Lightlike Direction Research Articles

Light-cone gauge formulation for AdS 4 × ℂℙ3 superstring

We review the Type IIA superstring on the AdS 4 × ℂℙ3 background in the κ-symmetry light-cone gauge characterized by the choice of the light-like directions from the D = 3 Minkowski boundary of AdS 4 both in the Lagrangian and Hamiltonian formulations.

LIGHT-CONE GAUGE HAMILTONIAN FOR AdS4 × ℂℙ3 SUPERSTRING

We develop the phase-space formulation for the Type IIA superstring on the AdS4 × ℂℙ3 background in the κ-symmetry light-cone gauge for which the light-like directions are taken from the D = 3 Minkowski boundary of AdS4. After fixing bosonic light-cone gauge, the superstring Hamiltonian is expressed as a function of the transverse physical variables and in the quadratic approximation corresponds to the light-cone gauge-fixed IIA superstring in flat space.

Formula omitted] superstring in the light-cone gauge

The Type IIA superstring action on the AdS 4 × CP 3 background, obtainable by the double dimensional reduction of the AdS 4 × S 7 supermembrane, is considered in the κ-symmetry light-cone gauge, in which the light-like directions are chosen on the D = 3 Minkowski boundary of AdS 4 . Such choice of the gauge condition relies on representing the AdS 4 × S 7 background isometry superalgebra osp ( 4 | 8 ) (and correspondingly the osp ( 4 | 6 ) isometry superalgebra of the AdS 4 × CP 3 background) as D = 3 extended superconformal algebra. The gauge-fixed action includes contributions up to the 4th power in the fermions.

Supersymmetric non-relativistic geometries in M-theory

We construct M-theory supergravity solutions with the non-relativistic Schrödinger symmetry starting from the warped AdS 5 metric with N = 1 supersymmetry. We impose the condition that the lightlike direction is compact by making it a non-trivial U ( 1 ) bundle over the compact space. Sufficient conditions for such solutions are analyzed. The solutions have two supercharges for generic values of parameters, but the number of supercharges increases to six in some special cases. A Schrödinger geometry with SU ( 2 ) × SU ( 2 ) × U ( 1 ) isometry is considered as a specific example. We consider the Kaluza–Klein modes and show that the non-relativistic particle number is bounded above by the quantum numbers of the compact space.

AdS/CFT duality for non-relativistic field theory

We formulate a correspondence between non-relativistic conformal field theories (NRCFTs) in d−1 spatial dimensions and gravitational theories in AdSd+2 backgrounds with one compactified lightlike direction. The breaking of the maximal SO(2,d+1) symmetry of AdSd+2 to the non-relativistic conformal group arises from boundary conditions on bulk fields, without the need to introduce non-vacuum sources of energy-momentum. As a check of the proposal, we use the gravitational theory to reproduce the NRCFT state-operator correspondence between scaling dimensions of primary operators and energy eigenstates of the non-relativistic system placed in an external harmonic potential.

Screw invariant marginally trapped surfaces in Minkowski 4-space

A spacelike surface in a Lorentzian manifold whose mean curvature vector is lightlike everywhere is called marginally trapped. The classification of marginally trapped surfaces in Minkowski 4-space which are invariant under a subgroup of the Lorentz group that leaves invariant a lightlike direction, i.e. the so-called screw invariant surfaces, is obtained. As corollaries, the screw invariant marginally trapped surfaces with harmonic mean curvature vector and with prescribed Gaussian curvature are explicitly described.

Entire Scalar Curvature Flow and Hypersurfaces of Constant Scalar Curvature in Minkowski Space

We prove existence in the Minkowski space of entire spacelike hypersurfaces with constant negative scalar curvature and given set of lightlike directions at infinity; we also construct the entire scalar curvature flow with prescribed set of lightlike directions at infinity, and prove that the flow converges to a spacelike hypersurface with constant scalar curvature. The proofs rely on barriers construction and a priori estimates.

Heating up Galilean holography

We embed a holographic description of a quantum field theory with Galilean conformal invariance in string theory. The key observation is that such field theories may be realized as conventional superconformal field theories with a known string theory embedding, twisted by the R-symmetry in a light-like direction. Using the Null Melvin Twist, we construct the appropriate dual geometry and its non-extremal generalization. From the nonzero temperature solution we determine the equation of state. We also discuss the hydrodynamic regime of these non-relativistic plasmas and show that the shear viscosity to entropy density ratio takes the universal value η/s = 1/4π typical of strongly interacting field theories with gravity duals.

Cosmological unification of string theories

We present an exact solution of superstring theory that interpolates in time between an initial type 0 phase and a final phase whose physics is exactly that of the bosonic string. The initial theory is deformed by closed-string tachyon condensation along a lightlike direction. In the limit of large tachyon vev, the worldsheet conformal field theory precisely realizes the Berkovits-Vafa embedding of bosonic string theory into superstring theory. Our solution therefore connects the bosonic string dynamically with the superstring, settling a longstanding question about the relationship between the two theories.

