This article is an introduction to newly discovered relations between volumes of moduli spaces of Riemann surfaces or super Riemann surfaces, simple models of gravity or supergravity in two dimensions, and random matrix ensembles. (The article is based on a lecture at the conference on the Mathematics of Gauge Theory and String Theory, University of Auckland, January 2020. It has been submitted to a special issue of the Quarterly Journal of Mathematics in memory of Michael Atiyah.)
The timelike tube theorem asserts that in quantum field theory without gravity, the algebra of observables in an open set $${\mathcal {U}}$$ is the same as
No abstract is provided for this article.
Superconducting cosmic strings are a plausible consequence of symmetry breaking in grand unified gauge theories. The luminosity in electromagnetic radiation of an oscillating current-carrying loop may substantially exceed the luminosity in gravitational radiation. In the typical case considered, the energy released electromagnetically is 1049 erg s−1, or 1066 erg in toto. Several consequences follow from this, the most interesting of which is the possibility that such loops may heat their surroundings, generating large, dense spherical shells of gas. Galaxies forming on these gravitationally unstable shells at moderate redshift will be seen at the present epoch to lie on bubbles having radii in the range 10–20h −1 Mpc if the initial ratio of luminosity in electromagnetic waves to that in gravitational wavess is > 10−3 for mass/length 1022 g cm−1. The required primordial energy density in magnetic fields is > 3 × 10−9 of the radiation energy density, if the charge carriers are bosons or superheavy fermions. Since these shells fill up space, the galaxies will have a distribution similar to that found in a recent survey of the northern sky. When the current saturates, a loop will emit particles copiously, and may be seen as an X-ray source at z ∼ 10–50. Such loops may also contribute significantly to the hard X-ray and γ-ray backgrounds and to 1020 eV cosmic rays.
No abstract is provided for this article.
We discuss high energy properties of states for (possibly interacting) quantum fields in curved spacetimes. In particular, if the spacetime is real analytic, we show that an analogue of the timelike tube theorem and the Reeh-Schlieder property hold with respect to states satisfying a weak form of microlocal analyticity condition. The former means the von Neumann algebra of observables of a spacelike tube equals the von Neumann algebra of observables of a significantly bigger region, that is obtained by deforming the boundary of the tube in a timelike manner. This generalizes theorems by Borchers and Araki to curved spacetimes.
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String theory is a quantum theory that reproduces the results of General Relativity at long distances but is completely different at short distances. Mathematically, string theory is based on a very new -- and little understood -- framework for geometry that reduces to ordinary differential geometry when the curvature is asymptotically small. In the 1990's, many interesting results were obtained about the behavior of string theory in spacetimes that develop singularities. In many cases, the physics at the singularity is governed by an effective Lagrangian constructed using an interesting bit of classical geometry such as the association of A-D-E groups with certain hypersurface singularities or the ADHM construction of instantons. In other examples, the physics at the singularity cannot be described in classical terms but involves a non-Gaussian conformal field theory.
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.
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 relation between approaches to integrability in superconformal Yang-Mills theory
Our discussions of string scattering amplitudes in the first volume of this book were limited to tree diagrams. These are the lowest-order approximations to string scattering amplitudes. In principle, quantum corrections to the tree level or classical results should be obtained by a perturbation expansion derived from string quantum field theory. Our present state of knowledge does not make this possible. Historically, loop diagrams were constructed by using unitarity to construct loop diagrams from tree diagrams. This unitarization of the tree diagrams led, in time, to the topological expansion, as sketched in chapter 1.
No abstract is provided for this article.
Geometric Langlands duality relates a representation of a simple Lie group $G^\vee$ to the cohomology of a certain moduli space associated with the dual group $G$. In this correspondence, a principal $SL_2$ subgroup of $G^\vee$ makes an unexpected appearance. Why this happens can be explained using gauge theory, as we will see in this article, with the help of the equations of Nahm and Bogomolny. (Based on a lecture at Geometry and Physics: Atiyah 80, Edinburgh, April 2009.)
We show that the genus zero correlation functions of an arbitrary topological field theory coupled to two-dimensional topological gravity are determined by an appropriate Landau-Ginzburg potential. We determine the potentials that arise for topological sigma models with CP1 or a Calabi-Yau manifold for target space. We present substantial evidence that the multi-matrix models that have been studied recently are equivalent to certain topological field theories coupled to topological gravity. We also describe a topological version of the general “string equation”.
It is shown how coupling to gauge fields can be used to explain the basic facts concerning holomorphic factorization of the WZW model of two dimensional co
In string theory, just as in other theories, it is necessary to understand the free theory well before trying to describe interactions. Our first task in a systematic exposition of string theory is to understand thoroughly the propagation of a single free string in space-time at both the classical and quantum levels. We begin in this chapter with a study of bosonic strings. In the course of this discussion, we will approach the bosonic string from many different points of view, corresponding to many different formalisms that have been developed over the years. These include various approaches to covariant and to light-cone quantization. Each adds important ingredients to an overall understanding of string theory, so it really is useful to become familiar with all of them.
