Vertex algebras and the formal loop space
Kapranov, Mikhail ; Vasserot, Eric
Publications Mathématiques de l'IHÉS, Tome 99 (2004), p. 209-269 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

We construct a certain algebro-geometric version ℒ(X) of the free loop space for a complex algebraic variety X. This is an ind-scheme containing the scheme ℒ 0 (X) of formal arcs in X as studied by Kontsevich and Denef-Loeser. We describe the chiral de Rham complex of Malikov, Schechtman and Vaintrob in terms of the space of formal distributions on ℒ(X) supported in ℒ 0 (X). We also show that ℒ(X) possesses a factorization structure: a certain non-linear version of a vertex algebra structure. This explains the heuristic principle that “all” linear constructions applied to the free loop space produce vertex algebras.

@article{PMIHES_2004__100__209_0,
     author = {Kapranov, Mikhail and Vasserot, Eric},
     title = {Vertex algebras and the formal loop space},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     volume = {99},
     year = {2004},
     pages = {209-269},
     doi = {10.1007/s10240-004-0023-9},
     mrnumber = {2102701},
     zbl = {1106.17038},
     language = {en},
     url = {http://dml.mathdoc.fr/item/PMIHES_2004__100__209_0}
}

Kapranov, Mikhail; Vasserot, Eric. Vertex algebras and the formal loop space. Publications Mathématiques de l'IHÉS, Tome 99 (2004) pp. 209-269. doi : 10.1007/s10240-004-0023-9. http://gdmltest.u-ga.fr/item/PMIHES_2004__100__209_0/

1. M. Artin, B. Mazur, Etale Homotopy, Lect. Notes Math. 100, Springer, 1970. | MR 883959 | Zbl 0182.26001

2. B. Bakalov, Beilinson-Drinfeld's definition of a chiral algebra, available from http://www.math.berkeley.edu/∼bakalov/.

3. A. Beilinson, I. Bernstein, A proof of the Jantzen conjectures, in: S. Gelfand, S. Gindikin (eds.), I. M. Gelfand Seminar 1, 1-50, Adv. Soviet Math. 16, Amer. Math. Soc., Providence, RI, 1993. | Zbl 0790.22007

4. A. Beilinson, V. Drinfeld, Chiral algebras, available from http://zaphod.uchicago.edu/∼benzvi/.

5. A. Beilinson, V. Drinfeld, Quantization of Hitchin's integrable system and Hecke eigensheaves, available from http://zaphod.uchicago.edu/∼benzvi/.

6. S. Bosch, W. Lütkebohmert, M. Raynaud, Néron models, Springer, 1990. | MR 1045822 | Zbl 0705.14001

7. C. Contou-Carrère, Jacobienne locale, groupe de bivecteurs de Witt universel et symbole modéré, C.R. Acad. Sci. Paris, Sér. I, Math., 318 (1994), 743-746. | MR 1272340 | Zbl 0840.14031

8. J. Denef, F. Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math., 135 (1999), 201-232. | MR 1664700 | Zbl 0928.14004

9. A. Grothendieck, J. Dieudonné, Eléments de géométrie algébrique, Grund. Math. Wiss. 166, Boston, Basel, Berlin: Springer, 1971. | Zbl 0203.23301

10. A. Grothendieck, Éléments de géométrie algébrique IV (rédigés avec la collaboration de Jean Dieudonné), Publ. Math., Inst. Hautes Étud. Sci., 24 (1965), 5-231, 28 (1966), 5-255, 32 (1967), 5-361.

11. E. Frenkel, Vertex algebras and algebraic curves, Séminaire Bourbaki, 875 (2000), Astérisque 276 (2002), 299-339. | Numdam | MR 1886764 | Zbl 0997.17015

12. I. B. Frenkel, J. Lepowsky, A. Meurman, Vertex Operator Algebras and the Monster, Pure Appl. Math. 134, Boston: Academic Press, 1988. | MR 996026 | Zbl 0674.17001

13. D. Gaitsgory, Notes on 2d conformal field theory and string theory, in: P. Deligne et al. (eds), Quantum fields and strings: a course for mathematicians, vol. 2, pp. 1017-1089, Providence, RI: Am. Math. Soc., 1999. | MR 1701613 | Zbl pre01735149

14. I. M. Gelfand, D. A. Kazhdan, D. B. Fuks, The actions of infinite-dimensional Lie algebras, Funct. Anal. Appl., 6 (1972), 9-13. | MR 301767 | Zbl 0267.18023

15. A. Grothendieck, J.-L. Verdier, Théorie des topos et cohomologie étale des schémas, SGA IV, Exp. I, Lect. Notes Math. 269, Springer, 1970. | MR 463174 | Zbl 0256.18008

16. Y.-Z. Huang, J. Lepowsky, On the 𝒟-module and the formal variable approachs to vertex algebras, Topics in geometry, pp. 175-202, Birkhäuser, 1996. | MR 1390314 | Zbl 0871.17023

17. V. Kac, Vertex algebras for beginners, Univ. Lect. Ser. 10, Providence, RI: Am. Math. Soc., 1997. | MR 1417941 | Zbl 0861.17017

18. M. Kapranov, Double affine Hecke algebras and 2-dimensional local fields, J. Am. Math. Soc., 14 (2001), 239-262. | MR 1800352 | Zbl 0958.20005

19. K. Kato, Existence theorem for higher local fields, Invitation to higher local fields (Münster, 1999), pp. 165-195 (electronic), Geom. Topol. Monogr. 3, Geom. Topol. Publ., Coventry, 2000. | MR 1804933 | Zbl 1008.11061

20. M. Kashiwara, T. Tanisaki, Kazhdan-Lusztig Conjecture for Symmetrizable Kac-Moody Lie Algebra. II Intersection Cohomologies of Schubert varieties, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989), pp. 159-195, Progr. Math., 92, Birkhäuser, 1990. | MR 1103590 | Zbl 0743.17024

21. D. A. Leites, Introduction to the theory of supermanifolds. Uspekhi Mat. Nauk., 35 (1980), 3-57. | MR 565567 | Zbl 0439.58007

22. A. Malikov, V. Schechtman, A. Vaintrob, Chiral De Rham complex, Comm. Math. Phys., 204 (1999), 439-473. | MR 1704283 | Zbl 0952.14013

23. A. N. Parshin, Higher-dimensional local fields and L-functions, Invitation to higher local fields (Münster, 1999), pp. 199-213 (electronic), Geom. Topol. Monogr. 3, Geom. Topol. Publ., Coventry, 2000. | MR 1804934 | Zbl 1008.11060

24. R. W. Thomason, T. Trobaugh, Higher algebraic K-theory of schemes and of derived categories, in: P. Cartier et al. (eds.), Grothendieck Festschrift, vol. III, Progr. Math., 88, pp. 247-435, Birkhäuser, 1990. | MR 1106918 | Zbl 0731.14001