An Octahedron of Complex Null Rays, and Conformal Symmetry Breaking
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D-Instantons and Twistors
Home Search Collections Journals About Contact us My IOPscience D-instantons and twistors This article has been downloaded from IOPscience. Please scroll down to see the full text article. JHEP03(2009)044 (http://iopscience.iop.org/1126-6708/2009/03/044) The Table of Contents and more related content is available Download details: IP Address: 132.166.22.147 The article was downloaded on 26/02/2010 at 16:55 Please note that terms and conditions apply. Published by IOP Publishing for SISSA Received: January 5, 2009 Accepted: February 11, 2009 Published: March 6, 2009 D-instantons and twistors JHEP03(2009)044 Sergei Alexandrov,a Boris Pioline,b Frank Saueressigc and Stefan Vandorend aLaboratoire de Physique Th´eorique & Astroparticules, CNRS UMR 5207, Universit´eMontpellier II, 34095 Montpellier Cedex 05, France bLaboratoire de Physique Th´eorique et Hautes Energies, CNRS UMR 7589, Universit´ePierre et Marie Curie, 4 place Jussieu, 75252 Paris cedex 05, France cInstitut de Physique Th´eorique, CEA, IPhT, CNRS URA 2306, F-91191 Gif-sur-Yvette, France dInstitute for Theoretical Physics and Spinoza Institute, Utrecht University, Leuvenlaan 4, 3508 TD Utrecht, The Netherlands E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: Finding the exact, quantum corrected metric on the hypermultiplet moduli space in Type II string compactifications on Calabi-Yau threefolds is an outstanding open problem. We address this issue by relating the quaternionic-K¨ahler metric on the hy- permultiplet moduli space to the complex contact geometry on its twistor space. In this framework, Euclidean D-brane instantons are captured by contact transformations between different patches. -
Open Dissertation-Final.Pdf
The Pennsylvania State University The Graduate School The Eberly College of Science CORRELATIONS IN QUANTUM GRAVITY AND COSMOLOGY A Dissertation in Physics by Bekir Baytas © 2018 Bekir Baytas Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2018 The dissertation of Bekir Baytas was reviewed and approved∗ by the following: Sarah Shandera Assistant Professor of Physics Dissertation Advisor, Chair of Committee Eugenio Bianchi Assistant Professor of Physics Martin Bojowald Professor of Physics Donghui Jeong Assistant Professor of Astronomy and Astrophysics Nitin Samarth Professor of Physics Head of the Department of Physics ∗Signatures are on file in the Graduate School. ii Abstract We study what kind of implications and inferences one can deduce by studying correlations which are realized in various physical systems. In particular, this thesis focuses on specific correlations in systems that are considered in quantum gravity (loop quantum gravity) and cosmology. In loop quantum gravity, a spin-network basis state, nodes of the graph describe un-entangled quantum regions of space, quantum polyhedra. We introduce Bell- network states and study correlations of quantum polyhedra in a dipole, a pentagram and a generic graph. We find that vector geometries, structures with neighboring polyhedra having adjacent faces glued back-to-back, arise from Bell-network states. The results present show clearly the role that entanglement plays in the gluing of neighboring quantum regions of space. We introduce a discrete quantum spin system in which canonical effective methods for background independent theories of quantum gravity can be tested with promising results. In particular, features of interacting dynamics are analyzed with an emphasis on homogeneous configurations and the dynamical building- up and stability of long-range correlations. -
