A Note on the Supersymmetries of the Self-Dual Supermembrane

In this letter we discuss the supersymmetry issue of the self dual supermembranes in (8+1) and (4+1)-dimensions. We find that all genuine solutions of the (8+1)-dimensional supermembrane, based on the exceptional group G_2, preserve one of the sixteen supersymmetries while all solutions in (4+1)-dimensions preserve eight of them.Comment: Latex file, 12pages, no figure

Seven Dimensional Octonionic Yang-Mills Instanton and its Extension to an Heterotic String Soliton

We construct an octonionic instanton solution to the seven dimensional Yang-Mills theory based on the exceptional gauge group $G_2$ which is the automorphism group of the division algebra of octonions. This octonionic instanton has an extension to a solitonic two-brane solution of the low energy effective theory of the heterotic string that preserves two of the sixteen supersymmetries and hence corresponds to $N=1$ space-time supersymmetry in the (2+1) dimensions transverse to the seven dimensions where the Yang-Mills instanton is defined.Comment: 7 pages, Latex document. This is the final version that appeared in Phys. Lett. B that includes an extra paragraph about the physical properties of the octonionic two-brane. We have also put an addendum regarding some related references that were brought to our attention recentl

General Solution of 7D Octonionic Top Equation

The general solution of a 7D analogue of the 3D Euler top equation is shown to be given by an integration over a Riemann surface with genus 9. The 7D model is derived from the 8D $Spin(7)$ invariant self-dual Yang-Mills equation depending only upon one variable and is regarded as a model describing self-dual membrane instantons. Several integrable reductions of the 7D top to lower target space dimensions are discussed and one of them gives 6, 5, 4D descendants and the 3D Euler top associated with Riemann surfaces with genus 6, 5, 2 and 1, respectively.Comment: 13 pages, Latex, 3 eps.files. Minor changes, eq.(4) adde

Octonionic Yang-Mills Instanton on Quaternionic Line Bundle of Spin(7) Holonomy

The total space of the spinor bundle on the four dimensional sphere S^4 is a quaternionic line bundle that admits a metric of Spin(7) holonomy. We consider octonionic Yang-Mills instanton on this eight dimensional gravitational instanton. This is a higher dimensional generalization of (anti-)self-dual instanton on the Eguchi-Hanson space. We propose an ansatz for Spin(7) Yang-Mills field and derive a system of non-linear ordinary differential equations. The solutions are classified according to the asymptotic behavior at infinity. We give a complete solution, when the gauge group is reduced to a product of SU(2) subalgebras in Spin(7). The existence of more general Spin(7) valued solutions can be seen by making an asymptotic expansion.Comment: A reference added; 22 pages,a final version to appear J.Geom.Phy

Higher Dimensional Analogues of Donaldson-Witten Theory

We present a Donaldson-Witten type field theory in eight dimensions on manifolds with $Spin(7)$ holonomy. We prove that the stress tensor is BRST exact for metric variations preserving the holonomy and we give the invariants for this class of variations. In six and seven dimensions we propose similar theories on Calabi-Yau threefolds and manifolds of $G_2$ holonomy respectively. We point out that these theories arise by considering supersymmetric Yang-Mills theory defined on such manifolds. The theories are invariant under metric variations preserving the holonomy structure without the need for twisting. This statement is a higher dimensional analogue of the fact that Donaldson-Witten field theory on hyper-K\"ahler 4-manifolds is topological without twisting. Higher dimensional analogues of Floer cohomology are briefly outlined. All of these theories arise naturally within the context of string theory.Comment: 23 Pages, Latex. Our statement that these theories are independent of the metric is corrected to the statement that the theories are invariant under deformations that preserve the holonomy structure of the manifold. We also include more details of the construction of a higher dimensional analogue of Floer theory. Three references are adde

