[Paper Review] Hodge and Prym tau functions, Jenkins-Strebel differentials and combinatorial model of $\mathcal M_{g,n}$

Marco Bertola, D. Korotkin|arXiv (Cornell University)|Apr 7, 2018

Algebraic Geometry and Number Theory30 references4 citations

TL;DR

This paper constructs explicit sections of line bundles over the combinatorial moduli space $\mathcal{M}_{g,n}$ using Hodge and Prym tau-functions derived from Jenkins-Strebel differentials. By analyzing the monodromy of the phase of these tau-functions around codimension-2 cycles, it identifies the Poincaré duals of Chern classes as combinations of Witten's cycle $W_5$ and Kontsevich's boundary $W_{1,1}$, providing combinatorial analogues of Mumford and Penner's relations.

ABSTRACT

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.

Motivation & Objective

• To construct explicit sections of line bundles over the combinatorial model of $\mathcal{M}_{g,n}$ using tau-functions from integrable systems.
• To interpret the phase of Hodge and Prym tau-functions as sections of circle bundles, linking them to tautological classes.
• To identify the Poincaré dual cycles of $\lambda$, $\lambda_P$, and $\kappa_1$ classes in the combinatorial model via monodromy analysis.
• To establish combinatorial analogues of Mumford's and Penner's relations in the context of Jenkins-Strebel differentials.
• To interpret topological operations like pentagon moves and Dehn twists as monodromy actions around key cycles $W_5$ and $W_{1,1}$.

Proposed method

• Utilizes Jenkins-Strebel differentials to parametrize the combinatorial model $\mathcal{M}_{g,n}[\mathbf{p}]$ via period coordinates.
• Defines Hodge ($\tau_+$) and Prym ($\tau_-$) tau-functions on spaces of meromorphic quadratic differentials $\mathcal{Q}_{g,n}^{\mathbf{k},\mathbf{l}}$.
• Applies the theory of isomonodromic deformations to derive differential equations for $\tau_\pm$ and analyze their monodromy.
• Identifies the phase of $\tau_\pm$ as a section of a circle bundle, with monodromy around codimension-2 sub-complexes yielding Poincaré duals.
• Expresses Kontsevich's symplectic form as a period integral and shows it coincides with the intersection pairing on odd homology of the double cover.
• Uses Darboux coordinates (homological coordinates) to realize the symplectic structure and relate it to the combinatorial model.

Experimental results

Research questions

• RQ1 What is the combinatorial cycle Poincaré dual to the $\psi_i$-class in the Jenkins-Strebel model?
• RQ2 How can the Kontsevich boundary $W_{1,1}$ be completed to yield a one-to-one correspondence with the Deligne-Mumford compactification $\overline{\mathcal{M}}_{g,n}$?
• RQ3 Can all $\lambda$-classes be expressed directly in terms of cycles in the JS combinatorial model without relying on $\kappa$-class relations?
• RQ4 Is there a self-contained derivation of the KP hierarchy generating function for linear Hodge integrals using only the flat combinatorial model?
• RQ5 How do pentagon moves and combinatorial Dehn twists correspond to monodromy of the phase of $\tau_\pm$ around $W_5$ and $W_{1,1}$?

Key findings

• The Poincaré dual of the Hodge class $\lambda + \frac{1}{12}\sum \psi_i$ is $\frac{1}{144}W_5 + \frac{13}{144}W_{1,1}$ in $\mathrm{Pic}(\overline{\mathcal{M}}_{g,n}[\mathbf{p}], \mathbb{Q})$.
• The Poincaré dual of the Prym class $\lambda_P + \frac{1}{12}\sum \psi_i$ is $\frac{13}{144}W_5 + \frac{25}{144}W_{1,1}$, providing a combinatorial analogue of Prym class relations.
• The $\kappa_1$-class satisfies $12\kappa_1 = W_5 + W_{1,1}$, offering a new proof of a known relation in the combinatorial model.
• The ratio $\Theta_- / \Theta_+ = \left( \tau_- / |\tau_-| \right)^{48} / \left( \tau_+ / |\tau_+| \right)^{48}$ gives a section of the $48$-fold circle bundle $S[(\chi_\kappa)^{48}]$.
• The monodromy of $\Phi_+$ (phase of $\tau_+$) around $W_5$ corresponds to a pentagon move, and around $W_{1,1}$ to a combinatorial Dehn twist.
• The monodromy of $\Phi_-$ (phase of $\tau_-$) around $W_5$ and $W_{1,1}$ confirms the same topological interpretation, validating the geometric role of these cycles.