AN INTEGRABLE MODEL WITH SOLITON SOLUTIONS WHICH HAVE APPLICATIONS TO TWO-DIMENSIONAL GRAVITY

2005 ◽  
Vol 20 (07) ◽  
pp. 1503-1514 ◽  
Author(s):  
PAUL BRACKEN
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Integrable Model ◽  
Soliton Solutions ◽  
Integrable Equation ◽  
Two Dimensions ◽  
Spin Connection ◽  
Two Dimensional ◽  
Dimensional Gravity ◽  
Chern Simons

The equations of motion for a theory described by a Chern–Simons type of action in two dimensions are obtained and investigated. The equation for the classical, continuous Heisenberg model is used as a form of gauge constraint to obtain a result which provides a completely integrable dynamics and which partially fixes the gauge degrees of freedom. Under a particular form of the spin connection, an integrable equation which can be analytically extended to a form of the nonlinear Schrödinger equation is obtained. Some explicit solutions are presented, and in particular a soliton solution is shown to lead to an integrable two-dimensional model of gravity.

EQUIVALENCE OF TWO-DIMENSIONAL GRAVITIES

1990 ◽  
Vol 05 (16) ◽  
pp. 1251-1258 ◽  
Author(s):  
NOUREDDINE MOHAMMEDI
Keyword(s):  
Gauge Theory ◽  
Explicit Solution ◽  
Light Cone ◽  
Equations Of Motion ◽  
Two Dimensional ◽  
Induced Gravity ◽  
Light Cone Gauge ◽  
Dimensional Gravity ◽  
The Relationship ◽  
Chern Simons

We find the relationship between the Jackiw-Teitelboim model of two-dimensional gravity and the SL (2, R) induced gravity. These are shown to be related to a two-dimensional gauge theory obtained by dimensionally reducing the Chern-Simons action of the 2+1 dimensional gravity. We present an explicit solution to the equations of motion of the auxiliary field of the Jackiw-Teitelboim model in the light-cone gauge. A renormalization of the cosmological constant is also given.

scholarly journals A NON-DEGENERATE SEMICLASSICAL LAGRANGIAN FOR DILATON-GRAVITY IN TWO DIMENSIONS

1992 ◽  
Vol 07 (33) ◽  
pp. 3071-3079 ◽  
Author(s):  
NOUREDDINE MOHAMMEDI
Keyword(s):  
Black Holes ◽  
Exact Solutions ◽  
Equations Of Motion ◽  
Two Dimensions ◽  
Singular Solutions ◽  
Two Dimensional ◽  
Semiclassical Theory ◽  
Dilaton Gravity ◽  
Dimensional Gravity ◽  
Matter Fields

An action for two-dimensional gravity conformally coupled to two dilaton-type fields is analyzed. Classically, the theory has some exact solutions. These include configurations representing black holes. A semiclassical theory is obtained by assuming that these singular solutions are caused by the collapse of some matter fields. The semiclassical equations of motion reveal then that any generic solution must have a flat geometry.

scholarly journals Averaging over Narain moduli space

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Alexander Maloney ◽  
Edward Witten
Keyword(s):  
Einstein Gravity ◽  
Two Dimensions ◽  
Three Dimensions ◽  
One Dimension ◽  
Simons Theory ◽  
Two Dimensional ◽  
Dual Theory ◽  
Recent Developments ◽  
Chern Simons ◽  
Chern Simons Theory

Abstract Recent developments involving JT gravity in two dimensions indicate that under some conditions, a gravitational path integral is dual to an average over an ensemble of boundary theories, rather than to a specific boundary theory. For an example in one dimension more, one would like to compare a random ensemble of two-dimensional CFT’s to Einstein gravity in three dimensions. But this is difficult. For a simpler problem, here we average over Narain’s family of two-dimensional CFT’s obtained by toroidal compactification. These theories are believed to be the most general ones with their central charges and abelian current algebra symmetries, so averaging over them means picking a random CFT with those properties. The average can be computed using the Siegel-Weil formula of number theory and has some properties suggestive of a bulk dual theory that would be an exotic theory of gravity in three dimensions. The bulk dual theory would be more like U(1)2D Chern-Simons theory than like Einstein gravity.

