This is the sixth of a series of lectures on F-theory. The lectures will present the theory of elliptically fibered Calabi-Yau manifolds in considerable detail, explain how these manifolds are used to produce string vacua by means of the ``F-theory'' construction, and how various properties of these string vacua are determined by the corresponding Calabi-Yau manifolds.

The talk will begin with a review of some recent progress in non-supersymmetric string theory, motivating a careful study of such backgrounds. Many non-supersymmetric closed string theories have negative modes (``tachyons''). I will describe an approach to understanding the result of tachyon condensation using configurations in string theory of effectively negative but finite tension (such as orientifold planes and their S-duals).

This is the fifth of a series of lectures on F-theory. The lectures will present the theory of elliptically fibered Calabi-Yau manifolds in considerable detail, explain how these manifolds are used to produce string vacua by means of the ``F-theory'' construction, and how various properties of these string vacua are determined by the corresponding Calabi-Yau manifolds.

We argue that vacua of string theory which asymptote at weak coupling to linear dilaton backgrounds are holographic (like AdS backgrounds). The full string theory in such vacua is ``dual'' to a theory without gravity in fewer dimensions. The dual theory is generically not a local quantum field theory. Excitations of the string vacuum, which can be studied in the weak coupling region using worldsheet methods, give rise to observables in the dual theory. An interesting example is string theory in the near-horizon background of parallel NS5-branes, the CHS model, which is dual to the decoupled NS5-brane theory (``little string theory''). This duality can be used to study some of the observables in this theory and some of their correlation functions.

This is the fourth of a series of lectures on F-theory. The lectures will present the theory of elliptically fibered Calabi-Yau manifolds in considerable detail, explain how these manifolds are used to produce string vacua by means of the ``F-theory'' construction, and how various properties of these string vacua are determined by the corresponding Calabi-Yau manifolds.

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.

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.

Some of the old problems in algebraic geometry, as well as relatively new problems in the theory of quantization, were solved using topological sigma models. The sigma models describe maps of a manifold M to a target space X. It is very well-known that no sensible theory exists when the dimension of M is greater than two. In my talk I will try to argue in favor of the existence of an interesting theory of maps in the case where M is a four-dimensional Riemannian manifold and X is a classifying space of some compact Lie group (or its finite-dimensional model). To get there we will need to introduce & develop certain aspects of Donaldson theory and higher-dimensional analogues of Whitman hierarchies. No knowledge of Donaldson theory or Whitman hierarchies is necessary.

This is the second of a series of lectures on F-theory. The lectures will present the theory of elliptically fibered Calabi-Yau manifolds in considerable detail, explain how these manifolds are used to produce string vacua by means of the ``F-theory'' construction, and how various properties of these string vacua are determined by the corresponding Calabi-Yau manifolds.

A smooth one-parameter family F₀ : Mⁿx [0,T) ---> (Nⁿ⁺¹,g) of hypersurfaces in a Riemannian manifold (N⁽ⁿ⁺¹⁾,g) is said to move by its curvature if it satisfies an evolution equation of the form
(d/dt) F(p,t) = f(p,t) p Mⁿ, t [0,T),
such that at each point of the surface its speed in normal direction is a function $f$ of the extrinsic curvature of the hypersurface. Examples such as the flow by mean curvature, flow by Gauss curvature or flow by inverse mean curvature arise naturally both in Differential Geometry, where they exhibit fascinating interactions between the extrinsic curvature of the surfaces and intrinsic geometric properties of the ambient manifold, and in Mathematical Physics, where they serve as models for the evolution of interfaces in phase transitions. The first lecture gives a general introduction to the main examples and phenomena and highlights some recent results. The second lecture shows how parabolic rescaling techniques can be combined with a priori estimates to study and in some cases classify possible singularities of the mean curvature flow. The series concludes with applications of hypersurface families in General relativity, including a recent proof of an optimal lower bound for the total energy of an isolated gravitating system by Huisken and Ilmanen.

