Microsheaves from Hitchin fibers via Floer theory
Vivek Shende
TL;DR
This work constructs microsheaves on the global nilpotent cone from Hitchin fibers by transporting smooth Hitchin fibers into the Floer-theoretic microsheaf framework of a conic hyperkähler integrable system $h:W\to A$. For a collection of smooth Hitchin fibers $L_i=h^{-1}(b_i)$, it produces microsheaves $F_i\in\mu sh_{h^{-1}(0)}(W)$ that are pairwise orthogonal and satisfy $\mathrm{End}(F_i)\cong H^*(L_i)$, via a diagonal-resolution argument that extends the GPS Fukaya–microsheaf correspondence to nonexact Lagrangians. This provides a Floer-theoretic pathway to potential Hecke eigensheaves in the Betti geometric Langlands program, while carefully addressing foundational issues such as coefficient fields (Novikov) and the stability locus; it also develops a nonexact-extension of the Fukaya framework and analyzes the interaction with hyperkähler geometry. The results illuminate how Hitchin fibers can be encoded as microsheaves and connect their Floer-theoretic invariants to the topology of the Hitchin fibers, offering new avenues toward a direct geometric Langlands program independent of representation-theoretic input. The discussion identifies current obstacles (e.g., nonconic coefficients, Hecke actions on the stable locus) and outlines future directions involving spectral networks and brane quantization to strengthen the Langlands-geometric interpretations.
Abstract
Fix a non-stacky component of the moduli of stable Higgs bundles, on which the Hitchin fibration is proper. We show that any smooth Hitchin fiber determines a microsheaf on the global nilpotent cone, that distinct fibers give rise to orthogonal microsheaves, and that the endomorphisms of the microsheaf is isomorphic to the cohomology of the Hitchin fiber. These results are consequences of recent advances in Floer theory. Natural constructions on our microsheaves provide plausible candidates for Hecke eigensheaves for the geometric Langlands correspondence.