Dual-BRST symmetry: 6D Abelian 3-form gauge theory

Within the framework of the Becchi–Rouet–Stora–Tyutin (BRST) formalism, we demonstrate the existence of the novel off-shell nilpotent (anti-)dual-BRST symmetries in the context of a six (5+1)-dimensional (6D) free Abelian 3-form gauge theory. Under these local and continuous symmetry transformations, the total gauge-fixing term of the Lagrangian density remains invariant. This observation should be contrasted with the off-shell nilpotent (anti-)BRST symmetry transformations, under which, the total kinetic term of the theory remains invariant. The anticommutator of the above nilpotent (anti-)BRST and (anti-)dual-BRST transformations leads to the derivation of a bosonic symmetry in the theory. There exists a discrete symmetry transformation in the theory which provides a thread of connection between the nilpotent (anti-)BRST and (anti-)dual-BRST transformations. This theory is endowed with a ghost-scale symmetry, too. We discuss the algebra of these symmetry transformations and show that the structure of the algebra is reminiscent of the algebra of de Rham cohomological operators of differential geometry.     

The relationship between the gauged BRST symmetry and the ghost number symmetry

A general method of gauging the BRST algebra by combining a (local) ghost number symmetry with the standard (global) BRST algebra, is displayed. This method enables us to construct an automatically nilpotent local BRST algebra and to obtain in a straightforward way the corresponding versions of the action and the eventual anomalies. To illustrate the procedure we study the Yang-Mills and the free bosonic string theories (including the “conformal” Beltrami parametrization) and show that it reproduces the results discussed in the literature. Two major outcomes of this scheme are briefly discussed: a possible connection between ghost number and BRST currents, arising from the Slavnov identities and the implications of the ghost number anomaly for the BRST localization program in theories such as string theory.     

Gauge invariant Lagrangian formulation of higher spin massive bosonic field theory in AdS space

In this work we develop the BRST approach to Lagrangian construction for the massive integer higher spin fields in an arbitrary dimensional AdS space. The theory is formulated in terms of auxiliary Fock space. Closed nonlinear symmetry algebra of higher spin bosonic theory in AdS space is found and a method of deriving the BRST operator for such an algebra is proposed. A general procedure of Lagrangian construction, describing the dynamics of a bosonic field with any spin is given on the base of the BRST operator. No off-shell constraints on the fields and the gauge parameters are used from the very beginning. As an example of general procedure, we derive the Lagrangians for massive bosonic fields with spin 0, 1 and 2, containing the total set of auxiliary fields and gauge symmetries.     

Rigid rotor as a toy model for Hodge theory

We apply the superfield approach to the toy model of a rigid rotor and show the existence of the nilpotent and absolutely anticommuting Becchi–Rouet–Stora–Tyutin (BRST) and anti-BRST symmetry transformations, under which, the kinetic term and the action remain invariant. Furthermore, we also derive the off-shell nilpotent and absolutely anticommuting (anti-) co-BRST symmetry transformations, under which, the gauge-fixing term and the Lagrangian remain invariant. The anticommutator of the above nilpotent symmetry transformations leads to the derivation of a bosonic symmetry transformation, under which, the ghost terms and the action remain invariant. Together, the above transformations (and their corresponding generators) respect an algebra that turns out to be a physical realization of the algebra obeyed by the de Rham cohomological operators of differential geometry. Thus, our present model is a toy model for the Hodge theory.     

Existence,uniqueness and cohomology of the classical BRST charge with ghosts of ghosts

A complete canonical formulation of the BRST theory of systems with redundant gauge symmetries is presented. These systems includep-form gauge fields, the superparticle, and the superstring. We first define the Koszul-Tate differential and explicitly show how the introduction of the momenta conjugate to the ghosts of ghosts makes it acyclic. The global existence of the BRST generator is then demonstrated, and the BRST charge is proved to be unique up to canonical transformations in the extended phase space, which includes the ghosts. Finally, the BRST cohomology in classical mechanics is investigated and shown to be equal to the cohomology of the exterior derivative along the gauge orbits, as in the irreducible case. This is done by re-expressing the exterior algebra along the gauge orbits as a free differential algebra containing generators of higher degree, which are identified with the ghosts of ghosts. The quantum cohomology is not dealt with.Aspirant du Fonds National de la Recherche Scientifique (Belgium)Chercheur qualifié au Fonds National de la Recherche Scientifique (Belgium)     

Local BRST Cohomology and Covariance

The paper provides a framework for a systematic analysis of the local BRST cohomology in a large class of gauge theories. The approach is based on the cohomology of s+d in the jet space of fields and antifields, s and d being the BRST operator and exterior derivative respectively. It relates the BRST cohomology to an underlying gauge covariant algebra and reduces its computation to a compactly formulated problem involving only suitably defined generalized connections and tensor fields. The latter are shown to provide the building blocks of physically relevant quantities such as gauge invariant actions, Noether currents and gauge anomalies, as well as of the equations of motion. Received: 25 July 1996 / Accepted: 23 April 1997     