Twistor-Strings, Grassmannians and Leading Singularities
Twistor-Strings, Grassmannians and Leading Singularities Mathew Bullimore1, Lionel Mason2 and David Skinner3 1Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford, OX1 3NP, United Kingdom 2Mathematical Institute, 24-29 St. Giles', Oxford, OX1 3LB, United Kingdom 3Perimeter Institute for Theoretical Physics, 31 Caroline St., Waterloo, ON, N2L 2Y5, Canada Abstract We derive a systematic procedure for obtaining explicit, `-loop leading singularities of planar = 4 super Yang-Mills scattering amplitudes in twistor space directly from their momentum spaceN channel diagram. The expressions are given as integrals over the moduli of connected, nodal curves in twistor space whose degree and genus matches expectations from twistor-string theory. We propose that a twistor-string theory for pure = 4 super Yang-Mills | if it exists | is determined by the condition that these leading singularityN formulæ arise as residues when an unphysical contour for the path integral is used, by analogy with the momentum space leading singularity conjecture. We go on to show that the genus g twistor-string moduli space for g-loop Nk−2MHV amplitudes may be mapped into the Grassmannian G(k; n). For a leading singularity, the image of this map is a 2(n 2)-dimensional subcycle of G(k; n) and, when `primitive', it is of exactly the type found from− the Grassmannian residue formula of Arkani-Hamed, Cachazo, Cheung & Kaplan. Based on this correspondence and the Grassmannian conjecture, we deduce arXiv:0912.0539v3 [hep-th] 30 Dec 2009 restrictions on the possible leading singularities of multi-loop NpMHV amplitudes. In particular, we argue that no new leading singularities can arise beyond 3p loops. -
Twistor Theory and Differential Equations
IOP PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICAL J. Phys. A: Math. Theor. 42 (2009) 404004 (19pp) doi:10.1088/1751-8113/42/40/404004 Twistor theory and differential equations Maciej Dunajski Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK E-mail: [email protected] Received 31 January 2009, in final form 17 March 2009 Published 16 September 2009 Online at stacks.iop.org/JPhysA/42/404004 Abstract This is an elementary and self-contained review of twistor theory as a geometric tool for solving nonlinear differential equations. Solutions to soliton equations such as KdV,Tzitzeica, integrable chiral model, BPS monopole or Sine–Gordon arise from holomorphic vector bundles over T CP1. A different framework is provided for the dispersionless analogues of soliton equations, such as dispersionless KP or SU(∞) Toda system in 2+1 dimensions. Their solutions correspond to deformations of (parts of) T CP1, and ultimately to Einstein– Weyl curved geometries generalizing the flat Minkowski space. A number of exercises are included and the necessary facts about vector bundles over the Riemann sphere are summarized in the appendix. PACS number: 02.30.Ik (Some figures in this article are in colour only in the electronic version) 1. Introduction Twistor theory was created by Penrose [19] in 1967. The original motivation was to unify general relativity and quantum mechanics in a non-local theory based on complex numbers. The application of twistor theory to differential equations and integrability has been an unexpected spin off from the twistor programme. -
Causal Dynamical Triangulations and the Quest for Quantum Gravity?
