Semi-infinite cohomology in conformal field theory and 2D gravity

We discuss various techniques for computing the semi-infinite cohomology of highest weight modules which arise in the BRST quantization of two dimensional field theories. In particular, we concentrate on two such theories - the G/H coset models and 2D gravity coupled to c ≤ 1 conformal matter.     

On the unitarity of gauged non-compact world-sheet supersymmetric WZNW models

In this paper we generalize our investigation of the unitarity of non-compact WZNW models connected to Hermitian symmetric spaces to the N=1$N=1$ world-sheet supersymmetric extension of these models. We will prove that these models have a unitary spectrum in a BRST approach for antidominant highest weight representations if the level and weights of the gauged subalgebra are integers. We will find new critical string theories in 7 and 9 space–time dimensions.     

Unitary representations of non-compact supergroups

We give a general theory for the construction of oscillator-like unitary irreducible representations (UIRs) of non-compact supergroups in a super Fock space. This construction applies to all non-compact supergroupsG whose coset spaceG/K with respect to their maximal compact subsupergroupK is “Hermitean supersymmetric”. We illustrate our method with the example of SU(m, p/n+q) by giving its oscillator-like UIRs in a “particle state” basis as well as “supercoherent state basis”. The same class of UIRs can also be realized over the “super Hilbert spaces” of holomorphic functions of aZ variable labelling the coherent states.     

On the unitarity of gauged non-compact WZNW strings

In this paper we investigate the unitarity of gauged non-compact WZNW strings, i.e., string theories formulated as G/H^′$G/H^{\prime }$ WZNW models, where G   is a non-compact group. These models represent string theories on non-trivial curved space–times with one time-like component. We will prove that for the class of models connected to Hermitian symmetric spaces, and a natural set of discrete highest weight representations, the BRST formulation, in which the gauging is defined through a BRST condition, yields unitarity. Unitarity requires the level times the Dynkin index to be an integer, as well as integer valued highest weights w.r.t. the compact subalgebra. We will also show that the BRST formulation is not equivalent to the conventional GKO coset formulation, defined by imposing a highest weight condition w.r.t. H^′$H^{\prime }$. The latter leads to non-unitary physical string states. This is, to our knowledge, the first example of a fundamental difference between the two formulations.     

Explicit construction of the classical BRST charge for nonlinear algebras

We give an explicit formula for the Becchi-Rouet-Stora-Tyutin (BRST) charge associated with Poisson superalgebras. To this end, we split the master equation for the BRST charge into a pair of equations such that one of themis equivalent to the original one and find a solution to this equation. The solution possesses a graphical representation in terms of diagrams.     

Notoph gauge theory: Superfield formalism

We derive absolutely anticommuting Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry transformations for the 4D free Abelian 2-form gauge theory by exploiting the superfield approach to BRST formalism. The antisymmetric tensor gauge field of the above theory was christened as the “notoph” (i.e. the opposite of “photon”) gauge field by Ogievetsky and Palubarinov way back in 1966–67. We briefly outline the problems involved in obtaining the absolute anticonimutativity of the (anti-) BRST transformations and their resolution within the framework of geometrical superfield approach to BRST formalism. One of the highlights of our results is the emergence of a Curci-Ferrari type of restriction in the context of 4D Abelian 2-form (notoph) gauge theory which renders the nilpotent (anti-) BRST symmetries of the theory to be absolutely anticommutative in nature.     

Interacting physical states of the bosonic string. I

A preliminary discussion on the importance of the contact terms in relation with the existence of many string vacua is presented. The problem of the BRST invariance of the string states transition amplitudes is considered in the sigma model approach to the string. It is shown that the BRST invariance is controlled by a gravitational (Diff) anomaly, and its connections with the standard Virasoro constraints are illustrated. The tachyon amplitudes are analysed and the BRST invariant dilaton vertex operator for non flat world-sheet is constructed.     

A FIELD THEORY MODEL FOR CLOSED BOSONIC STRINGS

Besed on the BRST approach to the first quantization of closed bosonic strings, we propose a gauge covariant field theory model for both free and interacting closed bosonic strings. The action is of the Chexn-Simons type and coincides with the cohomology of the BRST operators.     

