Euclidean Actions, Instantons, Solitons and Supersymmetry

TL;DR

This paper explores different Euclidean formulations of theories with axionic scalars, analyzing their relations, instanton actions, and supersymmetric properties, revealing conditions under which these formulations yield consistent physical results.

Contribution

It clarifies the relations between various Euclidean formulations of axionic theories and establishes conditions for consistent instanton actions and BPS solutions.

Findings

Semi-classical amplitudes are equivalent in two axionic actions.

Hodge dualized version yields different instanton action unless specific conditions are met.

A Euclidean BPS condition and a geometric criterion for BPS solutions are identified.

Abstract

Theories with axionic scalars admit three different Euclidean formulations, obtained by Wick rotation, Wick rotation combined with analytic continuation of the axionic scalars, and Wick rotation combined with Hodge dualization. We investigate the relation between these formulations for a class of theories which contains the sigma models of N=2 vector multiplets as a special case. It is shown that semi-classical amplitudes can be expressed equivalently using the two types of axionic actions, while the Hodge dualized version gives a different value for the instanton action unless the integration constants associated with the axion fields are chosen in a particular way. With this choice the instanton action is equal to the mass of the soliton or black hole obtained by dimensional lifting with respect to time. For supersymmetric models we use the Euclidean supersymmetry algebra to derive a Euclidean BPS condition, and we identify a geometrical criterion which distinguishes BPS from non-BPS extremal solutions.