Infinite-dimensional Symmetry Group Research Articles

Infinite Dimensional Symmetry Group Reductions and Conservation Laws of Lin–Reissner–Tsien Equation

Infinite Dimensional Symmetry Group Reductions and Conservation Laws of Lin–Reissner–Tsien Equation

Painlevé analysis, infinite dimensional symmetry group and symmetry reductions for the (2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani equation

The (2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani (KdVSKR) equation is studied by the singularity structure analysis. It is proven that it admits the Painlevé property. The Lie algebras which depend on three arbitrary functions of time t are obtained by the Lie point symmetry method. It is shown that the KdVSKR equation possesses an infinite-dimensional Kac–Moody–Virasoro symmetry algebra. By selecting first-order polynomials in t, a finite-dimensional subalgebra of physical transformations is studied. The commutation relations of the subalgebra, which have been established by selecting the Laurent polynomials in t, are calculated. This symmetry constitutes a centerless Virasoro algebra which has been widely used in the field of physics. Meanwhile, the similarity reduction solutions of the model are studied by means of the Lie point symmetry theory.

Infinite-dimensional symmetry group, Kac–Moody–Virasoro algebras and integrability of Kac–Wakimoto equation

An eighth-order equation in $$(3+1)$$ dimension is studied for its integrability. Its symmetry group is shown to be infinite-dimensional and is checked for Virasoro-like structure. The equation is shown to have no Painlev $$\acute{\mathrm{e}}$$ property. One- and two-dimensional classifications of infinite-dimensional symmetry algebra are also given.

E 9 exceptional field theory. Part II. The complete dynamics

We construct the first complete exceptional field theory that is based on an infinite-dimensional symmetry group. E9 exceptional field theory provides a unified description of eleven-dimensional and type IIB supergravity covariant under the affine Kac-Moody symmetry of two-dimensional maximal supergravity. We present two equivalent formulations of the dynamics, which both rely on a pseudo-Lagrangian supplemented by a twisted self-duality equation. One formulation involves a minimal set of fields and gauge symmetries, which uniquely determine the entire dynamics. The other formulation extends {mathfrak{e}}_9 by half of the Virasoro algebra and makes direct contact with the integrable structure of two-dimensional supergravity. Our results apply directly to other affine Kac-Moody groups, such as the Geroch group of general relativity.

Swing surfaces and holographic entanglement beyond AdS/CFT

We propose a holographic entanglement entropy prescription for general states and regions in two models of holography beyond AdS/CFT known as flat3/BMSFT and (W)AdS3/WCFT. Flat3/BMSFT is a candidate of holography for asymptotically flat three- dimensional spacetimes, while (W)AdS3/WCFT is relevant in the study of black holes in the real world. In particular, the boundary theories are examples of quantum field theories that feature an infinite dimensional symmetry group but break Lorentz invariance. Our holographic entanglement entropy proposal is given by the area of a swing surface that consists of ropes, which are null geodesics emanating from the entangling surface at the boundary, and a bench, which is a spacelike geodesic connecting the ropes. The proposal is supported by an extension of the Lewkowycz-Maldacena argument, reproduces previous results based on the Rindler method, and satisfies the first law of entanglement entropy.

Infinite dimensional symmetry groups of the Friedmann equations

We find the symmetry generators for the Friedman equations emanating from a perfect fluid source, in the presence of a cosmological constant term. The relevant dynamics is seen to be governed by two coupled, first order ordinary differential equations, the continuity and the quadratic constraint equation. Arbitrary functions appear in the components of the symmetry vector, indicating the infinity of the group. When the equation of state is considered as arbitrary but ab initio given, previously known results are recovered and/or generalized. When the pressure is considered among the dynamical variables, solutions for models with different equations of state are mapped to each other; thus enabling the presentation of solutions to models with complicated equations of state starting from simple known cases.

Large gauge symmetries of an asymptotically de Sitter horizon: An extended first law of thermodynamics

In this paper, we show that a universe with a dynamical cosmological constant approaching pure de Sitter at timelike infinity, enjoys an infinite dimensional symmetry group at its horizon. This group is larger than the usual $SO(4,1)$ of pure de Sitter. The charges associated with the asymptotic symmetry generators are non-integrable, and we demonstrate that they promote an extended version of the first law of thermodynamics. This contains four pairs of conjugate variables. The pair $(\Theta,\Lambda)$ corresponding to the change in the cosmological constant and its conjugate volume $\Theta$. The contribution of the surface tension of the horizon and its conjugate parameter surface area make a pair $(\sigma, A)$. The usual conjugate variables $ (T, S) $, $ (\Omega, J) $ and a term $ \partial_v \delta S $ corresponding to entropy production, are included. In addition, this extended first law describes the non-conservative behaviour of the asymptotic charges in non-equilibrium.

