We revisit the singular IIB supergravity solution describing M fractional 3-branes on the conifold (hep-th/0002159). Its 5-form flux decreases, which we explain by showing that the relevant N=1 SUSY SU(N+M) x SU(N) gauge theory undergoes repeated Seiberg-duality transformations in which N -> N-M. Far in the IR the gauge theory confines; its chiral symmetry breaking removed the singularity of hep-th/0002159 by deforming the conifold. We propose a non-singular pure-supergravity background dual to the field theory on all scales, with small curvature everywhere if the 't Hooft coupling (g_s M) is large. In the UV it approaches that of hep-th/0002159, incorporating the logarithmic flow of couplings. In the IR the deformation of the conifold gives a geometrical realization of chiral symmetry breaking and confinement. We suggest that pure N=1 Yang-Mills may be dual to strings propagating at small (g_s M) on a warped deformed conifold. We note also that the standard model itself may lie at the base of a duality cascade.

I review the homological mirror symmetry conjecture of Kontsevich and its relation with topological D-branes. This conjecture is nontrivial even for tori, which allows one to perform some interesting checks. It turns out that in the presence of a general B-field Kontsevich's conjecture must be modified. In particular, on the B-side, coherent sheaves on a Calabi-Yau must be replaced by coherent sheaves on a certain `noncommutative deformation' of the Calabi-Yau. By contrast, an algebraic description of the A-model branes, if it exists, should involve a noncommutative algebra even for a vanishing B-field.

We study the physics of D6 branes wrapped on supersymmetric three-cycles in type IIA string theory. In particular, we argue that although the superpotential in the resulting field theories vanishes to all orders in \alpha', nonperturbative contributions (from disk instantons) should generically be present. We discuss how mirror symmetry can relate these to tree-level sigma model computations in some circumstances. We also present an illustration of the role that background closed string moduli play in the brane worldvolume theory.

I discuss recent work attempting to construct stable vacua of string theory without supersymmetry.

The Holographic principle---in the form of the AdS/CFT correspondence---suggests a relation between processes in quantum gravity and phenomena in ordinary quantum field theory. The old `sum over histories' semi-classical approach to quantum gravity can be revisited in this light, by studying various gravitational instantons in AdS. We show that the AdS/CFT correspondence may be extended to spaces which are only locally asymptotically AdS, by examining the properties of the AdS-Taub-NUT and AdS-Taub-Bolt spacetimes. We also use holography to show that spacetime topology change in quantum gravity is a unitary process, in contrast to suggestions in the old `sum over histories' literature.

General conjectures about the SL(2,Z) modular transformation properties of N=4 super-Yang-Mills correlation functions are presented. It is shown how these modular transformation properties arise from the conjectured duality with IIB string theory on AdS₅ x S⁵. We discuss in detail a prediction of the AdS duality: that N=4 field theory, in an appropriate limit, must exhibit bonus symmetries, corresponding to the enhanced symmetries of IIB string theory in its supergravity limit.

Abstract: A great deal of the physics of perturbative open string theory can be understood by constructing weakly coupled 2D field theories with boundary which flow via the RG to the theory of the open string worldsheet in the background of interest. Among the applications are computing branes monodromies, and understanding the qualitative behavior of brane physics as open and closed string moduli vary and tachyons are condensed. I will discuss a general construction for (generically nonsupersymmetric) open string backgrounds with 'B-type' branes. I will also discuss work in progress on analogous constructions for 'A-type' branes, and relations to closed string tachyon condensation.

D-branes are usually treated as classical geometric objects in string theory. They can be defined more abstractly in conformal field theory using the boundary state formalism. Some general issues involving this formalism will be discussed and illustrated with examples involving D-branes on both symmetric and asymmetric orbifolds.

Motivated by phenomenological applications we consider (warped) compactifications of type IIA string theory on Calabi-Yau fourfolds in the presence of Ramond-Ramond fluxes. The main goal of the talk is a description of low-energy physics, including a new superspace formulation of two-dimensional N = (2,2) dilaton supergravity coupled to matter, and computation of the effective superpotentials induced by fluxes.

I will introduce generalized Kahler geometry and explain its relation to supersymmetric bihermitian geometry. I will then explain the behaviour of this geometry under T-duality transformations (which I will also define). Finally I will discuss topological D-branes in this geometry.

Non-Lagrangian branes with bundles carrying a non-trivial field strength are allowed in the A-model. A stability condition involving the field strength was derived by current-matching requirement. By considering the action of Hamiltonian vector fields, I relate this condition to a moment map, thus unifying the non-Lagrangian A-branes with the Lagrangian ones in a geometric framework. This leads to an extension to bundles of previously known mirror-symmetric exchanges.

The connection between large N gauge theories and string theories is typically made perturbatively. Namely, the Taylor expansion in 1/N corresponds to the genus expansion in string theory. Instanton effects in the gauge theory, however, go like exp(-N) and correspond to non-perturbative effects in the string theory. I will argue, in the context of the N=4 super Yang-Mills theory, how very non-trivial large N non-renormalization theorems in this instanton sector follow simply from the dual string description. The argument assumes little other than the existence of a derivative expansion in string theory. As a particular example, one can explain the striking agreement found by Dorey et al between certain weak coupling instanton calculations and the strong coupling predictions of supergravity.

