String Theory
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Arxiv:2009.05574V4 [Hep-Th] 9 Nov 2020 Predict a New Massless Spin One Boson [The ‘Lorentz’ Boson] Which Should Be Looked for in Experiments
Trace dynamics and division algebras: towards quantum gravity and unification Tejinder P. Singh Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India e-mail: [email protected] Accepted for publication in Zeitschrift fur Naturforschung A on October 4, 2020 v4. Submitted to arXiv.org [hep-th] on November 9, 2020 ABSTRACT We have recently proposed a Lagrangian in trace dynamics at the Planck scale, for unification of gravitation, Yang-Mills fields, and fermions. Dynamical variables are described by odd- grade (fermionic) and even-grade (bosonic) Grassmann matrices. Evolution takes place in Connes time. At energies much lower than Planck scale, trace dynamics reduces to quantum field theory. In the present paper we explain that the correct understanding of spin requires us to formulate the theory in 8-D octonionic space. The automorphisms of the octonion algebra, which belong to the smallest exceptional Lie group G2, replace space- time diffeomorphisms and internal gauge transformations, bringing them under a common unified fold. Building on earlier work by other researchers on division algebras, we propose the Lorentz-weak unification at the Planck scale, the symmetry group being the stabiliser group of the quaternions inside the octonions. This is one of the two maximal sub-groups of G2, the other one being SU(3), the element preserver group of octonions. This latter group, coupled with U(1)em, describes the electro-colour symmetry, as shown earlier by Furey. We arXiv:2009.05574v4 [hep-th] 9 Nov 2020 predict a new massless spin one boson [the `Lorentz' boson] which should be looked for in experiments. -
Orthogonal Symmetric Affine Kac-Moody Algebras
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 367, Number 10, October 2015, Pages 7133–7159 http://dx.doi.org/10.1090/tran/6257 Article electronically published on April 20, 2015 ORTHOGONAL SYMMETRIC AFFINE KAC-MOODY ALGEBRAS WALTER FREYN Abstract. Riemannian symmetric spaces are fundamental objects in finite dimensional differential geometry. An important problem is the construction of symmetric spaces for generalizations of simple Lie groups, especially their closest infinite dimensional analogues, known as affine Kac-Moody groups. We solve this problem and construct affine Kac-Moody symmetric spaces in a series of several papers. This paper focuses on the algebraic side; more precisely, we introduce OSAKAs, the algebraic structures used to describe the connection between affine Kac-Moody symmetric spaces and affine Kac-Moody algebras and describe their classification. 1. Introduction Riemannian symmetric spaces are fundamental objects in finite dimensional dif- ferential geometry displaying numerous connections with Lie theory, physics, and analysis. The search for infinite dimensional symmetric spaces associated to affine Kac-Moody algebras has been an open question for 20 years, since it was first asked by C.-L. Terng in [Ter95]. We present a complete solution to this problem in a series of several papers, dealing successively with the functional analytic, the algebraic and the geometric aspects. In this paper, building on work of E. Heintze and C. Groß in [HG12], we introduce and classify orthogonal symmetric affine Kac- Moody algebras (OSAKAs). OSAKAs are the central objects in the classification of affine Kac-Moody symmetric spaces as they provide the crucial link between the geometric and the algebraic side of the theory. -
String Theory. Volume 1, Introduction to the Bosonic String
