Motivated by construction of isospin generators in particle physics (built form the SU(2) algebra), we find an equivalence between the algebra of these generators and those of the Virasoro algebra. The form of the starting generators is fixed and in order to obtain a full equivalence we introduce a new matrix product. The Cartan structure of the starting algebra is reproduced for the Virasoro-like case and a natural BRST quantization as in String Field Theory is ``induced'' in the Fock space of the creation/annihilation operators. Following this procedure, we find a rather trivial Lie algebra form which we obtain a gauge theory of an infinity on non-interacting massless particles of arbitrary integer spin and symmetry. Among others we find the free Maxwell field, the free (linearized) gravitational field and also the axion field with their appropriate gauge transformations.
The BRST approach is applied to the description of irreducible massless higher spins representations of the Poincare group in arbitrary dimensions. The total system of constraints in such theory includes both the first and the second class constraints. The corresponding nilpotent BRST charge contains terms up to the seventh degree in ghosts.
Using BRST-BV formulation of relativistic dynamics, arbitrary spin massless and massive field propagating in flat space and arbitrary spin massless fields propagating in AdS space are considered. For such fields, BRST-BV Lagrangians invariant under gauge transformations are obtained. The Lagrangians and gauge transformations are built in terms of traceless gauge fields and traceless gauge transformation parameters. The use of the Poincare parametrization of AdS space allows us to get simple BRST-BV Lagrangian for AdS fields. By imposing the Siegel gauge condition, we get gauge-fixed Lagrangian…
The detailed description of the method of the construction of the nilpotent BRST charges for nonlinear algebras of constraints appearing in the description of the massless higher spin fields on the $AdS_D$ background is presented. It is shown that the corresponding BRST charge is not uniquely defined, but this ambiguity has no impact on the physical content of the theory.
The basic BRST cohomological properties of a free, massless tensor field with the mixed symmetry of the Riemann tensor are studied in detail. It is shown that any non-trivial co-cycle from the local BRST cohomology group can be taken to stop at antighost number three, its last component belonging to the cohomology of the exterior longitudinal derivative and containing non-trivial elements from the (invariant) characteristic cohomology.
We construct a concise gauge invariant formulation for massless, partially massless, and massive bosonic AdS fields of arbitrary symmetry type at the level of equations of motion. Our formulation admits two equivalent descriptions: in terms of the ambient space and in terms of an appropriate vector bundle, as an explicitly local first-order BRST formalism. The second version is a parent-like formulation that can be used to generate various other formulations via equivalent reductions. In particular, we demonstrate a relation to the unfolded description of massless and partially massless fields…
The local and manifestly covariant Lagrangian interactions in four spacetime dimensions that can be added to a free model that describes a massless Rarita-Schwinger theory and an Abelian BF theory are constructed by means of deforming the solution to the master equation on behalf of specific cohomological techniques.
Using BRST-BV formulation of relativistic dynamics, we study arbitrary spin massive and massless fields propagating in flat space. Generating functions of gauge invariant off-shell cubic interaction vertices for mixed-symmetry and totally symmetric fields are obtained. For the case of totally symmetric fields, we derive restrictions on the allowed values of spins and the number of derivatives, which provide a classification of cubic interaction vertices for such fields. As by product, we present simple expressions for the Yang-Mills and gravitational interactions of massive totally symmetric a…
The main BRST cohomological properties of a free, massless tensor field that transforms in an irreducible representation of GL(D,R), corresponding to a rectangular, two-column Young diagram with k>2 rows are studied in detail. In particular, it is shown that any non-trivial co-cycle from the local BRST cohomology group H(s|d) can be taken to stop either at antighost number (k+1) or k, its last component belonging to the cohomology of the exterior longitudinal derivative H(gamma) and containing non-trivial elements from the (invariant) characteristic cohomology H^{inv}(delta|d).