Asymptotic symmetries and charges at null infinity: from low to high spins

Weinberg’s celebrated factorisation theorem holds for soft quanta of arbitrary integer spin. The same result, for spin one and two, has been rederived assuming that the infinite-dimensional asymptotic symmetry group of Maxwell’s equations and of asymptotically flat spaces leave the S-matrix invariant. For higher spins, on the other hand, no such infinite-dimensional asymptotic symmetries were known and, correspondingly, no a priori derivation of Weinberg’s theorem could be conjectured. In this contribution we review the identification of higher-spin supertranslations and superrotations in D = 4 as well as their connection to Weinberg’s result. While the procedure we follow can be shown to be consistent in any D, no infinite-dimensional enhancement of the asymptotic symmetry group emerges from it in D > 4, thus leaving a number of questions unanswered.

Linearizability for third order evolution equations

The problem of linearization for third order evolution equations is considered. Criteria for testing equations for linearity are presented. A class of linearizable equations depending on arbitrary functions is obtained by requiring presence of an infinite-dimensional symmetry group. Linearizing transformations for this class are found using symmetry structure and local conservation laws. A number of special cases as examples are discussed. Their transformation to equations within the same class by differential substitutions and connection with KdV and mKdV equations is also reviewed in this framework.

Structure Preserving Discretizations of the Liouville Equation and their Numerical Tests

The main purpose of this article is to show how symmetry structures in partial differential equations can be preserved in a discrete world and reflected in difference schemes. Three different structure preserving discretizations of the Liouville equation are presented and then used to solve specific boundary value problems. The results are compared with exact solutions satisfying the same boundary conditions. All three discretizations are on four point lattices. One preserves linearizability of the equation, another the infinite-dimensional symmetry group as higher symmetries, the third one preserves the maximal finite-dimensional subgroup of the symmetry group as point symmetries. A 9-point invariant scheme that gives a better approximation of the equation, but significantly worse numerical results for solutions is presented and discussed.

New symmetries of QED

The soft photon theorem in U(1) gauge theories with only massless charged particles has recently been shown to be the Ward identity of an infinite-dimensional asymptotic symmetry group. This symmetry group is comprised of gauge transformations which approach angle-dependent constants at null infinity. In this paper, we extend the analysis to all U(1) theories, including those with massive charged particles such as QED.

Invariant discretization of partial differential equations admitting infinite-dimensional symmetry groups

This paper is concerned with the invariant discretization of differential equations admitting infinite-dimensional symmetry groups. By way of example, we first show that there are differential equations with infinite-dimensional symmetry groups that do not admit enough joint invariants preventing the construction of invariant finite difference approximations. To solve this shortage of joint invariants we propose to discretize the pseudo-group action. Computer simulations indicate that the numerical schemes constructed from the joint invariants of discretized pseudo-group can produce better numerical results than standard schemes.

EASY HOLOGRAPHY

It is argued that the role of infinite-dimensional asymptotic symmetry groups in gravity theories are essential for a holographic description of gravity and possibly to a resolution of the black hole information paradox. I present a simple toy model in two-dimensional hyperbolic/anti-de Sitter (AdS) space and describe, by very elementary considerations, how the asymptotic symmetry group is responsible for the entropy area law. Similar results apply also in three-dimensional AdS space. The failure of the approach in higher-dimensional AdS spaces is explained and resolved by considering other asymptotically noncompact homogeneous spaces.

Identities from infinite-dimensional symmetries of Herglotz variational functional

This paper formulates and proves a theorem which provides the identities corresponding to infinite-dimensional symmetry groups of the functional defined by the generalized variational principle of Herglotz in the case of several independent variables. It contains the classical second Noether theorem as a special case. The equations satisfied by the extrema of this functional, when the Lagrangian density depends on first and second order partial derivatives of the argument functions, are found. We apply the theorem to find two new identities satisfied by the four-potential of the electromagnetic field propagating in a conductive medium. We also obtain an identity corresponding to a gauge symmetry of the Klein-Gordon equation with dissipation/generation. This identity becomes a continuity law when the wave function is a solution of this equation.

