A geometric approach to scalar field theories on the supersphere

TL;DR

This paper develops a geometric framework for constructing globally supersymmetric scalar field theories on the supersphere, deriving invariant geometric structures and formulating supersymmetric actions with explicit component field decompositions.

Contribution

It introduces a geometric method to build supersymmetric scalar field theories on the supersphere, including invariant vielbein, spin connection, and explicit actions.

Findings

Derived invariant vielbein and spin connection for the supersphere

Constructed a supersymmetric scalar field action on the supersphere

Obtained Lagrange equations and Noether's theorem for the superscalar field

Abstract

Following a strictly geometric approach we construct globally supersymmetric scalar field theories on the supersphere, defined as the quotient space $S^{2|2} = UOSp(1|2)/\mathcal{U}(1)$. We analyze the superspace geometry of the supersphere, in particular deriving the invariant vielbein and spin connection from a generalization of the left-invariant Maurer-Cartan form for Lie groups. Using this information we proceed to construct a superscalar field action on $S^{2|2}$, which can be decomposed in terms of the component fields, yielding a supersymmetric action on the ordinary two-sphere. We are able to derive Lagrange equations and Noether's theorem for the superscalar field itself.