## Electric-magnetic duality and the geometric Langlands program.(English)Zbl 1128.22013

The paper under review aims to describe the geometric Langlands program for complex Riemann surfaces within quantum field theory. Actually, it develops the following key ideas. The first one is to construct a family of four-dimensional topological field theories based on some twisted version of $$N = 4$$ supersymmetric Yang-Mills theory in four dimensions. The second consists in developing ”generalized $$B$$-models”, a family of topological sigma-models which are obtained from compactification on a Riemann surface $$C$$ in two dimensions. The third one is based on the fact that Wilson and t’Hooft line operators are topological operators which naturally act on the branes of the two-dimensional sigma-model and these operators map a brane to, roughly speaking, a multiple of itself that is usually called an electric or magnetic eigenbrane which both are automatically exchanged by $$S$$-duality. The fourth key idea is that the electric eigenbranes are in the natural correspondence with homomorphisms of $$\pi_1(C)$$ to the complexification of the Langlands dual group. The fifth one is related to the fact that t’Hooft operators correspond to geometric Hecke operators, which are similar to those of the geometric Langlands program, as though they act on Higgs bundles instead of ordinary $$G$$-bundles. The last idea is to relate the present work to the usual formulation of the geometric Langlands correspondence.

### MSC:

 22E46 Semisimple Lie groups and their representations 14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) 32G13 Complex-analytic moduli problems 81T45 Topological field theories in quantum mechanics 81T60 Supersymmetric field theories in quantum mechanics
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