Contents

# Contents

## Idea

Tate K-theory is the elliptic cohomology theory associated with the Tate curve (the Tate elliptic curve over the Laurent series ring $\mathbb{Z}((q))$ ) (AHS 01, 2.7, Lurie 09, 4.3)

The corresponding elliptic genus is the Witten genus (AHS 01, Sec. 2.7).

### Plain version

The underlying cohomology theory is given by Laurent series in topological K-theory, and equivalently by completion of circle group-equivariant K-theory of free loop spaces $\mathcal{L}(-)$ (KM 04, Section 5, see also Lurie 09, Section 5.2):

\begin{aligned} Ell_{Tate}(X) & \simeq \; K(X)((q)) \\ & \simeq \; K_{S^1}(\mathcal{L}X) \otimes_{K_{S^1}(\ast)} \mathbb{Z}((q)) \,. \end{aligned}

### Equivariant version

The equivariant version of Tate K-theory is a form of equivariant elliptic cohomology. For $G$ a finite group and $X$ a topological G-space it comes down (Ganter 07, Def. 3.1, Ganter 13, Def. 2.6, review in Dove 19, Def. 6.16) to the sub-ring

$Ell_{Tate} \big( \prec (X \sslash G) \big) \;\subset\; \underset{[g]}{\bigoplus} K_{C_g}(X^g)(( q^{1/\left\vert g\right\vert} ))$

of the direct sum, over conjugacy classes of group elements $g$, of Laurent polynomials with coefficients in equivariant K-theory-groups on the $g$-fixed loci for equivariance group the centralizer of $g$

on those elements which satisfy the rotation condition:

Rotation condition. The $C_g$-equivariant vector bundles $V_j$ which form the coefficient of $q^{j/\left\vert g \right\vert }$ are such that $g$ acts on them by multiplication with $\exp\big( 2 \pi i \frac{j}{\left\vert g \right\vert} \big)$.

This rotation condition may be understood more intrinsically (this is made explicit on p. 63 of Dove 19) as that in implied by the orbifold K-theory on Huan's inertia orbifold (Huan 18), induced by the nature of the quotient groups

$\Lambda_g \;\coloneqq\; \frac{C_g \times \mathbb{R}}{ \langle (g^{-1},1) \rangle }$

appearing there. This quotient exactly equates the action of $g \in G$ with that of a rotation of $S^1$.

Hence, in generalization of twisted ad-equivariant K-theory there is twisted ad-equivariant Tate K-theory (an equivariant elliptic cohomology theory) relating to the Verlinde ring of positive energy loop group representations (Lurie 09, Sec. 5.2, Luecke 19, Cor. 3.2.5, Dove 19).

## Properties

### Relation to elliptic genus

The Ochanine genus lifts to a homomorphism of ring spectra $M Spin \to KO((q))$ from spin structure cobordism cohomology theory to Tate K-theory (Kreck-Stolz 93, lemma 5.8, lemma 5.4). This is the spin-orientation of elliptic cohomology

## References

As elliptic cohomology over the Tate sphere:

As completed $S^1$-equivariant K-theory of free loop space

with a streamlined account in

For simply-connected compact Lie groups:

• Kiran Luecke, Completed K-theory and Equivariant Elliptic Cohomology (arXiv:1904.00085)

For finite groups:

Last revised on July 9, 2021 at 06:16:02. See the history of this page for a list of all contributions to it.