Hodge和Prym tau函数、Jenkins-Strebel微分与$\mathcal M_{g,n}$的组合模型
Hodge and Prym tau functions, Jenkins-Strebel differentials and combinatorial model of $\mathcal M_{g,n}$
摘要：
本文的主要目的是应用可积系统理论的启发，基于Jenkins-Strebel微分，构造点Riemann曲面的模空间的组合模型上的线丛的显式截面.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，而截面是用τ函数构造的.组合模型被解释为一个复杂的二次微分解析模空间的实切片，其中每个τ函数的相位提供了一个圆丛的一部分。Prym和Hodge tau函数之比的相位给出了$κ_1$ -圆丛的一部分。通过计算余维$2$子复形周围相位的增量，我们将Poincaré对偶圈与相应线丛的Chern类区分开来：它们被明确地表示为Witten圈$W_5$和Kontsevich边界的组合.这提供了组合类似物的芒福德的关系$\mathcal M_{g,n}$和彭纳的关系在双曲组合模型。围绕$W_5$的环的自由同伦类被解释为五边形移动，而围绕Kontsevich边界的环的自由同伦类被解释为组合Dehn扭曲。在整个文件中，我们利用经典描述的组合模型的詹金斯-Strebel微分，参数化的{ \it同调坐标}方面，我们还表明，他们提供达布坐标的辛结构介绍Kontsevich。我们还表示后者在明确的几何条款作为交叉配对的奇同调的规范双覆盖。
英文摘要：
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 $κ_1$-circle bundle. By evaluating the increment of the phase around co-dimension $2$ sub-complexes, we identify the Poincaré\ 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.
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