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We show that it is possible to construct a well-defined effective field theory incorporating string winding modes without using strong constraint in double field theory. We show that X-ray (Radon) transform on a torus is well-suited for describing weakly constrained double fields, and any weakly constrained fields are represented as a sum of strongly constrained fields. Using inverse X- ray transform we define a novel binary operation which is compatible with the level matching constraint. Based on this formalism, we construct a consistent gauge transform and gauge invariant action without using strong constraint. We then discuss the relation of our result to the closed string field theory. Our construction suggests that there exists an effective field theory description for massless sector of closed string field theory on a torus in an associative truncation., 33 pages

The two-loop (order $\alpha'$) $\beta$-function equations, which are equivalent to the equations of motion of $\alpha'$-corrected string effective action, are considered for anisotropic homogeneous space-times. These equations are solved for all Bianchi-type models in two schemes of effective action, namely $R^2$ and Gauss-Bonnet schemes with zero cosmological constant and then the metric, dilaton and $B$-field are found at $\alpha'$ perturbative corrections., Comment: 33 pages

String theory gives S matrix elements om which is not possible to read any gauge information. Using factorization we go off shell in the simplest and most naive way and we read which are the vertices suggested by string. To compare with the associated Effective Field Theory it is natural to use color ordered vertices. The alpha'=0 color ordered vertices suggested by string theory are more efficient than the usual ones since the three gluon color ordered vertex has three terms instead of six and the four gluon one has one term instead of three. They are written in the so called Gervais-Neveu gauge. The full Effective Field Theory is in a generalization of the Gervais-Neveu gauge with alpha') corrections. Moreover a field redefinition is required to be mapped to the field used by string theory. We also give an intuitive way of understanding why string choose this gauge in terms of the minimal number of couplings necessary to reproduce the non abelian amplitudes starting from color ordered ones., Comment: 37 pages, 9 figures

We quantize the bosonic part of the D 1 string with closed boundary conditions on the light cone and we consider the U ( 1 ) worldsheet gauge field a dynamical variable. We compute also 3-Reggeon vertex by the overlapping technique. We find that the Fock space is the sum of sectors characterized by the momentum of the U ( 1 ) Wilson line and that these sectors do not interact among them. Each sector has exactly the same spectrum of the usual bosonic string when expressed in properly sector dependent rescaled variables. Rescaling is forced by factorization of the string amplitudes. We are also able to determine the relative string coupling constant of the different sectors. It follows a somewhat unexpected picture in which the effective action is always the same independently on the sector but string amplitudes are only the same when expressed in sector dependent rescaled variables.

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., Latex file, 40 pages.Some typos corrected and cosmetic changes

We construct a ``pseudo-supersymmetric" fermionic extension of the effective action of the bosonic string in arbitrary spacetime dimension D. The theory is invariant under pseudo-supersymmetry transformations up to the quadratic fermion order, which is sufficient in order to be able to derive Killing spinor equations in bosonic backgrounds, and hence to define BPS type solutions determined by a system of first-order equations. The pseudo-supersymmetric theory can be extended by coupling it to a Yang-Mills pseudo-supermultiplet. This also allows us to construct ``\alpha' corrections'' involving quadratic curvature terms. An exponential dilaton potential term, associated with the conformal anomaly for a bosonic string outside its critical dimension, can also be pseudo-supersymmetrised., Comment: Latex, 16 pages

The open string on the plane-wave limit of dSn × Sn with constant B2 and dilaton background fields is canonically quantized. This entails solving the classical equations of motion for the string, computing the symplectic form, and defining from its inverse the canonical commutation relations. Canonical quantization is proved to be perfectly suited for this task, since the symplectic form is unambiguously defined and non-singular. The string position and the string momentum operators are shown to satisfy equal-time canonical commutation relations. Noticeably the string position operators define non-commutative spaces for all values of the string world-sheet parameter σ, thus extending non-commutativity outside the branes on which the string endpoints may be assumed to move. The Minkowski space–time limit is smooth and reproduces the results in the literature, in particular non-commutativity gets confined to the endpoints.

