Quantum Spectral Curve of $\gamma$-twisted ${\cal N}=4$ SYM theory and fishnet CFT

TL;DR

This paper reviews the quantum spectral curve formalism for integrable models, applies it to a special fishnet CFT limit of gamma-twisted ${ m N}=4$ SYM, and demonstrates its effectiveness in deriving spectral equations and analyzing specific Feynman graphs.

Contribution

It formulates a minimal algebraic and analytic structure for the QSC applicable to various integrable models, including the fishnet CFT limit, and demonstrates its utility in deriving spectral data.

Findings

QSC formalism can be applied to gamma-twisted ${ m N}=4$ SYM and fishnet CFTs.

The QSC degenerates into a $Q$-system for the bi-scalar fishnet theory.

Numerical and analytic results for 'wheel' fishnet graphs are reviewed.

Abstract

We review the quantum spectral curve (QSC) formalism for anomalous dimensions of planar ${\cal\ N}=4$ SYM, including its $\gamma$-deformation. Leaving aside its derivation, we concentrate on formulation of the "final product" in its most general form: a minimal set of assumptions about the algebraic structure and the analyticity of the $Q$-system -- the full system of Baxter $Q$-functions of the underlying integrable model. The algebraic structure of the $Q$-system is entirely based on (super)symmetry of the model and is efficiently described by Wronskian formulas for $Q$-functions organized into the Hasse diagram. When supplemented with analyticity conditions on $Q$-functions, it fixes completely the set of physical solutions for spectrum of an integrable model. First we demonstrate the spectral equations on the example of $gl(N)$ and $gl(K|M)$ Heisenberg (super)spin chains. Supersymmetry $gl(K|M)$ occurs as a "rotation" of the Hasse diagram for a $gl(K+M)$ system. This picture helps us to construct the QSC formalism for spectrum of AdS$_5$/CFT$_4$-duality, with more complicated analyticity constraints on $Q$-functions which involve an infinitely branching Riemann surface and a set of Riemann-Hilbert conditions. As an example of application of QSC, we consider a special double scaling limit of $\gamma$-twisted ${\cal\ N}=4$ SYM, combining weak coupling and strong imaginary twist. This leads to a new type of non-unitary CFT dominated by particular integrable, and often computable, 4D fishnet Feynman graphs. For the simplest of such models -- the bi-scalar theory -- the QSC degenerates into the $Q$-system for integrable non-compact Heisenberg spin chain with conformal, $SU(2,2)$ symmetry. We apply the QSC for derivation of Baxter equation and the quantization condition for particular, "wheel" fishnet graphs, and review numerical and analytic results for them.