No abstract is provided for this article.
No abstract is provided for this article.
For vacua of string theory which leave unbroken N = 1 supersymmetry in the leading approximation, non-renormalization theorems guarantee that higher order (string or sigma model) perturbative corrections do not modify the space-time superpotential. If the low energy gauge group contains U(1) factors, this is not enough to ensure that the low energy physics is perturbatively stable. It is still possible to generate a Fayet-Iliopoulos D-term. We show that under certain conditions such a term is generated, and destabilizes the vacuum. This is a counterexample to various claims about the universal absence of quantum tadpoles when expanding around a classical solution with unbroken supersymmetry. Whether, in a given model, the generation of a D-term will destabilize the vacuum (or merely change the pattern of symmetry breaking) can be determined from properties of the low energy effective action.
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 <italic>q</italic>. 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 <italic>q</italic> 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.
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.
We consider the problem of identifying the CFT's that may be dual to pure gravity in three dimensions with negative cosmological constant. The c-theorem indicates that three-dimensional pure gravity is consistent only at certain values of the coupling constant, and the relation to Chern-Simons gauge theory hints that these may be the values at which the dual CFT can be holomorphically factorized. If so, and one takes at face value the minimum mass of a BTZ black hole, then the energy spectrum of three-dimensional gravity with negative cosmological constant can be determined exactly. At the most negative possible value of the cosmological constant, the dual CFT is very likely the monster theory of Frenkel, Lepowsky, and Meurman. The monster theory may be the first in a discrete series of CFT's that are dual to three-dimensional gravity. The partition function of the second theory in the sequence can be determined on a hyperelliptic Riemann surface of any genus. We also make a similar analysis of supergravity.
In toroidal compactification of string theory, enlarged symmetry groups arise at special radii of the compactified dimensions. Here an orbifold-like construction is considered in which the degrees of freedom describing the compact spatial dimensions are twisted by symmetries which exist only for special values of the radius. Constraints on the possible twists arising from the requirements of worldsheet supersymmetry and modular invariance are described, and some simple examples are given. One motivation for this work is that in twisting by a symmetry which exists only at a special value of the radius, one of the dilaton-like fields can acquire a mass, since the radius of the compact space is fixed.
A framework for background-independent open-string field theory is proposed. The approach involves using the Batalin-Vilkovisky formalism, in a way suggested by recent developments in closed-string field theory, to implicitly define a gauge-invariant Lagrangian in a hypothetical "space of all open-string world-sheet theories." It is built into the formalism that classical solutions of the string field theory are Becchi-Rouet-Stora-Tyutin- (BRST-) invariant open-string world-sheet theories and that, when expanding around a classical solution, the infinitesimal gauge transformations are generated by the world-sheet BRST operator.
Just as D-brane charge of type-IIA and type-IIB superstrings is clas- sied, respectively, by K 1 (X)and K( X), Ramond-Ramondelds in these theories are classied, respectively, by K(X)and K 1 ( X). Byanalyzing a recent proposal for how to interpret quantum self-duality of RRelds, we show that the Dirac quanti- zation formula for the RR p-forms, when properly formulated, receives corrections thatreflectcurvature,lowerbranecharges,andananomalyofD-braneworld-volume fermions. TheK-theoryframeworkisimportanthere,becausetheterminvolvingthe fermion anomaly cannot be naturally expressed in terms of cohomology and dier- ential forms.
No abstract is provided for this article.
I sketch what it is supposed to mean to quantize gauge theory, and how this can be made more concrete in perturbation theory and also by starting with a finite-dimensional lattice approximation. Based on real experiments and computer simulations, quantum gauge theory in four dimensions is believed to have a mass gap. This is one of the most fundamental facts that makes the Universe the way it is. This article is the written form of a lecture presented at the conference Geometric Analysis: Past and Future (Harvard University, August 27-September 1, 2008), in honor of the 60th birthday of S.-T. Yau.
No abstract is provided for this article.
We generalize the recently discovered relationship between JT gravity and double-scaled random matrix theory to the case that the boundary theory may have time-reversal symmetry and may have fermions with or without supersymmetry. The matching between variants of JT gravity and matrix ensembles depends on the assumed symmetries. Time-reversal symmetry in the boundary theory means that unorientable spacetimes must be considered in the bulk. In such a case, the partition function of JT gravity is still related to the volume of the moduli space of conformal structures, but this volume has a quantum correction and has to be computed using Reidemeister-Ray-Singer torsion. Presence of fermions in the boundary theory (and thus a symmetry $(-1)^F$) means that the bulk has a spin or pin structure. Supersymmetry in the boundary means that the bulk theory is associated to JT supergravity and is related to the volume of the moduli space of super Riemann surfaces rather than of ordinary Riemann surfaces. In all cases we match JT gravity or supergravity with an appropriate random matrix ensemble. All ten standard random matrix ensembles make an appearance -- the three Dyson ensembles and the seven Altland-Zirnbauer ensembles. To facilitate the analysis, we extend to the other ensembles techniques that are most familiar in the case of the original Wigner-Dyson ensemble of hermitian matrices. We also generalize Mirzakhani's recursion for the volumes of ordinary moduli space to the case of super Riemann surfaces.
Do the elementary particles known as neutrinos have mass? Yes, according to recent experiments. But how much? A surprising — and controversial — result suggests that the answer is not what we thought.
A variant of the usual supersymmetric nonlinear sigma model is described, governing maps from a Riemann surfaceΣ to an arbitrary almost complex manifo
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.
Certain aspects of the antifield-antibracket formalism for quantization of gauge theories are clarified. In particular, we discuss the geometrical meaning of the antifields, the geometric meaning of the antibracket, and the geometric meaning of the operator Δ that appears in the quantum correction to the master equation. Finally, we point out that the antibracket formalism contains most of the ingredients that would be needed to formulate an abstract Chern-Simons Lagrangian, as in open string field theory.
No abstract is provided for this article.
It has long been known that in principle, the genus g vacuum amplitude for bosonic strings or superstrings in 26 or 10 dimensions can be entirely determined from conditions of holomorphy. Moreover, this has been done in practice for bosonic strings of low genus. Here we describe in a unified way how to determine the bosonic string and superstring vacuum amplitude in genus 1 and 2 via holomorphy. The main novelty is the superstring analysis in genus 2, where we use holomorphy to get a new understanding of some of the results that previously have been obtained by more explicit calculations.