We discuss a semiclassical string description to circular Wilson loops without/with local operator insertions. By considering a semiclassical approximation of type IIB string theory on AdS_5 X S^5 around the corresponding classical solutions, quadratic actions with respect to fluctuations are computed. Then the dual corresponding operators describing the fluctuations are discussed from the point of view of a small deformation of the Wilson loops. The result gives new evidence for AdS/CFT correspondence., Comment: 23pp, 1 figure; typos fixed, version to be published in NPB

We investigate N -point string scattering amplitudes in AdS 3 space. Based on recent observations on the solutions of KZ and BPZ-type differential equations, we discuss how to describe the string theory in AdS 3 as a marginal deformation of a (flat) linear dilaton background. This representation resembles the called “discrete light-cone Liouville” realization as well as the FZZ dual description in terms of the sine-Liouville field theory. Consequently, the connection and differences between those and this realization are discussed. The free field representation presented here permits to understand the relation between correlators in both Liouville and WZNW theories in a very simple way. Within this framework, we discuss the spectrum and interactions of strings in Lorentzian AdS 3 .

The FZZT and ZZ branes in ( p , p + 1 ) minimal string theory are studied in terms of continuum loop equations. We show that systems in the presence of ZZ branes (D-instantons) can be easily investigated within the framework of the continuum string field theory developed by Yahikozawa and one of the present authors [M. Fukuma, S. Yahikozawa, Phys. Lett. B 396 (1997) 97, hep-th/9609210 ]. We explicitly calculate the partition function of a single ZZ brane for arbitrary p. We also show that the annulus amplitudes of ZZ branes are correctly reproduced.

We study the implication of decoupling zero-norm states in the high-energy limit, for the 26 dimensional bosonic open string theory. Infinitely many linear relations among 4-point functions are derived algebraically, and their unique solution is found. Equivalent results are also obtained by taking the high-energy limit of Virasoro constraints, and as an independent check, we compute all 4-point functions of 3 tachyons and an arbitrary massive state by saddle-point approximation., 30 pages, one figure, further typos corrected

Recent explorations of the AdS/CFT correspondence have unveiled integrable structures underlying both planar N = 4 super-Yang-Mills theory and type IIB string theory on AdS_5 x S^5. Integrability in the gauge theory emerges from the fact that the dilatation generator can be identified with the Hamiltonian of an integrable quantum spin chain, and the classical string theory has been shown to contain infinite towers of hidden currents, a typical signature of integrability. Efforts to match the integrable structures of various classical string configurations to those of corresponding gauge theory quantum spin chains have been largely successful. By studying a semiclassical expansion about a class of point-like solitonic solutions to the classical string equations of motion on AdS_5 x S^5, we take a step toward demonstrating that integrability in the string theory survives quantum corrections beyond tree level. Quantum fluctuations are chosen to align with background curvature corrections to the pp-wave limit of AdS_5 x S^5, and we present evidence for an infinite tower of local bosonic charges that are conserved by the quantum theory to quartic order in the expansion. We explicitly compute several higher charges based on a Lax representation of the worldsheet sigma model and provide a prescription for matching the eigenvalue spectra of these charges with corresponding quantities descending from the integrable structure of the gauge theory., Comment: v2: references and typos corrected; v3: minor corrections and comments, 23 pages

We explore the string spectrum in the Witten QCD model by considering classical string configurations, thereby obtaining energy formulas for quantum states with large excitation quantum numbers representing glueballs and Kaluza-Klein states. In units of the string tension, the energies of all states increase as the 't Hooft coupling $\lambda $ is decreased, except the energies of glueballs corresponding to strings lying on the horizon, which remain constant. We argue that some string solutions can be extrapolated to the small $\lambda $ regime. We also find the classical mechanics description of supergravity glueballs in terms of point-like string configurations oscillating in the radial direction, and reproduce the glueball energy formula previously obtained by solving the equation for the dilaton fluctuation., Comment: 20 pages, harvmac. References added

