scholarly journals LOOP VARIABLES AND THE (FREE) OPEN STRING IN A CURVED BACKGROUND

2005 ◽  
Vol 20 (04) ◽  
pp. 227-242 ◽  
Author(s):  
B. SATHIAPALAN
Keyword(s):  
Gauge Invariance ◽  
Sigma Model ◽  
Equations Of Motion ◽  
Flat Space ◽  
Open String ◽  
Target Space ◽  
Systematic Method ◽  
Curved Space ◽  
Generally Covariant ◽  
Space Equation

Using the loop variable formalism as applied to a sigma model in curved target space, we give a systematic method for writing down gauge and generally covariant equations of motion for the modes of the free open string in curved space. The equations are obtained by covariantizing the flat space equation and then demanding gauge invariance, which introduces additional curvature couplings. As an illustration of the procedure, the spin-two case is worked out explicitly.

scholarly journals LOOP VARIABLES AND THE INTERACTING OPEN STRING IN A CURVED BACKGROUND

2005 ◽  
Vol 20 (14) ◽  
pp. 1037-1045
Author(s):  
B. SATHIAPALAN
Keyword(s):  
Gauge Invariance ◽  
Equations Of Motion ◽  
Interaction Strength ◽  
Flat Space ◽  
Open String ◽  
Target Space ◽  
Gauge Invariant ◽  
Free Case ◽  
New Feature ◽  
Generally Covariant

Applying the loop variable proposal to a sigma model (with boundary) in a curved target space, we give a systematic method for writing the gauge and generally covariant interacting equations of motion for the modes of the open string in a curved background. As in the free case described in an earlier paper, the equations are obtained by covariantizing the flat space (gauge invariant) interacting equations and then demanding gauge invariance in the curved background. The resulting equation has the form of a sum of terms that would individually be gauge invariant in flat space or at zero interaction strength, but mix amongst themselves in curved space when interactions are turned on. The new feature is that the loop variables are deformed so that there is a mixing of modes. Unlike the free case, the equations are coupled, and all the modes of the open string are required for gauge invariance.

Feldtheoretische Konstruktion der Jordan—Brans—Dicke-Theorie / Fieldtheoretic Construction of the Jordan-Brans-Dicke-Theory

1971 ◽  
Vol 26 (4) ◽  
pp. 599-622
Author(s):  
H. von Grünberg
Keyword(s):  
Gauge Group ◽  
Tensor Field ◽  
Mathematical Proof ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Flat Space ◽  
Momentum Tensor ◽  
Field Equations ◽  
Tensor Theory ◽  
Generally Covariant

Abstract In the framework of Lorentz invariant theories of gravitation the fieldtheoretic approach of the generally covariant Jordan-Brans-Dicke-theory is investigated.It is shown that a slight restriction of the gauge group of Einstein's linear tensor theory leads to the linearized Jordan-Brans-Dicke-theory. The problem of the inconsistency of the field equations and the equations of motion is solved by introducing the Landau-Lifschitz energy momentum tensor of the gravitational field as an additional source term into the field equations. The second order of the theory together with the corresponding gauge group are calculated explicitly. By means of the structure of the gauge group of the tensor field it is possible to identify the successive orders of the scalar-tensor theory as an expansion of the Jordan-Brans-Dicke-theory in flat space-time. The question of the uniqueness of the procedure is answered by showing that the structure of the gauge group of the tensor field is predetermined by the linear equations of motion. The mathematical proof of this fact confirms formally the meaning of the equations of motion for the geometry of space.

scholarly journals ACTION FOR (FREE) OPEN STRING MODES IN AdS SPACE USING THE LOOP VARIABLE APPROACH

2007 ◽  
Vol 22 (02) ◽  
pp. 107-117 ◽  
Author(s):  
B. SATHIAPALAN
Keyword(s):  
Equations Of Motion ◽  
Flat Space ◽  
Open String ◽  
Closed String ◽  
Group Method ◽  
Open Strings ◽  
Variable Approach ◽  
Massive Spin ◽  
Complete Treatment ◽  
Space Case

