Invariant integration on orthosymplectic and unitary supergroups

TL;DR

This paper develops a new approach to invariant integration on orthosymplectic and unitary supergroups by linking sheaf-theoretical and differential geometric methods, resulting in explicit formulas for integrals.

Contribution

It introduces a novel framework connecting Harish-Chandra pairs with supermanifold theory to derive explicit invariant integral formulas on supergroups.

Findings

Explicit invariant integral formulas for OSp(m|2n) and U(p|q)

New expression for Lie superalgebra generators as invariant derivations

Bridging sheaf-theoretical and differential geometric approaches

Abstract

The orthosymplectic supergroup OSp(m|2n) and unitary supergroup U(p|q) are studied following a new approach that starts from Harish-Chandra pairs and links the sheaf-theoretical supermanifold approach of Berezin and others with the differential geometry approach of Rogers and others. The matrix elements of the fundamental representation of the Lie supergroup G are expressed in terms of functions on the product supermanifold G_0 x R^{0|N}, with G_0 the underlying Lie group and N the odd dimension of G. This product supermanifold is isomorphic to the supermanifold of G. This leads to a new expression for the standard generators of the corresponding Lie superalgebra g as invariant derivations on G. Using these results a new and transparent formula for the invariant integrals on OSp(m|2n) and U(p|q) is obtained.