The derived moduli stack of logarithmic flat connections

TL;DR

This paper constructs an explicit finite-dimensional model for the derived moduli stack of flat connections with logarithmic singularities along certain divisors, explores specific cases like plane curves, and discusses their shifted Poisson geometry.

Contribution

It provides a new finite-dimensional model for the derived moduli stack of logarithmic flat connections and investigates their shifted Poisson structures, especially for plane curve cases.

Findings

Explicit finite-dimensional model for the derived moduli stack.

Relation of moduli spaces to the Grothendieck-Springer resolution.

Conjecture and construction of shifted Poisson structures.

Abstract

We give an explicit finite-dimensional model for the derived moduli stack of flat connections on $\mathbb{C}^k$ with logarithmic singularities along a weighted homogeneous Saito free divisor. We investigate in detail the case of plane curves of the form $x^p = y^q$ and relate the moduli spaces to the Grothendieck-Springer resolution. We also discuss the shifted Poisson geometry of these moduli spaces. Namely, we conjecture that the map restricting a logarithmic connection to the complement of the divisor admits a shifted coisotropic structure and we construct a shifted Poisson structure on the formal neighborhood of a canonical connection in the case of plane curves $x^p = y^q$.