The Action of Instantons with Nut Charge

TL;DR

This paper investigates how nut charge influences the action of four-dimensional instantons with U(1) symmetry, revealing explicit contributions and boundary condition effects, with implications for the instanton Hamiltonian.

Contribution

It demonstrates that nut charge contributes explicitly to instanton action when calculated via dimensional reduction and relates boundary conditions to the action's expression.

Findings

Nut charge adds a non-zero term to the instanton action.

The action can be expressed using quantities on the fixed point set of the isometry.

Presence of nut charge implies a Misner string singularity affecting the Hamiltonian.

Abstract

We examine the effect of a non-trivial nut charge on the action of non-compact four-dimensional instantons with a U(1) isometry. If the instanton action is calculated by dimensionally reducing along the isometry, then the nut charge is found to make an explicit non-zero contribution. For metrics satisfying AF, ALF or ALE boundary conditions, the action can be expressed entirely in terms of quantities (including the nut charge) defined on the fixed point set of the isometry. A source (or sink) of nut charge also implies the presence of a Misner string coordinate singularity, which will have an important effect on the Hamiltonian of the instanton.