Chern-Simons flows on Aloff-Wallach spaces and Spin(7)-instantons

TL;DR

This paper studies the Chern-Simons flow and Spin(7)-instantons on Aloff-Wallach spaces, revealing explicit solutions and their relation to G_2-instantons and gradient flows on complex spaces.

Contribution

It introduces the analysis of Chern-Simons flows between G_2-instantons on Aloff-Wallach spaces and provides explicit solutions for specific cases.

Findings

Explicit instanton solution with tanh-like shape

Gradient-flow equations on C^3 x R^2 derived from instanton equations

Connection between Chern-Simons flow and Spin(7)-instantons

Abstract

Due to their explicit construction, Aloff-Wallach spaces are prominent in flux compactifications. They carry G_2-structures and admit the G_2-instanton equations, which are natural BPS equations for Yang-Mills instantons on seven-manifolds and extremize a Chern-Simons-type functional. We consider the Chern-Simons flow between different G_2-instantons on Aloff-Wallach spaces, which is equivalent to Spin(7)-instantons on a cylinder over them. For a general SU(3)-equivariant gauge connection, the generalized instanton equations turn into gradient-flow equations on C^3 x R^2, with a particular cubic superpotential. For the simplest member of the Aloff-Wallach family (with 3-Sasakian structure) we present an explicit instanton solution of tanh-like shape.