DOCSLIB.ORG

C6 the Superspace Method for Supergravity

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
C6 the Superspace Method for Supergravity PROC. 19th INT. CONF. HIGH ENERGY PHYSICS TOKYO, 1978 C6 The Superspace Method for Supergravity B. ZUMINO CERN, Geneva Superspace1"3 was introduced by Salam and Minkowski space. Just as Einstein's theory is Strathdee as a technique for the study of obtained when one generalizes from flat representations of the supersymmetry algebra Minkowski space to urved space-time, one in terms of supermultiplets of fields. The expects supergravity (the supersymmetric theory points of superspace are parametrized by coord­ of gravitation) to arise from the differential inates zA = (xa, 6a) where xa are the commuting geometry of curved superspace. Indeed this space-time coordinates while 6a are anticom- superfield approach to supergravity preceded muting variables (whcih are Majorana spinors). that45 based on ordinary fields. Supersymmetry transformations can be realized The first work on curved superspace was as rigid motions in superspace that of Arnowitt and Nath,6 who developed the differential geometry of a super-Riemannian superspace. However, the flat superspace of rigid supersymmetry does not fit naturally as where £a are the infinitesimal anticommuting a special case of a super-Riemannian geometry. spinorial (Majorana) parameters. The com­ For this reason a different geometry of super- mutator of two transformations (1) of param­ space was introduced by Wess and the eters Ci and C 7-13 2 author and by Akulov, Volkov and Soro- ka.14 In this geometry the tangent space (2) group at each point of superspace is not taken to be the graded Lorentz group (orthosimplec- is a translation of parameter —li^yj^^ as tic (4, 4)), but just the ordinary Lorentz group. required by the supersymmetry algebra. A Furthermore the superspace is allowed to have superfield V(x, 6) is a function in superspace. a torsion. More recently this geometry has Since the four components 6 anticommute, in been adopted by other authors.15 In this lec­ particular the square of each one is zero, a a ture we shall follow our own work, as deve­ power series in 0 is at most a polynomial of loped in refs.7"*13. Ours is a second order fourth degree. Therefore a superfield is equi­ formulation, in that we impose on the super- valent to a finite collection of ordinary fields torsion certain conditions ("constraints"), (a supermultiplet), carrying various spinorial which are sufficient to express the superconnec- indices and satisfying corresponding statistics. tion in terms of the supervielbein.* These When the superfield transforms like a scalar conditions also restrict to a certain extent the a a a (da=d/dx ) dV{^{didd )~xr oda)v (3) supervielbein itself. The dynamics is given in the coefficients of its expanision in 6a trans­ terms of a superspace action which must be form into each other. The "covariant deri­ varied keeping the constraints on the torsion vatives" satisfied. Our formulation is exactly equival­ ent to the recently developed formalism for supergravity with a minimal set of auxiliary commute with the infinitesimal operator in (3) and can be used to impose supersymmetric * It would be preferable to have a first order action constraints on superfields. They satisfy in superspace, from which both constraints and equa­ tions of motion follow. A possible choice for a first order action was suggested in ref. 8 and is also dis­ showing that superspace has torsion, even cussed in the lecture by J. H. Schwarz in these Proceed­ 19 though its curvature vanishes. This "flat" ings. However, dimensional arguments indicate that 6 integration of that superspace action does not superspace has been extensively used in the reproduce the correct supergravity action in coordinate study of supersymmetric field theories in space. 556 B. ZUMINO fields.16-18 Indeed this auxiliary field formali­ denote vectorial (bosonic), Greek letters spi­ sm can be derived from our geometrical norial (fermionic) indices. The supervielbein 13 A formulation. matrix EM (z), where A=(a, a), and its inverse M A non-geometric description of supergravity EA (z), can be used to transform world in superspace is favored by Ogievestsky and tensors into tangent space tensors and vice Sokatchev20 and by Warren Siegel.21 The versa (early alphabet letters denote tangent supergravity fields are contained in a superfield space indices). Covariant derivatives involve m B U (x, 6) endowed with a vector index m. The the superconnection 0M A , as in theory, based directly on this superfield, has a certain appeal because it is free of constraints. However, the lack of a complete geometrical Covariant derivatives with tangent space in­ M structure in such a theory makes the con­ dices, &'A=EA £/-x9 satisfy the graded com­ struction of invariants a difficult task. On mutation relations the other hand, from our point of view, the superfield Um emerges in solving the constraints C on the torsion. The supervielbein and the where TAB 9 are the supertorsion coefficients superconnection