Communications in Number Theory and Physics

Volume 7 (2013)

Number 3

Multi-point virtual structure constants and mirror computation of $CP^2$-model

Pages: 411 – 468



Masao Jinzenji (Department of Mathematics, Graduate School of Science, Hokkaido University, Sapporo, Japan)

Masahide Shimizu (Institute for the Advancement of Higher Education, Hokkaido University, Sapporo, Japan)


In this paper, we propose a geometrical approach to mirror computation of genus $0$ Gromov-Witten invariants of $CP^2$. We use multipoint 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.

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