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$\mathcal Q^{\mathbb R}_0([-3]^2)$\mathrm{.}Thespace$\mathcal Q_0^{\mathbb R}(-7)$ 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 ${\mathcal{Q}}_0^{\mathbb{R}}([-3{]}^2)$; 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 ${\mathcal{Q}}_0^{\mathbb{R}}([-3{]}^2)$, also called the spaces of Boutroux curves, in detail, together with corresponding circle bundles.