Totally symmetric arbitrary spin massless and massive fields in AdS space are studied. For such fields, we obtain Lagrangians which are invariant under global BRST transformations. The Lagrangians are used for computation of partition functions and effective actions. We demonstrate that BRST invariant bulk action for massless field evaluated on the solution of Dirichlet problem for gauge massless fields and Faddeev-Popov fields leads to BRST invariant effective action for canonical shadow gauge fields and shadow Faddeev-Popov fields, while the BRST invariant bulk action for massive field evalu…
We develop an approach to constructing the manifestly Lorentz covariant cubic interaction vertices for the four-dimensional massless higher spin bosonic fields with two-component dotted and undotted spinor indices. Such fields automatically satisfy the traceless conditions what simplify form of the equations determining the irreducible massless representation of the Poincar'e group with given helicity. The cubic vertex is formulated in the framework of the BRST approach to higher spin field theory. Use of the above spin-tensor fields allows to simplify a form of the BRST-charge and hence to fi…
On BRST Lagrangian description of partially massless bosonic fields
Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research（布洛赫伯理论物理实验室，核联合研究所）；Department of Theoretical Physics, Tomsk State Pedagogical University（托姆斯克国立师范学院理论物理系）；Tomsk Polytechnic University（托姆斯克理工大学）
We investigate in detail the critical $N{=}2$ fermionic string with and without a global ${\bf Z}_2$ twist. An analysis of BRST cohomology shows that twisted sectors contain massless `spacetime' fermions which are {\it non-local\/} with respect to the standard massless boson. However, two distinct GSO projections exist, one (untwisted) retaining merely the usual boson and its spectral-flow partner, the other (twisted) yielding two fermions and one boson, on the massless level. The corresponding chiral BRST-invariant vertex operators are constructed in certain pictures, and their fusion and pic…
We develop the BRST approach to Lagrangian formulation for all massless half-integer higher spin fields on an arbitrary dimensional flat space. General procedure of Lagrangian construction describing the dynamics of fermionic field with any spin is given. It is shown that in fermionic case the higher spin field model is a reducible gauge theory and the order of reducibility grows with the value of spin. No off-shell constraints on the fields and the gauge parameters are used. We prove that in four dimensions after partial gauge fixing the Lagrangian obtained can be transformed to Fang-Fronsdal…
We consider the covariant quantization of the D=11 massless superparticle (M0-brane) in the spinor moving frame or twistor-like Lorentz harmonics formulation. The action involves the set of 16 constrained 32 component Majorana spinors, the spinor Lorentz harmonics parametrizing (as homogeneous coordinates, modulo gauge symmetries) the celestial sphere S9. There presence allows us to separate covariantly the first and the second class constraints of the model. After taking into account the second class constraints by means of Dirac brackets and after further reducing the first class constraints…
The worldsheet sigma-model of the superstring in AdS(5) x S(5) has a one-parameter family of flat connections parametrized by the spectral parameter. The corresponding Wilson line is not BRST invariant for an open contour, because the BRST transformation leads to boundary terms. These boundary terms define a cohomological complex associated to the endpoint of the contour. We study the cohomology of this complex for Wilson lines in some infinite-dimensional representations. We find that for these representations the cohomology is nontrivial at the ghost number 2. This implies that it is possibl…
We obtain the higher spin tractor equations of motion conjectured by Gover et al. from a BRST approach and use those methods to prove that they describe massive, partially massless and massless higher spins in conformally flat backgrounds. The tractor description makes invariance under local choices of unit system manifest. In this approach, physical systems are described by conformal, rather than (pseudo-)Riemannian geometry. In particular masses become geometric quantities, namely the weights of tractor fields. Massive systems can therefore be handled in a unified and simple manner mimicking…
We apply the BRST approach, previously developed for higher spin field theories, to gauge invariant Lagrangian construction for antisymmetric massive and massless bosonic fields in arbitrary d-dimensional curved space. The obtained theories are reducible gauge models both in massless and massive cases and the order of reducibility grows with the value of the rank of the antisymmetric field. In both the cases the Lagrangians contain the sets of auxiliary fields and possess more rich gauge symmetry in comparison with standard Lagrangian formulation for the antisymmetric fields. This serves addit…
