Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence

Abstract

The spectral curve is the key ingredient in the modern theory of classical integrable systems. We develop a construction of the ``quantum spectral curve'' and argue that it takes the analogous structural and unifying role on the quantum level also. In the simplest, but essential case the ``quantum spectral curve'' is given by the formula "det"(L(z)-dz) [Talalaev04] (hep-th/0404153). As an easy application of our constructions we obtain the following: quite a universal receipt to define quantum commuting hamiltonians from the classical ones, in particular an explicit description of a maximal commutative subalgebra in U(gl(n)[t])/t^N and in U(\g[t^{-1}])\otimes U(t\g[t]); its relation with the center on the of the affine algebra; an explicit formula for the center generators and a conjecture on W-algebra generators; a receipt to obtain the q-deformation of these results; the simple and explicit construction of the Langlands correspondence; the relation between the ``quantum spectral curve'' and the Knizhnik-Zamolodchikov equation; new generalizations of the KZ-equation; the conjecture on rationality of the solutions of the KZ-equation for special values of level. In the simplest cases we observe the coincidence of the ``quantum spectral curve'' and the so-called Baxter equation. Connection with the KZ-equation offers a new powerful way to construct the Baxter's Q-operator.

54 pp. minor changes

References in corpus (23)

• Wilson-'t Hooft operators in four-dimensional gauge theories and S-duality
• Mirror symmetry, Langlands duality, and the Hitchin system
• Spectral determinants for Schroedinger equation and Q-operators of Conformal Field Theory
• Noncommutative symmetric functions
• Off-Shell Bethe Ansatz Equation and N-point Correlators in the SU(2) WZNW Theory
• Affine Algebras, Langlands Duality and Bethe Ansatz
• Combinatorial nature of ground state vector of O(1) loop model
• Quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials
• Quantum Group Representations and Baxter Equation
• Separation of variables for the classical and quantum Neumann model
• Unified description of correlators in non-Gaussian phases of Hermitean matrix model
• Reduction of bihamiltonian systems and separation of variables: an example from the Boussinesq hierarchy
• Quantization of the Gaudin System
• Spectrum and Bethe ansatz equations for the U_ {q}(gl(N)) closed and open spin chains in any representation
• Differential Geometry of Gerbes
• Classical limit of the Knizhnik-Zamolodchikov-Bernard equations as hierarchy of isomonodromic deformations. Free fields approach
• Cayley-Hamilton-Newton identities and quasitriangular Hopf algebras
• Spectrum of the quantum Neumann model
• Quantum line bundles on noncommutative sphere
• Dualities in integrable systems: geometrical aspects
• Quasideterminants and Casimir elements for the general Lie superalgebra
• Classical elliptic hypergeometric functions and their applications
• Uniqueness of higher Gaudin hamiltonians

Cited by in corpus (47)

• Gaudin models with irregular singularities
• Opers with irregular singularity and spectra of the shift of argument subalgebra
• Pseudo-differential equations, and the Bethe Ansatz for the classical Lie algebras
• On higher order Sugawara operators
• Quantum curves for Hitchin fibrations and the Eynard-Orantin theory
• Separation of Variables in AdS/CFT: Functional Approach for the Fishnet CFT
• QQ-system and Weyl-type transfer matrices in integrable SO(2r) spin chains
• Quantisation and nilpotent limits of Mishchenko-Fomenko subalgebras
• Limits of Gaudin algebras, quantization of bending flows, Jucys--Murphy elements and Gelfand--Tsetlin bases
• A review of the AdS/CFT Quantum Spectral Curve
• Center of the quantum affine vertex algebra in type A
• Gaudin Hamiltonians generate the Bethe algebra of a tensor power of vector representation of gl_N
• 1+1 Gaudin Model
• Argument Shift Method and Gaudin Model
• Higher Sugawara operators for the quantum affine algebras of type A
• Quantization of the shift of argument subalgebras in type A
• Manin Matrices, Quantum Elliptic Commutative Families and Characteristic Polynomial of Elliptic Gaudin Model
• The MacMahon Master Theorem for right quantum superalgebras and higher Sugawara operators for $\widehat{\mathfrak{gl}}(m|n)$
• Classical W-algebras in types A, B, C, D and G
• Yangian characters and classical W-algebras
• Segal-Sugawara vectors for the Lie algebra of type $G_2$
• Limits of Gaudin Systems: Classical and Quantum Cases
• Extended systems of Baxter Q-functions and fused flags I: simply-laced case
• Cyclotomic Gaudin models with irregular singularities
• A Family of ${\mathrm{GL}}_r$ Multiplicative Higgs Bundles on Rational Base
• Gauging Spacetime Symmetries On The Worldsheet And The Geometric Langlands Program -- II
• Veneziano amplitudes, spin chains and Abelian reduction of QCD
• Eigenvalues of Bethe vectors in the Gaudin model
• Rational solutions of Knizhnik-Zamolodchikov system
• On Segal--Sugawara vectors and Casimir elements for classical Lie algebras
• Center at the critical level for centralizers in type $A$
• Quantum generic Toda system
• Quasi modules for the quantum affine vertex algebra in type $A$
• A new formula for Pfaffian-type Segal-Sugawara vector
• Gauging Spacetime Symmetries on the Worldsheet and the Geometric Langlands Program
• $({\mathfrak{gl}}_M,{\mathfrak{gl}}_N)$-Dualities in Gaudin Models with Irregular Singularities
• Quasi-Topological Gauged Sigma Models, The Geometric Langlands Program, And Knots
• KZ equation, G-opers, quantum Drinfeld-Sokolov reduction and quantum Cayley-Hamilton identity
• Quantisation of bending flows
• Rational solutions of KZ equation, case $S_4$
• Pfaffian-type Sugawara operators
• About the Rationality of the Knizhnik-Zamolodchikov Equation Solution
• Decomplexification of the Capelli identities and holomorphic factorization
• Cohomology of a flag variety as a Bethe algebra
• Superopers revisited
• Bethe ansatz and Isomonodromic deformations
• Explicit Rational Solutions of Knizhnik-Zamolodchikov Equation