A quantization procedure of fields based on geometric Langlands correspondence (Q963524)

Summary: 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 \(^LG\). 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 \(^LG\). After that we quantize the vector bundles with connection over the coadjoint orbits of the symmetry group \(^LG\). 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 \(\mathfrak g =\mathrm{Lie}(G)\). 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.