Appendices: Mathematical Methods for Basic and Foundational Quantum Gravity Unstarred Appendices support Part I’s basic account. Starred Appendices support Parts II and III on interferences between Problem of Time facets. Double starred ones support the Epilogues on global aspects and deeper levels of mathematical structure being contemplated as Background Independent. If an Appendix is starred, the default is that all of its sections are starred likewise; a few are marked with double stars. Appendix A Basic Algebra and Discrete Mathematics A.1 Sets and Relations For the purposes of this book, take a set X to just be a collection of distinguishable objects termed elements. Write x ∈ X if x is an element of X and Y ⊂ X for Y a subset of X, ∩ for intersection, ∪ for union and Yc = X\Y for the complement of Y in X. Subsets Y1 and Y2 are mutually exclusive alias disjoint if Y1 ∩ Y2 =∅: the empty set. In this case, write Y1 ∪ Y2 as Y1 Y2: disjoint union.Apartition of a set X is a splitting of its elements into subsets pP that are mutually exclusive = and collectively exhaustive: P pP X. Finally, the direct alias Cartesian product of sets X and Z, denoted X × Z, is the set of all ordered pairs (x, z) for x ∈ X, z ∈ Z. For sets X and Z,afunction alias map ϕ : X → Z is an assignation to each x ∈ X of a unique image ϕ(x) = z ∈ Z. Such a ϕ is injective alias 1to1if ϕ(x1) = ϕ(x2) ⇒ x1 = x2, surjective alias onto if given z ∈ Z there is an x ∈ X such that ϕ(x) = z, and bijective if it is both injective and surjective. -
Twistor Theory at Fifty: from Rspa.Royalsocietypublishing.Org Contour Integrals to Twistor Strings Michael Atiyah1,2, Maciej Dunajski3 and Lionel Review J
Downloaded from http://rspa.royalsocietypublishing.org/ on November 10, 2017 Twistor theory at fifty: from rspa.royalsocietypublishing.org contour integrals to twistor strings Michael Atiyah1,2, Maciej Dunajski3 and Lionel Review J. Mason4 Cite this article: Atiyah M, Dunajski M, Mason LJ. 2017 Twistor theory at fifty: from 1School of Mathematics, University of Edinburgh, King’s Buildings, contour integrals to twistor strings. Proc. R. Edinburgh EH9 3JZ, UK Soc. A 473: 20170530. 2Trinity College Cambridge, University of Cambridge, Cambridge http://dx.doi.org/10.1098/rspa.2017.0530 CB21TQ,UK 3Department of Applied Mathematics and Theoretical Physics, Received: 1 August 2017 University of Cambridge, Cambridge CB3 0WA, UK Accepted: 8 September 2017 4The Mathematical Institute, Andrew Wiles Building, University of Oxford, Oxford OX2 6GG, UK Subject Areas: MD, 0000-0002-6477-8319 mathematical physics, high-energy physics, geometry We review aspects of twistor theory, its aims and achievements spanning the last five decades. In Keywords: the twistor approach, space–time is secondary twistor theory, instantons, self-duality, with events being derived objects that correspond to integrable systems, twistor strings compact holomorphic curves in a complex threefold— the twistor space. After giving an elementary construction of this space, we demonstrate how Author for correspondence: solutions to linear and nonlinear equations of Maciej Dunajski mathematical physics—anti-self-duality equations e-mail: [email protected] on Yang–Mills or conformal curvature—can be encoded into twistor cohomology. These twistor correspondences yield explicit examples of Yang– Mills and gravitational instantons, which we review. They also underlie the twistor approach to integrability: the solitonic systems arise as symmetry reductions of anti-self-dual (ASD) Yang–Mills equations, and Einstein–Weyl dispersionless systems are reductions of ASD conformal equations. -
The Pauli Exclusion Principle and SU(2) Vs. SO(3) in Loop Quantum Gravity
The Pauli Exclusion Principle and SU(2) vs. SO(3) in Loop Quantum Gravity John Swain Department of Physics, Northeastern University, Boston, MA 02115, USA email: [email protected] (Submitted for the Gravity Research Foundation Essay Competition, March 27, 2003) ABSTRACT Recent attempts to resolve the ambiguity in the loop quantum gravity description of the quantization of area has led to the idea that j =1edgesof spin-networks dominate in their contribution to black hole areas as opposed to j =1=2 which would naively be expected. This suggests that the true gauge group involved might be SO(3) rather than SU(2) with attendant difficulties. We argue that the assumption that a version of the Pauli principle is present in loop quantum gravity allows one to maintain SU(2) as the gauge