Structure, Classification, and Conformal Symmetry, of Elementary
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Quantum Groups and Algebraic Geometry in Conformal Field Theory
QUANTUM GROUPS AND ALGEBRAIC GEOMETRY IN CONFORMAL FIELD THEORY DlU'KKERU EI.INKWIJK BV - UTRECHT QUANTUM GROUPS AND ALGEBRAIC GEOMETRY IN CONFORMAL FIELD THEORY QUANTUMGROEPEN EN ALGEBRAISCHE MEETKUNDE IN CONFORME VELDENTHEORIE (mrt em samcnrattint] in hit Stdirlands) PROEFSCHRIFT TER VERKRIJGING VAN DE GRAAD VAN DOCTOR AAN DE RIJKSUNIVERSITEIT TE UTRECHT. OP GEZAG VAN DE RECTOR MAGNIFICUS. TROF. DR. J.A. VAN GINKEI., INGEVOLGE HET BESLUIT VAN HET COLLEGE VAN DE- CANEN IN HET OPENBAAR TE VERDEDIGEN OP DINSDAG 19 SEPTEMBER 1989 DES NAMIDDAGS TE 2.30 UUR DOOR Theodericus Johannes Henrichs Smit GEBOREN OP 8 APRIL 1962 TE DEN HAAG PROMOTORES: PROF. DR. B. DE WIT PROF. DR. M. HAZEWINKEL "-*1 Dit proefschrift kwam tot stand met "•••; financiele hulp van de stichting voor Fundamenteel Onderzoek der Materie (F.O.M.) Aan mijn ouders Aan Saskia Contents Introduction and summary 3 1.1 Conformal invariance and the conformal bootstrap 11 1.1.1 Conformal symmetry and correlation functions 11 1.1.2 The conformal bootstrap program 23 1.2 Axiomatic conformal field theory 31 1.3 The emergence of a Hopf algebra 4G The modular geometry of string theory 56 2.1 The partition function on moduli space 06 2.2 Determinant line bundles 63 2.2.1 Complex line bundles and divisors on a Riemann surface . (i3 2.2.2 Cauchy-Riemann operators (iT 2.2.3 Metrical properties of determinants of Cauchy-Ricmann oper- ators 6!) 2.3 The Mumford form on moduli space 77 2.3.1 The Quillen metric on determinant line bundles 77 2.3.2 The Grothendieck-Riemann-Roch theorem and the Mumford -
Noncommutative Geometry and the Spectral Model of Space-Time
S´eminaire Poincar´eX (2007) 179 – 202 S´eminaire Poincar´e Noncommutative geometry and the spectral model of space-time Alain Connes IHES´ 35, route de Chartres 91440 Bures-sur-Yvette - France Abstract. This is a report on our joint work with A. Chamseddine and M. Marcolli. This essay gives a short introduction to a potential application in physics of a new type of geometry based on spectral considerations which is convenient when dealing with noncommutative spaces i.e. spaces in which the simplifying rule of commutativity is no longer applied to the coordinates. Starting from the phenomenological Lagrangian of gravity coupled with matter one infers, using the spectral action principle, that space-time admits a fine structure which is a subtle mixture of the usual 4-dimensional continuum with a finite discrete structure F . Under the (unrealistic) hypothesis that this structure remains valid (i.e. one does not have any “hyperfine” modification) until the unification scale, one obtains a number of predictions whose approximate validity is a basic test of the approach. 1 Background Our knowledge of space-time can be summarized by the transition from the flat Minkowski metric ds2 = − dt2 + dx2 + dy2 + dz2 (1) to the Lorentzian metric 2 µ ν ds = gµν dx dx (2) of curved space-time with gravitational potential gµν . The basic principle is the Einstein-Hilbert action principle Z 1 √ 4 SE[ gµν ] = r g d x (3) G M where r is the scalar curvature of the space-time manifold M. This action principle only accounts for the gravitational forces and a full account of the forces observed so far requires the addition of new fields, and of corresponding new terms SSM in the action, which constitute the Standard Model so that the total action is of the form, S = SE + SSM . -
Connections Between Graph Theory, Additive Combinatorics, and Finite
