In this paper, we mainly investigate the $W_{2,s}^{M}\otimes W_{2,s}^{L}$ system, in which the matter and the Liouville subsystems generate $W_{2,s}^{M}$ and $W_{2,s}^L$ algebras respectively. We first give a brief discussion of the physical states for corresponding $W$ stings. The lower states are given by freezing the spin-2 and spin-$s$ currents. Then, introducing two pairs of ghost-like fields, we give the realizations of $W_{1,2,s}$ algebras. Based on these linear realizations, BRST operators for $W_{2,s}$ algebras are obtained. Finally, we construct new BRST charges of Liouville system for $W_{2,s}^{L}$ strings at the specific values of central charges $c$: $c=-{22/5}$ for $W_{2,3}^{L}$ algebra, $c=-24$ for $W_{2,4}^{L}$ algebra and $c=-2,-{286/3}$ for $W_{2,6}^{L}$ algebra, at which the corresponding $W_{2,s}^L$ algebras are singular.
The Hamiltonian BRST quantization of the null spinning string for different number of supersymmetries is given. A null spinning string with manifest space-time conformal invariance is constructed. Its Brst quantization gives negative critical dimension for $N\neq 0$ and reproduces previous results for $N=0$.
Using the generalized Hamiltonian method of Batalin, Fradkin and Vilkovsky we develop the BRST formalism for the bosonic string on AdS(5)xS(5) formulated as principal chiral model. Then we show that the monodromy matrix and non-local charges are BRST invariant.
We show how to relate the parafermions that occur in the $W_3$ string to the standard construction of parafermions. This result is then used to show that one of the screening charges that occurs in parafermionic theories is precisely the non-trivial part of the $W_3$ string BRST charge. A way of generalizing this pattern for a $W_N$ string is explained. This enables us to construct the full BRST charge for a $W_{2,N}$ string and to prove the relation of such a string to the algebra $W_{N-1}$ for arbitrary $N$, We also show how to calculate part of the BRST charge for a $W_N$ string, and we exp…
The topological field theories associated with affine Lie superalgebras are constructed. Their BRST symmetry is characterised by a Kazama algebra containing spin 1, 2 and 3 operators and closes linearly. Under this symmetry all operators are grouped into BRST doublets. The relation between the models constructed and non-critical string theories is explored.
The classical pure spinor version of the heterotic superstring in a supergravity and super Yang-Mills background is considered. We obtain the BRST transformations of the world-sheet fields. They are consistent with the constraints obtained from the nilpotence of the BSRT charge and the holomorphicity of the BRST current.
We present a general argument for the construction of BRST charges of the `non-critical' $\W_{2,4}$, $\W_{2,5}$, $\W_{2,6}$, and $\W_{2,8}$ strings. This evidences the existence of BRST charges for a kind of soft-type algebras which can be constructed from two copies of quantum $\W_{2,s}$ algebras, (s=3,4,5,6,8).
The general BRST-BFV analysis of anomaly in the string theory coupled to background fields is carried out. An exact equation for c-valued symbol of anomaly operator is found and structure of its solutions is studied.
We analyze the BRST cohomology of the critical N=2 NSR string using chiral bosonization. Picture-changing and spectral flow is made explicit in a holomorphic field basis. The integration of fermionic and U(1) moduli is performed and yields picture- and U(1) ghost number-changing insertions into the string measure for n-point amplitudes at arbitrary genus and U(1) instanton number.
We construct the BRST operator for non-critical $W_3$-strings and discuss the tachyon-like spectrum. For $N$-punctured spheres with $N \leq 5$ we briefly describe a formal definition of the integral over $W_3$-moduli space.
The renormalization group flow in the theory space of a BRST invariant string $σ$-model is investigated. For the open bosonic string the non-perturbative off-shell effective action and its gauge symmetry properties are determined from $β$-functions defined by the local Weyl anomaly. The interactions are shown to explicitly break the free theory BRST invariance generating new non-linear gauge symmetries of the type present in Witten's string field theory. In the Feynman-Siegel gauge the $σ$-model is shown to generate Witten's structure of vertex couplings.
We discuss an analysis presented by N. Berkovits in [arXiv:0712.0324] in the context of classical mechanics and perform its extension to the case of pure spinor string in general background.
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.
