Lorentz Invariant and Supersymmetric Interpretation of Noncommutative Quantum Field Theory

Abstract

In this paper, using a Hopf-algebraic method, we construct deformed Poincaré SUSY algebra in terms of twisted (Hopf) algebra. By adapting this twist deformed superPoincaré algrebra as our fundamental symmetry, we can see the consistency between the algebra and non(anti)commutative relation among (super)coordinates and interpret that symmetry of non(anti)commutative QFT is in fact twisted one. The key point is validity of our new twist element that guarantees non(anti)commutativity of space. It is checked in this paper for N = 1 case. We also comment on the possibility of noncommutative central charge coordinate. Finally, because our twist operation does not break the original algebra, we can claim that (twisted) SUSY is not broken in contrast to the string inspired N = 1/2 SUSY in N = 1 non(anti)commutative superspace.

Twists of quantum groups and noncommutative field theory

“…The exponent reproduces a Poisson tensor defining superbrackets (see e.g. [27]), and can be used to construct noncommutative superspace preserving super-Poincaré covariance [28,29]. The algebraic sector of the twisted Hopf superalgebra U t (sP) is not changed, as well as the coproduct of the abelian subalgebra of (super)translations with the generators P µ , Q α .…”

$ N = \frac{1}{2} $ deformations of chiral superspaces from new quantum Poincaré and Euclidean superalgebras

“…The technique of twisted Poincaré supersymmetry extending to SUSY theories the results of [11,12] has been already studied for Minkowski (see e.g. [21,22]) as well as for Euclidean (see e.g. [23]- [26]) supersymmetry.…”

Field theory on nonanticommutative superspace

“…This can be done in different ways. One possibility is to use a different ⋆-product, the one which preserves chirality [13]. However, chirality-preserving ⋆-product implies working in Euclidean space whereθ = (θ) * .…”

“…There the product of two chiral superfields is not a chiral superfield but the model is invariant under the full supersymmetry. The Hopf algebra of SUSY transformations is deformed by using the twist approach in [13]. Examples of deformation that introduce nontrivial commutation relations between chiral and fermionic coordinates are discussed in [14].…”

“…However, our choice of the twist is different from that in [13] since we want to work in Minkowski space-time. As undotted and dotted spinors are related by the usual complex conjugation, we obtain…”