Chen-Ruan Cohomology and Stringy Orbifold K-Theory for Stable Almost Complex Orbifolds

Abstract

Comparing to the construction of stringy cohomology ring of equivariant stable almost complex manifolds and its relation with the Chen-Ruan cohomology ring of the quotient almost complex orbifolds, the authors construct in this note a Chen-Ruan cohomology ring for a stable almost complex orbifold. The authors show that for a finite group G and a G-equivariant stable almost complex manifold X, the G-invariant part of the stringy cohomology ring of (X, G) is isomorphic to the Chen-Ruan cohomology ring of the global quotient stable almost complex orbifold [X/G]. Similar result holds when G is a torus and the action is locally free. Moreover, for a compact presentable stable almost complex orbifold, they study the stringy orbifold K-theory and its relation with Chen-Ruan cohomology ring.

Keywords

Stable almost complex orbifolds / Chen-Ruan cohomology / Orbifold K-theory / Stringy product

References

[1] Adem A, Leida J, Ruan Y. Orbifolds and Stringy Topology, 2007, Cambridge: Cambridge University Press

[2] Adem A, Ruan Y. Twisted orbifold K-theory. Commun. Math. Phys., 2003, 237(3): 533-556

[3] Adem A, Ruan Y, Zhang B. A stringy product on twisted orbifold K-theory. Morfismos, 2007, 11(2): 33-64

[4] Becerra E, Uribe B. Stringy product on twisted orbifold K-theory for abelian quotients. Trans. Amer. Math. Soc., 2009, 361(11): 5781-5803

[5] Brender A. Stable complex manifolds. Bull. Amer. Math. Soc., 1966, 72: 978-979

[6] Chen B, Du C-Y, Li T. A new equivariant cohomology ring. Math. Z., 2020, 295: 1163-1182

[7] Chen B, Hu S. A deRham model of Chen-Ruan cohomology ring of abelian orbifolds. Math. Ann., 2006, 336(1): 51-71

[8] Chen W, Ruan Y. Orbifold Gromov-Witten theory. Cont. Math., 2002, 310: 25-85

[9] Chen W, Ruan Y. A new cohomology theory for orbifold. Commun. Math. Phys., 2004, 248: 1-31

[10] Ding F, Jiang Y, Pan J. The Chen-Ruan cohomology of almost contact orbifolds. Acta Math. Sin. (Eng. Ser.), 2010, 26(1): 77-88

[11] Edidin D, Jarvis T, Kimura T. Logarithmic trace and orbifold products. Duke Math. J., 2010, 153(3): 427-473

[12] Edidin D, Jarvis T, Kimura T. A plethora of inertial products. Ann. K-Theory, 2016, 1(1): 85-108

[13] Edidin D, Jarvis T, Kimura T. Chern classes and compatible power operations in inertial K-theory. Ann. K-theory, 2017, 2(1): 73-130

[14] Fantechi B, Gottsche L. Orbifold cohomology for global quotients. Duke Math. J., 2003, 117(2): 197-227

[15] Goldin R, Holm T, Knutson A. Torsion in the full orbifold K-theory of abelian symplectic quotients. Geom. Dedicata, 2012, 157: 187-204

[16] Hu J, Wang B-L. Delocalized Chern character for stringy orbifold K-theory. Trans. Amer. Math. Soc., 2013, 365: 6309-6341

[17] Jarvis T, Kaufmann R, Kimura T. Stringy K-theory and the Chern character. Invent. Math., 2007, 168: 23-81

[18] Lin Y. Twisted orbifold K-theory for global quotient. Topology Appl., 2017, 225: 9-26

[19] Lupercio E, Uribe B. Gerbes over orbifolds and twisted K-Theory. Commun. Math. Phys., 2004, 245: 449-489

[20] Quillen D. Elementary proof of some results of cobordism theory using Steenord operators. Adv. Math., 1971, 7: 29-56