(1) After a general description of quiver gauge theories (QGT) which may possess 4-dimensional conformal invariance and which arise from orbifolded AdS/CFT some successful comparisons to the nonsupersymmetric Minimal Standard Model (MSM) are made. (2) A QGT scheme where grand unification of the MSM occurs at about 4 TeV is discussed. This has consequences for the next big collider, the Large Hadron Collider at CERN. (3) It is explained how nonsupersymmetric QGT can subsume an MSM which has no quadratic divergence in its Higgs scalar propagator. The necessary and sufficient condition for such naturalness is that the progenitor QGT possess chiral fermions.

It is proposed that conformality at the TeV scale be used to solve the hierarchy problem and to restrict fields additional to those of the standard model. This idea provides rigid predictions.

We present a framework for the study of D-branes on Calabi-Yau compactifications, valid throughout moduli space. The main elements of this framework are the identification of D-branes as objects in a category, and a new notion of stability. We apply these ideas to the study of the spectrum of BPS branes on a specific Calabi-Yau and the algebra of BPS states.

Using the generalized Konishi anomaly equation in the chiral ring of a four dimensional N=1 SU(N) supersymmetric gauge theory with fundamental and anti-fundamental matters, we derive the large N_f effective superpotentials of the theory. These superpotentials are singular when expressed in terms of local gauge-invariant chiral fields. We show that these effective superpotentials, despite being singular, are perfectly sensible, i.e. they correctly describe the moduli space of vacua, are consistent under RG flow to fewer flavors upon turning on masses, and also (in the case of SU(2) gauge group) reproduce by a tree-level calculation all the higher-derivative F-terms recently calculated by Beasley and Witten by instanton methods.

Strings are sensitive to geometric features of a target space in a way different from point particles. T-duality maps IIA/B theory in one background to IIB/A in a geometrically different looking background, which leads to an isomorphic worldsheet conformal field theory. If a target space has a U(1) isometry without fixed points, the Buscher rules yield a dual target space. It too has a U(1) isometry. We consider (quotients of) weighted homogeous affine hypersurfaces as local models of Calabi-Yau singularities. These have a U(1) action with a fixed point. Worldsheet instantons crucially change the Buscher rules, breaking the symmetry. We propose a simple worldsheet picture to take into account the worldsheet instanton contributions. In some cases, the dual cft has a related sigma model interpretation; it can be regarded as the `throat' region of a configuration of fivebranes. Interesting relations exist between isometries of one target space and affine symmetries of the dual.

D-branes on orbifolds with discrete torsion can be defined using projective representations of the orbifold group. We discuss the examples C³ / Z_n × Z_n in detail.

The talk will be focused on the relation between K theory and M theory and it's generalization to nonzero $B$ field. The main idea is a comparison of the long distance partition functions of the two theories. In the process we will outline a derivation of the (twisted) Atiyah - Hirzebruch spectral sequence from M theory. We will also discuss the relation between $SL(2,Z)$ duality and K theory.

A Calabi-Yau orbifold in complex dimension 3 is locally modeled on C^3/G where G is a finite subgroup of SL(3,C). From a geometrical perspective I view this as an orbifold with boundary S^5/G. To the geometry of the boundary we can associate the $\eta$-invariant which measures the spectral asymmetry of the Dirac operator. It turns out that this analytical invariant is already encoded in the algebraic package describing the variety C^3/G. In the first part of my talk I will describe this very curious relation. In the second part, I will focus on the case of isolated singularities and their crepant resolutions. Via the Atiyah-Patodi-Singer index theorem, the $\eta$-invariant is used to gain information about multiplicative structure in cohomology. This is expressed in terms of the representation theory of the finite group.

Recent developments have suggested that the classical physics of gravity in a given number of dimensions can be related to a non-gravitational theory in a lower number of dimensions. I will argue that the radial direction of five-dimensional AdS space can be generated by renormalization group (RG) flow from a dual four-dimensional theory. First, I will show that integrating out the physics outside finite balls in AdS spacetime generates a family of actions that satisfies an RG-like equation. Next, I will argue that the trace of the AdS spacetime stress tensor contains analogues of field theory beta functions and gives a gravitational analogue of an RG equation. Turning things around, I will show that the conformal anomaly computed from the four-dimensional field theory dual to AdS space completely encodes the gravitational physics of the skin of the five-dimensional spacetime. Finally, I will argue that recovering the deep interior of the five-dimensional spacetime from the four-dimensional field theory will require exotic new approaches to the renormalization group.

Abstract: The derived category of coherent sheaves almost certainly plays a role of fundamental importance in string theory on an algebraic variety. We review the various ideas that have led to this belief and outline the resulting consequences for geometry and physics.

A limit of a suitably-chosen string theory compactified on an elliptic Calabi-Yau threefold is believed to be equivalent to a Yang-Mills field theory with gravity in six dimensions. The Lie group associated to this Yang-Mills theory is encoded in the geometry of the Calabi-Yau space --- the Cartan subalgebra is generated by ruled surfaces and the weights of representations of fields appear as rational curves living inside the threefold. Cancellation of anomalies in the six-dimensional field theory predicts rather peculiar constraints on the configurations of such ruled surfaces and rational curves.

I will report on progress and problems with the classification of four-dimensional N=2 superconformal field theories with two complex-dimensional Coulomb branches.