We consider the pre-big bang scenario with a massive axion field which starts to dominate energy density when oscillating in an instanton-induced potential and subsequently reheats the universe as it decays into photons, thus creating adiabatic CMB perturbations. We find that the fluctuations in the axion field can give rise to a nearly flat spectrum of adiabatic perturbations with a spectral tilt $\Delta n$ in the range $-0.1 \lesssim \Delta n \lesssim 1$., Comment: 15 pages, 1 figure; Version to appear in Nucl.Phys.B. Figure improved, discussion and one reference added

This paper starts by building the energy–momentum tensor of a perfect fluid of p -branes coupled to ( p+4 )-dimensional general relativity. Having three homogeneous and isotropic macroscopical spatial dimensions, the system gravity/fluid can be reduced to an effective theory over the branes. For the string fluid ( p=1 ) the effective theory is nothing but the 2D dilaton gravity where the potential for the scalar field, which is the scale factor of the macroscopical space, is fixed by the state equation and the three-dimensional geometry. This theory can be solved allowing us to compare some relevant aspects in our homogeneous and isotropic string cosmologies with those of the Robertson–Walker ones. In particular, unlike the point-particle models, the existence of an initial singularity is strongly sensitive to the state equation, and it is remarkable that this model picks out the radiation state equation as the canonical case where the big-bang is kinematically forbidden. Moreover, we cannot reduce the Robertson–Walker cosmologies to the limit when the string size approaches to zero, because the existence of an upper bound on the string size is not compatible with the big-bang. Some examples are presented.

We construct a locally supersymmetric worldsheet formulation of a non-Abelian Ramond-Neveu-Schwarz (NARNS) string theory where the string coordinates are noncommuting matrices in a group U(N). This is described by the two dimensional supergravity coupled to supersymmetric Yang-Mills fields and adjoint matters in the gauge group U(N). We show that our NARNS string theory has a free string limit where it becomes N-copies of usual RNS string which can be described by the orbifold conformal field theory corresponding to the covariant worldsheet version of the Matrix string theory of Dijkgraaf, Verlinde and Verlinde. In the weak coupling limit, i.e. $g_s \to 0$ where $g_s$ is the coupling constant of our theory related with the Yang-Mills coupling as $g_{YM}^{-2}=\alpha' g_s^2$, a new additional dimension appears in the string spectrum and it can be speculatively interpreted as the compactified eleven dimensional coordinate whose dynamics is given by an orbifold O(N) sigma model., Comment: 37 pages, RevTex, An appendix on our conventions and some identities is added. Added references. Corrected typos

We find explicit expression for the one-loop four-graviton amplitude in eleven-dimensional supergravity compactified on a circle. Represented in terms of the string coupling (proportional to the compactification radius) it takes the form of an infinite sum of perturbative string loop corrections. We also compute the amplitude in the case of compactification on a 2-torus, which is given by an SL(2, Z ) invariant expansion in powers of the torus area. We discuss the structure of quantum corrections in eleven-dimensional theory and their relation to string theory.

