802 publications from this institution
String theories with two-dimensional space-time target spaces are characterized by the existence of a “ground ring” of operators of spin (0, 0). By understanding this ring, one can understand the symmetries of the theory and illuminate the relation of the critical string theory to matrix models. The symmetry groups that arise are, roughly, the area-preserving diffeomorphisms of a two-dimensional phase space that preserve the Fermi surface (of the matrix model) and the volume-preserving diffeomorphisms of a three-dimensional cone. The three dimensions in question are the matrix eigenvalue, its canonical momentum, and the time of the matrix model.
This volume is based on lecture courses and seminars given at the LMS Durham Symposium on the geometry of low-dimensional manifolds. This area has been one of intense research recently, with major breakthroughs that have illuminated the way a number of different subjects interact (for example: topology, differential and algebraic geometry and mathematical physics). The workshop brought together a number of distinguished figures to give lecture courses and seminars in these subjects; the volume that has resulted is the only expository source for much of the material, and will be essential for all research workers in geometry and mathematical physics.
In the gauge theory approach to the geometric Langlands program, ramification can be described in terms of "surface operators," which are supported on two-dimensional surfaces somewhat as Wilson or 't Hooft operators are supported on curves.We describe the relevant surface operators in N = 4 super Yang-Mills theory, and the parameters they depend on, and analyze how S-duality acts on these parameters.Then, after compactifying on a Riemann surface, we show that the hypothesis of S-duality for surface operators leads to a natural extension of the geometric Langlands program for the case of tame ramification.The construction involves an action of the affine Weyl group on the cohomology of the moduli space of Higgs bundles with ramification, and an action of the affine braid group on A-branes or B-branes on this space.Contents 3.7.Action Of The Affine Weyl Group 97 3.8.Nahm's Equations And Local Singularity Of M H 102 3.9.The Hitchin Fibration 108 4. Geometric Langlands With Tame Ramification 110 4.1.Review Of Unramified Case 110 4.2.Sigma Model With Ramification 111 4.3.Branes 114 4.4.Twisted D-Modules 116 4.5.Line Operators And Monodromies 124 4.6.Representations And Branes 131 5. Line Operators And Ramification 135 5.1.General Framework 136 5.2.The B-Model 139 5.3.The A-Model 141 6. Local Models And Realizations By String Theory 153 6.1.Overview 153 6.2.Linear Sigma Model For G C = SL(2, C) 155 6.3.Instantons And The Local Singularity 161 6.4.
No abstract is provided for this article.
The chiral ring of classical supersymmetric Yang-Mills theory with gauge group $Sp(N)$ or SO(N) is computed, extending previous work (of Cachazo, Douglas, Seiberg, and the author) for SU(N). The result is that, as has been conjectured, the ring is generated by the usual glueball superfield $S\sim \Tr W_αW^α$, with the relation $S^h=0$, $h$ being the dual Coxeter number. Though this proposition has important implications for the behavior of the quantum theory, the statement and (for the most part) the proofs amount to assertions about Lie groups with no direct reference to gauge theory.
We introduce a "web-based formalism" for describing the category of half-supersymmetric boundary conditions in $1+1$ dimensional massive field theories with ${\cal N}=(2,2)$ supersymmetry and unbroken $U(1)_R$ symmetry. We show that the category can be completely constructed from data available in the far infrared, namely, the vacua, the central charges of soliton sectors, and the spaces of soliton states on $\mathbb{R}$, together with certain "interaction and boundary emission amplitudes". These amplitudes are shown to satisfy a system of algebraic constraints related to the theory of $A_\infty$ and $L_\infty$ algebras. The web-based formalism also gives a method of finding the BPS states for the theory on a half-line and on an interval. We investigate half-supersymmetric interfaces between theories and show that they have, in a certain sense, an associative "operator product." We derive a categorification of wall-crossing formulae. The example of Landau-Ginzburg theories is described in depth drawing on ideas from Morse theory, and its interpretation in terms of supersymmetric quantum mechanics. In this context we show that the web-based category is equivalent to a version of the Fukaya-Seidel $A_\infty$-category associated to a holomorphic Lefschetz fibration, and we describe unusual local operators that appear in massive Landau-Ginzburg theories. We indicate potential applications to the theory of surface defects in theories of class S and to the gauge-theoretic approach to knot homology.