are expressed in terms of it and RAB is the Lie algebra valued super- and we can use the powerful techniques of curvature operator. The bracket in eq. (7) is differential geometry for the construction of an anticommutator if both indices A and B are invariants. spinorial, a commutator otherwise. Torsion The geometric superspace approach gives a and curvature satisfy the Bianchi identities complete and satisfactory method for the study of simple supergravity and its couplings to supersymmetric matter11,12 (both for the and vector-spinor and for the spinor-scalar-pseu- doscalar multiplets). All possible invariants of given dimension are easily constructed and 16 their knowledge can be used to discuss the The cyclic sums are taken with appropriate divergences which arise in perturbation theory. signs: a permutation of two vectorial or one The situation is quige different in the case of vectorial and one spinorial index gives rise to a extended supergravity. Clearly one must use change in sign, while for two spinorial indices here a larger superspace in which the fermionic there is no change in sign. coordinates carry, in addition to the spinorial index, an internal symmetry index, and the Specification of the geometry (kinematics) tangent space group should be the product of We now restrict the tangent space group the Lorentz group with the internal symmetry from the general graded linear group to the 0(AO. The difficulty consists in finding the (x and 0 dependent) Lorentz group. Since the correct separation of the equations into curvature operator belongs to the algebra of kinematical constraints (which are part of the the tangent space group, this implies restric­ specification of the geometry) and equations tions on the curvature coefficients (2a& = of motion (which should follow from the variation of an action, subject to the con­ straints). The results obtained so far22 for Ar=3 extended supergravity in superspace do not make this separation. These difficulties in the superspace method correspond to our (similarly for the superconnection). We also present ignorance as to the correct auxiliary impose the constraints on the supertorsion fields for extended supergravity. Affine superspace r 7 The points of superspace are parametrized but leave Tab, and Tab, undetermined. Using by coordinates zM = (xm, 6ft). Latin letters the Bianchi identities (8) and (9) one can show Super symmetry (Including Super gravity) 557 that (10) and (11) imply that all components chiral infinitesimal parameter function 2 of the supercurvature and supertorsion are expressible in terms of the three superfields and are (Howe and Tucker23) Ga$, WapT, R and their conjugates. Here we have used two-component spinor notation, G„p is hermitean, Wafi7 totally symmetric. For instance For the determinant of the supervielbein, (17) imolv The remaining content of the Bianchi identities while curvature and torsion behave as given by is expressed by the differential relations The superfield Ga& has a simple physical mean­ ing, which can be exhibited16 by considering Since the constraints are invariant under (17) the coefficients of its expansion in 6. For and (18), we can say that the geometry (kine­ instance, at the 66 level one finds a tensor which matics) is invariant under conformal trans­ contains the Einstein tensor, at the 666 level a formations. The dynamics may or may not spinor which is the Rarita-Schwinger ope­ be, depending on the choice of Lagrangian. rator. Similarly, R contains the contracted For instance, the action of supergravity (21), given below, is only invariant under general (scalar) curvature tensor, at the 66 level. Wan contains the Weyl conformal spinor, at the 6 coordinate transformations in superspace and level; for 6=0, it contains the Rarita-Schwinger under local Lorentz transformations, but not field strength. Observe that some components under conformal transformation. The action of conformal supergravity (28), instead, is of R are obtained from those of Gad by tracing over certain indices: in superspace this is ex­ also conformally invariant. pressed by the differential identities (13) in 6. 11 Dynamics Finally we observe that the constraints (11) The supergravity equations of motion are on the torsion imply derived from the action for any vector vA, where Varying (21) subject to the constraints (11) gives11 the equations is the graded determinant of the supervielbein as defined in the first of ref. 6. This formula Because of the identities (\3) thev eive permits integration by parts with covariant derivatives carrying tangent space indices. and this, in turn, implies Conformal transformations The correct conformal transformations in superspace must preserve the constraints (11). On the supergravity mass shell all components These transformation involve a covariantly of the supertorsion and of the supercurvature 558 B. ZUMINO are expressed in terms of the symmetric, chiral spinor superfield Wa^r which now also satisfies It satisfies identically (23), and of its derivatives (and, of course, of the complex conjugate quantities).
