We review the generalization of field theory to space-time with noncommuting coordinates, starting with the basics and covering most of the active directions of research. Such theories are now known to emerge from limits of M theory and string theory, and to describe quantum Hall states. In the last few years they have been studied intensively, and many qualitatively new phenomena have been discovered, both on the classical and quantum level. Submitted to Reviews of Modern Physics.

this paper, we will find a third conformal field theory of the conifold -- call it the C model. In the C model, there is a Z 2 discrete torsion sitting at the "singularity," which is in fact not a singularity in the conformal field theory sense, but just a region in which stringy effects are essential. The C model has no marginal operators (in particular the analogs of H

Some relationships between string theories and eleven-dimensional supergravity are discussed and reviewed. We see how some relationships can be derived from others. The cases of N = 2 supersymmetry in nine dimensions and N = 4 supersymmetry in four dimensions are discussed in some detail. The latter case leads to consideration of quotients of a K3 surface times a torus and to a possible peculiar relationship between eleven-dimensional supergravity and the heterotic strings in ten dimensions. Recent ideas concerning duality in string theory (such as [1, 2, 3]) have given hope to gaining insights into some non-perturbative form of string theory. Given the current status of string theory it is not easy to see how to prove such statements about duality. Rather, one can take the attitude that such dualities could be used, in part, as a defining property of string

In this paper the relationship between the classical description of the resolution of quotient singularities and the string picture is discussed in the context of N=(2,2) superconformal field theories. A method for the analysis of quotients locally of the form C d /G where G is abelian is presented. Methods derived from mirror symmetry are used to study the moduli space of the blowing-up process. The case C 2 /Zn is analyzed explicitly. One of the earliest manifolds considered for a superstring target space was a blown-up orbifold [1]. In this case the orbifold (the “Z-orbifold”) was constructed by modding out a torus of 6 real dimensions by a group isomorphic to Z3. This group action has 27 fixed points leading to 27 isolated quotient singularities in the resulting space. Algebraic geometry then

We study the D-brane spectrum of N = 2 string orbifold theories using the boundary state formalism. The construction is carried out for orbifolds with isolated singularities, non-isolated singularities and orbifolds with discrete torsion. Our results agree with the corresponding K-theoretic predictions when they are available and generalize them when they are not. This suggests that the classification of boundary states provides a sort of ”quantum K-theory ” just as chiral rings in CFT provide ”quantum ” generalizations of cohomology. We discuss the identification of these states with D-branes wrapping holomorphic cycles in the large radius limit of the CFT moduli space. The example C3 /Z3 is worked out in full detail using local mirror symmetry techniques. We find a precise correspondence between fractional branes at the orbifold point and configurations of D-branes described by vector bundles on the exceptional P2 cycle.

Abstract not found

Abstract not found

We present a nonsupersymmetric orbifold of type II string theory and show that it has vanishing cosmological constant at the one and two loop level. We argue heuristically that the cancellation persists at higher loops. July

We find the orbifold analog of the topological relation recently found by Freed and Witten which restricts the allowed D-brane configurations of Type II vacua with a topologically non-trivial flat B-field. The result relies in Douglas proposal – which we derive from worldsheet consistency conditions – of embedding projective representations on open string Chan-Paton factors when considering orbifolds with discrete torsion. The orbifold action on open strings gives a natural definition of the algebraic K-theory group – using twisted cross products – responsible for measuring Ramond-Ramond charges in orbifolds with discrete torsion. We show that the correspondence between fractional branes and Ramond-Ramond fields follows in an interesting fashion from the way that discrete torsion is implemented on open and closed strings. January

Most intersecting D-brane vacua in the literature contain additional massless adjoint fields in their low energy spectrum. The existence of these additional fields make it difficult to obtain negative beta functions and, eventually, asymptotic freedom. We address this important issue for N = 1 intersecting D-brane models, rephrasing the problems in terms of (open string) moduli stabilization. In particular, we consider a Z2 × Z2 orientifold construction where D6-branes wrap rigid 3-cycles and such extra adjoint fields do not arise. We derive the model building rules and consistency conditions for intersecting branes in this background, and provide N = 1 chiral vacua free of adjoint fields. More precisely, we construct a Pati-Salam-like model whose SU(4) gauge group is asymptotically free. We also comment on the application of these results for obtaining gaugino condensation in chiral D-brane models. Finally, we embed our constructions in the framework of flux compactification, and construct new classes of N = 1 and N = 0 chiral flux vacua. Permanent addressContents