Geometric Endoscopy and Mirror Symmetry
Edward Frenkel, Edward Witten
TL;DR
This work develops a geometric endoscopy program by analyzing how orbifold (finite-automorphism) singularities on Hitchin fibrations influence the geometric Langlands correspondence. By exploiting mirror symmetry between the dual Hitchin fibrations for $G$ and {}^LG, the authors show that endoscopic local systems correspond to fractional A-branes supported on singular Hitchin fibers, which translate into twisted ${ m D}$-modules and fractional Hecke eigensheaves in the Langlands framework. They provide explicit genus-one examples (SL$_2$ vs SO$_3$) illustrating the decomposition of B-branes into irreducible components and the consequent division of Hecke eigensheaves, predicting fractional eigenproperties and transfer phenomena. The paper then extends these ideas to higher genus and broader groups, linking to Ngô’s fundamental lemma, Seiberg–Witten theory, and domain-wall interpretations in QFT, while proposing a precise categorical and automorphic picture for endoscopy via fractional eigenbranes and their Hecke actions. These results offer concrete geometric mechanisms for endoscopy and furnish consistency checks with the classical trace formula and Grothendieck's correspondence between sheaves and automorphic functions over finite fields.
Abstract
The geometric Langlands correspondence has been interpreted as the mirror symmetry of the Hitchin fibrations for two dual reductive groups. This mirror symmetry, in turn, reduces to T-duality on the generic Hitchin fibers, which are smooth tori. In this paper we study what happens when the Hitchin fibers on the B-model side develop orbifold singularities. These singularities correspond to local systems with finite groups of automorphisms. In the classical Langlands Program local systems of this type are called endoscopic. They play an important role in the theory of automorphic representations, in particular, in the stabilization of the trace formula. Our goal is to use the mirror symmetry of the Hitchin fibrations to expose the special role played by these local systems in the geometric theory. The study of the categories of A-branes on the dual Hitchin fibers allows us to uncover some interesting phenomena associated with the endoscopy in the geometric Langlands correspondence. We then follow our predictions back to the classical theory of automorphic functions. This enables us to test and confirm them. The geometry we use is similar to that which is exploited in recent work by B.-C. Ngo, a fact which could be significant for understanding the trace formula.