Finite energy/action solutions of $p_1$ Yang-Mills equations on $p_2$ Schwarzschild and deSitter backgrounds for dimension $d \ge 4$

TL;DR

This paper explores finite energy solutions of generalized Yang-Mills equations on Schwarzschild and deSitter backgrounds, revealing new non-self-dual solutions and comparing their actions with self-dual solutions across various dimensions.

Contribution

It introduces solutions to $p_1$ Yang-Mills equations on $p_2$ Einstein-Hilbert backgrounds, expanding understanding of gauge fields in complex geometries.

Findings

Existence of finite energy solutions in various dimensions.

Identification of non-self-dual Yang-Mills solutions.

Comparison of actions between self-dual and non-self-dual solutions.

Abstract

Physically relevant gauge and gravitational theories can be seen as special members of hierarchies of more elaborate systems. The Yang-Mills (YM) system is the first member of a hierarchy of Lagrangians which we will index by $p_1$, and the Einstein-Hilbert (EH) system of general relativity is the first member of another hierarchy which we index by $p_2$. In this paper, we study the classical equations of the $p_1 = 1,2$ YM hierarchy considered in the background of special geometries (Schwarzschild, deSitter,anti-deSitter) of the $p_2=1,2,3$ EH hierarchy. Solutions are obtained in various dimensions and lead to several examples of non-self-dual YM fields. When $p_1=p_2$ self-dual solutions exist in addition. Their action is equal to the Chern-Pontryagin charge and can be compared with that of the non-self-dual solutions.