802 publications from this institution
Geometric Langlands duality is usually formulated as a statement about Riemann surfaces, but it can be naturally understood as a consequence of electric-magnetic duality of four-dimensional gauge theory. This duality in turn is naturally understood as a consequence of the existence of a certain exotic supersymmetric conformal field theory in six dimensions. The same six-dimensional theory also gives a useful framework for understanding some recent mathematical results involving a counterpart of geometric Langlands duality for complex surfaces. (This article is based on a lecture at the Raoul Bott celebration, Montreal, June 2008.)
The 1/N expansion in atomic physics is explained in order to motivate consideration of the 1/N expansion in QCD. Some comments are made concerning possible future development of the 1/N expansion in QCD.
We re-examine quantization via branes with the goal of understanding its relation to geometric quantization. If a symplectic manifold $M$ can be quantized in geometric quantization using a polarization ${\mathcal P}$, and in brane quantization using a complexification $Y$, then the two quantizations agree if ${\mathcal P}$ can be analytically continued to a holomorphic polarization of $Y$. We also show, roughly, that the automorphism group of $M$ that is realized as a group of symmetries in brane quantization of $M$ is the group of symplectomorphisms of $M$ that can be analytically continued to holomorphic symplectomorphisms of $Y$. We describe from the point of view of brane quantization several examples in which geometric quantization with different polarizations gives equivalent results.
Let $G$ be a simple and simply connected complex Lie group. We discuss the moduli space of holomorphic semistable principal $G$ bundles over an elliptic curve $E$. In particular we give a new proof of a theorem of Looijenga and Bernshtein-Shvartsman, that the moduli space is a weighted projective space. The method of proof is to study the deformations of certain unstable bundles coming from special maximal parabolic subgroups of $G$. We also discuss the associated automorphism sheaves and universal bundles, as well as the relation between various universal bundles and spectral covers.
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.
Von Neumann entropy has a natural extension to the case of an arbitrary semifinite von Neumann algebra, as was considered by I. E. Segal. We relate this entropy to the relative entropy and show that the entropy increase for an inclusion of von Neumann factors is bounded by the logarithm of the Jones index. The bound is optimal if the factors are infinite dimensional.
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We study four dimensional $N=2$ supersymmetric gauge theories on $R^3 \times S^1$ with a circle of radius $R$. They interpolate between four dimensional gauge theories ($R=\infty$) and $N=4$ supersymmetric gauge theories in three dimensions ($R=0$). The vacuum structure can be determined quite precisely as a function of $R$, agreeing with three and four-dimensional results in the two limits.
No abstract is provided for this article.
We analyze the u-plane contribution to Donaldson invariants of a four-manifold X. For $b_2^+(X)>1$, this contribution vanishes, but for $b_2^+=1$, the Donaldson invariants must be written as the sum of a u-plane integral and an SW contribution. The u-plane integrals are quite intricate, but can be analyzed in great detail and even calculated. By analyzing the u-plane integrals, the relation of Donaldson theory to N=2 supersymmetric Yang-Mills theory can be described much more fully, the relation of Donaldson invariants to SW theory can be generalized to four-manifolds not of simple type, and interesting formulas can be obtained for the class numbers of imaginary quadratic fields. We also show how the results generalize to extensions of Donaldson theory obtained by including hypermultiplet matter fields.
Modulo some highly plausible assumptions, we show that vector-like global symmetries (like isospin or baryon number) are not spontaneously broken in vector-like gauge theories with θ = 0 (like QCD). We also show that in these theories massless bound states do not form from massive constituents.
It is shown that in compactification of superstrings on manifolds of SU(3) holonomy, the superpotential receives no string theoretic corrections from the form it takes in the field theoretic limit, at least to all finite orders in sigma model perturbation theory. Modulo nonperturbative effects, this implies that those manifolds do indeed obey the exact classical equations of superstring theory, as has been argued on other grounds. Also, it is pointed out that the superpotential — even in the field theory limit — contains terms coupling charged fields to E6 singlets as well as self-couplings of the charged fields. A slightly tentative argument is given that on certain manifolds of SU(3) holonomy — though not all — it is possible to find conformally invariant sigma models that, while keeping unbroken supersymmetry, break E8 to SO(10) or SU(5) rather than E6. Including the effects of Wilson lines this would mean that E8 could be broken precisely to SU(3) × SU(2) × U(1) while keeping unbroken supersymmetry. These facts may open avenues for solving the problems associated with neutrino masses, proton decay, and renormalization group calculations of coupling constants. They also may lead to models with fewer unknown parameters than have been present in previous quasi-realistic models.
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.
These are notes on the theory of supermanifolds and integration on them, aiming to collect results that are useful for a better understanding of superstring perturbation theory in the RNS formalism.
