[Paper Review] Tautological equations in genus 2 via invariance conjectures

Daniele Arcara, Y. P. Lee|ArXiv.org|Feb 23, 2005

Nonlinear Waves and Solitons1 references3 citations

TL;DR

This paper verifies the Invariance Conjectures for tautological equations in genus 2, providing a uniform derivation of all known genus two tautological equations—Mumford–Getzler’s, Getzler’s, and Belorousski–Pandharipande’s equations—using the $ \mathfrak{r}_l$-invariance condition and Getzler’s Hodge number computations. The method confirms the conjectures for $(g,n,k) = (2,1,2), (2,2,2), (2,3,2)$, establishing a complete and systematic proof framework for genus 2 tautological relations.

ABSTRACT

We verify the Invariance Conjectures of tautological equations in genus two. In particular, a uniform derivation of all known genus two equations is given.

Motivation & Objective

• To verify the Invariance Conjectures for tautological equations in genus 2, specifically for $(g,n,k) = (2,1,2), (2,2,2), (2,3,2)$.
• To provide a uniform derivation of all known genus two tautological equations using the $ \mathfrak{r}_l$-invariance condition.
• To confirm that the invariance condition generates all tautological relations in genus 2, completing the inductive framework proposed in prior conjectures.
• To establish that the tautological relations in genus 2 are fully captured by the $ \mathfrak{r}_l$-invariance and Betti number constraints from Getzler’s calculations.

Proposed method

• The authors use the $ \mathfrak{r}_l$-operators on decorated graphs (gwis) to compute the image of a general tautological class in codimension 2 under the $ \mathfrak{r}_1$ and $ \mathfrak{r}_2$ maps.
• They apply the TRR (Tensor-Reduction Rule) to express $ \psi$-classes in terms of boundary strata, reducing the number of independent tautological classes to six in ${\overline{\mathcal{M}}}_{2,1}$.
• The method relies on symmetrization and anti-symmetrization of labels $i,j$ depending on the parity of $l$ to handle the $ \mathfrak{r}_l$-action on gwi graphs.
• Coefficients of the resulting $ \mathfrak{r}_1(E)$ and $ \mathfrak{r}_2(E)$ expressions are equated to zero, yielding a system of linear equations in the unknowns $c_1, \dots, c_6$.
• The system is solved using the Betti number data from Getzler’s computation, which confirms the rank of $R^2({\overline{\mathcal{M}}}_{g,n})$ and ensures no missing relations.
• The vanishing of all $ \mathfrak{r}_l(E)$ for $l \geq 1$ implies $E=0$ via Conjecture 2, proving the tautological equation.

Experimental results

Research questions

• RQ1 Does the $ \mathfrak{r}_l$-invariance condition generate all known tautological equations in genus 2?
• RQ2 Can the Invariance Conjectures be verified for $(g,n,k) = (2,1,2), (2,2,2), (2,3,2)$ using gwi calculus and Hodge data?
• RQ3 Is the system of equations derived from $ \mathfrak{r}_1(E)=0$ and $ \mathfrak{r}_2(E)=0$ sufficient to determine all tautological relations in genus 2?
• RQ4 Does the absence of additional relations beyond those derived from $ \mathfrak{r}_l$-invariance confirm that the tautological ring is fully captured in genus 2?

Key findings

• The Invariance Conjectures hold for all $(g,n,k) = (2,1,2), (2,2,2), (2,3,2)$, confirming the $ \mathfrak{r}_l$-invariance condition as a complete criterion for tautological relations in genus 2.
• A uniform derivation of Mumford–Getzler’s, Getzler’s, and Belorousski–Pandharipande’s equations is achieved via the $ \mathfrak{r}_l$-invariance condition.
• The system of equations from $ \mathfrak{r}_1(E)=0$ and $ \mathfrak{r}_2(E)=0$ yields a unique solution up to the known tautological relations, confirming no missing equations exist.
• Getzler’s Betti number computation confirms that the tautological ring $R^2({\overline{\mathcal{M}}}_{g,n})$ has the expected rank, validating the completeness of the derived equations.
• The method proves all known tautological equations in genus 2, completing the inductive framework for tautological relations across genera.
• The results support the Virasoro conjecture in the semisimple case and Witten’s conjecture on spin curves and Gelfand–Dickey hierarchies in genus 2.