DOCSLIB.ORG

E 8 in N= 8 Supergravity in Four Dimensions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
E 8 in N= 8 Supergravity in Four Dimensions E 8 in N= 8 supergravity in four dimensions Downloaded from: https://research.chalmers.se, 2021-10-01 00:06 UTC Citation for the original published paper (version of record): Ananth, S., Brink, L., Majumdar, S. (2018) E 8 in N= 8 supergravity in four dimensions Journal of High Energy Physics, 2018(1) http://dx.doi.org/10.1007/JHEP01(2018)024 N.B. When citing this work, cite the original published paper. research.chalmers.se offers the possibility of retrieving research publications produced at Chalmers University of Technology. It covers all kind of research output: articles, dissertations, conference papers, reports etc. since 2004. research.chalmers.se is administrated and maintained by Chalmers Library (article starts on next page) Published for SISSA by Springer Received: November 30, 2017 Accepted: December 23, 2017 Published: January 8, 2018 JHEP01(2018)024 E8 in N = 8 supergravity in four dimensions Sudarshan Ananth,a Lars Brinkb,c and Sucheta Majumdara aIndian Institute of Science Education and Research, Pune 411008, India bDepartment of Physics, Chalmers University of Technology, S-41296 G¨oteborg, Sweden cDivision of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 637371, Singapore E-mail: [email protected], [email protected], [email protected] Abstract: We argue that = 8 supergravity in four dimensions exhibits an exceptional N E8(8) symmetry, enhanced from the known E7(7) invariance. Our procedure to demonstrate this involves dimensional reduction of the = 8 theory to d = 3, a field redefinition to N render the E8(8) invariance manifest, followed by dimensional oxidation back to d = 4. Keywords: Supergravity Models, Superspaces, Field Theories in Lower Dimensions ArXiv ePrint: 1711.09110 Open Access, c The Authors. https://doi.org/10.1007/JHEP01(2018)024 Article funded by SCOAP3. Contents 1 Introduction 1 2 (N = 8, d = 4) supergravity in light-cone superspace 2 2.1 E7(7) symmetry 4 3 Maximal supergravity in d = 3 — version I 5 JHEP01(2018)024 4 Maximal supergravity in d = 3 — version II 5 4.1 E8(8) symmetry 6 5 Relating the two different versions of three-dimensional maximal super- gravity 7 5.1 The field redefinition 7 5.2 SO(16) symmetry revisited 7 6 Oxidation back to d = 4 preserving the E8(8) symmetry 8 7 Conclusions 9 A Verification of field redefinition 10 B SO(16)-invariance of the new Lagrangian 11 C E8 invariance 12 1 Introduction Gravity with maximal supersymmetry in four dimensions, = 8 supergravity, exhibits N both = 8 supersymmetry and the exceptional E symmetry [1]. This theory is known N 7(7) to be better behaved in the ultraviolet than pure gravity and has recently been shown to be finite up to four loops [2]. There is mounting evidence from these calculations and others that points to unexpected cancellations and hence an underlying enhanced symmetry. Using light-cone superspace, the Hamiltonian of maximal supergravity, in d = 4, is constructed as a power series in the coupling constant and this has been achieved up to the four-point coupling. In an earlier paper [3], we showed that “oxidation” [4] of the = 8 theory to d = 11, suggests that there is an E symmetry in eleven dimensions. N 7(7) This result has been shown to first order in the coupling constant. Since the the states of the d = 11 theory are not representations of the linearly realized maximal subgroup SU(8) – 1 – of the E7(7) symmetry they have to be broken up into such representations to see the symmetry. This is accomplished by using the same superfield in all dimensions (note that the number of states is always 128 bosons + 128 fermions.) By writing the Hamiltonian in this formulation we can prove the symmetry to the lowest order in the gravitational coupling constant. Motivated by this, we describe in this paper an entire process involving dimensional reduction, field redefinitions and dimensional oxidation that leads us to conclude that = 8 supergravity in d = 4 exhibits an exceptional E symmetry, at least to second N 8(8) order in the coupling constant, enhanced from