Chiral de Rham complex and orbifolds
Authors:
Edward Frenkel and Matthew Szczesny
Journal:
J. Algebraic Geom. 16 (2007), 599-624
DOI:
https://doi.org/10.1090/S1056-3911-07-00466-3
Published electronically:
May 1, 2007
MathSciNet review:
2357685
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Abstract |
References |
Additional Information
Abstract: Suppose that a finite group $G$ acts on a smooth complex variety $X$. Then this action lifts to the Chiral de Rham complex $\Omega ^{\operatorname {ch}}_{X}$ of $X$ and to its cohomology by automorphisms of the vertex algebra structure. We define twisted sectors for $\Omega ^{\operatorname {ch}}_{X}$ (and their cohomologies) as sheaves of twisted vertex algebra modules supported on the components of the fixed-point sets $X^{g}, g \in G$. Each twisted sector sheaf carries a BRST differential and is quasi-isomorphic to the de Rham complex of $X^{g}$. Putting the twisted sectors together with the vacuum sector and taking $G$-invariants, we recover the additive and graded structures of Chen-Ruan orbifold cohomology. Finally, we show that the orbifold elliptic genus is the partition function of the direct sum of the cohomologies of the twisted sectors.
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Additional Information
Edward Frenkel
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720
MR Author ID:
257624
ORCID:
0000-0001-6519-8132
Email:
frenkel@math.berkeley.edu
Matthew Szczesny
Affiliation:
Department of Mathematics, Boston University, 111 Cummington Street, Boston, Massachusetts 02215
Email:
szczesny@math.upenn.edu
Received by editor(s):
January 1, 2004
Received by editor(s) in revised form:
November 6, 2006
Published electronically:
May 1, 2007
Additional Notes:
The first author was partially supported by grants from the Packard Foundation and the NSF