The loop variable technique (for open strings in flat space) is a gauge-invariant generalization of the renormalization group method for obtaining equations of motion. Unlike the beta functions, which are only proportional to the equations of motion, here it gives the full equation of motion. In an earlier paper, a technique was described for adapting this method to open strings in gravitational backgrounds. However, unlike the flat space case, these equations cannot be derived from an action and are therefore not complete. This is because there are ambiguities in the method that involve curvature couplings that cannot be fixed by appealing to gauge invariance alone but need a more complete treatment of the closed string background. An indirect method to resolve these ambiguities is to require symmetricity of the second derivatives of the action. In general this will involve modifying the equations by terms with arbitrarily high powers of curvature tensors. This is illustrated for the massive spin-two field. It is shown that in the special case of an AdS or dS background, the exact action can easily be determined in this way.

scholarly journals LOOP VARIABLES AND GAUGE INVARIANCE IN (OPEN) BOSONIC STRING THEORY

2002 ◽  
Vol 17 (18) ◽  
pp. 1175-1190 ◽  
Author(s):  
B. SATHIAPALAN
Keyword(s):  
String Theory ◽  
Gauge Invariance ◽  
Equations Of Motion ◽  
Standard Form ◽  
Open String ◽  
Finite Width ◽  
Massive Mode ◽  
Gauge Transformations ◽  
Variable Approach ◽  
Free Case

We give a simplified and more complete description of the loop variable approach for writing down gauge-invariant equations of motion for the fields of the open string. A simple proof of gauge invariance to all orders is given. In terms of loop variables, the interacting equations look exactly like the free equations, but with a loop variable depending on an extra parameter, thus making it a band of finite width. The arguments for gauge invariance work exactly as in the free case. We show that these equations are Wilsonian RG equations with a finite worldsheet cutoff and that in the ir limit, equivalence with the Callan–Symanzik β-functions should ensure that they reproduce the on-shell scattering amplitudes in string theory. It is applied to the tachyon–photon system and the general arguments for gauge invariance can be easily checked to the order calculated. One can see that when there is a finite worldsheet cutoff in place, even the U(1) invariance of the equations for the photon, involves massive mode contributions. A field redefinition involving the tachyon is required to get the gauge transformations of the photon into the standard form.

scholarly journals Jacobi sigma models

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Francesco Bascone ◽  
Franco Pezzella ◽  
Patrizia Vitale
Keyword(s):  
Sigma Model ◽  
Open String ◽  
Target Space ◽  
Two Dimensional ◽  
Hamiltonian Approach ◽  
Metric Tensor ◽  
Finite Dimensional ◽  
Poisson Sigma Model ◽  
Jacobi Manifold ◽  
Polyakov Action

Abstract We introduce a two-dimensional sigma model associated with a Jacobi manifold. The model is a generalisation of a Poisson sigma model providing a topological open string theory. In the Hamiltonian approach first class constraints are derived, which generate gauge invariance of the model under diffeomorphisms. The reduced phase space is finite-dimensional. By introducing a metric tensor on the target, a non-topological sigma model is obtained, yielding a Polyakov action with metric and B-field, whose target space is a Jacobi manifold.

scholarly journals A non-relativistic limit of NS-NS gravity

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
E. A. Bergshoeff ◽  
J. Lahnsteiner ◽  
L. Romano ◽  
J. Rosseel ◽  
C. Şimşek
Keyword(s):  
Scale Invariance ◽  
Sigma Model ◽  
Equations Of Motion ◽  
Geometric Constraints ◽  
Target Space ◽  
Kinetic Term ◽  
Scale Invariant ◽  
Relativistic Limit ◽  
Cartan Geometry ◽  
Form Field

Abstract We discuss a particular non-relativistic limit of NS-NS gravity that can be taken at the level of the action and equations of motion, without imposing any geometric constraints by hand. This relies on the fact that terms that diverge in the limit and that come from the Vielbein in the Einstein-Hilbert term and from the kinetic term of the Kalb-Ramond two-form field cancel against each other. This cancelling of divergences is the target space analogue of a similar cancellation that takes place at the level of the string sigma model between the Vielbein in the kinetic term and the Kalb-Ramond field in the Wess-Zumino term. The limit of the equations of motion leads to one equation more than the limit of the action, due to the emergence of a local target space scale invariance in the limit. Some of the equations of motion can be solved by scale invariant geometric constraints. These constraints define a so-called Dilatation invariant String Newton-Cartan geometry.

