Integrability of Conformal Fishnet Theory

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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.