A Shortcut to the Q-Operator

Vladimir V. Bazhanov, Tomasz Lukowski , Carlo Meneghelli , Matthias Staudacher

MSC 2000

16W30 Coalgebras, bialgebras, Hopf algebras ; rings, modules, etc. on which these act

82B23 Exactly solvable models; Bethe ansatz

Abstract
Baxter's Q-operator is generally believed to be the most powerful tool for the exact diagonalization of integrable models. Curiously, it has hitherto not yet been properly constructed in the simplest such system, the compact spin-1/2 Heisenberg-Bethe XXX spin chain. Here we attempt to fill this gap and show how two linearly independent operatorial solutions to Baxter's TQ equation may be constructed as commuting transfer matrices if a twist field is present. The latter are obtained by tracing over infinitely many oscillator states living in the auxiliary channel of an associated monodromy matrix. We furthermore compare and differentiate our approach to earlier articles addressing the problem of the construction of the Q-operator for the XXX chain. Finally we speculate on the importance of Q-operators for the physical interpretation of recent proposals for the Y-system of AdS/CFT.