SO(9) supergravity in two dimensions

Abstract We present maximal supergravity in two dimensions with gauge group SO(9). The construction is based on selecting the proper embedding of the gauge group into the infinite-dimensional symmetry group of the ungauged theory. The bosonic part of the Lagrangian is given by a (dilaton-)gravity coupled non-linear gauged σ-model with Wess-Zumino term. We give explicit expressions for the fermionic sector, the Yukawa couplings and the scalar potential which supports a half-supersymmetric domain wall solution. The theory is expected to describe the low-energy effective action upon reduction on the D0-brane near-horizon warped AdS 2 ×S 8 geometry, dual to the supersymmetric (BFSS) matrix quantum mechanics.

Superposition of the Nadai and prandtl solutions for the system of two-dimensional ideal plasticity

A general algorithm is presented for transforming the exact solutions of the system of plane ideal plasticity of the Mises medium by using the superposition principle for solutions which arises as a corollary of the fact that the original system admits an infinite-dimensional symmetry group. As an example, there is considered the relation between the known exact solutions: the Prandtl solution for a thin layer compressed by rough solid plates and the Nadai solution for the radial distribution of stresses in a convergent channel in the shape of a flat wedge.

Localization and the interface between quantum mechanics, quantum field theory and quantum gravity II: The search of the interface between QFT and QG

The main topics of this second part of a two-part essay are some consequences of the phenomenon of vacuum polarization as the most important physical manifestation of modular localization. Besides philosophically unexpected consequences, it has led to a new constructive “outside-inwards approach” in which the pointlike fields and the compactly localized operator algebras which they generate only appear from intersecting much simpler algebras localized in noncompact wedge regions whose generators have extremely mild almost free field behavior. Another consequence of vacuum polarization presented in this essay is the localization entropy near a causal horizon which follows a logarithmically modified area law in which a dimensionless area (the area divided by the square of dR where dR is the thickness of a light-sheet) appears. There are arguments that this logarithmically modified area law corresponds to the volume law of the standard heat bath thermal behavior. We also explain the symmetry enhancing effect of holographic projections onto the causal horizon of a region and show that the resulting infinite dimensional symmetry groups contain the Bondi–Metzner–Sachs group. This essay is the second part of a partitioned longer paper.

Reduction of exterior differential systems with infinite dimensional symmetry groups

A symmetry-based method for constructing solutions to systems of differential equations founded on the reduction of exterior differential systems invariant under the action of an infinite dimensional pseudogroup is proposed. One can associate to any system of differential equations Δ=0 with a symmetry group \(\mathcal{G}\) an exterior differential system \(\mathcal{I}\) invariant under \(\mathcal{G}\) so that solutions of Δ=0 correspond to integral manifolds \(\mathcal{I}\). The \(\mathcal{G}\)-invariant exterior differential system gives rise to a reduced system \(\overline{\mathcal{I}}\) specified on a cross section to the pseudogroup orbits, and it is shown that solutions to Δ=0 can be reconstructed from integral manifolds \(\overline{\mathcal{I}}\) by solving an equation of generalized Lie type for the jets of pseudogroup transformations. In particular, as opposed to the classical method of symmetry reduction, every solution to the system of differential equations can, under some mild regularity assumptions, be constructed by the present algorithm.

New infinite-dimensional multiple symmetry groups for the EMDA theory with two vector fields

The symmetry structures of stationary axisymmetric Einstein-Maxwell-Dilaton-Axion theory with 2 vector fields (EMDA-2 theory, for brevity) 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. Two pairs of ED RH transformations are constructed and they are verified to give infinite-dimensional multiple symmetry groups of the EMDA-2 theory, each of these symmetry groups has the structure of semidirect product of Kac-Moody group S U ( 2 , 2 ) ^ and Virasoro group. Moreover, the infinitesimal forms of the given ED RH transformations are calculated out, and these infinitesimal RH transformations give exactly the same infinitesimal symmetry transformations of the EMDA-2 theory as constructed in one of our previous papers. These results demonstrate that the two pairs of ED RH transformations in this paper provide us with exponentiations of all the infinitesimal symmetries constructed 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.

Lie point symmetry and some new soliton-like solutions of the Konopelchenko–Dubrovsky equations

In this paper, the infinite-dimensional symmetry groups of the ( 2 + 1 ) -dimensional Konopelchenko–Dubrovsky (KD) equation is found by the classical Lie group method. The symmetry groups are used to perform symmetry reduction and together with tanh function method we obtain soliton-like solutions. Moreover, a characterization of the corresponding Lie algebra is given.