Global geometry on moduli of local systems for surfaces with boundary

TL;DR

This paper proves that moduli spaces of rank two local systems with fixed boundary traces on surfaces with boundary are log Calabi-Yau, revealing a new symmetry in multicurve counting series.

Contribution

It establishes the log Calabi-Yau property for these moduli spaces and uncovers a novel symmetry in multicurve enumeration.

Findings

Moduli spaces are log Calabi-Yau with trivial log canonical divisor.

Identifies a new symmetry in generating series for essential multicurves.

Provides a compactification framework for these moduli spaces.

Abstract

We show that every coarse moduli space, parametrizing complex special linear rank two local systems with fixed boundary traces on a surface with nonempty boundary, is log Calabi-Yau in that it has a normal projective compactification with trivial log canonical divisor. We connect this to a novel symmetry of generating series for counts of essential multicurves on the surface.