DOCSLIB.ORG

Four Dimensional Sigma Model Coupled to the Metric Tensor Field

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Four Dimensional Sigma Model Coupled to the Metric Tensor Field Zogjomt \] COMITETUL ME STAT PENTRU E NE ROIA NUCLEARA INSTITUTUL. CENTRAL DC FIZICA •UCUftf ŞTI - MAOURILI ROMANIA CENTRAL INSTITUTE OF PHYSICS INSTITUTE FOR PHYSICS AND NUCLEAR ENGINEERING SuchaieAt, P.0.8. 5206, R0MANIA fT-ISI-tO Feb/tua-xy Pour dimensional sigma model coupled to the metric tensor field G.Ghlka and U.VlAlneAcu ABSTRACT: We dlACuAA the ţouK dimensional nonllneax Algma model with an Internal (Tin) In variance coupled to the. metxlc tin A 01 ileld AatlA tying llnAteln equa - tlonA. We dexlve a bound on the coupling constant between the Algma ţleld and the metxlc tenAox, u&lng the theony oţ haxmonlc map*. A special attention IA paid to BlnAtein Apace A and Aome new explicit Aolu - tlonA o&' the model axe conAtxucted. INTRODUCTION In recent years there has been much interest in the finite action solutions of the Euclidean sigma model in two dimensions fl] . This model bears many similarities to a four dimensional non-Abelian gauge theory like scale lnva - r lance, asymptotic freedom, ins canton and rearon solutions , action bounded by a multiple of a topological charge» etc... When trying to generalise this model to a four dl - mensionai one there are difficulties. As long as we use a traditional Lagrangian, the sigma field $ has non-zero dl- mension and the constraint 2($*j =; ^ must introduce a A. dimensional constant "X which destroys the conformai inva - rlance, on the other hand a dimension ess $ implies a higher derivative Lagrangian with many troubles it tha quan­ tum level. A very elegant way to avoid these difficulties was found by de Alfaro, Pubini and Purlan [2] scnstructing a model in which the sigma field is coupled to the metric ten­ sor g v in a generally invariant fashion. Their model, having an internal Cf(iC) in variance, is characterized by the following Lagrangian density where M-,i> «1,2,3,4 , a - 1,2,...,n is "n internal index and the field $ is defined on a four dimensional Riemannian - 2 - manifold N taking values in an Euclidean sphere In Bq. (1,1) K is the coupling constant of the gravitational 2 field with the usual dimensions of a (length) , G ii metric ullr tensor, G « det(G v), R is the scalar curvature of M and F is a coupling constant. For generality a cosmologicei term (^A) was added but we shall deal mainly with the case A -0.Introduc- ing as in paper [2] a new set of quantities g -K~ G ,.X -KA , f *- KF we get _2 in these notations, the metric field has dimensions, of a (length) ^ and f are pure numbers and ţ is dimensionless. In the usual sigma model F is a positive constant and Interpreting K as the usual gravitational constant it follows that f is nonnegatlve. But wishing to keep the model as general as possible we shall not Impose a restriction on the sign of f from the beginning. The authors of [2"] were able to find explicite ly an in- Stanton solution for f " 3 and a naron solution for f - 2 in the 2 cate \ «0. it t rns out that for these values of f the order ot magnitude of the coupling F is not in -the typical range of high energy physics, interpreting X as the usual gravitational constant. But in spite of this problem we consider that the nodal deserves a closer examination. Concerning the cosmological constant we mention that in the last tins many authors reevaluated Its status. One argues [3] that the quantisation of gravity with a cosnologi - - 3 - cai tex» Sa both neoeeeary and feasible and the empirical •aloe of the constant ( very sasll and poasible sero) do not necessarily imply that one should set X - 0 in the bare lim- atein Lagrangian. On the other hand, in the " apacetlme foaai" program [4,5] , the apacetime looks nearly flat when viewed on large length scales, but suffers large fluctuations of the metric and topology on acales on the order of the Planck length. The foam-like structure can be described by introdu­ cing a cosmologicei tem as a Lagrange multiplier for the 4-volume in the path integral approach£6]. In the present paper we shall' use the theory of the harmonic mapa [7} aa a valuable tool to handle the nonlinear sigma model in four dimensions. The role of the harmonic mapa in different fields of particle physics and general relativity was pointed out by Misner [8} . In the last time many papara use the theory of the harmonic mapa in order to investigate the nonlinear sigma model [9] , Grlbov ambiguity in non-Abeli- an gauge theory [loj , general