Discrete torsion for the supersingular orbifold sigma genus

TL;DR

This paper explores the relationship between equivariant and orbifold elliptic genera, introduces an orbifold genus for supersingular elliptic curves, and shows how changing BU<6>-structures induces discrete torsion effects.

Contribution

It defines an orbifold genus for supersingular elliptic curves using character theory and relates it to existing orbifold genera, also analyzing the impact of BU<6>-structure variations.

Findings

Orbifold genus matches the Dijkgraaf et al. formula.

Varying BU<6>-structure induces discrete torsion.

Analytic equivariant genus coincides with the orbifold two-variable genus.

Abstract

The first purpose of this paper is to examine the relationship between equivariant elliptic genera and orbifold elliptic genera. We apply the character theory of Hopkins et. al. to the Borel-equivariant genus associated to the sigma orientation of Ando-Hopkins-Strickland to define an orbifold genus for certain total quotient orbifolds and supersingular elliptic curves. We show that our orbifold genus is given by the same sort of formula as the orbifold ``two-variable'' genus of Dijkgraaf et al. In the case of a finite cyclic orbifold group, we use the characteristic series for the two-variable genus to define an analytic equivariant genus in Grojnowski's equivariant elliptic cohomology, and we show that this gives precisely the orbifold two-variable genus. The second purpose of this paper is to study the effect of varying the BU<6>-structure in the Borel-equivariant sigma orientation. We show that varying the BU<6>-structure by a class in the third cohomology of the orbifold group produces discrete torsion in the sense of Vafa. This result was first obtained by Sharpe, for a different orbifold genus and using different methods.