The Index Theorem for Homogeneous Differential Operators on Supermanifolds

TL;DR

This paper extends Bott's index theorem for homogeneous elliptic operators to the setting of supermanifolds with the supergroup U(p|q), highlighting differences in character formulas for atypical representations.

Contribution

It formulates the index theorem for homogeneous differential operators on supermanifolds, specifically for the supergroup U(p|q), addressing atypical representation cases.

Findings

Index theorem formulated for supermanifolds and supergroups

Atypical representations have different character formulas

Contradicts previous statements on supermanifold index theory

Abstract

In mid 60s Bott proved that (1) the index theorem for homogeneous, G-invariant, elliptic differential operators acting in the spaces of sections of induced representations of G over G/H reduces to the Weyl character formula and (2) the index of an equivariant elliptic operator does not depend on the operator, but on the representations. Here the same theorem is formulated for the unitary supergroup G=U(p|q). For atypical representations the character formula does not reduce to that for the Lie group underlying the supergroup G and this contradicts a statement of Rempel and Schmitt on index on supermanifolds (Pseudodifferential operators and the index theorem on supermanifolds. Seminar Analysis, 1981/82, 92--131, Akad. Wiss. DDR, Berlin, 1982; id., Pseudodifferential operators and the index theorem on supermanifolds. Math. Nachr. 111 (1983), 153--175).