Publications

Topology and Supersymmetric Nonlinear Sigma-Models

Ads/CFT correspondence and symmetry breaking

We study, using the dual AdS description, the vacua of field theories where some of the gauge symmetry is broken by expectation values of scalar fields. In such vacua, operators built out of the scalar fields acquire expectation values, and we show how to calculate them from the behavior of perturbations to the AdS background near the boundary. Specific examples include the N = 4 SYM theory, and theories on D3-branes placed on orbifolds and conifolds. We also clarify some subtleties of the AdS/CFT correspondence that arise in this analysis. In particular, we explain how scalar fields in AdS space of sufficiently negative mass-squared can be associated with CFT operators of two possible dimensions. All dimensions are bounded from below by (d−2)/2; this is the unitarity bound for scalar operators in d-dimensional field theory. We further argue that the generating functional for correlators in the theory with one choice of operator dimension is a Legendre transform of the generating functional in the theory with the other choice.

Anti-de Sitter Space, Thermal Phase Transition, And Confinement In Gauge Theories

The correspondence between supergravity (and string theory) on $AdS$ space and boundary conformal field theory relates the thermodynamics of ${\cal N}=4$ super Yang-Mills theory in four dimensions to the thermodynamics of Schwarzschild black holes in Anti-de Sitter space. In this description, quantum phenomena such as the spontaneous breaking of the center of the gauge group, magnetic confinement, and the mass gap are coded in classical geometry. The correspondence makes it manifest that the entropy of a very large $AdS$ Schwarzschild black hole must scale ``holographically'' with the volume of its horizon. By similar methods, one can also make a speculative proposal for the description of large $N$ gauge theories in four dimensions without supersymmetry.

Axion topological field theory of topological superconductors

Topological superconductors are gapped superconductors with gapless and topologically robust quasiparticles propagating on the boundary. In this paper, we present a topological field theory description of three-dimensional time-reversal invariant topological superconductors. In our theory the topological superconductor is characterized by a topological coupling between the electromagnetic field and the superconducting phase fluctuation, which has the same form as the coupling of ``axions'' with an Abelian gauge field. As a physical consequence of our theory, we predict the level crossing induced by the crossing of special ``chiral'' vortex lines, which can be realized by considering $s$-wave superconductors in proximity with the topological superconductor. Our theory can also be generalized to the coupling with a gravitational field.

Chern-Simons Gauge Theory As A String Theory

Certain two dimensional topological field theories can be interpreted as string theory backgrounds in which the usual decoupling of ghosts and matter does not hold. Like ordinary string models, these can sometimes be given space-time interpretations. For instance, three-dimensional Chern-Simons gauge theory can arise as a string theory. The world-sheet model in this case involves a topological sigma model. Instanton contributions to the sigma model give rise to Wilson line insertions in the space-time Chern-Simons theory. A certain holomorphic analog of Chern-Simons theory can also arise as a string theory.

MINIINTRODUCTION TO STRINGS

No abstract is provided for this article.

An SYK-Like Model Without Disorder

Making use of known facts about tensor models, it is possible to construct a quantum system without quenched disorder that has the same large $n$ limit for its correlation functions and thermodynamics as the SYK model. This might be useful in further probes of this approach to holographic duality.

$N=2$ JT Supergravity and Matrix Models

Generalizing previous results for $N=0$ and $N=1$, we analyze $N=2$ 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 $N=2$ 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 $N=2$ analog of the recursion relations for Weil-Petersson volumes originally discovered by Mirzakhani in the bosonic case.

Vistas in Theoretical Physics

No abstract is provided for this article.

Computing the complete massless spectrum of a Landau-Ginzburg orbifold

We develop techniques to compute the complete massless spectrum in heterotic string compactification on N = 2 supersymmetric Landau-Ginzburg orbifolds. This includes not just the familiar charged fields, but also the gauge singlets. The number of gauge singlets can vary in the moduli space of a given compactification and can differ from what it would be in the large radius limit of the corresponding Calabi-Yau. Comparison with exactly soluble Gepner models provides a confirmation of our results at Gepner points. Our methods carry over straightforwardly to (0, 2) Landau-Ginzburg models.

Heineman Prize Recipient: Magic, Mystery, and Matrix

No abstract is provided for this article.

Conformal anomaly of submanifold observables in AdS/CFT correspondence

We analyze the conformal invariance of submanifold observables associated with k-branes in the AdS/CFT correspondence. For odd k, the resulting obsrvables are conformally invariant, and for even k, they transform with a conformal anomaly that is given by a local expression which we analyze in detail for k = 2.

Fermion path integrals and topological phases

Symmetry-protected phases of matter have been at the forefront of condensed matter physics in recent years. Bosonic symmetry-protected phases have been interpreted in terms of anomalies and group cohomology. The present article aims to develop an analogous description of fermionic symmetry-protected phases, such as the topological insulators that have been seen experimentally in 2 or 3 space dimensions. The relevant mathematical concepts include the Atiyah-Singer index theorem and the Atiyah-Patodi-Singer eta invariant.

Why Does Quantum Field Theory In Curved Spacetime Make Sense? And What Happens To The Algebra of Observables In The Thermodynamic Limit?

This article aims to explain some of the basic facts about the questions raised in the title, without the technical details that are available in the literature. We provide a gentle introduction to some rather classical results about quantum field theory in curved spacetime and about the thermodynamic limit of quantum statistical mechanics. We also briefly explain that these results have an analog in the large N limit of gauge theory.

Gapped Boundary Phases of Topological Insulators via Weak Coupling

The standard boundary state of a topological insulator in 3+1 dimensions has gapless charged fermions. We present model systems that reproduce this standard gapless boundary state in one phase, but also have gapped phases with topological order. Our models are weakly coupled and all the dynamics is explicit. We rederive some known boundary states of topological insulators and construct new ones. Consistency with the standard spin/charge relation of condensed matter physics places a nontrivial constraint on models.

Twistor Space Structure Of One-Loop Amplitudes In Gauge Theory

No abstract is provided for this article.

Conformal Field Theory In Four And Six Dimensions

The goal of these notes is to give a brief explanation of how electric-magnetic duality in four dimensions is related to the existence of an unusual conformal field theory in six dimensions.

Direct Proof of the Tree-Level Scattering Amplitude Recursion Relation in Yang-Mills Theory

Recently, by using the known structure of one-loop scattering amplitudes for gluons in Yang-Mills theory, a recursion relation for tree-level scattering amplitudes has been deduced. Here, we give a short and direct proof of this recursion relation based on properties of tree-level amplitudes only.