Supermetrics on supermanifolds

TL;DR

This paper extends a classical theorem about principal bundles and their reductions to the setting of supermanifolds, linking supergroup reductions to supermetrics on G-supermanifolds.

Contribution

It generalizes the reduction theorem to principal superbundles in the category of G-supermanifolds, connecting supergroup reductions with supermetrics.

Findings

Reduction of supergroup to orthogonal-symplectic subgroup corresponds to a supermetric.

Extension of classical bundle reduction theorem to supermanifolds.

Establishment of conditions for supermetric existence on G-supermanifolds.

Abstract

By virtue of the well-known theorem, a structure Lie group K of a principal bundle $P$ is reducible to its closed subgroup H iff there exists a global section of the quotient bundle P/K. In gauge theory, such sections are treated as Higgs fields, exemplified by pseudo-Riemannian metrics on a base manifold of P. Under some conditions, this theorem is extended to principal superbundles in the category of G-supermanifolds. Given a G-supermanifold M and a graded frame superbundle over M with a structure general linear supergroup, a reduction of this structure supergroup to an orthgonal-symplectic supersubgroup is associated to a supermetric on a G-supermanifold M.