Non-commutative derived analytic moduli functors
J. P. Pridham
TL;DR
This work develops a non-commutative extension of derived analytic geometry using free entire functional calculus (FEFC) algebras, relating FEFC to pro-Banach topologies and to derived dagger analytic stacks. It builds a comprehensive homotopical framework for non-commutative analytic moduli via FEFC-DGAs and stacky DGAs, establishing model structures, ∞-category equivalences, and cotangent complexes. The paper then constructs derived moduli of hyperconnections and pro-étale local systems, together with Riemann–Hilbert-type maps, and introduces analytic Hodge and twistor structures in the non-commutative setting, including shifted bisymplectic and double Poisson structures. Overall, it provides foundational tools to study non-commutative analytic moduli stacks, their triangulated and higher-categorical structures, and their connections to classical Betti, de Rham, and Dolbeault moduli via analytification and Riemann–Hilbert-type correspondences.
Abstract
We develop a formulation for non-commutative derived analytic geometry built from differential graded (dg) algebras equipped with free entire functional calculus (FEFC), relating them to simplicial FEFC algebras and to locally multiplicatively convex complete topological dg algebras. The theory is optimally suited for accommodating analytic morphisms between functors of algebraic origin, and we establish forms of Riemann-Hilbert equivalence in this setting. We also investigate classes of topological dg algebras for which moduli functors of analytic origin tend to behave well, and relate their homotopy theory to that of FEFC algebras. Applications include the construction of derived non-commutative analytic moduli stacks of pro-étale local systems and non-commutative derived twistor moduli functors, both equipped with shifted analytic bisymplectic structures, and hence shifted analytic double Poisson structures.