Self-Duality in D <= 8-dimensional Euclidean Gravity

In the context of D-dimensional Euclidean gravity, we define the natural generalisation to D-dimensions of the self-dual Yang-Mills equations, as duality conditions on the curvature 2-form of a Riemannian manifold. Solutions to these self-duality equations are provided by manifolds of SU(2), SU(3), G_2 and Spin(7) holonomy. The equations in eight dimensions are a master set for those in lower dimensions. By considering gauge fields propagating on these self-dual manifolds and embedding the spin connection in the gauge connection, solutions to the D-dimensional equations for self-dual Yang-Mills fields are found. We show that the Yang-Mills action on such manifolds is topologically bounded from below, with the bound saturated precisely when the Yang-Mills field is self-dual. These results have a natural interpretation in supersymmetric string theory.Comment: 9 pages, Latex, factors in eqn. (6) corrected, acknowledgement and reference added, typos fixe

Integrable Symplectic Trilinear Interaction Terms for Matrix Membranes

Cubic interactions are considered in 3 and 7 space dimensions, respectively, for bosonic membranes in Poisson Bracket form. Their symmetries and vacuum configurations are discussed. Their associated first order equations are transformed to Nahm's equations, and are hence seen to be integrable, for the 3-dimensional case, by virtue of the explicit Lax pair provided. The constructions introduced also apply to commutator or Moyal Bracket analogues.Comment: 11 pages, LaTe

Self-Dual N=(1,0) Supergravity in Eight Dimensions with Reduced Holonomy Spin(7)

We construct chiral N=(1,0) self-dual supergravity in Euclidean eight-dimensions with reduced holonomy Spin(7), including all the higher-order interactions in a closed form. We first establish the non-chiral N=(1,1) superspace supergravity in eight-dimensions with SO(8) holonomy without self-duality, as the foundation of the formulation. In order to make the whole computation simple, and the generalized self-duality compatible with supersymmetry, we adopt a particular set of superspace constraints similar to the one originally developed in ten-dimensional superspace. The intrinsic properties of octonionic structure constants make local supersymmetry, generalized self-duality condition, and reduced holonomy Spin(7) all consistent with each other.Comment: 14 pages, no figures. Some missing references, typos and grammatical errors have been corrected with other relatively minor improvement

Octonionic representations of Clifford algebras and triality

The theory of representations of Clifford algebras is extended to employ the division algebra of the octonions or Cayley numbers. In particular, questions that arise from the non-associativity and non-commutativity of this division algebra are answered. Octonionic representations for Clifford algebras lead to a notion of octonionic spinors and are used to give octonionic representations of the respective orthogonal groups. Finally, the triality automorphisms are shown to exhibit a manifest \perm_3 \times SO(8) structure in this framework.Comment: 33 page

Self-dual non-Abelian N = 1 tensor multiplet in D = 2+ 2 dimensions

We present a self-dual non-Abelian N=1 supersymmetric tensor multiplet in D=2+2 space-time dimensions. Our system has three on-shell multiplets: (i) The usual non-Abelian Yang-Mills multiplet (A_\mu{}^I, \lambda{}^I) (ii) A non-Abelian tensor multiplet (B_{\mu\nu}{}^I, \chi^I, \varphi^I), and (iii) An extra compensator vector multiplet (C_\mu{}^I, \rho^I). Here the index I is for the adjoint representation of a non-Abelian gauge group. The duality symmetry relations are G_{\mu\nu\rho}{}^I = - \epsilon_{\mu\nu\rho}{}^\sigma \nabla_\sigma \varphi^I, F_{\mu\nu}{}^I = + (1/2) \epsilon_{\mu\nu}{}^{\rho\sigma} F_{\rho\sigma}{}^I, and H_{\mu\nu}{}^I = +(1/2) \epsilon_{\mu\nu}{\rho\sigma} H_{\rho\sigma}{}^I, where G and H are respectively the field strengths of B and C. The usual problem with the coupling of the non-Abelian tensor is avoided by non-trivial Chern-Simons terms in the field strengths G_{\mu\nu\rho}{}^I and H_{\mu\nu}{}^I. For an independent confirmation, we re-formulate the component results in superspace. As applications of embedding integrable systems, we show how the {\cal N} = 2, r = 3 and {\cal N} = 3, r = 4 flows of generalized Korteweg-de Vries equations are embedded into our system.Comment: 21 pages, 0 figure