Period sheaves via perverse pullbacks

TL;DR

The paper develops a comprehensive framework to realize period sheaves for Hamiltonian spaces using perverse pullbacks in derived geometry. By proving a dimensional reduction theorem for relative (−1)-shifted cotangent bundles, it shows that perverse pullbacks refine ordinary pullbacks and relate to microstalks, enabling global Fourier–Sato/Kashiwara-type dualities. The construction recovers known period-sheaf structures in cotangent and Whittaker cases and provides an induction formalism that yields normalized period sheaves and Eisenstein-like functors, linking to Coulomb/Higgs-branch phenomena and geometric Langlands-style dualities. The work thus unifies microlocal, symplectic, and representation-theoretic aspects to produce a robust toolkit for studying period sheaves on Hamiltonian spaces with broad implications for cohomological DT theory and moduli problems. It also sets the stage for further extensions to Coulomb branches and semi-infinite geometries in the broader Langlands program.

Abstract

We construct period sheaves for Hamiltonian spaces, as conjectured in the work of Ben-Zvi, Sakellaridis and Venkatesh, using the perverse pullback functors introduced in the authors' previous work. We prove a dimensional reduction isomorphism (generalizing the results of Davison and Kinjo in cohomological Donaldson--Thomas theory) which implies that perverse pullbacks refine the ordinary pullback functors in constructible sheaf theory, and relate perverse pullbacks to microstalk functors. These results imply that our period sheaves recover the known constructions in the cotangent and Whittaker cases.

Theorems & Definitions (127)

• Theorem 1: \ref{['prop:microsupportestimate']}
• Theorem 2: \ref{['thm:dimensionalreduction']}
• Theorem 3
• Remark 1.1
• Example 1.2
• Example 1.3
• Example 1.4
• Proposition 1.5
• Proposition 1.6
• Example 1.7