Global stringy orbifold cohomology, K-theory and de Rham theory
type: Item
instance of: scholarly article
title: Global stringy orbifold cohomology, K-theory and de Rham theory
zbMATH Open document ID: 1217.14041
author: Ralph M. Kaufmann
published in: Letters in Mathematical Physics
publication date: 2010-11-12T00:00:00Z
review text: There are basically two approaches to stringy multiplications for global quotients by finite group actions. The first one, the so-called `push-after-pull' approach, has been used to define stringy extensions of the cohomology ring [\textit{W.~Chen} and \textit{Y.~Ruan}, Commun. Math. Phys. 248, No. 1, 1--31 (2004; Zbl 1063.53091); \textit{B.~Fantechi} and \textit{L.~Göttsche}, Duke Math. J. 117, No. 2, 197--227 (2003; Zbl 1086.14046)], of the Chow ring [\textit{D.~Abramovich, T.~Graber} and \textit{A.~Vistoli}, Contemp. Math. 310, 1--24 (2002; Zbl 1067.14055)], and of the Grothendieck ring and of topological K-theory [\textit{T.~J.~Jarvis, R.~Kaufmann} and \textit{T.~Kimura}, Invent. Math. 168, No. 1, 23--81 (2007; Zbl 1132.14047)] of a smooth projective \(G\)-manifold or of a smooth \(G\)-manifold endowed with a \(G\)-equivariant stably almost complex structure. The second one, the so called `pull-after-push' approach, has been applied by the author in the study of the cyclic twisted sector and in the study of singularities with symmetries; in this approach, the stringy multiplications are described in terms of the solutions of an algebraic cocycle equation [\textit{R.~M.~Kaufmann}, Int. J. Math. 14, No. 6, 573--617 (2003; Zbl 1083.57037); Commun. Math. Phys. 248, No. 1, 33--83 (2004; Zbl 1099.16010); Contemporary Mathematics 403, 67--116 (2006; Zbl 1116.14037)]. The `pull-after-push' approach has also been used by \textit{B. Chen} and \textit{S. Hu} to describe a de Rham model for the Chen-Ruan cohomology ring of an abelian orbifold [Math. Ann. 336, No. 1, 51--71 (2006; Zbl 1122.14018)]. In the present paper the author gives a rigorous formulation of the de Rham theory for any (i.e., not necessarily abelian) global quotient by a finite group, from both the push-after-pull and the pull-after-push point of view. It is also shown how in the cyclic twisted sector the push-after-push formula of Jarvis-Kaufmann-Kimura gives a solution of the pull-after-push equation, wh…
reviewed by: Domenico Fiorenza
Mathematics Subject Classification ID: 14N35
Mathematics Subject Classification ID: 53D45
Mathematics Subject Classification ID: 81T45
Mathematics Subject Classification ID: 14F40
Mathematics Subject Classification ID: 14C35
Mathematics Subject Classification ID: 19E08
zbMATH DE Number: 5815662
zbMATH Keywords: orbifolds
zbMATH Keywords: stringy geometry
zbMATH Keywords: K-theory
zbMATH Keywords: twist fields
zbMATH Keywords: orbifold field theory
zbMATH Keywords: de Rham theory
zbMATH Keywords: singularity theory
zbMATH Keywords: Landau-Ginzburg theory
MaRDI profile type: Publication
full work available at URL: s11005-010-0427-z
OpenAlex ID: W2013061051
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DOI: 10.1007/S11005-010-0427-Z
description: scientific article; zbMATH DE number 5815662