# Integrability of Conformal Fishnet Theory

## TL;DR

This paper demonstrates the integrability of fishnet Feynman graphs in a bi-scalar conformal theory, linking it to an $SU(2,2)$ spin chain and Quantum Spectral Curve, enabling precise computation of operator dimensions at any coupling.

## Contribution

It establishes the integrability of fishnet graphs via a conformal spin chain and develops exact methods to compute anomalous dimensions for operators in the theory.

## Key findings

- Derived Baxter equations for conformal spin chain Q-functions.
- Computed anomalous dimensions for operators with J=3 at high precision.
- Linked integrability of fishnet graphs to classical string solutions at strong coupling.

## Abstract

We study integrability of fishnet-type Feynman graphs arising in planar four-dimensional bi-scalar chiral theory recently proposed in arXiv:1512.06704 as a special double scaling limit of gamma-deformed $\mathcal{N}=4$ SYM theory. We show that the transfer matrix "building" the fishnet graphs emerges from the $R-$matrix of non-compact conformal $SU(2,2)$ Heisenberg spin chain with spins belonging to principal series representations of the four-dimensional conformal group. We demonstrate explicitly a relationship between this integrable spin chain and the Quantum Spectral Curve (QSC) of $\mathcal{N}=4$ SYM. Using QSC and spin chain methods, we construct Baxter equation for $Q-$functions of the conformal spin chain needed for computation of the anomalous dimensions of operators of the type $\text{tr}(\phi_1^J)$ where $\phi_1$ is one of the two scalars of the theory. For $J=3$ we derive from QSC a quantization condition that fixes the relevant solution of Baxter equation. The scaling dimensions of the operators only receive contributions from wheel-like graphs. We develop integrability techniques to compute the divergent part of these graphs and use it to present the weak coupling expansion of dimensions to very high orders. Then we apply our exact equations to calculate the anomalous dimensions with $J=3$ to practically unlimited precision at any coupling. These equations also describe an infinite tower of local conformal operators all carrying the same charge $J=3$. The method should be applicable for any $J$ and, in principle, to any local operators of bi-scalar theory. We show that at strong coupling the scaling dimensions can be derived from semiclassical quantization of finite gap solutions describing an integrable system of noncompact $SU(2,2)$ spins. This bears similarities with the classical strings arising in the strongly coupled limit of $\mathcal{N}=4$ SYM.