FLRW COSMOLOGY FROM YANG–MILLS GRAVITY WITH TRANSLATIONAL GAUGE SYMMETRY

2013 ◽  
Vol 28 (07) ◽  
pp. 1350018 ◽  
Author(s):  
DANIEL KATZ
Keyword(s):  
Gauge Invariance ◽  
Equations Of Motion ◽  
Flat Space ◽  
Metric Tensor ◽  
Yang Mills ◽  
Effective Metric ◽  
Starting Point ◽  
Special Cases ◽  
The Subject ◽  
Flrw Cosmology

We extend to basic cosmology the subject of Yang–Mills gravity — a theory of gravity based on local translational gauge invariance in flat space–time. It has been shown that this particular gauge invariance leads to tensor factors in the macroscopic limit of the equations of motion of particles which plays the same role as the metric tensor of general relativity (GR). The assumption that this "effective metric" tensor takes on the standard FLRW form is our starting point. Equations analogous to the Friedmann equations are derived and then solved in closed form for the three special cases of a universe dominated by (1) matter, (2) radiation and (3) dark energy. We find that the solutions for the scale factor are similar to, but distinct from, those found in the corresponding GR based treatment.

scholarly journals NONLINEAR SIGMA MODEL ON CONIFOLDS

2002 ◽  
Vol 17 (09) ◽  
pp. 517-533 ◽  
Author(s):  
R. PARTHASARATHY ◽  
K. S. VISWANATHAN
Keyword(s):  
Sigma Model ◽  
Equations Of Motion ◽  
Target Space ◽  
Warp Factor ◽  
Nonlinear Sigma Model ◽  
Explicit Solutions ◽  
Type Ii ◽  
Ricci Flat ◽  
Complex Dimension ◽  
Gauge Connection

Explicit solutions to the conifold equations with complex dimension n = 3, 4 in terms of complex coordinates (fields) are employed to construct the Ricci-flat Kähler metrics on these manifolds. The Kähler two-forms are found to be closed. The complex realization of these conifold metrics are used in the construction of two-dimensional nonlinear sigma model with the conifolds as target spaces. The action for the sigma model is shown to be bounded from below. By a suitable choice of the "integration constants", arising in the solution of Ricci flatness requirement, the metric and the equations of motion are found to be non-singular. As the target space is Ricci-flat, the perturbative one-loop counterterms being absent, the model becomes topological. The inherent U(1) fiber over the base of the conifolds is shown to correspond to a gauge connection in the sigma model. The same procedure is employed to construct the metric for the resolved conifold, in terms of complex coordinates and the action for a nonlinear sigma model with resolved conifold as target space, is found to have a minimum value, which is topological. The metric is expressed in terms of the six real coordinates and compared with earlier works. The harmonic function, which is the warp factor in Type II-B string theory, is obtained and the ten-dimensional warped metric has the AdS5 × X5 geometry.

SIGMA MODELS WITH (2, 2) WORLD SHEET SUPERSYMMETRY

1994 ◽  
Vol 08 (25n26) ◽  
pp. 3725-3739 ◽  
Author(s):  
F. DELDUC ◽  
E. SOKATCHEV
Keyword(s):  
Sigma Models ◽  
Gauge Invariance ◽  
Degrees Of Freedom ◽  
Sigma Model ◽  
Space Dimension ◽  
Target Space ◽  
World Sheet ◽  
Chiral Superfields

We propose a new mechanism for constructing (2, 2) superspace actions for WZWN sigma models. We use the already known semi-chiral superfields, but we choose the Lagrangian in a special way, so that it has a gauge invariance capable of eliminating half of the initially propagating degrees of freedom. Thus we avoid the doubling of the target space dimension. The nonlinear differential conditions on the Lagrangian are derived and it is shown that this type of action corresponds to the general (2, 2) sigma model.

TWO-LOOP BETA FUNCTION IN THE OPEN STRING SIGMA MODEL AND EQUIVALENCE WITH STRING EFFECTIVE EQUATIONS OF MOTION

1988 ◽  
Vol 03 (14) ◽  
pp. 1349-1359 ◽  
Author(s):  
O.D. ANDREEV ◽  
A.A. TSEYTLIN
Keyword(s):  
Vector Field ◽  
Scattering Amplitudes ◽  
Effective Action ◽  
Sigma Model ◽  
Equations Of Motion ◽  
Beta Function ◽  
Open String ◽  
Equation Of Motion ◽  
Β Function ◽  
The Given

We consider the sigma-model on the disc corresponding to the open bose string in external abelian vector field and compute the order ∂3F3-terms in its two-loop β-function. The vanishing of the two-loop β-function (modulo diffeomorphism and gauge terms) is shown to be equivalent (to the given order) to the equation of motion for the string theory effective action reconstructed from the string scattering amplitudes.

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