String Theory, Superstring Theory and Beyond
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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. -
Arxiv:1910.05796V2 [Math-Ph] 26 Oct 2020 5
Towards a conformal field theory for Schramm-Loewner evolutions Eveliina Peltola [email protected] Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, D-53115 Bonn, Germany We discuss the partition function point of view for chordal Schramm-Loewner evolutions and their relation- ship with correlation functions in conformal field theory. Both are closely related to crossing probabilities and interfaces in critical models in two-dimensional statistical mechanics. We gather and supplement previous results with different perspectives, point out remaining difficulties, and suggest directions for future studies. 1. Introduction 2 2. Schramm-Loewner evolution in statistical mechanics and conformal field theory4 A. Schramm-Loewner evolution (SLE) 5 B. SLE in critical models – the Ising model6 C. Conformal field theory (CFT) 7 D. Martingale observables for interfaces 9 3. Multiple SLE partition functions and applications 12 A. Multiple SLEs and their partition functions 12 B. Definition of the multiple SLE partition functions 13 C. Properties of the multiple SLE partition functions 16 D. Relation to Schramm-Loewner evolutions and critical models 18 4. Fusion and operator product expansion 20 A. Fusion and operator product expansion in conformal field theory 21 B. Fusion: analytic approach 22 C. Fusion: systematic algebraic approach 23 D. Operator product expansion for multiple SLE partition functions 26 arXiv:1910.05796v2 [math-ph] 26 Oct 2020 5. From OPE structure to products of random distributions? 27 A. Conformal bootstrap 27 B. Random tempered distributions 28 C. Bootstrap construction? 29 Appendix A. Representation theory of the Virasoro algebra 31 Appendix B. Probabilistic construction of the pure partition functions 32 Appendix C. -
Stephen Hawking: 'There Are No Black Holes' Notion of an 'Event Horizon', from Which Nothing Can Escape, Is Incompatible with Quantum Theory, Physicist Claims
NATURE | NEWS Stephen Hawking: 'There are no black holes' Notion of an 'event horizon', from which nothing can escape, is incompatible with quantum theory, physicist claims. Zeeya Merali 24 January 2014 Artist's impression VICTOR HABBICK VISIONS/SPL/Getty The defining characteristic of a black hole may have to give, if the two pillars of modern physics — general relativity and quantum theory — are both correct. Most physicists foolhardy enough to write a paper claiming that “there are no black holes” — at least not in the sense we usually imagine — would probably be dismissed as cranks. But when the call to redefine these cosmic crunchers comes from Stephen Hawking, it’s worth taking notice. In a paper posted online, the physicist, based at the University of Cambridge, UK, and one of the creators of modern black-hole theory, does away with the notion of an event horizon, the invisible boundary thought to shroud every black hole, beyond which nothing, not even light, can escape. In its stead, Hawking’s radical proposal is a much more benign “apparent horizon”, “There is no escape from which only temporarily holds matter and energy prisoner before eventually a black hole in classical releasing them, albeit in a more garbled form. theory, but quantum theory enables energy “There is no escape from a black hole in classical theory,” Hawking told Nature. Peter van den Berg/Photoshot and information to Quantum theory, however, “enables energy and information to escape from a escape.” black hole”. A full explanation of the process, the physicist admits, would require a theory that successfully merges gravity with the other fundamental forces of nature. -
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).......................... -
Anomalies, Dualities, and Topology of D = 6 N = 1 Superstring Vacua