relativity [ll] , etc. He observe that the field $ is in fact a harmonic map from a Riemannlan four dimensional manifold M into a (n-1) dimensional sphere (for an internal (/ (n) lnvarlance). On the other h**d, from the equations of motion we find that the metric on M is such that the Ricci tensor is related to the "pull-back" of the metric of the final sphere S .These observations will permit us to find some constraints on the parameter f if the space M is compact as we shall mainly assume in our paper. - 4 - Of course the nee of the compact spaces is a techni­ cal restriction which allows as to find solutions with finite action. This doss not swan necessarily that tarn actual space- tine is cospact and one can interpret it as a normalization prescription like introducing a box in ordinary quanta» meche- nics. On the other hand it is known that starting with a non- coapact manifold N It Is difficult to obtain finite action so­ lutions. Por example Garber et al [9] proved that the d-di - menslonal T -model ( <$: f\ —• S^ ) with d>2 has no re­ gular finite solution. We must mention also that starting with a compact manifold it is essential to search solutions satisfy- w ing the constraint (1.2). This can be put in connection with the fact that compact, orientable Riemannlan manifolds without boundary do not support non trivial harmonic functions £12J Hence this kind of manifolds is not interesting for an Eucli­ dean relat1vistic theory with energy-momentum tensor construc­ ted from a free scalar field without mass. In Section 2 we derive the general constraints imposed by the harmonic property of the map <§ and by the Einstein equations for the metric. We are able to derive a bound on the coupling f which might allow a hope to obtain a solution with acceptable value for F , at least for a vanishing cos­ mologicei term. Special attention is paid to Einstein mani - folds which form an important class of possible solutions if the metric tensor of II is proportional to the pull-back of the metric of the final sphere. Por this class of solutions we express the coupling f in a geometrical way» namely as the ratio of the scalar curvature of M and an eigenvalue "f - 5 - tha Laplace - Beltrami oparattor on thia manifold. In Section 3 wa construct eeveral aolutiona with finita action. They involve for * various topologia* lika S4, s2 x S2, 2 CP which appaar alao in tha study of tha gravitational lnatan- tona . Binatain aquations on conpact «anifold have solutions with multiple topologia* which can play an inportant rola in tha path integral in tha spaca of metric* [5, 6] . Finally wa discuss the meren type solution (infinita action) assuming that tha aanlfold N la conformally flat. In tha last section we aake son» abort convent* concer­ ning the aodel and it* possible generalisations. A collection of useful formulae about the harmonic maps is presented in the Appendix. I. GENERAL PROPERTIES In this «ection w* «hall derive soma general properties of tha solution* of the model described in the ItUKoduvUon assuming that the four dimensional manifold M is compact without boundary. Tha loler - Lagrange equations corresponding to the Lagranglan (1.3) are where the left hand side of Eq.(2.2) is the usual Laplace - ! - 6 Beltrami operator defined on K. As it was observed in paper [2] for A - 0 the field Bqs. (2.1) and (2.2) are invariant under the simple resettling. and consequently any solution of the metric field g is defined in this case op to a multiplicative constant. It is important to observe that any finite action solution of Bq. (2.2) is a harmonic map betMeen the manifold K with the metric g ^v. anad aman Euclideaniirr 11 mmn spherinneiei JS* isome trically immersed in R n. In fact Bq. (2.2) is just Bq. (A.tO) in which the field $ is the composition of a map <f tM^s""1 and of th« Riemannlan immersion ^ s Sn~ -•R,". In our case the energy density of the harmonic map ^ is (see Eqs. (A.5) (A.10)). (2.4) Taking into account this observation» we can use the general properties of the harmonic maps to obtain some features of th solutions of the model Sells and Sampson ţ.13^ , using the general method ini­ tiated by Bonhner [l2*] and the Ricci identities, were able to obtain *he following equation satisfied by the energy den­ sity of an harmonic map u> ; M + H 7 - 2 (2.5) ACW- W(Acf)) + QU^ With i> (r where R'^^d is the Rlemann - Christoffel curvature tensor on N (with the metric h_. ) and R ^, is the Ricci tensor on M (with the metric g ). Assuming that the Rleir.annian manifold M ti? compact without boundary from Stokes' theorem we have 4.
