regularized action functional and CFT arguments.

arXiv comment62 pages, minor corrections, references added

References in corpus (78)

• Liouville Correlation Functions from Four-dimensional Gauge Theories
• Loop and surface operators in N=2 gauge theory and Liouville modular geometry
• Instantons and recursion relations in N=2 Susy gauge theory
• Nekrasov Functions and Exact Bohr-Sommerfeld Integrals
• Quantization of Integrable Systems and Four Dimensional Gauge Theories
• Instanton counting on blowup. I. 4-dimensional pure gauge theory
• Gauge Theory Loop Operators and Liouville Theory
• Quantum Geometry of Refined Topological Strings
• Defects and Quantum Seiberg-Witten Geometry
• Small Instantons, Little Strings and Free Fermions
• On AGT conjecture
• Duality in Integrable Systems and Gauge Theories
• Painleve III and 2D Polymers
• Deformed Prepotential, Quantum Integrable System and Liouville Field Theory
• Classical Conformal Blocks and Painleve VI
• Matone's Relation in the Presence of Gravitational Couplings
• Resurgence and the Nekrasov-Shatashvili Limit: Connecting Weak and Strong Coupling in the Mathieu and Lam'e Systems
• How instanton combinatorics solves Painlevé VI, V and III's
• Isomonodromic tau-functions from Liouville conformal blocks
• Conformal field theory of Painlevé VI
• On Painlevé/gauge theory correspondence
• Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions
• Non-perturbative approaches to the quantum Seiberg-Witten curve
• Differential equation for four-point correlation function in Liouville field theory and elliptic four-point conformal blocks
• Nekrasov prepotential with fundamental matter from the quantum spin chain
• Bilinear equations on Painleve tau functions from CFT
• Isomonodromy, Painlevé Transcendents and Scattering off of Black Holes
• The dependence on the monodromy data of the isomonodromic tau function
• Quantum Painleve-Calogero Correspondence
• Connection problem for the sine-Gordon/Painlevé III tau function and irregular conformal blocks
• Q-deformed Painleve tau function and q-deformed conformal blocks
• Kerr-de Sitter Quasinormal Modes via Accessory Parameter Expansion
• Poles of Integrale Tritronquee and Anharmonic Oscillators. A WKB Approach
• Transformations of Spherical Blocks
• Scalar quasinormal modes of Kerr-AdS${}_5$
• TBA for the Toda chain
• Irregular conformal blocks, with an application to the fifth and fourth Painlevé equations
• Blowup Equations for Refined Topological Strings
• Confluent conformal blocks and the Teukolsky master equation
• Riemann-Hilbert correspondence and blown up surface defects
• Elliptic Blowup Equations for 6d SCFTs. III: E-strings, M-strings and Chains
• Isomonodromic $\tau$-functions and $W_N$ conformal blocks
• Exact Quantization Conditions, Toric Calabi-Yau and Nonperturbative Topological String
• Classical conformal blocks from TBA for the elliptic Calogero-Moser system
• ODE/IM correspondence and the Argyres-Douglas theory
• Poles of Integrale Tritronquee and Anharmonic Oscillators. Asymptotic localization from WKB analysis
• Argyres-Douglas theories, Painlevé II and quantum mechanics
• Classical Conformal Blocks and Accessory Parameters from Isomonodromic Deformations
• Cluster Toda chains and Nekrasov functions
• BPS relations from spectral problems and blowup equations
• Painleve equations from Nakajima-Yoshioka blowup relations
• Elliptic Calogero-Moser Systems and Isomonodromic Deformations
• Accessory parameters in confluent Heun equations and classical irregular conformal blocks
• $\mathcal{N}=2^{\ast }$ gauge theory, free fermions on the torus and Painlevé VI
• Critical values of the Yang-Yang functional in the quantum sine-Gordon model
• Poles of Painlevé IV Rationals and their Distribution
• Pure $SU(2)$ gauge theory partition function and generalized Bessel kernel
• Backlund transformation of Painleve III($D_8$) tau function
• On the connection problem for Painlevé I
• Quantum periods and prepotential in $\mathcal{N}=2$ SU(2) SQCD
• Free fermions, W-algebras and isomonodromic deformations
• Coupling of two conformal field theories and Nakajima-Yoshioka blow-up equations
• Nonlinear eigenvalue problems for generalized Painlevé equations
• Isomonodromic tau functions on a torus as Fredholm determinants, and charged partitions
• Combinatorial expressions for the tau functions of $q$-Painlevé V and III equations
• On the Existence of Monodromies for the Rabi model
• On the large $\Omega$-deformations in the Nekrasov-Shatashvili limit of $\mathcal{N}=2^{\ast }$ SYM
• Higher rank isomonodromic deformations and W-algebras
• A Riemann-Hilbert Approach to the Heun Equation
• The Elliptic Painlevé Lax Equation vs. van Diejen's 8-Coupling Elliptic Hamiltonian
• W(E_10) Symmetry, M-Theory and Painleve Equations
• Combinatorial approach to Mathieu and Lamé equations
• SU(N) q-Toda equations from mass deformed ABJM theory
• Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformations
• Irregular conformal blocks, Painlevé III and the blow-up equations
• Quantum Representation of Affine Weyl Groups and Associated Quantum Curves
• Exact quantization conditions for the elliptic Ruijsenaars-Schneider model
• Crossing invariant correlation functions at $c=1$ from isomonodromic $\tau$ functions

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