The algebraic structure of Galilean superconformal symmetries

TL;DR

This paper explores the algebraic structure of Galilean superconformal symmetries across various dimensions, constructing superalgebras and linking them to superconformal mechanics models.

Contribution

It introduces a systematic method to construct Galilean superconformal algebras for multiple dimensions, including supersymmetrization and contraction techniques.

Findings

Constructed Galilean superconformal algebras for d=1 to 5.

Linked d=3 case to N=4 superconformal mechanics.

Presented alternative derivations via Inonu-Wigner contraction.

Abstract

The semisimple part of d-dimensional Galilean conformal algebra g^(d) is given by h^(d)=O(2,1)+O(d), which after adding via semidirect sum the 3d-dimensional Abelian algebra t^(d) of translations, Galilean boosts and constant accelerations completes the construction. We obtain Galilean superconformal algebra G^(d) by firstly defining the semisimple superalgebra H^(d) which supersymmetrizes h^(d), and further by considering the expansion of H^(d) by tensorial and spinorial graded Abelian charges in order to supersymmetrize the Abelian generators of t^(d). For d=3 the supersymmetrization of h^(3) is linked with specific model of N=4 extended superconformal mechanics, which is described by the superalgebra D(2,1;\alpha) if \alpha=1. We shall present as well the alternative derivations of extended Galilean superconformal algebras for d=1,2,3,4,5 by employing the Inonu-Wigner contraction method.