Separation of Variables in AdS/CFT: Functional Approach for the Fishnet CFT

Abstract

The major simplification in a number of quantum integrable systems is the existence of special coordinates in which the eigenstates take a factorised form. Despite many years of studies, the basis realising the separation of variables (SoV) remains unknown in N=4 SYM and similar models, even though it is widely believed they are integrable. In this paper we initiate the SoV approach for observables with nontrivial coupling dependence in a close cousin of N=4 SYM - the fishnet 4D CFT. We develop the functional SoV formalism in this theory, which allows us to compute non-perturbatively some nontrivial observables in a form suitable for numerical evaluation. We present some applications of these methods. In particular, we discuss the possible SoV structure of the one-point correlators in presence of a defect, and write down a SoV-type expression for diagonal OPE coefficients involving an arbitrary state and the Lagrangian density operator. We believe that many of the findings of this paper can be applied in the N=4 SYM case, as we speculate in the last part of the article.

53 pages + appendices, 5 figures. v2: references added, small changes to the text. Accepted for publication in JHEP

References in corpus (9)

• $\mathcal{N}=1$ supersymmetric indices and the four-dimensional A-model
• What is an integrable quench?
• On three-point correlation functions in the gauge/gravity duality
• Structure Constants and Integrable Bootstrap in Planar N=4 SYM Theory
• The Quantum Spectral Curve of the ABJM theory
• Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence
• The One-Loop Spectral Problem of Strongly Twisted $\mathcal{N}$=4 Super Yang-Mills Theory
• Completeness of SoV Representation for $\mathrm{SL}(2,\mathbb{R})$ Spin Chains
• Conformal quantum mechanics & the integrable spinning Fishnet

Cited by in corpus (19)

• Integrability and Conformal Bootstrap: One Dimensional Defect CFT
• Weak Gravity Conjecture from Conformal Field Theory: A Challenge from Hyperscaling Violating and Kerr-Newman-AdS Black Holes
• Gaudin Models and Multipoint Conformal Blocks II: Comb channel vertices in 3D and 4D
• Crosscap States in Integrable Field Theories and Spin Chains
• On exact overlaps for $\mathfrak{gl}(N)$ symmetric spin chains
• Duality Relations for Overlaps of Integrable Boundary States in AdS/dCFT
• Mirror channel eigenvectors of the $d$-dimensional fishnets
• Structure constants of short operators in planar $\mathcal{N}=4$ SYM theory
• Form-factors and complete basis of observables via separation of variables for higher rank spin chains
• Hexagonalization of Fishnet integrals I: mirror excitations
• Large Charges on the Wilson Loop in $\mathcal{N}=4$ SYM: II. Quantum Fluctuations, OPE, and Spectral Curve
• Checkerboard CFT
• Bootstrability in Defect CFT: Integrated Correlators and Sharper Bounds
• Correlation functions of determinant operators in conformal fishnet theory
• A Large Twist Limit for Any Operator
• Three-Point Functions in ABJM and Bethe Ansatz
• Separation of variables for higher rank integrable models: review of recent progress
• Bethe ansatz inside Calogero-Sutherland models
• Orthogonality of Q-Functions up to Wrapping in Planar N=4 Super Yang-Mills Theory