Gauge invariant Lagrangian construction for massive bosonic higher spin fields in D dimensions

We develop the BRST approach to Lagrangian formulation for massive higher integer spin fields on a flat space–time of arbitrary dimension. General procedure of gauge invariant Lagrangian construction describing the dynamics of massive bosonic field 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 representation for the closed algebra generated by the constraints defining an irreducible massive bosonic representation of the Poincaré group. We also construct Lagrangian describing propagation of all massive bosonic fields simultaneously. As an example of the general procedure, we derive the Lagrangians for spin-1, spin-2 and spin-3 fields containing total set of auxiliary fields and gauge symmetries of free massive bosonic higher spin field theory.     

Gauge invariant Lagrangian construction for massive higher spin fermionic fields

We formulate a general gauge invariant Lagrangian construction describing the dynamics of massive higher spin fermionic fields in arbitrary dimensions. Treating the conditions determining the irreducible representations of Poincaré group with given spin as the operator constraints in auxiliary Fock space, we built the BRST charge for the model under consideration and find the gauge invariant equations of motion in terms of vectors and operators in the Fock space. It is shown that like in massless case [I.L. Buchbinder, V.A. Krykhtin, A. Pashnev, Nucl. Phys. B 711 (2005) 367, hep-th/0410215], the massive fermionic higher spin field models are the reducible gauge theories and the order of reducibility grows with the value of spin. In compare with all previous approaches, no off-shell constraints on the fields and the gauge parameters are imposed from the very beginning, all correct constraints emerge automatically as the consequences of the equations of motion. As an example, we derive a gauge invariant Lagrangian for massive spin 3/2 field.     

On Lagrangian formulations for arbitrary bosonic HS fields on Minkowski backgrounds

We review the details of unconstrained Lagrangian formulations for Bose particles propagated on an arbitrary dimensional flat space-time and described by the unitary irreducible integer higher-spin representations of the Poincare group subject to Young tableaux Y(s ₁, ..., s _k ) with k rows. The procedure is based on the construction of scalar auxiliary oscillator realizations for the symplectic sp(2k) algebra which encodes the second-class operator constraints subsystem in the HS symmetry algebra. Application of an universal BRST approach reproduces gauge-invariant Lagrangians with reducible gauge symmetries describing the free dynamics of both massless and massive bosonic fields of any spin with appropriate number of auxiliary fields.     

CFT adapted gauge invariant formulation of arbitrary spin fields in AdS and modified de Donder gauge

Using Poincaré parametrization of AdS space, we study totally symmetric arbitrary spin massless fields in AdS space of dimension greater than or equal to four. CFT adapted gauge invariant formulation for such fields is developed. Gauge symmetries are realized similarly to the ones of Stueckelberg formulation of massive fields. We demonstrate that the curvature and radial coordinate contributions to the gauge transformation and Lagrangian of the AdS fields can be expressed in terms of ladder operators. Realization of the global AdS symmetries in the conformal algebra basis is obtained. Modified de Donder gauge leading to simple gauge fixed Lagrangian is found. The modified de Donder gauge leads to decoupled equations of motion which can easily be solved in terms of the Bessel function. Interrelations between our approach to the massless AdS fields and the Stueckelberg approach to massive fields in flat space are discussed.     

Loop variables and gauge invariant exact renormalization group equations for (open) string theory

An exact renormalization group equation is written down for the world sheet theory describing the bosonic open string in general backgrounds. Loop variable techniques are used to make the equation gauge invariant. This is worked out explicitly up to level 3. The equation is quadratic in the fields and can be viewed as a proposal for a string field theory equation. As in the earlier loop variable approach, the theory has one extra space dimension and mass is obtained by dimensional reduction. Being based on the sigma model RG, it is background independent. It is intriguing that in contrast to BRST string field theory, the gauge transformations are not modified by the interactions up to the level calculated. The interactions can be written in terms of gauge invariant field strengths for the massive higher spin fields and the non-zero mass is essential for this. This is reminiscent of Abelian Born–Infeld action (along with derivative corrections) for the massless vector field, which is also written in terms of the field strength.     

On the Gauge Invariant Lagrangian for Seiberg--Witten Topological Monopoles

A topological gauge invariant Lagrangian for Seiberg--Witten monopoleequations is constructed. The action is invariant under a huge classof gauge transformations which after BRST fixing leads to the BRSTinvariant action associated to Seiberg--Witten monopole topologicaltheory. The supersymmetric transformation of the fields involved inthe construction is obtained from the nilpotent BRST algebra.     

Massive higher spin fields coupled to a scalar: Aspects of interaction and causality

We consider in detail the most general cubic Lagrangian which describes an interaction between two identical higher spin fields in a triplet formulation with a scalar field, all fields having the same values of the mass. After performing the gauge fixing procedure we find that for the case of massive fields the gauge invariance does not guarantee the preservation of the correct number of propagating physical degrees of freedom. In order to get the correct number of degrees of freedom for the massive higher spin field one should impose some additional conditions on parameters of the vertex. Further independent constraints are provided by the causality analysis, indicating that the requirement of causality should be imposed in addition to the requirement of gauge invariance in order to have a consistent propagation of massive higher spin fields.     

