Octonions, G 2 Symmetry, Generalized Self-Duality and Supersymmetries in Dimensions D   8 (original) (raw)

Octonions, G_2 Symmetry, Generalized Self-Duality and Supersymmetries in Dimensions D le 8

2002

We establish N=(1/8,1) supersymmetric Yang-Mills vector multiplet with generalized self-duality in Euclidian eight-dimensions with the original full SO(8) Lorentz covariance reduced to SO(7). The key ingredient is the usage of octonion structure constants made compatible with SO(7) covariance and chirality in 8D. By a simple dimensional reduction together with extra constraints, we derive N=1/8+7/8 supersymmetric self-dual vector multiplet in 7D with the full SO(7) Lorentz covariance reduced to G_2. We find that extra constraints needed on fields and supersymmetry parameter are not obtained from a simple dimensional reduction from 8D. We conjecture that other self-dual supersymmetric theories in lower dimensions D =6 and 4 with respective reduced global Lorentz covariances such as SU(3) \subset SO(6) and SU(2) \subset SO(4) can be obtained in a similar fashion.

Octonions and supersymmetry in three dimensions

Classical and Quantum Gravity, 2008

We present two N = 1 locally supersymmetric sigma-models in 3D with the seven and eight dimensional target spaces with the reduced G 2 and Spin(7) holonomies instead of the maximal SO(7) and SO(8), respectively. It turns out that particular Wess-Zumino-Novikov-Witten-like terms are possible with the octonion structure constant for these σ-models. As a by-product, we show that a new N = 7 vector multiplet also exists in 3D with non-trivial interactions with the octonion structure constant. A non-trivial solution for the interacting vectorial field is also given.

Self-dual N=(1,0) supergravity in eight dimensions with reduced holonomy Spin(7)

Physics Letters B, 2003

We construct chiral N = (1, 0) self-dual supergravity in Euclidean eightdimensions 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.

Self-dual Yang-Mills fields in eight dimensions

Letters in Mathematical Physics, 1996

Strongly self-dual Yang-Mills fields in even dimensional spaces are characterised by a set of constraints on the eigenvalues of the Yang-Mills fields F µν . We derive a topological bound on R 8 , M (F, F ) 2 ≥ k M p 2 1 where p 1 is the first Pontrjagin class of the SO(n) Yang-Mills bundle and k is a constant. Strongly self-dual Yang-Mills fields realise the lower bound.

Octonionic Realizations of One-Dimensional Extended Supersymmetries: A Classification

Modern Physics Letters A, 2003

The classification of the octonionic realizations of the one-dimensional extended supersymmetries is here furnished. These are non-associative realizations which, albeit inequivalent, are put in correspondence with a subclass of the already classified associative representations for 1D extended supersymmetries. Examples of dynamical systems invariant under octonionic realizations of the extended supersymmetries are given. We cite among the others the octonionic spinning particles, the N = 8 KdV, etc. Possible applications to supersymmetric systems arising from dimensional reduction of the octonionic superstring and M-theory are mentioned.

7D bosonic higher spin gauge theory: symmetry algebra and linearized constraints

Nuclear Physics B, 2002

We construct the minimal bosonic higher spin extension of the 7D AdS algebra SO(6, 2), which we call hs(8 * ). The generators, which have spin s = 1, 3, 5, ..., are realized as monomials in Grassmann even spinor oscillators. Irreducibility, in the form of tracelessness, is achieved by modding out an infinite dimensional ideal containing the traces. In this a key role is played by the tree bilinear traces which form an SU (2) K algebra. We show that gauging of hs(8 * ) yields a spectrum of physical fields with spin s = 0, 2, 4, ... which make up a UIR of hs(8 * ) isomorphic to the symmetric tensor product of two 6D scalar doubletons. The scalar doubleton is the unique SU (2) K invariant 6D doubleton. The spin s ≥ 2 sector comes from an hs(8 * )-valued one-form which also contains the auxiliary gauge fields required for writing the curvature constraints in covariant form. The physical spin s = 0 field arises in a separate zero-form in a 'quasi-adjoint' representation of hs(8 * ). This zero-form also contains the spin s ≥ 2 Weyl tensors, i.e. the curvatures which are non-vanishing on-shell. We suggest that the hs(8 * ) gauge theory describes the minimal bosonic, massless truncation of M theory on AdS 7 × S 4 in an unbroken phase where the holographic dual is given by N free (2, 0) tensor multiplets for large N .

Self-dual supergravity in seven dimensions with reduced holonomy G2

Physics Letters B, 2003

We present self-dual N = 2 supergravity in superspace for Euclidean seven dimensions with the reduced holonomy G 2 ⊂ SO(7), including all higher-order terms. As its foundation, we first establish N = 2 supergravity without self-duality in Euclidean seven dimensions. We next show how the generalized self-duality in terms of octonion structure constants can be consistently imposed on the superspace constraints. We found two self-dual N = 2 supergravity theories possible in 7D, depending on the representations of the two spinor charges of N = 2. The first formulation has both of the two spinor charges in the 7 of G 2 with 24 + 24 on-shell degrees of freedom. The second formulation has both charges in the 1 of G 2 with 16 + 16 on-shell degrees of freedom.

Self-duality and octonionic analyticity of S7-valued antisymmetric fields in eight dimensions

Nuclear Physics B, 1986

We exhibit two octonionic extensions of the Kalb-Ramond type fields with rank two and four in eight dimensions. By analogy to the d = 4 (anti-) self-duality of the SU(2) = S 3 quaternionic gauge field we consider the respective d = 8 (anti-) self-duality equations for these nonlinear, ST-valued antisymmetric fields. By way of an octonionic 't Hooft ansatz these equations reduce to the same generalized Fueter-Cauchy-Riemann equations over S 8. Explicit (9n + 8) parameter $7~ S 7 mapping solutions, n being a winding number, are found in terms of holomorphic functions of the spacetime octonion. An infinite number of local continuity equations results.

Supersymmetric gauge theories in five and six dimensions

Physics Letters B, 1997

We investigate consistency conditions for supersymmetric gauge theories in higher dimensions. First, we give a survey of Seiberg's necessary conditions for the existence of such theories with simple groups in five and six dimensions. We then make some comments on how theories in different dimensions are related. In particular, we discuss how the Landau pole can be avoided in theories that are not asymptotically free in four dimensions, and the mixing of tensor and vector multiplets in dimensional reduction from six dimensions.

Extended supersymmetry and self-duality in 2 + 2 dimensions

Physics Letters B, 1992

The N= 2 supersymmetric self-dual Yang-Mills theory and the N= 4 and N= 2 self-dual supergravities in 2 + 2 space-time • dimensions are formulated for the first time. These formulations utilize solutions of the Bianchi identities subject to the super-Yang-Mills or supergravity constraints in the relevant N-extended superspace with the space-time signature (2,2). The selfduality condition on the Yang-Mills field strength or on the Riemann tensor is not modified, but is accompanied by the field equations of other superpartners. These self-dual systems are conjectured to generate supersymmetric exactly soluble models in lower dimensions.