group while still naturally achieving the desired suppression of spin-1/2 punctures. Areas come from j = 1 punctures rather than j =1=2 punctures for much the same reason that photons lead to macroscopic classically observable fields while electrons do not. 1 I. INTRODUCTION The recent successes of the approach to canonical quantum gravity using the Ashtekar variables have been numerous and significant. Among them are the proofs that area and volume operators have discrete spectra, and a derivation of black hole entropy up to an overall undetermined constant [1]. An excellent recent review leading directly to this paper is by Baez [2], and its influence on this introduction will be clear. The basic idea is that a basis for the solution of the quantum constraint equations is given by spin-network states, which are graphs whose edges carry representations j of SU(2). -
Spin Networks and Sl(2, C)-Character Varieties
SPIN NETWORKS AND SL(2; C)-CHARACTER VARIETIES SEAN LAWTON AND ELISHA PETERSON Abstract. Denote the free group on 2 letters by F2 and the SL(2; C)-representation variety of F2 by R = Hom(F2; SL(2; C)). The group SL(2; C) acts on R by conjugation, and the ring of in- variants C[R]SL(2;C) is precisely the coordinate ring of the SL(2; C)- character variety of a three-holed sphere. We construct an iso- morphism between the coordinate ring C[SL(2; C)] and the ring of matrix coe±cients, providing an additive basis of C[R]SL(2;C). Our main results use a spin network description of this basis to give a strong symmetry within the basis, a graphical means of computing the product of two basis elements, and an algorithm for comput- ing the basis elements. This provides a concrete description of the regular functions on the SL(2; C)-character variety of F2 and a new proof of a classical result of Fricke, Klein, and Vogt. 1. Introduction The purpose of this work is to demonstrate the utility of a graph- ical calculus in the algebraic study of SL(2; C)-representations of the fundamental group of a surface of Euler characteristic -1. Let F2 be a rank 2 free group, the fundamental group of both the three-holed sphere and the one-holed torus. The set of representations R = Hom(F2; SL(2; C)) inherits the structure of an algebraic set from SL(2; C). The subset of representations that are completely reducible, denoted by Rss, have closed orbits under conjugation. -
Twistor Theory of Symplectic Manifolds
Twistor Theory of Symplectic Manifolds R. Albuquerque [email protected] Departamento de Matem´atica Universidade de Evora´ 7000 Evora´ Portugal J. Rawnsley [email protected] Mathematics Institute University of Warwick Coventry CV47AL England November 2004 Abstract This article is a contribuition to the understanding of the geometry of the twistor space of a symplectic manifold. We consider the bundle l with fibre the Siegel domain Sp(2n, R)/U(n) Z existing over any given symplectic manifold M. Then, while recalling the construction of the celebrated almost complex structure induced on l by a symplectic connection on M, we Z arXiv:math/0405516v2 [math.SG] 17 Oct 2005 study and find some specific properties of both. We show a few examples of twistor spaces, develop the interplay with the symplectomorphisms of M, find some results about a natural almost-hermitian structure on l and finally discuss the holomorphic completeness of the Z respective “Penrose transform”. Let (M,ω) be a smooth symplectic manifold of dimension 2n. Then we may consider 2 the bundle π : l M (0.1) Z −→ of all complex structures j on the tangent spaces to M compatible with ω. Having fibre a cell, the bundle becomes interesting if it is seen with a particular and well known l almost complex structure, denoted ∇, — with which we start to treat by the name J Z of “Twistor Space” of the symplectic manifold M. The almost complex structure is induced by a symplectic connection on the base manifold and its integrability equation has already been studied ([13, 14, 6, 21]). -
A Speculative Ontological Interpretation of Nonlocal Context-Dependency in Electron Spin
