International Journal of Mathematics and Mathematical Sciences
Volume 2009 (2009), Article ID 749631, 14 pages
Research Article

A Quantization Procedure of Fields Based on Geometric Langlands Correspondence

Department of Geometry and Topology, Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet Road Cau Giay District 10307, Hanoi, Vietnam

Received 29 September 2009; Accepted 16 November 2009

Academic Editor: Palle Jorgensen

Copyright © 2009 Do Ngoc Diep. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We expose a new procedure of quantization of fields, based on the Geometric Langlands Correspondence. Starting from fields in the target space, we first reduce them to the case of fields on one-complex-variable target space, at the same time increasing the possible symmetry group GL. Use the sigma model and momentum maps, we reduce the problem to a problem of quantization of trivial vector bundles with connection over the space dual to the Lie algebra of the symmetry group GL. After that we quantize the vector bundles with connection over the coadjoint orbits of the symmetry group GL. Use the electric-magnetic duality to pass to the Langlands dual Lie group G. Therefore, we have some affine Kac-Moody loop algebra of meromorphic functions with values in Lie algebra 𝔤 =Lie(G). Use the construction of Fock space reprsentations to have representations of such affine loop algebra. And finally, we have the automorphic representations of the corresponding Langlands-dual Lie groups G.