Oscillator-like unitary representations of non-compact groups with a jordan structure and the non-compact groups of supergravity

We give a general bosonic construction of oscillator-like unitary irreducible representations (UIR) of non-compact groups whose coset spaces with respect to their maximal compact subgroups are Hermitian symmetric. With the exception of E₇₍₇₎, they include all the non-compact invariance groups of extended supergravity theories in four dimensions. These representations have the remarkable property that each UIR is uniquely determined by an irreducible representation of the maximal compact subgroup. We study the connection between our construction, the Hermitian symmetric spaces and the Tits-Koecher construction of the Lie algebras of corresponding groups. We then give the bosonic construction of the Lie algebra ofE ₇₍₇₎ in SU(8), SO(8) and U(7) bases and study its properties. Application of our method toE ₇₍₇₎ leads to reducible unitary representations.Dedicated to Feza Gürsey on the occasion of his 60th birthdayAlexander von Humboldt Fellow, on leave from Physics Dept., Bogaziçi University, Istanbul/Turkey: work supported in part by TBTAK, The National Science and Technology Council of Turkey     

Canonical BRST quantisation of worldsheet gravities

We reformulate the BRST quantisation of chiral Virasoro and W₃ worldsheet gravities. Our approach follows directly the classic BRST formulation of Yang-Mills theory in employing a derivative gauge condition instead of the conventional conformal gauge condition, supplemented by an introduction of momenta in order to put the ghost action back into first-order form. The consequence of these simple changes is a considerable simplification of the BRST formulation, the evaluation of anomalies and the expression of Wess-Zumino consistency conditions. In particular, the transformation rules of all fields now constitute a canonical transformation generated by the BRST operator Q, and we obtain in this reformulation a new result that the anomaly in the BRST Ward identity is obtained by application of the anomalous operator Q², calculated using operator products, to the gauge fermion.     

A Gauge Invariant Description for the General Conic Constrained Particle from the FJBW Iteration Algorithm

We consider a second-degree algebraic curve describing a general conic constraint imposed on the motion of a massive spinless particle. The problem is trivial at classical level but becomes involved and interesting concerning its quantum counterpart with subtleties in its symplectic structure and symmetries. We start with a second-class version of the general conic constrained particle, which encompasses previous versions of circular and elliptical paths discussed in the literature. By applying the symplectic FJBW iteration program, we proceed on to show how a gauge invariant version for the model can be achieved from the originally second-class system. We pursue the complete constraint analysis in phase space and perform the Faddeev-Jackiw symplectic quantization following the Barcelos-Wotzasek iteration program to unravel the essential aspects of the constraint structure. While in the standard Dirac-Bergmann approach there are four second-class constraints, in the FJBW they reduce to two. By using the symplectic potential obtained in the last step of the FJBW iteration process, we construct a gauge invariant model exhibiting explicitly its BRST symmetry. We obtain the quantum BRST charge and write the Green functions generator for the gauge invariant version. Our results reproduce and neatly generalize the known BRST symmetry of the rigid rotor, clearly showing that this last one constitutes a particular case of a broader class of theories.     

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.      

BRST Quantization of Unimodular Gravity

We study the quantization of two versions of unimodular gravity, namely fully diffeomorphism-invariant unimodular gravity and unimodular gravity with fixed metric determinant, utilizing standard path integral approach. We derive the BRST symmetry of effective actions corresponding to several relevant gauge conditions. We observe that for some gauge conditions, the restricted gauge structure may complicate the formulation and effective actions, in particular, if the chosen gauge conditions involve the canonical momentum conjugate to the induced metric on the spatial hypersurface. The BRST symmetry is extended further to the finite field-dependent BRST transformation, in order to establish the mapping between different gauge conditions in each of the two versions of unimodular gravity.     

BRST symmetries in SU(3) linear sigma model

We study the BRST symmetries in the SU(3) linear sigma model which is constructed through the introduction of a novel matrix for the Goldstone boson fields satisfying geometrical constraints embedded in a SU(2) subgroup. To treat these constraints we exploit the improved Dirac quantization scheme. We also discuss phenomenological aspects in the mean field approach to this model. Received: 16 November 2001 / Revised version: 23 February 2002 / Published online: 26 July 2002     

ON GEOMETRIC QUANTIZATION FOR BOSONIC STRING (Ⅱ)

The Hilbert space and the representation of the generators of Virasoro algebra for bosonic string under a holomorphic polarization are given in this paper,It is shown that the contre term of Virasoro algebra may be interpreted as curvature of a holomorphic vector bundle (holomorphic Fock bundle) on coset space G¹¹=G/H where G denotes the conformal transformation group and H the one-parameter subgroup generated by the generator L₀.The condition of the conformal anomaly cancellation may be expressed as the vanishing curvature of the bundle which is obtained by the product of the holomorphic Fock bundle and the holomorphic ghost vacuum bundle.The geometric interpretations of both classical and quantized BRST operators,ghost and antighost operators are also discussed.     

