Chiral de Rham Complex and Orbifolds

2003/07/14 by Edward Frenkel, Frenkel, Edward, Matthew Szczesny +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #math.AG #math.QA

paper · pdf · doi:10.48550/arxiv.math/0307181

openalex publication_date 2003/07/14 · arxiv created 2003/07/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Suppose that a finite group G acts on a smooth complex variety X. Then this action lifts to the Chiral de Rham Complex of X and to its cohomology by automorphisms of the vertex algebra structure. We define twisted sectors for the Chiral de Rham Complex (and their cohomologies) as sheaves of twisted vertex algebra modules supported on the components of the fixed-point sets X^g, g ∈ 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.

Citations

• Elliptic Genera of singular varieties, orbifold elliptic genus and chiral deRham complex

Cited by

• Pointed admissible G-covers and G-equivariant cohomological field theories
• (Log) Chiral de Rham complex and A_n-singular surfaces

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