Intrinsic Parameterization of Supermanifold Morphisms from $\mathbb{R}^{0|2}$ via Decomposable Bivector Bundles

TL;DR

This paper provides an intrinsic geometric classification of supermanifold maps from $R^{0|2}$ to smooth manifolds, relating them to decomposable bivector bundles without auxiliary structures, and unifies different perspectives in supersymmetric theories.

Contribution

It introduces a canonical, intrinsic framework for classifying supermanifold morphisms from $R^{0|2}$, avoiding auxiliary structures and connecting to decomposable bivector bundles.

Findings

Established an isomorphism between the space of maps and pullback of bivector bundles.

Derived algebraic constraints for linear dependence of odd vectors.

Unified topological and algebraic views in a canonical supersymmetric framework.

Abstract

We present an intrinsic geometric classification of the supermanifold of maps from $\mathbb{R}^{0|2}$ to any smooth manifold $S$, avoiding auxiliary structures. The key isomorphism relates this space to the pullback of the decomposable bivector bundle over $S$, shown via algebraic constraints forcing odd vectors to be linearly dependent. The reduced manifold has fiber dimension $\dim S + 1$, unifying topological or algebraic views for a canonical framework in supersymmetric theories, distinct from prior works using connections.