Spectrum of D=6, N=4b Supergravity on AdS_3 x S^3

The complete spectrum of D=6, N=4b supergravity with n tensor multiplets compactified on AdS_3 x S^3 is determined. The D=6 theory obtained from the K_3 compactification of Type IIB string requires that n=21, but we let n be arbitrary. The superalgebra that underlies the symmetry of the resulting supergravity theory in AdS_3 coupled to matter is SU(1,1|2)_L x SU(1,1|2)_R. The theory also has an unbroken global SO(4)_R x SO(n) symmetry inherited from D=6. The spectrum of states arranges itself into a tower of spin-2 supermultiplets, a tower of spin-1, SO(n) singlet supermultiplets, a tower of spin-1 supermultiplets in the vector representation of SO(n) and a special spin-1/2 supermultiplet also in the vector representation of SO(n). The SU(2)_L x SU(2)_R Yang-Mills states reside in the second level of the spin-2 tower and the lowest level of the spin-1, SO(n) singlet tower and the associated field theory exhibits interesting properties.Comment: 37 pages, latex, 5 tables and 3 figures, typos corrected, a reference adde

M-Fivebrane from the Open Supermembrane

Covariant field equations of M-fivebrane in eleven dimensional curved superspace are obtained from the requirement of kappa-symmetry of an open supermembrane ending on a fivebrane. The worldvolume of the latter is a (6|16) dimensional supermanifold embedded in the (11|32) dimensional target superspace. The kappa-symmetry of the system imposes a constraint on this embedding, and a constraint on a modified super 3-form field strength on the fivebrane worldvolume. These constraints govern the dynamics of the M-fivebrane.Comment: 11 pages, Latex, references and appendix adde

The supermembrane revisited

The M2-brane is studied from the perspective of superembeddings. We review the derivation of the M2-brane dynamics and the supergravity constraints from the standard superembedding constraint and we discuss explicitly the induced d=3, N=8 superconformal geometry on the worldvolume. We show that the gauged supermembrane, for a target space with a U(1) isometry, is the standard D2-brane in a type IIA supergravity background. In particular, the D2-brane action, complete with the Dirac-Born-Infeld term, arises from the gauged Wess-Zumino worldvolume 4-form via the brane action principle. The discussion is extended to the massive D2-brane considered as a gauged supermembrane in a massive D=11 superspace background. Type IIA supergeometry is derived using Kaluza-Klein techniques in superspace.Comment: Latex, 46 pages, clarifying remarks and references adde

On Brane Actions and Superembeddings

Actions for branes, with or without worldsurface gauge fields, are discussed in a unified framework. A simple algorithm is given for constructing the component Green-Schwarz actions. Superspace actions are also discussed. Three examples are given to illustrate the general procedure: the membrane in D=11 and the D2-brane, which both have on-shell worldsurface supermultiplets, and the membrane in D=4, which has an off-shell multiplet.Comment: 19 pages, late

The unrestricted blocking number in convex geometry

Let K be a convex body in \mathbb{R}^n. We say that a set of translates \left \{ K + \underline{u}_i \right \}_{i=1}^{p} block K if any other translate of K which touches K, overlaps one of K + \underline{u}_i, i = 1, . . . , p. The smallest number of non-overlapping translates (i.e. whose interiors are disjoint) of K, all of which touch K at its boundary and which block any other translate of K from touching K is called the Blocking Number of K and denote it by B(K). This thesis explores the properties of the blocking number in general but the main purpose is to study the unrestricted blocking number B_\alpha(K), i.e., when K is blocked by translates of \alpha K, where \alpha is a fixed positive number and when the restrictions that the translates are non-overlapping or touch K are removed. We call this number the Unrestricted Blocking Number and denote it by B_\alpha(K). The original motivation for blocking number is the following famous problem: Can a rigid material sphere be brought into contact with 13 other such spheres of the same size? This problem was posed by Kepler in 1611. Although this problem was raised by Kepler, it is named after Newton since Newton and Gregory had a dispute over the solution which was eventually settled in Newton’s favour. It is called the Newton Number, N(K) of K and is defined to be the maximum number of non-overlapping translates of K which can touch K at its boundary. The well-known dispute between Sir Isaac Newton and David Gregory concerning this problem, which Newton conjectured to be 12, and Gregory thought to be 13, was ended 180 years later. In 1874, the problem was solved by Hoppe in favour of Newton, i.e., N(\beta^3) = 12. In his proof, the arrangement of 12 unit balls is not unique. This is thought to explain why the problem took 180 years to solve although it is a very natural and a very simple sounding problem. As a generalization of the Newton Number to other convex bodies the blocking number was introduced by C. Zong in 1993. “Another characteristic of mathematical thought is that it can have no success where it cannot generalize.” C. S. Pierce As quoted above, in mathematics generalizations play a very important part. In this thesis we generalize the blocking number to the Unrestricted Blocking Number. Furthermore; we also define the Blocking Number with negative copies and denote it by B_(K). The blocking number not only gives rise to a wide variety of generalizations but also it has interesting observations in nature. For instance, there is a direct relation to the distribution of holes on the surface of pollen grains with the unrestricted blocking number