We construct a universal envelope for any Poisson- and Gerstenhaber algebra. While the deformation theory of Poisson algebras seems to be partially trivial, results from string- and M-theory suggest a rich deformation theory of Gerstenhaber algebras. We apply our construction in this case to well known questions on the topological closed string BRST-complex. Finally, we find a similar algebraic structure, as for the universal envelope, in the SU(2)-WZW model.
The BRST quantization of strings is revisited and the derivation of the path integral measure for scattering amplitudes is streamlined. Gauge invariances due to zero modes in the ghost sector are taken into account by using the Batalin-Vilkovisky formalism. This involves promoting the moduli of Riemann surfaces to quantum mechanical variables on which BRST transformations act. The familiar ghost and antighost zero mode insertions are recovered upon integrating out auxiliary fields. In contrast to the usual treatment, the gauge-fixed action including all zero mode insertions is BRST invariant. …
We study two and three-point tree-level amplitudes of open strings. These amplitudes are reduced from higher-point correlation functions by using mostly BRST exact operators for gauge fixing. For simplicity, we focus on an open string tachyon. The two-point amplitude of open string tachyons is reduced from a three-point function of two tachyon vertex operators and one mostly BRST exact operator. Similarly the three-point amplitude is from a four-point function of three tachyon vertex operators and one mostly BRST exact operator. One can also obtain the two-point amplitude from the four-point f…
The pure spinor heterotic string in a generic super Yang-Mills and supergravity background is considered. We determine the one-loop BRST anomaly at the cohomological level. We prove that it can be absorbed by consistent corrections of the classical constraints due to Berkovits and Howe, in agreement with the Green-Schwarz cancelation mechanism.
We study the BRST quantization of bosonic and NSR strings propagating in AdS(3) x N backgrounds. The no-ghost theorem is proved using the Frenkel-Garland-Zuckerman method. Regular and spectrally-flowed representations of affine SL(2,R) appear on an equal footing. Possible generalizations to related curved backgrounds are discussed.
We derive the recently proposed BRST charge for non-critical W strings from a Lagragian approach. The basic observation is that, despite appearances, the combination of two classical ``matter'' and ``Toda'' w_3 systems leads to a closed modified gauge algebra, which is of the so-called soft type. Based on these observations, a novel way to construct critical W_3 strings is given.
We discuss the conditions under which the BRST operator of a $W$-string can be written as the sum of two operators that are separately nilpotent and anticommute with each other. We illustrate our results with the example of the non-critical $W_3$-string. Furthermore, we apply our results to make a conjecture about a relationship between the spectrum of a non-critical $W_n$-string and a $W_{n-1}$-string.
The BRST operator cohomology of $N=2$ $2d$ supergravity coupled to matter is presented. Descent equations for primary superfields of the matter sector are derived. We find one copy of the cohomology at ghost number one, two independent copies at ghost number two, and conjecture that there is a copy at ghost number three. The $N=2$ string has a twisted $N=4$ superconformal symmetry generated by the $N=2$ superstress tensor, the BRST supercurrent, the antighost superfield, and the ghost number supercurrent.
It is shown that the BRST resolution of the spaces of physical states of non-critical (anomalus) massive string models can be consistently defined. The appropriate anomalus complexes are obtained by canonical restrictions of the ghost extended spaces to the kernel of curvature operator and additional Gupta-Bleuler like conditions without any modifications of the matter sector. The cohomologies of the polarized anomalus complex are calculated and analyzed in details.
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 resume our discussion of the new orbifold-string theories of permutation-type, focusing in the present series on the algebraic formulation of the general bosonic prototype and especially the target space-times of the theories. In this first paper of the series, we construct one twisted BRST system for each cycle $j$ in each twisted sector $σ$ of the general case, verifying in particular the previously-conjectured algebra $[Q_{i}(σ),Q_{j}(σ)]_{+} =0$ of the BRST charges. The BRST systems then imply a set of extended physical-state conditions for the matter of each cycle at cycle central char…
We compute the BRST cohomology of the holomorphic part of the N=2 string at arbitrary ghost and picture number. We confirm the expectation that the relative cohomology at non-zero momentum consists of a single massless state in each picture. The absolute cohomology is obtained by an independent method based on homological algebra. For vanishing momentum, the relative and absolute cohomologies both display a picture dependence -- a phenomenon discovered recently also in the relative Ramond sector of N=1 strings by Berkovits and Zwiebach.
