1,630 research outputs found

    A-D-E Classification of Conformal Field Theories

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    The ADE classification scheme is encountered in many areas of mathematics, most notably in the study of Lie algebras. Here such a scheme is shown to describe families of two-dimensional conformal field theories.Comment: 19 pages, 4 figures, 4 tables; review article to appear in Scholarpedia, http://www.scholarpedia.org

    Chiral Supersymmetric Standard Model Spectra from Orientifolds of Gepner Models

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    We construct d=4,N=1 orientifolds of Gepner models with just the chiral spectrum of the standard model. We consider all simple current modular invariants of c=9 tensor products of N=2 minimal models. For some very specific tensor combinations, and very specific modular invariants and orientifold projections, we find a large number of such spectra. We allow for standard model singlet (dark) matter and non-chiral exotics. The Chan-Paton gauge group is either U(3) x Sp(2) x U(1) x U(1) or U(3) x U(2) x U(1) x U(1). In many cases the standard model hypercharge U(1) has no coupling to RR 2-forms and hence remains massless; in some of those models the B-L gauge boson does acquire a mass.Comment: 16 pages, LaTeX, minor corrections, references added Link added to updated and almost complete result

    Topological dualities in the Ising model

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    We relate two classical dualities in low-dimensional quantum field theory: Kramers-Wannier duality of the Ising and related lattice models in 22 dimensions, with electromagnetic duality for finite gauge theories in 33 dimensions. The relation is mediated by the notion of boundary field theory: Ising models are boundary theories for pure gauge theory in one dimension higher. Thus the Ising order/disorder operators are endpoints of Wilson/'t Hooft defects of gauge theory. Symmetry breaking on low-energy states reflects the multiplicity of topological boundary states. In the process we describe lattice theories as (extended) topological field theories with boundaries and domain walls. This allows us to generalize the duality to non-abelian groups; finite, semi-simple Hopf algebras; and, in a different direction, to finite homotopy theories in arbitrary dimension.Comment: 62 pages, 22 figures; v2 adds important reference [S]; v2 has reworked introduction, additional reference [KS], and minor changes; v4 for publication in Geometry and Topology has all new figures and a few minor changes and additional reference

    Definable sets of Berkovich curves

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    In this article, we functorially associate definable sets to kk-analytic curves, and definable maps to analytic morphisms between them, for a large class of kk-analytic curves. Given a kk-analytic curve XX, our association allows us to have definable versions of several usual notions of Berkovich analytic geometry such as the branch emanating from a point and the residue curve at a point of type 2. We also characterize the definable subsets of the definable counterpart of XX and show that they satisfy a bijective relation with the radial subsets of XX. As an application, we recover (and slightly extend) results of Temkin concerning the radiality of the set of points with a given prescribed multiplicity with respect to a morphism of kk-analytic curves. In the case of the analytification of an algebraic curve, our construction can also be seen as an explicit version of Hrushovski and Loeser's theorem on iso-definability of curves. However, our approach can also be applied to strictly kk-affinoid curves and arbitrary morphisms between them, which are currently not in the scope of their setting.Comment: 53 pages, 1 figure. v2: Section 7.2 on weakly stable fields added and other minor changes. Final version. To appear in Journal of the Institute of Mathematics of Jussie

    Analytic cell decomposition and analytic motivic integration

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    The main results of this paper are a Cell Decomposition Theorem for Henselian valued fields with analytic structure in an analytic Denef-Pas language, and its application to analytic motivic integrals and analytic integrals over \FF_q((t)) of big enough characteristic. To accomplish this, we introduce a general framework for Henselian valued fields KK with analytic structure, and we investigate the structure of analytic functions in one variable, defined on annuli over KK. We also prove that, after parameterization, definable analytic functions are given by terms. The results in this paper pave the way for a theory of \emph{analytic} motivic integration and \emph{analytic} motivic constructible functions in the line of R. Cluckers and F. Loeser [\emph{Fonctions constructible et int\'egration motivic I}, Comptes rendus de l'Acad\'emie des Sciences, {\bf 339} (2004) 411 - 416]
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