Super unitary representations revisited

TL;DR

This paper proposes a revised definition of super Hilbert spaces that ensures all connected super Lie groups have super unitary representations, introducing new structures and decompositions involving supermetrics and Fourier transforms.

Contribution

It introduces a weakened super Hilbert space framework enabling super unitary representations for all connected super Lie groups and develops related supermetric and Fourier transform tools.

Findings

Super unitary representations exist for all connected super Lie groups under the new framework.

A supermetric on supermanifolds relates to the Hodge-star and Fermionic Fourier transform.

Decomposition of super unitary representations over odd parameters is achieved.

Abstract

With the usual definition of a super Hilbert space and a super unitary representation, it is easy to show that there are lots of super Lie groups for which the left-regular representation is not super unitary. I will argue that weakening the definition of a super Hilbert space (by allowing the super scalar product to be non-homogeneous, not just even) will allow the left-regular representation of all (connected) super Lie groups to be super unitary (with an adapted definition). Along the way I will introduce a (super) metric on a supermanifold that will allow me to define super and non-super scalar products on function spaces and I will show that the former are intimately related to the Hodge-star operation and the Fermionic Fourier transform. The latter also allows me to decompose certain super unitary representations as a direct integral over odd parameters of a family of super unitary representations depending on these odd parameters.