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Moduli spaces of contact instantons on Sasakian 5-manifolds with transverse Calabi-Yau structures and orbifold K3 surfaces

Tomohiro Arai, Kurando Baba

Abstract

We study anti-self-dual contact instantons on 5-dimensional Sasakian manifolds with transverse Calabi-Yau structures. In this case, the leaf space is a Calabi-Yau orbifold, and the moduli space of irreducible anti-self-dual contact instantons is a hyperkahler manifold. Using the singularity data of the leaf spaces, we prove that the transverse Levi-Civita connection gives an irreducible anti-self-dual contact instanton in the case when the leaf space is one of the 95 orbifold $K3$ surfaces classified by Reid. Moreover, we compute explicitly the complex dimension of the corresponding moduli spaces in all 95 cases.

Moduli spaces of contact instantons on Sasakian 5-manifolds with transverse Calabi-Yau structures and orbifold K3 surfaces

Abstract

We study anti-self-dual contact instantons on 5-dimensional Sasakian manifolds with transverse Calabi-Yau structures. In this case, the leaf space is a Calabi-Yau orbifold, and the moduli space of irreducible anti-self-dual contact instantons is a hyperkahler manifold. Using the singularity data of the leaf spaces, we prove that the transverse Levi-Civita connection gives an irreducible anti-self-dual contact instanton in the case when the leaf space is one of the 95 orbifold surfaces classified by Reid. Moreover, we compute explicitly the complex dimension of the corresponding moduli spaces in all 95 cases.

Paper Structure

This paper contains 6 sections, 2 theorems, 24 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sasakian manifold
  4. Transverse Calabi-Yau structures
  5. Contact instantons and their moduli spaces
  6. Proof of Main theorem

Key Result

Theorem 3.1

Suppose that the leaf space $X$ is a Reid orbifold $K3$ surface with singularity data $\{m_{1},\dotsc,m_{k}\}$. Then, $\nabla^{*}$ is irreducible if the following relation holds:

Theorems & Definitions (10)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 3.1: BH
  • Example 3.2: Label 33 in Table \ref{['table:main_result']}
  • Theorem 3.3: BH
  • Example 3.4: continued in Example \ref{['ex:label33']}