Skip to main content
SpringerLink
Log in

Localization of Affine W-Algebras

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We introduce the notion of an asymptotic algebra of chiral differential operators. We then construct, via a chiral Hamiltonian reduction, one such algebra over a resolution of the intersection of the Slodowy slice with the nilpotent cone. We compute the space of global sections of this algebra, thereby proving a localization theorem for affine W-algebras at the critical level.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arakawa T.: A remark on the C 2-finiteness condition for vertex algebras. Math. Z. 270(1–2), 559–575 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  2. Arakawa, T.: Associated varieties of modules over Kac–Moody algebras and C 2-cofiniteness of W-algebras. arXiv:1004.1554

  3. Arakawa T.: W-algebras at the critical level. Contemp. Math. 565, 1–14 (2012)

    Article  MathSciNet  Google Scholar 

  4. Arakawa, T., Chebotarov, D., Malikov, F.: Algebras of twisted chiral differential operators and affine localization of \({\mathfrak{g}}\)-modules. Sel. Math. (N.S.), 17(1), 1–46 (2011). arXiv:0810.4964

  5. Arakawa, T., Malikov, F.: A chiral Borel–Weil–Bott theorem. Adv. Math. 229(5), 2908–2949 (2012). arXiv:0903.1281

  6. Beilinson A., Bernstein J.: Localisation de \({\mathfrak{g}}\)-modules(French). C. R Acad. Sci. Paris Se’r. I Math. 292(1), 15–18 (1981)

    MATH  MathSciNet  Google Scholar 

  7. Beilinson A., Bernstein, J.: A proof of Jantzen conjectures. I. M. Gelfand Seminar, Adv. Soviet Math. 16, Part 1, pp. 1–50. Amer. Math. Soc., Providence (1993)

  8. Beilinson, A., Drinfeld, V.: Chiral algebras. American Mathematical Society Colloquium Publications, vol. 51. American Mathematical Society, Providence (2004)

  9. Beilinson, A., Feigin, B., Mazur, B.: Introduction to algebraic field theory on curves (unpublished)

  10. Bershadsky M.: Conformal field theories via Hamiltonian reduction. Commun. Math. Phys. 139(1), 71–82 (1991)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  11. Bellamy, G., Kuwabara, T.: On deformation quantizations of hypertoric varieties. Pac. J. Math. 260(1), 89–127 (2012). arXiv:math/1005.4645v2

  12. Brylinski J.-L., Kashiwara M.: Kazhdan–Lusztig conjecture and holonomic systems. Inv. Math. 64(3), 387–410 (1981)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  13. Bressler P.: The first Pontryagin class. Compos. Math. 143(5), 1127–1163 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Brieskorn, E.: Singular elements of semi-simple algebraic groups, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, Gauthier-Villars, Paris, pp. 279–284 (1971)

  15. De Sole A., Kac V.G.: Finite vs affine W-algebras, Japan. J. Math. 1, 137–261 (2006)

    MATH  MathSciNet  Google Scholar 

  16. Dodd, C., Kremnizer, K.: A localization theorem for finite W-algebras. arXiv:math/0911.2210

  17. Feigin B.: Semi-infinite homology of Lie, Kac–Moody and Virasoro algebras (Russian). Uspekhi Mat. Nauk 39(2), 155–156 (1984)

    MATH  MathSciNet  Google Scholar 

  18. Feigin B., Frenkel E. (1992) Affine Kac–Moody algebras at the critical level and Gelfand-Dikii algebras. In: Tsuchiya A., Eguchi T., Jimbo M. (eds.) Infinite Analysis, Adv. Series in Math. Phys., vol. 16, pp. 197–215. World Scientific, Singapore

  19. Frenkel, E., Gaitsgory, D.: Weyl modules and opers without monodromy, Arithmetic and geometry around quantization. Progr. Math., vol. 279, pp. 101–121. Birkhäuser Boston, Inc., Boston (2010). arXiv:math/0706.3725

  20. Fu, B.: A survey on symplectic singularities and resolutions. Ann. Math. Blaise Pascal 13(2), 209–236 (2006). arXiv:math.AG/0510346

