Power operations in orbifold Tate $K$-theory

Abstract

We formulate the axioms of an orbifold theory with power operations. We define orbifold Tate K-theory, by adjusting Devoto's definition of the equivariant theory, and proceed to construct its power operations. We calculate the resulting symmetric powers, exterior powers and Hecke operators and put our work into context with orbifold loop spaces, level structures on the Tate curve and generalized Moonshine.

Quasi-elliptic cohomology I

“…in the same way as (2.11). L orb (X/ /G) has the same objects as the orbifold loop space in [17] and has more morphisms with the T−action added.…”

“…In this section 3.4 we introduce the induction formula for quasi-elliptic cohomology. The induction formula for Tate K-theory is constructed in Section 2.3.3, [18].…”

“…In Section 4, we present the construction of orbifold quasi-elliptic cohomology via the loop space of bibundles. Moreover, we give another construction motivated by Ganter's construction of orbifold Tate K-theory in [18]. The two constructions of orbifold quasi-elliptic cohomology are equivalent.…”

Quasi-theories and their equivariant orthogonal spectra

“…In addition, Tate K-theory has the closest ties to Witten's original insight that the elliptic cohomology of a space X is related to the T−equivariant K-theory of the free loop space LX = C ∞ (S 1 , X) with the circle T acting on LX by rotating loops. Ganter gave a careful interpretation in Section 2, [5] of this statement that the definition of G−equivariant Tate K-theory for finite groups G is modelled on the loop space of a global quotient orbifold.…”

Almost Global Homotopy Theory

“…Example 6.4 (Generalized Tate K-theory and generalized quasi-elliptic cohomology). In Section 2 [8] Ganter gave an interpretation of G−equivariant Tate K-theory for finite groups G by the loop space of a global quotient orbifold. Apply the loop construction n times, we can get the n−th generalized Tate K-theory.…”