The principal goal of the paper is to apply the approach inspired by the theory of integrable systems to construct explicit sections of line bundles over the combinatorial model of the moduli space of pointed Riemann surfaces based on Jenkins-Strebel differentials. The line bundles are tensor products of the determinants of the Hodge or Prym vector bundles with the standard tautological line bundles $\mathcal L_j$ and the sections are constructed in terms of tau functions. The combinatorial model is interpreted as the real slice of a complex analytic moduli space of quadratic differentials where the phase of each tau-function provides a section of a circle bundle. The phase of the ratio of the Prym and Hodge tau functions gives a section of the $\kappa_1$-circle bundle. By evaluating the increment of the phase around co-dimension $2$ sub-complexes, we identify the Poincare\ dual cycles to the Chern classes of the corresponding line bundles: they are expressed explicitly as combination of Witten's cycle $W_5$ and Kontsevich's boundary. This provides combinatorial analogues of Mumford's relations on $\mathcal M_{g,n}$ and Penner's relations in the hyperbolic combinatorial model. The free homotopy classes of loops around $W_5$ are interpreted as pentagon moves while those of loops around Kontsevich's boundary as combinatorial Dehn twists. Throughout the paper we exploit the classical description of the combinatorial model in terms of Jenkins--Strebel differentials, parametrized in terms of {\it homological coordinates}; we also show that they provide Darboux coordinates for the symplectic structure introduced by Kontsevich. We also express the latter in clear geometric terms as the intersection pairing in the odd homology of the canonical double cover.

中文翻译：

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$${\mathcal {M}}_{g,n}$$ 的 Hodge 和 Prym Tau 函数、Strebel 微分和组合模型

本文的主要目标是应用受可积系统理论启发的方法，在基于 Jenkins-Strebel 微分的尖黎曼曲面模空间的组合模型上构造线丛的显式截面。线丛是 Hodge 或 Prym 向量丛的行列式与标准重言式线丛 $\mathcal L_j$ 的张量积，并且截面是根据 tau 函数构造的。组合模型被解释为二次微分的复解析模空间的实切片，其中每个 tau 函数的相位提供圆丛的一部分。Prym 和 Hodge tau 函数之比的相位给出了 $\kappa_1$-圆束的一部分。通过评估围绕共维 $2$ 子复合体的相位增量，我们将 Poincare\ 双循环识别为相应线丛的陈类：它们明确表示为 Witten 循环 $W_5$ 和 Kontsevich 边界的组合。这提供了 Mumford 在 $\mathcal M_{g,n}$ 上的关系和 Penner 在双曲组合模型中的关系的组合类似物。围绕$W_5$ 的循环的自由同伦类被解释为五边形移动，而围绕Kontsevich 边界的那些循环被解释为组合Dehn 扭曲。在整篇论文中，我们利用 Jenkins--Strebel 微分对组合模型的经典描述，根据 {\it 同调坐标}参数化；我们还表明，它们为 Kontsevich 引入的辛结构提供了 Darboux 坐标。