Gauge theory and the analytic form of the geometric Langlands program. (English) Zbl 07802655
Summary: We present a gauge-theoretic interpretation of the “analytic” version of the geometric Langlands program, in which Hitchin Hamiltonians and Hecke operators are viewed as concrete operators acting on a Hilbert space of quantum states. The gauge theory ingredients required to understand this construction – such as electric-magnetic duality between Wilson and ’t Hooft line operators in four-dimensional gauge theory – are the same ones that enter in understanding via gauge theory the more familiar formulation of geometric Langlands, but now these ingredients are organized and applied in a novel fashion.
MSC:
| 81P05 | General and philosophical questions in quantum theory |
| 70H05 | Hamilton’s equations |
| 11F60 | Hecke-Petersson operators, differential operators (several variables) |
| 46C05 | Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) |
| 70S15 | Yang-Mills and other gauge theories in mechanics of particles and systems |
| 78A30 | Electro- and magnetostatics |
| 32C37 | Duality theorems for analytic spaces |
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