nLab orbifold K-theory

Idea

Orbifold K-theory is K-theory (typically: topological K-theory) of orbifolds, usually meant to reduce to $G$-equivariant K-theory on $G$-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 by Pronk & Scull 2007). 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 Adem & Ruan 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

Early discussion via $C^\ast$-algebras:

• Matilde Marcolli, Varghese Mathai: Twisted higher index theory on good orbifolds and fractional quantum numbers [arXiv:math/9803051]

• Matilde Marcolli, Varghese Mathai: Twisted index theory on good orbifolds, I: noncommutative Bloch theory, Communications in Contemporary Mathematics 1 4 (1999) 553-587 [arXiv:math/9911102v1, doi:10.1142/S0219199799000213]

• Matilde Marcolli, Varghese Mathai: Twisted higher index theory on good orbifolds, II: fractional quantum numbers, Commun. Math. Phys. 217 1 (2001) 55-87 [arXiv:math/9911103, doi:10.1007/s002200000351]

Incorporating the twist via bundle gerbes and bundle gerbe modules on Lie groupoids:

• 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]

• Jean-Louis Tu, Ping Xu, Camille Laurent-Gengoux: Twisted K-theory of differentiable stacks, Annales Scientifiques de l’École Normale Supérieure 37 6 (2004) 841-910 [arXiv:math/0306138, doi:10.1016/j.ansens.2004.10.002]

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

• 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]

• Alejandro Adem, Johann Leida, Yongbin Ruan: Orbifold K-Theory, chapter 3 of: Orbifolds and Stringy Topology, Cambridge Tracts in Mathematics 171 (2007) [doi:10.1017/CBO9780511543081.004, pdf]

The general observation that this is well-defined (independent of the realization of the orbifold as a global quotient):

• 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 (2016) [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]

• Kiyonori Gomi: Freed-Moore K-theory [arXiv:1705.09134, spire:1601772]

• Valentin Zakharevich, Sections 2.2, 2.3 of: K-Theoretic Computation of the Verlinde Ring, PhD 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 Prop. 3.5 in Freed, Hopkins & Teleman 2007.

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 (2007) 23–81 [arXiv:math/0502280, doi:10.1007/s00222-006-0026-x]

proven there to coincide with Adem & Ruan 2001 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 8 2 (2011) 339-362 [arXiv:0812.4964, doi:10.1017/is010005021jkt118]

A construction of twisted orbifold K-theory using classifying stacks:

• Hisham Sati, Urs Schreiber: Section 3.1 of: Orientations of Orbi-K-Theory measuring Topological Phases and Brane Charges, Part I of: Geometric Orbifold Cohomology, CRC Press (2026)

On orbifold Tate K-theory:

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

See also:

• Carla Farsi: K-Theoretical Index Theorems for Good Orbifolds, Proceedings of the AMS 115 3 (1992) 769-773 [doi:10.2307/2159225, jstor:2159225]

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]