Unitary representations of super Lie groups and applications to the classification and multiplet structure of super particles

TL;DR

This paper develops a framework for unitary representations of super Lie groups, extending classical theories, and applies it to classify super particles and super multiplets across various dimensions.

Contribution

It introduces a category of unitary representations for super Lie groups and extends classical induction and Mackey theorems to this setting.

Findings

Classified irreducible unitary representations of super Poincaré groups.

Extended classical induction and Mackey theorems to super Lie groups.

Compared mathematical results with physical super multiplet classifications.

Abstract

It is well known that the category of super Lie groups (SLG) is equivalent to the category of super Harish-Chandra pairs (SHCP). Using this equivalence, we define the category of unitary representations (UR's) of a super Lie group. We give an extension of the classical inducing construction and Mackey imprimitivity theorem to this setting. We use our results to classify the irreducible unitary representations of semidirect products of super translation groups by classical Lie groups, in particular of the super Poincar\'e groups in arbitrary dimension. Finally we compare our results with those in the physical literature on the structure and classification of super multiplets.