Integrability and the spectrum of two-dimensional fishnet CFT

TL;DR

This paper develops a comprehensive set of equations for the spectrum of 2D fishnet CFT, combining numerical and analytical methods inspired by the Quantum Spectral Curve framework.

Contribution

It introduces a novel operatorial derivation of spectral equations and extends the analysis to twisted cases, advancing non-perturbative understanding of the theory.

Findings

Numerical solutions reveal complex energy levels and state collisions.

New analytical method for deriving Asymptotic Bethe Ansatz equations.

Extended results to twisted models for future correlation function analysis.

Abstract

We formulate a closed set of equations for the spectrum of two-dimensional bi-scalar fishnet conformal field theory, comprising Baxter equations and quantisation conditions, which we derive operatorially from the underlying sl(2) spin chain. These equations are reminiscent of the Quantum Spectral Curve (QSC) framework found in other holographic CFTs and are expected to provide a complete non-perturbative description of the spectrum at arbitrary coupling. We solve the QSC numerically at finite coupling and uncover a rich analytic structure, including state collisions and complex energy levels. Analytically, we introduce a new method to derive the Asymptotic Bethe Ansatz equations, which control the spectrum up to wrapping order and incorporate spinning states. We further extend our results to the twisted case, which may be particularly useful for future separation of variables analyses of correlation functions in this theory.