Load more

Recommended publications
  • Particles-Versus-Strings.Pdf
    Particles vs. strings http://insti.physics.sunysb.edu/~siegel/vs.html In light of the huge amount of propaganda and confusion regarding string theory, it might be useful to consider the relative merits of the descriptions of the fundamental constituents of matter as particles or strings. (More-skeptical reviews can be found in my physics parodies.A more technical analysis can be found at "Warren Siegel's research".) Predictability The main problem in high energy theoretical physics today is predictions, especially for quantum gravity and confinement. An important part of predictability is calculability. There are various levels of calculations possible: 1. Existence: proofs of theorems, answers to yes/no questions 2. Qualitative: "hand-waving" results, answers to multiple choice questions 3. Order of magnitude: dimensional analysis arguments, 10? (but beware hidden numbers, like powers of 4π) 4. Constants: generally low-energy results, like ground-state energies 5. Functions: complete results, like scattering probabilities in terms of energy and angle Any but the last level eventually leads to rejection of the theory, although previous levels are acceptable at early stages, as long as progress is encouraging. It is easy to write down the most general theory consistent with special (and for gravity, general) relativity, quantum mechanics, and field theory, but it is too general: The spectrum of particles must be specified, and more coupling constants and varieties of interaction become available as energy increases. The solutions to this problem go by various names -- "unification", "renormalizability", "finiteness", "universality", etc. -- but they are all just different ways to realize the same goal of predictability.
    [Show full text]
  • 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.
    [Show full text]
  • Introductory Lectures on Quantum Field Theory
    Introductory Lectures on Quantum Field Theory a b L. Álvarez-Gaumé ∗ and M.A. Vázquez-Mozo † a CERN, Geneva, Switzerland b Universidad de Salamanca, Salamanca, Spain Abstract In these lectures we present a few topics in quantum field theory in detail. Some of them are conceptual and some more practical. They have been se- lected because they appear frequently in current applications to particle physics and string theory. 1 Introduction These notes are based on lectures delivered by L.A.-G. at the 3rd CERN–Latin-American School of High- Energy Physics, Malargüe, Argentina, 27 February–12 March 2005, at the 5th CERN–Latin-American School of High-Energy Physics, Medellín, Colombia, 15–28 March 2009, and at the 6th CERN–Latin- American School of High-Energy Physics, Natal, Brazil, 23 March–5 April 2011. The audience on all three occasions was composed to a large extent of students in experimental high-energy physics with an important minority of theorists. In nearly ten hours it is quite difficult to give a reasonable introduction to a subject as vast as quantum field theory. For this reason the lectures were intended to provide a review of those parts of the subject to be used later by other lecturers. Although a cursory acquaintance with the subject of quantum field theory is helpful, the only requirement to follow the lectures is a working knowledge of quantum mechanics and special relativity. The guiding principle in choosing the topics presented (apart from serving as introductions to later courses) was to present some basic aspects of the theory that present conceptual subtleties.
    [Show full text]
  • 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.
    [Show full text]
  • Andrew Thomson Dissertation.Pdf
    PHENOMENOLOGY OF COMPACTIFICATIONS OF M-THEORY ON MANIFOLDS OF G2 HOLONOMY by Andrew James Thomson Supervisor: Kelly Stelle A dissertation submitted in partial fulfilment of the requirements for the degree of Master of Science of Imperial College London 24 September 2010 Contents I. Introduction 3 II. Holonomy and G2 manifolds 7 III. Chirality and singular manifolds 11 IV. Moduli stabilization and the hierarchy problem 14 V. Conclusion 17 References 19 2 I. Introduction The Standard Model is a phenomenally successful theory of particle physics, well tested at quantum mechanical level up to the TeV energy scale, and renormalizable (notwithstanding that the Higgs particle has not been observed yet) [1]. However there are several issues (or at least perceived ones) with the model. First of all, it only describes three of the four fundamental forces (the strong, electromagnetic and weak forces) in a quantum manner, leaving gravity still described only in a classical fashion; however we will not be concerned with this any further during this dissertation. The primary issue with which we will be concerned is the so-called hierarchy problem (also known as the weak-scale instability) – why there are two (electroweak and Planck) mass scales and why they are so far apart, why the electroweak scale is not modified at loop level by the Planck scale and whether there are any other scales between them. [1,2] One wonders why we should be concerned with as yet purely theoretical imperfections which have not yet manifested themselves in any experimental observations. One only has to look to the history of the standard model itself for a precedent for looking for physics beyond it – the earliest theories of the weak interaction, which were based on the interaction of four fermions at a point, broke down when calculated to higher order at the then unimaginably high energies of 300 GeV (Heisenberg 1939) before they had been violated by any observation.
