Infinite-dimensional Symmetry Group Research Articles

New infinite-dimensional quadruple symmetry groups for the general symplectic gravity model

The symmetry structures of the general symplectic gravity model are further studied. By using the so-called extended double (ED)-complex function method, the usual Riemann–Hilbert (RH) problem is extended to an ED-complex formulation. For any fixed non-negative integer n, two pairs of ED RH transformations are constructed and they are verified to give infinite-dimensional quadruple symmetry groups of the general symplectic gravity model, each of these symmetry groups has the structure of semidirect product of Kac–Moody group Sp(2(n+1̂),R) and Virasoro group. Moreover, the infinitesimal forms of these RH transformations are calculated out and they are found to give exactly the same results as previous; these demonstrate that the two pairs of ED RH transformations in this paper provide exponentiations of all the infinitesimal symmetries in our previous paper. The finite forms of symmetry transformations given in the present paper are more important and useful for theoretic studies and new solution generation, etc.

NEW INFINITE-DIMENSIONAL DOUBLE SYMMETRY GROUPS FOR THE EINSTEIN–KALB–RAMOND THEORY

The symmetry structures of the dimensionally reduced Einstein–Kalb–Ramond (EKR) theory is further studied. By using a so-called extended double (ED)-complex method, the usual Riemann–Hilbert (RH) problem is extended to an ED-complex formulation. A pair of ED RH transformations are constructed and they are verified to give infinite-dimensional double symmetry groups of the EKR theory, each of these symmetry groups has the structure of semidirect product of Kac–Moody group [Formula: see text] and Virasoro group. Moreover, the infinitesimal forms of these RH transformations are calculated out and they are found to give exactly the same results as previous, these demonstrate that the pair of ED RH transformations in this paper provide exponentiations of all the infinitesimal symmetries in our previous paper. The finite forms of symmetry transformations given in the present paper are more important and useful for theoretic studies and new solution generation, etc.

New infinite-dimensional symmetry groups for heterotic string theory

Some Riemann-Hilbert (RH) problems are introduced for carrying out symmetry transformations of the 2-dimensional heterotic string theory. A pair of RH transformations are constructed, and they are verified to give an infinite-dimensional symmetry group of the considered theory. This symmetry group has the structure of the semidirect product of the Kac-Moody group $\stackrel{^}{O(d,d+n)}$ and Virasoro group. Moreover, the infinitesimal forms of these RH transformations are calculated out, and they are found to give exactly the same results as in my previous paper. These demonstrate that the pair of RH transformations in the current paper provide exponentiations of all the infinitesimal symmetries in my previous paper. The finite forms of symmetry transformations given in the present paper are more important and useful for theoretic studies and new solution generation, etc.

Symmetry Analysis and Exact Solutions of (2+1)-Dimensional Sawada–Kotera Equation

Based on the symbolic computation system Maple, the infinite-dimensional symmetry group of the (2+1)-dimensional Sawada–Kotera equation is found by the classical Lie group method and the characterization of the group properties is given. The symmetry groups are used to perform the symmetry reduction. Moreover, with Lou's direct method that is based on Lax pairs, we obtain the symmetry transformations of the Sawada–Kotera and Konopelchenko–Dubrovsky equations, respectively.

On the Strong Coupling Limit of the Faddeev-Hopf Model

The variational calculus for the Faddeev-Hopf model on a general Riemannian domain, with general Kähler target space, is studied in the strong coupling limit. In this limit, the model has key similarities with pure Yang-Mills theory, namely conformal invariance in dimension 4 and an infinite dimensional symmetry group. The first and second variation formulae are calculated and several examples of stable solutions are obtained. In particular, it is proved that all immersive solutions are stable. Topological lower energy bounds are found in dimensions 2 and 4. An explicit description of the spectral behaviour of the Hopf map $${S^3 \rightarrow S^2}$$ is given, and a conjecture of Ward concerning the stability of this map in the full Faddeev-Hopf model is proved.

THE ROLE OF A DYNAMICAL MEASURE AND DYNAMICAL TENSION IN BRANE CREATION AND GROWTH

The use of a non-Riemannian measure of integration in the action of strings and branes allows the possibility of dynamical tension. In particular, lower dimensional objects living in the string/brane can induce discontinuities in the tension: the effect of pair creation on the string tension is studied. We then investigate the role that these new features can play in string and brane creation and growth. A mechanism is studied by means of which a scalar field can transfer its energy to the tension of strings and branes. An infinite-dimensional symmetry group of this theory is discussed. Creation and growth of bubbles in a formulation that requires mass generation for the bulk gauge fields coupled to the branes is also discussed.

