[Paper Review] Polynomial structure of Gromov-Witten potential of quintic $3$-folds via NMSP

Huai-Liang Chang, Shuai Guo+1 more|arXiv (Cornell University)|Sep 28, 2018

Algebraic Geometry and Number Theory15 references11 citations

TL;DR

This paper establishes the polynomial structure of the Gromov-Witten potential for quintic 3-folds using the NMSP (Non-Linear Sigma Model with Projective Twisting) framework. By analyzing equivariant localization in the $$\mathscr{O}(5)$-twisted $\mathbb{P}^{N+4}$ theory and constructing an $R$-matrix from stabilization of localization graphs, the authors prove that the genus $g$ Gromov-Witten potential lies in the Yamaguchi-Yau ring generated by five specific invariants, confirming a deep algebraic structure in mirror symmetry.

ABSTRACT

We use stable graphs to package the $\mathrm{NMSP}$ relations. Our tools are the $S$ matrix of the $\mathcal{O}(5)$-twisted $ \mathbb P^{\mathrm N+4}$ equivariant GW theory, and the $R$ matrix obtained from the stablization of the theory's localization formula.

Motivation & Objective

• To establish the algebraic structure of the Gromov-Witten potential for quintic Calabi-Yau threefolds.
• To prove that the genus $g$ Gromov-Witten potential lies in the Yamaguchi-Yau ring generated by five specified invariants.
• To use the NMSP formalism and equivariant localization to derive a graph sum formula for invariants.
• To construct and analyze the $R$-matrix from stabilization of localization graphs in the $\mathscr{O}(5)$-twisted $\mathbb{P}^{N+4}$ theory.
• To connect the NMSP potential to the quintic’s quantum differential equation and mirror map.

Proposed method

• The authors use stable graphs to package NMSP relations and define $\mathrm{NMSP}$ correlators via virtual cycles in the $G=(\mathbb{C}^*)^N$-equivariant $\mathscr{O}(5)$-twisted $\mathbb{P}^{N+4}$ theory.
• They apply virtual localization to decompose the virtual cycle into contributions from fixed loci indexed by flat decorated graphs $\Theta \in G_{g,n,\mathbb{d}}^{\mathrm{fl}}$, with a focus on the regular subclass $G_{g,n,\mathbb{d}}^{\mathrm{reg}}$.
• The $S$-matrix of the $\mathscr{O}(5)$-twisted theory is computed via equivariant integration, and the $R$-matrix is derived from stabilization of the localization formula.
• The theory is structured using the base field $\mathbb{F} = \mathbb{Q}(\zeta_N)(t)$ and the coefficient ring $\mathbb{A} = \mathbb{F}[[q]]$, with equivariant parameters $t_\alpha$ substituted via $t_\alpha = -\zeta_N^\alpha t$.
• The quantum differential equation (QDE) for the quintic is solved using the mirror map, and the $S$-matrix is computed explicitly in terms of $I$-functions and their derivatives.
• The $R$-matrix is constructed by composing the localization $R^\mathrm{loc}$ with the GRR formula, enabling the identification of the quintic potential with the Yamaguchi-Yau ring.

Experimental results

Research questions

• RQ1 Does the Gromov-Witten potential of the quintic 3-fold exhibit a polynomial structure in terms of a finite set of generators?
• RQ2 Can the NMSP formalism be used to derive a graph sum formula for Gromov-Witten invariants of the quintic?
• RQ3 Is the genus $g$ Gromov-Witten potential of the quintic contained in the Yamaguchi-Yau ring generated by five specified invariants?
• RQ4 How does the $R$-matrix from localization stabilize to yield the correct quantum cohomology structure?
• RQ5 What is the precise relationship between the $S$-matrix of the $\mathscr{O}(5)$-twisted $\mathbb{P}^{N+4}$ theory and the quantum differential equation of the quintic?

Key findings

• The genus $g$ Gromov-Witten potential $F_g$ of the quintic 3-fold lies in the Yamaguchi-Yau ring $\mathscr{R} = \mathbb{Q}[A, B, B_2, B_3, Y]$, confirming a finite algebraic structure.
• The $S$-matrix of the $\mathscr{O}(5)$-twisted $\mathbb{P}^{N+4}$ theory is computed explicitly at the mirror point $\tau_Q$, yielding a matrix expression involving $J_1, J_2', J_3$ and their ratios.
• The $R$-matrix is constructed via stabilization of localization graphs, ensuring compatibility with the GRR formula and enabling the identification of the quintic potential with the NMSP potential.
• The quantum differential equation (QDE) for the quintic is solved using the mirror map, and the resulting $S^Q(z)^*$ matrix is expressed as a series in $1/z$ with coefficients in terms of $I$-function derivatives.
• The $R$-matrix derived from localization and GRR provides a bridge between the NMSP theory and the quintic’s quantum cohomology, proving the polynomial structure of the potential.
• The $\mathrm{NMSP}$ correlators are shown to be polynomials in $q' = -q/t^N$ with coefficients in $\mathbb{F}$, after substitution of equivariant parameters, confirming the finitely generated nature of the potential.