RU-96-16 NSF-ITP-96-21 CALT-68-2057 IASSNS-HEP-96/53 hep-th/9605184 Anomalies, Dualities, and Topology of D =6 N =1 Superstring Vacua Micha Berkooz,1 Robert G. Leigh,1 Joseph Polchinski,2 John H. Schwarz,3 Nathan Seiberg,1 and Edward Witten4 1Department of Physics, Rutgers University, Piscataway, NJ 08855-0849 2Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106-4030 3California Institute of Technology, Pasadena, CA 91125 4Institute for Advanced Study, Princeton, NJ 08540 Abstract We consider various aspects of compactifications of the Type I/heterotic Spin(32)/Z2 theory on K3. One family of such compactifications in- cludes the standard embedding of the spin connection in the gauge group, and is on the same moduli space as the compactification of arXiv:hep-th/9605184v1 25 May 1996 the heterotic E8 × E8 theory on K3 with instanton numbers (8,16). Another class, which includes an orbifold of the Type I theory re- cently constructed by Gimon and Polchinski and whose field theory limit involves some topological novelties, is on the moduli space of the heterotic E8 × E8 theory on K3 with instanton numbers (12,12). These connections between Spin(32)/Z2 and E8 × E8 models can be demonstrated by T duality, and permit a better understanding of non- perturbative gauge fields in the (12,12) model. In the transformation between Spin(32)/Z2 and E8 × E8 models, the strong/weak coupling duality of the (12,12) E8 × E8 model is mapped to T duality in the Type I theory. The gauge and gravitational anomalies in the Type I theory are canceled by an extension of the Green-Schwarz mechanism. -
Towards a String Dual of SYK Arxiv:2103.03187V1 [Hep-Th] 4 Mar
Towards A String Dual of SYK Akash Goel and Herman Verlinde Department of Physics, Princeton University, Princeton, NJ 08544, USA Abstract: We propose a paradigm for realizing the SYK model within string theory. Using the large N matrix description of c < 1 string theory, we show that the effective theory on a large number Q of FZZT D-branes in (p; 1) minimal string theory takes the form of the disorder averaged SYK model with J p interaction. The SYK fermions represent open strings between the FZZT branes and the ZZ branes that underly the matrix model. The continuum SYK dynamics arises upon taking the large Q limit. We observe several qualitative and quantitative links between the SYK model and (p; q) minimal string theory and propose that the two describe different phases of a single system. We comment on the dual string interpretation of double scaled SYK and on the relevance of our results to the recent discussion of the role of ensemble averaging in holography. arXiv:2103.03187v2 [hep-th] 24 Aug 2021 Contents 1 Introduction2 2 SYK from the Two Matrix Model4 2.1 FZZT-branes in the two matrix model4 2.2 Kontsevich matrix model from FZZT branes6 2.3 SYK matrix model from FZZT branes7 3 Towards the Continuum SYK model8 3.1 Non-commutative SYK8 4 From Minimal Strings to SYK 10 4.1 SYK and the (p; q) spectral curve 11 4.2 FZZT brane correlation function 12 4.3 Minimal String-SYK phase diagram 13 5 SYK as a Non-Critical String 14 6 Conclusion 16 A D-branes in Minimal String Theory 18 A.1 Matrices and the non-commutative torus 23 A.2 Non-commutative SYK 24 A.3 Matrix SYK 26 A.4 Mapping between Matrix SYK and Non-Commutative SYK 27 { 1 { 1 Introduction The SYK model is the prototype of a maximally chaotic quantum system with low energy dynamics given by near-AdS2 gravity. -
Eternal Inflation and Its Implications
IOP PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICAL J. Phys. A: Math. Theor. 40 (2007) 6811–6826 doi:10.1088/1751-8113/40/25/S25 Eternal inflation and its implications Alan H Guth Center for Theoretical Physics, Laboratory for Nuclear Science, and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA E-mail: [email protected] Received 8 February 2006 Published 6 June 2007 Online at stacks.iop.org/JPhysA/40/6811 Abstract Isummarizetheargumentsthatstronglysuggestthatouruniverseisthe product of inflation. The mechanisms that lead to eternal inflation in both new and chaotic models are described. Although the infinity of pocket universes produced by eternal inflation are unobservable, it is argued that eternal inflation has real consequences in terms of the way that predictions are extracted from theoretical models. The ambiguities in defining probabilities in eternally inflating spacetimes are reviewed, with emphasis on the youngness paradox that results from a synchronous gauge regularization technique. Although inflation is generically eternal into the future, it is not eternal into the past: it can be proven under reasonable assumptions that the inflating region must be incomplete in past directions, so some physics other than inflation is needed to describe the past boundary of the inflating region. PACS numbers: 98.80.cQ, 98.80.Bp, 98.80.Es 1. Introduction: the successes of inflation Since the proposal of the inflationary model some 25 years ago [1–4], inflation has been remarkably successful in explaining many important qualitative and quantitative properties of the universe. In this paper, I will summarize the key successes, and then discuss a number of issues associated with the eternal nature of inflation. -
Pitp Lectures
