Baxter Q-Operators and Representations of Yangians

Abstract

We develop a new approach to Baxter Q-operators by relating them to the theory of Yangians, which are the simplest examples for quantum groups. Here we open up a new chapter in this theory and study certain degenerate solutions of the Yang-Baxter equation connected with harmonic oscillator algebras. These infinite-state solutions of the Yang-Baxter equation serve as elementary, "partonic" building blocks for other solutions via the standard fusion procedure. As a first example of the method we consider sl(n) compact spin chains and derive the full hierarchy of operatorial functional equations for all related commuting transfer matrices and Q-operators. This leads to a systematic and transparent solution of these chains, where the nested Bethe equations are derived in an entirely algebraic fashion, without any reference to the traditional Bethe ansatz techniques.

27 pages, 5 figures; v2: typos fixed, references updated and added

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• Oscillator Construction of su(n|m) Q-Operators
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• Review of AdS/CFT Integrability, Chapter III.1: Bethe Ansätze and the R-Matrix Formalism
• Separation of variables for rational gl(n) spin chains in any compact representation, via fusion, embedding morphism and Backlund flow
• QQ-system and Weyl-type transfer matrices in integrable SO(2r) spin chains
• Yangian of AdS3/CFT2 and its deformation
• String Bits and the Spin Vertex
• Non-compact quantum spin chains as integrable stochastic particle processes
• Oscillator versus prefundamental representations II. Arbitrary higher ranks
• Quasi-classical expansion of the star-triangle relation and integrable systems on quad-graphs
• Q-operators for the open Heisenberg spin chain
• Finite type modules and Bethe ansatz equations
• Quantum groups and functional relations for higher rank
• A review of the AdS/CFT Quantum Spectral Curve
• Quantum Spectral Curve of $\gamma$-twisted $\mathcal{N}=4$ SYM theory and fishnet CFT
• Asymptotic representations of augmented q-Onsager algebra and boundary K-operators related to Baxter Q-operators
• Baxter operators for arbitrary spin II
• Dissipation-driven integrable fermionic systems: from graded Yangians to exact nonequilibrium steady states
• Asymptotic representations and q-oscillator solutions of the graded Yang-Baxter equation related to Baxter Q-operators
• Completeness of Wronskian Bethe equations for rational gl(m|n) spin chains
• The universal R-matrix and factorization of the L-operators related to the Baxter Q-operators
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• A note on $q$-oscillator realizations of $U_q(gl(M|N))$ for Baxter $Q$-operators
• 't Hooft lines of ADE-type and Topological Quivers
• The equilibrium landscape of the Heisenberg spin chain
• Rational Lax matrices from antidominantly shifted extended Yangians: BCD types
• One-Dimensional Vertex Models Associated with a Class of Yangian Invariant Haldane-Shastry Like Spin Chains
• Correlators on the wall and ${\mathfrak{sl}}_n$ spin chain
• Toroidal q-Opers
• Universal Baxter TQ-relations for open boundary quantum integrable systems
• Baxter operators and asymptotic representations
• Length-two representations of quantum affine superalgebras and Baxter operators
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• A Family of ${\mathrm{GL}}_r$ Multiplicative Higgs Bundles on Rational Base
• From Baxter Q-Operators to Local Charges
• Evaluation of the operatorial Q-system for non-compact super spin chains
• Yangians and Baxter's relations
• Superspin Chains Solutions from 4D Chern-Simons Theory
• The fate of the Konishi multiplet in the β-deformed Quantum Spectral Curve
• Drinfeld basis for string-inspired Baxter operators
• Quantum groups and functional relations for arbitrary rank
• Extended T-systems, Q matrices and T-Q relations for $s\ell (2)$ models at roots of unity
• Transfer matrices of rational spin chains via novel BGG-type resolutions
• The Zoo of Opers and Dualities
• Integrable systems and AdS/CFT duality
• Determinant Form of Correlators in High Rank Integrable Spin Chains via Separation of Variables
• Q-operators for higher spin eight vertex models with an even number of sites
• Complex D($2,1;\zeta$) and spin chain solutions from Chern-Simons theory
• Line Operators in 4d Chern-Simons Theory and Cherkis Bows
• Topological 4D gravity and gravitational defects
• Jordanian spin chains for twisted strings in $Ad{S}_5\times S^5$
• On Exceptional 't Hooft Lines in 4D-Chern-Simons Theory
• Orthogonality of Q-Functions up to Wrapping in Planar N=4 Super Yang-Mills Theory