Load more

Recommended publications
  • Arxiv:2007.12543V4 [Hep-Th] 15 Mar 2021 Contents
    Jacobi sigma models F. Basconea;b Franco Pezzellaa Patrizia Vitalea;b aINFN - Sezione di Napoli, Complesso Universitario di Monte S. Angelo Edificio 6, via Cintia, 80126 Napoli, Italy bDipartimento di Fisica “E. Pancini”, Università di Napoli Federico II, Complesso Universitario di Monte S. Angelo Edificio 6, via Cintia, 80126 Napoli, Italy E-mail: [email protected], [email protected], [email protected] Abstract: We introduce a two-dimensional sigma model associated with a Jacobi mani- fold. The model is a generalisation of a Poisson sigma model providing a topological open string theory. In the Hamiltonian approach first class constraints are derived, which gen- erate gauge invariance of the model under diffeomorphisms. The reduced phase space is finite-dimensional. By introducing a metric tensor on the target, a non-topological sigma model is obtained, yielding a Polyakov action with metric and B-field, whose target space is a Jacobi manifold. Keywords: Sigma Models, Topological Strings arXiv:2007.12543v4 [hep-th] 15 Mar 2021 Contents 1 Introduction1 2 Poisson sigma models3 3 Jacobi sigma models5 3.1 Jacobi brackets and Jacobi manifold5 3.1.1 Homogeneous Poisson structure on M × R from Jacobi structure7 3.2 Poisson sigma model on M × R 7 3.3 Action principle on the Jacobi manifold8 3.3.1 Hamiltonian description, constraints and gauge transformations9 4 Metric extension and Polyakov action 15 5 Jacobi sigma model on SU(2) 16 6 Conclusions and Outlook 18 1 Introduction Jacobi sigma models are here introduced as a natural generalisation of Poisson sigma mod- els.
    [Show full text]
  • Gauged Sigma Models and Magnetic Skyrmions Abstract Contents
    SciPost Phys. 7, 030 (2019) Gauged sigma models and magnetic Skyrmions Bernd J. Schroers Maxwell Institute for Mathematical Sciences and Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, UK [email protected] Abstract We define a gauged non-linear sigma model for a 2-sphere valued field and a SU(2) connection on an arbitrary Riemann surface whose energy functional reduces to that for critically coupled magnetic skyrmions in the plane, with arbitrary Dzyaloshinskii-Moriya interaction, for a suitably chosen gauge field. We use the interplay of unitary and holo- morphic structures to derive a general solution of the first order Bogomol’nyi equation of the model for any given connection. We illustrate this formula with examples, and also point out applications to the study of impurities. Copyright B. J. Schroers. Received 23-05-2019 This work is licensed under the Creative Commons Accepted 28-08-2019 Check for Attribution 4.0 International License. Published 10-09-2019 updates Published by the SciPost Foundation. doi:10.21468/SciPostPhys.7.3.030 Contents 1 Introduction1 2 Gauged sigma models on a Riemann surface3 2.1 Conventions3 2.2 Energy and variational equations4 2.3 The Bogomol’nyi equation5 2.4 Boundary terms6 3 Solving the Bogomol’nyi equation7 3.1 Holomorphic versus unitary structures7 3.2 Holomorphic structure of the gauged sigma model9 3.3 A general solution 11 4 Applications to magnetic skyrmions and impurities 12 4.1 Critically coupled magnetic skyrmions with any DM term 12 4.2 Axisymmetric DM interactions 13 4.3 Rank one DM interaction 15 4.4 Impurities as non-abelian gauge fields 16 5 Conclusion 17 References 18 1 SciPost Phys.
    [Show full text]
  • Free Lunch from T-Duality
    Free Lunch from T-Duality Ulrich Theis Institute for Theoretical Physics, Friedrich-Schiller-University Jena, Max-Wien-Platz 1, D{07743 Jena, Germany [email protected] We consider a simple method of generating solutions to Einstein gravity coupled to a dilaton and a 2-form gauge potential in n dimensions, starting from an arbitrary (n m)- − dimensional Ricci-flat metric with m commuting Killing vectors. It essentially consists of a particular combination of coordinate transformations and T-duality and is related to the so-called null Melvin twists and TsT transformations. Examples obtained in this way include two charged black strings in five dimensions and a finite action configuration in three dimensions derived from empty flat space. The latter leads us to amend the effective action by a specific boundary term required for it to admit solutions with positive action. An extension of our method involving an S-duality transformation that is applicable to four-dimensional seed metrics produces further nontrivial solutions in five dimensions. 1 Introduction One of the most attractive features of string theory is that it gives rise to gravity: the low-energy effective action for the background fields on a string world-sheet is diffeomor- phism invariant and includes the Einstein-Hilbert term of general relativity. This effective action is obtained by integrating renormalization group beta functions whose vanishing is required for Weyl invariance of the world-sheet sigma model and can be interpreted as a set of equations of motion for the background fields. Solutions to these equations determine spacetimes in which the string propagates.
