Analysis on Superspace: an Overview
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The Pursuit of Quantum Gravity
The Pursuit of Quantum Gravity Cécile DeWitt-Morette The Pursuit of Quantum Gravity Memoirs of Bryce DeWitt from 1946 to 2004 123 Cécile DeWitt-Morette Department of Physics Center for Relativity University of Texas at Austin Austin Texas USA [email protected] ISBN 978-3-642-14269-7 e-ISBN 978-3-642-14270-3 DOI 10.1007/978-3-642-14270-3 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011921724 c Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is con- cerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publica- tion or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protec- tive laws and regulations and therefore free for general use. Cover design:WMXDesignGmbH,Heidelberg Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Dedicated to our daughters, Nicolette, Jan, Christiane, Abigail Preface This book is written for the curious reader. I hope it will also be a good read for the professional physicist. -
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. -
Jhep01(2020)007
Published for SISSA by Springer Received: March 27, 2019 Revised: November 15, 2019 Accepted: December 9, 2019 Published: January 2, 2020 Deformed graded Poisson structures, generalized geometry and supergravity JHEP01(2020)007 Eugenia Boffo and Peter Schupp Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany E-mail: [email protected], [email protected] Abstract: In recent years, a close connection between supergravity, string effective ac- tions and generalized geometry has been discovered that typically involves a doubling of geometric structures. We investigate this relation from the point of view of graded ge- ometry, introducing an approach based on deformations of graded Poisson structures and derive the corresponding gravity actions. We consider in particular natural deformations of the 2-graded symplectic manifold T ∗[2]T [1]M that are based on a metric g, a closed Neveu-Schwarz 3-form H (locally expressed in terms of a Kalb-Ramond 2-form B) and a scalar dilaton φ. The derived bracket formalism relates this structure to the generalized differential geometry of a Courant algebroid, which has the appropriate stringy symme- tries, and yields a connection with non-trivial curvature and torsion on the generalized “doubled” tangent bundle E =∼ TM ⊕ T ∗M. Projecting onto TM with the help of a natural non-isotropic splitting of E, we obtain a connection and curvature invariants that reproduce the NS-NS sector of supergravity in 10 dimensions. Further results include a fully generalized Dorfman bracket, a generalized Lie bracket and new formulas for torsion and curvature tensors associated to generalized tangent bundles. -
Some Results in Supergeometry
Some results in supergeometry Harmonic maps from super Riemann surfaces and Automorphism supergroups of supermanifolds Inaugural-Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakult¨at der Universit¨atzu K¨oln vorgelegt von Dominik Ostermayr aus Dachau 2017 Berichterstatter: PD Dr. Alexander Alldridge Prof. Dr. George Marinescu Prof. Dr. Tilmann Wurzbacher Tag der m¨undlichen Pr¨ufung: 20. Januar 2017 Kurzzusammenfassung Die vorliegende Arbeit besteht aus zwei unabh¨angigenund eigenst¨andigen Teilen. Gegenstand des ersten Teils sind harmonische Abbildungen von super-Riemannschen Fl¨achen nach komplex-projektiven R¨aumenund projektiven R¨aumenbez¨uglich des Super- schiefk¨orpers D: In beiden F¨allenwird die Theorie der Gauß-Transformierten entwickelt und der Begriff der Isotropie studiert, insbesondere mit Hinblick auf den Zusammenhang zu holomorphen Differentialen auf der super-Riemannschen Fl¨ache. Uberdies¨ geben wir eine Definition f¨urharmonische Abbildungen endlichen Typs f¨ureine spezielle Klasse von n n+1 Abbildungen nach CP j und erhalten so eine Klassifikation bestimmter harmonischer super-Tori. Ferner untersuchen wir die Gleichungen, die von den unterliegenden Objek- ten erf¨ulltwerden und geben ein Beispiel eines harmonischen super-Torus in DP 2 dessen unterliegende Abbildung nicht harmonisch ist. Im zweiten Teil studieren wir einen klassischen Satz, der besagt, dass die Gruppe der Automorphismen einer Mannigfaltigkeit, die eine G-Struktur endlichen Typs erhalten, eine Lie-Gruppe bildet, im Kontext von Supermannigfaltigkeiten. Wir verallgemeinern dieses Theorem auf die Kategorie der cs Mannigfaltigkeiten und illustrieren es anhand einiger, sowohl klassische Objekte verallgemeinernder als auch genuin supergeometrischer, Beispiele. Insbesondere ist es n¨otigeine neue Klasse von Supermannigfaltigkeiten einzuf¨uhren - gemischte Supermannigfaltigkeiten. iii Abstract This thesis consists of two independent and self-contained parts. -