The unified theory of string and two-dimensional quantum gravity is considered. We introduce nontrivial dynamics for the two-dimensional metric gμν from the very beginning and calculate the path integral over the string coordinates and gμν without taking into account the order of integrations. Throughout the paper we use two different kinds of gauges - the covariant one of the harmonic type and also the conformal gauge, where the original (D + 1)-dimensional sigma model with quantum gravity becomes the (D+2)-dimensional sigma model on the classical background of gμν The general symmetries of the theory consist in the reparametrizations of the target space coordinates, in the conformal transformations of the metric and in the usual 2d diffeomorphisms. These symmetries do not disturb the structure of the background fields in the (D+2) -dimensional formulation. On the other hand the related arbitrariness of the renormalization does not affect the qualitative structure of the loop contributions to the Weyl anomaly. In the theory with quantum gravity the parameter a′ does not play as the parameter of the loop expansion. That is why the one-loop conditions of the Weyl invariance differs from the well known effective equations which arise in the standard approach when gμν is not quantized simultaneously with the string coordinates. Therefore, despite the new conditions of the Weyl invariance for the background fields are different from the standard effective equations, our result does not contradict to the standard approach. The new one-loop conditions of the Weyl invariance are much more complicated and contain the higher derivatives in the dilaton sector.

Noncompact groups, similar to those that appeared in various supergravity theories in the 1970's, have been turning up in recent studies of string theory. First it was discovered that moduli spaces of toroidal compactification are given by noncompact groups modded out by their maximal compact subgroups and discrete duality groups. Then it was found that many other moduli spaces have analogous descriptions. More recently, noncompact group symmetries have turned up in effective actions used to study string cosmology and other classical configurations. This paper explores these noncompact groups in the case of toroidal compactification both from the viewpoint of low-energy effective field theory, using the method of dimensional reduction, and from the viewpoint of the string theory world sheet. The conclusion is that all these symmetries are intimately related. In particular, we find that Chern--Simons terms in the three-form field strength $H_{\mu\nu\rho}$ play a crucial role., Comment: 40 pages, phyzzx, CALT-68-1790

We discuss the relation between space-time renormalization group equation for closed string field theory and world-sheet renormalization group equation for first-quantized strings. Restricting our attention to massless states we argue that there is a one-to-one correspondence between the fixed point solutions of the two renormalization group equations. In particular, we show how to extract the Fischler-Susskind mechanism from the string field theory equation in the case of the bosonic string.

We consider those tachyon backgrounds in 1 + 1 dimensional string theory which are invariant under one linear combination of time and space translations. These correspond to world-sheet theories which are a product of a free theory and a Liouville theory, where the Liouville central charge can take any value. The nonlinear tachyon field theory derived from the random matrix model gives exact information about the Liouville theory. We use this to study the Liouville theory in the “strong gravity” region, the critical string tachyon instability, and quantum cosmology.

If a conformal field theory is perturbed by a nearly marginal operator of dimension ( 1 h, 1 h ), then, under certain conditions and for a certain value of the perturbing parameter, we get a new conformal field theory. Starting from a string field theory based on the original conformal field theory, we show how to construct a classical solution in this string field theory to all orders in h , which represents string field theory formulated around the perturbed conformal field theory. We also show that the equation of motion of string field theory forces us to have a time-dependent dilaton background of the form ax 0 to order h 3 , which compensates for the change in the central charge of the conformal field theory under the perturbation.

Given a solution ψcl of the classical equations of motion in either closed or open string field theory based on a certain conformal field theory (CFT) we may define a shifted string field ψ=ψ − ψcl. The string field theory action expressed interms of the shifted field is then given by S(ψ) = S(ψ + ψcl − S(ψcl), where S(ψ) is the original action of the string field theory. In S(ψ) the coefficient QB of the quadratic term in ψ has the property that (QB)2 = 0. It was shown in a previous paper that in the limit when the background ψcl is weak, QB can be identified with the BRST charge of a new conformal field theory (CFT″), which is obtained by perturbing the orginal conformal field theory (CTF) by a marginal operator. In this paper we compute various on-shell S-matrix elements using the action S(ψ) and show that they are identical to the on-shell S-matrix in the string based on the new conformal field theory (CTF″). This in turn shows that S(ψ) describes string theory formulated around the background of this new conformal field theory.

