arXiv subjects

George Jorjadze

Publications and source records attributed to George Jorjadze.

At least 19 recordsLinked to original sources

arXiv2025

Recursive relations for the S-matrix of Liouville theory

We analyze the relation between the vertex operators of the in and out fields in Liouville theory. This is used to derive equations for the S-matrix, from which a recursive relation for the normal symbol of the S-matrix for discrete center-of-mass momenta is obtained. Its solution is expressed as multiple contour-integrals of a generalized Dotsenko-Fateev type. This agrees with the functional integral representation of the scattering matrix of Liouville theory which we had proposed previously.

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arXiv2020

On the S-matrix of Liouville theory

The S-matrix for each chiral sector of Liouville theory on a cylinder is computed from the loop expansion of correlation functions of a one-dimensional field theory on a circle with a non-local kinetic energy and an exponential potential. This action is the Legendre transform of the generating function of semiclassical scattering amplitudes. It is derived from the relation between asymptotic in- and out-fields. Its relevance for the quantum scattering process is demonstrated by comparing explicit loop diagrams computed from this action with other methods of computing the S-matrix, which are also developed.

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arXiv2020

Canonical maps and integrability in $T\bar T$ deformed 2d CFTs

We study $T\bar T$ deformations of 2d CFTs with periodic boundary conditions. We relate these systems to string models on $\mathbb{R}\times {S}^1\times{\cal M}$, where $\cal M$ is the target space of a 2d CFT. The string model in the light cone gauge is identified with the corresponding 2d CFT and in the static gauge it reproduces its $T\bar T$ deformed system. This relates the deformed system and the initial one by a worldsheet coordinate transformation, which becomes a time dependent canonical map in the Hamiltonian treatment. The deformed Hamiltonian defines the string energy and we express it in terms of the chiral Hamiltonians of the initial 2d CFT. This allows exact quantization of the deformed system, if the spectrum of the initial 2d CFT is known. The generalization to non-conformal 2d field theories is also discussed.

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arXiv2017

Coset construction of AdS particle dynamics

We analyze dynamics of the AdS$_{N+1}$ particle realized on the coset SO$(2,N)/$SO$(1,N)$. Hamiltonian reduction provides the physical phase space in terms of the coadjoint orbit obtained by boosting a timelike element of ${\frak so}(2,N)$. We show equivalence of this approach to geometric quantization and to the SO$(N)$ covariant oscillator description, for which the boost generators entail a complicated operator ordering. As an alternative scheme, we introduce dual oscillator variables and derive their algebra at the classical and the quantum level. This simplifies the calculations of the commutators for the boost generators and leads to unitary irreducible representations of ${\frak so}(2,N)$ for all admissible values of the mass parameter. We furthermore discuss an SO$(N)$ covariant supersymmetric extensions of the oscillator quantization, with its realization for superparticles in AdS$_2$ and AdS$_3$ given by recent works.

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arXiv2016

Quantization of the ${\rm AdS}_3$ Superparticle on ${\rm OSP}(1|2)^2/{\rm SL}(2,\mathbb{R})$

We analyze ${\rm AdS}_3$ superparticle dynamics on the coset ${\rm OSP}(1|2) \times {\rm OSP}(1|2)/{\rm SL}(2,\mathbb{R})$. The system is quantized in canonical coordinates obtained by gauge invariant Hamiltonian reduction. The left and right Noether charges of a massive particle are parametrized by coadjoint orbits of a timelike element of $\frak{osp}(1|2)$. Each chiral sector is described by two bosonic and two fermionic canonical coordinates corresponding to a superparticle with superpotential $W=q-m/q$, where $m$ is the particle mass. Canonical quantization then provides a quantum realization of $\frak{osp}(1|2)\oplus\frak{osp}(1|2)$. For the massless particle the chiral charges lie on the coadjoint orbit of a nilpotent element of $\frak{osp}(1|2)$ and each of them depends only on one real fermion, which demonstrates the underlying $κ$-symmetry. These remaining left and right fermionic variables form a canonical pair and the system is described by four bosonic and two fermionic canonical coordinates. Due to conformal invariance of the massless particle, the $\frak{osp}(1|2)\oplus\frak{osp} (1|2)$ extends to the corresponding superconformal algebra $\frak{osp}(2|4)$. Its 19 charges are given by all real quadratic combinations of the canonical coordinates, which trivializes their quantization.

