Quantum Spectral Curve of γ-Twisted 𝒩 = 4 SYM Theory and Fishnet CFT

Abstract

We review the quantum spectral curve (QSC) formalism for the spectrum of anomalous dimensions of [Formula: see text] SYM, including its [Formula: see text]-deformation. Leaving aside its derivation, we concentrate on the formulation of the “final product” in its most general form: a minimal set of assumptions about the algebraic structure and the analyticity of the [Formula: see text]-system — the full system of Baxter [Formula: see text]-functions of the underlying integrable model. The algebraic structure of the [Formula: see text]-system is entirely based on (super)symmetry of the model and is efficiently described by Wronskian formulas for [Formula: see text]-functions organized into the Hasse diagram. When supplemented with analyticity conditions on [Formula: see text]-functions, it fixes completely the set of physical solutions for the spectrum of an integrable model. First, we demonstrate the spectral equations on the example of [Formula: see text] and [Formula: see text] Heisenberg (super)spin chains. Supersymmetry [Formula: see text] occurs as a simple “rotation” of the Hasse diagram for a [Formula: see text] system. Then we apply this method to the spectral problem of [Formula: see text]/CFT4-duality, describing the QSC formalism. The main difference with the spin chains consists in more complicated analyticity constraints on [Formula: see text]-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 [Formula: see text]-twisted [Formula: see text] 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 [Formula: see text]-system for integrable non-compact Heisenberg spin chain with conformal, [Formula: see text] symmetry. We describe the QSC derivation of Baxter equation and the quantization condition for particular fishnet graphs — wheel graphs, and review numerical and analytic results for them.