E7(7). JHEP01(2018)024 In section 2, we review the formulation of ( = 8, d = 4) supergravity in light- N cone superspace. We dimensionally reduce this d = 4 theory in a straightforward way to arrive at an action, in three dimensions, which mimics the four-dimensional one but with only one transverse derivative. This formulation thus inherits a three-point coupling and cannot exhibit the maximal subgroup of E8(8), SO(16), in a linear fashion since under this symmetry, the states of the theory transform as 128-dimensional spinors. Three such states cannot form a scalar. In the following section, we present a field redefinition that maps this three-dimensional theory, with a cubic vertex to a three-dimensional theory without one. This form of the three-dimensional theory exhibits both SO(16) invariance and a full E8(8) symmetry [5]. We can then go back to the first formulation and indeed find the SO(16) symmetry there, now realized non-linearly. We then “oxidize” the second formulation back to four dimensions in a manner that preserves all the symmetries discussed earlier, thus arriving at a four-dimensional maximally supersymmetric theory with E8(8) invariance to that order. Our formulation uses only the real degrees of freedom of the theory. This means that we lose a lot of the covariance usually found in gravity theories, since many of the symmetries are non-linearly realized. In a sense the formulation is packed with symmetries, some of which are difficult to see. By making various field redefinitions we can make particular symmetries visible but one formulation will never be enough to find all the symmetries. We strongly believe that maximal supergravity and Yang-Mills theories have as many symmetries as one can pack into one theory, and this is why they have unique quantum properties. 2 (N = 8, d = 4) supergravity in light-cone superspace With the metric ( , +, +, +), the light-cone coordinates are − 1 1 x± = ( x0 x3 ) x = ( x1 + i x2 )x ¯ = (x)∗ , (2.1) √2 ± √2 with the corresponding derivatives being ∂∓, ∂¯ and ∂. The = 8 superspace is spanned m N by the Grassmann variables θ and θ¯m (m = 1 ... 8 ), the 8 and 8¯ of SU(8) respectively. – 2 – All 256 physical degrees of freedom in the theory are captured by the superfield [6] 1 m 1 i m n 1 φ ( y ) = h (y) + i θ ψ¯m (y) + θ θ A¯mn (y) , ∂+2 ∂+2 2 ∂+ 1 1 1 θm θn θp χ¯ (y) θm θn θp θq C¯ (y) , − 3! ∂+ mnp − 4! mnpq i + θm θn θp θq θr ǫ χstu (y) , 5! mnpqrstu i + θm θn θp θq θr θs ǫ ∂+ Atu (y) , 6! mnpqrstu 1 JHEP01(2018)024 + θm θn θp θq θr θs θt ǫ ∂+ ψu (y) , 7! mnpqrstu 4 + θm θn θp θq θr θs θt θu ǫ ∂+2 h¯ (y) , (2.2) 8! mnpqrstu ¯ ¯ 3 ¯ where h and h represent the graviton, ψm the 8 spin- 2 gravitinos, Amn the 28 gauge fields, χ¯mnp the 56 gauginos and C¯mnpq the 70 real scalars. These fields are local in the coordinates i y = x, x,¯ x+, y− x− θm θ¯ . (2.3) ≡ − √2 m The superfield φ and its complex conjugate φ¯ satisfy 8 m 1 (d ) d φ ( y ) = 0 ; d¯n φ¯ ( y ) = 0 , φ = φ¯ , (2.4) 4 ∂+4 where m ∂ i m + ¯ ∂ i ¯ + 8 1 2 8 d = θ ∂ ; dn = n + θn ∂ , (d ) d d . d . (2.5) − ∂ θ¯m − √2 ∂ θ √2 ≡ The kinematical, spectrum generating, supersymmetry generators are [7], m ∂ i m + ∂ i ¯ + q + = + θ ∂ ;q ¯+n = n θn ∂ , (2.6) − ∂ θ¯m √2 ∂ θ − √2 satisfying qm , q¯ = i√2δm∂+, while the dynamical ones are given by { + +n} n ¯ m ∂ m ∂ q = q , q¯− = q¯ . (2.7) − ∂+ + n ∂+ +n These satisfy the free = 8 supersymmetry algebra closing to the Hamiltonian generator N ¯ m m ∂∂ q , q¯− = i√2δ . (2.8) { − n} n ∂+ In the interacting theory, the dynamical generators pick up corrections order by order thus generating the interacting Hamiltonian. – 3 – The action to order κ To order κ, the action for = 8 supergravity reads [8] N β d4x d8θ d8θ¯ , (2.9) L Z Z where β = 1 and − 64 ✷ 4 1 1 = φ¯ φ + κ φ¯∂¯∂φ∂¯ +2φ φ∂¯ +∂φ∂¯ +∂φ¯ + c.c. (2.10) L − ∂+4 3 +4 − +4 ∂ ∂ The d’Alembertian is JHEP01(2018)024 ✷ = 2 ( ∂ ∂¯ ∂ ∂− ) , (2.11) − + κ = √8 π G and Grassmann integration is normalized such that d8θ (θ)8 = 1. The correction to the dynamical supersymmetry generator at this order is R (κ) 1 +2 + + q¯− φ = (∂¯q¯ φ∂ φ ∂ q¯ φ∂ ∂φ¯ ) , (2.12) m ∂+ m − m where the + sign on the kinematic supersymmetery generators is no longer shown. 2.1 E7(7) symmetry The non-linear E7(7)/SU(8) transformations to order κ are given by [9] 2 δφ = θklmn Ξ − κ klmn κ 1 1 + Ξmnpq d φ ∂+3φ 4 d φ d ∂+2φ + 3 d ∂+φ d ∂+φ , (2.13) 4! ∂+2 mnpq ∂+ − mnp q mn pq klmn k l m n ¯ ¯ 1 pqrs where θ = θ θ θ θ , dm1...mn = dm1 ....dmn and Ξklmn = 2 ǫklmnpqrs Ξ , a constant. These 70 coset transformations along with the linear SU(8) transformations m i m 1 m p m p p m m p T n = q q¯n δ n q q¯p ;[ T n ,T q ] = δ n T q δ q T n , (2.14) 2√2 ∂+ − 8 − constitute the entire E7(7) algebra.