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arXiv2016

Orbit method quantization of the AdS$_2$ superparticle

We consider the Hamiltonian reduction and canonical quantization of a massive AdS$_2$ superparticle realized on the coset OSP(1|2)/SO(1,1). The phase space of the massive superparticle is represented as a coadjoint orbit of a timelike element of $\mathfrak{osp}$(1|2). This orbit has a well defined symplectic structure and the OSP(1|2) 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 $\mathfrak{osp}$(1|2). We also perform a similar analysis and discuss new features and inconsistencies in the massless case.

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arXiv2015

Isometry Group Orbit Quantization of Spinning Strings in AdS_3 x S^3

Describing the bosonic AdS_3 x S^3 particle and string in SU(1,1) x SU(2) group variables, we provide a Hamiltonian treatment of the isometry group orbits of solutions via analysis of the pre-symplectic form. For the particle we obtain a one-parameter family of orbits parameterized by creation-annihilation variables, which leads to the Holstein-Primakoff realization of the isometry group generators. The scheme is then applied to spinning string solutions characterized by one winding number in AdS_3 and two winding numbers in S^3. We find a two-parameter family of orbits, where quantization again provides the Holstein-Primakoff realization of the symmetry generators with an oscillator type energy spectrum. Analyzing the minimal energy at strong coupling we verify the spectrum of short strings at special values of winding numbers.

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arXiv2013

Static Gauge and Energy Spectrum of Single-mode Strings in AdS_5xS^5

Motivated by the notorious difficulties in determining the first quantum corrections to the spectrum of short strings in AdS_5xS^5 from first 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 AdS_5 string mode excitation next to an arbitrary particle like zero-mode contribution in the full AdS_5xS^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 field 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 AdS_6, classical integrability and preservation of the isometries of AdS_5xS^5 at the quantum level.

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arXiv2012

Quantization of AdS x S particle in static gauge

We quantize the particle dynamics in AdS_{N+1} x S^M spacetime in static gauge, which leads to the coordinate representation with wave functions depending only on spatial coordinates. The energy square operator is quadratic in canonical momenta and contains a scalar curvature term. We analyze the self-adjointness of this operator and calculate its spectrum. We then construct unitary representations of the isometry group SO(2,N) x SO(M+1) and calculate the quantum relation between the Casimir numbers.

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arXiv2012

On particle type string solutions in AdS_3 x S^3

The AdS_3 x S^3 string dynamics is described in a conformal gauge using the SL(2,R) and SU(2) group variables as the target space coordinates. A subclass of string surfaces with constant induced metric tensor on both AdS_3 and S^3 projections is considered. The general solution of string equations on this subclass is presented and the corresponding conserved charges related to the isometry transformations are calculated. The subclass of solutions is characterized by a finite number of parameters. The Poisson bracket structure on the space of parameters is calculated, its connection to the particle dynamics in SL(2,R) x SU(2) is analyzed and a possible way of quantization is discussed.

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arXiv2012

Bosonic String Quantization in Static Gauge

The bosonic string in D dimensional Minkowski space-time is quantized in static gauge. It is shown that the system can be described by D-1 massless free fields constrained on the surface L_m = 0, for m \neq 0, where L_m are the generators of conformal transformations. The free fields are quantized and the physical states are selected by the conditions L_m|phys>=0, for m>0. The Poincaré 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.

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arXiv2011

Coordinate representation of particle dynamics in AdS and in 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 E^2, 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 E^2 and in the Klein-Gordon type equation have the same coefficient equal to (N-1)/(4N).

