A Quantization Procedure of Fields Based on Geometric Langlands Correspondence

Abstract

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 =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.