Generalized vector products, duality, and octonionic identities in <i>D</i>=8 geometry

Dündarer, Resit; Gürsey, Feza; Tze, Chia‐Hsiung

doi:10.1063/1.526321

Cited by 85 publications

(97 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…This new construction in field theory and string theory is intimately linked with the existence of octonions or Cayley numbers. SO(8) may be decomposed into G 2 × S 7 L × S 7 R where the left and right seven spheres are left and right multiplication by octonions [2]. The eight dimensional duality condition is related to this decomposition in precisely the way that the four-dimensional 6 Note that by embedding the spin connection in the gauge connection, each metric on M D automatically gives a solution of (1) in flat space.…”

Section: Discussionmentioning

confidence: 99%

“…It is natural to ask if the four-dimensional equations for self-duality have an analogue in higher dimensions. This question was considered in [1] and a natural set of equations for self-duality in dimensions four through eight were found (the eight dimensional equations were independently discovered in [2]). These equations are:…”

mentioning

confidence: 99%

“…These equations imply Ricci flatness and thus their solutions obey Einstein's equations with zero cosmological constant. It is thus natural to consider the higher dimensional equations for self-duality [1,2] in the context of Euclidean gravity. That is the purpose of this note 3 .…”

mentioning

confidence: 99%

“…Here, the duality operator φ µνλρ is related to the structure constants of the octonions [1,2]. The self-dual Yang-Mills equations were considered in Euclidean gravity by replacing the Yang-Mills curvature with the curvature of the Riemannian 4-manifold.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Self-duality inD<~8-dimensional Euclidean gravity

Acharya

O’Loughlin

1997

Phys. Rev. D

View full text Add to dashboard Cite

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.

show abstract

Section: Discussionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Self-duality inD<~8-dimensional Euclidean gravity

Acharya

O’Loughlin

1997

Phys. Rev. D

View full text Add to dashboard Cite

show abstract

“…[35][36][37][38][39][40][41][42][43][44][45][46][47][48][49], since they play important role in the present paper. Exceptional groups recently appeared also as symmetries of Freudenthal dual Lagrangians, as investigated, e.g., in [50].…”

Section: Introductionmentioning

confidence: 99%

8. Invariant Differential Operators for Noncompact Lie Algebras Parabolically Related to Conformal Lie Algebras

2016

Noncompact Semisimple Lie Algebras and Groups

View full text Add to dashboard Cite

In the present paper we continue the project of systematic construction of invariant differential operators for non-compact semisimple Lie groups. Our starting points is the class of algebras, which we call 'conformal Lie algebras' (CLA), which have very similar properties to the conformal algebras of Minkowski space-time, though our aim is to go beyond this class in a natural way. For this we introduce the new notion of parabolic relation between two non-compact semisimple Lie algebras G and G ′ that have the same complexification and possess maximal parabolic subalgebras with the same complexification. Thus, we consider the exceptional algebra E 7(7) which is parabolically related to the CLA E 7(−25) , the parabolic subalgebras including E 6(6) and E 6(−26) . Other interesting examples are the orthogonal algebras so(p, q) all of which are parabolically related to the conformal algebra so(n, 2) with p + q = n + 2, the parabolic subalgebras including the Lorentz subalgebra so(n − 1, 1) and its analogs so(p − 1, q − 1). We consider also E 6(6) and E 6(2) which are parabolically related to the hermitian symmetric case E 6(−14) , the parabolic subalgebras including real forms of sl(6). We also give a formula for the number of representations in the main multiplets valid for CLAs and all algebras that are parabolically related to them. In all considered cases we give the main multiplets of indecomposable elementary representations including the necessary data for all relevant invariant differential operators. In the case of so(p, q) we give also the reduced multiplets. We should stress that the multiplets are given in the most economic way in pairs of shadow fields. Furthermore we should stress that the classification of all invariant differential operators includes as special cases all possible conservation laws and conserved currents, unitary or not.

show abstract

Flux compactification of M‐theory on compact manifolds with Spin(7) holonomy

Constantin

2005

Fortschritte der Physik

View full text Add to dashboard Cite

At the leading order, M-theory admits minimal supersymmetric compactifications if the internal manifold has exceptional holonomy. The inclusion of non-vanishing fluxes in M-theory and string theory compactifications induce a superpotential in the lower dimensional theory, which depends on the fluxes. In this work, we check the conjectured form of this superpotential in the case of warped M-theory compactifications on Spin(7) holonomy manifolds. We perform a Kaluza-Klein reduction of the eleven-dimensional supersymmetry transformation for the gravitino and we find by direct comparison the superpotential expression. We check the conjecture for the heterotic string compactified on a Calabi-Yau three-fold as well. The conjecture can be checked indirectly by inspecting the scalar potential obtained after the compactification of M-theory on Spin (7) holonomy manifolds with non-vanishing fluxes. The scalar potential can be written in terms of the superpotential and we show that this potential stabilizes all the moduli fields describing deformations of the metric except for the radial modulus.All the above analyses require the knowledge of the minimal supergravity action in three dimensions. Therefore we calculate the most general causal N = 1 three-dimensional, gauge invariant action coupled to matter in superspace and derive its component form using Ectoplasmic integration theory. We also show that the three-dimensional theory which results from the compactification is in agreement with the more general supergravity construction.The compactification procedure takes into account higher order quantum correction terms in the low energy effective action. We analyze the properties of these terms on a Spin (7) background. We derive a perturbative set of solutions which emerges from a warped compactification on a Spin(7) holonomy manifold with non-vanishing flux for the M-theory field strength and we show that in general the Ricci flatness of the internal manifold is lost, which means that the supergravity vacua are deformed away from the exceptional holonomy. Using the superpotential form we identify the supersymmetric vacua out of this general set of solutions.

show abstract

Generalized vector products, duality, and octonionic identities in D=8 geometry

Cited by 85 publications

References 19 publications

Self-duality inD<~8-dimensional Euclidean gravity

Self-duality inD<~8-dimensional Euclidean gravity

8. Invariant Differential Operators for Noncompact Lie Algebras Parabolically Related to Conformal Lie Algebras

Flux compactification of M‐theory on compact manifolds with Spin(7) holonomy

Contact Info

Product

Resources

About