Harmonic Forms and Deformation of ALC metrics with Spin(7) holonomy

TL;DR

This paper studies how certain special geometric structures called ALC metrics with Spin(7) holonomy can be deformed, focusing on changes in the M theory circle size related to string coupling, using explicit cohomogeneity one examples.

Contribution

It identifies specific harmonic self-dual four forms that deform Spin(7) ALC metrics, especially affecting the asymptotic circle radius in M theory compactifications.

Findings

Deformation solutions are governed by first order differential equations.

An L^2-normalizable solution alters the asymptotic M theory circle radius.

The analysis links harmonic forms to geometric deformations of Spin(7) metrics.

Abstract

Asymptotically locally conical (ALC) metric of exceptional holonomy has an asymptotic circle bundle structure that accommodates the M theory circle in type IIA reduction. Taking Spin(7) metrics of cohomogeneity one as explicit examples, we investigate deformations of ALC metrics, in particular that change the asymptotic S^1 radius related to the type IIA string coupling constant. When the canonical four form of Spin(7) holonomy is taken to be anti-self-dual, the deformations of Spin(7) metric are related to the harmonic self-dual four forms, which are given by solutions to a system of first order differential equations, due to the metric ansatz of cohomogeneity one. We identify the L^2-normalizable solution that deforms the asymptotic radius of the M theory circle.