UNIVERSITY OF CALIFORNIA, SAN DIEGO Connections between graph theory, additive combinatorics, and finite incidence geometry A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Michael Tait Committee in charge: Professor Jacques Verstra¨ete,Chair Professor Fan Chung Graham Professor Ronald Graham Professor Shachar Lovett Professor Brendon Rhoades 2016 Copyright Michael Tait, 2016 All rights reserved. The dissertation of Michael Tait is approved, and it is acceptable in quality and form for publication on microfilm and electronically: Chair University of California, San Diego 2016 iii DEDICATION To Lexi. iv TABLE OF CONTENTS Signature Page . iii Dedication . iv Table of Contents . .v List of Figures . vii Acknowledgements . viii Vita........................................x Abstract of the Dissertation . xi 1 Introduction . .1 1.1 Polarity graphs and the Tur´annumber for C4 ......2 1.2 Sidon sets and sum-product estimates . .3 1.3 Subplanes of projective planes . .4 1.4 Frequently used notation . .5 2 Quadrilateral-free graphs . .7 2.1 Introduction . .7 2.2 Preliminaries . .9 2.3 Proof of Theorem 2.1.1 and Corollary 2.1.2 . 11 2.4 Concluding remarks . 14 3 Coloring ERq ........................... 16 3.1 Introduction . 16 3.2 Proof of Theorem 3.1.7 . 21 3.3 Proof of Theorems 3.1.2 and 3.1.3 . 23 3.3.1 q a square . 24 3.3.2 q not a square . 26 3.4 Proof of Theorem 3.1.8 . 34 3.5 Concluding remarks on coloring ERq ........... 36 4 Chromatic and Independence Numbers of General Polarity Graphs . -
Lie Group and Geometry on the Lie Group SL2(R)
INDIAN INSTITUTE OF TECHNOLOGY KHARAGPUR Lie group and Geometry on the Lie Group SL2(R) PROJECT REPORT – SEMESTER IV MOUSUMI MALICK 2-YEARS MSc(2011-2012) Guided by –Prof.DEBAPRIYA BISWAS Lie group and Geometry on the Lie Group SL2(R) CERTIFICATE This is to certify that the project entitled “Lie group and Geometry on the Lie group SL2(R)” being submitted by Mousumi Malick Roll no.-10MA40017, Department of Mathematics is a survey of some beautiful results in Lie groups and its geometry and this has been carried out under my supervision. Dr. Debapriya Biswas Department of Mathematics Date- Indian Institute of Technology Khargpur 1 Lie group and Geometry on the Lie Group SL2(R) ACKNOWLEDGEMENT I wish to express my gratitude to Dr. Debapriya Biswas for her help and guidance in preparing this project. Thanks are also due to the other professor of this department for their constant encouragement. Date- place-IIT Kharagpur Mousumi Malick 2 Lie group and Geometry on the Lie Group SL2(R) CONTENTS 1.Introduction ................................................................................................... 4 2.Definition of general linear group: ............................................................... 5 3.Definition of a general Lie group:................................................................... 5 4.Definition of group action: ............................................................................. 5 5. Definition of orbit under a group action: ...................................................... 5 6.1.The general linear -
Finite Projective Geometries 243
FINITE PROJECTÎVEGEOMETRIES* BY OSWALD VEBLEN and W. H. BUSSEY By means of such a generalized conception of geometry as is inevitably suggested by the recent and wide-spread researches in the foundations of that science, there is given in § 1 a definition of a class of tactical configurations which includes many well known configurations as well as many new ones. In § 2 there is developed a method for the construction of these configurations which is proved to furnish all configurations that satisfy the definition. In §§ 4-8 the configurations are shown to have a geometrical theory identical in most of its general theorems with ordinary projective geometry and thus to afford a treatment of finite linear group theory analogous to the ordinary theory of collineations. In § 9 reference is made to other definitions of some of the configurations included in the class defined in § 1. § 1. Synthetic definition. By a finite projective geometry is meant a set of elements which, for sugges- tiveness, are called points, subject to the following five conditions : I. The set contains a finite number ( > 2 ) of points. It contains subsets called lines, each of which contains at least three points. II. If A and B are distinct points, there is one and only one line that contains A and B. HI. If A, B, C are non-collinear points and if a line I contains a point D of the line AB and a point E of the line BC, but does not contain A, B, or C, then the line I contains a point F of the line CA (Fig. -
