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

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.