Publications

Our basic ideas about physics went through several upheavals early this century. Quantum mechanics taught us that the classical notions of the position and velocity of a particle were only approximations of the truth. With general relativity, spacetime became a dynamical variable, curving in response to mass and energy. Contemporary developments in theoretical physics suggest that another revolution may be in progress, through which a new source of “fuzziness” may enter physics, and spacetime itself may be reinterpreted as an approximate, derived concept. (See figure 1.) In this article I survey some of these developments.

It has been shown recently that by turning on a large noncommutativity parameter, the description of tachyon condensation in string theory can be drastically simplified. We reconsider these issues from the standpoint of string field theory, showing that, from this point of view, the key fact is that in the limit of a large B-field, the string field algebra factors as the product of an algebra that acts on the string center of mass only and an algebra that acts on all other degrees of freedom carried by the string.

Most particle physicists now believe that protons, neutrons, and other strongly interacting particles are built from more basic constituents known as “quarks” and “gluons,” which interact according to the rules of a relativistic quantum field theory known as “quantum chromodynamics.”

These notes provide an introduction to recent work by Kevin Costello in which integrable lattice models of classical statistical mechanics in two dimensions are understood in terms of quantum gauge theory in four dimensions. This construction will be compared to the more familiar relationship between quantum knot invariants in three dimensions and Chern-Simons gauge theory. (Based on a Whittaker Colloquium at the University of Edinburgh and a lecture at Strings 2016 in Beijing.)

Twistor Geometry And Field Theory. By R. S. Ward. and Raymond O. Wells Jr. Cambridge University Press: 1990. Pp.520. £50, $79.50.

We investigate the differential geometry of the moduli space of instantons on S^3 x S^1. Extending previous results, we show
that a sigma-model with this target space can be expected to possess a large N=4 superconformal symmetry, supporting speculations that this sigma-model may be dual to Type IIB superstring theory on AdS_3 x S^3 x S^3 x S^1. The sigma-model is parametrized by three integers -- the rank of the gauge
group, the instanton number, and a ``level'' (the integer coefficient of a topologically nontrivial B-field, analogous to a WZW level). 
These integers are expected to correspond to two five-brane charges and
a one-brane charge. The sigma-model is weakly coupled when the level, conjecturally corresponding to one of the five-brane changes, becomes very large, keeping the
other parameters fixed. The central charges of the large N=4 algebra agree, at least semiclassically, with expectations from the duality.

An attempt is made to construct a realistic model of particle physics based on eleven-dimensional supergravity with seven dimensions compactified. It is possible to obtain an SU(3) × SU(2) × U(1) gauge group, but the proper fermion quantum numbers are difficult to achieve.

More general constructions are given of six-dimensional theories that look at low energy like six-dimensional super Yang-Mills theory.The constructions start with either parallel fivebranes in Type IIB, or M-theory on (C 2 x § 1 )/r for T a suitable finite group.Via these constructions, one can obtain six-dimensional theories with any simple gauge group, and SU(r) theories with any rational theta angle.A matrix construction of these theories is also possible.

It has been shown that the N matrix model two-dimensional gravity is related to certain topological field theories obtained by twisting the N = 2 minimal models. In this paper, the latter theories are studied by realizing them as gauged WZW models. This leads to an algebrogeometric description of the topological correlation functions from which many of their standard properties can be recovered. We find that in a certain sense the model at level k, with k analytically continued to −3, is equivalent to the Penner model (which computes the Euler characteristic of the moduli space of Riemann surfaces). We also gain better understanding of formulas of Lerche, Vafa, and Warner; Gepner; and Spiegelglas.

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We analyze proton decay via dimension-six operators in certain GUT-like models derived from Type IIA orientifolds with D6-branes. The amplitude is parametrically enhanced by a factor of α GUT −1/3 relative to the corresponding result in four-dimensional GUTs. Nonetheless, even assuming a plausible enhancement from the threshold corrections, we find little overall enhancement of the proton decay rate from dimension-six operators, so that the predicted lifetime from this mechanism remains close to 1036 years.

N = 2 supersymmetric gauge theories in four dimensions are studied by formulating them as the quantum field theories derived from configurations of fourbranes, fivebranes, and sixbranes in Type IIA superstrings, and then reinterpreting those configurations in M-theory. This approach leads to explicit solutions for the Coulomb branch of a large family of four-dimensional N = 2 field theories with zero or negative beta function.

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.

We compute static properties of baryons in an SU(2) × SU(2) chiral theory (the Skyrme model) whose solitons can be interpreted as the baryons of QCD. Our results are generally within about 30% of experimental values. We also derive some relations that hold generally in soliton models of baryons, and therefore, serve as tests of the 1 N expansion.

We build up local, time translation covariant Boundary Quantum Field Theory nets of von Neumann algebras $${\mathcal A_V}$$ on the Minkowski half-plane M +

The recent discovery of an explicit conformal field theory description of Type II p-branes makes it possible to investigate the existence of bound states of such objects. In particular, it is possible with reasonable precision to verify the prediction that the Type IIB superstring in ten dimensions has a family of soliton and bound state strings permuted by SL(2,Z). The space-time coordinates enter tantalizingly in the formalism as non-commuting matrices.

The SYK model is a quantum mechanical model that has been proposed to be holographically dual to a $1+1$-dimensional model of a quantum black hole. An emergent "gravitational" mode of this model is governed by an unusual action that that has been called the Schwarzian action. It governs a reparametrization of a circle. We show that the path integral of the Schwarzian theory is one-loop exact. The argument uses a method of fermionic localization, even though the model itself is purely bosonic.

The effect of modular transformations which change the spin structure of the string world-sheet is discussed. Invariance of string perturbation theory under such transformations forces us to sum over different spin structures with well defined coefficients. The sum over spin structures amounts to performing the GSO projections. We explain why the GSO projections is compatible with unitarit; that is, why wrong G-parity states are not produced in pairs. On the contrary, we show that unitarity requires either the GSO projection, which gives a supersymmetric spectrum, or another projection which removes the massless spin- 3 2 particle from the spectrum. This is in agreement with the familiar idea that a consistent theory with a massless spin- 3 2 particle must be supersymmetric. These considerations lead us to find four new string theories in ten dimensions. Although they all have tachyons, they are otherwise consistent, modular invariant and unitary. One of these theories is particularly interesting since it has anomaly free chiral fermions in ten dimensions.