nLab Tate K-theory

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):

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

(For more exposition see also Dove 19, Sec. 6.2).

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

Related concepts

• elliptic genus

  • Ochanine genus

  • Witten genus

• elliptic Chern character

• topological K-theory

• twisted ad-equivariant K-theory

References

As elliptic cohomology over the Tate sphere:

• Matthew Ando, Michael Hopkins, Neil Strickland, Section 2.7 of: Elliptic spectra, the Witten genus and the theorem of the cube, Invent. math. 146, 595–687 (2001) [doi:10.1007/s002220100175]

• Jacob Lurie, A Survey of Elliptic Cohomology, in: Algebraic Topology Abel Symposia 4 (2009) 219-277 [pdf, doi:10.1007/978-3-642-01200-6_9]

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

• Nitu Kitchloo, Jack Morava, Section 5 of: Thom Prospectra for Loopgroup representations [arXiv:math/0404541]

As a form of equivariant elliptic cohomology (twisted ad-equivariant Tate K-theory):

• Nora Ganter, Section 3.1 in: Stringy power operations in Tate K-theory [arXiv:math/0701565]

• Nora Ganter, Power operations in orbifold Tate K-theory, Homology Homotopy Appl. 15 1 (2013) 313-342 [arXiv:1301.2754, euclid:hha/1383943680]

• Charles Rezk, Quasi-Elliptic Cohomology (2014) [pdf]

• Zhen Huan, Quasi-elliptic cohomology (2017) [hdl:2142/97268]

• Zhen Huan, Quasi-Elliptic Cohomology and its Power Operations, J. Homotopy and Related Structures 13 (2018) 715–767 [arXiv:1612.00930, doi:10.1007/s40062-018-0201-y]

• Zhen Huan, Quasi-elliptic cohomology and its Spectrum [arXiv:1703.06562]

• Zhen Huan, Quasi-Elliptic Cohomology I, Advances in Mathematics, 337 (2018) 107-138 [arXiv:1805.06305, doi:10.1016/j.aim.2018.08.007]

with a streamlined account in

• Zhen Huan, Quasi-theories [arXiv:1809.06651]

• Zhen Huan, Quasi-theories and their equivariant orthogonal spectra [arXiv:1809.07622]

and some general abstract clarification on Huan's inertia orbifold in:

• Hisham Sati, Urs Schreiber: Cyclification of Orbifolds, Comm. Math. Phys. (2023) [arXiv:2212.13836, talk]

Review:

• Zhen Huan, Quasi-elliptic cohomology theory and the twisted, twisted Real theories, talk at Perimeter Institute (2020) [video: doi:10.48660/20050059]

For simply-connected compact Lie groups:

• Kiran Luecke, Completed K-theory and Equivariant Elliptic Cohomology, Advances in Mathematics 410 B (2022) 108754 [arXiv:1904.00085, doi:10.1016/j.aim.2022.108754]

For finite groups:

• Thomas Dove, Twisted Equivariant Tate K-Theory [arXiv:1912.02374]

Generalization to twisted equivariant K-theory

• Zhen Huan, Matthew Spong, Twisted Quasi-elliptic cohomology and twisted equivariant elliptic cohomology (arXiv:2006.00554)

and further to KR-theory:

• Zhen Huan, Matthew B. Young, Twisted Real quasi-elliptic cohomology [arXiv:2210.07511]

  Zhen Huan, Twisted Real quasi-elliptic cohomology, talk at CQTS (Nov 2022) [video]

On quasi-elliptic cohomology of representation spheres as an approximation to equivariant Cohomotopy:

• Zhen Huan: Twisted Quasi-elliptic cohomology and twisted equivariant elliptic cohomology [arXiv:2006.00554]

• Zhen Huan: Quasi-elliptic cohomology of 4-spheres [arXiv:2408.02278]