The integrand and integration region for the N-point amplitude in the open oriented bosonic string are obtained to all orders in perturbation theory. The result is derived from the Witten covariant string field theory by using on-shell and off-shell conformal methods and Riemann surface mathematics. Although only the off-shell g-loop tachyon amplitudes are computed explicitly, the methods generalize to other external states. We derive the g-loop ghost-Jacobi identity in which the ghost correlation function cancels the jacobian factor in changing from second-quantized to first-quantized variables. Moduli space is discussed from several viewpoints and it is shown that string field theory provides an algorithm for its determination.

In a previous paper we wrote down the gauge-invariant action for open string field theory in an arbitrary background field describing a two-dimensional conformal field theory on the upper half-plane. In this paper we discuss the gauge fixing and the calculation of tree-level four-point amplitude in this string field theory, and show that the result agrees with the answer expected from the appropriate four-point correlation function in conformal field theory.

In the context of covariant string field theory, we investigate the existence and properties of nonperturbative vacua for the open bosonic string. The study is performed using successive orders in a level-truncation method. We find a candidate nonperturbative vacuum and obtain numerical values for nonzero expectation values of all fields to level two. Perturbation theory about this vacuum is shown to be well defined. The occurence and location of poles in the propagators of all fields to level two are established. We observe that string theories necessarily have running couplings at tree level and that the open bosonic string is asymptotically free. The nontrivial momentum dependence means that certain poles seen in the canonical vacuum are absent in the nonperturbative vacuum. This result follows from the nature of the string as an extended object and is likely to hold for general string theories.

We explicitly calculate the one-loop mass shifts for the gravitation, the Kalb-Ramond field and the dilaton in bosonic theory. These are found by computing the regularizedtwo-point functions on the torus and then subtracting the tadpole contributions. The results from here are consistent with general covariance and gauge invariance.

The Lorentz covariance of the light-cone string field theory of the bosonic orientable open-closed mixed system in 26 dimensions is studied. The Lorentz algebra defined by the classical Poisson bracket does not close. The quantum commutators of the Lorentz generators and the hamiltonian are defined by using the Bjorken-Johnson-Low limit and are computed in the perturbation theory. It is shown that the algebra closes at the quantum level to all orders of the perturbation theory up to possible effects of divergences in loop amplitudes. The non-vanishing terms at the tree level, which violate the covariance, are cancelled by anomalies of the open string one-loop terms and there is no higher-loop anomaly.

We explore the possibility of expressing classical bosonic p-brane field equations as the Wilson equation of a p + 1 dimensional scalar field theory. We find consistent results only for p = 1, in which case we recover bosonic string theory.

We argue that vacuum configuration of open strings correspond to Caldeira-Leggett models of dissipative quantum mechanics (DQM) evaluated at a delocalization critical point. This connection reveals that critical DQM can be expected to manifest reparametrization invariance (inherited from the conformal invariance of string theory) rather than just scale invariance. This connection should open up new ways of constructing analytic and approximate solutions of open string theory (in particular, topological solitons such as monopoles and instantons).

We give an operator formulation of the string field theory proposed by E. Witten using half-string oscillator modes. This formalism, identifying the physical string states as infinite matrices, is developed in such a way that interactions at the level of vertices can be calculated in terms of products and traces of these matrices without relying on ill-defined manipulations of functional integrals.

We consider Witten's formulation of the interacting open bosonic string field theory. In particular, we study the Fock space representation of the vertex, including the fields associated with the bosonized two-dimensional ghost oscillators. We calculate the vertex using conformal mapping techniques, and also via a decomposition of the constraints implied by the string overlap. The conformal mapping takes the overlap world sheet to an anticone, where the Neumann function may be found. We show how this is related to the mappings with branch cut singularities, which take this world sheet to the upper half plane.

The field theoretic equation of motion of string is interpreted as the exact renormalization group equation. The generators of string gauge invariance are the redundant operators of the RG; gauge equivalence is related to universality. We give a geometric interpretation of the theory and show in principle how to derive the S-matrix.

