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Deformations of Clarke-Oliveira's Instantons on Bryant-Salamon $Spin(7)$-Manifold

Tathagata Ghosh

TL;DR

This work computes deformations and virtual dimensions for Clarke--Oliveira's $Spin(7)$-instantons on the Bryant--Salamon AC $Spin(7)$-manifold by leveraging an AC deformation theory framework that uses twisted Dirac operators on the AC end and their spectral data on the squashed $S^7$ link. The authors classify Clarke--Oliveira instantons as invariant connections, derive an exact one-parameter family $A_{y_0}$ and a limiting instanton $A_{ ext{lim}}$, and analyze deformation spaces via index theory and boundary eta invariants. They determine the virtual dimensions to be 1 for $A_{y_0}$ and -1 for $A_{ ext{lim}}$, with a zero index at weight $- frac52$ and a single spectral-flow crossing at $ u=-2$ contributing to the deformation count. The results hinge on precise eigenvalue analyses of twisted/untwisted Dirac operators on the squashed sphere, APS index computations for the AC end, and connected-sum Pontryagin-class considerations, providing a rigorous account of obstructions and possible deformations for AC $Spin(7)$-instantons in this setting.

Abstract

In this paper we compute the deformations of Clarke-Oliveira's instantons on the Bryant-Salamon $Spin(7)$-Manifold. The Bryant-Salamon $Spin(7)$-Manifold -- the negative spinor bundle of $S^4$ -- is an asymptotically conical manifold where the link is the squashed $7$-sphere. We use the deformation theory developed by the author in a previous paper to calculate the deformations of Clarke-Oliveira's instantons and calculate the virtual dimensions of the moduli spaces.

Deformations of Clarke-Oliveira's Instantons on Bryant-Salamon $Spin(7)$-Manifold

TL;DR

This work computes deformations and virtual dimensions for Clarke--Oliveira's -instantons on the Bryant--Salamon AC -manifold by leveraging an AC deformation theory framework that uses twisted Dirac operators on the AC end and their spectral data on the squashed link. The authors classify Clarke--Oliveira instantons as invariant connections, derive an exact one-parameter family and a limiting instanton , and analyze deformation spaces via index theory and boundary eta invariants. They determine the virtual dimensions to be 1 for and -1 for , with a zero index at weight and a single spectral-flow crossing at contributing to the deformation count. The results hinge on precise eigenvalue analyses of twisted/untwisted Dirac operators on the squashed sphere, APS index computations for the AC end, and connected-sum Pontryagin-class considerations, providing a rigorous account of obstructions and possible deformations for AC -instantons in this setting.

Abstract

In this paper we compute the deformations of Clarke-Oliveira's instantons on the Bryant-Salamon -Manifold. The Bryant-Salamon -Manifold -- the negative spinor bundle of -- is an asymptotically conical manifold where the link is the squashed -sphere. We use the deformation theory developed by the author in a previous paper to calculate the deformations of Clarke-Oliveira's instantons and calculate the virtual dimensions of the moduli spaces.

Paper Structure

This paper contains 16 sections, 12 theorems, 130 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Bryant--Salamon Spin(7)-Manifold
  3. The Squashed 7-Sphere
  4. The Bryant--Salamon Metric
  5. Clarke-Oliveira's Instantons
  6. Eigenvalues of the Dirac Operators on Squashed 7-Sphere
  7. Dirac operators on Homogeneous Nearly G2-Manifolds
  8. Eigenvalue Bounds for the Twisted Dirac Operators on Squashed 7-Sphere
  9. Eigenvalues of the Untwisted Dirac operator
  10. Eigenvalues of the Twisted Dirac Operator
  11. Deformations of Clarke--Oliveira's Instantons
  12. Index of the Twisted Dirac Operator
  13. Eta Invariant of the Boundary
  14. Index of the Dirac Operator Twisted by the One Parameter Family of Instantons
  15. Index of the Dirac Operator Twisted by the Limiting Instantons
  16. ...and 1 more sections

Key Result

Theorem 1.1

me2023paper The twisted Dirac operator is Fredholm if $\nu+\frac{5}{2} \in \mathbb{R} \setminus\mathop{\mathrm{Spec}}\nolimits\slashed{\mathfrak{D}}_{A_\Sigma}$. For two weights $\nu, \nu'$ with $\nu \leq \nu'$ for which the corresponding Dirac operators are Fredholm, we have

Figures (7)

  • Figure 1: $\slashed{S}^-(S^4)$ with cigar metric.
  • Figure 2: The cylinder $\slashed{C}$.
  • Figure 3: The cylinder $C_\Sigma$.
  • Figure 4: $\slashed{S}^-(S^4) = B_\Sigma \amalg \slashed{C}$.
  • Figure 5: $B_\Sigma \amalg \overline{C}_\Sigma \cong M_\Sigma$.
  • ...and 2 more figures

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 3.1
  • Proposition 3.2: clarke-oli2020instantons
  • Theorem 3.3: clarke-oli2020instantons
  • Remark 4.1
  • Theorem 4.2
  • Corollary 4.3
  • Theorem 4.4
  • proof
  • ...and 8 more