Supersymmetric Quantum Spherical Model: A Model for Hodge Theory

TL;DR

This paper explores the symmetry properties of a supersymmetric quantum spin model, establishing a connection with differential geometry's Hodge theory and demonstrating its role as a physical realization of mathematical dualities.

Contribution

It introduces a supersymmetric quantum spherical model that maps symmetry transformations to de Rham cohomological operators, serving as a novel toy model for Hodge theory.

Findings

Mapping between symmetry transformations and de Rham operators

Existence of discrete symmetries realizing Hodge duality

Model serves as a physical realization of Hodge theory

Abstract

We discuss various symmetry properties of the N = 2 supersymmetric quantum spin model in one (0 + 1)-dimension of spacetime and provide their relevance in the realm of the mathematics of differential geometry. We show one-to-one mapping between the continuous symmetry transformations (and corresponding generators) and de Rham cohomological operators of differential geometry. One of the novel observations is the existence of discrete symmetry transformations which play a crucial role in providing the physical realization of the Hodge duality ($\star$) operation. Thus, the present model provides a toy model for the Hodge theory.