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arXiv2010

On spacelike and timelike minimal surfaces in $AdS_n$

We discuss timelike and spacelike minimal surfaces in $AdS_n$ 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 $AdS_n, n\geq 4$ beyond the four cusp surfaces used in the Alday-Maldacena conjecture. Furthermore, we give a parameterization of flat timelike minimal surfaces in $AdS_5$ in terms of two chiral fields.

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arXiv2010

Space-like minimal surfaces in AdS x S

We present a four parameter family of classical string solutions in AdS_3 x S^3, which end along a light-like tetragon at the boundary of AdS_3 and carry angular momentum along two cycles on the sphere. The string surfaces are space-like and their projections on AdS_3 and on S^3 have constant mean curvature. The construction is based on the Pohlmeyer reduction of the related sigma model. After embedding in AdS_5 x S^5, 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.

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arXiv2010

Vacuum type space-like string surfaces in AdS_3 x S^3

We construct and classify all space-like minimal surfaces in AdS_3 x S^3 which globally admit coordinates with constant induced metric on both factors. Up to O(2,2) x O(4) transformations all these surfaces, except one class, are parameterized by four real parameters. The classes of surfaces correspond to different regions in this parameter space and show quite different boundary behavior. Our analysis uses a direct construction of the string coordinates via a group theoretical treatment based on the map of AdS_3 x S^3 to SL(2,R) x SU(2). This is complemented by a cross check via standard Pohlmeyer reduction. After embedding in AdS_5 x S^5 we calculate the regularized area for solutions with a boundary spanned by a four point scattering s-channel momenta configuration.

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arXiv2009

Singular Liouville fields and spiky strings in $\rr^{1,2}$ and $SL(2,\rr)$

The closed string dynamics in $\rr^{1,2}$ and $SL(2,\rr)$ is studied within the scheme of Pohlmeyer reduction. In both spaces two different classes of string surfaces are specified by the structure of the fundamental quadratic forms. The first class in $\rr^{1,2}$ is associated with the standard lightcone gauge strings and the second class describes spiky strings and their conformal deformations on the Virasoro coadjoint orbits. These orbits correspond to singular Liouville fields with the monodromy matrixes $\pm I$. The first class in $SL(2,\rr)$ is parameterized by the Liouville fields with vanishing chiral energy functional. Similarly to $\rr^{1,2}$, the second class in $SL(2,\rr)$ describes spiky strings, related to the vacuum configurations of the $SL(2,\rr)/U(1)$ coset model.

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arXiv2008

Operator Approach to Boundary Liouville Theory

We propose new methods for calculation of the discrete spectrum, the reflection amplitude and the correlation functions of boundary Liouville theory on a strip with Lorentzian signature. They are based on the structure of the vertex operator $V=e^{-ϕ}$ in terms of the asymptotic operators. The methods first are tested for the particle dynamics in the Morse potential, where similar structures appear. Application of our methods to boundary Liouville theory reproduces the known results obtained earlier in the bootstrap approach, but there can arise a certain extension when the boundary parameters are near to critical values. Namely, in this case we have found up to four different equidistant series of discrete spectra, and the reflection amplitude is modified respectively.

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arXiv2007

Boundary Liouville Theory: Hamiltonian Description and Quantization

The paper is devoted to the Hamiltonian treatment of classical and quantum properties of Liouville field theory on a timelike strip in 2d Minkowski space. We give a complete description of classical solutions regular in the interior of the strip and obeying constant conformally invariant conditions on both boundaries. Depending on the values of the two boundary parameters these solutions may have different monodromy properties and are related to bound or scattering states. By Bohr-Sommerfeld quantization we find the quasiclassical discrete energy spectrum for the bound states in agreement with the corresponding limit of spectral data obtained previously by conformal bootstrap methods in Euclidean space. The full quantum version of the special vertex operator $e^{-ϕ}$ in terms of free field exponentials is constructed in the hyperbolic sector.

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