802 publications from this institution
We re-examine quantization via branes with the goal of understanding its relation to geometric quantization. If a symplectic manifold $M$ can be quantized in geometric quantization using a polarization ${\mathcal P}$, and in brane quantization using a complexification $Y$, then the two quantizations agree if ${\mathcal P}$ can be analytically continued to a holomorphic polarization of $Y$. We also show, roughly, that the automorphism group of $M$ that is realized as a group of symmetries in brane quantization of $M$ is the group of symplectomorphisms of $M$ that can be analytically continued to holomorphic symplectomorphisms of $Y$. We describe from the point of view of brane quantization several examples in which geometric quantization with different polarizations gives equivalent results.
We describe a mechanism for using discrete symmetries to solve the doublet-triplet splitting problem of four-dimensional supersymmetric GUT's. We present two versions of the mechanism, one via ``deconstruction,'' and one in terms of M-theory compactification to four dimensions on a manifold of G_2 holonomy.
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.
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.)
No abstract is provided for this article.
The bosonic string theory described in chapters 2 and 3, despite all its beautiful features, has a number of shortcomings. The most obvious of these are the absence of fermions and the presence of tachyons. It is conceivable that the latter feature merely indicates that the vacuum has been incorrectly identified, and that (as in a Higgs theory) there is some other stable vacuum that does not give rise to tachyons. Despite considerable effort over the years, this remains a conjecture. Another route, which has proved more fruitful, is to try to formulate another string theory instead. Progress in this direction has involved the introduction of internal degrees of freedom propagating along the string.
We argue that multi-trace interactions in quantum field theory on the boundary of AdS space can be incorporated in the AdS/CFT correspondence by using a more general boundary condition for the bulk fields than has been considered hitherto. We illustrate the procedure for a renormalizable four-dimensional field theory with a $(\Tr Φ^2)^2$ interaction. In this example, we show how the AdS fields with the appropriate boundary condition reproduce the renormalization group effects found in the boundary field theory. We also construct in related examples a line of fixed points with a nonperturbative duality, and a flow between two methods of quantization.
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”.
The modifications of the classical equations of motion of the gravitational field in type II string theory are derived by studying tree-level gravitational scattering amplitudes. The effective gravitational action is determined through quartic order in the Riemann tensor. It is shown that generic Ricci-flat manifolds do not solve the modified equations, unless in addition the manifolds are Kähler (2 N-dimensional manifolds of SU(N) holonomy). Translated into sigma model language, this calculation would indicate that the N = 1 supersymmetric sigma model in 2 dimensions, with a Ricci-flat target space, is not conformally invariant, but has a nonzero beta function at four-loop order.
No abstract is provided for this article.
No abstract is provided for this article.
The title of this article refers to analytic continuation of three-dimensional Chern-Simons gauge theory away from integer values of the usual coupling parameter k, to explore questions such as the volume conjecture, or analytic continuation of three-dimensional quantum gravity (to the extent that it can be described by gauge theory) from Lorentzian to Euclidean signature. Such analytic continuation can be carried out by rotating the integration cycle of the Feynman path integral. Morse theory or Picard-Lefschetz theory gives a natural framework for describing the appropriate integration cycles. An important part of the analysis involves flow equations that turn out to have a surprising four-dimensional symmetry. After developing a general framework, we describe some specific examples (involving the trefoil and figure-eight knots in S^3). We also find that the space of possible integration cycles for Chern-Simons theory can be interpreted as the "physical Hilbert space" of a twisted version of N=4 super Yang-Mills theory in four dimensions.
No abstract is provided for this article.
No abstract is provided for this article.
We explore the consequences for macroscopic effective Lagrangians of assuming that the momentum density is proportional to the flow of a conserved current. Such a relation holds in models that microscopically contain only one type of charge carrier. We show that such a relation leads to certain universal corrections to the usual effective Lagrangians considered for the description of superfluids. A fully consistent derivation of the superconducting Hall effect predicted by London is given. Some related issues that arise in anyon superconductivity are discussed: the linear Hall effect is derived in a transparent way, and the vanishing of the b term (under appropriate assumptions) is demonstrated formally. Some other possible applications are sketched.
We compute the non-perturbative superpotential in $F$-theory compactification to four dimensions on a complex three-fold $\P^1\times S$, where $S$ is a rational elliptic surface. In contrast to examples considered previously, the superpotential in this case has interesting modular properties; it is essentially an $E_8$ theta function.
The self-dual Yang—Mills equations in four dimensions are integrable, for any gauge group, via the twistor transform. Integrability of the Yang—Mills equations proper in four dimensions with a finite dimensional gauge group cannot reasonably be hoped for. However, longstanding questions about the large N limit of QCD suggest that a new form of integrability might conceivably emerge in the limit of an infinite dimensional gauge group.
