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.

CONFORMALLY INVARIANT FIELD THEORY IN TWO DIMENSIONS AND STRINGS IN CURVED SPACETIME

1988 ◽  
Vol 03 (08) ◽  
pp. 1759-1846 ◽  
Author(s):  
SANJAY JAIN
Keyword(s):  
Field Theory ◽  
Equations Of Motion ◽  
Virasoro Algebra ◽  
Functional Integral ◽  
Two Dimensions ◽  
Intrinsic Metric ◽  
Existence Of A Solution ◽  
Conformally Invariant ◽  
Brst Charge ◽  
Spacetime Metric

The formalism of conformally invariant field theory on a 2-dimensional real manifold with an intrinsic metric is developed in the functional integral framework. This formalism is used to study the relationships between reparametrization, Weyl, conformal and BRST invariances for strings in generic backgrounds. Conformal invariance of string amplitudes in the presence of backgrounds is formulated in terms of the Virasoro conditions, i.e., that physical vertex operators generate (1,1) representations of the Virasoro algebra, or, equivalently, the condition Q|Ψ〉=0 on physical states |Ψ〉, where Q is the BRST charge. The consequences of these conditions are investigated in the case of specific backgrounds. Strings in group manifolds are discussed exactly. For a generic slowly varying spacetime metric and dilaton field, a perturbatively renormalized vertex operator solution to the Virasoro conditions is constructed. It is shown that the existence of a solution to the Virasoro conditions or the equation Q|Ψ〉=0 requires the spacetime metric to satisfy Einstein’s equations. These conditions therefore constitute equations of motion for both the spectrum and backgrounds of string theory.

scholarly journals 2D QUANTUM GRAVITY COUPLED TO RENORMALIZABLE MATTER FIELDS

1994 ◽  
Vol 09 (32) ◽  
pp. 5689-5709 ◽  
Author(s):  
JAN AMBJØRN ◽  
KAZUO GHOROKU
Keyword(s):  
Renormalization Group ◽  
Quantum Gravity ◽  
Effective Action ◽  
Two Dimensional ◽  
Conformally Invariant ◽  
Renormalization Group Equations ◽  
Β Function ◽  
Matter Fields

We consider two-dimensional quantum gravity coupled to matter fields which are renormalizable but not conformally invariant. Questions concerning the β function and the effective action are addressed, and the effective action and the dressed renormalization group equations are determined for various matter potentials.

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.

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.

scholarly journals STRING THEORETICAL INTERPRETATION FOR FINITE N YANG–MILLS THEORY IN TWO DIMENSIONS

2005 ◽  
Vol 20 (01) ◽  
pp. 29-41 ◽  
Author(s):  
TOSHIHIRO MATSUO ◽  
SO MATSUURA
Keyword(s):  
Asymptotic Expansion ◽  
Closed String ◽  
Two Dimensions ◽  
Perturbative Expansion ◽  
Target Space ◽  
Theoretical Interpretation ◽  
Two Dimensional ◽  
Yang Mills ◽  
Large N ◽  
Residual Part

We discuss the equivalence between a string theory and the two-dimensional Yang–Mills theory with SU (N) gauge group for finite N. We find a sector which can be interpreted as a sum of covering maps from closed string worldsheets to the target space, whose covering number is less than N. This gives an asymptotic expansion of 1/N whose large N limit becomes the chiral sector defined by Gross and Taylor. We also discuss that the residual part of the partition function provides the nonperturbative corrections to the perturbative expansion.

scholarly journals D = (0|2) DIRAC–MAXWELL–EINSTEIN THEORY AS A WAY FOR DESCRIBING SUPERSYMMETRIC QUARTIONS

1994 ◽  
Vol 09 (09) ◽  
pp. 1555-1568 ◽  
Author(s):  
DMITRIJ P. SOROKIN ◽  
DMITRIJ V. VOLKOV
Keyword(s):  
Gravitational Field ◽  
Dimensional Space ◽  
Equations Of Motion ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Dirac Theory ◽  
Einstein Theory ◽  
Abelian Gauge ◽  
Dimensional Space Time

Drawing an analogy with the Dirac theory of fermions interacting with electromagnetic and gravitational field we write down supersymmetric equations of motion and construct a superfield action for particles with spin [Formula: see text] and [Formula: see text] (quartions), where the role of quartion momentum in effective (2+1)-dimensional space–time is played by an Abelian gauge superfield propagating in a basic two-dimensional Grassmann-odd space with a cosmological constant showing itself as the quartion mass. So, the (0|2) (0 even and 2 odd) dimensional model of quartions interacting with the gauge and gravitational field manifests itself as an effective (2 + 1)-dimensional supersymmetric theory.

scholarly journals CORRELATION FUNCTIONS OF MINIMAL MODELS COUPLED TO TWO-DIMENSIONAL QUANTUM SUPERGRAVITY

1992 ◽  
Vol 07 (04) ◽  
pp. 333-343 ◽  
Author(s):  
KENICHIRO AOKI ◽  
ERIC D’HOKER
Keyword(s):  
Correlation Functions ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Minimal Models ◽  
Point Correlation ◽  
Bosonic Case

We compute the three-point functions of Neveu-Schwarz primary fields of the minimal models on the sphere when coupled to supergravity in two dimensions. The results show that the three-point correlation functions are determined by the scaling dimensions of the fields, as in the bosonic case.

TWO DIMENSIONAL QUANTUM GRAVITY AND STRINGS

1990 ◽  
Vol 05 (10) ◽  
pp. 1833-1859 ◽  
Author(s):  
A.A. TSEYTLIN
Keyword(s):  
String Theory ◽  
Quantum Gravity ◽  
Conformal Factor ◽  
Target Space ◽  
Effective Theory ◽  
Two Dimensional ◽  
Conformal Gauge

We discuss some recent suggestions about a relation between 2-d quantum gravity and string theory. We consider the general 2-d σ model with D-dimensional target space coupled to 2-d quantum gravity and give arguments in favor of the conjecture that the effective theory which describes this system in the conformal gauge may be interpreted as a a model with a D+1-dimensional target space with the conformal factor of the metric playing the role of the D+1 coordinate. The latter σ model must be Weyl invariant while the original one may be arbitrary. The conformal “split” invariance which must be present in the conformal gauge imposes no restrictions on the original σ model couplings, but restricts the additional (D+1-dimensional) couplings which appear in the D+1-dimensional σ model.

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.

STOCHASTIC DIFFERENTIAL EQUATIONS ON TWO-DIMENSIONAL THEORY SPACE AND MORSE THEORY

1989 ◽  
Vol 04 (08) ◽  
pp. 745-756 ◽  
Author(s):  
SUMIT R. DAS ◽  
GAUTAM MANDAL ◽  
SPENTA R. WADIA
Keyword(s):  
Morse Theory ◽  
Morse Function ◽  
Saddle Points ◽  
Two Dimensional ◽  
Minimal Models ◽  
Dimensional Theory ◽  
Topological Characterization ◽  
Renormalization Group Equations ◽  
Kähler Potential

The renormalization group equations are shown to be saddle points of the action of a superparticle moving in the presence of a Kähler potential. This allows us to view the “theory space” in the language of Morse theory where the Kähler potential in the Morse function. In the case of two-dimensional field theory, Zamolodchikov’s c function can be used as a Morse function. The Morse polynomial tr (tF) can be computed entirely from the universal Virasoro data at the various fixed points and this provides a topological characterization of the space of potential functions. This idea is applied to the two-dimensional theory space which contains the c<1 minimal models.

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