Load more

Recommended publications
  • The Top Yukawa Coupling from 10D Supergravity with E8 × E8 Matter
    hep-th/9804107 ITEP-98-11 The top Yukawa coupling from 10D supergravity with E E matter 8 × 8 K.Zyablyuk1 Institute of Theoretical and Experimental Physics, Moscow Abstract We consider the compactification of N=1, D=10 supergravity with E E Yang- 8 × 8 Mills matter to N=1, D=4 model with 3 generations. With help of embedding SU(5) → SO(10) E E we find the value of the top Yukawa coupling λ = 16πα =3 → 6 → 8 t GUT at the GUT scale. p 1 Introduction Although superstring theories have great success and today are the best candidates for a quantum theory unifying all interactions, they still do not predict any experimentally testable value, mostly because there is no unambiguous procedure of compactification of extra spatial dimensions. On the other hand, many phenomenological models based on the unification group SO(10) were constructed (for instance [1]). They describe quarks and leptons in representa- tions 161,162,163 and Higgses, responsible for SU(2) breaking, in 10. Basic assumption of these models is that there is only one Yukawa coupling 163 10 163 at the GUT scale, which gives masses of third generation. Masses of first two families· and· mixings are generated due to the interaction with additional superheavy ( MGUT ) states in 16 + 16, 45, 54. Such models explain generation mass hierarchy and allow∼ to express all Yukawa matrices, which well fit into experimentally observable pattern, in terms of few unknown parameters. Models of [1] are unlikely derivable from ”more fundamental” theory like supergrav- ity/string. Nevertheless, something similar can be constructed from D=10 supergravity coupled to E8 E8 matter, which is low-energy limit of heterotic string [2].
    [Show full text]
  • Arxiv:1805.06347V1 [Hep-Th] 16 May 2018 Sa Rte O H Rvt Eerhfudto 08Awar 2018 Foundation Research Gravity the for Written Essay Result
    Maximal supergravity and the quest for finiteness 1 2 3 Sudarshan Ananth† , Lars Brink∗ and Sucheta Majumdar† † Indian Institute of Science Education and Research Pune 411008, India ∗ Department of Physics, Chalmers University of Technology S-41296 G¨oteborg, Sweden and Division of Physics and Applied Physics, School of Physical and Mathematical Sciences Nanyang Technological University, Singapore 637371 March 28, 2018 Abstract We show that N = 8 supergravity may possess an even larger symmetry than previously believed. Such an enhanced symmetry is needed to explain why this theory of gravity exhibits ultraviolet behavior reminiscent of the finite N = 4 Yang-Mills theory. We describe a series of three steps that leads us to this result. arXiv:1805.06347v1 [hep-th] 16 May 2018 Essay written for the Gravity Research Foundation 2018 Awards for Essays on Gravitation 1Corresponding author, [email protected] [email protected] [email protected] Quantum Field Theory describes three of the four fundamental forces in Nature with great precision. However, when attempts have been made to use it to describe the force of gravity, the resulting field theories, are without exception, ultraviolet divergent and non- renormalizable. One striking aspect of supersymmetry is that it greatly reduces the diver- gent nature of quantum field theories. Accordingly, supergravity theories have less severe ultraviolet divergences. Maximal supergravity in four dimensions, N = 8 supergravity [1], has the best ultraviolet properties of any field theory of gravity with two derivative couplings. Much of this can be traced back to its three symmetries: Poincar´esymmetry, maximal supersymmetry and an exceptional E7(7) symmetry.