Superconformal Theories in Six Dimensions
Thesis for the degree of Doctor of Philosophy Superconformal Theories in Six Dimensions P¨ar Arvidsson arXiv:hep-th/0608014v1 2 Aug 2006 Department of Fundamental Physics CHALMERS UNIVERSITY OF TECHNOLOGY G¨oteborg, Sweden 2006 Superconformal Theories in Six Dimensions P¨ar Arvidsson Department of Fundamental Physics Chalmers University of Technology SE-412 96 G¨oteborg, Sweden Abstract This thesis consists of an introductory text, which is divided into two parts, and six appended research papers. The first part contains a general discussion on conformal and super- conformal symmetry in six dimensions, and treats how the corresponding transformations act on space-time and superspace fields. We specialize to the case with chiral (2, 0) supersymmetry. A formalism is presented for incorporating these symmetries in a manifest way. The second part of the thesis concerns the so called (2, 0) theory in six dimensions. The different origins of this theory in terms of higher- dimensional theories (Type IIB string theory and M-theory) are treated, as well as compactifications of the six-dimensional theory to supersym- metric Yang-Mills theories in five and four space-time dimensions. The free (2, 0) tensor multiplet field theory is introduced and discussed, and we present a formalism in which its superconformal covariance is made manifest. We also introduce a tensile self-dual string and discuss how to couple this string to the tensor multiplet fields in a way that respects superconformal invariance. Keywords Superconformal symmetry, Field theories in higher dimensions, String theory. This thesis consists of an introductory text and the following six ap- pended research papers, henceforth referred to as Paper I-VI: I. -
Arxiv:2103.10700V2 [Gr-Qc] 27 Jun 2021 Omlg Sol H Onro Agrsmer Tutr Hc M Which Structure Symmetry Diffeomorphisms
Proper time reparametrization in cosmology: M¨obius symmetry and Kodama charges Jibril Ben Achour 1, 2 1 Arnold Sommerfeld Center for Theoretical Physics, Munich, Germany 2 Yukawa Center for Theoretical Physics, Kyoto University, Kyoto, Japan Abstract It has been noticed that for a large class of cosmological models, the gauge fixing of the time-reparametrization invariance does not completely fix the clock. Instead, the system enjoys a surprising residual Noether symmetry under M¨obius reparametrization of the proper time. In this work, we provide a unified treatment of this hidden conformal symmetry and its realization in the homogeneous and isotropic sector of the Einstein-Scalar-Λ system. We consider the flat Friedmann-Robertson-Walker (FRW) model, the (A)dS cosmology and provide a first treatment of the model with spatial constant curvature. We derive the general condition relating the choice of proper time and the conformal weight of the scale factor and provide a detailed analysis of the conserved Noether charges generating this symmetry which can be viewed as evolving constants of motion of these cosmological systems. Our approach allows us to identify new realizations of this symmetry while recovering previous results in a unified manner. We also present the general mapping onto the conformal particle and discuss the solution-generating nature of the transformations beyond the M¨obius symmetry. Finally, we show that, at least in a restricted context, this hidden conformal symmetry is intimately related to the so called Kodama charges of spherically symmetric gravity. This new connection suggests that the M¨obius invariance of cosmology is only the corner of a larger symmetry structure which might find an interesting realization in term of volume-preserving diffeomorphisms. -
An Introduction to Finite Geometry
