Separation of variables and scalar products at any rank

Abstract

Separation of variables (SoV) is a special property of integrable models which ensures that the wavefunction has a very simple factorised form in a specially designed basis. Even though the factorisation of the wavefunction was recently established for higher rank models by two of the authors and G. Sizov, the measure for the scalar product was not known beyond the case of rank one symmetry. In this paper we show how this measure can be found, bypassing an explicit SoV construction. A key new observation is that the measure for spin chains in a highest-weight infinite dimensional representation of sl(N) couples Q-functions at different nesting levels in a non-symmetric fashion. We also managed to express a large number of form factors as ratios of determinants in our new approach. We expect our method to be applicable in a much wider setup including the problem of computing correlators in integrable CFTs such as the fishnet theory, N=4 SYM and the ABJM model.

29 pages; v2: references added, minor corrections

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Cited by in corpus (30)

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• Exactly solvable single-trace four point correlators in $\chi$CFT${}_4$
• Weak integrability breaking and level spacing distribution
• A review of the AdS/CFT Quantum Spectral Curve
• The Holographic Dual of Strongly γ-deformed N=4 SYM Theory: Derivation, Generalization, Integrability and Discrete Reparametrization Symmetry
• Mirror channel eigenvectors of the $d$-dimensional fishnets
• Open Fishchain in N=4 Supersymmetric Yang-Mills Theory
• On Scalar Products in Higher Rank Quantum Separation of Variables
• Form-factors and complete basis of observables via separation of variables for higher rank spin chains
• Separation of variables bases for integrable $g{l}_{\mathcal{M}|\mathcal{N}}$ and Hubbard models
• Functional Equations and Separation of Variables for Exact g-Function
• Hexagonalization of Fishnet integrals I: mirror excitations
• Wrapping corrections for long range spin chains
• Correlation functions by Separation of Variables: the XXX spin chain
• Correlation functions for open XXZ spin 1/2 quantum chains with unparallel boundary magnetic fields
• Current mean values in the XYZ model
• Semi-classical quantisation of magnetic solitons in the anisotropic Heisenberg quantum chain
• Bootstrability in Defect CFT: Integrated Correlators and Sharper Bounds
• Correlation functions of determinant operators in conformal fishnet theory
• Determinant Form of Correlators in High Rank Integrable Spin Chains via Separation of Variables
• Separation of variables for higher rank integrable models: review of recent progress
• Bethe ansatz inside Calogero-Sutherland models
• On correlation functions for the open XXZ chain with non-longitudinal boundary fields : the case with a constraint
• The Gelfand-Tsetlin basis for infinite-dimensional representations of $g{l}_n(\mathbb{C})$
• Orthogonality of Q-Functions up to Wrapping in Planar N=4 Super Yang-Mills Theory
• BC-type open $SL(2,\mathbb{C})$ spin chain
• The open XYZ spin 1/2 chain: Separation of Variables and scalar products for boundary fields related by a constraint