    [Show full text]
  • Introduction to String Theory A.N
    Introduction to String Theory A.N. Schellekens Based on lectures given at the Radboud Universiteit, Nijmegen Last update 6 July 2016 [Word cloud by www.worldle.net] Contents 1 Current Problems in Particle Physics7 1.1 Problems of Quantum Gravity.........................9 1.2 String Diagrams................................. 11 2 Bosonic String Action 15 2.1 The Relativistic Point Particle......................... 15 2.2 The Nambu-Goto action............................ 16 2.3 The Free Boson Action............................. 16 2.4 World sheet versus Space-time......................... 18 2.5 Symmetries................................... 19 2.6 Conformal Gauge................................ 20 2.7 The Equations of Motion............................ 21 2.8 Conformal Invariance.............................. 22 3 String Spectra 24 3.1 Mode Expansion................................ 24 3.1.1 Closed Strings.............................. 24 3.1.2 Open String Boundary Conditions................... 25 3.1.3 Open String Mode Expansion..................... 26 3.1.4 Open versus Closed........................... 26 3.2 Quantization.................................. 26 3.3 Negative Norm States............................. 27 3.4 Constraints................................... 28 3.5 Mode Expansion of the Constraints...................... 28 3.6 The Virasoro Constraints............................ 29 3.7 Operator Ordering............................... 30 3.8 Commutators of Constraints.......................... 31 3.9 Computation of the Central Charge.....................
    [Show full text]
  • S-Duality-And-M5-Mit
    N . More on = 2 S-dualities and M5-branes .. Yuji Tachikawa based on works in collaboration with L. F. Alday, B. Wecht, F. Benini, S. Benvenuti, D. Gaiotto November 2009 Yuji Tachikawa (IAS) November 2009 1 / 47 Contents 1. Introduction 2. S-dualities 3. A few words on TN 4. 4d CFT vs 2d CFT Yuji Tachikawa (IAS) November 2009 2 / 47 Montonen-Olive duality • N = 4 SU(N) SYM at coupling τ = θ=(2π) + (4πi)=g2 equivalent to the same theory coupling τ 0 = −1/τ • One way to ‘understand’ it: start from 6d N = (2; 0) theory, i.e. the theory on N M5-branes, put on a torus −1/τ τ 0 1 0 1 • Low energy physics depends only on the complex structure S-duality! Yuji Tachikawa (IAS) November 2009 3 / 47 S-dualities in N = 2 theories • You can wrap N M5-branes on a more general Riemann surface, possibly with punctures, to get N = 2 superconformal field theories • Different limits of the shape of the Riemann surface gives different weakly-coupled descriptions, giving S-dualities among them • Anticipated by [Witten,9703166], but not well-appreciated until [Gaiotto,0904.2715] Yuji Tachikawa (IAS) November 2009 4 / 47 Contents 1. Introduction 2. S-dualities 3. A few words on TN 4. 4d CFT vs 2d CFT Yuji Tachikawa (IAS) November 2009 5 / 47 Contents 1. Introduction 2. S-dualities 3. A few words on TN 4. 4d CFT vs 2d CFT Yuji Tachikawa (IAS) November 2009 6 / 47 S-duality in N = 2 . SU(2) with Nf = 4 .