  21. Gan, W.L., Ginzburg, V.: Quantization of Slodowy slices. Int. Math. Res. Not. 2, 243–255 (2004)

  22. Ginzburg V.: Harish–Chandra bimodules for quantized Slodowy slices. Represent. Theory 13, 236–271 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  23. Gorbounov V., Malikov F., Schechtman V.: Gerbes of chiral differential operators. II. Vertex algebroids. Invent. Math. 155(3), 605–680 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  24. Gorbounov V., Malikov F., Schechtman V.: On chiral differential operators over homogeneous spaces. Int. J. Math. Math. Sci. 26(2), 83–106 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  25. Kac V.: Vertex algebras for beginners. University Lecture Series, vol. 10. American Mathematical Society, Providence (1997)

    Google Scholar 

  26. Kapranov M.: Noncommutative geometry based on commutator expansions. J. Reine Angew. Math. 505, 73–118 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  27. Kashiwara, M., Rouquier, R.: Microlocalization of rational Cherednik algebras. Duke Math. J. 144(3), 525–573 (2008). arXiv:0705.1245

  28. Kronheimer P.B.: The construction of ALE spaces as hyper-Kähler quotients. J. Differ. Geom. 29, 665–683 (1989)

    MATH  MathSciNet  Google Scholar 

  29. Kac V., Roan S., Wakimoto M.: Quantum reduction for affine superalgebras. Commun. Math. Phys. 241(2-3), 307–342 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  30. Kuwabara, T.: Representation theory of the rational Cherednik algebras of type \({\mathbb{Z}/l\mathbb{Z}}\) via microlocal analysis. Publ. Res. Inst. Math. Sci. 49(1), 87–110 (2013). arXiv:1003.3407v2

  31. Kostant B., Sternberg S.: Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras. Ann. Phys. 176(1), 49–113 (1987)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  32. Kac V., Wakimoto M.: Quantum reduction and representation theory of superconformal, algebras. Adv. Math. 185(2), 400–458 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  33. Li H.: Vertex algebras and vertex Poisson algebras. Contemp. Math. 6(1), 61–110 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  34. Losev I.: Quantized symplectic actions and W-algebras. J. Am. Math. Soc. 23(1), 35–59 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  35. Malikov F., Schechtman V., Vaintrob A.: Commun. Math. Phys. 204, 439–473 (1999)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  36. Mustată, M.: Jet schemes of locally complete intersection canonical singularities. Inv. Math. 145(3), 397–424. With an appendix by D. Eisenbud and E. Frenkel (2001)

  37. Polyakov A.M.: Gauge transformations and diffeomorphisms. Internat. J. Modern Phys. A 5(5), 833–842 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  38. Premet, A.: Special transverse slices and their enveloping algebras. Adv. Math. 170(1), 397-424 (2002). With an appendix by Serge Scryabin

  39. Slodowy P.: Simple singularities and simple algebraic groups. Lecture Notes in Mathematics, vol. 815. Springer, Berlin (1980)

    Google Scholar 

  40. Vaisman I.: Lectures on the geometry of Poisson manifolds. Progress in Mathematics, vol. 118. Birkhäuser Verlag, Basel (1994)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. RIMS, Kyoto University, Kyoto, 606-8502, Japan

    T. Arakawa

  2. Department of Mathematics and International Laboratory of Representation Theory and Mathematical Physics, National Research University Higher School of Economics, 20 Myasnitskaya Ulitsa, Moscow, 101000, Russia

    T. Kuwabara

  3. Department of Mathematics, University of Southern California, Los Angeles, CA, 90089, USA

    F. Malikov

Authors
  1. T. Arakawa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. Kuwabara
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. F. Malikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. Kuwabara.

Additional information

Communicated by Y. Kawahigashi

T. Arakawa is partially supported by the JSPS Grant-in-Aid for Scientific Research (B) No. 20340007 and the JSPS Grant-in-Aid for challenging Exploratory Research No. 23654006.

T. Kuwabara was partially supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST)(2011-0027952).

F. Malikov is partially supported by an NSF grant.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Arakawa, T., Kuwabara, T. & Malikov, F. Localization of Affine W-Algebras. Commun. Math. Phys. 335, 143–182 (2015). https://doi.org/10.1007/s00220-014-2183-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-014-2183-x

Keywords

Search

Navigation