This page intentionally left blank String Theory, An Introduction to the Bosonic String The two volumes that comprise String Theory provide an up-to-date, comprehensive, and pedagogic introduction to string theory. Volume I, An Introduction to the Bosonic String, provides a thorough introduction to the bosonic string, based on the Polyakov path integral and conformal field theory. The first four chapters introduce the central ideas of string theory, the tools of conformal field theory and of the Polyakov path integral, and the covariant quantization of the string. The next three chapters treat string interactions: the general formalism, and detailed treatments of the tree-level and one loop amplitudes. Chapter eight covers toroidal compactification and many important aspects of string physics, such as T-duality and D-branes. Chapter nine treats higher-order amplitudes, including an analysis of the finiteness and unitarity, and various nonperturbative ideas. An appendix giving a short course on path integral methods is also included. Volume II, Superstring Theory and Beyond, begins with an introduction to supersym- metric string theories and goes on to a broad presentation of the important advances of recent years. The first three chapters introduce the type I, type II, and heterotic superstring theories and their interactions. The next two chapters present important recent discoveries about strongly coupled strings, beginning with a detailed treatment of D-branes and their dynamics, and covering string duality, M-theory, and black hole entropy. A following chapter collects many classic results in conformal field theory. The final four chapters are concerned with four-dimensional string theories, and have two goals: to show how some of the simplest string models connect with previous ideas for unifying the Standard Model; and to collect many important and beautiful general results on world-sheet and spacetime symmetries. -
Spectra of Conformal Sigma Models
Spectra of Conformal Sigma Models Dissertation zur Erlangung des Doktorgrades an der Fakultät für Mathematik, Informatik und Naturwissenschaften Fachbereich Physik der Universität Hamburg vorgelegt von Václav Tlapák aus Berlin Hamburg 2014 Tag der Disputation: 30. Januar 2015 Folgende Gutachter empfehlen die Annahme der Dissertation: Prof. Dr. Volker Schomerus Prof. Dr. Gleb Arutyunov Zusammenfassung Die vorliegende Arbeit beschäftigt sich mit den Spektren von konformen Sigma Modellen, die auf (verallgemeinerten) symmetrischen Räumen defi- niert sind. Die Räume, auf denen Sigma Modelle ohne Wess-Zumino-Term konform sind, sind Supermannigfaltigkeiten, also Mannigfaltigkeiten, die fermionische Richtungen aufweisen. Wir stellen die Konstruktion von Ver- tex Operatoren vor, gefolgt von der Hintergrundfeld-Entwicklung. Für semi- symmetrische Räume berechnen wir anschließend die diagonalen Terme der anomalen Dimensionen dieser Operatoren in führender Ordnung. Das Resul- tat stimmt mit dem für symmetrische Räume überein, jedoch treten nicht- diagonale Terme auf, die hier nicht weiter betrachtet werden. Anschließend präsentieren wir eine detaillierte Analyse des Spectrums des Supersphären S3j2 Sigma Modells. Dies ist eins der einfachsten Beispie- le für konforme Sigma Modelle auf symmetrischen Räumen und dient als Illustration für die Mächtigkeit der vorgestellten Methoden. Wir verwenden die erhaltenen Daten, um eine Dualität mit dem OSP(4j2) Gross-Neveu Modell zu untersuchen, die von Candu und Saleur vorgeschlagen wurde. Wir verwenden dazu ein Resultat, welches die anomalen Dimensionen von 1 2 BPS Operatoren zu allen Ordnungen berechnet. Wir finden das gesamte Grundzustandsspektrum des Sigma Modells. Darüber hinaus legen wir dar, dass sowohl die Zwangsbedingungen als auch die Bewegungsgleichungen des Sigma Modells korrekt vom Gross-Neveu Modell implementiert werden. Die Dualität wird weiterhin durch ein neues exaktes Resultat für die anomalen Dimensionen der Grundzustände unterstützt. -
Durham Research Online
Durham Research Online Deposited in DRO: 30 January 2017 Version of attached le: Published Version Peer-review status of attached le: Peer-reviewed Citation for published item: Bhattacharya, Jyotirmoy and Lipstein, Arthur E. (2016) '6d dual conformal symmetry and minimal volumes in AdS.', Journal of high energy physics., 2016 (12). p. 105. Further information on publisher's website: https://doi.org/10.1007/JHEP12(2016)105 Publisher's copyright statement: Open Access, c The Authors. Article funded by SCOAP3. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Additional information: Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full DRO policy for further details. Durham University Library, Stockton Road, Durham DH1 3LY, United Kingdom Tel : +44 (0)191 334 3042 | Fax : +44 (0)191 334 2971 https://dro.dur.ac.uk Published for SISSA by Springer Received: November 16, 2016 Revised: December 8, 2016 Accepted: December 12, 2016 Published: December 20, 2016 6d dual conformal symmetry and minimal volumes JHEP12(2016)105 in AdS Jyotirmoy Bhattacharya and Arthur E. -