This is the first of a series of lectures on F-theory. The lectures will present the theory of elliptically fibered Calabi-Yau manifolds in considerable detail, explain how these manifolds are used to produce string vacua by means of the ``F-theory'' construction, and how various properties of these string vacua are determined by the corresponding Calabi-Yau manifolds.

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.

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

The dynamics of D branes moving in certain string theory backgrounds can be used to learn about supersymmetric field theories in several dimensions. In this talk I will center on the realization of four-dimensional N=2 and N=1 gauge field theories, by using two types of backgrounds. The first (`brane configurations') involves D branes in the presence of NS fivebranes, whereas the second involves D branes sitting at singular points. A wide variety of conformal theories with marginal couplings, and of finite N=1 theories arises from these constructions. I will also emphasize the equivalence of different constructions by T duality.

The AdS/CFT duality buys us a consistent non-perturbative description of gravity (in some class of backgrounds) at the price of manifest spacetime causal structure and locality. In this talk I will describe the bare beginnings of an attempt to understand these issues by studying the duality in Lorentzian signature and showing how states, operators, and classical fields are mapped between the dual theories. I will then describe two examples of classical D-brane and string probes of AdS backgrounds and show that locality in AdS shows up as scale locality for the dual boundary configurations. I will close with some suggestions for what the horizon and singularity of the AdS black hole might mean in the boundary CFT.

Supersymmetry can severely constrain higher derivative terms in the effective action of both Yang-Mills theories and string theories. I will explain how to obtain these constraints.

A surprising anomaly in diffeomorphisms acting on the normal bundle of the M-theory fivebrane remained a puzzle for almost two years. We present a cancellation mechanism based on a careful treatment of the fivebrane by smoothing it out through coupling to gravity which results in a relation between antisymmetric tensor gauge transformations and diffeomorphisms and leads to a complete cancellation of the anomalies. This formalism is used for derivation of the R-symmetry anomalies of the A_N (0,2) tensor-multiplet theories and for understanding the role of anomalies in the AdS/CFT correspondence. The Kaluza-Klein origin of Chern-Simons terms in lower-dimensional supergravity theories is displayed via modification of the Freund-Rubin ansatz. The results imply the existence of interesting 1/N corrections in the AdS/CFT correspondence. A simple derivation of black hole entropy in d=4, N=2 compactifications of M-theory is presented.

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.

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

To define a consistent perturbative geometric heterotic compactification one must specify not only a Calabi-Yau manifold M but also a bundle E on M. The bundle E is required to satisfy a subtle constraint known as ``stability,'' which depends upon the Kähler form. This dependence upon the Kähler form is highly nontrivial---the Kähler cone splits into subcones, with a distinct moduli space of bundles in each subcone---and has long been overlooked by physicists. In this talk we shall describe this behavior and its physical manifestation.

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.

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.

We study the conformal field theory describing the extreme low-energy excitations of parallel D3-branes located at a singular point of the transverse space. For quotient singularities such a description is known. Using the fact that partial resolutions of quotient singularities contain other kinds of singular points, as well as a mapping of the moduli space of a singularity onto the parameter space of the corresponding field theory, we compute the worldvolume field theory for branes at more general singularities. We compare our results to the predictions of an extended version of Maldacena's AdS/CFT correspondence as presented last week by D. Morrison.

The ``horizon'' of N coincident branes in Minkowski space is the unit sphere in the transverse directions; in a certain scaling limit, string or M-theory in the presence of branes is dual to a compactification on the product of an anti-de Sitter space with the horizon. We study the analogous limit when N branes are placed at a singular point, so that the horizon becomes the so-called ``link'' of the singularity, and is no longer a sphere. A similar scaling argument leads to a natural extension of Maldacena's celebrated ``AdS/CFT correspondence conjecture'' to this situation. The conformal field theories in question have less supersymmetry than the cases studied by Maldacena, with the amount being determined by the Killing spinors on the horizon manifold. An important mathematical tool is the relationship -- derived some years ago by C. Bär -- between Killing spinors on a manifold and covariantly constant spinors on the cone over that manifold.