Publications

No abstract is provided for this article.

These notes are devoted to explaining aspects of the mirror manifold problem that can be naturally understood from the point of view of topological field theory. Basically this involves studying the topological field theories made by twisting $N=2$ sigma models. This is mainly a review of old results, except for the discussion in \S7 of certain facts that may be relevant to constructing the ``mirror map'' between mirror moduli spaces.

Some properties of the anomaly-free O(32) superstring theory recently discovered by Green and Schwarz are discussed. With proper choice of ground state, the theory leads in four dimensions to an SU(5) theory with any desired number of standard generations (and no exotic or mirror fermions). It predicts axions and stable Nielsen-Olesen vortex lines. It can be consistently compactified only if certain topological conditions are imposed.

For various reasons, it seems necessary to include complex saddle points in the "Euclidean" path integral of General Relativity. But some sort of restriction on the allowed complex saddle points is needed to avoid various unphysical examples. In this article, a speculative proposal is made concerning a possible restriction on the allowed saddle points in the gravitational path integral. The proposal is motivated by recent work of Kontsevich and Segal on complex metrics in quantum field theory, and earlier work of Louko and Sorkin on topology change from a real time point of view.

We consider the behavior of a slowly moving classical point particle in a magnetic field in two dimensions, and show that, although energy conservation would permit the particle to escape to infinity, it in fact does not escape but is permanently trapped in the field. For any given magnetic field, this is true for particles of slow enough velocity. For such motion the magnetic flux enclosed by the Larmor orbits is an adiabatic invariant. Our results may be described by saying the deviations from conservation of this invariant are not cumulative but remain bounded over arbitrary time intervals, and are small if the velocity is small.

Synthesizing older ideas about the 1/N expansion in gauge theory, the quantum mechanics of black holes, and quantum field theory in Anti de Sitter space, a new correspondence between gauge theory and quantum gravity has illuminated both subjects.

Twenty-five years ago, Michael Green, John Schwarz, and Edward Witten wrote two volumes on string theory. Published during a period of rapid progress in this subject, these volumes were highly influential for a generation of students and researchers. Despite the immense progress that has been made in the field since then, the systematic exposition of the foundations of superstring theory presented in these volumes is just as relevant today as when first published. A self-contained introduction to superstrings, Volume 1 begins with an elementary treatment of the bosonic string, before describing the incorporation of additional degrees of freedom: fermionic degrees of freedom leading to supersymmetry and internal quantum numbers leading to gauge interactions. A detailed discussion of the evaluation of tree-approximation scattering amplitudes is also given. Featuring a new preface setting the work in context in light of recent advances, this book is invaluable for graduate students and researchers in general relativity and elementary particle theory.

We study the restrictions imposed by cancellation of the tadpoles for two-, three-, and four-form gauge fields in string theory, M-theory and F-theory compactified to two, three and four dimensions, respectively. For a large class of supersymmetric vacua, turning on a sufficient number of strings, membranes and three-branes, respectively, can cancel the tadpoles, and preserve supersymmetry. However, there are cases where the tadpole cannot be removed in this way, either because the tadpole is fractional, or because of its sign. For M-theory and F-theory compactifications, we also explore the relation of the membranes and three-branes to the non-perturbative space-time superpotential.

The Gopakumar-Vafa (GV) formula expresses certain couplings that arise in Type IIA compactication to four dimensions on a Calabi-Yau manifold in terms of a counting of BPS states in M-theory. The couplings in question have applications to topological strings and supersymmetric black holes. In this paper, we reconsider the GV formula, taking a close look at the Schwinger-like computation that was suggested in the original GV work. The goal is to understand the background that must be used in this computation, the role played by the extended supersymmetry of this background, and how the computation gives a holomorphic result though supercially depending only on particle masses. We also examine in a similar way the Ooguri-Vafa (OV) formula, which is an extension of the GV formula to include D4-branes.

This lecture surveys a few loosely related topics, ranging from the scarcity of quantum field theories -- and the role that this has played, and still plays, in physics -- to paradoxes involving black holes in soluble two dimensional string theory and the question of whether naked singularities might be of even greater interest to string theorists than black holes.

I discuss certain surface operators in gauge theory and their relevance to the ramified case of the geometric Langlands program.

The Feynman rules of gauge invariant open string field theory are shown to produce a triangulation of the moduli space of Riemann surfaces of arbitrary topology. If follows that the gauge invariant open string field theory is modular invariant.

No abstract is provided for this article.

It has recently been observed that 2 + 1 dimensional gravity is a well-defined, finite, and soluble theory, and is particularly simple if the cosmological constant is zero. In this paper it is shown that in the latter case, it is possible to compute the topology-changing amplitudes rather explicitly. One finds that if the quantum theory is formulated on space-times that do not admit classical solutions, it is dominated by planckian distances. If formulated on space-times on which a classical solution is possible, the quantum theory escapes from the planckian domain into the classical regime. The cosmological constant, if zero in the classical lagrangian, is subject to neither finite nor infinite renormalization and remains zero in the quantum theory. We also briefly discuss the coupling to point masses, and some generalizations to include fermionic symmetries, including a super Chern-Simons action that is related to the Casson invariant.

No abstract is provided for this article.

The GUT-based approach to physics has been attractive since it was first put forward close to thirty years ago; it has been enriched by new ideas, notably supersymmetry and strings; and there are real hints that it is on the right track, notably from measurements of the weak mixing angle and neutrino masses. In this article (based on my Heinrich Hertz lecture at SUSY 2002 at DESY, June, 2002), some of the arguments for grand unification will be reviewed.