Note on integrability of marginally deformed ABJ(M) theories

Abstract

We study the anomalous dimensions of operators in the scalar sector of planar β-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 β-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 γdeformation of ABJ(M) theories and find that the corresponding Hamiltonian is the same as the one in β-deformed case at two-loop level in the scalar sector.

Y-system for γ-deformed ABJM theory

“…This result is consistent with Appendix A, where direct computations through a deformed R-matrix were performed. For γ 1 = γ 2 = 0, γ 3 = −β, this comes back to the result in [40] for β-deformed case. 3…”

“…The integrability of γ-deformed ABJM theory in [41] in the scalar sector at two-loop level can be proved in a similar way as it was done for β-deformed theory [40]. In this paper we make the very natural assumption that the planar γ-deformed ABJM theory is integrable for all the sectors and to all loop orders, and compute the twist matrix as in [26].…”

“…The proof of integrability of planar γ-deformed ABJM theory in the scalar sector at two loop level is almost the same as the one for β-deformed theory in [40], the only difference is that now we need to use the star product defined in eq. (2.1).…”

“…In three-dimensional case, similar aspects have been less studied. In [40], integrability of planar β-deformed ABJM theories was established at two-loop order in the scalar sector. 1 Recently, integrability of orbifold ABJM theories was studied in details in [42].…”

“…Several applications of our method in the sl(2)-like sector and some important examples in β-deformed ABJM are presented as well. * chenhh@ihep.ac.cn † liup51@ihep.ac.cn ‡ wujb@ihep.ac.cn 1 Integrability of a γ-deformation different from the ones in [41] was also studied in [40].…”