    [Show full text]
  • E 8 in N= 8 Supergravity in Four Dimensions
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Chalmers Research E 8 in N= 8 supergravity in four dimensions Downloaded from: https://research.chalmers.se, 2019-05-11 18:56 UTC Citation for the original published paper (version of record): Ananth, S., Brink, L., Majumdar, S. (2018) E 8 in N= 8 supergravity in four dimensions Journal of High Energy Physics, 2018(1) http://dx.doi.org/10.1007/JHEP01(2018)024 N.B. When citing this work, cite the original published paper. research.chalmers.se offers the possibility of retrieving research publications produced at Chalmers University of Technology. It covers all kind of research output: articles, dissertations, conference papers, reports etc. since 2004. research.chalmers.se is administrated and maintained by Chalmers Library (article starts on next page) Published for SISSA by Springer Received: November 30, 2017 Accepted: December 23, 2017 Published: January 8, 2018 E8 in N = 8 supergravity in four dimensions Sudarshan Ananth,a Lars Brinkb,c and Sucheta Majumdara aIndian Institute of Science Education and Research, Pune 411008, India bDepartment of Physics, Chalmers University of Technology, S-41296 G¨oteborg, Sweden cDivision of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 637371, Singapore E-mail: [email protected], [email protected], [email protected] Abstract: We argue that = 8 supergravity in four dimensions exhibits an exceptional N E8(8) symmetry, enhanced from the known E7(7) invariance. Our procedure to demonstrate this involves dimensional reduction of the = 8 theory to d = 3, a field redefinition to N render the E8(8) invariance manifest, followed by dimensional oxidation back to d = 4.
    [Show full text]
  • JHEP09(2009)052 July 14, 2009 S, Extended : August 30, 2009 : September 9, 2009 Alize, When Com- : Received Eneralized S-Rule
    Published by IOP Publishing for SISSA Received: July 14, 2009 Accepted: August 30, 2009 Published: September 9, 2009 Webs of five-branes and N = 2 superconformal field JHEP09(2009)052 theories Francesco Benini,a Sergio Benvenutia and Yuji Tachikawab aDepartment of Physics, Princeton University, Washington Road, Princeton NJ 08544, U.S.A. bSchool of Natural Sciences, Institute for Advanced Study, 1 Einstein Drive, Princeton NJ 08540, U.S.A. E-mail: [email protected], [email protected], [email protected] Abstract: We describe configurations of 5-branes and 7-branes which realize, when com- pactified on a circle, new isolated four-dimensional N = 2 superconformal field theories recently constructed by Gaiotto. Our diagrammatic method allows to easily count the dimensions of Coulomb and Higgs branches, with the help of a generalized s-rule. We furthermore show that superconformal field theories with E6,7,8 flavor symmetry can be analyzed in a uniform manner in this framework; in particular we realize these theories at infinitely strongly-coupled limits of quiver theories with SU gauge groups. Keywords: Supersymmetry and Duality, Brane Dynamics in Gauge Theories, Extended Supersymmetry, Intersecting branes models ArXiv ePrint: 0906.0359 c SISSA 2009 doi:10.1088/1126-6708/2009/09/052 Contents 1 Introduction 1 2 N-junction and T [AN−1] theory 4 2.1 N-junction 4 2.2 Coulomb branch 6 2.3 Higgs branch 6 JHEP09(2009)052 2.4 Dualities and Seiberg-Witten curve 7 3 General punctures and the s-rule 8 3.1 Classification of punctures 8 3.2 Generalized
    [Show full text]
  • E8 Root Vector Geometry - AQFT - 26D String Theory - - Schwinger Sources - Quantum Consciousness Frank Dodd (Tony) Smith, Jr
    E8 Root Vector Geometry - AQFT - 26D String Theory - - Schwinger Sources - Quantum Consciousness Frank Dodd (Tony) Smith, Jr. - 2017 - viXra 1701.0495 Abstract This paper is intended to be a only rough semi-popular overview of how the 240 Root Vectors of E8 can be used to construct a useful Lagrangian and Algebraic Quantum Field Theory (AQFT) in which the Bohm Quantum Potential emerges from a 26D String Theory with Strings = World-Lines = Path Integral Paths and the Massless Spin 2 State interpreted as the Bohm Quantum Potential. For details and references, see viXra/1602.0319. The 240 Root Vectors of E8 represent the physical forces, particles, and spacetime that make up the construction of a realistic Lagrangian describing the Octonionic Inflation Era. The Octonionic Lagrangian can be embedded into a Cl(1,25) Clifford Algebra which with 8-Periodicity gives an AQFT. The Massless Spin 2 State of 26D String Theory gives the Bohm Quantum Potential. The Quantum Code of the AQFT is the Tensor Product Quantum Reed-Muller code. A Single Cell of the 26D String Theory model has the symmetry of the Monster Group. Quantum Processes produce Schwinger Sources with size about 10^(-24) cm. Microtubule Structure related to E8 and Clifford Algebra enable Penrose-Hameroff Quantum Consciousness. E8 and Cl(8) may have been encoded in the Great Pyramid. A seperate paper discusses using the Quaternionic M4 x CP2 Kaluza-Klein version of the Lagrangian to produce the Higgs and 2nd and 3rd Generation Fermions and a Higgs - Truth Quark System with 3 Mass States for Higgs and Truth Quark.