String theory exhibits a number of string miracles. These are “accidental” relations between couplings in the low-energy effective action which cannot be understood in terms of the symmetries of that action. It is shown that the existence of (0, 2) models need not be a string miracle, but can often be understood in terms of a discrete R symmetry. This understanding points to many new (0, 2) models, including some with gauge group SU(3) × SU(2) × U(1). Discrete R symmetries also protect some massless particles from acquiring a mass at special points in the moduli space of the conformal field theory. From a two dimensional point of view, this can be understood in terms of index theorems not only in the R but also in the NS sector. An important tool in the study of string compactification and conformal field theory is the perturbative expansion around a conformal field theory. We prove that the expansion of the space time superpotential around a space-time supersymmetric vacuum does not miss non-perturbative effects (as long as we do not expand around a boundary of moduli space). This shows that certain results which have been obtained to all orders in this expansion are exact. Also, it shows that the construction of the (0, 2) models in perturbation theory around (2, 2) models is exact as well.

Mechanisms for the preferential expansion of three of the spatial dimensions of a ten-dimensional universe are considered, in the context of the ten-dimensional supertring derived field theory with the gravitational field, the dilaton field and the three-index tensor field H . Two models in which the H field defines spatial three-spaces are investigated. One is quasi-chaotic at early times and leads to uniform expansion of all dimensions at later times; the other gives preferential expansion of one three-space. One possibility for the transition to the observed universe, through particle production by the dilaton field, is studied.

Gauge-invariant actions for free bosonic string fields are derived from the corresponding BRST-invariant gauge-fixed action. The auxiliary components needed to make the action local and maintain unconstrained gauge invariance arise from an SU(1,1)-invariant sector of a pair of opposite-metric fermionic string coordinates or of an extra timelike bosonic one.

A simple method for constructing gauge invariant actions for both free massive bosonic fields of arbitrary spin as well as for string fields is described. The method is based on the use of Lagrance multipliers and the introduction of auxiliary fields and results in expressions that are straightforward generalizations of the Stueckelberg action for massive vector fields. An application of the method to the free bosonic string gives the corresponding gauge invariant action for the string field functional in a rather pedestrian way. Although the role of the critical dimensions is apparent in the derivation we show how to extend the result outside of 26 dimensions.

We set up the Polyakov path integral formalism for the open and unoriented bosonic string at tree and one-loop level of string perturbation theory. The relevant Teichmuller parameters, conformal Killing vectors, “large” diffeomorphisms, and functional determinants are given explicitly. As a toy application the one-loop vacuum energy for the open and closed sectors of the unoriented bosonic string is computed.

The static potentials from bosonic string models are examined to the two-loop level. Available MC data for 4-dimensional SU(3) are in approximate agreement with the universal leading order predictions but fail to reproduce the Nambu string when the two-loop terms are included in the analysis. We also confront Nambu string model predictions for T c σ with SU(2), SU(3) and SU(N → ∞) MC data. In the SU(2) case, impressive agreement is found both in 3 and 4 dimensions. In contrast, the data for SU(3) and SU(N → ∞) differ from the string model prediction. These theories have identical values for T c σ , indicating that N → ∞ is a good approximation to N = 3.

We formulate string field theory geometrically by writing each term in the field theory action. as an expectation value in the 2-dimensional conformal field theory on the world surface. We show how the symmetries of the theory can be analyzed and the gauge invariance demonstrated from this point of view. As an application, we give a complete proof of the gauge invariance of Witten's open string field theory.

We develop the point of view that the string vertex is nothing but the Bogoliubov-transformed SL(2, R)-invariant out-ground state induced by the conformal (Mandelstam) mapping. We explicitly demonstrate this idea, using the open string field theory of Witten. The notion of asymptotic string fields is introduced and the vertex is constructed without referring to the conventional delta-function overlap equation. The ghost insertion factors are understood as a consequence of the presence of the kernels in the ghost Bogoliubov transformations. We prove the vanishing of the anomaly of the BRST charge under the Bogoliubov transformations at d = 26, from which the BRST invariance of the vertex associated with any Mandelstam map follows, immediately. Underlying physical pictures are contemplated.

