802 publications from this institution
The non-perturbative superpotential can be effectively calculated in M-theory compactification to three dimensions on a Calabi-Yau four-fold X. For certain X, the superpotential is identically zero, while for other X, a non-perturbative superpotential is generated. Using F-theory, these results carry over to certain Type IIB and heterotic string compactifications to four dimensions with N = 1 supersymmetry. In the heterotic string case, the non-perturbative superpotential can be interpreted as coming from space-time and world-sheet instantons; in many simple cases contributions come only from finitely many values of the instanton numbers.
This paper is devoted to a systematic discussion of the supersymmetric index Tr (-1)^F for the minimal supersymmetric Yang-Mills theory -- with any simple gauge group G -- primarily in four spacetime dimensions. The index has refinements that probe confinement and oblique confinement and the possible spontaneous breaking of chiral symmetry and of global symmetries, such as charge conjugation, that are derived from outer automorphisms of the gauge group. Predictions for the index and its refinements are obtained on the basis of standard hypotheses about the infrared behavior of gauge theories. The predictions are confirmed via microscopic calculations which involve a Born-Oppenheimer computation of the spectrum as well as mathematical formulas involving triples of commuting elements of G and the Chern-Simons invariants of flat bundles on the three-torus.
Surface operators in gauge theory are analogous to Wilson and 't Hooft line operators except that they are supported on a two-dimensional surface rather than a one-dimensional curve. In a previous paper, we constructed a certain class of half-BPS surface operators in N=4 super Yang-Mills theory, and determined how they transform under S-duality. Those surface operators depend on a relatively large number of freely adjustable parameters. In the present paper, we consider the opposite case of half-BPS surface operators that are ``rigid'' in the sense that they do not depend on any parameters at all. We present some simple constructions of rigid half-BPS surface operators and attempt to determine how they transform under duality. This attempt is only partially successful, suggesting that our constructions are not the whole story. The partial match suggests interesting connections with quantization. We discuss some possible refinements and some string theory constructions which might lead to a more complete picture.
No abstract is provided for this article.
The Nahm pole boundary condition for certain gauge theory equations in four and five dimensions is defined by requiring that a solution should have a specified singularity along the boundary. In the present paper, we show that this boundary condition is elliptic and has regularity properties analogous to more standard elliptic boundary conditions. We also establish a uniqueness theorem for the solution of the relevant equations on a half-space with Nahm pole boundary conditions. These results are expected to have a generalization involving knots, with applications to the Jones polynomial and Khovanov homology.
Knowing that a four-dimensional theory with gauge group G0 is unified in a theory with gauge group G puts restrictions on what global symmetries are possible in the low-energy world. Here we analyze those restrictions assuming that unification in G occurs in four dimensions and assuming that unification occurs only in a higher-dimensional theory. There are possibilities for global symmetries which are not possible in the former case, so in principle indirect evidence for higher dimensions might be found by finding peculiar global symmetries in the low-energy world.
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.
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.
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.
An attempt is made to interpret the interactions of bosonic open strings as defining a non-cummulative, associative algebra, and to formulate the classical non-linear field theory of such strings in the language of non-commulative geometry. The point of departure is the BRST approach to string field theory. A setting is given in which there is a unique gauge invariant action, whose linearized approximation reproduces the conventional Veneziano spectrum. A derivation of conventional Veneziano model amplitudes from this gauge invariant action is sketched. Some brief comments are made about attempts to extend these results to open superstrings and to closed strings.
No abstract is provided for this article.
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.)
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.
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.
