We present a manifestly T-dual formulation of curved spaces such as an AdS space. For group manifolds related by the orthogonal vielbein fields the three form H = dB in the doubled space is universal at least locally. We construct an affine nondegenerate doubled bosonic AdS algebra to define the AdS space with the Ramond-Ramond flux. The non-zero commutator of the left and right momenta leads to that the left momentum is in an AdS space while the right momentum is in a dS space. Dimensional reduction constraints and the physical AdS algebra are shown to preserve all the doubled coordinates.

We present affine Lie algebras generated by the supercovariant derivatives and the supersymmetry generators for the left and right moving modes in the doubled space. Chirality is manifest in our doubled space as well as the T-duality symmetry. We present gauge invariant bosonic and superstring actions preserving the two-dimensional diffeomorphism invariance and the κ-symmetry where doubled spacetime coordinates are chiral fields. The doubled space becomes the usual space by dimensional reduction constraints.

A superspace with manifest T-duality including Ramond-Ramond gauge fields is presented. The superspace is defined by the double nondegenerate super-Poincaré algebras where Ramond-Ramond charges are introduced by central extension. This formalism allows a simple treatment that all the supergravity multiplets are in a vielbein superfield and all torsions with dimension 1 and less are trivial. A Green-Schwarz superstring action is also presented where the Wess-Zumino term is given in a bilinear form of local currents. Equations of motion are separated into left and right modes in a suitable gauge.

We consider the dynamics of a particle of mass M and spin J in AdS(3). The study reveals the presence of different dynamical sectors depending on the relative values of M, J and the AdS(3) radius R. For the subcritical (MR2)-R-2 - J(2) > 0 and supercritical (MR2)-R-2 - J(2) < 0 cases, it is seen that the equations of motion give the geodesics of AdS(3). For the critical case (MR2)-R-2 = J(2) there exist extra gauge transformations which further reduce the physical degrees of freedom, and the motion corresponds to the geodesics of AdS(2). This result should be useful in the holographic interpretation of the entanglement entropy for two-dimensional conformal field theories with gravitational anomalies.

A superspace formulation of type II superstring background with manifest T-duality symmetry is presented. This manifestly T-dual formulation is constructed in a space spanned by two sets of nondegenerate super-Poincaré algebras. Supertorsion constraints are obtained from consistency of the κ-symmetric Virasoro constraints. All superconnections and vielbein fields are solved in terms of a prepotential which is one of the vielbein components. AdS5×S5 background is explained in this formulation.

We reconsider a model of two relativistic particles interacting via a multiplicative potential, as an example of a simple dynamical system with sectors, or branches, with different dynamics and degrees of freedom. The presence or absence of sectors depends on the values of rest masses. Some aspects of the canonical quantization are described. The model could be interpreted as a bigravity model in one dimension.

We study all the symmetries of the free Schrodinger equation in the non-commutative plane. These symmetry transformations form an infinite-dimensional Weyl algebra that appears naturally from a two-dimensional Heisenberg algebra generated by Galilean boosts and momenta. These infinite high symmetries could be useful for constructing non-relativistic interacting higher spin theories. A finite-dimensional subalgebra is given by the Schrodinger algebra which, besides the Galilei generators, contains also the dilatation and the expansion. We consider the quantization of the symmetry generators in both the reduced and extended phase spaces, and discuss the relation between both approaches.

We consider the non-linear realizations of the Poincare group for p-branes with local subgroup SO(1,p)*SO(D-(p+1)). The Nambu-Goto p-brane action is constructed using the Maurer Cartan forms of the unbroken translations. We perform a throughout phase space analysis of the action and show that it leads to the canonical action of a p-brane. We also construct some higher order derivative terms of the effective p-brane action using the MC forms of the broken Lorentz transformations.

The Maxwell algebra is the result of enlarging the Poincaré algebra by six additional tensorial Abelian generators that make the fourmomenta non-commutative. We present a local gauge theory based on the Maxwell algebra with vierbein, spin connection and six additional geometric Abelian gauge fields. We apply this geometric framework to the construction of Maxwell gravity, which is described by the Einstein action plus a generalized cosmological term. We mention a Friedman-Robertson-Walker cosmological approximation to the Maxwell gravity field equations, with two scalar fields obtained from the additional gauge fields. Finally, we outline further developments of the Maxwell symmetries framework.

We present "M5 algebra" to derive Courant brackets of the generalized geometry of T ⊕ Λ2 T ∗ ⊕ Λ5 T ∗: the Courant bracket generates the generalized diffeomorphism including gauge transformations of three and six form gauge fields. The Dirac bracket between selfdual gauge fields on a M5-brane gives a C [3]-twisted contribution to the Courant brackets. For M-theory compactified on a five dimensional torus the U-duality symmetry is SO(5,5) and the M5 algebra basis is in the 16-dimensional spinor representation. The M5 worldvolume diffeomorphism constraints can be written as bilinear forms of the basis and transform as a SO(5,5) vector. We also present an extended space spanned by the 16-dimensional coordinates with section conditions determined from the M5 worldvolume diffeomorphism constraints.

We show how the SL(5) duality in M-theory is explained from a canonical analysis of M2-brane mechanics. Diffeomorphism constraints for a M2-brane coupled to supergravity background in d = 4 are reformulated in a SL(5) covariant form, in which spatial diffeomorphism constraints are recast into a SL(5) vector and the generalized metric in the Hamiltonian constraint is quartic in the SL(5) generalized vielbein. The Hamiltonian for a M2 brane has the SL(5) duality symmetry in a background dependent gauge.