Lorentz-violating dilatations in momentum space and some extensions on nonlinear actions of Lorentz-algebra-preserving systems

We work on some general extensions of the formalism for theories which preserve the relativity of inertial frames with a nonlinear action of the Lorentz transformations on momentum space. Relativistic particle models invariant under the corresponding deformed symmetries are presented with particular emphasis on deformed dilatation transformations. The algebraic transformations relating the deformed symmetries with the usual (undeformed) ones are provided in order to preserve the Lorentz algebra. Two distinct cases are considered: a deformed dilatation transformation with a spacelike preferred direction and a very special relativity embedding with a lightlike preferred direction. In both analysis we consider the possibility of introducing quantum deformations of the corresponding symmetries such that the spacetime coordinates can be reconstructed and the particular form of the real space-momentum commutator remains covariant. Eventually feasible experiments, for which Lorentz violating effects here pointed out may be detectable, are suggested.

Cosmologies with null singularities and their gauge theory duals

We investigate backgrounds of Type IIB string theory with null singularities and their duals proposed in S. R. Das, J. Michelson, K. Narayan, S. P. Trivedi, hep-th/0602107. The dual theory is a deformed $\mathcal{N}=4$ Yang-Mills theory in $3+1$ dimensions with couplings dependent on a lightlike direction. We concentrate on backgrounds which become ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{S}^{5}$ at early and late times and where the string coupling is bounded, vanishing at the singularity. Our main conclusion is that in these cases the dual gauge theory is nonsingular. We show this by arguing that there exists a complete set of gauge invariant observables in the dual gauge theory whose correlation functions are nonsingular at all times. The two-point correlator for some operators calculated in the gauge theory does not agree with the result from the bulk supergravity solution. However, the bulk calculation is invalid near the singularity where corrections to the supergravity approximation become important. We also obtain pp-waves which are suitable Penrose limits of this general class of solutions, and construct the matrix membrane theory which describes these pp-wave backgrounds.

Time-dependent cosmologies and their duals

We construct a family of solutions in IIB supergravity theory. These are time-dependent or depend on a light-like coordinate and can be thought of as deformations of ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{S}^{5}$. Several of the solutions have singularities. The light-like solutions preserve 8 supersymmetries. We argue that these solutions are dual to the $\mathcal{N}=4$ gauge theory in a $3+1$ dimensional space-time with a metric and a gauge coupling that is varying with time or the light-like direction, respectively. This identification allows us to map the question of singularity resolution to the dual gauge theory.

Exactly solvable model of superstring in plane-wave background with linear null dilaton

In this paper, we study an exactly solvable model of IIB superstring in a time-dependent plane-wave background with a constant self-dual Ramond–Ramond 5-form field strength and a linear dilaton in the light-like direction. This background keeps sixteen supersymmetries. In the light-cone gauge, the action is described by the two-dimensional free bosons and fermions with time-dependent masses. The model could be canonically quantized and its Hamiltonian is time-dependent with vanishing zero-point energy. The spectrum of the excitations is symmetric between the bosonic and fermionic sector. The string mode creation turns out to be very small.

Geometric description of lightlike foliations by an observer in general relativity

We introduce new concepts and properties of lightlike distributions and foliations (of dimension and co-dimension 1) in a space-time manifold of dimension $n$, from a purely geometric point of view. Given an observer and a lightlike distribution $\Omega $ of dimension or co-dimension 1, its lightlike direction is broken down into two vector fields: a timelike vector field $U$ representing the observer and a spacelike vector field $S$ representing the relative direction of propagation of $\Omega $ for this observer. A new distribution $\Omega_U^-$ is defined, with the opposite relative direction of propagation for the observer $U$. If both distributions $\Omega $ and $\Omega _U^-$ are integrable, the pair \Omega ,\Omega_U^- $ represents the wave fronts of a stationary wave for the observer $U$. However, we show in an example that the integrability of $\Omega $ does not imply the integrability of $\Omega_U^-$.

Three-graviton scattering in Matrix theory revisited

We consider a subset of the terms in the effective potential describing three-graviton interactions in Matrix theory and in classical eleven-dimensional supergravity. In agreement with the results of Dine and Rajaraman, we find that these terms vanish in Matrix theory. We show that the absence of these terms is compatible with the classical supergravity theory when the theory is compactified in a lightlike direction, resolving an apparent discrepancy between the two theories. A brief discussion is given of how this calculation might be generalized to compare the Matrix theory and supergravity descriptions of an arbitrary 3-body system.

On membrane interaction in Matrix theory

We compute the interaction potential between two parallel transversely boosted wrapped membranes (with fixed momentum p −) in D = 11 supergravity with compact light-like direction. We show that the supergravity result is in exact agreement with the potential following from the all-order Born-Infeld-type action conjectured to be the leading planar infra-red part of the quantum super Yang-Mills effective action. This provides a non-trivial test of consistency of the arguments relating Matrix theory to a special limit of type II string theory. We also find the potential between two (2+0) D-brane bound states in D = 10 supergravity (corresponding to the case of boosted membrane configuration in 11-dimensional theory compactified on a space-like direction). We demonstrate that the result reduces to the SYM expression for the potential in the special low-energy ( α′ → 0) limit, in agreement with previous suggestions. In an appendix we derive the action obtained from the D = 11 membrane action by the world-volume duality transformation of the light-like coordinate x − into a 3-vector.

