[Paper Review] Coordinate Change of Gauss-Manin System and Generalized Mirror Transformation

Masao Jinzenji|arXiv (Cornell University)|Oct 15, 2003

Algebraic Geometry and Number Theory16 references19 citations

TL;DR

This paper establishes a coordinate change framework for the virtual Gauss-Manin system to derive the generalized mirror transformation for quantum cohomology of general type projective hypersurfaces $M^k_N$ with $k > N$. By extending the virtual Gauss-Manin connection to include deformation parameters $x_j$ corresponding to $O^{e_j}$, the authors compute the Jacobian $\partial t_j / \partial x_i$ from matrix elements, integrate it to obtain coordinate transformations $x_i(t_1, \dots, t_{N-2})$, and reproduce known genus 0 Gromov-Witten invariants up to degree 5. The method is validated by exact agreement with fixed-point computations for degree 6 curves in $M^{14}_{13}$, confirming the algorithm's predictive power for arbitrary genus 0 invariants.

ABSTRACT

In this paper, we explicitly derive the generalized mirror transformation of quantum cohomology of general type projective hypersurfaces, proposed in our previous article, as an effect of coordinate change of the virtual Gauss-Manin system.

Motivation & Objective

• To provide an explicit, algorithmic method to compute K"ahler Gromov-Witten invariants of general type hypersurfaces $M^k_N$ with $k > N$ for rational curves of arbitrary degree.
• To resolve the gap in previous mirror symmetry constructions by explicitly deriving the generalized mirror transformation as a coordinate change of the virtual Gauss-Manin system.
• To validate the method by reproducing known invariants up to degree 5 and predicting new ones, including a confirmed result for degree 6 invariants in $M^{14}_{13}$.
• To unify the virtual structure constants and mirror transformation within a coherent framework based on Iritani's theory of equivariant Givental-style quantization.

Proposed method

• Construct a virtual Gauss-Manin system with matrix elements defined by virtual structure constants derived from recursive formulas in previous works.
• Extend the system to include deformation parameters $x_j$ ($j = 1, \dots, N-2$) corresponding to $O^{e_j}$, and define virtual Gauss-Manin connections $\partial_{x_j} \vec{\psi} = \tilde{C}_j(e^{x_1}) \vec{\psi}$ with commutativity and normalization conditions.
• Derive the Jacobian matrix $\partial t_j / \partial x_i$ from the matrix elements $\tilde{C}_i(e^{x_1})$ using the flat coordinate condition $\partial \psi_0 / \partial t_j = \psi_j$, leading to $\partial t_j / \partial x_i = (\tilde{C}_i(e^{x_1}))_{0j}$.
• Integrate the Jacobian to obtain the coordinate transformation $x_i = x_i(t_1, \dots, t_{N-2})$, with explicit expressions derived from $\exp(j t_1)$-weighted virtual structure constants.
• Apply the associativity equation and a modified K"ahler equation (derived from Iritani's framework) to perturb the rank-3 tensor $\bar{C}_{ijm}(e^{x_1})$ in $x_2, \dots, x_{N-2}$, enabling systematic computation of higher-point Gromov-Witten invariants.
• Expand the perturbed tensor in $x_2, \dots, x_{N-2}$ to recover the generalized mirror transformation as a power series, matching known results up to degree 5 and predicting new ones.

Experimental results

Research questions

• RQ1 How can the generalized mirror transformation for $M^k_N$ with $k > N$ be systematically derived from the virtual Gauss-Manin system?
• RQ2 What is the precise coordinate transformation $x_i(t_1, \dots, t_{N-2})$ that maps B-model deformation parameters to A-model flat coordinates?
• RQ3 How can the virtual Gauss-Manin connection be extended to include higher-degree insertions $O^{e_j}$ to enable computation of higher-genus invariants?
• RQ4 To what extent do the associativity and modified K"ahler equations allow reconstruction of higher-point Gromov-Witten invariants from lower-point data?
• RQ5 Can the proposed method predict and verify genus 0 Gromov-Witten invariants for rational curves of degree 6 in $M^{14}_{13}$?

Key findings

• The generalized mirror transformation for $M^k_N$ with $k > N$ is derived as a coordinate change of the virtual Gauss-Manin system, explicitly linking B-model parameters to A-model flat coordinates.
• The Jacobian $\partial t_j / \partial x_i$ is computed from matrix elements of the virtual Gauss-Manin connection $\tilde{C}_i(e^{x_1})$, enabling full reconstruction of the coordinate transformation.
• The method successfully reproduces all known genus 0 Gromov-Witten invariants up to degree 5, confirming consistency with prior results from [8] and [12].
• For degree 6 rational curves in $M^{14}_{13}$, the method predicts $L^{13,14,6}_8 = 3895919811389645033770563942661264371465474526956421097691226041140067266620858637887736545388962432/9375$, which exactly matches numerical results from Kontsevich's fixed-point theorem.
• The framework naturally incorporates Iritani's modified K"ahler equation and associativity constraints, providing a systematic and predictive method for arbitrary genus 0 invariants.