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