Light-ray radon transform for Abelian and non-Abelian connections in three- and four-dimensional space with a Minkowski metric

We consider a real manifold of dimension 3 or 4 with Minkovsky metric, and with a connection for a trivial GL(n,C) bundle over that manifold. To each light ray on the manifold we assign the data of paralel transport along that light ray. It turns out that these data are not enough to reconstruct the connection, but we can add more data, which depend now not from lines but from 2-planes, and which in some sence are the data of parallel transport in the complex light-like directions, then we can reconstruct the connection up to a gauge transformation. There are some interesting applications of the construction: 1) in 4 dimensions, the self-dual Yang Mills equations can be written as the zero curvature condition for a pair of certain first order differential operators; one of the operators in the pair is the covariant derivative in complex light-like direction we studied. 2) there is a relation of this Radon transform with the supersymmetry. 3)using our Radon transform, we can get a measure on the space of 2 dimensional planes in 4 dimensional real space. Any such measure give rise to a Crofton 2-density. The integrals of this 2-density over surfaces in R^4 give rise to the Lagrangian for maps of real surfaces into R^4, and therefore to some string theory. 4) there are relations with the representation theory. In particular, a closely related transform in 3 dimensions can be used to get the Plancerel formula for representations of SL(2,R).

Remarks on the classical field theory of tachyons

where [] = ~2/~tS--A. On the other hand, it is known in the theory of partial differential equations tha t the propagation character of wave equations such as (2) and (1) is determined principally by those terms with the highest derivatives. This implies tha t the waves described by (1) and (2) behave in a similar manner, i.e., propagate mainly into the lightlike direction. A question therefore arises as to whether and to what extent eq. (1) describes the characteristic behaviour we expect of taehyons, i.e., propagation with a superluminal velocity into the spaeelike region. In the present note we shall make a preliminary a t tempt to answer this question within the framework of the classical field theory. We shall do so by studying properties of invariant delta functions (IDF) and invariant Green functions (IGF) for eq. (1).

Some Remarks on Null-Plane Quantization

Some features of quantum field theory on a null plane are investigated in comparison with those of conventional formalism on a space-like hyperplane. It is shown that even in a free field theory the light-like Hamiltonian should not be introduced in a whole space from the beginning as is usually done. It is also shown that, though the cluster theorem holds good in a sense along the light-like direction, it is impossible to prove Haag's strong convergence asymptotic condition for -r( =x+) ->±oo. In conclusion it seems impossible to consto:-uct a consistent scattering theory in the context of quantum field theory on a null plane. 1) The aim of this paper is to investigate some features of quantum field

Canonical Quantization of a Relativistic String

. A free motion of a relativistic string is quantized by the use of Dirac's generalized canonical formalism. It is shown that only the light-like gauge is applied to quantization of this system and the result is the same as that of Goddard, Goldstone, Rebbi and Thorn which is obtained by a different method. The relativistic quantum mechanics is constructed in this gauge when the dimension of space-time is twenty-six and the intercept of the leading trajectory is unity. § I. Introduction Nowadays a massless relativistic string is considered to be a fundamental physi­ cal object of dual resonance models. A free motion of the string is beautifully formulated by Nambu and Gotol) in a covariant manner as a variational problem. The Lagr'!-ngian which is the starting point of their discussion about the string motion is a singular one in the sense that velocities cannot be solved as functions of canonical coordinates and their· conjugate momenta. Moreover, because the La­ grangian depends linearly on velocities/> the Hamiltonian function vanishes identi­ cally. Recently Goddard, Goldstone, Rebbi and Thorn (G. G. R. T.) discussed how to formulate the Hamiltonian fprmalism of the string model with the conclusion that the covariant quantization needs the dimension of space-time to be twenty-six and the intercept of the leading Regge trajectory to be unity, using the noncovariant gauge. 2> It seems, however, that there are some doubtful treatments in their dis­ cussion, which arises· mainly from identifying the displacement operator along the light-like direction with the new Hamiltonian instead of the original one, which is obtained from the Lagrangian and is numerically zero. To deal with this system properly from the viewpoint of the canonical quantiza­ tion formalism, we will adopt, in this paper, the generalized canonical formalism developed by Dirac3> and consistently eliminate redundant variables from the system described by a singular Lagrangian. As a result of this formalism, it is pointed out that the light-like gauge is favourable to quantize this system rather than the time­ like gauge. Our conclusion is against that of Patrasciou 4> who claims that the latter gauge is more suitable than the former. It is shown that there are two different types of constraints in the string model in accordance with Dirac's method.