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

We study special “discriminant” circle bundles over two elementary moduli spaces of meromorphic quadratic differentials with real periods denoted by Q 0 ℝ (−7) and Q 0 ℝ ([−3]2). The space Q 0 ℝ (−7) is the moduli space of meromorphic quadratic differentials on the Riemann sphere with one pole of order seven with real periods; it appears naturally in the study of a neighborhood of the Witten cycle W5 in the combinatorial model based on Jenkins–Strebel quadratic differentials of Mg,n. The space Q 0 ℝ ([−3]2) is the moduli space of meromorphic quadratic differentials on the Riemann sphere with two poles of order at most three with real periods; it appears in the description of a neighborhood of Kontsevich’s boundary W1,1 of the combinatorial model. Applying the formalism of the Bergman tau function to the combinatorial model (with the goal of analytically computing cycles Poincaré dual to certain combinations of tautological classes) requires studying special sections of circle bundles over Q 0 ℝ (−7) and Q 0 ℝ ([−3]2). In the Q 0 ℝ (−7) case, a section of this circle bundle is given by the argument of the modular discriminant. We study the spaces Q 0 ℝ (−7) and Q 0 ℝ ([−3]2), also called the spaces of Boutroux curves, in detail together with the corresponding circle bundles.