We examine AdS Galileon Lagrangians using the method of non-linear realization. By contractions 1) flat curvature limit and 2) non-relativistic brane algebra limit and 3) (1)+(2) limits we obtain DBI, Newton-Hoock and Galilean Galileons respectively. We make clear how these Lagrangians appear as invariant 4-forms and/or pseudo-invariant Wess-Zumino terms using Maurer-Cartan equations on the coset $G/SO(3,1)$. We show the equations of motion are written in terms of the MC forms only and explain why the inverse Higgs condition is obtained as the equation of motion for all cases. The supersymmetric extension is also examined using SU(2,2|1)/(SO(3,1)x U(1)) supercoset and five WZ forms are constructed. They are reduced to the corresponding five Galileon WZ forms in the bosonic limit and are candidates of for supersymmetric Galileon.

The Maxwell algebra, an enlargement of Poincare algebra by Abelian tensorial generators, can be obtained in arbitrary dimension D by the suitable contraction of O(D - 1, 1) circle plus O(0 1,2) (Lorentz algebra circle plus AdS algebra). We recall that in D = 4 the Lorentz algebra 0(3.1) is described by the realification Sp(R)(2 vertical bar C) of complex algebra Sp(2 vertical bar C)similar or equal to Sl(2 vertical bar C) and O(3,2)similar or equal to Sp(4). We study various D = 4 N-extended Maxwell superalgebras obtained by the contractions of real superalgebras OSp(R)(2N - k: 2 vertical bar C) circle plus OSp(k;4) (k = 0, 1, 2,...,2N); (extended Lorentz superalgebra circle plus extended AdS superalgebra). If N =1 (k = 0, 1, 2) one arrives at three different versions of simple Maxwell superalgebra. For any fixed N we get 2N different superextensions of Maxwell algebra with n-extended Poincare superalgebras (1 <= n <= N) and the internal symmetry sectors obtained by suitable contractions of the real algebra O-R (2N - k vertical bar C) circle plus O(k). Finally the comments on possible applications of Maxwell superalgebras are presented. (C) 2011 Elsevier B.V. All rights reserved.

We consider Schrodinger equations for a nonrelativistic particle obeying an N + 1-th order higher derivative classical equation of motion. These equations are invariant under N (odd)-extended Galilean conformal algebras in general d + 1 dimensions. In 2 + 1 dimensions, the exotic Schrodinger equations are invariant under N (even)-Galilean conformal algebras.

We show how the SL(5) duality in M-theory is explained from a canonical analysis of M2-brane mechanics. Diffeomorphism constraints for a M2-brane coupled to supergravity background in d = 4 are reformulated in a SL(5) covariant form, in which spatial diffeomorphism constraints are recast into a SL(5) vector and the generalized metric in the Hamiltonian constraint is quartic in the SL(5) generalized vielbein. The Hamiltonian for a M2 brane has the SL(5) duality symmetry in a background dependent gauge.

By gauging the Maxwell spacetime algebra, the standard geometric framework of Einstein gravity with cosmological constant term is extended by adding six four-vector fields A(mu)(ab)(x) associated with the six Abelian tensorial charges in the Maxwell algebra. In the simplest Maxwell extension of Einstein gravity this leads to a generalized cosmological term that includes a contribution from these vector fields. We also consider going beyond the basic gravitational model by means of bilinear actions for the new Abelian gauge fields. Finally, an analogy with the supersymmetric generalization of gravity is indicated. In an appendix, we propose an equivalent description of the model in terms of a shift of the standard spin connection by the A(mu)(ab)(x) fields.

We construct a pseudoclassical particle model associated to the twisted [FORMULA] SUSY algebra in four dimensions. The particle model has four kappa symmetries. Three of them can be used to reduce the model to the vector supersymmetry particle case. The quantization of the model gives rise to two copies of the 4d Dirac equation. The kappa symmetries result to be associated to 4 TSUSY invariant bilinear odd operators who are null operators when a particular condition is satisfied. These operators are in correspondence one to one with analogous operators existing in the case of the [FORMULA] SUSY algebra, making both cases 1/2 BPS.

We construct a pseudoclassical particle model associated to the twisted N=2 SUSY algebra in four dimensions. The particle model has four kappa symmetries. Three of them can be used to reduce the model to the vector supersymmetry particle case. The quantization of the model gives rise to two copies of the 4d Dirac equation. The kappa symmetries result to be associated to 4 TSUSY invariant bilinear odd operators who are null operators when a particular condition is satisfied. These operators are in correspondence one to one with analogous operators existing in the case of the N=2 SUSY algebra, making both cases 1/2 BPS.

We construct a pseudoclassical particle model associated to the twisted \( \mathcal{N} = 2 \) SUSY algebra in four dimensions. The particle model has four kappa symmetries. Three of them can be used to reduce the model to the vector supersymmetry particle case. The quantization of the model gives rise to two copies of the 4d Dirac equation. The kappa symmetries result to be associated to 4 TSUSY invariant bilinear odd operators who are null operators when a particular condition is satisfied. These operators are in correspondence one to one with analogous operators existing in the case of the \( \mathcal{N} = 2 \) SUSY algebra, making both cases 1/2 BPS.