Introduction to Graded Geometry, Batalin-Vilkovisky Formalism and Their Applications
ARCHIVUM MATHEMATICUM (BRNO) Tomus 47 (2011), 415–471 INTRODUCTION TO GRADED GEOMETRY, BATALIN-VILKOVISKY FORMALISM AND THEIR APPLICATIONS Jian Qiu and Maxim Zabzine Abstract. These notes are intended to provide a self-contained introduction to the basic ideas of finite dimensional Batalin-Vilkovisky (BV) formalism and its applications. A brief exposition of super- and graded geometries is also given. The BV–formalism is introduced through an odd Fourier transform and the algebraic aspects of integration theory are stressed. As a main application we consider the perturbation theory for certain finite dimensional integrals within BV-formalism. As an illustration we present a proof of the isomorphism between the graph complex and the Chevalley-Eilenberg complex of formal Hamiltonian vectors fields. We briefly discuss how these ideas can be extended to the infinite dimensional setting. These notes should be accessible toboth physicists and mathematicians. Table of Contents 1. Introduction and motivation 416 2. Supergeometry 417 2.1. Idea 418 2.2. Z2-graded linear algebra 418 2.3. Supermanifolds 420 2.4. Integration theory 421 3. Graded geometry 423 3.1. Z-graded linear algebra 423 3.2. Graded manifold 425 4. Odd Fourier transform and BV-formalism 426 4.1. Standard Fourier transform 426 4.2. Odd Fourier transform 427 4.3. Integration theory 430 4.4. Algebraic view on the integration 433 5. Perturbation theory 437 5.1. Integrals in Rn-Gaussian Integrals and Feynman Diagrams 437 2010 Mathematics Subject Classification: primary 58A50; secondary 16E45, 97K30. Key words and phrases: Batalin-Vilkovisky formalism, graded symplectic geometry, graph homology, perturbation theory. -
Super-Higgs in Superspace
Article Super-Higgs in Superspace Gianni Tallarita 1,* and Moritz McGarrie 2 1 Departamento de Ciencias, Facultad de Artes Liberales, Universidad Adolfo Ibáñez, Santiago 7941169, Chile 2 Deutsches Elektronen-Synchrotron, DESY, Notkestrasse 85, 22607 Hamburg, Germany; [email protected] * Correspondence: [email protected] or [email protected] Received: 1 April 2019; Accepted: 10 June 2019; Published: 14 June 2019 Abstract: We determine the effective gravitational couplings in superspace whose components reproduce the supergravity Higgs effect for the constrained Goldstino multiplet. It reproduces the known Gravitino sector while constraining the off-shell completion. We show that these couplings arise by computing them as quantum corrections. This may be useful for phenomenological studies and model-building. We give an example of its application to multiple Goldstini. Keywords: supersymmetry; Goldstino; superspace 1. Introduction The spontaneous breakdown of global supersymmetry generates a massless Goldstino [1,2], which is well described by the Akulov-Volkov (A-V) effective action [3]. When supersymmetry is made local, the Gravitino “eats” the Goldstino of the A-V action to become massive: The super-Higgs mechanism [4,5]. In terms of superfields, the constrained Goldstino multiplet FNL [6–12] is equivalent to the A-V formulation (see also [13–17]). It is, therefore, natural to extend the description of supergravity with this multiplet, in superspace, to one that can reproduce the super-Higgs mechanism. In this paper we address two issues—first we demonstrate how the Gravitino, Goldstino, and multiple Goldstini obtain a mass. Secondly, by using the Spurion analysis, we write down the most minimal set of new terms in superspace that incorporate both supergravity and the Goldstino multiplet in order to reproduce the super-Higgs mechanism of [5,18] at lowest order in M¯ Pl. -
Download Report 2010-12
RESEARCH REPORt 2010—2012 MAX-PLANCK-INSTITUT FÜR WISSENSCHAFTSGESCHICHTE Max Planck Institute for the History of Science Cover: Aurora borealis paintings by William Crowder, National Geographic (1947). The International Geophysical Year (1957–8) transformed research on the aurora, one of nature’s most elusive and intensely beautiful phenomena. Aurorae became the center of interest for the big science of powerful rockets, complex satellites and large group efforts to understand the magnetic and charged particle environment of the earth. The auroral visoplot displayed here provided guidance for recording observations in a standardized form, translating the sublime aesthetics of pictorial depictions of aurorae into the mechanical aesthetics of numbers and symbols. Most of the portait photographs were taken by Skúli Sigurdsson RESEARCH REPORT 2010—2012 MAX-PLANCK-INSTITUT FÜR WISSENSCHAFTSGESCHICHTE Max Planck Institute for the History of Science Introduction The Max Planck Institute for the History of Science (MPIWG) is