Non-commutative derived analytic moduli functors

TL;DR

This paper develops a framework for non-commutative derived analytic geometry using differential graded algebras with functional calculus, enabling the study of moduli stacks with rich geometric structures.

Contribution

It introduces a novel formulation connecting dg algebras with functional calculus to analytic moduli functors, expanding the tools for non-commutative derived geometry.

Findings

Constructed derived non-commutative analytic moduli stacks.

Established Riemann-Hilbert equivalences in this setting.

Linked homotopy theory of topological dg algebras to FEFC algebras.

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-\'etale local systems and non-commutative derived twistor moduli functors, both equipped with shifted analytic bisymplectic structures, and hence shifted analytic double Poisson structures.