QUANTIZATION OF REDUCED CHERN–SIMONS GRAVITY WITH A SOURCE

2004 ◽  
Vol 19 (28) ◽  
pp. 4883-4897 ◽  
Author(s):  
A. V. NAZARENKO
Keyword(s):  
Continuous Spectrum ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Quantum States ◽  
Theory Of Constraints ◽  
Volume Operator ◽  
Modular Transformations ◽  
Probability Of Transition ◽  
Chern Simons

Using the Dirac's theory of constraints, procedure of reduction of field degrees of freedom, whose number is restricted by equations of motion and topological conditions, is proposed. Such a procedure is applied in the case of space with the topology of a torus to the Chern–Simons gravity generalized by inclusion of a source. It is shown that in this system some modular transformations preserving the volume do not lead to physically equivalent states. Such a breaking of modular symmetry reduces the degeneration of quantum states with preservation of continuous spectrum of the volume operator. Probability of transition between spaces of different volumes is computed.

CRITICAL BEHAVIOR IN TWO-DIMENSIONAL QUANTUM GRAVITY AND EQUATIONS OF MOTION OF THE STRING

1990 ◽  
Vol 05 (11) ◽  
pp. 799-813 ◽  
Author(s):  
SUMIT R. DAS ◽  
AVINASH DHAR ◽  
SPENTA R. WADIA
Keyword(s):  
Equations Of Motion ◽  
Two Dimensions ◽  
Target Space ◽  
Two Dimensional ◽  
Minimal Models ◽  
Conformally Invariant ◽  
Motion Time ◽  
Dynamical Degree ◽  
Renormalization Group Equations

We show how consistent quantization determines the renormalization of couplings in a quantum field theory coupled to gravity in two dimensions. The special status of couplings corresponding to conformally invariant matter is discussed. In string theory, where the dynamical degree of freedom of the two-dimensional metric plays the role of time in target space, these renormalization group equations are themselves the classical equations of motion. Time independent solutions, like classical vacuua, correspond to the situation in which matter is conformally invariant. Time dependent solutions, like tunnelling configurations between vacuua, correspond to special trajectories in theory space. We discuss an example of such a trajectory in the space containing the c < 1 minimal models. We also discuss the connection between this work and the recent attempts to construct non-perturbative string theories based on matrix models.

A Model of Affine Gravity in Two Dimensions and Plurality of Topology

1997 ◽  
Vol 12 (28) ◽  
pp. 5067-5080 ◽  
Author(s):  
Marco Ferraris ◽  
Mauro Francaviglia ◽  
Igor Volovich
Keyword(s):  
Vector Field ◽  
Equations Of Motion ◽  
Linear Connection ◽  
Functional Space ◽  
Conformal Group ◽  
Two Dimensions ◽  
General Mechanism ◽  
Classical Action ◽  
Nonlinear Lagrangians ◽  
Dimensional Gravity

A new model of two-dimensional gravity with an action depending only on a linear connection is suggested. This model is a topological one, in the sense that the classical action does not contain a metric or zweibein at all. A metric and an additional vector field are instead generated in the process of solving the equations of motion for the connection. The general solution of these equations of motion is given by an arbitrary Weyl connection which can be described by using the space of orbits under the action of the conformal group in the functional space containing all pairs formed by a metric and a vector field. By choosing a gauge one obtains a constant curvature equation. It is shown that this model admits an equivalent description by using a family of Lagrangians depending on the metric and the connection as independent variables. We show that nonlinear Lagrangians in the first order formalism lead to plurality of topology for the manifolds under consideration and give a simple general mechanism of governing topology change.