New infinite-dimensional symmetry groups for the stationary axisymmetric Einstein–Maxwell equations with multiple Abelian gauge fields

The so-called extended hyperbolic complex (EHC) function method is used to study further the stationary axisymmetric Einstein–Maxwell theory with p Abelian gauge fields (EM-p theory, for short). Two EHC structural Riemann–Hilbert (RH) transformations are constructed and are then shown to give an infinite-dimensional symmetry group of the EM-p theory. This symmetry group is verified to have the structure of semidirect product of Kac–Moody group SU(+1,1) and Virasoro group. Moreover, the infinitesimal forms of these two RH transformations are calculated and found to give exactly the same infinitesimal transformations as in previous author's paper by a different scheme. This demonstrates that the results obtained in the present paper provide some exponentiations of all the infinitesimal symmetry transformations obtained before.

A Commutative Model of a Representation of the Group O(n, 1)X and a Generalized Lebesgue Measure in the Space of Distributions

For an irreducible unitary representation of an O(n, 1) current group, we consider a commutative model obtained by diagonalization with respect to a maximal unipotent subgroup. This model leads to a new measure on the space of distributions. The measure is invariant with respect to an infinite-dimensional linear symmetry group.

Method of group foliation and non-invariant solutions of partial differential equations

Using the heavenly equation as an example, we propose the method of group foliation as a tool for obtaining non-invariant solutions of PDEs with infinite-dimensional symmetry groups. The method involves the study of compatibility of the given equations with a differential constraint, which is automorphic under a specific symmetry subgroup and therefore selects exactly one orbit of solutions. By studying the integrability conditions of this automorphic system, i.e. the resolving equations, one can provide an explicit foliation of the entire solution manifold into separate orbits. The new important feature of the method is the extensive use of the operators of invariant differentiation for the derivation of the resolving equations and for obtaining their particular solutions. Applying this method we obtain exact analytical solutions of the heavenly equation, non-invariant under any subgroup of the symmetry group of the equation.

Group foliation and non-invariant solutions of the heavenly equation

The main physical result of this paper are exact analytical solutions of the heavenly equation, of importance in the general theory of relativity. These solutions are not invariant under any subgroup of the symmetry group of the equation. The main mathematical result is a new method of obtaining noninvariant solutions of partial differential equations with infinite dimensional symmetry groups. The method involves the compatibility of the given equations with a differential constraint, which is automorphic under a specific symmetry subgroup, the latter acting transitively on the submanifold of the common solutions. By studying the integrability of the resulting conditions, one can provide an explicit foliation of the entire solution manifold of the considered equations.

SELF-DUAL YANG–MILLS: SYMMETRIES AND MODULI SPACE

Geometry of the solution space of the self-dual Yang–Mills (SDYM) equations in Euclidean four-dimensional space is studied. Combining the twistor and group-theoretic approaches, we describe the full infinite-dimensional symmetry group of the SDYM equations and its action on the space of local solutions to the field equations. It is argued that owing to the relation to a holomorphic analogue of the Chern–Simons theory, the SDYM theory may be as solvable as 2D rational conformal field theories, and successful nonperturbative quantization may be developed. An algebra acting on the space of self-dual conformal structures on a 4-space (an analogue of the Virasoro algebra) and an algebra acting on the space of self-dual connections (an analogue of affine Lie algebras) are described. Relations to problems of topological and N=2 strings are briefly discussed.

String theory on AdS3

It was shown by Brown and Henneaux that the classical theory of gravity on AdS_3 has an infinite-dimensional symmetry group forming a Virasoro algebra. More recently, Giveon, Kutasov and Seiberg (GKS) constructed the corresponding Virasoro generators in the first-quantized string theory on AdS_3. In this paper, we explore various aspects of string theory on AdS_3 and study the relation between these two works. We show how semi-classical properties of the string theory reproduce many features of the AdS/CFT duality. Furthermore, we examine how the Virasoro symmetry of Brown and Henneaux is realized in string theory, and show how it leads to the Virasoro Ward identities of the boundary CFT. The Virasoro generators of GKS emerge naturally in this analysis. Our work clarifies several aspects of the GKS construction: why the Brown-Henneaux Virasoro algebra can be realized on the first-quantized Hilbert space, to what extent the free-field approximation is valid, and why the Virasoro generators act on the string worldsheet localized near the boundary of AdS_3. On the other hand, we find that the way the central charge of the Virasoro algebra is generated is different from the mechanism proposed by GKS.

Extensions of Conformal Nets¶and Superselection Structures

Starting with a conformal Quantum Field Theory on the real line, we show that the dual net is still conformal with respect to a new representation of the Moebius group. We infer from this that every conformal net is normal and conormal, namely the local von Neumann algebra associated with an interval coincides with its double relative commutant inside the local von Neumann algebra associated with any larger interval. The net and the dual net give together rise to an infinite dimensional symmetry group, of which we study a class of positive energy irreducible representations. We mention how superselsection sectors extend to the dual net and we illustrate by examples how, in general, this process generates solitonic sectors. We describe the free theories associated with the lowest weight n representations of PSL(2,R), showing that they violate 3-regularity for n>2. When n>1, we obtain examples of non Moebius-covariant sectors of a 3-regular (non 4-regular) net.

