Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

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: 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
Email: frenkel@math.berkeley.edu

Matthew Szczesny
Affiliation: Department of Mathematics, Boston University, 111 Cummington Street, Boston, Massachusetts 02215
Email: szczesny@math.upenn.edu

DOI: https://doi.org/10.1090/S1056-3911-07-00466-3
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

American Mathematical Society