Gauge invariant and gauge fixed actions for various higher-spin fields from string field theory

We propose a systematic procedure for extracting gauge invariant and gauge fixed actions for various higher-spin gauge field theories from covariant bosonic open string field theory. By identifying minimal gauge invariant part for the original free string field theory action, we explicitly construct a class of covariantly gauge fixed actions with BRST and anti-BRST invariance. By expanding the actions with respect to the level N   of string states, the actions for various massive fields including higher-spin fields are systematically obtained. As illustrating examples, we explicitly investigate the level N?3$N?3$ part and obtain the consistent actions for massive graviton field, massive 3rd rank symmetric tensor field, or anti-symmetric field. We also investigate the tensionless limit of the actions and explicitly derive the gauge invariant and gauge fixed actions for general rank n symmetric and anti-symmetric tensor fields.     

Higher spin fields and gravity

We analyze the coupling to gravity of massless bosonic gauge fields of any spin starting from a free theory formulated in terms of a nilpotent BRST operator.     

Cubic interaction vertex of higher-spin fields with external electromagnetic field

We fulfill the detailed analysis of coupling the charged bosonic higher-spin fields to external constant electromagnetic field in first order in external field strength. Cubic interaction vertex of arbitrary massive and massless bosonic higher-spin fields with external field is found. Construction is based on deformation of free Lagrangian and free gauge transformations by terms linear in electromagnetic field strength. In massive case a formulation with Stueckelberg fields is used. We begin with the most general form of deformations for Lagrangian and gauge transformations, admissible by Lorentz covariance and gauge invariance and containing some number of arbitrary coefficients, and require the gauge invariance of the deformed theory in first order in strength. It yields the equations for the coefficients which are exactly solved. As a result, the complete interacting Lagrangian of arbitrary bosonic higher-spin fields with constant electromagnetic field in first order in electromagnetic strength is obtained. Causality of massive spin-2 and spin-3 fields propagation in the corresponding electromagnetic background is proved.     

Nilpotent symmetry invariance in the non-Abelian 1-form gauge theory: Superfield formalism

We demonstrate that the nilpotent Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry invariance of the Lagrangian density of a four (3 + 1)-dimensional (4D) non-Abelian 1-form gauge theory with Dirac fields can be captured within the framework of the superfield approach to BRST formalism. The above 4D theory, where there is an explicit coupling between the non-Abelian 1-form gauge field and the Dirac fields, is considered on a (4,2)-dimensional supermanifold, parametrized by the bosonic 4D spacetime variables and a pair of Grassmannian variables. We show that the Grassmannian independence of the super-Lagrangian density, expressed in terms of the (4,2)-dimensional superfields, is a clear signature of the presence of the (anti-)BRST invariance in the original 4D theory.      

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.     

Cubic interaction vertices for fermionic and bosonic arbitrary spin fields

Using the light-cone gauge approach to relativistic field dynamics, we study arbitrary spin fermionic and bosonic fields propagating in flat space of dimension greater than or equal to four. Generating functions of parity invariant cubic interaction vertices for totally symmetric and mixed-symmetry massive and massless fields are obtained. For the case of totally symmetric fields, we derive restrictions on the allowed values of spins and the number of derivatives. These restrictions provide a complete classification of parity invariant cubic interaction vertices for totally symmetric fermionic and bosonic fields. As an example of application of the light-cone formalism, we obtain simple expressions for the Yang–Mills and gravitational interactions of massive arbitrary spin fermionic fields. For some particular cases, using our light-cone cubic vertices, we discuss the corresponding manifestly Lorentz invariant and on-shell gauge invariant cubic vertices.     

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.     

BRST quantization in the Siegel gauge

From a formal generalization to N copies of the free open string field theory BRST-quantized in the Siegel gauge we reproduce the BRST quantization of the free closed bosonic string field theory and obtain the one of massless higher spin field theories.     

Parent Field Theory and Unfolding in BRST First-Quantized Terms

For free-field theories associated with BRST first-quantized gauge systems, we identify generalized auxiliary fields and pure gauge variables already at the first-quantized level as the fields associated with algebraically contractible pairs for the BRST operator. Locality of the field theory is taken into account by separating the space–time degrees of freedom from the internal ones. A standard extension of the first-quantized system, originally developed to study quantization on curved manifolds, is used here for the construction of a first-order parent field theory that has a remarkable property: by elimination of generalized auxiliary fields, it can be reduced both to the field theory corresponding to the original system and to its unfolded formulation. As an application, we consider the free higher-spin gauge theories of Fronsdal.Senior Research Associate of the National Fund for Scientific Research (Belgium).Postdoctoral Visitor of the National Fund for Scientific Research (Belgium).     

BRST Formalism in Self-Dual Chern-Simons Theory with Matter Fields

We apply BRST method to the self-dual Chern-Simons gauge theory with matter fields and the generators of symmetries of the system from an elegant Lie algebra structure under the operation of Poisson bracket. We discuss four different cases: abelian, nonabelian, relativistic, and nonrelativistic situations and extend the system to the whole phase space including ghost fields. In addition, we obtain the BRST charge of the field system and check its nilpotence of the BRST transformation which plays an important role such as in topological quantum field theory and string theory.