made up of three Departments, each administered by a Director, and several Independent Research Groups, each led for five years by an outstanding junior scholar. Since its foundation in 1994 the MPIWG has investigated fundamental questions of the history of knowl- edge from the Neolithic to the present. The focus has been on the history of the natu- ral sciences, but recent projects have also integrated the history of technology and the history of the human sciences into a more panoramic view of the history of knowl- edge. Of central interest is the emergence of basic categories of scientific thinking and practice as well as their transformation over time: examples include experiment, ob- servation, normalcy, space, evidence, biodiversity or force. -
SUSY and XD Notes
Supersymmetry and Extra Dimensions Flip Tanedo LEPP, Cornell University Ithaca, New York A pedagogical set of notes based on lectures by Fernando Quevedo, Adrian Signer, and Csaba Cs´aki,as well as various books and reviews. Last updated: January 9, 2009 ii iii Abstract This is a set of combined lecture notes on supersymmetry and extra dimensions based on various lectures, textbooks, and review articles. The core of these notes come from Professor Fernando Quevedo's 2006- 2007 Lent Part III lecture course of the same name [1]. iv v Acknowledgements Inspiration to write up these notes in LATEX came from Steffen Gielen's excel- lent notes from the Part III Advanced Quantum Field Theory course and the 2008 ICTP Introductory School on the Gauge Gravity Correspondence. Notes from Profes- sor Quevedo's 2005-2006 Part III Supersymmetry and Extra Dimensions course exist in TEXform due to Oliver Schlotterer. The present set of notes were written up indepen- dently, but simliarities are unavoidable. It is my hope that these notes will provide a broader pedagogical introduction supersymmetry and extra dimensions. vi vii Preface These are lecture notes. Version 1 of these notes are based on Fernando Quevedo's lecture notes and structure. I've also incorporated some relevant topics from my research that I think are important to round-out the course. Version 2 of these notes will also incorporate Csaba Cs´aki'sAdvanced Particle Physics notes. Framed text. Throughout these notes framed text will include parenthetical dis- cussions that may be omitted on a first reading. They are meant to provide a broader picture or highlight particular applications that are not central to the main purpose of the chapter. -
Supersymmetric R4 Actions and Quantum Corrections to Superspace Torsion Constraints
DAMTP-2000-91, NORDITA-2000/87 HE, SPHT-T00/117, hep-th/0010182 SUPERSYMMETRIC R4 ACTIONS AND QUANTUM CORRECTIONS TO SUPERSPACE TORSION CONSTRAINTS K. Peetersa, P. Vanhoveb and A. Westerbergc a CERN, TH-division 1211 Geneva 23, Switzerland b SPHT, Orme des Merisiers, CEA / Saclay, 91191 Gif-sur-Yvette, France c NORDITA, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark k.peeters, [email protected], [email protected] We present the supersymmetrisation of the anomaly-related R4 term in eleven dimensions and show that it induces no non-trivial modifications to the on-shell supertranslation algebra and the superspace torsion constraints before inclusion of gauge-field terms.1 1 Higher-derivative corrections and supersymmetry The low-energy supergravity limits of superstring theory as well as D-brane effective actions receive infinite sets of correction terms, proportional to in- 2 creasing powers of α0 = ls and induced by superstring theory massless and massive modes. At present, eleven-dimensional supergravity lacks a corre- sponding microscopic underpinning that could similarly justify the presence of higher-derivative corrections to the classical Cremmer-Julia-Scherk action [1]. Nevertheless, some corrections of this kind are calculable from unitarity argu- ments and super-Ward identities in the massless sector of the theory [2] or by anomaly cancellation arguments [3, 4]. Supersymmetry puts severe constraints on higher-derivative corrections. For example, it forbids the appearance of certain corrections (like, e.g, R3 corrections to supergravity effective actions [5]), and groups terms into var- ious invariants [6–9]. The structure of the invariants that contain anomaly- cancelling terms is of great importance due to the quantum nature of the 1Based on talks given by K.P. -
Interpreting Supersymmetry