MIFPA-10-34 PiTP Lectures Katrin Becker1 Department of Physics, Texas A&M University, College Station, TX 77843, USA [email protected] Contents 1 Introduction 2 2 String duality 3 2.1 T-duality and closed bosonic strings .................... 3 2.2 T-duality and open strings ......................... 4 2.3 Buscher rules ................................ 5 3 Low-energy effective actions 5 3.1 Type II theories ............................... 5 3.1.1 Massless bosons ........................... 6 3.1.2 Charges of D-branes ........................ 7 3.1.3 T-duality for type II theories .................... 7 3.1.4 Low-energy effective actions .................... 8 3.2 M-theory ................................... 8 3.2.1 2-derivative action ......................... 8 3.2.2 8-derivative action ......................... 9 3.3 Type IIB and F-theory ........................... 9 3.4 Type I .................................... 13 3.5 SO(32) heterotic string ........................... 13 4 Compactification and moduli 14 4.1 The torus .................................. 14 4.2 Calabi-Yau 3-folds ............................. 16 5 M-theory compactified on Calabi-Yau 4-folds 17 5.1 The supersymmetric flux background ................... 18 5.2 The warp factor ............................... 18 5.3 SUSY breaking solutions .......................... 19 1 These are two lectures dealing with supersymmetry (SUSY) for branes and strings. These lectures are mainly based on ref. [1] which the reader should consult for original references and additional discussions. 1 Introduction To make contact between superstring theory and the real world we have to understand the vacua of the theory. Of particular interest for vacuum construction are, on the one hand, D-branes. These are hyper-planes on which open strings can end. On the world-volume of coincident D-branes, non-abelian gauge fields can exist. -
Duality and the Nature of Quantum Space-Time
Duality and the nature of quantum space-time Sanjaye Ramgoolam Queen Mary, University of London Oxford, January 24, 2009 INTRODUCTION Strings provide a working theory of quantum gravity existing in harmony with particle physics Successes include computations of quantum graviton scattering, new models of beyond-standard-model particle physics based on branes, computation of black hole entropy A powerful new discovery : Duality I An unexpected equivalence between two systems, which a priori look completely unrelated, if not manifest opposites. I Large and Small (T-duality) I Gravitational and non-Gravitational. (Gauge-String duality) Unification and Duality I They are both powerful results of string theory. They allow us to calculate things we could not calculate before. I Unification : We asked for it and found it. We kind of know why it had to be there. I Duality : We didn’t ask for it. We use it. We don’t know what it really means. Unification v/s Duality I Unification has an illustrious history dating back to the days of Maxwell. I Before Maxwell, we thought magnets attracting iron on the one hand and lightning on the other had nothing to do with each other I After Maxwell : Magnets produce B-field. Electric discharge in lightning is caused by E-fields. The coupled equations of both allow fluctuating E, B-fields which transport energy travelling at the speed of light. In fact light is electromagnetic waves. Unification v/s Duality I Einstein tried to unify gravity with quantum phsyics. I String Theory goes a long way. I Computes graviton interaction probablities. -
Arxiv:0807.3472V1 [Cond-Mat.Stat-Mech] 22 Jul 2008 Conformal Field Theory and Statistical Mechanics∗
Conformal Field Theory and Statistical Mechanics∗ John Cardy July 2008 arXiv:0807.3472v1 [cond-mat.stat-mech] 22 Jul 2008 ∗Lectures given at the Summer School on Exact methods in low-dimensional statistical physics and quantum computing, les Houches, July 2008. 1 CFT and Statistical Mechanics 2 Contents 1 Introduction 3 2 Scale invariance and conformal invariance in critical behaviour 3 2.1 Scaleinvariance ................................. 3 2.2 Conformalmappingsingeneral . .. 6 3 The role of the stress tensor 7 3.0.1 An example - free (gaussian) scalar field . ... 8 3.1 ConformalWardidentity. 9 3.2 An application - entanglement entropy . ..... 12 4 Radial quantisation and the Virasoro algebra 14 4.1 Radialquantisation.............................. 14 4.2 TheVirasoroalgebra .............................. 15 4.3 NullstatesandtheKacformula . .. 16 4.4 Fusionrules ................................... 17 5 CFT on the cylinder and torus 18 5.1 CFTonthecylinder .............................. 18 5.2 Modularinvarianceonthetorus . ... 20 5.2.1 Modular invariance for the minimal models . .... 22 6 Height models, loop models and Coulomb gas methods 24 6.1 Heightmodelsandloopmodels . 24 6.2 Coulombgasmethods ............................. 27 6.3 Identification with minimal models . .... 28 7 Boundary conformal field theory 29 7.1 Conformal boundary conditions and Ward identities . ......... 29 7.2 CFT on the annulus and classification of boundary states . ......... 30 7.2.1 Example................................. 34 CFT and Statistical Mechanics 3 7.3 -