On the complete integrability of the static axially symmetric self-dual gauge field equations for an arbitrary group

We present an analysis of static axially symmetric gauge fields for an arbitrary gauge group G. Two ansätze are considered. The full ansatz involves a total of 4d(d = dim G), the reduced ansatz only 2d functions of (?, z). Imposing self-duality is shown to reduce the problem to a sigma model in the curved two-dimensional (?, z) space over the coset spaces G?/G for the full, and G¹/K for the reduced ansatz. G? is the complexification of G. ¹ is a particular non-compact form of G, and K the local form-preserving symmetry group of the reduced ansatz. We give explicitly the Lax-pair type representations (linear scattering problem) of the sigma model, indicating that the standard methods available for certain non-linear two-dimensional problems can be used to generate solutions. Our procedure has the advantage that only real fields over a real manifold enter the analysis.     

BRST charges for finite nonlinear algebras

Some ingredients of the BRST construction for quantum Lie algebras are applied to a wider class of quadratic algebras of constraints. We build the BRST charge for a quantum Lie algebra with three generators and ghost-anti-ghosts commuting with constraints. We consider a one-parametric family of quadratic algebras with three generators and show that the BRST charge acquires the conventional form after a redefinition of ghosts. The modified ghosts form a quadratic algebra. The family possesses a nonlinear involution, which implies the existence of two independent BRST charges for each algebra in the family. These BRST charges anticommute and form a double BRST complex.     

BRST model for equivariant cohomology and representatives for the equivariant thom class

In this paper the BRST formalism for topological field theories is studied in a mathematical setting. The BRST operator is obtained as a member of a one parameter family of operators connecting the Weil model and the Cartan model for equivariant cohomology. Furthermore, the BRST operator is identified as the sum of an equivariant derivation and its Fourier transform. Using this, the Mathai-Quillen representative for the Thom class of associated vector bundles is obtained as the Fourier transform of a simple BRST closed element.Supported by the SV FOM/SMC Mathematical Physics, The Netherlands     

Cosmological perturbations in SFT inspired non-local scalar field models

We clearly and consistently supersymmetrize the celebrated horizontality condition to derive the off-shell nilpotent and absolutely anticommuting Becchi?CRouet?CStora?CTyutin (BRST) and anti-BRST symmetry transformations for the supersymmetric system of a free spinning relativistic particle within the framework of superfield approach to BRST formalism. For the precise determination of the proper (anti-)BRST symmetry transformations for all the bosonic and fermionic dynamical variables of our system, we consider the present theory on a (1,2)-dimensional supermanifold parameterized by an even (bosonic) variable (??) and a pair of odd (fermionic) variables ?? and $\bar{\theta}$ (with $\theta^{2} = \bar{\theta}^{2} = 0$ , $\theta\bar{\theta}+ \bar{\theta}\theta= 0$ ) of the Grassmann algebra. One of the most important and novel features of our present investigation is the derivation of (anti-)BRST invariant Curci?CFerrari type restriction which turns out to be responsible for the absolute anticommutativity of the (anti-)BRST transformations and existence of the coupled (but equivalent) Lagrangians for the present theory of a supersymmetric system. These observations are completely new results for this model.     

On the reanimation of a local BRST invariance in the (Refined) Gribov–Zwanziger formalism

We localize a previously established nonlocal BRST invariance of the Gribov–Zwanziger (GZ) action by the introduction of additional fields. We obtain a modified GZ action with a corresponding local, albeit not nilpotent, BRST invariance. We show that correlation functions of the original elementary GZ fields do not change upon evaluation with the modified partition function. We discuss that for vanishing Gribov mass, we are brought back to the original Yang–Mills theory with standard BRST invariance.