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Hori's Equation for Gravitational Virtual Structure Constants of Calabi-Yau Hypersurface in CP^{N-1}

Masao Jinzenji, Kohki Matsuzaka

TL;DR

This work tackles the problem of computing gravitational intersection numbers on moduli spaces of quasimaps from $\mathbb{CP}^{1}$ to $\mathbb{CP}^{N-1}$ with Calabi–Yau hypersurface constraints. It develops the localization framework on the orbifold moduli space $\widetilde{Mp}_{0,2|1}(N,d)$ by constructing the orbi-bundle $\mathcal{E}'_d$ whose top Chern class enforces the Calabi–Yau condition, and derives an explicit fixed-point formula for the gravitational and non-gravitational insertions. The main result is a quasimap analogue of Hori's equation: for $j>0$, $w( \sigma_j(\mathcal{O}_{h^a}) \mathcal{O}_{h^b} | \mathcal{O}_{h} )_{0,d} = d \cdot w( \sigma_j(\mathcal{O}_{h^a}) \mathcal{O}_{h^b} )_{0,d} + w( \sigma_{j-1}(\mathcal{O}_{h^{a+1}}) \mathcal{O}_{h^b} )_{0,d}$, linking gravitational constants to their non-gravitational counterparts and aligning with mirror symmetry via the $I$-function. The approach provides a robust computational path for genus-0 Calabi–Yau Gromov–Witten data and clarifies how period integrals arise as generating functions of gravitational structure constants."

Abstract

In this paper, we give a proof of Hori's equation for intersection numbers of the moduli space of quasimaps from CP^{1} with (2+1) marked points to CP^{N-1} by using localization technique.

Hori's Equation for Gravitational Virtual Structure Constants of Calabi-Yau Hypersurface in CP^{N-1}

TL;DR

This work tackles the problem of computing gravitational intersection numbers on moduli spaces of quasimaps from to with Calabi–Yau hypersurface constraints. It develops the localization framework on the orbifold moduli space by constructing the orbi-bundle whose top Chern class enforces the Calabi–Yau condition, and derives an explicit fixed-point formula for the gravitational and non-gravitational insertions. The main result is a quasimap analogue of Hori's equation: for , , linking gravitational constants to their non-gravitational counterparts and aligning with mirror symmetry via the -function. The approach provides a robust computational path for genus-0 Calabi–Yau Gromov–Witten data and clarifies how period integrals arise as generating functions of gravitational structure constants."

Abstract

In this paper, we give a proof of Hori's equation for intersection numbers of the moduli space of quasimaps from CP^{1} with (2+1) marked points to CP^{N-1} by using localization technique.
Paper Structure (4 sections, 4 theorems, 84 equations)

This paper contains 4 sections, 4 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. The Moduli Space $\widetilde{Mp}_{0,2|1}(N,d)$
  3. The Orbi-Bundle ${\cal E}_{d}^{\prime}$
  4. Proof of the Main Theorem: Using Localization Technique

Key Result

Theorem 1

(Main Theorem) For $j > 0$, the following equality holds.

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • Remark 1
  • Remark 2