24 research outputs found
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 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 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 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 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 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