We show that two-dimensional non-linear sigma models that are finite to some order in perturbation theory need not be conformally invariant to that order, although they may be invariant under rigid scale transformations. We give the conditions sufficient for one-loop conformal invariance and discuss implications for string theory.

We set up a formulation of the theory of interacting strings based on the physical picture of a string in the infinite-momentum frame as the continuum limit of a chain of mass points moving in the transverse space and interacting via nearest-neighbor harmonic potentials. The theory is completely formulated for these “discrete strings” and the continuum limit is then studied in the context of perturbation theory in the interaction strength. We establish that this limit yields the bosonic dual string model, provided the space-time dimension is 26. This article completes for the bosonic string a program initiated nearly a decade ago by Giles and Thorn. Second-quantized field equations are derived for discrete strings, and a “classical” limit of these equations is discussed.

The relations betweeb a field F μν M satisfying the usual Maxwell equations and a field F μν D satisfying the symmetric Maxwell-Dirac equations, and the singular potential solving both of these is given. The action principle is formulated in both forms and the reality of the string is shown. A string with spin is constructed by placing electric charges at its end-points. The motion and interactions of the string, the relation between flux and angular momentum quantization and the passage to two-body Hamiltonians are examined.

A formulation of Witten's field theory of interacting open bosonic strings has been given based on a purely cubic action principle; however, the construction involves formal arguments which could in principle be invalidated by anomalies. We establish that the potential quantum anomalies cancel identically, making use of the recently developed operator formalism of Witten's theory. We first give an operator construction of the BRST current j(σ), determining its proper normal-ordering prescription through the requirement that j(σ) itself be BRST invariant. (As a by-product, we also obtain its anomalous current-current anticommutator.) The anomaly calculation then involves numerous nontrivial identities for the Neumann coefficients occurring in the oscillator form of the vertex, and gives a stringent consistency check upon the form of j(σ). Finally, we comment upon further issues to be resolved in pursuing this approach to string field theory.

By quantizing bosonic string mechanics in harmonic world-sheet coordinates, we obtain a BRST operator whose coordinates include the world-sheet metric. These gauges are covariant under an ISp(2) symmetry which includes not only BRST transformations but also anti-BRST and the Sp(2) which relates them. This symmetry of the gauge-fixed formalism is a subgroup of an OSp(1, 1|2) useful for gauge-covariant free string field theory.

Covariant canonical quantization of the bosonic string is performed, based on the BRS invariance principle. Defining a physical state by a modified form of Kugo-Ojima's subsidiary condition QB|phys〉 = 0 (QB = BRS charge), we prove a no-ghost theorem in a simple manner. There an interesting relation between critical dimension and BRS symmetry is noticed; namely, the conditions D = 26 and α0 = 1 for the space-time dimension D and the zero-intercept α0 of leading trajectory are required by the nilpotency QB2 = 0 of the BRS charge. In addition, ghost number fractionization is observed in this system.

We covariantly first-quantize, via the BRST approach, the closed bosonic string compactified on a torus — in particular the 18-dimensional theory with E8 × E8 gauge bosons. Further, we give the corresponding free field theory. Thus the model is ready for the introduction of string interactions, and may be very useful for the study of gauge interactions in string theories.

The theory of interacting heterotic strings is presented. Vertex operators are derived in both the bosonic and fermionic formulations of the theory and are shown to be consistent with gauge invariance, Lorentz invariance, and supersymmetry. Three- and four-point amplitudes for the scattering of massless string states are calculated and used to derive the low-energy field theory limit of the heterotic string. Divergences in string theories are discussed and it is shown that one-loop heterotic string amplitudes are finite and modular invariant only for gauge group E 8 ×E 8 or spin (32)/Z 2 .