From Jack to Double Jack Polynomials via the Supersymmetric Bridge

Abstract

The Calogero-Sutherland model occurs in a large number of physical contexts, either directly or via its eigenfunctions, the Jack polynomials. The supersymmetric counterpart of this model, although much less ubiquitous, has an equally rich structure. In particular, its eigenfunctions, the Jack superpolynomials, appear to share the very same remarkable combinatorial and structural properties as their non-supersymmetric version. These super-functions are parametrized by superpartitions with fixed bosonic and fermionic degrees. Now, a truly amazing feature pops out when the fermionic degree is sufficiently large: the Jack superpolynomials stabilize and factorize. Their stability is with respect to their expansion in terms of an elementary basis where, in the stable sector, the expansion coefficients become independent of the fermionic degree. Their factorization is seen when the fermionic variables are stripped off in a suitable way which results in a product of two ordinary Jack polynomials (somewhat modified by plethystic transformations), dubbed the double Jack polynomials. Here, in addition to spelling out these results, which were first obtained in the context of Macdonal superpolynomials, we provide a heuristic derivation of the Jack superpolynomial case by performing simple manipulations on the supersymmetric eigen-operators, rendering them independent of the number of particles and of the fermionic degree. In addition, we work out the expression of the Hamiltonian which characterizes the double Jacks. This Hamiltonian, which defines a new integrable system, involves not only the expected Calogero-Sutherland pieces but also combinations of the generators of an underlying affine ${\widehat{\mathfrak{sl}}}_2$ algebra.

References in corpus (23)

• Liouville Correlation Functions from Four-dimensional Gauge Theories
• Exact operator solution of the Calogero-Sutherland model
• On combinatorial expansion of the conformal blocks arising from AGT conjecture
• Collective fields, Calogero-Sutherland model and generalized matrix models
• Excited States of Calogero-Sutherland Model and Singular Vectors of the $W_N$ Algebra
• Deformed quantum Calogero-Moser problems and Lie superalgebras
• Properties of Non-Abelian Fractional Quantum Hall States at Filling $\nu =\frac{k}{r}$
• Super Lax Pairs and Infinite Symmetries in The $1/r^2$ System
• Density Correlation Functions in Calogero Sutherland Models
• Finalizing the proof of AGT relations with the help of the generalized Jack polynomials
• Calogero-Sutherland model and bulk-boundary correlations in conformal field theory
• Supersymmetric Calogero-Moser-Sutherland models and Jack superpolynomials
• Relaxed singular vectors, Jack symmetric functions and fractional level $\widehat{\mathfrak{sl}}(2)$ models
• Orthogonality of Jack polynomials in superspace
• Correspondence between conformal field theory and Calogero-Sutherland model
• The supersymmetric Ruijsenaars-Schneider model
• Superconformal field theory and Jack superpolynomials
• Superanalogs of the Calogero operators and Jack polynomials
• From Jack polynomials to minimal model spectra
• Hermite and Laguerre Symmetric Functions Associated with Operators of Calogero-Moser-Sutherland Type
• Supersymmetric Extensions of Calogero--Moser--Sutherland like Models: Construction and Some Solutions
• Calogero-Sutherland eigenfunctions with mixed boundary conditions and conformal field theory correlators
• Supersymmetry and Models for Two Kinds of Interacting Particles

Cited by in corpus (3)

• Yangian associated with 2D $\mathcal{N}=1$ SCFT
• Supersymmetric polynomials and algebro-combinatorial duality
• Symmetric functions in superspace: a compendium of results and open problems (including a SageMath worksheet)