Evaluation of the operatorial Q-system for non-compact super spin chains

Abstract

We present an approach to evaluate the full operatorial Q-system of all u(p, q|r + s)-invariant spin chains with representations of Jordan-Schwinger type. In particular, this includes the super spin chain of planar N = 4 super Yang-Mills theory at one loop in the presence of a diagonal twist. Our method is based on the oscillator construction of Q-operators. The Q-operators are built as traces over Lax operators which are degenerate solutions of the Yang-Baxter equation. For non-compact representations these Lax operators may contain multiple infinite sums that conceal the form of the resulting functions. We determine these infinite sums and calculate the matrix elements of the lowest level Q-operators. Transforming the Lax operators corresponding to the Q-operators into a representation involving only finite sums allows us to take the supertrace and to obtain the explicit form of the Q-operators in terms of finite matrices for a given magnon sector. Imposing the functional relations, we then bootstrap the other Q-operators from those of the lowest level. We exemplify this approach for non-compact spin −s spin chains and apply it to N = 4 at the one-loop level using the BMN vacuum as an example.

Non-compact Quantum Spin Chains as Integrable Stochastic Particle Processes

“…These can then be combined into a process of the type which we discussed here with particles hopping to the left and to the right. A similar relation between the local charges of the Q-operator, which also arise from two special points, and the spin chain Hamiltonian was obtained in [33, Appendix C] for the rational limit with s = 1 2 based on the oscillator construction of Q-operators [34][35][36], see also [37,38] for the Q-operator construction of supersymmetric spin chains including the one that underlies N = 4 SYM at weak coupling.…”

QQ-system and Weyl-type transfer matrices in integrable SO(2r) spin chains

“…It would be interesting to generalize our approach to the study of spin chains with open boundary conditions and to the non-compact, highest weight and principal series representations of D algebras. For a much better studied case of these aspects in A-type integrable system we refer the reader to [50,[78][79][80][81]. One encounters non-compact representations in sigma models [82] and spin chains [83] with principal series representations of the d-dimensional conformal group SO (2, d) [83].…”

New compact construction of eigenstates for supersymmetric spin chains

“…It would be important to generalize both the bosonic and supersymmetric constructions to arbitrary representations, in particular to the antisymmetric representation of su(4) relevant for 1-point functions in N = 4 SYM with a defect [65][66][67]), as well as to noncompact spin chains. Another curious direction is to look for relations with the construction of spin chain Q-operators [68][69][70]. It is also interesting to explore deformations of our construction corresponding to the trigonometric XXZ case and to the Gaudin models (either bosonic or supersymmetric [71]).…”