Global stringy orbifold cohomology, K-theory and de Rham theory
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, while in the general, not necessarily cyclic, case this can still be achieved by considering suitable ring extensions. In particular, by these extension, the stringy versions of all the functors \(H^*\), \(A^*\), \(K_0\), \(K^{\text{top}}\) can be treated from the pull-after-push point of view. In the final part of the paper, possible applications to orbifold Landau-Ginzburg theories are described.
scientific article; zbMATH DE number 5815662
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