Kaluza-Klein Gravity, Concentrating on the General Rel- Ativity, Rather Than Particle Physics Side of the Subject
Kaluza-Klein Gravity J. M. Overduin Department of Physics and Astronomy, University of Victoria, P.O. Box 3055, Victoria, British Columbia, Canada, V8W 3P6 and P. S. Wesson Department of Physics, University of Waterloo, Ontario, Canada N2L 3G1 and Gravity Probe-B, Hansen Physics Laboratories, Stanford University, Stanford, California, U.S.A. 94305 Abstract We review higher-dimensional unified theories from the general relativity, rather than the particle physics side. Three distinct approaches to the subject are identi- fied and contrasted: compactified, projective and noncompactified. We discuss the cosmological and astrophysical implications of extra dimensions, and conclude that none of the three approaches can be ruled out on observational grounds at the present time. arXiv:gr-qc/9805018v1 7 May 1998 Preprint submitted to Elsevier Preprint 3 February 2008 1 Introduction Kaluza’s [1] achievement was to show that five-dimensional general relativity contains both Einstein’s four-dimensional theory of gravity and Maxwell’s the- ory of electromagnetism. He however imposed a somewhat artificial restriction (the cylinder condition) on the coordinates, essentially barring the fifth one a priori from making a direct appearance in the laws of physics. Klein’s [2] con- tribution was to make this restriction less artificial by suggesting a plausible physical basis for it in compactification of the fifth dimension. This idea was enthusiastically received by unified-field theorists, and when the time came to include the strong and weak forces by extending Kaluza’s mechanism to higher dimensions, it was assumed that these too would be compact. This line of thinking has led through eleven-dimensional supergravity theories in the 1980s to the current favorite contenders for a possible “theory of everything,” ten-dimensional superstrings. -
Introduction to Supersymmetry(1)
Introduction to Supersymmetry(1) J.N. Tavares Dep. Matem¶aticaPura, Faculdade de Ci^encias,U. Porto, 4000 Porto TQFT Club 1Esta ¶euma vers~aoprovis¶oria,incompleta, para uso exclusivo nas sess~oesde trabalho do TQFT club CONTENTS 1 Contents 1 Supersymmetry in Quantum Mechanics 2 1.1 The Supersymmetric Oscillator . 2 1.2 Witten Index . 4 1.3 A fundamental example: The Laplacian on forms . 7 1.4 Witten's proof of Morse Inequalities . 8 2 Supergeometry and Supersymmetry 13 2.1 Field Theory. A quick review . 13 2.2 SuperEuclidean Space . 17 2.3 Reality Conditions . 18 2.4 Supersmooth functions . 18 2.5 Supermanifolds . 21 2.6 Lie Superalgebras . 21 2.7 Super Lie groups . 26 2.8 Rigid Superspace . 27 2.9 Covariant Derivatives . 30 3 APPENDIX. Cli®ord Algebras and Spin Groups 31 3.1 Cli®ord Algebras . 31 Motivation. Cli®ord maps . 31 Cli®ord Algebras . 33 Involutions in V .................................. 35 Representations . 36 3.2 Pin and Spin groups . 43 3.3 Spin Representations . 47 3.4 U(2), spinors and almost complex structures . 49 3.5 Spinc(4)...................................... 50 Chiral Operator. Self Duality . 51 2 1 Supersymmetry in Quantum Mechanics 1.1 The Supersymmetric Oscillator i As we will see later the \hermitian supercharges" Q®, in the N extended SuperPoincar¶eLie Algebra obey the anticommutation relations: i j m ij fQ®;Q¯g = 2(γ C)®¯± Pm (1.1) m where ®; ¯ are \spinor" indices, i; j 2 f1; ¢ ¢ ¢ ;Ng \internal" indices and (γ C)®¯ a bilinear form in the spinor indices ®; ¯. When specialized to 0-space dimensions ((1+0)-spacetime), then since P0 = H, relations (1.1) take the form (with a little change in notations): fQi;Qjg = 2±ij H (1.2) with N \Hermitian charges" Qi; i = 1; ¢ ¢ ¢ ;N. -
Off-Shell Interactions for Closed-String Tachyons