Superstrings from Hamiltonian reduction

In any string theory there is a hidden, twisted superconformal symmetry algebra, part of which is made up by the BRST current and the anti-ghost. We investigate how this algebra can be systematically constructed for strings with N − 2 supersymmetries, via quantum Hamiltonian reduction of the Lie superalgebras osp(N|2). The motivation is to understand how one could systematically construct generalized string theories from superalgebras. We also briefly discuss the BRST algebra of the topological string, which is a doubly twisted N = 4 superconformal algebra.     

The model for self-dual chiral bosons as a Hodge theory

We consider (1+1) dimensional theory for a single self-dual chiral boson as a classical model for gauge theory. Using the Batalin–Fradkin–Vilkovisky (BFV) technique, the nilpotent BRST and anti-BRST symmetry transformations for this theory have been studied. In this model other forms of nilpotent symmetry transformations like co-BRST and anti-co-BRST, which leave the gauge-fixing part of the action invariant, are also explored. We show that the nilpotent charges for these symmetry transformations satisfy the algebra of the de Rham cohomological operators in differential geometry. The Hodge decomposition theorem on compact manifold is also studied in the context of conserved charges.     

Conformally invariant Yang-Mills theory realizing the Virasoro-Kac-Moody algebra

We will discuss a conformally invariant Yang-Mills theory in two dimensions, which realizes the Virasoro-Kac-Moody algebra as the BRST symmetry. The theory will have the Virasoro anomaly with the central charge −26, but no anomaly for the Kac-Moody algebra.     

On BRST quantization of second class constraint algebras

A BRST quantization of second-class constraint algebras that avoids Dirac brackets is constructed, and the BRST operator is shown to be related to the BRST operator of first class algebra by a nonunitary canonical transformation. The transformation converts the second class algebra into an effective first class algebra with the help of an auxiliary second class algebra constructed from the dynamical Lagrange multipliers of the Dirac approach. The BRST invariant path integral for second class algebras is related to the path integral of the pertinent Dirac brackets, using the Parisi-Sourlas mechaism. As an application the possibility of string theories in subcritical dimensions is considered.     

Noether定理与黑洞的质量公式及黑洞熵

运用并发展了协变相空间的Noether荷方法,对于真空广义相对论稳态轴对称黑洞得到:黑洞质量公式是关于Killing向量场和完整Cauchy面上的零Noether荷以及黑洞力学第一定律.对于一大类向量场,利用零标架方法证明在视界附近的约化代数的中心项为零.这表明,Carlip用纯粹对称性分析的方法来解释黑洞熵的微观起源值得商榷.     

Chetaev型约束的相对运动动力学系统Nielsen方程的Noether对称性与Noether守恒量

研究Chetaev型约束的相对运动动力学系统Nielsen方程的Noether对称性与Noether守恒量. 对Chetaev型约束的相对运动力学系统Nielsen方程的运动微分方程、Noether对称性定义和判据进行具 体的研究, 得到了Noether对称性直接导致的Noether守恒量的表达式. 最后举例说明结果的应用.     

Classification of first-order Lagrangians on the line

We present a complete classification for first-order Lagrangians defined on the line according to the Noether point symmetry algebra they admit. All possible canonical forms of Lagrangians that admit Noether algebras are given.     

N = 1 superstring in 2 + 2 dimensions

In this paper we construct a (2,2) dimensional string theory with manifest N = 1 spacetime supersymmetry. We use Berkovits' approach of augmenting the spacetime supercoordinates by the conjugate momenta for the fermionic variables. The worldsheet symmetry algebra is a twisted and truncated “small” N = 4 superconformal algebra. The realisation of the symmetry algebra is reducible with an infinite order of reducibility. We study the physical states of the theory by two different methods. In one of them, we identify a subset of irreducible constraints, which is by itself critical. We construct the BRST operator for the irreducible constraints, and study the cohomology and interactions. This method breaks the SO(2,2) spacetime symmetry of the original reducible theory. In another approach, we study the theory in a fully covariant manner, which involves the introduction of infinitely many ghosts for ghosts.     

Noether symmetry and non-Noether conserved quantity of the relativistic holonomic nonconservative systems in general Lie transformations

For a relativistic holonomic nonconservative system, by using the Noether symmetry, a new non-Noether conserved quantity is given under general infinitesimal transformations of groups. On the basis of the theory of invariance of differential equations of motion under general infinitesimal transformations, we construct the relativistic Noether symmetry, Lie symmetry and the condition under which the Noether symmetry is a Lie symmetry under general infinitesimal transformations. By using the Noether symmetry, a new relativistic non-Noether conserved quantity is given which only depends on the variables $t$, $q_s $ and $\dot {q}_s $. An example is given to illustrate the application of the results.     

