NEW APPROACHES TO THE ANALYTIC LANGLANDS CORRESPONDENCE & DESCRIPTION OF WOBBLY BUNDLES

• Applicant: Dr. Duong Dinh
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 561911940

Let X be a Riemann surface with negative Euler characteristic, and G a complex reductive group. More than 30 years ago, Beilinson-Drinfeld showed that the classical integrable Hitchin system associated to (X, G), i.e. the moduli of G-Higgs bundles on X, admits a natural quantisation. The quantum Hitchin Hamiltonians are intimately related to an important class of special cases of the geometric Langlands correspondence: D-module on the moduli stack of G-bundles defined by quantum Hitchin Hamiltonians are classified by opers on X with the gauge group being the Langlands dual of G. The full geometric Langlands conjecture has been recently proved by Gaitsgory-Raskin et al. The analytic Langlands correspondence, conjectured by Teschner and developed by Etingof-Frenkel-Kazhdan, is a more recent development. This conjecture proposes that solutions to the spectral problem of these quantum Hitchin Hamiltonians are in particular classified by real opers. Furthermore, these solutions are expected to be eigenstates of the analogues of Hecke operators. Attempts to construct solutions to this spectral problem, and classify these solutions have been met with serious analytic challenges on the moduli stack of G-bundles. The main theme in this research program is the proposal of a new method to construct these solutions. As a start, in the rank-2 cases, we propose an analogue of Drinfeld’s approach to the geometric Langlands correspondence in the analytic setting. In the analytic geometric Langlands correspondence setting, starting from a real oper on X, we propose that a unitary integral transform can produce solutions to the spectral problem of quantum Hitchin Hamiltonians. The merit of this method is that one can transpose analytic challenges on the moduli stack to those on X, which should be more tractable. Another goal of this research program is to find an analogue in the analytic Langlands correspondence of Donagi-Pantev’s approach to the geometric Langlands correspondence. In the geometric Langlands correspondence, this approach makes extensive use of the nonabelian Hodge correspondence between Higgs bundles and local systems on varieties of arbitrary dimensions, and Fourier-Mukai transform of dual Hitchin fibrations associated to Langlands dual groups. Donagi-Pantev conjectured and proved in some cases that, starting from a local system on X, this procedure produces local systems on the moduli space of bundles that are the restriction of the corresponding D-module. We expect that an analogue of Donagi-Pantev’s approach exists in the analytic Langlands correspondence and will likewise help circumvent analytic challenges on the moduli stack/moduli space of bundles.

• DFG Programme: WBP Fellowship
• International Connection: USA
• Host: Professor Dr. Ron Donagi