## Stringy product on twisted orbifold K-theory for abelian quotients

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- by Edward Becerra and Bernardo Uribe PDF
- Trans. Amer. Math. Soc.
**361**(2009), 5781-5803 Request permission

## Abstract:

In this paper we present a model to calculate the stringy product on twisted orbifold K-theory of Adem-Ruan-Zhang for abelian complex orbifolds.

In the first part we consider the non-twisted case on an orbifold presented as the quotient of a manifold acted by a compact abelian Lie group. We give an explicit description of the obstruction bundle, we explain the relation with the product defined by Jarvis-Kaufmann-Kimura and, via a Chern character map, with the Chen-Ruan cohomology, we explicitly calculate the stringy product for a weighted projective orbifold.

In the second part we consider orbifolds presented as the quotient of a manifold acted by a finite abelian group and twistings coming from the group cohomology. We show a decomposition formula for twisted orbifold K-theory that is suited to calculate the stringy product and we use this formula to calculate two examples when the group is $(\mathbb {Z}/2)^3$.

## References

- Alejandro Adem and Yongbin Ruan,
*Twisted orbifold $K$-theory*, Comm. Math. Phys.**237**(2003), no. 3, 533–556. MR**1993337**, DOI 10.1007/s00220-003-0849-x - A. Adem, Y. Ruan, and B. Zhang. A stringy product on twisted orbifold $K$-theory. arxiv:math.AT/0605534.
- M. F. Atiyah and I. M. Singer,
*The index of elliptic operators. III*, Ann. of Math. (2)**87**(1968), 546–604. MR**236952**, DOI 10.2307/1970717 - M. F. Atiyah and G. B. Segal,
*Equivariant $K$-theory and completion*, J. Differential Geometry**3**(1969), 1–18. MR**259946** - Bohui Chen and Shengda Hu,
*A deRham model for Chen-Ruan cohomology ring of abelian orbifolds*, Math. Ann.**336**(2006), no. 1, 51–71. MR**2242619**, DOI 10.1007/s00208-006-0774-3 - Weimin Chen and Yongbin Ruan,
*A new cohomology theory of orbifold*, Comm. Math. Phys.**248**(2004), no. 1, 1–31. MR**2104605**, DOI 10.1007/s00220-004-1089-4 - Ali Nabi Duman. An example of a twisted fusion algebra. Preprint.
- Tyler J. Jarvis, Ralph Kaufmann, and Takashi Kimura,
*Stringy $K$-theory and the Chern character*, Invent. Math.**168**(2007), no. 1, 23–81. MR**2285746**, DOI 10.1007/s00222-006-0026-x - Wolfgang Lück and Bob Oliver,
*Chern characters for the equivariant $K$-theory of proper $G$-CW-complexes*, Cohomological methods in homotopy theory (Bellaterra, 1998) Progr. Math., vol. 196, Birkhäuser, Basel, 2001, pp. 217–247. MR**1851256** - Ernesto Lupercio and Bernardo Uribe,
*Loop groupoids, gerbes, and twisted sectors on orbifolds*, Orbifolds in mathematics and physics (Madison, WI, 2001) Contemp. Math., vol. 310, Amer. Math. Soc., Providence, RI, 2002, pp. 163–184. MR**1950946**, DOI 10.1090/conm/310/05403 - Ieke Moerdijk,
*Orbifolds as groupoids: an introduction*, Orbifolds in mathematics and physics (Madison, WI, 2001) Contemp. Math., vol. 310, Amer. Math. Soc., Providence, RI, 2002, pp. 205–222. MR**1950948**, DOI 10.1090/conm/310/05405 - Daniel Quillen,
*Elementary proofs of some results of cobordism theory using Steenrod operations*, Advances in Math.**7**(1971), 29–56 (1971). MR**290382**, DOI 10.1016/0001-8708(71)90041-7 - Graeme Segal,
*Equivariant $K$-theory*, Inst. Hautes Études Sci. Publ. Math.**34**(1968), 129–151. MR**234452**

## Additional Information

**Edward Becerra**- Affiliation: Departamento de Matemáticas, Universidad de los Andes, Carrera 1 N. 18A - 10, Bogotá, Colombia
- Email: es.becerra75@uniandes.edu.co
**Bernardo Uribe**- Affiliation: Departamento de Matemáticas, Universidad de los Andes, Carrera 1 N. 18A - 10, Bogotá, Colombia
- Email: buribe@uniandes.edu.co
- Received by editor(s): June 27, 2007
- Published electronically: June 4, 2009
- Additional Notes: Both authors acknowledge the support of COLCIENCIAS through the grant 120440520246 and of CONACYT-COLCIENCIAS throught contract number 376-2007

The second author was partially supported by the “Fondo de apoyo a investigadores jovenes” from Universidad de los Andes - © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**361**(2009), 5781-5803 - MSC (2000): Primary 14N35, 19L47; Secondary 55N15, 55N91
- DOI: https://doi.org/10.1090/S0002-9947-09-04760-6
- MathSciNet review: 2529914