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Chiral de Rham complex and orbifolds

Author(s): Edward Frenkel; Matthew Szczesny
Journal: J. Algebraic Geom. 16 (2007), 599-624.
Posted: May 1, 2007
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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.

References:

[BD]
A. Beilinson, V. Drinfeld, Chiral Algebras, AMS 2004. MR 2058353 (2005d:17007)

[BK]
B. Bakalov, V. G. Kac, Twisted Modules over Lattice Vertex Algebras, in Proc. V Internat. Workshop ``Lie Theory and Its Applications in Physics'' (Varna, June 2003), eds. H.-D. Doebner and V. K. Dobrev, World Scientific, Singapore, 2004. MR 2172171 (2006i:17039)

[Bo]
L. Borisov, Vertex Algebras and Mirror Symmetry, Comm. Math. Phys. 215, (2001) 517-557. MR 1810943 (2002f:17046)

[BL]
L. Borisov, A. Libgober, Elliptic genera of toric varieties and applications to mirror symmetry, Invent. Math. 140 (2000), no. 2, 453-485. MR 1757003 (2001j:58037)

[BL1]
L. Borisov, A. Libgober, Elliptic Genera of Singular Varieties, Duke Math. J. 116 (2003), no. 2, 319-351. MR 1953295 (2004k:58034)

[BL2]
L. Borisov, A. Libgober, Elliptic Genera of singular varieties, orbifold elliptic genus and chiral de Rham complex, Mirror symmetry, IV (Montreal, QC, 2000), 325-342, AMS/IP Stud. Adv. Math., 33, Amer. Math. Soc., Providence, RI, 2002. MR 1969036 (2004e:11039)

[CR]
W. Chen, Y. Ruan, A new cohomology theory for orbifolds, Preprint math.AG/ 0004129. MR 2104605 (2005j:57036)

[D]
C. Dong, Twisted modules for vertex algebras associated with even lattices, J. Algebra 165 (1994), 91-112. MR 1272580 (95i:17032)

[DLM]
C. Dong, H. Li and G. Mason, Twisted representations of vertex operator algebras, Math. Ann. 310 (1998), 571-600. MR 1615132 (99d:17030)

[FFR]
A. Feingold, I. Frenkel, J. Reis, Spinor construction of vertex operator algebras, triality, and $ E^{(1)}_8$, Contemp. Math. 121, AMS, 1991. MR 1123265 (92k:17041)

[FB]
E. Frenkel, D. Ben-Zvi, Vertex algebras and algebraic curves, Mathematical Surveys and Monographs 88, AMS, 2001. MR 1849359 (2003f:17036)

[FS]
E. Frenkel, M. Szczesny, Twisted modules over vertex algebras on algebraic curves, Adv. in Math. 187 (2004), 195-227. MR 2074176 (2005k:17035)

[FLM]
I. Frenkel, J. Lepowsky, A. Meurman, Vertex operator algebras and the monster, Academic Press, 1988. MR 996026 (90h:17026)

[FMS]
D. Friedan, E. Martinec, S. Shenker, Conformal invariance, supersymmetry and string theory, Nuclear Phys. B271 (1986) 93-165. MR 845945 (87i:81202)

[GMS]
V. Gorbounov, F. Malikov, V. Schechtman, Gerbes of chiral differential operators, Math. Res. Lett. 7 (2000), no. 1, 55-66. MR 1748287 (2002c:17040)

[GMS2]
V. Gorbounov, F. Malikov, V. Schechtman, Gerbes of chiral differential operators. II. Vertex algebroids, Invent. Math. 155 (2004), no. 3, 605-680. MR 2038198 (2005e:17047)

[GMS3]
V. Gorbounov, F. Malikov, V. Schechtman, Gerbes of chiral differential operators. III, Gerbes of chiral differential operators. III, The orbit method in geometry and physics (Marseille, 2000), 73-100, Progr. Math., 213, Birkhäuser Boston, Boston, MA, 2003. MR 1995376 (2005a:17028)

[K]
V. Kac, Vertex algebras for beginners, Second Edition, AMS, 1998. MR 1651389 (99f:17033)

[Li]
H. Li, Local systems of twisted vertex operators, vertex operator superalgebras and twisted modules, Contemp. Math 193 (1996), 203-236. MR 1372724 (96m:17050)

[LZ]
B. Lian, G. Zuckerman, New perspectives on the BRST-algebraic structure of string theory, Comm. Math. Phys. 154 (1993), 613-646. MR 1224094 (94e:81333)

[Le1]
J. Lepowsky, Calculus of twisted vertex operators, Proc. Nat. Acad. Sci. U.S.A. 82 (1985) 8295-8299. MR 820716 (88f:17030)

[MSV]
F. Malikov, V. Schechtman, A. Vaintrob, Chiral de Rham Complex, Comm. Math. Phys. 204 (1999), 439-473. MR 1704283 (2000j:17035a)

[MS]
F. Malikov, V. Schechtman, Chiral de Rham complex. II, Differential topology, infinite-dimensional Lie algebras, and applications, 149-188, Amer. Math. Soc. Transl. Ser. 2, 194, Amer. Math. Soc., Providence, RI, 1999. MR 1729362 (2000j:17035b)

[MS2]
F. Malikov, V. Schechtman, Chiral Poincare duality, preprint math.AG 9905008 MR 1739212 (2000m:14019)

[S]
V. Schechtman, Sur les algebres de vertex attachees aux varietes algebraiques. (French), Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 525-532, Higher Ed. Press, Beijing, 2002. MR 1957062 (2004d:17032)

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

PII: S 1056-3911(07)00466-3
Received by editor(s): January 1, 2004
Received by editor(s) in revised form: November 6, 2006
Posted: May 1, 2007
Additional Notes: The first author was partially supported by grants from the Packard Foundation and the NSF

Journal of Algebraic Geometry
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