Global space-time symmetries of quantized Euclidean and Minkowski superspaces

C Gonera,M Wodzislawski

doi:10.1007/jhep04(2012)088

Abstract

Abstract Starting with assumptions both simple and natural from “physical” point of view we present a direct construction of the transformations preserving wide class of (anti)commutation relations which describe Euclidean/Minkowski superspace quantizations. These generalized transformations act on deformed superspaces as the ordinary ones do on undeformed spaces but they depend on non(anti)commuting parameters satisfying some consistent (anti)commutation relations. Once the coalgebraic structure compatible with the algebraic one is introduced in the set of transformations we deal with quantum symmetry supergroup. This is the case for intensively studied so called $ N = \frac{1}{2} $ supersymmetry as well as its three parameter extension. The resulting symmetry transformations — supersymmetric extension of θ — Euclidean group can be regarded as global counterpart of appropriately twisted Euclidean superalgebra that has been shown to preserve $ N = \frac{1}{2} $ supersymmetry.

Highlights

(anti)commutation relations which describe Euclidean/Minkowski superspace quantizations
In the present paper, starting from the assumptions which are both simple and natural from ”physical” point of view we presented a direct construction of generalized Euclidean/Poincare transformations which preserve a wide class of commutation rules defining deformations of the relevant superspaces
If the algebraic sector of transformations defined by these relations is consistent with coalgebraic structure one deals with quantum symmetry supergroup

Summary

Global Symmetries of Deformed Superspaces

Speaking, (super)space deformations as well as construction of their symmetries can be considered as a procedure of replacing commuting coordinates of undeformed (super)spaces and commuting parameters of transformations acting on these coordinates by noncommuting quantities satisfying some well-defined, consistent commutation rules. In the Minkowski superspace case, dotted and undotted grassmannian quantities are required to be related by conjugation transformation This is additional, as compared to Euclidean case, condition determining the structure of both coordinates and parameters commutation rules. Consistency of Jacobi identities and commutation rules of space-time coordinates ( as well as those of parameters ) implies some complicated constraints on Cαβ, Eαβand Πmβ constants, as in Euclidean case. Contrary to the latter case there exist no matrices C and E with c-number elements which satisfy these constraints (see appendix). It is described by eqs. (2.31), (2.32) and (2.33) where the matrices B† and AT are replaced by A† and AT , defined by eqs. (2.41)

Summary

Euclidean superspace

Minkowski superspace