Quantum SU(2|1) supersymmetric Calogero-Moser spinning systems

Abstract

SU(2|1) supersymmetric multi-particle quantum mechanics with additional semi-dynamical spin degrees of freedom is considered. In particular, we provide an N = 4 supersymmetrization of the quantum U(2) spin Calogero-Moser model, with an intrinsic mass parameter coming from the centrally-extended superalgebra su(2|1). The full system admits an SU(2|1) covariant separation into the center-of-mass sector and the quotient. We derive explicit expressions for the classical and quantum SU(2|1) generators in both sectors as well as for the total system, and we determine the relevant energy spectra, degeneracies, and the sets of physical states.2 It was shown in [17], that the centrally extended superalgebra su(2|1) can be represented as a semi-direct sum of su(2|1) and an extra R-symmetry generator: su(2|1) ≃ su(2|1)+ ⊃ u(1). The central charge is a combination of the R-symmetry generator and the internal U(1) generator of su(2|1). In the models under consideration the central charge operator is identified with the canonical Hamiltonian.3 The kinetic term of the variables Z k a in the action (1.5) is of the first-order in the time derivatives, in contrast to the dynamical variable X a b with the second-order kinetic term. Just for this reason we call Z k a ,Z a k semi-dynamical variables. In the Hamiltonian (see below), they appear only in the interaction terms and enter through the SU(2) current S (ik) .

Supersymmetric hyperbolic Calogero-Sutherland models by gauging

“…An interesting question is as to whether the classical and quantum integrability of the bosonic Calogero-Sutherland model is preserved upon the supersymmetrization considered. Another option is the generalization of the N = 4 model to deformed SU(2|1) supersymmetry along the lines of [19,20]. The intrinsic mass parameter (deformation parameter) in this case will amount to an additional oscillator potential in the corresponding actions.…”

“…A different approach was put forward in [15] (see also [16,11]), where N = 1, 2, 4 supersymmetric extensions of the rational Calogero model were derived by a gauging procedure [17,18] applied to matrix superfield systems. Recently, such an approach was applied for obtaining new superconformal Calogero-Moser systems with deformed supersymmetry and intrinsic mass parameter [19,20]. A characteristic feature of the gauging approach is the presence of extra fermionic fields (as compared to a minimal extension) and, in the N = 4 case, of bosonic semi-dynamical spinning variables.…”

Supersymmetric many-body Euler–Calogero–Moser model

“…In recent years notable progress was achieved in the supersymmetrization of the bosonic matrix models [1,2,3,4,5,6]. It has been known for a long time that matrix models are an efficient tool of constructing conformally invariant systems (see e.g.…”

“…therein) For example, the Calogero model as well as its different extensions [8,9,10,11,12] are closely related to matrix models and can be obtained from them by a reduction procedure. The supersymmetrization of matrix models consists in replacing the bosonic matrix entries by superfields [1,2,3,4,5]. While this approach has been quite successful for N ≤ 4 extended supersymmetry, it seems to be less efficient or even inapplicable for N > 4 supersymmetric cases.…”

N-extended supersymmetric Calogero models

“…This idea is not new. It has successfully been employed in [16,17,18,19]. The resulting supersymmetric systems feature…”