[Paper Review] Gerby Localization, Z_3-Hodge Integrals and the GW Theory of C^3/Z_3

Charles Cadman, Renzo Cavalieri|ArXiv.org|May 15, 2007

Algebraic Geometry and Number Theory25 references8 citations

TL;DR

This paper establishes recursive relations that effectively compute all equivariant Gromov-Witten invariants of the orbifold $[\mathbb{C}^3/\mathbb{Z}_3]$ by interpreting them as $\mathbb{Z}_3$-Hodge integrals on moduli spaces of twisted stable maps. Using Atiyah-Bott localization on gerbes over $\mathbb{P}^1$, the authors derive a system of recursions and a corresponding system of PDEs, enabling explicit computation of invariants and verifying predictions from physics via mirror symmetry.

ABSTRACT

We exhibit a set of recursive relations that completely determine all equivariant Gromov-Witten invariants of the quotient orbifold C^3/Z_3. We interpret such invariants as G-Hodge Integrals, and produce relations among them via Atiyah-Bott localization on moduli spaces of twisted stable maps to gerbes over the projective line.

Motivation & Objective

• To compute all equivariant Gromov-Witten invariants of the orbifold $[\mathbb{C}^3/\mathbb{Z}_3]$ using recursive relations.
• To interpret these invariants as $\mathbb{Z}_3$-Hodge integrals on moduli spaces of $\mathbb{Z}_3$-admissible covers.
• To derive relations among these Hodge integrals using Atiyah-Bott localization on gerbes over $\mathbb{P}^1$.
• To translate the recursion relations into a system of partial differential equations (PDEs).
• To verify and compute invariants predicted by physicists via mirror symmetry, particularly those of Aganagic, Bouchard, and Klemm.

Proposed method

• The authors use Atiyah-Bott localization on moduli spaces of twisted stable maps to $\mathcal{B}\mathbb{Z}_3$, specifically gerbes over $\mathbb{P}^1$, to derive relations among $\mathbb{Z}_3$-Hodge integrals.
• They define $\mathcal{L}^\omega$ and $\mathcal{L}^{\bar{\omega}}$ generating functions encoding Gromov-Witten invariants and derive a system of PDEs from the localization relations.
• The method relies on interpreting Gromov-Witten invariants as integrals of $\lambda_i$ classes over moduli spaces $\mathcal{A}(n_1,n_2)$ of $\mathbb{Z}_3$-admissible covers of $\mathbb{P}^1$.
• The recursion relations are derived from the structure of the obstruction bundle and the $\psi$-class removal formulas in the localization computation.
• The authors relate their $\mathcal{L}$ functions to Givental's $J$-function, though a full reformulation in $J$-function terms remains elusive.
• They implement the recursions computationally to generate a table of invariants for $n_1 + n_2 \leq 24$, including non-equivariant cases.

Experimental results

Research questions

• RQ1 Can all equivariant Gromov-Witten invariants of $[\mathbb{C}^3/\mathbb{Z}_3]$ be effectively computed using recursive relations?
• RQ2 How can $\mathbb{Z}_3$-Hodge integrals on moduli spaces of $\mathbb{Z}_3$-admissible covers be related via localization?
• RQ3 What system of PDEs governs the generating functions of these invariants?
• RQ4 Do the computed invariants match the predictions of Aganagic, Bouchard, and Klemm from topological string theory?
• RQ5 Can the $\mathcal{L}$-functions be meaningfully reformulated in terms of Givental's $J$-function?

Key findings

• The paper provides a complete set of recursive relations that effectively compute all equivariant Gromov-Witten invariants of $[\mathbb{C}^3/\mathbb{Z}_3]$.
• All three-part $\mathbb{Z}_3$-Hodge integrals $\int_{\mathcal{A}(n_1,n_2)} \lambda_i \lambda_j \lambda_k$ are effectively computable via the derived recursions.
• The authors compute and tabulate invariants for $n_1 + n_2 \leq 24$, including non-equivariant cases such as $\langle \omega^3 \rangle = \frac{1}{3}$ and $\langle \omega^6 \rangle = -\frac{1}{27}$.
• The recursion system is encoded in a system of PDEs involving the generating functions $\mathcal{L}^\omega$ and $\mathcal{L}^{\bar{\omega}}$, with explicit relations for $\psi$-class removal.
• The results confirm the predictions of Aganagic, Bouchard, and Klemm for Gromov-Witten invariants of $[\mathbb{C}^3/\mathbb{Z}_3]$.
• The method establishes a bridge between orbifold Gromov-Witten theory, Hodge integrals, and localization, with potential connections to the Crepant Resolution Conjecture and tautological classes on moduli spaces of curves.