Note on Integrability of Marginally Deformed ABJ(M) Theories

TL;DR

This paper investigates the integrability of scalar sector operators in eta-deformed ABJ(M) theories, demonstrating that the two-loop anomalous dimension matrix corresponds to an integrable spin chain system and extending the analysis to \\gamma-deformations.

Contribution

It establishes the integrability of the two-loop anomalous dimension matrix in eta-deformed ABJ(M) theories and shows the Hamiltonian's equivalence in \\gamma-deformed cases at two loops.

Findings

Anomalous dimension matrix forms an integrable Hamiltonian.

Derived \\beta-deformed Bethe ansatz equations for the spin chain.

Hamiltonian remains the same in \\gamma-deformation at two loops.

Abstract

We study the anomalous dimensions of operators in the scalar sector of \beta-deformed ABJ(M) theories. We show that the anomalous dimension matrix at two-loop order gives an integrable Hamiltonian acting on an alternating SU(4) spin chain with the spins at odd lattice sides in the fundamental representation and the spins at even lattices in the anti-fundamental representation. We get a set of $\beta$-deformed Bethe ansatz equations which give the eigenvalues of Hamiltonian of this deformed spin chain system. Based on our computations, we also extend our study to non-supersymmetric three-parameter $\gamma$-deformation of ABJ(M) theories and find that the corresponding Hamiltonian is the same as the one in \beta-deformed case at two-loop level in the scalar sector.