Extended superconformal symmetry and Calogero-Marchioro model

We show that the two dimensional Calogero-Marchioro Model (CMM) without the harmonic confinement can naturally be embedded into an extended SU(1, 1|2) superconformal Hamiltonian. We study the quantum evolution of the superconformal Hamiltonian in terms of suitable compact operators of theN = 2 extended de Sitter superalgebra with central charge and discuss the pattern of supersymmetry breaking. We also study the arbitrary D dimensional CMM having dynamical OSp(2|2) supersymmetry and point out the relevance of this model in the context of the low energy effective action of the dimensionally reduced Yang-Mills theory. PACS numbers: 03.65.Fd, 11.30.Pb, 05.30.Pr, 71.10.Pm Typeset using REVTEX E-mail address: [email protected] 1 The Calogero-Moser-Sutherland (CMS) system is a class of exactly solvable models in one dimension [1–4]. These models have been studied extensively from the time of its inception more than thirty years ago and are well understood. There are many higher dimensional generalizations of these models [5–12]. Unfortunately, not a single of these models are known to be exactly solvable or integrable. Among all these systems, the two dimensional Calogero-Marchioro model (CMM) deserves a special attention for several reasons. First of all, for a certain value of the coupling constant, different npoint correlation functions can be calculated analytically by mapping this model to a complex Random Matrix Theory [6,12,13]. This model also has been studied extensively [12,14,15] in connection with several condensed matter systems like, Quantum Hall effect, Quantum Dots, two dimensional Bose systems etc., revealing many interesting features. The purpose of this letter is to unveil one more new feature of this model. We first study the D dimensional N -particle super-CMM with ND bosonic and ND fermionic degrees of freedom. We show how a small subset of the spectrum can be constructed, both in supersymmetry breaking and supersymmetry preserving phases, using the dynamical OSp(2|2) symmetry of the model. We then show, within the specific formalism, only the two dimensional CMM without the harmonic confinement can naturally be embedded into an extended superconformal Hamiltonian. In other words, we construct an extended N = 2 superconformal version of the two dimensional CMM. This construction is valid for arbitrary values of the coupling constant and also for arbitrary N number of particles. We study the quantum evolution in terms of suitable compact operators of the extended N = 2 de Sitter superalgebra. Though we are able to find an infinite number of exact eigenstates of these super-operators, the set is not complete and we are unable to find the complete spectrum. We also discuss the supersymmetry breaking pattern of the extended N = 2 de Sitter supersymmetry with central charge and show