A SPECULATIVE ONTOLOGICAL INTERPRETATION OF NONLOCAL CONTEXT-DEPENDENCY IN ELECTRON SPIN Martin L. Shough1 1st October 2002 Contents Abstract ................................................................................................................... .2 1. Spin Correlations as a Restricted Symmetry Group ............................................... 5 2. Ontological Basis of a Generalised Superspin Symmetry .....................................17 3. Epistemology & Ontology of Spin Measurement ................................................. 24 4. The Superspin Network. Symmetry-breaking and the Emergence of Local String Modes .......................................................................................... 39 5. Superspin Interpretation of the Field ................................................................... 57 6. Reflections and connections ................................................................................ 65 7. Cosmological implications .................................................................................. 74 References ........................................................................................................... 89 1 [email protected] Abstract Intrinsic spin is understood phenomenologically, as a set of symmetry principles. Inter-rotations of fermion and boson spins are similarly described by supersymmetry principles. But in terms of the standard quantum phenomenology an intuitive (ontological) understanding of spin is not to be expected, even though (or rather, -
Spin Foam Vertex Amplitudes on Quantum Computer—Preliminary Results
universe Article Spin Foam Vertex Amplitudes on Quantum Computer—Preliminary Results Jakub Mielczarek 1,2 1 CPT, Aix-Marseille Université, Université de Toulon, CNRS, F-13288 Marseille, France; [email protected] 2 Institute of Physics, Jagiellonian University, Łojasiewicza 11, 30-348 Cracow, Poland Received: 16 April 2019; Accepted: 24 July 2019; Published: 26 July 2019 Abstract: Vertex amplitudes are elementary contributions to the transition amplitudes in the spin foam models of quantum gravity. The purpose of this article is to make the first step towards computing vertex amplitudes with the use of quantum algorithms. In our studies we are focused on a vertex amplitude of 3+1 D gravity, associated with a pentagram spin network. Furthermore, all spin labels of the spin network are assumed to be equal j = 1/2, which is crucial for the introduction of the intertwiner qubits. A procedure of determining modulus squares of vertex amplitudes on universal quantum computers is proposed. Utility of the approach is tested with the use of: IBM’s ibmqx4 5-qubit quantum computer, simulator of quantum computer provided by the same company and QX quantum computer simulator. Finally, values of the vertex probability are determined employing both the QX and the IBM simulators with 20-qubit quantum register and compared with analytical predictions. Keywords: Spin networks; vertex amplitudes; quantum computing 1. Introduction The basic objective of theories of quantum gravity is to calculate transition amplitudes between configurations of the gravitational field. The most straightforward approach to the problem is provided by the Feynman’s path integral Z i (SG+Sf) hY f jYii = D[g]D[f]e } , (1) where SG and Sf are the gravitational and matter actions respectively. -
Entanglement on Spin Networks Loop Quantum
ENTANGLEMENT ON SPIN NETWORKS IN LOOP QUANTUM GRAVITY Clement Delcamp London - September 2014 Imperial College London Blackett Laboratory Department of Theoretical Physics Submitted in partial fulfillment of the requirements for the degree of Master of Science of Imperial College London I would like to thank Pr. Joao Magueijo for his supervision and advice throughout the writ- ing of this dissertation. Besides, I am grateful to him for allowing me to choose the subject I was interested in. My thanks also go to Etera Livine for the initial idea and for introducing me with Loop Quantum Gravity. I am also thankful to William Donnelly for answering the questions I had about his article on the entanglement entropy on spin networks. Contents Introduction 7 1 Review of Loop Quantum Gravity 11 1.1 Elements of general relativity . 12 1.1.1 Hamiltonian formalism . 12 1.1.2 3+1 decomposition . 12 1.1.3 ADM variables . 13 1.1.4 Connection formalism . 14 1.2 Quantization of the theory . 17 1.2.1 Outlook of the construction of the Hilbert space . 17 1.2.2 Holonomies . 17 1.2.3 Structure of the kinematical Hilbert space . 19 1.2.4 Inner product . 20 1.2.5 Construction of the basis . 21 1.2.6 Aside on the meaning of diffeomorphism invariance . 25 1.2.7 Operators on spin networks . 25 1.2.8 Area operator . 27 1.2.9 Physical interpretation of spin networks . 28 1.2.10 Chunks of space as polyhedra . 30 1.3 Explicit calculations on spin networks . 32 2 Entanglement on spin networks 37 2.1 Outlook .