Integrable reductions of Spin(7) and G_2 invariant self-dual Yang-Mills equations and gravity

TL;DR

This paper explores integrable reductions of Spin(7) and G_2 invariant self-dual Yang-Mills equations in higher dimensions, revealing connections to gravitational theories, string theory, and M-theory, and classifying solutions via algebraic curves.

Contribution

It introduces new integrable systems from higher-dimensional self-dual Yang-Mills equations and links them to gravitational theories and string/M-theory solutions.

Findings

Spin(7)-invariant system is completely integrable and classified by algebraic curves.

G_2-invariant reductions lead to integrable 6D systems related to known dynamical models.

Existence of solitonic solutions interpolating between different vacua.

Abstract

There is remarkable relation between self-dual Yang-Mills and self-dual Einstein gravity in four Euclidean dimensions. Motivated by this we investigate the Spin(7) and G_2 invariant self-dual Yang-Mills equations in eight and seven Euclidean dimensions and search for their possible analogs in gravitational theories. The reduction of the self-dual Yang-Mills equations to one dimension results into systems of first order differential equations. In particular, the Spin(7)-invariant case gives rise to a 7-dimensional system which is completely integrable. The different solutions are classified in terms of algebraic curves and are characterized by the genus of the associated Riemann surfaces. Remarkably, this system arises also in the construction of solutions in gauged supergravities that have an interpretation as continuous distributions of branes in string and M-theory. For the G_2 invariant case we perform two distinct reductions, both giving rise to 6-dimensional systems. The first reduction, which is a complex generalization of the 3-dimensional Euler spinning top system, preserves an SU(2) X SU(2) X Z_2 symmetry and is fully integrable in the particular case where an extra U(1) symmetry exists. The second reduction we employ, generalizes the Halphen system familiar from the dynamics of monopoles. Finally, we analyze massive generalizations and present solitonic solutions interpolating between different degenerate vacua.