Covariant quantization of string theories is developed in the context of conformal field theory and the BRST quantization procedure. The BRST method is used to covariantly quantize superstrings, and in particular to construct the vertex operators for string emission as well as the supersymmetry charge. The calculation of string loop diagrams is sketched. We discuss how conformal methods can be used to study string compactification and dynamics.

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A new type of superstring theory is constructed as a chiral combination of the closed D = 26 bosonic and D = 10 fermionic strings. The theory is supersymmetric, Lorentz invariant, and free of tachyons. Consistency requires the gauge group to be Spin(32)/Z 2 or £ 8 x E%.PACS numbers: ll.30. Pb, 11.30.Ly, 12.10.En Recent interest in superstring unified field theories has been sparked by the discovery of Green and Schwarz 1 that nonorientable (type I) open and closed superstrings 2 with iV= 1 supersymmetry are finite and free of anomalies if the gauge group is SO(32). Previously the only consistent, anomaly-free 3 superstring theory was that of orientable (type II) N = 2 supersymmetric closed strings. The new theory has the advantage of already containing a large (and unique) gauge group. It is much easier to contemplate this theory producing the low-energy gauge group, as well as families of chiral massless fermions, upon compactification of the original ten dimensions. Witten has discussed some of the phenomenology of this theory and has shown that it is easy to imagine compactifications that yield an SU(5) theory with any number of standard fermionic generations. 4 The anomaly cancellation mechanism of Green and Schwarz is based on group theoretical properties of SO(32) which are shared by only one other semisimple Lie group, namely E%x E s . Such a group, however, cannot appear in the standard form of open-string theory, in which gauge groups are introduced by attaching quantum numbers to the ends of the string and Chan-Paton factors 5 to the scattering amplitudes. This procedure yields only the gauge groups SO(A0 and Sp(2A0. 6 The correspondence between the lowenergy limit of existing supersymmetric string theories with anomaly-free, Z) = 10, supergravity field theories suggests the existence of a new kind of string theory whose low-energy limit would have an E s x E% gauge group. Eschewing the Chan-Paton route to gauge groups for open strings, one might try to obtain fgXfg by compactifying a higher-dimensional closedstring theory. An important clue to how such a theory might arise is provided by the work of Frenkel and Kac. 7 J In this Letter we shall outline the construction of a new kind of closed-string theory, whose low-energy limit is Z> = 10, N=l supergravity coupled to supersymmetric Yang-Mills theory with gauge group Spin(32)/Z 2 or E s x E%. This theory is constructed as a hybrid of the Z> = 10 fermionic string and the Z> = 26 bosonic string, which preserves the appealing features of both. We show that the orientable, closed heterotic 8 string has an N = 1 supersymmetric spectrum of states of positive metric, is free of tachyons and is Lorentz invariant. The requirement that gravitational and gauge anomalies be absent necessitates the compactification of the extra sixteen bosonic coordinates of the heterotic string on a maximal torus of determined radius, in a way that produces gauge groups Spin(32)/Z 2 or E s xE s .We further argue that the heterotic loop diagrams are free of all infinities-thus...

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We consider two dimensional supergravity coupled toĉ = 1 matter. This system can also be interpreted as noncritical type 0 string theory in a two dimensional target space. After reviewing and extending the traditional descriptions of this class of theories, we provide a matrix model description. The 0B theory is similar to the realization of two dimensional bosonic string theory via matrix quantum mechanics in an inverted harmonic oscillator potential; the difference is that we expand around a non-perturbatively stable vacuum, where the matrix eigenvalues are equally distributed on both sides of the potential. The 0A theory is described by a quiver matrix model. July 2003 made in [7].2 Equivalently, this model can be described as quantum mechanics of a unitary matrix U with potential Tr(U + U † ). After the double scaling limit is taken, both models are described by fermions in an upside down harmonic oscillator potential filling both sides of the potential to Fermi level −µ.3 The spectrum of D-branes in 10-d type 0 theories was studied in [21][22][23][24][25].3 the boundary states of the full quantum theory, and their relation to the minisuperspace wavefunctions. The annulus partition function is also considered. Section 8 discusses some aspects of the worldvolume dynamics of D-branes and their interactions with closed string fields. 4The precise nature of our proposal for a type 0B matrix model is presented in section 9, where we check the torus worldsheet calculation against a computation of the finite temperature matrix model free energy. Section 10 introduces the type 0A matrix model and performs the analogous check. In section 11 we study the relation of tachyon condensation and rolling matrix eigenvalues [7-9] through a calculation of the disk expectation value of the ground ring generators. We also explore the radiation of closed strings produced by the decay. Three appendices contain useful technical results.It is also possible to show thatĉ < 1 superconformal minimal models coupled to 2d supergravity are dual to unitary matrix models, 5 as one might expect on the basis of gravitational RG flow [30][31][32]. These matrix models were solved in the planar limit in [33], and in the double scaling limit in [34][35][36]. These models will be the subject of a separate publication.Note Added: While completing this manuscript, we learned of work by T. Takayanagi and N. Toumbas [37] where the matrix model for 2d type 0 strings is also proposed. The 2d Fermionic StringFermionic strings are described by N = 1 supersymmetric world sheet field theories coupled to worldsheet supergravity. The construction of type II string theories requires the existence of a non-anomalous chiral (−1) F L symmetry of the worldsheet theory. The generic background may only admit a nonchiral (−1) F ; the use of this class of (GSO) projection gives type 0 string theory (see [38] for a review).4 Sections 2-8 review known material but also describe many new results. The reader who is impatient to get to the matrix model can move directly to...

Covariant quantization of string theories is developed in the context of conformal field theory and the BRST quantization procedure. The BRST method is used to covariantly quantize superstrings, and in particular to construct the vertex operators for string emission as well as the supersymmetry charge. The calculation of string loop diagrams is sketched. We discuss how conformal methods can be used to study string compactification and dynamics.

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After the work of Seiberg and Witten, it has been seen that the dynamics of N=2 Yang-Mills theory is governed by a Riemann surface $\Sigma$. In particular, the integral of a special differential $\lambda_{SW}$ over (a subset of) the periods of $\Sigma$ gives the mass formula for BPS-saturated states. We show that, for each simple group $G$, the Riemann surface is a spectral curve of the periodic Toda lattice for the dual group, $G^\vee$, whose affine Dynkin diagram is the dual of that of $G$. This curve is not unique, rather it depends on the choice of a representation $\rho$ of $G^\vee$; however, different choices of $\rho$ lead to equivalent constructions. The Seiberg-Witten differential $\lambda_{SW}$ is naturally expressed in Toda variables, and the N=2 Yang-Mills pre-potential is the free energy of a topological field theory defined by the data $\Sigma_{\gg,\rho}$ and $\lambda_{SW}$.Comment: 20 pages, latex, 3 uuencoded figures (needs epsf.tex); minor errors corrected, references adde