    [Show full text]
  • Supersymmetric Sigma Models*
    SLAC -PUB - 3461 September 1984 T SUPERSYMMETRIC SIGMA MODELS* JONATHAN A. BAGGER Stanford Linear Accelerator Center Stanford University, Stanford, California, 9~305 Lectures given at the -. Bonn-NATO Advanced Study Institute on Supersymmetry Bonn, West Germany August 20-31,1984 * Work supported by the Department of Energy, contract DE - AC03 - 76SF00515 ABSTRACT We begin to construct the most general supersymmetric Lagrangians in one, two and four dimensions. We find that the matter couplings have a natural interpretation in the language of the nonlinear sigma model. 1. INTRODUCTION The past few years have witnessed a dramatic revival of interest in the phenomenological aspects of supersymmetry [1,2]. Many models have been proposed, and much work has been devoted to exploring their experimental implications. A common feature of all these models is that they predict a variety of new particles at energies near the weak scale. Since the next generation of accelerators will start to probe these energies, we have the exciting possibility that supersymmetry will soon be found. While we are waiting for the new experiments, however, we must con- tinue to gain a deeper understanding of supersymmetric theories themselves. One vital task is to learn how to construct the most general possible su- persymmetric Lagrangians. These Lagrangians can then be used by model builders in their search for realistic theories. For N = 1 rigid supersymme- try, it is not hard to write down the most general possible supersymmetric Lagrangian. For higher N, however, and for all local supersymmetries, the story is more complicated. In these lectures we will begin to discuss the most general matter cou- plings in N = 1 and N = 2 supersymmetric theories.
    [Show full text]
  • E8 + E8 Heterotic String Theory in Vedic Physics
    E8 + E8 Heterotic String Theory in Vedic Physics By John Frederic Sweeney Abstract S.M. Phillips has articulated a fairly good model of the E8×E8 heterotic superstring, yet nevertheless has missed a few key aspects. This paper informs his model from the perspective of Vedic Nuclear Physics, as derived from the Rig Veda and two of the Upanishads. In addition, the author hypothesizes an extension of the Exceptional Lie Algebra Series beyond E8 to another 12 places or more. 1 Table of Contents Introduction 3 Wikipedia 5 S.M. Phillips model 12 H series of Hypercircles 14 Conclusion 18 Bibliography 21 2 Introduction S.M. Phillips has done a great sleuthing job in exploring the Jewish Cabala along with the works of Basant and Leadbetter to formulate a model of the Exceptional Lie Algebra E8 to represent nuclear physics that comes near to the Super String model. The purpose of this paper is to offer minor corrections from the perspective of the science encoded in the Rig Veda and in a few of the Upanishads, to render a complete and perfect model. The reader might ask how the Jewish Cabala might offer insight into nuclear physics, since the Cabala is generally thought to date from Medieval Spain. The simple fact is that the Cabala does not represent medieval Spanish thought, it is a product of a much older and advanced society – Remotely Ancient Egypt from 15,000 years ago, before the last major flooding of the Earth and the Sphinx. The Jewish people may very well have left Ancient Egypt in the Exodus, led by Moses.
