George Jorjadze

On spacelike and timelike minimal surfaces in AdS(n)

We discuss timelike and spacelike minimal surfaces in AdSn using a Pohlmeyer type reduction. The differential equations for the reduced system are derived in a parallel treatment of both type of surfaces, with emphasis on their characteristic differences. In the timelike case we find a formulation corresponding to a complete gauge fixing of the torsion. In the spacelike case we derive three sets of equations, related to different parameterizations enforced by the Lorentzian signature of the metric in normal space. On the basis of these equations, we prove that there are no flat spacelike minimal surfaces in AdSn,n ≥ 4 beyond the four cusp surfaces used in the Alday-Maldacena conjecture. Furthermore, we give a parameterization of flat timelike minimal surfaces in AdS5 in terms of two chiral fields.

Space-like minimal surfaces in AdS × S

We present a four parameter family of classical string solutions in AdS3 ×S3, which end along a light-like tetragon at the boundary of AdS3 and carry angular momentum along two cycles on the sphere. The string surfaces are space-like and their projections on AdS3 and on S3 have constant mean curvature. The construction is based on the Pohlmeyer reduction of the related sigma model. After embedding in AdS5 × S5, we calculate the regularized area and analyze conserved charges. Comments on possible relations to scattering amplitudes are presented. We also sketch time-like versions of our solutions.

Coordinate representation of particle dynamics in AdS and generic static spacetimes

We discuss the quantum dynamics of a particle in static curved spacetimes in a coordinate representation. The scheme is based on the analysis of the squared energy operator E2, which is quadratic in momenta and contains a scalar curvature term. Our main emphasis is on AdS spaces, where this term is fixed by the isometry group. As a byproduct the isometry generators are constructed and the energy spectrum is reproduced. In the massless case the conformal symmetry is realized as well. We show the equivalence between this quantization and the covariant quantization, based on the Klein–Gordon-type equation in AdS. We further demonstrate that the two quantization methods in an arbitrary (N + 1)-dimensional static spacetime are equivalent to each other if the scalar curvature terms both in the operator E2 and in the Klein–Gordon-type equation have the same coefficient equal to .

Quantization of a relativistic particle on the SL(2, R) manifold based on Hamiltonian reduction

Abstract A quantum theory is constructed for the system of a relativistic particle with mass m moving freely on the SL(2, R ) group manifold. Applied to the cotangent bundle of SL(2, R ) , the method of Hamiltonian reduction allows us to split the reduced system into two coadjoint orbits of the group. We find that the Hilbert space consists of states given by the discrete series of the unitary irreducible representations of SL(2, R ) , and with a positive-definite, discrete spectrum.

Correlation functions and vertex operators of Liouville theory

We calculate correlation functions for vertex operators with negative integer exponentials of a periodic Liouville field, and derive the general case by continuing them as distributions. The path-integral based conjectures of Dorn and Otto prove to be conditionally valid only. We formulate integral representations for the generic vertex operators and indicate structures which are related to the Liouville S-matrix.

Bosonic string quantization in a static gauge

The bosonic string in D-dimensional Minkowski space–time is quantized in a static gauge. It is shown that the system can be described by D − 1 massless free fields constrained on the surface Lm = 0, for m ≠ 0, where Lm are the generators of conformal transformations. The free fields are quantized and the physical states are selected by the conditions Lm|Ψph〉 = 0, for m > 0. The Poincare group generators on the physical Hilbert space are constructed and the critical dimension D = 26 is recovered from the commutation relations of the boost operators. The equivalence with the covariant quantization is established. A possible generalization to the AdS string dynamics is discussed.

Oscillator quantization of the massive scalar particle dynamics on AdS spacetime

Abstract The set of trajectories for massive spinless particles on AdS N + 1 spacetime is described by the dynamical integrals related to the isometry group SO ( 2 , N ) . The space of dynamical integrals is mapped one to one to the phase space of the N-dimensional oscillator. Quantizing the system canonically, the classical expressions for the symmetry generators are deformed in a consistent way to preserve the so ( 2 , N ) commutation relations. This quantization thus yields new explicit realizations of the spin zero positive energy UIRs of SO ( 2 , N ) for generic N. The representations as usual can be characterized by their minimal energy α and are valid in the whole range of α allowed by unitarity.

Static gauge and energy spectrum of single-mode strings in AdS 5 × S 5

Motivated by the notorious diculties in determining the rst quantum corrections to the spectrum of short strings in AdS5 S 5 from rst principles, we study closed bosonic strings in this background employing a static gauge. In this gauge the world-sheet Hamiltonian density is constant along the extension of the string and directly proportional to the square of the spacetime energy. We quantize this system in a minisuperspace approach, in which we consider only a single AdS5 string mode excitation next to an arbitrary particle like zero-mode contribution in the full AdS5 S 5 background. We determine the quantum spectrum using this method to the next-to-next-to-leading order in the large ’t Hooft coupling expansion. We argue for an ordering prescription which should arise from supersymmetrization and indeed recover the integrability based predictions for the spectrum of the lightest excitation, dual to the Konishi eld scaling dimensions. The higher excitations fail to agree, but this is shown to be a consequence of the string mode truncation employed. Despite this simple setup, our system reveals intriguing features, such as a close connection to particles in AdS6, classical integrability and preservation of the isometries of AdS5 S 5 at the quantum level.

Orbit method quantization of the AdS2 superparticle

We consider the Hamiltonian reduction and canonical quantization of a massive AdS2 superparticle realized on the coset SO(1,1). The phase space of the massive superparticle is represented as a coadjoint orbit of a timelike element of . This orbit has a well defined symplectic structure and the symmetry is realized as the Poisson bracket algebra of the Noether charges. We then construct canonical coordinates given by one bosonic and one fermionic oscillator, whose quantization leads to the Holstein–Primakoff type realization of . We also perform a similar analysis and discuss new features and inconsistencies in the massless case.

Massless scalar particle on AdS spacetime: hamiltonian reduction and quantization

We investigate the massless scalar particle dynamics on