Semi-infinite wedges and the conformal limit of the fermionic Calogero-Sutherland Model with spin $\frac{1}{2}$

TL;DR

This paper derives the conformal limit of the fermionic spin-1/2 Calogero-Sutherland Model using wedge product formalism, revealing its state space structure and algebraic symmetries, and connecting it to the Haldane-Shastry Model.

Contribution

It introduces a wedge product formalism to analyze the conformal limit of the fermionic Calogero-Sutherland Model and describes its state space and algebraic symmetries in detail.

Findings

Identifies the state space with a Fock space of two complex fermions.

Expresses the Hamiltonian and Yangian generators in terms of $ ext{sl}_2$ currents and bosons.

Connects the model's algebraic structure to the Haldane-Shastry Model at special coupling values.

Abstract

The conformal limit over an anti-ferromagnetic vacuum of the fermionic spin $\frac{1}{2}$ Calogero-Sutherland Model is derived by using the wedge product formalism. The space of states in the conformal limit is identified with the Fock space of two complex fermions, or, equivalently, with a tensor product of an irreducible level-1 module of $\slt$ and a Fock space module of the Heisenberg algebra.The Hamiltonian and the Yangian generators of the Calogero-Sutherland Model are represented in terms of $\slt$ currents and bosons. At special values of the coupling constant they give rise to the Hamiltonian and the Yangian generators of the conformal limit of the Haldane-Shastry Model acting in an irreducible level-1 module of $\slt$. At generic values of the coupling constant the space of states is decomposed into irreducible representations of the Yangian.