On free 4D Abelian 2-form and anomalous 2D Abelian 1-form gauge theories

We demonstrate a few striking similarities and some glaring differences between (i) the free four- (3+1)-dimensional (4D) Abelian 2-form gauge theory, and (ii) the anomalous two- (1+1)-dimensional (2D) Abelian 1-form gauge theory, within the framework of Becchi–Rouet–Stora–Tyutin (BRST) formalism. We demonstrate that the Lagrangian densities of the above two theories transform in a similar fashion under a set of symmetry transformations even though they are endowed with a drastically different variety of constraint structures. With the help of our understanding of the 4D Abelian 2-form gauge theory, we prove that the gauge-invariant version of the anomalous 2D Abelian 1-form gauge theory is a new field-theoretic model for the Hodge theory where all the de Rham cohomological operators of differential geometry find their physical realizations in the language of proper symmetry transformations. The corresponding conserved charges obey an algebra that is reminiscent of the algebra of the cohomological operators. We briefly comment on the consistency of the 2D anomalous 1-form gauge theory in the language of restrictions on the harmonic state of the (anti-) BRST and (anti-) co-BRST invariant version of the above 2D theory.     

Chetaev型约束的相对运动动力学系统Nielsen方程的Noether对称性与Noether守恒量

研究Chetaev型约束的相对运动动力学系统Nielsen方程的Noether对称性与Noether守恒量. 对Chetaev型约束的相对运动力学系统Nielsen方程的运动微分方程、Noether对称性定义和判据进行具 体的研究, 得到了Noether对称性直接导致的Noether守恒量的表达式. 最后举例说明结果的应用.     

Generalized classical BRST cohomology and reduction of Poisson manifolds

In this paper, we formulate a generalization of the classical BRST construction which applies to the case of the reduction of a Poisson manifold by a submanifold. In the case of symplectic reduction, our procedure generalizes the usual classical BRST construction which only applies to symplectic reduction of a symplectic manifold by a coisotropic submanifold, i.e. the case of reducible first class constraints. In particular, our procedure yields a method to deal with second-class constraints. We construct the BRST complex and compute its cohomology. BRST cohomology vanishes for negative dimension and is isomorphic as a Poisson algebra to the algebra of smooth functions on the reduced Poisson manifold in zero dimension. We then show that in the general case of reduction of Poisson manifolds, BRST cohomology cannot be identified with the cohomology of vertical differential forms.Address after September 1992     

Reducible gauge algebra of BRST-invariant constraints

We show that it is possible to formulate the most general first-class gauge algebra of the operator formalism by only using BRST-invariant constraints. In particular, we extend a previous construction for irreducible gauge algebras to the reducible case. The gauge algebra induces two nilpotent, Grassmann-odd, mutually anti-commuting BRST operators that bear structural similarities with BRST/anti-BRST theories but with shifted ghost number assignments. In both cases we show how the extended BRST algebra can be encoded into an operator master equation. A unitarizing Hamiltonian that respects the two BRST symmetries is constructed with the help of a gauge-fixing boson. Abelian reducible theories are shown explicitly in full detail, while non-Abelian theories are worked out for the lowest reducibility stages and ghost momentum ranks.     

The geometric nature of Lie and Noether symmetries

It is shown that the Lie and the Noether symmetries of the equations of motion of a dynamical system whose equations of motion in a Riemannian space are of the form [(x)\ddot]ⁱ+G_jkⁱ[(x)\dot]^j[(x)\dot] ^k+f(xⁱ)=0{\ddot{x}^{i}+\Gamma_{jk}^{i}\dot{x}^{j}\dot{x} ^{k}+f(x^{i})=0} where f(x ⁱ ) is an arbitrary function of its argument, are generated from the Lie algebra of special projective collineations and the homothetic algebra of the space respectively. Therefore the computation of Lie and Noether symmetries of a given dynamical system in these cases is reduced to the problem of computation of the special projective algebra of the space. It is noted that the Lie and Noether symmetry vectors are common to all dynamical systems moving in the same background space. The selection of the vectors which are Lie/Noether symmetries for a given dynamical system is done by means of a set of differential conditions involving the vectors and the potential function defining the dynamical system. The general results are applied to a number of different applications concerning (a) The motion in Euclidean space under the action of a general central potential (b) The motion in a space of constant curvature (c) The determination of the Lie and the Noether symmetries of class A Bianchi type hypersurface orthogonal spacetimes filled with a scalar field minimally coupled to gravity (d) The analytic computation of the Bianchi I metric when the scalar field has an exponential potential.