A Quantization Procedure of Fields Based on Geometric Langlands Correspondence

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 . 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 . After that we quantize the vector bundles with connection over the coadjoint orbits of the symmetry group . Use the electric-magnetic duality to pass to the Langlands dual Lie group . Therefore, we have some affine Kac-Moody loop algebra of meromorphic functions with values in Lie algebra 𠔤 . 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 .