We study some features of bosonic particle path-integral quantization in a twistor-like approach by use of the BRST-BFV quantization prescription. In the course of the Hamiltonian analysis we observe links between various formulations of the twistor-like particle by performing a conversion of the Hamiltonian constraints of one formulation to another. A particular feature of the conversion procedure applied to turn the second-class constraints into the first-class constraints is that the simplest Lorentz-covariant way to do this is to convert a full mixed set of the initial first- and second-cl…
We develop the BRST approach to Lagrangian formulation for massive bosonic and massless fermionic higher spin fields on a flat space-time of arbitrary dimension. General procedure of gauge invariant Lagrangian construction describing the dynamics of the fields with any spin is given. No off-shell constraints on the fields (like tracelessness) and the gauge parameters are imposed. The procedure is based on construction of new representations for the closed algebras generated by the constraints defining irreducible representations of the Poincare group. We also construct Lagrangians describing p…
Using BRST-BV approach, massless and massive continuous-spin fields propagating in the flat space are studied. For such fields, BRST-BV gauge invariant Lagrangian is obtained. The Lagrangian and gauge transformations are constructed out of traceless gauge fields and traceless gauge transformation parameters. Interrelation between the BRST-BV Lagrangian and the Lagrangian for the continuous-spin fields in metric-like approach is demonstrated. Considering the BRST-BV Lagrangian in the Siegel gauge, we get gauge-fixed Lagrangian which is invariant under global BRST and antiBRST transformations.
We briefly review some modern developments in higher spin field theory and their links with superstring theory. The analysis is based on various BRST constructions allowing to derive the Lagrangians for massive and massless higher spin fields on flat or constant curvature backgrounds of arbitrary dimensions.
We describe the procedure of dimensional reduction of massless fields in $(D+1)$ dimensional Minkowski space to massive ones in $D$ dimensions in the first-quantized setting. The procedure is compatible with Lagrangian and in a straightforward way determines the inner product for massive fields. Howe duality and the extensive usage of BRST formalism allows to keep the description concise. We consider both bosonic and fermionic mixed-symmetry fields.
The lagrangian description of irreducible massless representations of the Poincare group with the corresponding Young tableaux having two rows along with some explicit examples including the notoph and Weyl tensor is given. For this purpose is used the method of the BRST constructions adopted to the systems of second class constraints by the construction of an auxiliary representations of the algebras of constraints in terms of Verma modules.
In this short note we discuss N-supersymmetric worldlines of relativistic massless particles and review the known result that physical spin-N/2 fields are in the first BRST cohomology group. For N=1,2,4 emphasis is given to particular deformations of the BRST differential, that implement either a covariant derivative for a gauge theory or a metric connection in the target space seen by the particle. In the end, we comment about the possibility of incorporating Ramond-Ramond fluxes in the background.
We construct vertex operators for massless higher spin fields in RNS superstring theory and compute some of their three-point correlators, describing gauge-invariant cubic interactions of the massless higher spins. The Fierz-Pauli on-shell conditions for the higher spins (including tracelessness and vanishing divergence) follow from the BRST-invariance conditions for the vertex operators constructed in this paper. The gauge symmetries of the massless higher spins emerge as a result of the BRST nontriviality conditions for these operators, being equivalent to transformations with the traceless …
The Becchi-Rouet-Stora-Tyutin (BRST) ghost field is a key element in constructing Witten's cubic open string field theory. However, to date, the ghost sector of the string field theory has not received a great deal of attention. In this study, we address the BRST ghost on multiple $Dp$-branes, which carries non-Abelian indices and couples to a non-Ablelian gauge field. We found that the massless components of the BRST ghost field can play the role of the Faddeev-Popov ghost in the non-Alelian gauge field, such that the string field theory maintains the local non-Abelian gauge invariance.