    [Show full text]
  • TASI 2008 Lectures: Introduction to Supersymmetry And
    TASI 2008 Lectures: Introduction to Supersymmetry and Supersymmetry Breaking Yuri Shirman Department of Physics and Astronomy University of California, Irvine, CA 92697. [email protected] Abstract These lectures, presented at TASI 08 school, provide an introduction to supersymmetry and supersymmetry breaking. We present basic formalism of supersymmetry, super- symmetric non-renormalization theorems, and summarize non-perturbative dynamics of supersymmetric QCD. We then turn to discussion of tree level, non-perturbative, and metastable supersymmetry breaking. We introduce Minimal Supersymmetric Standard Model and discuss soft parameters in the Lagrangian. Finally we discuss several mech- anisms for communicating the supersymmetry breaking between the hidden and visible sectors. arXiv:0907.0039v1 [hep-ph] 1 Jul 2009 Contents 1 Introduction 2 1.1 Motivation..................................... 2 1.2 Weylfermions................................... 4 1.3 Afirstlookatsupersymmetry . .. 5 2 Constructing supersymmetric Lagrangians 6 2.1 Wess-ZuminoModel ............................... 6 2.2 Superfieldformalism .............................. 8 2.3 VectorSuperfield ................................. 12 2.4 Supersymmetric U(1)gaugetheory ....................... 13 2.5 Non-abeliangaugetheory . .. 15 3 Non-renormalization theorems 16 3.1 R-symmetry.................................... 17 3.2 Superpotentialterms . .. .. .. 17 3.3 Gaugecouplingrenormalization . ..... 19 3.4 D-termrenormalization. ... 20 4 Non-perturbative dynamics in SUSY QCD 20 4.1 Affleck-Dine-Seiberg
    [Show full text]
  • Lessons from Perturbative Unitarity in Graviton Scattering Amplitudes
    Lessons from perturbative unitarity in graviton scattering amplitudes Yu-tin Huang National Taiwan University Nima Arkani-Hamed, Tzu-Chen Huang, Ellis Ye Yuan, Warren Siegel Strings and Fields 2016 YITP The cartoon story of string theory, The presence of world-sheet ! high energy softness, infinite excitations (solution to UV completion) There is much more to the story than the world-sheet Two apparently unrelated developments: • Constraints from/of perturbative completion: • Positivity: For any effective field theory, L = φrφ + ai Oi (φ) The existence of unitary, Lorentz invariant UV completion ! ai > 0 for certain operators Adams, Arkani-Hamed, Dubovsky, Nicolis, Rattazzi • Causality: For gravitation interactions, any perturbative correction to R produce time advancement interactions. It’s cure require infinite number of higher spin states Camanho, Edelstein, Maldacena, Zhiboedov • World-sheet based field theory amplitudes • Witten’s twistor string theory: Topological B-model in CP3j4. • “Scattering equations” Cachazo, He, Yuan X ki · ka Scattering Equations : Ea = = 0 σ − σ i6=a i a Massless kinematics parameterizes the moduli space of n-punctured Riemann spheres The “physical” origin of string theory is likely to have nothing to do with the world-sheet In this talk I will pursuit this line of thought by answering • What constraints does perturbative unitarity imposes on the S-matrix? For massive scalar, see Caron-Huot, Komargodski, Sever, Zhiboedov • Are these “world-sheet” field theory a limit of string theory ? We already have a well
    [Show full text]
  • Aspects of Supersymmetry and Its Breaking
    1 Aspects of Supersymmetry and its Breaking a b Thomas T. Dumitrescu ∗ and Zohar Komargodski † aDepartment of Physics, Princeton University, Princeton, New Jersey, 08544, USA bSchool of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey, 08540, USA We describe some basic aspects of supersymmetric field theories, emphasizing the structure of various supersym- metry multiplets. In particular, we discuss supercurrents – multiplets which contain the supersymmetry current and the energy-momentum tensor – and explain how they can be used to constrain the dynamics of supersym- metric field theories, supersymmetry breaking, and supergravity. These notes are based on lectures delivered at the Carg´ese Summer School 2010 on “String Theory: Formal Developments and Applications,” and the CERN Winter School 2011 on “Supergravity, Strings, and Gauge Theory.” 1. Supersymmetric Theories In this review we will describe some basic aspects of SUSY field theories, emphasizing the structure 1.1. Supermultiplets and Superfields of various supermultiplets – especially those con- A four-dimensional theory possesses = 1 taining the supersymmetry current. In particu- supersymmetry (SUSY) if it contains Na con- 1 lar, we will show how these supercurrents can be served spin- 2 charge Qα which satisfies the anti- used to study the dynamics of supersymmetric commutation relation field theories, SUSY-breaking, and supergravity. ¯ µ Qα, Qα˙ = 2σαα˙ Pµ . (1) We begin by recalling basic facts about super- { } multiplets and superfields. A set of bosonic and Here Q¯ is the Hermitian conjugate of Q . (We α˙ α fermionic operators B(x) and F (x) fur- will use bars throughout to denote Hermitian con- i i nishes a supermultiplet{O if these} operators{O satisfy} jugation.) Unless otherwise stated, we follow the commutation relations of the schematic form conventions of [1].
