Yang-Mills Flows on Nearly Kähler Manifolds and G 2-Instantons

Abstract

We consider Lie(G)-valued G-invariant connections on bundles over spaces G/H, R×G/H and R 2 ×G/H, where G/H is a compact nearly Kähler six-dimensional homogeneous space, and the manifolds R×G/H and R 2 ×G/H carry G 2 -and Spin(7)-structures, respectively. By making a G-invariant ansatz, Yang-Mills theory with torsion on R×G/H is reduced to Newtonian mechanics of a particle moving in a plane with a quartic potential. For particular values of the torsion, we find explicit particle trajectories, which obey first-order gradient or hamiltonian flow equations. In two cases, these solutions correspond to anti-self-dual instantons associated with one of two G 2 -structures on R×G/H. It is shown that both G 2 -instanton equations can be obtained from a single Spin(7)-instanton equation on R 2 ×G/H.

Yang-Mills instantons and dyons on homogeneous G 2-manifolds

“…which describes G 2 -invariant gauge fields on R × S 6 , where S 6 = G 2 /SU(3) [13]. All is consistent with the decomposition 14 (of G 2 ) = 8 adj + 3 +3 (of SU (3)) .…”

“…1 is a zeroenergy solution φ(τ ), as was already noticed in [13]. Vice versa, any solution of (5.12) gives a special solution to the equations (5.4), (5.5) and (4.7), (4.8).…”

“…Various solutions to these first-order equations were found e.g. in [10]- [13], mostly on flat space R d and various cosets.…”

Generalised Chern-Simons Theory and G2-Instantons over Associative Fibrations

“…On the other hand, if A = A t + σ i is a G 2 -instanton, then it minimises the Yang-Mills functional (5). This implies…”

“…In the high-energy physics community, solutions to a very similar problem in the context of G 2 -structures with torsion have been found eg. for cylinders over nearly-Kähler homogeneous spaces [5] and more generally for cones over nontrivial manifolds admitting real Killing spinors [7].…”