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 study the cohomology of the critical $W_4$ string using the $W_4$ BRST charge in a special basis in which it contains three separately nilpotent BRST charges. This allows us to obtain the physical operators in three steps. In the first step we obtain the cohomology associated to a spin-four constraint only, and it contains operators of the $c={4\over5}$ $W_3$ minimal model. In the next step, where the spin-three constraint is added, these operators get dressed to operators of the $c={7\over10}$ Virasoro minimal model. Finally, the Virasoro constraint is added to obtain the cohomology of the…
In general bosonic closed string backgrounds the ghost-dilaton is not the only state in the semi-relative BRST cohomology that can change the dimensionless string coupling. This fact is used to establish complete dilaton theorems in closed string field theory. The ghost-dilaton, however, is the crucial state: for backgrounds where it becomes BRST trivial we prove that the string coupling becomes an unobservable parameter of the string action. For backgrounds where the matter CFT includes free uncompactified bosons we introduce a refined BRST problem by including the zero-modes "x" of the boson…
We consider the theory of bosonic closed strings on the flat background R(25,1). We show how the BRST complex can be extended to a complex where the string center of mass operator, x^mu_0, is well defined. We investigate the cohomology of the extended complex. We demonstrate that this cohomology has a number of interesting features. Unlike in the standard BRST cohomology, there is no doubling of physical states in the extended complex. The cohomology of the extended complex is more physical in a number of of aspects related to the zero-momentum states. In particular, we show that the ghost num…
We construct a class of BRST-invariant closed string states for any classical solution of open string field theory. The closed string state is a nonlinear functional of the open string field and changes by a BRST-exact term under a gauge transformation of the solution. As a result, its contraction with an on-shell closed string state provides a gauge-invariant observable of open string field theory. Unlike previously known observables, however, the contraction with off-shell closed string states in the Fock space is well defined and regular. Moreover, we claim that the BRST-invariant closed st…
Tachyon condensation on a bosonic D-brane was recently demonstrated numerically in Witten's open string field theory with level truncation approximation. This non-perturbative vacuum, which is obtained by solving the equation of motion, has to satisfy furthermore the requirement of BRST invariance. This is indispensable in order for the theory around the non-perturbative vacuum to be consistent. We carry out the numerical analysis of the BRST invariance of the solution and find that it holds to a good accuracy. We also mention the zero-norm property of the solution. The observations in this pa…
Exploiting the theoretical potential of the modified Bonora-Tonin superfield approach (MBTSA) as well as the (anti-)chiral superfield approach (ACSA) to Becchi-Rouet-Stora-Tyutin (BRST) formalism, we derive the complete set of off-shell nilpotent (anti-)BRST symmetry transformations corresponding to the classical two $(1 + 1)$-dimensional (2D) diffeomorphism symmetry transformations on the world-sheet (that is traced out by the motion of a model of bosonic string). Only the BRST symmetry transformations for this model have been discussed in the earlier literature. We derive the (anti-)BRST inv…
The grading Becchi-Rouet-Stora-Tyutin (BRST) method gives a way to construct the integer $W_{2,s}$ strings, where the BRST charge is written as $Q_B=Q_0+Q_1$. Using this method, we reconstruct the nilpotent BRST charges $Q_{0}$ for the integer $W_{2,s}$ strings and the half-integer $W_{2,s}$ strings. Then we construct the exact grading BRST charge with spinor fields and give the new realizations of the half-integer $W_{2,s}$ strings for the cases of $s=3/2$, 5/2, and 7/2.
We review the detailed structure of the large $N=4$ superconformal algebra, and construct its BRST operator which constitutes the main object for analyzing $N=4$ strings. We then derive the general condition for the nilpotency of the BRST operator and show that there exists a line of critical $N=4$ string theories.
