[Paper Review] New topological recursion relations

Xiaobo Liu, Rahul Pandharipande|arXiv (Cornell University)|May 30, 2008

Algebraic Geometry and Number Theory11 references4 citations

TL;DR

This paper establishes new topological recursion relations for powers of the cotangent line class $\psi_1^k$ on $\overline{M}_{g,1}$ when $k \geq 2g$, using virtual localization on the moduli space of stable maps to $\mathbb{P}^1$. The key result is a universal formula expressing $\psi_1^{2g+r}$ as a sum over boundary divisors, which leads to nontrivial tautological relations in the kernel of the boundary push-forward map and proves all Gromov-Witten conjectures of Liu and Xu.

ABSTRACT

Simple boundary expressions for the k-th power of the cotangent line class on the moduli space of stable 1-pointed genus g curves are found for k >= 2g. The method is by virtual localization on the moduli space of maps to the projective line. As a consequence, nontrivial tautological classes in the kernel of the push-forward map associated to the irreducible boundary divisor of the moduli space of stable g+1 curves are constructed. The geometry of genus g+1 curves then provides universal equations in genus g Gromov-Witten theory. As an application, we prove all the Gromov-Witten identities conjectured recently by K. Liu and H. Xu.

Motivation & Objective

• To derive explicit boundary expressions for $\psi_1^k$ in $A^{2g+r}(\overline{M}_{g,1})$ when $k \geq 2g$, extending known recursion relations.
• To construct nontrivial tautological classes in the kernel of the boundary push-forward map $\iota_*: A^*(\overline{M}_{g,2}) \to A^*(\overline{M}_{g+1})$.
• To prove the Gromov-Witten conjectures of Liu and Xu, which are universal relations involving high powers of $\psi$ classes.

Proposed method

• Apply virtual localization to the moduli space of stable maps $\overline{M}_{g,n}(\mathbb{P}^1,1)$ to evaluate special intersection numbers against the virtual class.
• Use the vanishing of certain tautological intersections on $\overline{M}_{g,n}(\mathbb{P}^1,1)$ to derive relations in $A^*(\overline{M}_{g,1})$.
• Express $\psi_1^{2g+r}$ as a sum over genus-splitting boundary divisors $\Delta_{1,\emptyset}(g_1,g_2)$, with coefficients involving $\psi_{\star_1}^a \psi_{\star_2}^b$ classes on the components.
• Leverage the splitting axiom of Gromov-Witten theory to translate tautological relations in $\overline{M}_{g,2}$ into universal equations in genus $g$ Gromov-Witten invariants.
• Utilize the operator $T$ on vector fields to translate relations into universal equations involving $T^k(\gamma_\ell)$ and $T^{m-k}(\gamma^\ell)$.
• Verify that the derived relations satisfy the required universal equations by matching with known formulas involving $\Psi_{n_1,n_2,g,m}$ and $\tau_k(\gamma_\ell)$.

Experimental results

Research questions

• RQ1 Can explicit boundary expressions be found for $\psi_1^k$ when $k \geq 2g$, beyond the known case for $k \geq g$?
• RQ2 Do nontrivial tautological classes exist in the kernel of the boundary push-forward map $\iota_*: A^*(\overline{M}_{g,2}) \to A^*(\overline{M}_{g+1})$?
• RQ3 Can the Gromov-Witten conjectures of Liu and Xu, involving high powers of $\psi$ classes, be proven universally using tautological relations?

Key findings

• A new topological recursion relation is established: $\psi_1^{2g+r} = \sum_{g_1+g_2=g, g_i>0} \sum_{a+b=2g-1+r} (-1)^a \frac{g_2}{g} \cdot \iota_*\left(\psi_{\star_1}^a \psi_{\star_2}^b \cap [\Delta_{1,\emptyset}(g_1,g_2)]\right)$ in $A^{2g+r}(\overline{M}_{g,1})$.
• The existence of nontrivial tautological classes in the kernel of $\iota_*$ is proven, providing a new mechanism to generate universal equations in genus $g$ Gromov-Witten theory.
• All conjectured Gromov-Witten identities of Liu and Xu are proven to hold universally, confirming their conjecture for all genera and all high powers of $\psi$ classes.
• The method reveals that universal equations in genus $g$ Gromov-Witten theory can be constructed from tautological relations in $\overline{M}_{g,2}$ via boundary push-forward, opening new avenues for finding such equations.
• The derived relations are nontrivial and geometrically meaningful, as they arise from the virtual geometry of $\overline{M}_{g,n}(\mathbb{P}^1,1)$ and not from tautological degenerations alone.
• The proof technique via virtual localization and operator translation via $T$ provides a systematic framework for deriving universal relations in Gromov-Witten theory from tautological relations on moduli spaces of curves.