Algorithmic determination of infinite-dimensional symmetry groups for integrable systems in 1 + 1 dimensions

We present algorithms and describe CA-packages to compute the infinitesimal generators of infinite-dimensional symmetry groups for integrable PDEs (evolution equations) in one space and one time dimension. Here, integrable is meant in the sense that the vector field defining the equation is a member of the abelian part of some infinite-dimensional Virasoro algebra. The method of computation is completely different from the usual prolongation method, no determining equations are solved. Instead, all necessary generators of the finitely generated Virasoro algebra are computed from one given element by direct Lie algebra methods. The implementation of the algorithms in MuPAD is described. A sample session is included in which the recursion structures of the KdV and the Krichever-Novikov equations are computed.

Principle of nongravitating vacuum energy and some of its consequences.

For Einstein's general relativity (GR) or the alternatives suggested up to date, the vacuum energy gravitates. We present a model where a new measure is introduced for integration of the total action in D-dimensional spacetime. This measure is built from D scalar fields ${\mathrm{\ensuremath{\Phi}}}_{\mathit{a}}$. As a consequence of such a choice of the measure, the matter Lagrangian ${\mathit{L}}_{\mathit{m}}$ can be changed by adding a constant while no gravitational effects, such as a cosmological term, are induced. Such a nongravitating vacuum energy theory has an infinite dimensional symmetry group which contains volume-preserving diffeomorphisms in the internal space of scalar fields ${\mathrm{\ensuremath{\Phi}}}_{\mathit{a}}$. Other symmetries contained in this symmetry group suggest a deep connection of this theory with theories of extended objects. In general the theory is different from GR although for certain choices of ${\mathit{L}}_{\mathit{m}}$, which are related to the existence of an additional symmetry, solutions of GR are solutions of the model. This is achieved in four dimensions if ${\mathit{L}}_{\mathit{m}}$ is due to fundamental bosonic and fermionic strings. Other types of matter where this feature of the theory is realized, are, for example, scalars without potential or subjected to nonlinear constraints, massless fermions, and point particles. The point particle plays a special role, since it is a good phenomenological description of matter at large distances. de Sitter space is realized in an unconventional way, where the de Sitter metric holds, but such de Sitter space is supported by the existence of a variable scalar field which in practice destroys the maximal symmetry. The only spacetime where maximal symmetry is not broken, in a dynamical sense, is Minkowski space. The theory has nontrivial dynamics in 1+1 dimensions, unlike GR. \textcopyright{} 1996 The American Physical Society.

QUANTUM NONLOCAL FIELD THEORY: PHYSICS WITHOUT INFINITIES

It is argued that nonlocality is an essential ingredient in Relativistic Quantum Field Theory in order to have finite theory without recourse to a renormalisation program. A critical review of the physical constraints on the form the nonlocality can take is presented. The conclusion of this review is that nonlocality must be restricted to interactions with the vacuum sea of virtual particles. A successful formulation of such a theory, QNFT, is applied to scalar electrodynamics and serves to illustrate how gauge invariance and manifest finiteness can be achieved. The importance of the infinite dimensional symmetry groups that occur in QNFT are discussed as an alternative to supersymmetry, the ability to generate masses by breaking the nonlocal symmetry with a noninvariant functional measure is given a critical assessment. To demonstrate some of the many novel applications QNFT may make possible two examples are mooted, the existence of electroweak monopoles and the formulation of a finite perturbative theory of Quantum Gravity.

W? gravity ? A geometric approach

A brief review is given of an adaptation of the coadjoint orbit method appropriate for the study of models with infinite-dimensional symmetry groups. It is illustrated on several examples, including derivation of the WZNW action of induced D=2 (N,0) supergravity. As the main application, we present the geometric action on a generic coadjoint orbit of the deformed group of area-preserving diffeomorphisms. This action is precisely the anomalous effective WZNW action of D=2 matter fields coupled to a chiralW ∞ gravity background. Similar actions are given which produce the KP hierarchy as on-shell equations of motion.

Infinite dimensional algebras in Chern-Simons quantum mechanics

We study a charged particle in an electromagnetic field without kinetic or potential term. Although dynamically trivial, this system is interesting because it has an infinite dimensional symmetry group. We discuss the way in which this group behaves under quantization. In certain cases we find that it is related to higher spin algebras.

Continuous symmetries of discrete equations

Lie group techniques for solving differential equations are extended to differential-difference equations. As an application, it is shown that the two-dimensional Toda lattice has an infinite dimensional symmetry group with a Kac-Moody-Virasoro Lie algebra.

Analysis and applications of the symmetry group of the multidimensional three-wave resonant interaction problem

The Lie algebra of the symmetry group of the three-wave resonant interaction system in three (or more) space dimensions is shown to be infinite-dimensional and to have the structure of the direct sum of three Kac-Moody-Virasoro u ̂ (1) algebras. The symmetry group is obtained and its one- and two-dimensional subgroups are used to perform symmetry reduction. A wide range of new solutions is obtained. The methods presented in this article are applicable to any system of partial differential equations with an infinite-dimensional symmetry group.