Symplectic Geometry of Supersymmetry and Nonlinear Sigma Model

TL;DR

This paper explores the symplectic geometry underlying supersymmetric field theories, demonstrating that superloop space provides a natural Hamiltonian framework, especially for nonlinear sigma models with complex fermionic interactions.

Contribution

It establishes the symplectic structure of supersymmetric theories at the supermultiplet level using superloop space, highlighting the importance of nonconventional auxiliary fields.

Findings

Superloop space is essential for a Hamiltonian interpretation of supersymmetric actions.

The nonlinear sigma model's quartic fermionic term necessitates superloop variables.

The approach generalizes the symplectic structure to N=1 supermultiplets.

Abstract

Recently it has been argued, that Poincar\'{e} supersymmetric field theories admit an underlying loop space hamiltonian (symplectic) structure. Here shall establish this at the level of a general $N=1$ supermultiplet. In particular, we advocate the use of a superloop space and explain the necessity of using nonconventional auxiliary fields. As an example we consider the nonlinear $\sigma$-model. Due to the quartic fermionic term, we conclude that the use of superloop space variables is necessary for the action to have a hamiltonian loop space interpretation.