Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 086, 28 pages      arXiv:0807.1641      http://dx.doi.org/10.3842/SIGMA.2008.086
Contribution to the Special Issue on Kac-Moody Algebras and Applications

Vertex Algebroids over Veronese Rings

Fyodor Malikov
Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA

Received July 28, 2008, in final form December 07, 2008; Published online December 13, 2008

Abstract
We find a canonical quantization of Courant algebroids over Veronese rings. Part of our approach allows a semi-infinite cohomology interpretation, and the latter can be used to define sheaves of chiral differential operators on some homogeneous spaces including the space of pure spinors punctured at a point.

Key words: differential graded algebra; vertex algebra; algebroid.

pdf (393 kb)   ps (244 kb)   tex (33 kb)

References

  1. Arkhipov S., Gaitsgory D., Differential operators on the loop group via chiral algebras, Int. Math. Res. Not. 2002 (2002), no. 4, 165-210, math.AG/0009007.
  2. Behrend K., Differential graded schemes I: Perfect resolving algebras, math.AG/0212225.
  3. Backelin J., Fröberg R., Koszul algebras, Veronese subrings and rings with linear resolutions, Rev. Roumaine Math. Pures Appl. 30 (1985), no. 2, 85-97.
  4. Berkovits N., Nekrasov N., The character of pure spinors, Lett. Math. Phys. 74 (2005), 75-109, hep-th/0503075.
  5. Berkovits N., Super-Poincaré covariant quantization of the superstring, J. High Energy Phys. 2000 (2000), no. 4, 18, 17 pages, hep-th/0001035.
  6. Bezrukavnikov R., Koszul property and Frobenius splitting of Schubert varieties, alg-geom/9502021.
  7. Bressler P., The first Pontryagin class, Compos. Math. 143 (2007), 1127-1163, math.AT/0509563.
  8. Courant T.J., Dirac manifolds, Trans. Amer. Math. Soc. 319 (1990), 631-661.
  9. Dorfman I., Dirac structures of integrable evolution equations, Phys. Lett. A 125 (1987), no. 5, 240-246.
  10. Frenkel E., Private communication.
  11. Frenkel E., Wakimoto modules, opers and the center at the critical level, Adv. Math. 195 (2005), 297-404, math.QA/0210029.
  12. Frenkel E., Ben-Zvi D., Vertex algebras and algebraic curves, 2nd ed., Mathematical Surveys and Monographs, Vol. 88, American Mathematical Society, Providence, RI, 2004.
  13. Feigin B., Semi-infinite homology of Lie, Kac-Moody and Virasoro algebras, Uspekhi Mat. Nauk 39 (1984), no. 2, 195-196 (in Russian).
  14. Feigin B., Parkhomenko S., Regular representation of affine Kac-Moody algebras, in Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), Math. Phys. Stud., Vol. 19, Kluwer Acad. Publ., Dordrecht, 1996, 415-424, hep-th/9308065.
  15. Gorbounov V., Malikov F., Schechtman V., Gerbes of chiral differential operators. II. Vertex algebroids, Inv. Math. 155 (2004), 605-680, math.AG/0003170.
  16. Gorbounov V., Malikov F., Schechtman V., On chiral differential operators over homogeneous spaces, Int. J. Math. Math. Sci. 26 (2001), no. 2, 83-106, math.AG/0008154.
  17. Gorbounov V., Schechtman V., Homological algebra and divergent series, arXiv:0712.3670.
  18. Hinich V., Homological algebra of homotopy algebras, Comm. Algebra 25 (1997), 3291-3323, q-alg/9702015.
  19. Inamdar S.P., Mehta V.B., Frobenius splitting of Schubert varieties and linear syzygies, Amer. J. Math. 116 (1994), 1569-1586.
  20. Kac V., Vertex algebras for beginners, 2nd ed., University Lecture Series, Vol. 10, American Mathematical Society, Providence, RI, 1998.
  21. Li H., Abelianizing vertex algebras, Comm. Math. Phys. 259 (2005), 391-411, math.QA/0409140.
  22. Liu Z.-J., Weinstein A., Xu P., Manin triples for Lie bialgebroids, J. Differential Geom. 45 (1997), 547-574, dg-ga/9508013.
  23. Malikov F., Lagrangian approach to sheaves of vertex algebras, Comm. Math. Phys. 278 (2008), 487-548, math.AG/0604093.
  24. Nekrasov N., Lectures on curved beta-gamma systems, pure spinors, and anomalies, hep-th/0511008.
  25. Primc M., Vertex algebras generated by Lie algebras, J. Pure Appl. Algebra 135 (1999), 253-293, math.QA/9901095.

Previous article   Next article   Contents of Volume 4 (2008)