The BaxterQ-operator for the gradedSL(2|1) spin chain

Abstract

We study an integrable noncompact superspin chain model that emerged in recent studies of the dilatation operator in the N = 1 super-Yang-Mills theory. It was found that the latter can be mapped into a homogeneous Heisenberg magnet with the quantum space in all sites corresponding to infinite-dimensional representations of the SL(2|1) group. We extend the method of the Baxter Q−operator to spin chains with supergroup symmetry and apply it to determine the eigenspectrum of the model. Our analysis relies on a factorization property of the R−operators acting on the tensor product of two generic infinite-dimensional SL(2|1) representations. It allows us to factorize an arbitrary transfer matrix into a product of three 'elementary' transfer matrices which we identify as Baxter Q−operators. We establish functional relations between transfer matrices and use them to derive the TQ-relations for the Q−operators. The proposed construction can be generalized to integrable models based on supergroups of higher rank and, in distinction to the Bethe Ansatz, it is not sensitive to the existence of the pseudovacuum state in the quantum space of the model.

Quantum spectral curve for arbitrary state/operator in AdS5/CFT4

“…It reduces to the one for a non-compact supersymmetric Heisenberg spin chain. Q-operators are well understood and explained for non-compact [61][62][63][64] and supersymmetric [63,65]. Even though it is clear that at a finite coupling this problem literally is not very well posed due to a scheme dependence, in a relevant formulation, it still should contain a certain rational bit of information.…”

Baxter's Q-operators for supersymmetric spin chains

“…It was remarked many times [23,24,34,39,44,45,[49][50][51] that there are equivalent, but different, forms of the Bethe Ansatz in the model. In fact, it is easy to argue that there are precisely 3!…”

Universality of three gaugino anomalous dimensions in Script N = 4 SYM

“…Our proof builds on the results of [25,16] The states associated with the highest weight at the boundary Ñ Ñ ½ are trivially related to the states Tr´Ñ · Ä µ in the bosonic sl´¾µ sector. Those at the opposite boundary Ñ Ñ Ä ½ are associated with Ä-gaugino states Tr´Ñ · Ä µ.…”

“…A nested Bethe Ansatz valid in this sector is described in [25], according to the methods of [4,16]. Up to three loops, the Bethe Ansatz equations read…”

“…The all-loop asymptotic generalization of the Baxter equation appeared in [15]. All-loop extensions to the sl´¾ ½µ sector are described in [16,16].…”