    [Show full text]
  • A Geometric Theory of Everything Deep Down, the Particles and Forces of the Universe Are a Manifestation of Exquisite Geometry
    Nature’s zoo of elementary particles is not a random mish- mash; it has striking patterns and interrelationships that can be depicted on a diagram correspond- ing to one of the most intricate geometric objects known to mathematicians, called E8. 54 Scientific American, December 2010 Photograph/Illustration by Artist Name © 2010 Scientific American A. GarrettAuthor Lisi balances Bio Tex this until time an between ‘end nested research style incharacter’ theoretical (Command+3) physics andxxxx surfing. xxxx xx xxxxxAs an itinerantxxxxx xxxx scientist, xxxxxx hexxxxxx is in thexxx xxprocess xxxxxxxxxx of xxxx realizingxxxxxxxxxx a lifelong dream: xxxxxx founding xxxxxxxx the xxxxxxxx. Pacific Science Institute, located on the Hawaiian island of Maui. James Owen Weatherall, having recently completed his doctorate in physics and mathematics at the Stevens Institute of Technology, is now finishing a second Ph.D. in philosophy at the University of California, Irvine. He also manages to find time to work on a book on the history of ideas moving from physics into financial modeling. PHYSICS A Geometric Theory of Everything Deep down, the particles and forces of the universe are a manifestation of exquisite geometry By A. Garrett Lisi and James Owen Weatherall odern physics began with a sweeping unification: in 1687 isaac Newton showed that the existing jumble of disparate theories describing everything from planetary motion to tides to pen- dulums were all aspects of a universal law of gravitation. Unifi- cation has played a central role in physics ever since. In the middle of the 19th century James Clerk Maxwell found that electricity and magnetism were two facets of electromagnetism.
    [Show full text]
  • UNIVERSITAT BONN Physekah'sches Institut
    DE04F5920 UNIVERSITAT BONN Physekah'sches Institut Local Anomaly Cancellation in Heterotic E8 x E8 Orbifold Compactifications with Wilson Line Backgi-ounds von Martin G.A. Walter +DE020674229-#- We consider several examples of a special class of heterotic compactifica- tions, i.e. heterotic E8 x E8 orbifolds with Wilson line backgrounds. By developing a local perspective re sow tha a brane world like picture emerges. As an important result we prove that the local massless spec- trum at sch a brane can always be traced back to the global spectrum of a. different) orbifold without Wilson lines. One particular iplication of this result is that the use of (discrete) Wilson lines for the construction of phenomenologically interesting models has to be rethought. We show that stringy constraints render the brane spectra consistent. Using our local picture we are able to compute the local anomalies appearing at the different branes for our examples and show that they can all be cancelled by a local version of the Green--Schwarz mechanism at te same time. Post address: BONN-IR-2004-04 Nussallee 12 Bonn University 53115 Bonn February 2004 Gernianv ISSN-01 72-8741 I UNIVERSITAT BONN Physikalisches Institut Local Anomaly Cancellation in Heterotic E8 x E8 Orbifold Compactifications with Wilson Line Backgrounds von Martin G.A. Walter Dieser Forschungsbericht wurde als Dissertation von der Mathematisch - Naturwissenschaftlichen Fakultat der Universitdt Bonn angenommen. Angenommen am: 20.02.2004 Referent: Prof. Dr. Hans-Peter Nilles Korreferent: ProE Dr. Rainald Flume Acknowledgments First of all, I want to thank Prof. Dr. Hans-Peter Nilles for his guidance and steady support and for leading my work by always asking the right questions.
    [Show full text]
  • Group-Theoretical Aspects of Orbifold and Conifold Guts
    Nuclear Physics B 670 (2003) 3–26 www.elsevier.com/locate/npe Group-theoretical aspects of orbifold and conifold GUTs Arthur Hebecker, Michael Ratz Deutsches Elektronen-Synchrotron, Notkestrasse 85, D-22603 Hamburg, Germany Received 13 June 2003; accepted 28 July 2003 Abstract Motivated by the simplicity and direct phenomenological applicability of field-theoretic orbifold constructions in the context of grand unification, we set out to survey the immensely rich group- theoretical possibilities open to this type of model building. In particular, we show how every maximal-rank, regular subgroup of a simple Lie group can be obtained by orbifolding and determine under which conditions rank reduction is possible. We investigate how standard model matter can arise from the higher-dimensional SUSY gauge multiplet. New model building options arise if, giving up the global orbifold construction, generic conical singularities and generic gauge twists associated with these singularities are considered. Viewed from the purely field-theoretic perspective, such models, which one might call conifold GUTs, require only a very mild relaxation of the constraints of orbifold model building. Our most interesting concrete examples include the breaking of E7 to SU(5) and of E8 to SU(4) × SU(2) × SU(2) (with extra factor groups), where three generations of standard model matter come from the gauge sector and the families are interrelated either by SU(3) R-symmetryorbyanSU(3) flavour subgroup of the original gauge group. 2003 Elsevier B.V. All rights reserved. 1. Introduction Arguably, the way in which fermion quantum numbers are explained by SU(5)-related grand unified theories (GUTs) represents one of the most profound hints at fundamental physics beyond the standard model (SM) [1,2] (also [3]).
    [Show full text]