[Paper Review] Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves

Enrico Arbarello, Maurizio Cornalba|arXiv (Cornell University)|Jun 30, 1994

Algebraic Geometry and Number Theory15 references133 citations

TL;DR

This paper establishes a geometric correspondence between combinatorial cycles on the moduli space of curves and algebro-geometric cohomology classes, showing that intersection numbers of ψ-classes over combinatorial cycles $W_{m_*,n}$ equal those over explicit polynomials $X_{m_*,n}$ in Mumford-Morita-Miller classes. The key result is a precise duality between combinatorial cycles and algebraic cohomology classes on the Deligne-Mumford compactification $\overline{\mathcal{M}}_{g,n}$, verified in codimension one and for the first 11 weight cases.

ABSTRACT

Based on the combinatorial description of the moduli spaces of curves provided by Strebel differentials, Witten and Kontsevich have introduced combinatorial cohomology classes $W_{(m_0,m_1,m_2,\dots),n}$, and conjectured that these can be expressed in terms of Mumford-Morita-Miller classes. It is argued that this link should be provided by a theorem of Di Francesco, Itzykson and Zuber which relates the derivatives of the Witten-Kontsevich partition function with respect to one set of variables to the derivatives with respect to the other set of variables. Two things are shown. First of all that this works in complex codimension 1. Secondly that in all the cases when it has been possible to make the Di Francesco, Itzykson and Zuber correpondence explicit this translates into identities of the type $$ \int_{W_{(m_0,m_1,m_2,\dots),n}}\prodψ_i^{d_i} =\int_{\overline{\cal{M}}_{g,n}} X_{(m_0,m_1,m_2,\dots),n}\prodψ_i^{d_i} $$ where the $X_{(m_0,m_1,m_2,\dots),n}$ are explicit polynomials in the algebro-geometric classes and the $ψ_i$ are the Chern classes of the point bundles, for any choice of $d_1,\dots,d_n$.

Motivation & Objective

• To establish a geometric link between combinatorial cycles $W_{m_*,n}$ on the moduli space $\mathcal{M}_{g,n}$ and algebro-geometric cohomology classes.
• To verify Witten's conjecture that combinatorial cycles can be expressed in terms of Mumford-Morita-Miller classes.
• To show that intersection numbers of $\psi_i$-classes over $W_{m_*,n}$ equal those over explicit polynomials $X_{m_*,n}$ in the tautological ring of $\overline{\mathcal{M}}_{g,n}$.
• To extend the Di Francesco-Itzykson-Zuber correspondence to a geometric duality on the compactified moduli space.

Proposed method

• Uses Kontsevich's matrix model to relate generating series $F$ of $\psi$-class intersection numbers to asymptotic expansions of Hermitian matrix integrals.
• Applies the Di Francesco-Itzykson-Zuber correspondence to translate derivatives of $\exp F$ with respect to $s$-variables into combinations of derivatives with respect to $t$-variables.
• Interprets the resulting identities geometrically as cohomological dualities between combinatorial cycles $W_{m_*,n}$ and algebraic classes $X_{m_*,n}$.
• Analyzes the codimension one case using the stratification of $\overline{\mathcal{M}}_{g,n}$ by singular curves and the dual graph formalism.
• Employs the $\psi_i$-classes as Chern classes of cotangent line bundles at marked points to define intersection numbers.
• Computes explicit expressions for $X_{m_*,n}$ in terms of $\psi$-classes and other tautological classes, verified up to weight 11.

Experimental results

Research questions

• RQ1 Can combinatorial cycles $W_{m_*,n}$ on $\mathcal{M}_{g,n}$ be represented as algebraic cohomology classes in $\overline{\mathcal{M}}_{g,n}$?
• RQ2 Is there a geometric realization of the Di Francesco-Itzykson-Zuber correspondence in terms of cohomology classes on the compactified moduli space?
• RQ3 Do intersection numbers $\int_{W_{m_*,n}} \prod \psi_i^{d_i}$ equal $\int_{\overline{\mathcal{M}}_{g,n}} X_{m_*,n} \prod \psi_i^{d_i}$ for explicit polynomials $X_{m_*,n}$?
• RQ4 Can the duality between combinatorial cycles and algebraic classes be extended beyond codimension one?

Key findings

• The intersection numbers $\langle \tau_{d_1} \dots \tau_{d_n} \rangle_{m_*}$ over combinatorial cycles $W_{m_*,n}$ match those over explicit algebraic classes $X_{m_*,n}$ in the tautological ring of $\overline{\mathcal{M}}_{g,n}$.
• In codimension one, the subspace of cohomology on which the duality holds is maximal for $n > 1$, confirming the duality's strength.
• For the first 11 weight cases (as defined by degree of $F$), the Di Francesco-Itzykson-Zuber correspondence translates into exact cohomological identities.
• The generating series $F$ satisfies a system of partial differential equations that encode the correspondence between $s$- and $t$-derivatives, which are geometrically realized as cohomological identities.
• The correspondence is verified explicitly for $\partial^3 F / \partial s_2^3$, yielding a complex expression in terms of $t$-derivatives and $\psi$-class terms.
• The result supports the conjecture that the Di Francesco-Itzykson-Zuber correspondence underlies a deep geometric duality between combinatorial and algebro-geometric classes on $\overline{\mathcal{M}}_{g,n}$.