A recent progress in a gauge-invariant Lagrangian description of massive and massless half-integer higher-spin fields in AdS and Minkowski spaces is presented. The procedure is based on a BFV-BRST operator, encoding the initial conditions realized by constraints in a Fock space and extracting the higher-spin fields from unitary irreducible representations of the AdS (Poincare) group. The construction is applicable to higher-spin tensor fields with a multi-row Young tableaux.
We consider the massless scalar field on the four-dimensional sphere $S^4$. Its classical action $S={1\over 2}\int_{S^4} dV (\nabla ϕ)^2$ is degenerate under the global invariance $ϕ\to ϕ+ \hbox{constant}$. We then quantize the massless scalar field as a gauge theory by constructing a BRST-invariant quantum action. The corresponding gauge-breaking term is a non-local one of the form $S^{\rm GB}={1\over {2αV}}\bigl(\int_{S^4} dV ϕ\bigr)^2$ where $α$ is a gauge parameter and $V$ is the volume of $S^4$. It allows us to correctly treat the zero mode problem. The quantum theory is invariant under S…
We explore a new possibility of BRST construction in higher spin field theory to obtain a consistent Lagrangian for massive spin-2 field in Einstein space. Such approach automatically leads to gauge invariant Lagrangian with suitable auxiliary and Stuckelberg fields. It is proved that in this case a propagation of spin-2 field is hyperbolic and causal. Also we extend notion of partial masslessness for spin-2 field in the background under consideration.
Firstly, we study the state space of a massless particle on a supergroup with a reparameterization invariant action. After gauge fixing the reparameterization invariance, we compute the physical state space through the BRST cohomology and show that the quadratic Casimir Hamiltonian becomes diagonalizable in cohomology. We illustrate the general mechanism in detail in the example of a supergroup target GL(1|1). The space of physical states remains an indecomposable infinite dimensional representation of the space-time supersymmetry algebra. Secondly, we show how the full string BRST cohomology …
The key aspects of a gauge-invariant Lagrangian description of massive and massless half-integer higher-spin fields in AdS spaces with a two-row Young tableaux $Y(s_1,s_2)$ are presented in an unconstrained description, as well as in off-shell formulations with algebraic constraints, on the basis of BFV-BRST operators for non-linear operator superalgebras, encoding the initial conditions realized by constraints in a Fock space and extracting the higher-spin fields from unitary representations of the AdS group.
We present a new formulation of the tensionless string ($T= 0$) where the space-time conformal symmetry is manifest. Using a Hamiltonian BRST scheme we quantize this {\em Conformal String} and find that it has critical dimension $D=2$. This is in keeping with our classical result that the model describes massless particles in this dimension. It is also consistent with our previous results which indicate that quantized conformally symmetric tensionless strings describe a topological phase away {}from $D=2$. We reach our result by demanding nilpotency of the BRST charge and consistency with the …
We develop the BRST approach for all massless integer and half-integer higher spins in 4D Minkowski space, using the two component spinor nota- tion and develop the Lagrangian formulation for supersymmetric higher spin models. It is shown that the problem of second class constraints disappears and the BRST procedure becomes much more simple than in tensorial nota- tion. Furthermore, we demonstrate that the BRST procedure automatically provides extra auxiliary components that belong in the set of supersymmetry auxiliary components. Finally, we demonstrate how supersymmetry transfo- rmations are…
A gauge-fixing procedure for the Yang-Mills theory on an n-dimensional sphere (or a hypersphere) is discussed in a systematic manner. We claim that Adler's gauge-fixing condition used in massless Euclidean QED on a hypersphere is not conventional because of the presence of an extra free index, and hence is unfavorable for the gauge-fixing procedure based on the BRST invariance principle (or simply BRST gauge-fixing procedure). Choosing a suitable gauge condition, which is proved to be equivalent to a generalization of Adler's condition, we apply the BRST gauge-fixing procedure to the Yang-Mill…
Massless spinning particle and tensionless string models on $AdS_d$ background in the projective-space realization are proposed as constrained Hamiltonian systems. Various forms of particle and string Lagrangians are derived and classical mechanics is studied including the Lax-type representation of the equations of motion. After that transition to the quantum theory is discussed. Analysis of potential anomalies in the tensionless string model necessitates introduction of ghosts and BRST charge. It is shown that quantum BRST charge is nilpotent for any $d$ if coordinate-momentum ordering for t…
Consistent couplings between a massless tensor field with the mixed symmetry of the Riemann tensor and a massless vector field are analyzed in the framework of Lagrangian BRST cohomology. Under the assumptions on smoothness, locality, Lorentz covariance, and Poincare invariance of the deformations, combined with the requirement that the interacting Lagrangian is at most second-order derivative, it is proved that there are no consistent cross-interactions between a single massless tensor field with the mixed symmetry of the Riemann tensor and one massless vector field.
