Microsheaves from Hitchin fibers via Floer theory

Abstract

Consider a smooth component of the moduli space of stable Higgs bundles on which the Hitchin fibration is proper. We record in this note the following corollaries to existing results on Floer theory and Fukaya categories. (1) Any any smooth Hitchin fiber determines a microsheaf on the global nilpotent cone. (2) Distinct fibers give rise to orthogonal microsheaves. (3) The endomorphisms of the microsheaf is isomorphic to the cohomology of the Hitchin fiber.Any sheaf on the moduli stack of bundles which is microsupported in the nilpotent cone restricts (by definition) to a microsheaf on the locus of stable and nilpotent Higgs bundles. Conversely, such microsheaves can be restricted to the moduli of stable bundles, and then extended to the stack of all bundles. We expect (but do not prove) that our microsheaves should restrict from, and extend to, the Hecke eigensheaves of the geometric Langlands programme. HITCHIN FIBERS AND HECKE EIGENSHEAVESLet us recall what Hecke eigensheaves are, and why one might hope to get them from fibers of the Hitchin system. More detailed discussions can be found in e.g. [7,32,15]. This section is purely motivational and logically unrelated to the results presented in the remainder of the article.Let C be a smooth compact complex curve, G a reductive group, and Bun G (C) the moduli of G-bundles on C. We write [X] for a category of sheaves on X, whose precise nature is irrelevant here. There are 'Hecke operators'given by convolution with respect to a correspondence parameterizing pairs of bundles which are isomorphic away from a single point c ∈ C, and whose difference at this point is controlled by a cocharacter µ of G.The geometric Langlands conjecture asserts, in particular, the existence of certain F ∈ [Bun G (C)] which are 'Hecke eigensheaves' in the sense that H µ (F ) = F ρ µ (χ), where χ is a G ∨ local system on C, and ρ µ is the representation corresponding to the character µ of G ∨ .The Hecke action transforms microsupports (or characteristic cycles) by setwise convolution with the conic Lagrangian ss(H µ ) ⊂ T * Bun G (C) × T * Bun G (C) × T * C; which is roughly the conormal to the image of the Hecke correspondence. This conormal respects the fibers of Hitchin's integrable system h :is the spectral cover corresponding to the point b ∈ B and representation µ.The above geometry suggests that any sufficiently functorial procedure relating Hitchin fibers to sheaves on Bun G (C) -and spectral curves to sheaves on C -may be expected to yield Hecke eigensheaves. Quantization is one such procedure: the semiclassical limit of a quantum state on M determines a Lagrangian submanifold of T * M (see e.g. [6]). Taking M = Bun G (C), and the corresponding Lagrangians to be the Hitchin fibers, quantizing the Hitchin system may be expected to yield Hecke eigensheaves, and indeed does [7]. Related approaches include [5,14,32,9,15]. This note explains how to use Floer theory to produce sheaves on Bun G (C) from Hitchin fibers.

Invariance of microsheaves on stable Higgs bundles

“…[15,8,18], but we are not aware of any even conjectural answer to Question 1.3.) One can also use the relationship with Fukaya categories to produce objects from smooth Hitchin fibers, which presently cannot be done with microsheaf methods alone, as these fibers are nonexact [25].…”