Interpreting Supersymmetry David John Baker Department of Philosophy, University of Michigan [email protected] October 7, 2018 Abstract Supersymmetry in quantum physics is a mathematically simple phenomenon that raises deep foundational questions. To motivate these questions, I present a toy model, the supersymmetric harmonic oscillator, and its superspace representation, which adds extra anticommuting dimensions to spacetime. I then explain and comment on three foundational questions about this superspace formalism: whether superspace is a sub- stance, whether it should count as spatiotemporal, and whether it is a necessary pos- tulate if one wants to use the theory to unify bosons and fermions. 1 Introduction Supersymmetry{the hypothesis that the laws of physics exhibit a symmetry that transforms bosons into fermions and vice versa{is a long-standing staple of many popular (but uncon- firmed) theories in particle physics. This includes several attempts to extend the standard model as well as many research programs in quantum gravity, such as the failed supergravity program and the still-ascendant string theory program. Its popularity aside, supersymmetry (SUSY for short) is also a foundationally interesting hypothesis on face. The fundamental equivalence it posits between bosons and fermions is prima facie puzzling, given the very different physical behavior of these two types of particle. And supersymmetry is most naturally represented in a formalism (called superspace) that modifies ordinary spacetime by adding Grassmann-valued anticommuting coordinates. It 1 isn't obvious how literally we should interpret these extra \spatial" dimensions.1 So super- symmetry presents us with at least two highly novel interpretive puzzles. Only two philosophers of science have taken up these questions thus far. -
Personal Recollections of the Early Days Before and After the Birth of Supergravity
Personal recollections of the early days before and after the birth of Supergravity Peter van Nieuwenhuizen I March 29, 1976: Dan Freedman, Sergio Ferrara and I submit first paper on Supergravity to Physical Review D. I April 28: Stanley Deser and Bruno Zumino: elegant reformulation that stresses the role of torsion. I June 2, 2016 at 1:30pm: 5500 papers with Supergravity in the title have appeared, and 15000 dealing with supergravity. I Supergravity has I led to the AdS/CFT miracle, and I made breakthroughs in longstanding problems in mathematics. I Final role of supergravity? (is it a solution in search of a problem?) I 336 papers in supergravity with 126 collaborators I Now: many directions I can only observe in awe from the sidelines I will therefore tell you my early recollections and some anecdotes. I will end with a new research program that I am enthusiastic about. In 1972 I was a Joliot Curie fellow at Orsay (now the Ecole Normale) in Paris. Another postdoc (Andrew Rothery from England) showed me a little book on GR by Lawden.1 I was hooked. That year Deser gave lectures at Orsay on GR, and I went with him to Brandeis. In 1973, 't Hooft and Veltman applied their new covariant quantization methods and dimensional regularization to pure gravity, using the 't Hooft lemma (Gilkey) for 1-loop divergences and the background field formalism of Bryce DeWitt. They found that at one loop, pure gravity was finite: 1 2 2 ∆L ≈ αRµν + βR = 0 on-shell. But what about matter couplings? 1\An Introduction to Tensor Calculus and Relativity". -
Creative Discomfort: the Culture of the Gelfand Seminar at Moscow University Slava Gerovitch
Creative Discomfort: The Culture of the Gelfand Seminar at Moscow University Slava Gerovitch 1 Memory Israel Gelfand’s weekly seminar at Moscow State University, which ran continu- ously from 1943 to 1989, has gained a legendary status in the Russian mathematics community. It has been praised as “maybe the greatest seminar in the history of the Mechanical-Mathematical Faculty of Moscow University,”1 “probably the best seminar in the history of mathematics,”2 and even “one of the most productive seminars in the history of science.”3 According to seminar participants, the seminar “ardently followed all that was new in mathematics anywhere in the world”4 and “made a decisive impact on mathematical life in Moscow.”5 Many outstanding mathematicians remember the seminar fondly as their crucial coming-of-age experience. Before we conjure up an idyllic image of a harmonic chorus of great mathe- maticians conversing magnificently on topics of utmost scholarly importance, let us read a bit more from the memoirs of the same seminar participants. The seminar has 1Tikhomirov (2008), p. 10. 2Interview with Aleksei Sosinskii, 20 October 2009 (http://polit.ru/article/2009/10/20/absossinsky_ about_imgelfand/). 3Tikhomirov (2008), p. 25. 4Landis (2007), p. 69. 5Arnold (2009), p. 40. S. Gerovitch (&) Department of Mathematics, Massachusetts Institute of Technology, Massachusetts, USA e-mail: [email protected] © Springer International Publishing Switzerland 2016 51 B. Larvor (ed.), Mathematical Cultures, Trends in the History of Science, DOI 10.1007/978-3-319-28582-5_4