The first fundamental theorem of invariant theory for the orthosymplectic super group

TL;DR

This paper presents a new proof of the first fundamental theorem of invariant theory for the orthosymplectic super group, extending methods to related super groups and explaining properties of Sergeev's super Pfaffian.

Contribution

It introduces a novel proof technique inspired by Atiyah, Bott, and Patodi, applicable to super groups, and clarifies the polynomial nature of Sergeev's super Pfaffian.

Findings

New proof of the fundamental theorem for orthosymplectic super group

Extension of methods to periplectic super group

Explanation of Sergeev's super Pfaffian as polynomial

Abstract

We give a new proof, inspired by an argument of Atiyah, Bott and Patodi, of the first fundamental theorem of invariant theory for the orthosymplectic super group. We treat in a similar way the case of the periplectic super group. Lastly, the same method is used to explain the fact that Sergeev's super Pfaffian, an invariant for the special orthosymplectic super group, is polynomial.