Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Reciprocity Laws in the Verlinde Formulae
for the Classical Groups


Authors: W. M. Oxbury and S. M. J. Wilson
Journal: Trans. Amer. Math. Soc. 348 (1996), 2689-2710
MSC (1991): Primary 14D20, 14H15
MathSciNet review: 1340183
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Verlinde formula is computed for each of the simply-connected classical Lie groups, and it is shown that the resulting formula obeys certain reciprocity laws with respect to the exchange of the rank and the level. Some corresponding dualities between spaces of sections of theta line bundles over moduli spaces of $G$-bundles on curves are conjectured but not proved.


References [Enhancements On Off] (What's this?)

  • [B1] A. Beauville, Vector bundles on curves and generalised theta functions: recent results and open problems, preprint, 1994.
  • [B2] ------, Conformal blocks, fusion rules and the Verlinde formula, preprint, 1994.
  • [DT] Ron Donagi and Loring W. Tu, Theta functions for 𝑆𝐿(𝑛) versus 𝐺𝐿(𝑛), Math. Res. Lett. 1 (1994), no. 3, 345–357. MR 1302649, 10.4310/MRL.1994.v1.n3.a6
  • [F] Gerd Faltings, A proof for the Verlinde formula, J. Algebraic Geom. 3 (1994), no. 2, 347–374. MR 1257326
  • [FH] William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249
  • [K] Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219
  • [KNR] Shrawan Kumar, M. S. Narasimhan, and A. Ramanathan, Infinite Grassmannians and moduli spaces of 𝐺-bundles, Math. Ann. 300 (1994), no. 1, 41–75. MR 1289830, 10.1007/BF01450475
  • [O] W.M. Oxbury, Prym varieties and the moduli of spin bundles, to appear in proceedings of the Annual Europroj Conference, Barcelona, 1994.
  • [Sz] A. Szenes, The combinatorics of the Verlinde formulas (N.J. Hitchin et al., ed.), in Vector bundles in algebraic geometry, Cambridge, 1995.
  • [Z] D. Zagier, Elementary aspects of the Verlinde formula and of the Harder-Narasimhan-Atiyah-Bott formula, preprint, 1994.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 14D20, 14H15

Retrieve articles in all journals with MSC (1991): 14D20, 14H15


Additional Information

W. M. Oxbury
Affiliation: Department of Mathematical Sciences, Science Laboratories, South Road, Durham DH1 3LE, U.K.
Email: w.m.oxbury@durham.ac.uk

S. M. J. Wilson
Affiliation: Department of Mathematical Sciences, Science Laboratories, South Road, Durham DH1 3LE, U.K.
Email: s.m.j.wilson@durham.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-96-01563-2
Received by editor(s): March 6, 1995
Received by editor(s) in revised form: May 21, 1995
Article copyright: © Copyright 1996 American Mathematical Society