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Stringy product on twisted orbifold K-theory for abelian quotients


Authors: Edward Becerra and Bernardo Uribe
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
Published electronically: June 4, 2009
MathSciNet review: 2529914
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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 $ (\ensuremath{{\mathbb{Z}}}/2)^3$.


References [Enhancements On Off] (What's this?)

  • [AR03] A. Adem and Y. Ruan.
    Twisted orbifold $ K$-theory.
    Comm. Math. Phys., 237(3):533-556, 2003. MR 1993337 (2004e:19004)
  • [ARZ] A. Adem, Y. Ruan, and B. Zhang.
    A stringy product on twisted orbifold $ K$-theory.
    arxiv:math.AT/0605534.
  • [AS68] M. F. Atiyah and I. M. Singer.
    The index of elliptic operators. III.
    Ann. of Math. (2), 87:546-604, 1968. MR 0236952 (38:5245)
  • [AS69] M. F. Atiyah and G. B. Segal.
    Equivariant $ K$-theory and completion.
    J. Differential Geometry, 3:1-18, 1969. MR 0259946 (41:4575)
  • [CH06] B. Chen and S. Hu.
    A deRham model for Chen-Ruan cohomology ring of abelian orbifolds.
    Math. Ann., 336(1):51-71, 2006. MR 2242619 (2007d:14044)
  • [CR04] W. Chen and Y. Ruan.
    A new cohomology theory of orbifold.
    Comm. Math. Phys., 248(1):1-31, 2004. MR 2104605 (2005j:57036)
  • [Dum] Ali Nabi Duman.
    An example of a twisted fusion algebra.
    Preprint.
  • [JKK07] Tyler J. Jarvis, Ralph Kaufmann, and Takashi Kimura.
    Stringy $ K$-theory and the Chern character.
    Invent. Math., 168(1):23-81, 2007. MR 2285746 (2007i:14059)
  • [LO01] W. Lück and B. Oliver.
    Chern characters for the equivariant $ K$-theory of proper $ G$-CW-complexes.
    In Cohomological methods in homotopy theory (Bellaterra, 1998), volume 196 of Progr. Math., pages 217-247. Birkhäuser, Basel, 2001. MR 1851256 (2002m:55016)
  • [LU02] E. Lupercio and B. Uribe.
    Loop groupoids, gerbes, and twisted sectors on orbifolds.
    In Orbifolds in mathematics and physics (Madison, WI, 2001), volume 310 of Contemp. Math., pages 163-184. Amer. Math. Soc., Providence, RI, 2002. MR 1950946 (2004c:58043)
  • [Moe02] I. Moerdijk.
    Orbifolds as groupoids: An introduction.
    In Orbifolds in mathematics and physics (Madison, WI, 2001), volume 310 of Contemp. Math., pages 205-222. Amer. Math. Soc., Providence, RI, 2002. MR 1950948 (2004c:22003)
  • [Qui71] D. Quillen.
    Elementary proofs of some results of cobordism theory using Steenrod operations.
    Advances in Math., 7:29-56 (1971), 1971. MR 0290382 (44:7566)
  • [Seg68] G. Segal.
    Equivariant $ K$-theory.
    Inst. Hautes Études Sci. Publ. Math., (34):129-151, 1968. MR 0234452 (38:2769)

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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

DOI: https://doi.org/10.1090/S0002-9947-09-04760-6
Keywords: Stringy product, twisted orbifold K-theory, Chen-Ruan cohomology, inverse transgression map
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
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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