Non(anti)commutative Superspace

TL;DR

This paper explores the most general form of non(anti)commutative superspace geometries in N=1 and N=2 supersymmetry, analyzing their algebraic structures, associativity constraints, and explicit deformation series.

Contribution

It characterizes the full range of non(anti)commutative superspace geometries compatible with supertranslations, including nonassociative cases and associative deformations.

Findings

Existence of nontrivial non(anti)commutative superspaces with coordinate-dependent parameters.

Explicit series expansion of the associative *-product up to three terms.

More general supergeometries possible in N=2 Euclidean superspace due to conjugation relations.

Abstract

We investigate the most general non(anti)commutative geometry in N=1 four-dimensional superspace, invariant under the classical (i.e., undeformed) supertranslation group. We find that a nontrivial non(anti)commutative superspace geometry compatible with supertranslations exists with non(anti)commutation parameters which may depend on the spinorial coordinates. The algebra is in general nonassociative. Imposing associativity introduces additional constraints which however allow for nontrivial commutation relations involving fermionic coordinates. We obtain explicitly the first three terms of a series expansion in the deformation parameter for a possible associative *-product. We also consider the case of N=2 euclidean superspace where the different conjugation relations among spinorial coordinates allow for a more general supergeometry.