Discrete Torsion for the Supersingular Orbifold Sigma Genus

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 [HKR00] to the Borel-equivariant genus associated to the sigma orientation of [AHS01] 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 [DMVV97] and [BL02]. In the case of a finite cyclic orbifold group, we use the characteristic series for the two-variable genus in the formulae of [And03] 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 H(BG; Z), where G is the orbifold group, produces discrete torsion in the sense of [Vaf85]. This result was first obtained by Sharpe [Sha], for a different orbifold genus and using different methods.