    [Show full text]
  • 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.
    [Show full text]
  • Introduction to Supersymmetry
    Introduction to Supersymmetry Pre-SUSY Summer School Corpus Christi, Texas May 15-18, 2019 Stephen P. Martin Northern Illinois University [email protected] 1 Topics: Why: Motivation for supersymmetry (SUSY) • What: SUSY Lagrangians, SUSY breaking and the Minimal • Supersymmetric Standard Model, superpartner decays Who: Sorry, not covered. • For some more details and a slightly better attempt at proper referencing: A supersymmetry primer, hep-ph/9709356, version 7, January 2016 • TASI 2011 lectures notes: two-component fermion notation and • supersymmetry, arXiv:1205.4076. If you find corrections, please do let me know! 2 Lecture 1: Motivation and Introduction to Supersymmetry Motivation: The Hierarchy Problem • Supermultiplets • Particle content of the Minimal Supersymmetric Standard Model • (MSSM) Need for “soft” breaking of supersymmetry • The Wess-Zumino Model • 3 People have cited many reasons why extensions of the Standard Model might involve supersymmetry (SUSY). Some of them are: A possible cold dark matter particle • A light Higgs boson, M = 125 GeV • h Unification of gauge couplings • Mathematical elegance, beauty • ⋆ “What does that even mean? No such thing!” – Some modern pundits ⋆ “We beg to differ.” – Einstein, Dirac, . However, for me, the single compelling reason is: The Hierarchy Problem • 4 An analogy: Coulomb self-energy correction to the electron’s mass A point-like electron would have an infinite classical electrostatic energy. Instead, suppose the electron is a solid sphere of uniform charge density and radius R. An undergraduate problem gives: 3e2 ∆ECoulomb = 20πǫ0R 2 Interpreting this as a correction ∆me = ∆ECoulomb/c to the electron mass: 15 0.86 10− meters m = m + (1 MeV/c2) × .
    [Show full text]
  • Generalised Geometry and Supergravity Backgrounds
    Imperial College, London Department of Physics MSc dissertation Generalised geometry and supergravity backgrounds Author: Supervisor: Daniel Reilly Prof. Daniel Waldram September 30, 2011 Submitted in partial fulfilment of the requirements for the degree of Master of Science of Imperial College London Abstract We introduce the topcic of generalised geometry as an extension of differential geomentry on the bundle TM ⊕ T ∗M. An algebraic structure is built that leads to the formation of the generalised tangent bundle. We place a generalised Riemannian metric on this space and show that it is compatible with the metric for a generalised K¨ahlerstructure. We investigate the nature of supersymmetric backgrounds in type II supergravity by discussing the usual and generalised Calabi-Yau structure. From this we show how the formalism of generalised geometry may be used to describe a manifold with such a structure. It's unique appeal to physicists lies in its unification of the fields in NS-NS sector of supergravity. 1 Contents 1 Introduction 4 1.1 Early work . 4 1.2 Generalised geometry . 6 2 The Courant bracket 8 2.1 Lie algebroids . 8 2.2 Courant bracket . 10 2.3 Twisted Courant barcket . 12 3 Generalised tangent bundle 14 3.1 Linear structure . 14 3.2 Courant algebroid . 17 4 Metric on TM ⊕ T ∗M 19 4.1 Dirac structure . 19 4.2 Generalised metric . 20 4.3 Complex structure . 21 4.4 Generalised complex structure . 23 4.5 Generalised K¨ahlerstructure . 24 5 Spinors on TM ⊕ T ∗M 28 5.1 Clifford algebra . 28 5.2 Generalised Calbi-Yau structure .
    [Show full text]
  • 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.
    [Show full text]