Quantum-classical duality for Gaudin magnets with boundary

Abstract

We establish a remarkable relationship between the quantum Gaudin models with boundary and the classical many-body integrable systems of Calogero-Moser type associated with the root systems of classical Lie algebras (B, C and D). We show that under identification of spectra of the Gaudin Hamiltonians H G j with particles velocitiesq j of the classical model all integrals of motion of the latter take zero values. This is the generalization of the quantum-classical duality observed earlier for Gaudin models with periodic boundary conditions and Calogero-Moser models associated with the root system of the type A.

Quantum-classical correspondence for gl(1|1) supersymmetric Gaudin magnet with boundary

“…The proof for gl(2|0) with parameters (A.5) or (A.8) was given in [28]. Here we prove the cases of the superalgebras gl(1|1) and gl(0|2).…”

“…In this paper we study the quantum-classical duality appeared previously in a number of different contexts [1,11,13,22,23,27,28]. In the general case it is a certain relation between classical integrable many-body systems and quantum spin chains or Gaudin models.…”

“…The corresponding factorization formulas of the type (1.17) and (1.18) were derived in [29], see (A.7) and (A.10). Based on this knowledge we showed in [28] that (1.20) is quantum-classically dual to the boundary Gaudin magnet-the Gaudin limit of the XXX quantum spin chain with (some special) boundary conditions introduced by Sklyanin [26]. Using the algebraic Bethe ansatz for the boundary Gaudin models [16,20] we derived the underlying matrix identities (A.17 is related to three supersymmetric Gaudin magnets associated with superalgebras gl(2|0), gl(1|1) and gl(0|2).…”

“…Calogero-Moser systems and Gaudin models with boundary. In our previous paper [28] we studied the duality for Calogero-Moser models associated with the classical root systems of simple Lie algebras, i.e. to the root systems of B N , C N and D N types.…”

“…The eigenvalues and BE are described in the next section. It should be stressed that our previous construction for gl(2|0) [28] cannot be straightforwardly extended to the supersymmetric case. Some additional computational tricks are required for the proof.…”