CANONICAL DESCRIPTION OF A TWO-DIMENSIONAL GRAVITY

1992 ◽  
Vol 07 (19) ◽  
pp. 1757-1764 ◽  
Author(s):  
K.G. AKDENIZ ◽  
Ö.F. DAYI ◽  
A. KIZILERSÜ
Keyword(s):  
Degrees Of Freedom ◽  
Mass Shell ◽  
Gravity Theory ◽  
Two Dimensional ◽  
Lagrangian Methods ◽  
Hamiltonian Methods ◽  
Dimensional Gravity ◽  
Conformal Gauge

A two-dimensional gravity theory which was studied before within the Lagrangian methods, in the conformal gauge is investigated in terms of the Hamiltonian methods. Although the reparametrization invariant and the conformal gauge fixed Lagrangians lead to different number of physical degrees of freedom, it is shown that on mass-shell they are equivalent.

scholarly journals THE “DUAL” VARIABLES OF YANG-MILLS THEORY AND LOCAL GAUGE-INVARIANT VARIABLES

1996 ◽  
Vol 11 (32) ◽  
pp. 5701-5728 ◽  
Author(s):  
ORI GANOR ◽  
J. SONNENSCHEIN
Keyword(s):  
Degrees Of Freedom ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Gauge Invariant ◽  
Yang Mills ◽  
Six Degrees Of Freedom ◽  
Local Gauge ◽  
Dual Variables ◽  
Topological Part

After adding auxiliary fields and integrating out the original variables, the Yang-Mills action can be expressed in terms of local gauge-invariant variables. This method reproduces the known solution of the two-dimensional SU (N) theory. In more than two dimensions the action splits into a topological part and a part proportional to αs. We demonstrate the procedure for SU (2) in three dimensions where we reproduce a gravitylike theory. We discuss the four-dimensional case as well. We use a cubic expression in the fields as a space-time metric to obtain a covariant Lagrangian. We also show how the four-dimensional SU (2) theory can be expressed in terms of a local action with six degrees of freedom only.

scholarly journals Charged rotating BTZ black holes in noncommutative spaces and torsion gravity

2018 ◽  
Vol 2018 (4) ◽  
Author(s):  
Shoichi Kawamoto ◽  
Koichi Nagasaki ◽  
Wen-Yu Wen
Keyword(s):  
Black Holes ◽  
Equations Of Motion ◽  
Einstein Equations ◽  
Noncommutative Space ◽  
Simons Theory ◽  
Btz Black Holes ◽  
Dimensional Gravity ◽  
Cartan Theory ◽  
Suitable Framework ◽  
Chern Simons

Abstract We consider charged rotating BTZ black holes in noncommutative space using a Chern–Simons theory formulation of $(2+1)$-dimensional gravity. The noncommutativity between the radial and the angular variables is introduced through the Seiberg–Witten map for gauge fields, and the deformed geometry to the first order in the noncommutative parameter is derived. It is found that the deformation also induces nontrivial torsion, and Einstein–Cartan theory appears to be a suitable framework to investigate the equations of motion. Though the deformation is indeed nontrivial, the deformed and the original Einstein equations are found to be related by a rather simple coordinate transformation.

scholarly journals Classical instanton solutions in quantum field theory

2020 ◽  
pp. 78-85
Author(s):  
Roman G. Shulyakovsky ◽  
Alexander S. Gribowsky ◽  
Alexander S. Garkun ◽  
Maxim N. Nevmerzhitsky ◽  
Alexei O. Shaplov ◽  
...  
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Theory ◽  
Yukawa Potential ◽  
Equations Of Motion ◽  
Two Dimensions ◽  
Quantum Field ◽  
Two Dimensional ◽  
Yang Mills ◽  
The One

Instantons are non-trivial solutions of classical Euclidean equations of motion with a finite action. They provide stationary phase points in the path integral for tunnel amplitude between two topologically distinct vacua. It make them useful in many applications of quantum theory, especially for describing the wave function of systems with a degenerate vacua in the framework of the path integrals formalism. Our goal is to introduce the current situation about research on instantons and prepare for experiments. In this paper we give a review of instanton effects in quantum theory. We find in stanton solutions in some quantum mechanical problems, namely, in the problems of the one-dimensional motion of a particle in two-well and periodic potentials. We describe known instantons in quantum field theory that arise, in particular, in the two-dimensional Abelian Higgs model and in SU(2) Yang – Mills gauge fields. We find instanton solutions of two-dimensional scalar field models with sine-Gordon and double-well potentials in a limited spatial volume. We show that accounting of instantons significantly changes the form of the Yukawa potential for the sine-Gordon model in two dimensions.

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