Chern-Simons flows on Aloff-Wallach spaces and spin(7) instantons
Because of their explicit construction, Aloff-Wallach spaces are prominent in flux compactifications. They carry G{sub 2} structures and admit the G{sub 2}-instanton equations, which are natural Bogomol'nyi-Prasad-Sommerfeld equations for Yang-Mills instantons on seven-manifolds and extremize a Chern-Simons-type functional. We consider the Chern-Simons flow between different G{sub 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{sup 3}xR{sup 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.