scholarly journals Infinite pseudo-conformal symmetries of classical $T \bar T$, $J \bar T $ and $J T_a$ - deformed CFTs

2021 ◽  
Vol 11 (4) ◽  
Author(s):  
Monica Guica ◽  
Ruben Monten
Keyword(s):  
Poisson Bracket ◽  
Hamiltonian Formalism ◽  
Conformal Transformations ◽  
Two Dimensional ◽  
Witt Algebra ◽  
Moody Algebra ◽  
Field Dependent ◽  
Dependent Form ◽  
Infinite Set ◽  
Conformal Symmetries

We show that T\bar{T}, J\bar{T}TT‾,JT‾ and JT_aJTa - deformed classical CFTs posses an infinite set of symmetries that take the form of a field-dependent generalization of two-dimensional conformal transformations. If, in addition, the seed CFTs possess an affine U(1)U(1) symmetry, we show that it also survives in the deformed theories, again in a field-dependent form. These symmetries can be understood as the infinitely-extended conformal and U(1)U(1) symmetries of the underlying two-dimensional CFT, seen through the prism of the ``dynamical coordinates’’ that characterise each of these deformations. We also compute the Poisson bracket algebra of the associated conserved charges, using the Hamiltonian formalism. In the case of the J\bar{T}JT‾ and JT_{a}JTa deformations, we find two copies of a functional Witt - Kac-Moody algebra. In the case of the T\bar{T}TT‾ deformation, we show that it is also possible to obtain two commuting copies of the Witt algebra.

scholarly journals Finite BRST–anti-BRST transformations in generalized Hamiltonian formalism

2014 ◽  
Vol 29 (27) ◽  
pp. 1450159 ◽  
Author(s):  
Pavel Yu. Moshin ◽  
Alexander A. Reshetnyak
Keyword(s):  
Dynamical Systems ◽  
Path Integral ◽  
Hamiltonian Path ◽  
Hamiltonian Formalism ◽  
Lagrangian Formalism ◽  
Yang Mills ◽  
Gauge Dependence ◽  
Field Dependent ◽  
Gauge Fixing ◽  
Constrained Dynamical Systems

We introduce the notion of finite BRST–anti-BRST transformations for constrained dynamical systems in the generalized Hamiltonian formalism, both global and field-dependent, with a doublet λa, a = 1, 2, of anticommuting Grassmann parameters and find explicit Jacobians corresponding to these changes of variables in the path integral. It turns out that the finite transformations are quadratic in their parameters. Exactly as in the case of finite field-dependent BRST–anti-BRST transformations for the Yang–Mills vacuum functional in the Lagrangian formalism examined in our previous paper [arXiv:1405.0790 [hep-th]], special field-dependent BRST–anti-BRST transformations with functionally-dependent parameters λa= ∫ dt(saΛ), generated by a finite even-valued function Λ(t) and by the anticommuting generators saof BRST–anti-BRST transformations, amount to a precise change of the gauge-fixing function for arbitrary constrained dynamical systems. This proves the independence of the vacuum functional under such transformations. We derive a new form of the Ward identities, depending on the parameters λaand study the problem of gauge dependence. We present the form of transformation parameters which generates a change of the gauge in the Hamiltonian path integral, evaluate it explicitly for connecting two arbitrary Rξ-like gauges in the Yang–Mills theory and establish, after integration over momenta, a coincidence with the Lagrangian path integral [arXiv:1405.0790 [hep-th]], which justifies the unitarity of the S-matrix in the Lagrangian approach.

FOURIER ANALYSIS OVER HYPERBOLIC ALGEBRA, PSEUDO-DIFFERENTIAL OPERATORS, AND HYPERBOLIC DEFORMATION OF CLASSICAL MECHANICS

2007 ◽  
Vol 10 (03) ◽  
pp. 421-438 ◽  
Author(s):  
ANDREI KHRENNIKOV
Keyword(s):  
Quantum Mechanics ◽  
Fourier Analysis ◽  
Poisson Bracket ◽  
Commutative Algebra ◽  
Differential Operators ◽  
Classical Mechanics ◽  
Complex Numbers ◽  
Two Dimensional ◽  
Quantum Deformation ◽  
Pseudo Differential Operators

We develop Fourier analysis over hyperbolic algebra (the two-dimensional commutative algebra with the basis e1 = 1, e2 = j, where j2 = 1). We demonstrated that classical mechanics has, besides the well-known quantum deformation over complex numbers, another deformation — so-called hyperbolic quantum mechanics. The classical Poisson bracket can be obtained as the limit h → 0 not only of the ordinary Moyal bracket, but also a hyperbolic analogue of the Moyal bracket.

