The Loom for General Fishnet CFTs

We propose a broad class of $d$-dimensional conformal field theories of $SU(N)$ adjoint scalar fields generalising the 4$d$ Fishnet CFT (FCFT) discovered by \"O. G\"urdogan and one of the authors as a special limit of $\gamma$-deformed $\mathcal{N}=4$ SYM theory. In the planar $N\to\infty$ limit the FCFTs are dominated by the ``fishnet" planar Feynman graphs. These graphs are explicitly integrable, as was shown long ago by A. Zamolodchikov. The Zamolodchikov's construction, based on the dual Baxter lattice (straight lines on the plane intersecting at arbitrary slopes) and the star-triangle identities, can serve as a ``loom" for ``weaving" the Feynman graphs of these FCFTs, with certain types of propagators, at any $d$. The Baxter lattice with $M$ different slopes and any number of lines parallel to those, generates an FCFT consisting of $M(M-1)$ fields and a certain number of chiral vertices of different valences with distinguished couplings. These non-unitary, logarithmic CFTs enjoy certain reality properties for their spectrum due to a symmetry similar to the PT-invariance of non-hermitian hamiltonians proposed by C. Bender. We discuss in more detail the theories generated by a loom with $M=2,3,4$, and the generalisation of the loom FCFTs for spinning fields in 4$d$.