Betti Geometric Langlands

Abstract

Refined forms of the local Langlands correspondence seek to relate representations of reductive groups over local fields with sheaves on stacks of Langlands parameters. But what kind of sheaves? Conjectures in the spirit of Kazhdan-Lusztig theory (due to Vogan and Soergel) describe representations of a group and its pure inner forms with fixed central character in terms of constructible sheaves. Conjectures in the spirit of geometric Langlands (due to Fargues, Zhu and Hellmann) describe representations with varying central character of a large family of groups associated to isocrystals in terms of coherent sheaves. The latter conjectures also take place on a larger parameter space, in which Frobenius (or complex conjugation) is allowed a unipotent part.In this article we propose a general mechanism that interpolates between these two settings. This mechanism derives from the theory of cyclic homology, as interpreted through circle actions in derived algebraic geometry. We apply this perspective to categorical forms of the local Langlands conjectures for both archimedean and non-archimedean local fields. In the nonarchimedean case, we describe how circle actions relate coherent and constructible realizations of affine Hecke algebras and of all smooth representations of GLn, and propose a mechanism to relate the two settings in general. In the archimedean case, we explain how to use circle actions to derive the constructible local Langlands correspondence (in the form due to Adams-Barbasch-Vogan and Soergel) from a coherent form (a real counterpart to Fargues' conjecture): the tamely ramified geometric Langlands conjecture on the twistor line, which we survey.

Integrating quantum groups over surfaces

“…These quantize moduli spaces of local systems on surfaces and provide a unifying perspective on various constructions in quantum group theory. Quantum character varieties form the two‐dimensional part of a topological field theory which is a model for the Kapustin–Witten theory (GL‐twisted four‐dimensional super Yang–Mills theory), and provide the spectral side of the quantum Betti geometric Langlands conjecture . These connections (which are discussed further in Sections 1.4.2 and 1.4.4) suggest many rich structures for quantum character varieties, some of which we discuss in this paper, and many which we plan to explore in future papers.…”

“…The quantum Betti conjecture relates the categories constructed in this paper with their automorphic counterparts, which are given by twisted sheaves with nilpotent singular support on . This conjecture is motivated in turn by the work of Kapustin–Witten , in which Langlands duality is related to electric‐magnetic S‐duality in four‐dimensional topological field theory ( supersymmetric Yang–Mills theory in the GL twist).…”

“…The spectral side of the ‘de Rham’ geometric Langlands conjecture involves coherent sheaves on the space of flat ‐connections on an algebraic curve (that is, ‐bundles on the de Rham space of ). The Betti version of the conjecture replaces this category by coherent sheaves on the character variety of the underlying topological surface (that is, ‐bundles on the Betti version of ).…”

Stacky dualities for the moduli of Higgs bundles

“…Date: November 6, 2018. 1 For just the tip of the iceberg: Seiberg-Witten theory and Donaldson invariants [52,20,53]; 3d N = 4 gauge theories and symplectic duality [9,12,10]; 4d N = 4 Yang-Mills theories and the geometric Langlands program [36,29,18,21,7].…”

The topological chiral homology of the spherical category

“…In this section, we assume that is connected and reductive, with Langlands dual . Recall the rough statement of the Betti geometric Langlands conjecture (see for a thorough discussion): for any smooth complete curve , there is an equivalence …”