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

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.

31 pages, latex; v2: some references added

References in corpus (4)

• Gauge Theory at Large N and New G_2 Holonomy Metrics
• New Complete Non-compact Spin(7) Manifolds
• (Weak) G_2 Holonomy from Self-duality, Flux and Supersymmetry
• Gravitational domain walls and p-brane distributions

Cited by in corpus (12)

• RG flows from Spin(7), CY 4-fold and HK manifolds to AdS, Penrose limits and pp waves
• An eight-dimensional approach to G_2 manifolds
• Calibrations, Monopoles and Fuzzy Funnels
• Octonions, G_2 Symmetry, Generalized Self-Duality and Supersymmetries in Dimensions D \le 8
• Self-Dual N=(1,0) Supergravity in Eight Dimensions with Reduced Holonomy Spin(7)
• Self-Dual Supergravity in Seven Dimensions with Reduced Holonomy G_2
• Towards an Ashtekar Formalism in Twelve Dimensions
• Superfield Description of a Self-Dual Supergravity a la MacDowell-Mansouri
• Hamiltonian formulation and integrability of a complex symmetric nonlinear system
• Self-dual non-Abelian N = 1 tensor multiplet in D = 2+ 2 dimensions
• Self-Dual Yang-Mills and Vector-Spinor Fields, Nilpotent Fermionic Symmetry, and Supersymmetric Integrable Systems
• Non-Collapsing Membrane Instantons in Higher Dimensions