Multi-Point Virtual Structure Constants and Mirror Computation of CP^2-model

TL;DR

This paper introduces a geometric method using multi-point virtual structure constants to compute genus 0 Gromov-Witten invariants of CP^2, including open string cases, simplifying mirror transformation calculations.

Contribution

It develops a novel geometric framework with multi-point virtual structure constants for mirror computation of Gromov-Witten invariants, extending to open strings and non-nef hypersurfaces.

Findings

Generated mirror maps from multi-point constants.

Successfully computed open Gromov-Witten invariants of CP^2.

Simplified the process of generalized mirror transformation.

Abstract

In this paper, we propose a geometrical approach to mirror computation of genus 0 Gromov-Witten invariants of CP^2. We use multi-point virtual structure constants, which are defined as intersection numbers of a compact moduli space of quasi maps from CP^1 to CP^2 with 2+n marked points. We conjecture that some generating functions of them produce mirror map and the others are translated into generating functions of Gromov-Witten invariants via the mirror map. We generalize this formalism to open string case. In this case, we have to introduce infinite number of deformation parameters to obtain results that agree with some known results of open Gromov-Witten invariants of CP^2. We also apply multi-point virtual structure constants to compute closed and open Gromov-Witten invariants of a non-nef hypersurface in projective space. This application simplifies the computational process of generalized mirror transformation.