nLab orbifold K-theory

Idea

Orbifold K-theory is K-theory (typically: topological K-theory) of orbifolds, typically meant to reduce to equivariant K-theory on global quotient orbifolds.

Definition

In general it is subtle to decide whether a given orbifold cohomology-theory $E^\bullet(\mathcal{X})$ is equivalently the $G$-equivariant cohomology of a topological G-space $X$ for any realization of $\mathcal{X}$ as a global quotient orbifold of $X$ by $G$ (as highlighted inPronk-Scull 07). But for topological equivariant K-theory this is the case, by Pronk-Scull 07, Prop. 4.1.

Therefore it makes sense to define the K-theory of an orbifold $\mathcal{X}$ which is equivalent to a global quotient orbifold

$$\mathcal{X} \simeq \prec(X \!\sslash\! G)$$

to be the $G$-equivariant K-theory of $X$:

$$K^\bullet(\mathcal{X}) \;\coloneqq\; K_G^\bullet(X) \,.$$

This is the approach taken in AdemRuan 01, Def. 3.4

Related concepts

• equivariant K-theory

• twisted equivariant K-theory

• orbifold differential K-theory

• twisted ad-equivariant Tate K-theory

  • equivariant elliptic cohomology

References

General

The definition originates, via bundle gerbes and bundle gerbe modules on Lie groupoids, in:

• Ernesto Lupercio, Bernardo Uribe, Gerbes over Orbifolds and Twisted K-theory, Comm. Math. Phys. 245 (2004) 449-489. (arXiv:math/0105039, doi:10.1007/s00220-003-1035-x)

The definition of the K-theory of global quotient orbifolds as the twisted equivariant K-theory of the universal covering space appears in

• Alejandro Adem, Yongbin Ruan, Twisted Orbifold K-Theory, Commun. Math. Phys. 237 (2003) 533-556 $[$arXiv:math/0107168, doi:10.1007/s00220-003-0849-x$]$

Review in

• Alejandro Adem, Johann Leida, Yongbin Ruan, Section 3 of: Orbifolds and Stringy Topology, Cambridge Tracts in Mathematics 171 (2007) (doi:10.1017/CBO9780511543081, pdf)

The general proof that this is well-defined (independent of the realization of the orbifold as a global quotient) is due to

• Dorette Pronk, Laura Scull, Section 4 of: Translation Groupoids and Orbifold Bredon Cohomology, Canad. J. Math. 62(2010), 614-645 (arXiv:0705.3249, doi:10.4153/CJM-2010-024-1)

A more geometric model of orbifold K-theory in terms of bundles of Fredholm operators over Lie groupoids/differentiable stacks:

• Daniel Freed, Michael Hopkins, Constantin Teleman, Loop groups and twisted K-theory I, Journal of Topology, 4 4 [arXiv:0711.1906, doi:10.1112/jtopol/jtr019]

Review in:

• Daniel Freed, Lecture 1 of: Lectures on twisted K-theory and orientifolds, lecures at K-Theory and Quantum Fields, ESI 2012 (pdf)

• Joost Nuiten, Section 3.2.2 of: Cohomological quantization of local prequantum boundary field theory MSc thesis, Utrecht, August 2013 (pdf)

• Valentin Zakharevich, Sections 2.2, 2.3 of: K-Theoretic Computation of the Verlinde Ring, thesis 2018 (hdl:2152/67663, pdf, pdf)

The claim that these two definitions are equivalent, in that this groupoid K-theory reduces to equivariant K-theory on global quotient orbifolds, is Freed-Hopkins-Teleman 07, Prop. 3.5.

Another definition of K-theory of orbifolds (“full orbifold K-theory”) is due to

• Tyler J. Jarvis, Ralph Kaufmann, Takashi Kimura, Stringy K-theory and the Chern character, Invent. math. 168, 23–81 (2007) (arXiv:math/0502280, doi:10.1007/s00222-006-0026-x)

proven there to coincide with Adem-Ruan 01 on global quotients.

More on this in:

• Rebecca Goldin, Megumi Harada, Tara S. Holm, Takashi Kimura, The full orbifold K-theory of abelian symplectic quotients, Journal of K-theory, Volume 8, Issue 2 (2011) pp. 339-362 (arXiv:0812.4964, doi:10.1017/is010005021jkt118)

Via global equivariant homotopy theory

The suggestion (Schwede 17, Intro, Schwede 18, p. ix-x) that orbifolds should be regarded as orbispaces in global equivariant homotopy theory and then their orbifold cohomology be given by equivariant cohomology with coefficients in global equivariant spectra is worked out for (Bredon cohomology and) orbifold K-theory in:

• Branko Juran, Orbifolds, Orbispaces and Global Homotopy Theory (arXiv:2006.12374)

Example 5.31 there shows that on global quotient orbifolds this is again equivalent to the previous definitions.

Stringy product

• Alejandro Adem, Yongbin Ruan, Bin Zhang, A Stringy Product on Twisted Orbifold K-theory, Morfismos (10th Anniversary Issue), Vol. 11, No 2 (2007), 33-64. (arXiv:math/0605534, Morfismos pdf)

• Edward Becerra, Bernardo Uribe, Stringy product on twisted orbifold K-theory for abelian quotients, Trans. Amer. Math. Soc. 361 (2009), 5781-5803 (arXiv:0706.3229, doi:10.1090/S0002-9947-09-04760-6)

• Jianxun Hu, Bai-Ling Wang, Delocalized Chern character for stringy orbifold K-theory, Trans. Amer. Math. Soc. 365 (2013), 6309-6341 (arXiv:1110.0953, doi:10.1090/S0002-9947-2013-05834-5)