Preprint typeset in JHEP style - PAPER VERSION hep-th/0403238 KIAS-P04017 SLAC-PUB-10384 SU-ITP-04-11 TIFR-04-04 Off-Shell Interactions for Closed-String Tachyons Atish Dabholkarb,c,d, Ashik Iqubald and Joris Raeymaekersa aSchool of Physics, Korea Institute for Advanced Study, 207-43, Cheongryangri-Dong, Dongdaemun-Gu, Seoul 130-722, Korea bStanford Linear Accelerator Center, Stanford University, Stanford, CA 94025, USA cInstitute for Theoretical Physics, Department of Physics, Stanford University, Stanford, CA 94305, USA dDepartment of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India E-mail:[email protected], [email protected], [email protected] Abstract: Off-shell interactions for localized closed-string tachyons in C/ZN super- string backgrounds are analyzed and a conjecture for the effective height of the tachyon potential is elaborated. At large N, some of the relevant tachyons are nearly massless and their interactions can be deduced from the S-matrix. The cubic interactions be- tween these tachyons and the massless fields are computed in a closed form using orbifold CFT techniques. The cubic interaction between nearly-massless tachyons with different charges is shown to vanish and thus condensation of one tachyon does not source the others. It is shown that to leading order in N, the quartic contact in- teraction vanishes and the massless exchanges completely account for the four point scattering amplitude. This indicates that it is necessary to go beyond quartic inter- actions or to include other fields to test the conjecture for the height of the tachyon potential. Keywords: closed-string tachyons, orbifolds. -
Quantum Field Theory*
Quantum Field Theory y Frank Wilczek Institute for Advanced Study, School of Natural Science, Olden Lane, Princeton, NJ 08540 I discuss the general principles underlying quantum eld theory, and attempt to identify its most profound consequences. The deep est of these consequences result from the in nite number of degrees of freedom invoked to implement lo cality.Imention a few of its most striking successes, b oth achieved and prosp ective. Possible limitation s of quantum eld theory are viewed in the light of its history. I. SURVEY Quantum eld theory is the framework in which the regnant theories of the electroweak and strong interactions, which together form the Standard Mo del, are formulated. Quantum electro dynamics (QED), b esides providing a com- plete foundation for atomic physics and chemistry, has supp orted calculations of physical quantities with unparalleled precision. The exp erimentally measured value of the magnetic dip ole moment of the muon, 11 (g 2) = 233 184 600 (1680) 10 ; (1) exp: for example, should b e compared with the theoretical prediction 11 (g 2) = 233 183 478 (308) 10 : (2) theor: In quantum chromo dynamics (QCD) we cannot, for the forseeable future, aspire to to comparable accuracy.Yet QCD provides di erent, and at least equally impressive, evidence for the validity of the basic principles of quantum eld theory. Indeed, b ecause in QCD the interactions are stronger, QCD manifests a wider variety of phenomena characteristic of quantum eld theory. These include esp ecially running of the e ective coupling with distance or energy scale and the phenomenon of con nement. -
Notes on Statistical Field Theory
Lecture Notes on Statistical Field Theory Kevin Zhou [email protected] These notes cover statistical field theory and the renormalization group. The primary sources were: • Kardar, Statistical Physics of Fields. A concise and logically tight presentation of the subject, with good problems. Possibly a bit too terse unless paired with the 8.334 video lectures. • David Tong's Statistical Field Theory lecture notes. A readable, easygoing introduction covering the core material of Kardar's book, written to seamlessly pair with a standard course in quantum field theory. • Goldenfeld, Lectures on Phase Transitions and the Renormalization Group. Covers similar material to Kardar's book with a conversational tone, focusing on the conceptual basis for phase transitions and motivation for the renormalization group. The notes are structured around the MIT course based on Kardar's textbook, and were revised to include material from Part III Statistical Field Theory as lectured in 2017. Sections containing this additional material are marked with stars. The most recent version is here; please report any errors found to [email protected]. 2 Contents Contents 1 Introduction 3 1.1 Phonons...........................................3 1.2 Phase Transitions......................................6 1.3 Critical Behavior......................................8 2 Landau Theory 12 2.1 Landau{Ginzburg Hamiltonian.............................. 12 2.2 Mean Field Theory..................................... 13 2.3 Symmetry Breaking.................................... 16 3 Fluctuations 19 3.1 Scattering and Fluctuations................................ 19 3.2 Position Space Fluctuations................................ 20 3.3 Saddle Point Fluctuations................................. 23 3.4 ∗ Path Integral Methods.................................. 24 4 The Scaling Hypothesis 29 4.1 The Homogeneity Assumption............................... 29 4.2 Correlation Lengths.................................... 30 4.3 Renormalization Group (Conceptual).......................... -