An introduction to Finite Geometry Geertrui Van de Voorde Ghent University, Belgium Pre-ICM International Convention on Mathematical Sciences Delhi INCIDENCE STRUCTURES EXAMPLES I Designs I Graphs I Linear spaces I Polar spaces I Generalised polygons I Projective spaces I ... Points, vertices, lines, blocks, edges, planes, hyperplanes . + incidence relation PROJECTIVE SPACES Many examples are embeddable in a projective space. V : Vector space PG(V ): Corresponding projective space FROM VECTOR SPACE TO PROJECTIVE SPACE FROM VECTOR SPACE TO PROJECTIVE SPACE The projective dimension of a projective space is the dimension of the corresponding vector space minus 1 PROPERTIES OF A PG(V ) OF DIMENSION d (1) Through every two points, there is exactly one line. PROPERTIES OF A PG(V ) OF DIMENSION d (2) Every two lines in one plane intersect, and they intersect in exactly one point. (3) There are d + 2 points such that no d + 1 of them are contained in a (d − 1)-dimensional projective space PG(d − 1, q). DEFINITION If d = 2, a space satisfying (1)-(2)-(3) is called a projective plane. WHICH SPACES SATISFY (1)-(2)-(3)? THEOREM (VEBLEN-YOUNG 1916) If d ≥ 3, a space satisfying (1)-(2)-(3) is a d-dimensional PG(V ). WHICH SPACES SATISFY (1)-(2)-(3)? THEOREM (VEBLEN-YOUNG 1916) If d ≥ 3, a space satisfying (1)-(2)-(3) is a d-dimensional PG(V ). DEFINITION If d = 2, a space satisfying (1)-(2)-(3) is called a projective plane. PROJECTIVE PLANES Points, lines and three axioms (a) ∀r 6= s ∃!L (b) ∀L 6= M ∃!r (c) ∃r, s, t, u If Π is a projective plane, then interchanging points and lines, we obtain the dual plane ΠD. -
Lectures on Conformal Field Theory Arxiv:1511.04074V2 [Hep-Th] 19
Prepared for submission to JHEP Lectures on Conformal Field Theory Joshua D. Quallsa aDepartment of Physics, National Taiwan University, Taipei, Taiwan E-mail: [email protected] Abstract: These lectures notes are based on courses given at National Taiwan University, National Chiao-Tung University, and National Tsing Hua University in the spring term of 2015. Although the course was offered primarily for graduate students, these lecture notes have been prepared for a more general audience. They are intended as an introduction to conformal field theories in various dimensions working toward current research topics in conformal field theory. We assume the reader to be familiar with quantum field theory. Familiarity with string theory is not a prerequisite for this lectures, although it can only help. These notes include over 80 homework problems and over 45 longer exercises for students. arXiv:1511.04074v2 [hep-th] 19 May 2016 Contents 1 Lecture 1: Introduction and Motivation2 1.1 Introduction and outline2 1.2 Conformal invariance: What?5 1.3 Examples of classical conformal invariance7 1.4 Conformal invariance: Why?8 1.4.1 CFTs in critical phenomena8 1.4.2 Renormalization group 12 1.5 A preview for future courses 16 1.6 Conformal quantum mechanics 17 2 Lecture 2: CFT in d ≥ 3 22 2.1 Conformal transformations for d ≥ 3 22 2.2 Infinitesimal conformal transformations for d ≥ 3 24 2.3 Special conformal transformations and conformal algebra 26 2.4 Conformal group 28 2.5 Representations of the conformal group 29 2.6 Constraints of Conformal -
Star Gazing in Affine Planes
STAR GAZING IN AFFINE PLANES . D.R. Breach No.1'1 August 1981 Reprinted December 1981 r r I STAR GAZING IN AFFINE PLANES D.R. Breach No. 17 August 1981 Reprinted December 1981 This report is the text of an invited address given at the Ninth Australian Conference on Combinatorial Mathematics, the University of Queensland, Brisbane Queensland, Australia; August 1981. This reprinting has allowed the correction of some minor misprints in the original and also for the redrawing of some of the diagrams; December 1981. l STAR GAZING IN AFFINE PLANES It is however better to assume as a postulate the fact, inseparably connected with the idea of a straight line, that there exists onZy one straight Zine containing two given points, or, if two straight Zines have two points in common~ they coincide throughout. Sir Thomas L. Heath commenting on Euclid's postulate 1, (1908). 