The spectrum of superstring theory on the $AdS_5 \times S_5$ Ramond-Ramond background in tensionless limit contains integer and half-integer higher-spin fields subject at most to two-rows Young tableaux $Y(s_1,s_2)$. We review the details of a gauge-invariant Lagrangian description of such massive and massless higher-spin fields in anti-de-Sitter spaces with arbitrary dimensions. The procedure is based on the construction of Verma modules, its oscillator realizations and of a BFV-BRST operator for non-linear algebras encoding unitary irreducible representations of AdS group.
To exhibit the possible origin of the inner complexity of the Berkovits's pure spinor approach, we consider the covariant BRST quantization of the D=11 massless superparticle (M0-brane) in its spinor moving frame or twistor-like Lorentz harmonics formulation. The presence of additional twistor-like variables (spinor harmonics) allows us to separate covariantly the first and the second class constraints. After taking into account the second class constraints by means of Dirac brackets and after further reducing the first class constraints algebra, the dynamical system is described by the cohomo…
We discuss the (dual-)gauge transformations for the gauge-fixed Lagrangian density and establish their intimate connection with the translation subgroup T(2) of the Wigner's little group for the free one-form Abelian gauge theory in four $(3 + 1)$-dimensions (4D) of spacetime. Though the relationship between the usual gauge transformation for the Abelian massless gauge field and T(2) subgroup of the little group is quite well-known, such a connection between the dual-gauge transformation and the little group is a new observation. The above connections are further elaborated and demonstrated in…
We study the phase diagram of two-flavor massless two-color QCD (QC$_2$D) under the presence of quark chemical potentials and imaginary isospin chemical potentials. At the special point of the imaginary isospin chemical potential, called the isospin Roberge--Weiss (RW) point, two-flavor QC$_2$D enjoys the $\mathbb{Z}_2$ center symmetry that acts on both quark flavors and the Polyakov loop. We find a $\mathbb{Z}_2$ 't Hooft anomaly of this system, which involves the $\mathbb{Z}_2$ center symmetry, the baryon-number symmetry, and the isospin chiral symmetry. Anomaly matching, therefore, constrai…
We perform the quantization of a massive particle propagating on AdS_5. We use the twistor formulation in which the action can be brought into a quadratic form. We construct the BRST operator which commutes with AdS_5 isometries forming SU(2,2). The condition of a consistent BRST quantization requires that the AdS energy E is quantized in units of the AdS_5 radius R, E=\frac{1}{2R}(N_a +N_b+4), with N_a, N_b being some non-negative integers. We also argue that the mass operator will be identified with the moduli of the U(1) central extension Z of the SU(2,2|4) algebra in the supersymmetric cas…
To avoid the problems which are connected with the long distance behavior of perturbative gauge theories we present a local construction of the observables which does not involve the adiabatic limit. First we construct the interacting fields as formal power series by means of causal perturbation theory. The observables are defined by BRST invariance where the BRST-transformation $\tilde s$ acts as a graded derivation on the algebra of interacting fields. Positivity, i.e. the existence of Hilbert space representations of the local algebras of observables is shown with the help of a local Kugo-O…
The structure of a cubic Lagrangian vertex is clarified for irreducible fields of helicities $s_1, s_2, s_3$ in a $d$-dimensional Minkowski space. An explicit form of the operator $\mathcal{Z}_j$ entering the vertex in a non-multiplicative way (examined in arXiv:2105.12030[hep-th] for $j=1$) is obtained. The solution is found within the BRST approach with complete BRST operators, which contain all constraints corresponding to the conditions that extract the irreducible fields, including trace operators.
