Title: Discriminant circle bundles over local models of Strebel graphs and Boutroux curves,
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.