802 publications from this institution
In the first of these two lectures, I describe a gauge theory approach to understanding quantum knot invariants as Laurent polynomials in a complex variable q. The two main steps are to reinterpret three-dimensional Chern-Simons gauge theory in four dimensional terms and then to apply electric-magnetic duality. The variable q is associated to instanton number in the dual description in four dimensions. In the second lecture, I describe how Khovanov homology can emerge upon adding a fifth dimension. (Based on lectures presented at the Clay Research Conference at Oxford University, and also at the Galileo Galilei Institute in Florence, the University of Milan, Harvard University, and the University of Pennsylvania.)
The geometric Langlands correspondence was described some years ago in terms of $S$-duality of $\N=4$ super Yang-Mills theory. Some additional matters relevant to this story are described here. The main goal is to explain directly why an $A$-brane of a certain simple kind can be an eigenbrane for the action of 't Hooft operators. To set the stage, we review some facts about Higgs bundles and the Hitchin fibration. We consider only the simplest examples, in which many technical questions can be avoided.
A semiclassical model of baryon-monopole interactions is presented. The model, which could serve as the basis for practical calculations of catalysis cross sections, interpolates between a soliton model of baryons at large distances and a radial field theory near the monopole core.
No abstract is provided for this article.
A study is made of the implications of heterotic string T-duality and extended gauge symmetry for the conjectured equivalence of heterotic and Type I superstrings. While at first sight heterotic string world-sheet dynamics appears to conflict with Type I perturbation theory, a closer look shows that Type I perturbation theory “miraculously” breaks down, in some cases via novel mechanisms, whenever the heterotic string has massless particles not present in Type I perturbation theory. This strongly suggests that the two theories actually are equivalent. As further evidence in the same direction, we show that the Dirichlet one-brane of Type I string theory has the same world-sheet structure as the heterotic string.
I consider the algebra of operators along the world line of an observer as a background independent algebra in quantum gravity.
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.
These notes are based on lectures at the PSSCMP/PiTP summer school that was held at Princeton University and the Institute for Advanced Study in July, 2015. They are devoted largely to topological phases of matter that can be understood in terms of free fermions and band theory. They also contain an introduction to the fractional quantum Hall effect from the point of view of effective field theory.
Let $M$ be a complete Einstein manifold of negative curvature, and assume that (as in the AdS/CFT correspondence) it has a Penrose compactification with a conformal boundary $N$ of positive scalar curvature. We show that under these conditions, $H_n(M;Z)=0$ and in particular $N$ must be connected. These results resolve some puzzles concerning the AdS/CFT correspondence.
We derive a holomorphic anomaly equation for the Vafa-Witten partition function for twisted four-dimensional \mathcal{N} =4 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mstyle mathvariant="script"><mml:mi>𝒩</mml:mi></mml:mstyle><mml:mo>=</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:math> super Yang-Mills theory on \mathbb{CP}^{2} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mstyle mathvariant="double-struck"><mml:mi>ℂ</mml:mi><mml:mi>ℙ</mml:mi></mml:mstyle><mml:mn>2</mml:mn></mml:msup></mml:math> for the gauge group SO(3) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>S</mml:mi><mml:mi>O</mml:mi><mml:mo stretchy="false" form="prefix">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false" form="postfix">)</mml:mo></mml:mrow></mml:math> from the path integral of the effective theory on the Coulomb branch. The holomorphic kernel of this equation, which receives contributions only from the instantons, is not modular but ‘mock modular’. The partition function has correct modular properties expected from S <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>S</mml:mi></mml:math> -duality only after including the anomalous nonholomorphic boundary contributions from anti-instantons. Using M-theory duality, we relate this phenomenon to the holomorphic anomaly of the elliptic genus of a two-dimensional noncompact sigma model and compute it independently in two dimensions. The anomaly both in four and in two dimensions can be traced to a topological term in the effective action of six-dimensional (2,0) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mo stretchy="false" form="prefix">(</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false" form="postfix">)</mml:mo></mml:mrow></mml:math> theory on the tensor branch. We consider generalizations to other manifolds and other gauge groups to show that mock modularity is generic and essential for exhibiting duality when the relevant field space is noncompact.
Just as parallel threebranes on a smooth manifold are related to string theory on AdS 5 × S 5 , parallel threebranes near a conical singularity are related to string theory on AdS5 × X 5, for a suitable X 5. For the example of the conifold singularity, for which X 5 = (SU(2) × SU(2))/U(1), we argue that string theory on AdS5 × X 5 can be described by a certain N = 1 supersymmetric gauge theory which we describe in detail.
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.
In most field theories that enter physics, the energy is defined as the integral of a gauge invariant physically meaningful energy density T00(x):
The large-$N$ limit of the two-dimensional $\mathrm{U}(N)$ (Wilson) lattice gauge theory is explicitly evaluated for all fixed $\ensuremath{\lambda}={g}^{2}N$ by steepest-descent methods. The $\ensuremath{\lambda}$ dependence is discussed and a third-order phase transition, at $\ensuremath{\lambda}=2$, is discovered. The possible existence of such a weak- to strong-coupling third-order phase transition in the large-$N$ four-dimensional lattice gauge theory is suggested, and its meaning and implications are discussed.
Some non-perturbative constraints on supersymmetry breaking are derived. It is demonstrated that dynamical supersymmetry breaking does not occur in certain interesting classes of theories.
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.
A non-abelian generalization of the usual formulas for bosonization of fermions in 1+1 dimensions is presented. Any fermi theory in 1+1 dimensions is equiv
The Langlands program of number theory, or what we might call Langlands duality, was proposed in more or less its present form by Robert Langlands, in the late 1960s. It is a kind of unified scheme for many results in number theory ranging from quadratic reciprocity, which is hundreds of years old, to modern results such as Andrew Wiles’ proof of Fermat’s last theorem, which involved a sort of special case of the Langlands program. For today, however, I will not assume any prior knowledge of the Langlands program.
In these lecture notes from Strings `91, I briefly sketch the analogy between two dimensional black holes and the s-wave sector of four dimensional black holes, and the physical interest of the latter, particularly in the magnetically charged case.