There are hints of a novel object ("Planet 9") with a mass $5-10$ $M_\oplus$ in the outer Solar System, at a distance of order 500 AU. If it is a relatively conventional planet, it can be found in telescopic searches. Alternatively, it has been suggested that this body might be a primordial black hole (PBH). In that case, conventional searches will fail. A possible alternative is to probe the gravitational field of this object using small, laser-launched spacecraft, like the ones envisioned in the Breakthrough Starshot project. With a velocity of order $.001~c$, such spacecraft can reach Planet 9 roughly a decade after launch and can discover it if they can report timing measurements accurate to $10^{-5}$ seconds back to Earth.
No abstract is provided for this article.
Vertex operators for AdS 3 background with Ramond Ramond flux
In this paper, I will be considering conformal field theory (CFT) mainly in four and six dimensions, occasionally recalling facts about two dimensions. The notion of conformal field theory is familiar to physicists. From a mathematical point of view, we can keep in mind Graeme Segal's definition of conformal field theory. Instead of just summarizing the definition here, I will review how physicists actually study examples of quantum field theory, as this will make clear the motivation for the definition.
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.
By looking at phase transitions which occur as parameters are varied in supersymmetric gauge theories, a natural relation is found between sigma models based on Calabi-Yau hypersurfaces in weighted projective spaces and Landau-Ginzburg models. The construction permits one to recover the known correspondence between these types of models and to greatly extend it to include new classes of manifolds and also to include models with (0,2) world-sheet supersymmetry. The construction also predicts the possibility of certain physical processes involving a change in the topology of space-time.
We present a gauge-theoretic interpretation of the "analytic" version of the geometric Langlands program, in which Hitchin Hamiltonians and Hecke operators are viewed as concrete operators acting on a Hilbert space of quantum states. The gauge theory ingredients required to understand this construction -- such as electric-magnetic duality between Wilson and 't Hooft line operators in four-dimensional gauge theory -- are the same ones that enter in understanding via gauge theory the more familiar formulation of geometric Langlands, but now these ingredients are organized and applied in a novel fashion.
Superstring perturbation theory is traditionally carried out by using picture-changing operators (PCO's) to integrate over odd moduli. Naively the PCO's can be inserted anywhere on a string worldsheet, but actually a constraint must be placed on PCO insertions to avoid spurious singularities. Accordingly, it has been long known that the simplest version of the PCO procedure is valid only locally on the moduli space of Riemann surfaces, and that a correct PCO-based algorithm to compute scattering amplitudes must be based on piecing together local descriptions. Recently, was proposed as a relatively simple method to do this. Here, we spell out in detail what vertical integration means if carried out systematically. This involves a hierarchical procedure with corrections of high order. One might anticipate such a structure from the viewpoint of super Riemann surfaces.
Recent developments in the understanding of N = 2 supersymmetric Yang-Mills theory in four dimensions suggest a new point of view about Donaldson theory of four manifolds: instead of defining fourmanifold invariants by counting SU (2) instantons, one can define equivalent four-manifold invariants by counting solutions of a non-linear equation with an abelian gauge group.This is a "dual" equation in which the gauge group is the dual of the maximal torus of SU (2).The new viewpoint suggests many new results about the Donaldson invariants.
No abstract is provided for this article.
We study boundary conditions in N=4 super Yang-Mills theory that preserve one-half the supersymmetry. The obvious Dirichlet boundary conditions can be modified to allow some of the scalar fields to have a ``pole'' at the boundary. The obvious Neumann boundary conditions can be modified by coupling to additional fields supported at the boundary. The obvious boundary conditions associated with orientifolds can also be generalized. In preparation for a separate study of how electric-magnetic duality acts on these boundary conditions, we explore moduli spaces of solutions of Nahm's equations that appear in the presence of a boundary. Though our main interest is in boundary conditions that are Lorentz-invariant (to the extent possible in the presence of a boundary), we also explore non-Lorentz-invariant but half-BPS deformations of Neumann boundary conditions. We make preliminary comments on the action of electric-magnetic duality, deferring a more serious study to a later paper.