Codimension One Branes

We study codimension one branes, i.e. p-branes in (p+2)-dimensions, in the superembedding approach for the cases where the worldvolume superspace is embedded in a minimal target superspace with half supersymmetry breaking. This singles out the cases p=1,2,3,5,9. For p=3,5,9 the superembedding geometry naturally involves a fundamental super 2-form potential on the worldvolume whose generalised field strength obeys a constraint deducible from considering an open supermembrane ending on the p-brane. This constraint, together with the embedding constraint, puts the system on-shell for p=5 but overconstrains the 9-brane in D=11 such that the Goldstone superfield is frozen. For p=3 these two constraints give rise to an off-shell linear multiplet on the worldvolume. An alternative formulation of this case is given in which the linear multiplet is dualised to an off-shell scalar multiplet. Actions are constructed for both cases and are shown to give equivalent equations of motion. After gauge fixing a local Sp(1) symmetry associated with shifts in the Sp(1)_R Goldstone modes, we find that the auxiliary fields in the scalar multiplet parametrise a two-sphere. For completeness we also discuss briefly the cases p=1,2 where the equations of motion (for off-shell multiplets) are obtained from an action principle.Comment: 38 pages, latex, cover page correcte

Dilatonic p-brane solitons

We find new 4-brane and 5-brane solitons in massive gauged $D=6$, $N=2$ and $D=7$, $N=1$ supergravities. In each case, the solutions preserve half of the original supersymmetry. These solutions make use of the metric and dilaton fields only. We also present more general dilatonic $(D-2)$-branes in $D$ dimensions.Comment: 9 pages, Latex, no figure

Supersymmetric Higher Spin Theories

We revisit the higher spin extensions of the anti de Sitter algebra in four dimensions that incorporate internal symmetries and admit representations that contain fermions, classified long ago by Konstein and Vasiliev. We construct the $dS_4$, Euclidean and Kleinian version of these algebras, as well as the corresponding fully nonlinear Vasiliev type higher spin theories, in which the reality conditions we impose on the master fields play a crucial role. The ${\cal N}=2$ supersymmetric higher spin theory in $dS_4$, on which we elaborate further, is included in this class of models. A subset of Konstein-Vasiliev algebras are the higher spin extensions of the $AdS_4$ superalgebras $osp(4|{\cal N})$ for ${\cal N}=1,2,4$ mod 4 and can be realized using fermionic oscillators. We tensor the higher superalgebras of the latter kind with appropriate internal symmetry groups and show that the ${\cal N}=3$ mod 4 higher spin algebras are isomorphic to those with ${\cal N}=4$ mod 4. We describe the fully nonlinear higher spin theories based on these algebras as well, and we elaborate further on the ${\cal N}=6$ supersymmetric theory, providing two equivalent descriptions one of which exhibits manifestly its relation to the ${\cal N}=8$ supersymmetric higher spin theory.Comment: 30 pages. Contribution to J. Phys. A special volume on "Higher Spin Theories and AdS/CFT" edited by M. R. Gaberdiel and M. Vasilie

Representations of p-brane topological charge algebras

The known extended algebras associated with p-branes are shown to be generated as topological charge algebras of the standard p-brane actions. A representation of the charges in terms of superspace forms is constructed. The charges are shown to be the same in standard/extended superspace formulations of the action.Comment: 22 pages. Typos fixed, refs added. Minor additions to comments sectio

Higher Spins in AdS and Twistorial Holography

In this paper we simplify and extend previous work on three-point functions in Vasiliev's higher spin gauge theory in AdS4. We work in a gauge in which the space-time dependence of Vasiliev's master fields is gauged away completely, leaving only the internal twistor-like variables. The correlation functions of boundary operators can be easily computed in this gauge. We find complete agreement of the tree level three point functions of higher spin currents in Vasiliev's theory with the conjectured dual free O(N) vector theory.Comment: 23 pages. v3: minor errors fixed, added comments and reference