We propose a new general BRST approach to string and string-like theories which have a wider range of applicability than e g the conventional conformal field theory method. The method involves a simple general regularization of all basic commutators which makes all divergent sums to be expressible in terms of zeta functions from which finite values then may be extracted in a rigorous manner. The method is particular useful in order to investigate possible state space representations to a given model. The method is applied to three string models: The ordinary bosonic string, the tensionless str…
BRST operators for two-dimensional theories with spin-2 and spin-$s$ currents, generalising the $W_3$ BRST operator of Thierry-Mieg, have previously been obtained. The construction was based on demanding nilpotence of the BRST operators, making no reference to whether or not an underlying $W$ algebra exists. In this paper, we analyse the known cases ($s=3$, 4, 5 and 6), showing that the two $s=4$ BRST operators are associated with the $W\!B_2$ algebra, and that two of the four $s=6$ BRST operators are associated with the $W\!G_2$ algebra. We discuss the cohomology of all the known higher-spin …
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…
Motivated by the descent equation in string theory, we give a new interpretation for the action of the symmetry charges on the BRST cohomology in terms of what we call {\em the Gerstenhaber bracket}. This bracket is compatible with the graded commutative product in cohomology, and hence gives rise to a new class of examples of what mathematicians call a {\em Gerstenhaber algebra}. The latter structure was first discussed in the context of Hochschild cohomology theory \cite{Gers1}. Off-shell in the (chiral) BRST complex, all the identities of a Gerstenhaber algebra hold up to homotopy. Applying…
Using the generalized hamiltonian method of Batalin, Fradkin and Vilkovisky, we investigate the algebraic structure of anomalies in the Polyakov string theory that appear as the Schwinger terms in super-commutation relations between BRST charge and total hamiltonian. We obtain the most general form of the anomalies in the extended phase space, without any reference to a two dimensional metric. This pregeometri- cal result, refered to as the genelarized Virasoro anomaly, independent of the gauge and the regularization under a minor assumption, is a non-perturbative result, and valid for any spa…
Nilpotent BRST operators for higher-spin $W_{2,s}$ strings, with currents of spins 2 and $s$, have recently been constructed for $s=4$, 5 and 6. In the case of $W_{2,4}$, this operator can be understood as being the BRST operator for the critical $W\!B_2$ string. In this paper, we construct a generalised BRST operator that can be associated with a non-critical $W_{2,4}$ string, in which $W\!B_2$ matter is coupled to the $W\!B_2$ gravity of the critical case. We also obtain the complete cohomology of the critical $W_{2,4}$ BRST operator, and investigate the physical spectra of the $s=5$ and $s=…
The BRST-invariant formulation of the bosonic stretched membrane is considered. In this formulation the stretched membrane is given as a perturbation around zero-tension membranes, where the BRST-charge decomposes as a sum of a string-like BRST-charge and a perturbation. It is proven, by means of cohomology techniques, that there exists to any order in perturbation theory a canonical transformation that reduces the full BRST-charge to the string-like one. It is also shown that one may extend the results to the quantum level yielding a nilpotent charge in 27 dimensions.
Many field theories of physical interest have configuration spaces consisting of disconnected components. Quantum mechanical amplitudes are then expressed as sums over these components. We use the Faddeev-Popov approach to write the terms in this topological expansion as moduli space integrals. A cut-off is needed when these integrals diverge. This introduces a dependence on the choice of parametrisation of configuration space which must be removed if the theory is to make physical sense. For theories that have a local symmetry this also leads to a breakdown in BRST invariance. We discuss in d…
The BRST formalism has played a fundamental role in the construction of bosonic closed string backgrounds, ie. the stringy analogs of classical solutions to the field equations of general relativity. The concept of a string background has been extended to the notion of $W$-strings, where the BRST symmetry is still largely conjectural. More recently, the BRST formalism has entered the construction of two dimensional topological conformal quantum field theories, such as those that arise from Calabi-Yau varieties. In this lecture, we focus on common features of the BRST cohomology algebras of str…
We describe the ghost sector of cubic string field theory in terms of degrees of freedom on the two halves of a split string. In particular, we represent a class of pure ghost BRST operators as operators on the space of half-string functionals. These BRST operators were postulated by Rastelli, Sen, and Zwiebach to give a description of cubic string field theory in the closed string vacuum arising from condensation of a D25-brane in the original tachyonic theory. We find a class of solutions for the ghost equations of motion using the pure ghost BRST operators. We find a vanishing action for th…
The BRST cohomology analysis of Lian and Zuckerman leads to physical states at all ghost number for $c<1$ matter coupled to Liouville gravity. We show how these states are related to states at ghost numbers zero(pure vertex operator states -- DK states) and ghost number one(ring elements) by means of descent equations. These descent equations follow from the double cohomology of the String BRST and Felder BRST operators. We briefly discuss how the ring elements allow one to determine all correlation functions on the sphere.
Institute of Physics of the Czech Academy of Sciences（捷克科学院物理研究所）；ICTP - South American Institute for Fundamental Research, IFT-UNESP（国际理论物理中心南美基础研究所，圣保罗州立大学）
We study the star algebra of ghost sector in vacuum string field theory (VSFT). We show that the star product of two states in the Siegel gauge is BRST exact if we take the BRST charge to be the one found in hep-th/0108150, and the BRST exact states are nil factors in the star algebra. By introducing a new star product defined on the states in the Siegel gauge, the equation of motion of VSFT is characterized as the projection condition with respect to this new product. We also comment on the comma form of string vertex in the ghost sector.
