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.
Vertex operators for AdS 3 background with Ramond Ramond flux
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.
The connection between instantons and the breaking of supersymmetry and ordinary symmetries is studied in a variety of (2+1)-dimensional gauge theories.
Einstein was one of the founders of quantum mechanics, yet he disliked the randomness that lies at the heart of the theory. God does not, he famously said, play dice. However, quantum theory has survived a century of experimental tests, although it has yet to be reconciled with another of Einstein's great discoveries – the general theory of relativity. Below four theorists – Gerard 't Hooft, Edward Witten, Fay Dowker and Paul Davies– outline their views on the current status of quantum theory and the way forward
This paper addresses various questions involving global anomalies in particle theory and string theory. It is shown that the question of whether a manifold is a spin manifold is equivalent to a question about global anomalies in the propagation of a point particle. In the superstring case, it is shown that the measure of the heterotic theory has no global anomalies on any Riemann surface. This generalizes known one loop results. Also, a topological condition is derived which restricts the possible choices of Wilson lines for grand unified symmetry breaking. It is argued that the long-term development of global anomalies in string theory will involve eventual study of global anomalies in the determinant of the Dirac-Ramond operator.
No abstract is provided for this article.
► Witten replies: In my article I tried to explain in a succinct way some of the exciting highlights of string theory, and I assumed for readers only a basic comfort level with Feynman diagrams and general relativity. The points in question should be widely understandable, but I am not sure where they have been explained in quite as elementary yet substantive a way as I aimed for.I certainly did not claim that everything has been understood; there are plenty of unsolved problems, as George Chapline points out, and that is one reason that the subject remains exciting. It was not possible in a short article to explain all the fascinating things that have been discovered and the many interesting ways that string theory interacts with other topics in physics and mathematics. Some of that has been covered in the past in other articles in Physics Today (see, for example, the article by Steve Giddings, April 2013, page 30, and the Quick Study by Hong Liu, June 2012, page 68).I have worked on the specific subject of twistor theory quite a lot, as Chapline probably realizes. Actually, one reason that I suspect string theory is on the right track is that when critics have had good ideas—whether involving black hole entropy, noncommutative geometry, or twistor theory—those ideas have tended to be absorbed into string theory.I regret that Peter Hansen did not find my article compelling, and I hope other readers thought otherwise. Many circumstantial clues suggest that string theory is on the right track. If that is the case, it is reasonable to hope that it will become clear, probably through a combination of theoretical and observational progress.© 2016 American Institute of Physics.
No abstract is provided for this article.
No abstract is provided for this article.
We analyze the dynamics of M-theory on a manifold of G_2 holonomy that is developing a conical singularity. The known cases involve a cone on CP^3, where we argue that the dynamics involves restoration of a global symmetry, SU(3)/U(1)^2, where we argue that there are phase transitions among three possible branches corresponding to three classical spacetimes, and S^3 x S^3 and its quotients, where we recover and extend previous results about smooth continuations between different spacetimes and relations to four-dimensional gauge theory.
M-theory compactification on a manifold X of $G_2$ holonomy can give chiral fermions in four dimensions only if X is singular. A number of examples of conical singularities that give chiral fermions are known; the present paper is devoted to describing some additional examples. In some of them, the physics can be determined but the metric is not known explicitly, while in others the metric can be described explicitly but the physics is more challenging to understand.
The partition function of Ramond-Ramond p-form fields in Type IIA supergravity on a ten-manifold X contains subtle phase factors that are associated with T-duality, self-duality, and the relation of the RR fields to K-theory. The analogous partition function of M-theory on X x S1 contains subtle phases that are similarly associated with E8 gauge theory. We analyze the detailed phase factors on the two sides and show that they agree, thereby testing M-theory/Type IIA duality as well as the K-theory formalism in an interesting way. We also show that certain D-brane states wrapped on nontrivial homology cycles are actually unstable, that (-1)^{F_L} symmetry in Type IIA superstring theory depends in general on a cancellation between a fermion anomaly and an anomaly of RR fields, and that Type IIA superstring theory with no wrapped branes is well-defined only on a spacetime with W_7=0.
This problem exemplifies the difference between the topological anomaly and the geometric anomaly: the topological isomorphism class of this line bundle is determined by the third Stiefel-Whitney class W3(i/), and this may vanish even if the holonomy is nontrivial.
Correlation functions in Liouville theory are meromorphic functions of the Liouville momenta, as is shown explicitly by the DOZZ formula for the three-poin
This paper addresses various questions involving global anomalies in particle theory and string theory. It is shown that the question of whether a manifold is a spin manifold is equivalent to a question about global anomalies in the propagation of a point particle. In the superstring case, it is shown that the measure of the heterotic theory has no global anomalies on any Riemann surface. This generalizes known one loop results. Also, a topological condition is derived which restricts the possible choices of Wilson lines for grand unified symmetry breaking. It is argued that the long-term development of global anomalies in string theory will involve eventual study of global anomalies in the determinant of the Dirac-Ramond operator.