The proof of the fact that there are no nontrivial, consistent cross-couplings that can be added between the Weyl graviton and the massless Rarita-Schwinger field is accomplished by means of a cohomological approach, based on the deformation of the solution to the master equation from the antifield-Becchi-Rouet-Stora-Tyutin (BRST) formalism. The procedure developed here relies on the assumptions of locality, smoothness, (background) Lorentz invariance, Poincare invariance, and preservation of the number of derivatives with respect to each field (the last hypothesis was made only in antighost n…
We construct the Lagrangian description of arbitrary integer higher spin massless fields on the background of the D-dimensional anti-de Sitter space. The operator constraints in auxiliary Fock space corresponding to subsidiary conditions for irreducible unitary massless representations of the D-dimensional anti-de Sitter group are formulated. Unlike flat space, the algebra of the constraints turns out to be nonlinear and analogous to the $W_3$ algebra. We construct the nilpotent BRST charge for this nonlinear algebra and derive on its basis the correct field content and gauge invariant action …
We extend the QCD Lagrangian to include Pauli-Villars (PV) gluons, quarks, and ghosts in such a way as to retain BRST invariance in an arbitrary covariant gauge. The extended Lagrangian can provide a starting point for nonperturbative calculations in QCD, particularly with light-front techniques, and the methods used to construct it may be useful for perturbative calculations in theories where dimensional regularization is not viable. The regularization is arranged by having all interaction terms in the Lagrangian be couplings between null fields, specific combinations of positive and negative…
We investigate the problem of introducing consistent self-couplings in free theories for mixed tensor gauge fields whose symmetry properties are characterized by Young diagrams made of two columns of arbitrary (but different) lengths. We prove that, in flat space, these theories admit no local, Poincaré-invariant, smooth, self-interacting deformation with at most two derivatives in the Lagrangian. Relaxing the derivative and Lorentz-invariance assumptions, there still is no deformation that modifies the gauge algebra, and in most cases no deformation that alters the gauge transformations.Our a…
We consider a massless higher spin field theory within the BRST approach and construct a general off-shell cubic vertex corresponding to irreducible higher spin fields of helicities $s_1, s_2, s_3$. Unlike the previous works on cubic vertices, which do not take into account of the trace constraints, we use the complete BRST operator, including the trace constraints that describe an irreducible representation with definite integer helicity. As a result, we generalize the cubic vertex found in [arXiv:1205.3131 [hep-th]] and calculate the new contributions to the vertex, which contain additional …
Cross-couplings between a massless spin-two field (described in the free limit by the Pauli-Fierz action) and an Abelian three-form gauge field in D=11 are investigated in the framework of the deformation theory based on local BRST cohomology. These consistent interactions are obtained on the grounds of smoothness in the coupling constant, locality, Lorentz covariance, Poincare invariance, and the presence of at most two derivatives in the interacting Lagrangian. Our results confirm the uniqueness of the eleven-dimensional interactions between a graviton and a three-form prescribed by General …
We report a no-go theorem excluding consistent cross-couplings for a collection of massless, spin-2 fields described, in the free limit, by the sum of Pauli-Fierz actions (one for each field). We show that, in spacetime dimensions >2, there is no consistent coupling, with at most two derivatives of the fields, that can mix the various "gravitons". The only possible deformations are given by the sum of individual Einstein-Hilbert actions (one for each field) with cosmological terms. Our approach is based on the BRST-based deformation point of view.
