Stringy product on twisted orbifold Ktheory for abelian quotients
Authors:
Edward Becerra and Bernardo Uribe
Journal:
Trans. Amer. Math. Soc. 361 (2009), 57815803
MSC (2000):
Primary 14N35, 19L47; Secondary 55N15, 55N91
Published electronically:
June 4, 2009
MathSciNet review:
2529914
Fulltext PDF Free Access
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Abstract: In this paper we present a model to calculate the stringy product on twisted orbifold Ktheory of AdemRuanZhang for abelian complex orbifolds. In the first part we consider the nontwisted 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 JarvisKaufmannKimura and, via a Chern character map, with the ChenRuan 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 Ktheory that is suited to calculate the stringy product and we use this formula to calculate two examples when the group is .
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Twisted orbifold theory. Comm. Math. Phys., 237(3):533556, 2003. MR 1993337 (2004e:19004)
 [ARZ]
 A. Adem, Y. Ruan, and B. Zhang.
A stringy product on twisted orbifold theory. arxiv:math.AT/0605534.
 [AS68]
 M. F. Atiyah and I. M. Singer.
The index of elliptic operators. III. Ann. of Math. (2), 87:546604, 1968. MR 0236952 (38:5245)
 [AS69]
 M. F. Atiyah and G. B. Segal.
Equivariant theory and completion. J. Differential Geometry, 3:118, 1969. MR 0259946 (41:4575)
 [CH06]
 B. Chen and S. Hu.
A deRham model for ChenRuan cohomology ring of abelian orbifolds. Math. Ann., 336(1):5171, 2006. MR 2242619 (2007d:14044)
 [CR04]
 W. Chen and Y. Ruan.
A new cohomology theory of orbifold. Comm. Math. Phys., 248(1):131, 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 theory and the Chern character. Invent. Math., 168(1):2381, 2007. MR 2285746 (2007i:14059)
 [LO01]
 W. Lück and B. Oliver.
Chern characters for the equivariant theory of proper CWcomplexes. In Cohomological methods in homotopy theory (Bellaterra, 1998), volume 196 of Progr. Math., pages 217247. Birkhäuser, Basel, 2001. MR 1851256 (2002m:55016)
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 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 163184. Amer. Math. Soc., Providence, RI, 2002. MR 1950946 (2004c:58043)
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 I. Moerdijk.
Orbifolds as groupoids: An introduction. In Orbifolds in mathematics and physics (Madison, WI, 2001), volume 310 of Contemp. Math., pages 205222. 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., 7:2956 (1971), 1971. MR 0290382 (44:7566)
 [Seg68]
 G. Segal.
Equivariant theory. Inst. Hautes Études Sci. Publ. Math., (34):129151, 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:
http://dx.doi.org/10.1090/S0002994709047606
PII:
S 00029947(09)047606
Keywords:
Stringy product,
twisted orbifold Ktheory,
ChenRuan 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 CONACYTCOLCIENCIAS throught contract number 3762007
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.
