Non(anti)commutative superspace

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.

N=2σ -model action on non(anti)commutative superspace

“…(9) and show that it is identical to the result in eqn. (10). We comment more on this issue later on.…”

“…However, it turns out that such a superspace deformation is only possible in a Euclidean space [10]. Following the work in [8], various aspects of supersymmetric theories with non(anti)commutative deformation are being actively investigated.…”

A renormalizableN= 1/2 SYM theory with interacting matter

“…In the supersymmetric case, the appearance of the RR flux F αβ modifies the superspace geometry through the appearance of a nontrivial anticommutator {θ α , θ β } = F αβ [3][4][5][6][7][8]. The effect on field theories defined in nonanticommutative (NAC) superspace is that the multiplication among superfields is no longer commutative but described by the so-called * -product.…”

Non-singlet Q-deformed N=(1,0) and N=(1Castro1, Quevedo2 2006Physics Letters B8000View full textAdd to dashboardCite

“…Nilpotent deformations in extended supersymmetric field theories were first analyzed in superspace [9,10] and later on in harmonic superspace [11,12]. In this paper we work in Euclidean harmonic superspace in four dimensions [13], where (θ α i ) * =θα i .…”