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

TL;DR

This paper develops a functional separation of variables approach for the fishnet 4D CFT, enabling non-perturbative computation of complex observables and providing insights potentially applicable to N=4 SYM.

Contribution

It introduces a novel functional SoV formalism for the fishnet CFT, facilitating non-perturbative calculations of observables with potential extensions to N=4 SYM.

Findings

Developed a functional SoV formalism for fishnet CFT.

Derived a SoV-type expression for diagonal OPE coefficients.

Discussed the potential application of methods to N=4 SYM.

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.