Statistical Field Theory University of Cambridge Part III Mathematical Tripos
Preprint typeset in JHEP style - HYPER VERSION Michaelmas Term, 2017 Statistical Field Theory University of Cambridge Part III Mathematical Tripos David Tong Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 OBA, UK http://www.damtp.cam.ac.uk/user/tong/sft.html [email protected] –1– Recommended Books and Resources There are a large number of books which cover the material in these lectures, although often from very di↵erent perspectives. They have titles like “Critical Phenomena”, “Phase Transitions”, “Renormalisation Group” or, less helpfully, “Advanced Statistical Mechanics”. Here are some that I particularly like Nigel Goldenfeld, Phase Transitions and the Renormalization Group • Agreatbook,coveringthebasicmaterialthatwe’llneedanddelvingdeeperinplaces. Mehran Kardar, Statistical Physics of Fields • The second of two volumes on statistical mechanics. It cuts a concise path through the subject, at the expense of being a little telegraphic in places. It is based on lecture notes which you can find on the web; a link is given on the course website. John Cardy, Scaling and Renormalisation in Statistical Physics • Abeautifullittlebookfromoneofthemastersofconformalfieldtheory.Itcoversthe material from a slightly di↵erent perspective than these lectures, with more focus on renormalisation in real space. Chaikin and Lubensky, Principles of Condensed Matter Physics • Shankar, Quantum Field Theory and Condensed Matter • Both of these are more all-round condensed matter books, but with substantial sections on critical phenomena and the renormalisation group. Chaikin and Lubensky is more traditional, and packed full of content. Shankar covers modern methods of QFT, with an easygoing style suitable for bedtime reading. Anumberofexcellentlecturenotesareavailableontheweb.Linkscanbefoundon the course webpage: http://www.damtp.cam.ac.uk/user/tong/sft.html. -
Euclidean Field Theory
February 2, 2011 Euclidean Field Theory Kasper Peeters & Marija Zamaklar Lecture notes for the M.Sc. in Elementary Particle Theory at Durham University. Copyright c 2009-2011 Kasper Peeters & Marija Zamaklar Department of Mathematical Sciences University of Durham South Road Durham DH1 3LE United Kingdom http://maths.dur.ac.uk/users/kasper.peeters/eft.html [email protected] [email protected] 1 Introducing Euclidean Field Theory5 1.1 Executive summary and recommended literature............5 1.2 Path integrals and partition sums.....................7 1.3 Finite temperature quantum field theory.................8 1.4 Phase transitions and critical phenomena................ 10 2 Discrete models 15 2.1 Ising models................................. 15 2.1.1 One-dimensional Ising model................... 15 2.1.2 Two-dimensional Ising model................... 18 2.1.3 Kramers-Wannier duality and dual lattices........... 19 2.2 Kosterlitz-Thouless model......................... 21 3 Effective descriptions 25 3.1 Characterising phase transitions..................... 25 3.2 Mean field theory.............................. 27 3.3 Landau-Ginzburg.............................. 28 4 Universality and renormalisation 31 4.1 Kadanoff block scaling........................... 31 4.2 Renormalisation group flow........................ 33 4.2.1 Scaling near the critical point................... 33 4.2.2 Critical surface and universality................. 34 4.3 Momentum space flow and quantum field theory........... 35 5 Lattice gauge theory 37 5.1 Lattice action................................. 37 5.2 Wilson loops................................. 38 5.3 Strong coupling expansion and confinement.............. 39 5.4 Monopole condensation.......................... 39 3 4 1 Introducing Euclidean Field Theory 1.1. Executive summary and recommended literature This course is all about the close relation between two subjects which at first sight have very little to do with each other: quantum field theory and the theory of criti- cal statistical phenomena.