Publications

A bstract Generalizing previous results for $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 0 and $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 1, we analyze $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 JT supergravity on asymptotically AdS 2 spaces with arbitrary topology and show that this theory of gravity is dual, in a holographic sense, to a certain random matrix ensemble in which supermultiplets of different R -charge are statistically independent and each is described by its own $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 random matrix ensemble. We also analyze the case with a time-reversal symmetry, either commuting or anticommuting with the R -charge. In order to compare supergravity to random matrix theory, we develop an $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 analog of the recursion relations for Weil-Petersson volumes originally discovered by Mirzakhani in the bosonic case.

This article is an introduction to causal properties of General Relativity. Topics include the Raychaudhuri equation, singularity theorems of Penrose and Hawking, the black hole area theorem, topological censorship, and the Gao-Wald theorem. The article is based on lectures at the 2018 summer program Prospects in Theoretical Physics that was held at the IAS as well as the New Zealand Mathematical Research Institute summer school held in Nelson in January, 2020.

Surface operators in gauge theory are analogous to Wilson and 't Hooft line operators except that they are supported on a two-dimensional surface rather than a one-dimensional curve.In a previous paper, we constructed a certain class of half-BPS surface operators in N = 4 super Yang-Mills theory, and determined how they transform under S-duality.Those surface operators depend on a relatively large number of freely adjustable parameters.In the present paper, we consider the opposite case of half-BPS surface operators that are "rigid" in the sense that they do not depend on any parameters at all.We present some simple constructions of rigid half-BPS surface operators and attempt to determine how they transform under duality.This attempt is only partially successful, suggesting that our constructions are not the whole story.The partial match suggests interesting connections with quantization.We discuss some possible refinements and some string theory constructions which might lead to a more complete picture.

In his later years, Einstein sought a unified theory that would extend general relativity and provide an alternative to quantum theory. There is now talk of a ‘theory of everything’ (although Einstein himself never used the phrase). Fifty years after his death, how close are we to such a theory?

We reconsider Chern-Simons gauge theory on a Seifert manifold M (the total space of a nontrivial circle bundle over a Riemann surface). When M is a Seifert manifold, Lawrence and Rozansky have shown from the exact solution of Chern-Simons theory that the partition function has a remarkably simple structure and can be rewritten entirely as a sum of local contributions from the flat connections on M. We explain how this empirical fact follows from the technique of non-abelian localization as applied to the Chern-Simons path integral. In the process, we show that the partition function of Chern-Simons theory on M admits a topological interpretation in terms of the equivariant cohomology of the moduli space of flat connections on M.

We show that the nonlinear sigma model can be coupled to supergravity only if Newton's constant is an integral multiple of 1/F π 2.

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.)

We present a conformal field theory - obtained from a gauged WZW model - that describes a closed, inhomogeneous expanding and recollapsing universe in 3 + 1 dimensions. A possible violation of cosmic censorship is avoided because the universe recollapses just when a naked singularity was about to form. The model has been chosen to have c = 4 (or c ̂ = 4 in the supersymmetric case), just like four dimensional Minkowski space.

We show that conformally invariant gravity in three dimensions is equivalent to the Yang-Mills gauge theory of the conformal group in three dimensions, with a Chern-Simons action. This means that conformal gravity is finite and exactly soluble.

It was recently proposed (by Candelas, Horowitz, Strominger and the author) that compact manifolds of SU(3) holonomy are an interesting starting point for superstring phenomenology. Some aspects of such models are explored in this paper. Possible low-energy gauge groups are classified. It is shown that these theories have a mechanism to produce extra massless Higgs doublets unaccompanied by light color triplets. The SU(3) × SU(2) × U(1) couplings obey the standard relations after E6 breaking, so the standard computation of sin2 θ is valid. These models lead typically to discrete global symmetries, but a model in which these symmetries forbid all baryon-violating dimension-four operators may be hard to find. An alternative possibility (involving O(6) holonomy) is discussed in the last section.

Compactification of ten-dimensional supergravity on Calabi—Yau manifolds (as recently proposed by Candelas, Horowitz, Strominger, and the author) gives n = 1 supergravity theories in four dimensions. This paper is devoted to working out the Kähler potential and superpotential which arise.

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The Gopakumar-Vafa (GV) formula expresses certain couplings that arise in Type IIA compactification 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 superficially 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.

It is argued that large gauge hierarchies occur naturally in some theories with supersymmetry spontaneously broken at the three level. Such theories may also lead to time-dependent values of the natural “constants”.

Perturbative superstring theory is revisited, with the goal of giving a simpler and more direct demonstration that multi-loop amplitudes are gauge-invariant (apart from known anomalies), satisfy space-time supersymmetry when expected, and have the expected infrared behavior. The main technical tool is to make the whole analysis, including especially those arguments that involve integration by parts, on supermoduli space, rather than after descending to ordinary moduli space.

If superstrings are to describe nature they must account not only for general coordinate invariance and local supersymmetry, but also for the local gauge symmetries that underly the other forces. Indeed, nonabelian gauge symmetry is more obviously needed than local supersymmetry! One possibility is that the gauge symmetries are not present at all in the tendimensional world, but arise only upon reduction to four dimensions. This idea, which seems to be forced upon us if we try to describe nature with type II superstrings, proves to have enormous difficulties. We will discuss these issues to some extent in chapter 14.