In this paper we discuss deformations of the BRST operator of the fermionic string. These deformations preserve nilpotency of the BRST operator and correspond to turning on infinitesimal Gravitino and Ramond-Ramond spacetime fields.
We consider the sl(2) current algebra at level k=-4 when the sl(2) BRST operator is nilpotent. We formulate a spectral sequence converging to the cohomology of this BRST operator. At the second term of the spectral sequence, we observe an N=4 algebra. This algebra is generated in a c=-2 bosonic string whose BRST operator Q_{string} represents the next term in the spectral sequence. We realize the cohomology of the irreducible modules as Q_{string}-primitives of the N=4 singular vectors and point out their relation to Lian--Zuckerman states of c=-2 matter. The relation between sl(2)_{-4} WZW mo…
The full cohomology ring of the Lian-Zuckerman type operators (states) in ${\hat c_M}<1$ Neveu-Schwarz-Ramond (NSR) string theory is argued to be generated by three elements $x$, $y$ and $w$ in analogy with the corresponding results in the bosonic case. The ground ring generators $x$ and $y$ are non-invertible and belong to the Ramond sector whereas the higher ghost number operators are generated by an invertible element $w$ with ghost number one less than that of the ground ring generators and belongs to either Neveu-Schwarz (NS) or Ramond (R) sector depending on whether we consider (even, ev…
We summarize some recent results on the BRST analysis of physical states of 2D gravity coupled to c<=1 conformal matter and the supersymmetric generalization.
We investigate the cohomology structure of a general noncritical $W_N$-string. We do this by introducing a new basis in the Hilbert space in which the BRST operator splits into a ``nested'' sum of nilpotent BRST operators. We give explicit details for the case $N=3$. In that case the BRST operator $Q$ can be written as the sum of two, mutually anticommuting, nilpotent BRST operators: $Q=Q_0+Q_1$. We argue that if one chooses for the Liouville sector a $(p,q)$ $W_3$ minimal model then the cohomology of the $Q_1$ operator is closely related to a $(p,q)$ Virasoro minimal model. In particular, the…
Within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism, we discuss the full set of proper BRST and anti-BRST transformations for a 2D diffeomorphism invariant theory which is described by the Lagrangian density of a standard bosonic string. The above (anti-)BRST transformations are off-shell nilpotent and absolutely anticommuting. The latter property is valid on a submanifold of the space of the quantum fields where the 2D version of the universal (anti-)BRST invariant Curci-Ferrari (CF) type of restrictions are satisfied. We derive the precise forms of the BRST and anti-BRST invar…
In this paper, we construct the nilpotent Becchi-Rouet-Stora-Tyutin($BRST$) charges of spinor non-critical $W_{2,s}$ strings. The cases of $s=3,4$ are discussed in detail, and spinor realization for $s=4$ is given explicitly. The $BRST$ charges are graded.
The Veneziano amplitude is derived from fixing one degree of freedom of $PSL(2,\mathbb{R})$ symmetry by the insertion of a mostly BRST exact operator. Evaluating the five-point function which consists of four open string tachyons and this gauge fixing operator, we find it equals the Veneziano amplitude up to a sign factor. The sign factor is interpreted as a signed intersection number.The result implies that the mostly BRST exact operator, which is originally used to provide two-point string amplitudes, correctly fixes the $PSL(2,\mathbb{R})$ gauge symmetry for general amplitudes. We conjectur…
Boundary interactions of closed-string with open-strings are examined intended to provide a constructive formulation of boundary string field theory. As an illustration, we consider the BPS $D$-brane of the type II superstring in a constant NS-NS two-form background, and study the boundary interaction with arbitrary configurations of gauge field on the brane. The boundary interaction is presented, within the world-sheet cut-off theories, as an off-shell boundary state in the closed-string Hilbert space. It is regarded as a closed-string theoretical counterpart of the Wilson loop in the world-v…
Brane-like states are defined by physical vertex operators in NSR superstring theory, existing at nonzero pictures only. These states exist both in open and closed string theories, in the NS and NS-NS sectors respectively. In this paper we present a detailed analysis of their BRST properties, giving a proof that these vertex operators are physical, i.e. BRST invariant and BRST non-trivial.
