Towards a self-dual geometric Langlands program
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This thesis is comprised of two logically separate but conjecturally related parts. In the first part of the thesis I study theories of class S  via the formalism of relative quantum field theories . From this physical formalism, and by analogy to the physical derivation of usual geometric Langlands [45, 86], I conjecture the existence of a self-dual version of the geometric Langlands program. In the second part of the thesis I study shifted Cartier duality for the moduli of Higgs bundles. The main results are: (1) a criteria for ramification of L-valued cameral covers, (2) a generalisation of the Langlands duality/mirror symmetry results for the moduli of Higgs bundles of [24, 37], and (3) the existence of a self-dual version of the moduli of Higgs bundles. This self-dual space is conjecturally the target space for a theory of class S compactified on a torus, and provides positive evidence for the self-dual geometric Langlands program.