Effective Field Theories, Reductionism and Scientific Explanation Stephan
To appear in: Studies in History and Philosophy of Modern Physics Effective Field Theories, Reductionism and Scientific Explanation Stephan Hartmann∗ Abstract Effective field theories have been a very popular tool in quantum physics for almost two decades. And there are good reasons for this. I will argue that effec- tive field theories share many of the advantages of both fundamental theories and phenomenological models, while avoiding their respective shortcomings. They are, for example, flexible enough to cover a wide range of phenomena, and concrete enough to provide a detailed story of the specific mechanisms at work at a given energy scale. So will all of physics eventually converge on effective field theories? This paper argues that good scientific research can be characterised by a fruitful interaction between fundamental theories, phenomenological models and effective field theories. All of them have their appropriate functions in the research process, and all of them are indispens- able. They complement each other and hang together in a coherent way which I shall characterise in some detail. To illustrate all this I will present a case study from nuclear and particle physics. The resulting view about scientific theorising is inherently pluralistic, and has implications for the debates about reductionism and scientific explanation. Keywords: Effective Field Theory; Quantum Field Theory; Renormalisation; Reductionism; Explanation; Pluralism. ∗Center for Philosophy of Science, University of Pittsburgh, 817 Cathedral of Learning, Pitts- burgh, PA 15260, USA (e-mail: [email protected]) (correspondence address); and Sektion Physik, Universit¨at M¨unchen, Theresienstr. 37, 80333 M¨unchen, Germany. 1 1 Introduction There is little doubt that effective field theories are nowadays a very popular tool in quantum physics. -
Arxiv:Hep-Th/0006117V1 16 Jun 2000 Oadmarolf Donald VERSION HYPER - Style
Preprint typeset in JHEP style. - HYPER VERSION SUGP-00/6-1 hep-th/0006117 Chern-Simons terms and the Three Notions of Charge Donald Marolf Physics Department, Syracuse University, Syracuse, New York 13244 Abstract: In theories with Chern-Simons terms or modified Bianchi identities, it is useful to define three notions of either electric or magnetic charge associated with a given gauge field. A language for discussing these charges is introduced and the proper- ties of each charge are described. ‘Brane source charge’ is gauge invariant and localized but not conserved or quantized, ‘Maxwell charge’ is gauge invariant and conserved but not localized or quantized, while ‘Page charge’ conserved, localized, and quantized but not gauge invariant. This provides a further perspective on the issue of charge quanti- zation recently raised by Bachas, Douglas, and Schweigert. For the Proceedings of the E.S. Fradkin Memorial Conference. Keywords: Supergravity, p–branes, D-branes. arXiv:hep-th/0006117v1 16 Jun 2000 Contents 1. Introduction 1 2. Brane Source Charge and Brane-ending effects 3 3. Maxwell Charge and Asymptotic Conditions 6 4. Page Charge and Kaluza-Klein reduction 7 5. Discussion 9 1. Introduction One of the intriguing properties of supergravity theories is the presence of Abelian Chern-Simons terms and their duals, the modified Bianchi identities, in the dynamics of the gauge fields. Such cases have the unusual feature that the equations of motion for the gauge field are non-linear in the gauge fields even though the associated gauge groups are Abelian. For example, massless type IIA supergravity contains a relation of the form dF˜4 + F2 H3 =0, (1.1) ∧ where F˜4, F2,H3 are gauge invariant field strengths of rank 4, 2, 3 respectively.