# Discriminant circle bundles over local models of Strebel graphs and Boutroux curves

## TL;DR

This paper investigates special circle bundles over moduli spaces of meromorphic quadratic differentials with real periods, focusing on their structure and sections, to facilitate the application of the Bergman tau-function in algebraic geometry.

## Contribution

It provides a detailed study of circle bundles over Boutroux curve spaces, linking them to combinatorial models and tau-functions for computing tautological classes.

## Key findings

- Analysis of circle bundles over $ ext{Q}_0^{ ext{R}}(-7)$ and $ ext{Q}_0^{ ext{R}}([-3]^2)$
- Identification of sections of these bundles via modular discriminant argument
- Connection to combinatorial models and tau-function formalism

## Abstract

We study special circle bundles over two elementary moduli spaces of meromorphic quadratic differentials with real periods denoted by $\mathcal Q_0^{\mathbb R}(-7)$ and $\mathcal Q^{\mathbb R}_0([-3]^2)$. The space $\mathcal Q_0^{\mathbb R}(-7)$ is the moduli space of meromorphic quadratic differentials on the Riemann sphere with one pole of order 7 with real periods; it appears naturally in the study of a neighbourhood of the Witten's cycle $W_1$ in the combinatorial model based on Jenkins-Strebel quadratic differentials of $\mathcal M_{g,n}$. The space $\mathcal Q^{\mathbb R}_0([-3]^2)$ is the moduli space of meromorphic quadratic differentials on the Riemann sphere with two poles of order at most 3 with real periods; it appears in description of a neighbourhood of Kontsevich's boundary $W_{-1,-1}$ of the combinatorial model. The application of the formalism of the Bergman tau-function to the combinatorial model (with the goal of computing analytically Poincare dual cycles to certain combinations of tautological classes) requires the study of special sections of circle bundles over $\mathcal Q_0^{\mathbb R}(-7)$ and $\mathcal Q^{\mathbb R}_0([-3]^2)$; in the case of the space $\mathcal Q_0^{\mathbb R}(-7)$ a section of this circle bundle is given by the argument of the modular discriminant. We study the spaces $\mathcal Q_0^{\mathbb R}(-7)$ and $\mathcal Q^{\mathbb R}_0([-3]^2)$, also called the spaces of Boutroux curves, in detail, together with corresponding circle bundles.