Electric-field-dependent absorption of ZnSe-based quantum wells: The transition from two-dimensional to three-dimensional behavior

1998 ◽  
Vol 58 (16) ◽  
pp. 10709-10720 ◽  
Author(s):  
D. Merbach ◽  
E. Schöll ◽  
W. Ebeling ◽  
P. Michler ◽  
J. Gutowski
Keyword(s):  
Electric Field ◽  
Quantum Wells ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Field Dependent ◽  
Dependent Absorption

A Linearized Two-Dimensional Theory for High-Speed Hydrofoils Near the Free Surface

1966 ◽  
Vol 10 (01) ◽  
pp. 25-48
Author(s):  
Richard P. Bernicker
Keyword(s):  
Direct Problem ◽  
High Speed ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Pressure Loading ◽  
Dimensional Theory ◽  
Linear Algebraic Equations ◽  
The Matrix ◽  
Sectional Shape ◽  
Infinite Set

A linearized two-dimensional theory is presented for high-speed hydrofoils near the free surface. The "direct" problem (hydrofoil shape specified) is attacked by replacing the actual foil with vortex and source sheets. The resulting integral equation for the strength of the singularity distribution is recast into an infinite set of linear algebraic equations relating the unknown constants in a Glauert-type vorticity expansion to the boundary condition on the foil. The solution is achieved using a matrix inversion technique and it is found that the matrix relating the known and unknown constants is a function of depth of submergence alone. Inversion of this matrix at each depth allows the vorticity constants to be calculated for any arbitrary foil section by matrix multiplication. The inverted matrices have been calculated for several depth-to-chord ratios and are presented herein. Several examples for specific camber and thickness distributions are given, and results indicate significant effects in the force characteristics at depths less than one chord. In particular, thickness effects cause a loss of lift at shallow submergences which may be an appreciable percentage of the total design lift. The second part treats the "indirect" problem of designing a hydrofoil sectional shape at a given depth to achieve a specified pressure loading. Similar to the "direct" problem treated in the first part, integral equations are derived for the camber and thickness functions by replacing the actual foil by vortex and source sheets. The solution is obtained by recasting these equations into an infinite set of linear algebraic equations relating the constants in a series expansion of the foil geometry to the known pressure boundary conditions. The matrix relating the known and unknown constants is, again, a function of the depth of submergence alone, and inversion techniques allow the sectional shape to be determined for arbitrary design pressure distributions. Several examples indicate the procedure and results are presented for the change in sectional shape for a given pressure loading as the depth of submergence of the foil is decreased.

scholarly journals Hamiltonian Formulation of Two-Dimensional Gyroviscous MHD

1984 ◽  
Vol 39 (11) ◽  
pp. 1023-1027 ◽  
Author(s):  
Philip J. Morrison ◽  
I. L. Caldas ◽  
H. Tasso
Keyword(s):  
Field Theory ◽  
Poisson Bracket ◽  
Lie Algebra ◽  
Hamiltonian Formulation ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Poisson Type

Gyroviscous MHD in two dimensions is shown to be a Hamiltonian field theory in terms of a non-canonical Poisson bracket. This bracket is of the Lie-Poisson type, but possesses an unfamiliar inner Lie algebra. Analysis of this algebra motivates a transformation that enables a Clebsch-like potential decomposition that makes Lagrangian and canonical Hamiltonian formulations possible.

scholarly journals THE CONSERVED CHARGES AND INTEGRABILITY OF THE CONFORMAL AFFINE TODA MODELS

1994 ◽  
Vol 09 (30) ◽  
pp. 2783-2801 ◽  
Author(s):  
H. ARATYN ◽  
L. A. FERREIRA ◽  
J. F. GOMES ◽  
A. H. ZIMERMAN
Keyword(s):  
Poisson Bracket ◽  
Equations Of Motion ◽  
Curvature Form ◽  
Two Dimensional ◽  
Conserved Charges ◽  
Zero Curvature ◽  
Algebraic Properties ◽  
Toda Model ◽  
Infinite Sets ◽  
Poisson Commute

We construct infinite sets of local conserved charges for the conformal affine Toda model. The technique involves the abelianization of the two-dimensional gauge potentials satisfying the zero-curvature form of the equations of motion. We find two infinite sets of chiral charges and apart from two lowest spin charges, all the remaining ones do not possess chiral densities. Charges of different chiralities Poisson commute among themselves. We discuss the algebraic properties of these charges and use the fundamental Poisson bracket relation to show that the charges conserved in time are in involution. Connections to other Toda models are established by taking particular limits.

THE N = 2 SUPER-KAC-MOODY ALGEBRA AND THE WZW-MODEL IN (2,0) SUPERSPACE

1990 ◽  
Vol 05 (24) ◽  
pp. 4753-4767 ◽  
Author(s):  
GUSTAV W. DELIUS
Keyword(s):  
Poisson Bracket ◽  
Coadjoint Orbits ◽  
Moody Algebra ◽  
Bracket Algebra

The N = 2 super-Kac-Moody algebra is defined as the algebra corresponding to an N = 2 superloop group. Using the method of coadjoint orbits, which is explained, an action is derived which has the N = 2 super-Kac-Moody algebra as its Poisson bracket algebra and is invariant under the N = 2 superloop group. It is invariant under N = 2 superconformal transformations. This action is thus the generalization of the WZW model to (2,0) superspace. We study the relation to the known WZW model in (1,0) superspace after imposing a constraint on the (2,0) superfield.

scholarly journals GAUGING OF CHERN-SIMONS p-BRANES

1994 ◽  
Vol 09 (19) ◽  
pp. 3367-3375 ◽  
Author(s):  
RAIKO P. ZAIKOV
Keyword(s):  
Phase Space ◽  
Poisson Bracket ◽  
Target Space ◽  
Closed Algebra ◽  
Chern Simons ◽  
Infinite Set

The Chern-Simons membranes and in general the Chern-Simons p-branes moving in D-dimensional target space admit an infinite set of secondary constraints. With respect to the Poisson bracket these constraints form a closed algebra which contains the classical W1+∞ algebra in p dimensions as a subalgebra. A corresponding gauged theory in the phase space is constructed in a Hamiltonian gauge as an analog of the ordinary W gravity.

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