1. THE AUTHOR'S COMPLAINT Even in the study of modern geometry it is as well to remember that geometry has its origin in the measurement of the earth, that is to say surveying, and that figures and diagrams are the very heart of the subject both for the transmission and preservation of information and for the development of new ideas and methods. Therefore, while accepting that a diagram in itself is not a proof, one should not be reluctant to introduce them into one's formal presentations. After all, if a figure has been helpful to you in composing your thoughts why selfishly deny that figure to others? In teaching a course on combinatorics I have found students doubting the existence of a finite projective plane geometry with thirteen points on the grounds that they could not draw it (with 'straight' lines) on paper although they had tried to do so. -
Finite Geometries for Those with a Finite Patience for Mathematics 1
Finite Geometries for Those with a Finite Patience for Mathematics Michael Greenberg September 13, 2004 1 Introduction 1.1 Objective When my friends ask me what I’ve been studying this past summer and I tell them that I’ve been looking at finite geometries, it is disappointing that they all seem to groan at simply the name of the subject. They seem more than willing to tell me about how poorly they did in their high school geometry courses, and that they believe they could never begin understand what I’m studying. The most disappointing part of their response is that most of them will never see some of the beautiful and elementary results that stem from basic finite geometries simply because they assume it is too complicated. Although some of the textbooks on the subject might be difficult for a non math-major to sift through, my intent is to simplify some of the basic results so that any novice can understand them. 1.2 Method My intent is to compare the axiomatic system of Euclidean geometry that everyone is familiar with to the simpler axiomatic systems of finite geometries. Whereas it is a very intimidating task to show how Euclid’s postulates lead to many of the theorems in plane geometry, it is much easier to look at simple cases in finite geometry to show how the axioms shape the geometries. I will prove some simple theorems using these basic axioms to show the reader how axiomatic systems can produce nontrivial theorems, and I will touch on projective planes and affine planes to show examples of more constrained axiomatic systems that produce very elegant results. -
Introduction to Conformal Field Theory, Second Edition
Introduction to conformal field theory Torben Skrzypek and Jannik Fehre July 9, 2018 Abstract This summary contains two talks held in the seminar on "Holography and large-N dualities" in summer 2018. The first two sections summarise the talk held by Torben Skrzypek who explained the mathematical prerequisites to conformal quantum field theory, as they are needed in the following. He worked out the specific form of conformal transformations, identified the conformal group and showed, how in the quantisa- tion process the Lie algebra has to be extended to find a unitary represention. The first part mainly follows Schottenloher [1] but has also taken inspiration from Blumenhagen and Plauschinns book [2]. The last two sections outline the talk given by Jannik Fehre on the key techniques and features of two dimensional conformal field theory. As an example, he applied these insights to the free boson on the cylinder. The second part focusses on a discussion close to Blumenhagen and Plauschinn [2] but also uses some details from the lecture notes by Qualls [3]. Contents 1 Conformal transformations and the conformal group 2 1.1 Local conformal transformations . .2 1.2 Conformal compactification and the conformal group . .3 1.3 Special case of d = 2 ...........................................4 2 Central extensions and the Virasoro algebra 6 2.1 Central extensions in quantising symmetries . .6 2.2 Quantising conformal symmetry - the Virasoro algebra . .8 3 Conformal field theory in 2 dimensions 9 3.1 Radial quantisation . .9 3.2 2- and 3-point function . 10 3.3 The energy-momentum tensor . 11 3.4 Operator product expansion .