    [Show full text]
  • Einstein Manifolds As Yang-Mills Instantons
    arXiv:1101.5185 Einstein Manifolds As Yang-Mills Instantons a b John J. Oh ∗ and Hyun Seok Yang † a Division of Computational Sciences in Mathematics, National Institute for Mathematical Sciences, Daejeon 305-340, Korea b Institute for the Early Universe, Ewha Womans University, Seoul 120-750, Korea b Center for Quantum Spacetime, Sogang University, Seoul 121-741, Korea ABSTRACT It is well-known that Einstein gravity can be formulated as a gauge theory of Lorentz group where spin connections play a role of gauge fields and Riemann curvature tensors correspond to their field strengths. One can then pose an interesting question: What is the Einstein equation from the gauge theory point of view? Or equivalently, what is the gauge theory object corresponding to Einstein manifolds? We show that the Einstein equations in four dimensions are precisely self-duality equa- tions in Yang-Mills gauge theory and so Einstein manifolds correspond to Yang-Mills instantons in SO(4) = SU(2) SU(2) gauge theory. Specifically, we prove that any Einstein manifold L × R with or without a cosmological constant always arises as the sum of SU(2)L instantons and SU(2)R anti-instantons. This result explains why an Einstein manifold must be stable because two kinds of arXiv:1101.5185v4 [hep-th] 2 Jul 2013 instantons belong to different gauge groups, instantons in SU(2)L and anti-instantons in SU(2)R, and so they cannot decay into a vacuum. We further illuminate the stability of Einstein manifolds by showing that they carry nontrivial topological invariants. Keywords: Einstein manifold, Yang-Mills instanton, Self-duality May 30, 2018 ∗[email protected] †[email protected] 1 Introduction It seems that the essence of the method of physics is inseparably connected with the problem of interplay between local and global aspects of the world’s structure, as saliently exemplified in the index theorem of Dirac operators.
    [Show full text]
  • The Grassmannian Sigma Model in SU(2) Yang-Mills Theory
    The Grassmannian σ-model in SU(2) Yang-Mills Theory David Marsh Undergraduate Thesis Supervisor: Antti Niemi Department of Theoretical Physics Uppsala University Contents 1 Introduction 4 1.1 Introduction ............................ 4 2 Some Quantum Field Theory 7 2.1 Gauge Theories .......................... 7 2.1.1 Basic Yang-Mills Theory ................. 8 2.1.2 Faddeev, Popov and Ghosts ............... 10 2.1.3 Geometric Digression ................... 14 2.2 The nonlinear sigma models .................. 15 2.2.1 The Linear Sigma Model ................. 15 2.2.2 The Historical Example ................. 16 2.2.3 Geometric Treatment ................... 17 3 Basic properties of the Grassmannian 19 3.1 Standard Local Coordinates ................... 19 3.2 Plücker coordinates ........................ 21 3.3 The Grassmannian as a homogeneous space .......... 23 3.4 The complexied four-vector ................... 25 4 The Grassmannian σ Model in SU(2) Yang-Mills Theory 30 4.1 The SU(2) Yang-Mills Theory in spin-charge separated variables 30 4.1.1 Variables .......................... 30 4.1.2 The Physics of Spin-Charge Separation ......... 34 4.2 Appearance of the ostensible model ............... 36 4.3 The Grassmannian Sigma Model ................ 38 4.3.1 Gauge Formalism ..................... 38 4.3.2 Projector Formalism ................... 39 4.3.3 Induced Metric Formalism ................ 40 4.4 Embedding Equations ...................... 43 4.5 Hodge Star Decomposition .................... 48 2 4.6 Some Complex Dierential Geometry .............. 50 4.7 Quaternionic Decomposition ................... 54 4.7.1 Verication ........................ 56 5 The Interaction 58 5.1 The Interaction Terms ...................... 58 5.1.1 Limits ........................... 59 5.2 Holomorphic forms and Dolbeault Cohomology ........ 60 5.3 The Form of the interaction ..................
    [Show full text]
  • Eienstein Field Equations and Heisenberg's Principle Of
    International Journal of Scientific and Research Publications, Volume 2, Issue 9, September 2012 1 ISSN 2250-3153 EIENSTEIN FIELD EQUATIONS AND HEISENBERG’S PRINCIPLE OF UNCERTAINLY THE CONSUMMATION OF GTR AND UNCERTAINTY PRINCIPLE 1DR K N PRASANNA KUMAR, 2PROF B S KIRANAGI AND 3PROF C S BAGEWADI ABSTRACT: The Einstein field equations (EFE) or Einstein's equations are a set of 10 equations in Albert Einstein's general theory of relativity which describe the fundamental interaction (e&eb) of gravitation as a result of space time being curved by matter and energy. First published by Einstein in 1915 as a tensor equation, the EFE equate spacetime curvature (expressed by the Einstein tensor) with (=) the energy and momentum tensor within that spacetime (expressed by the stress–energy tensor).Both space time curvature tensor and energy and momentum tensor is classified in to various groups based on which objects they are attributed to. It is to be noted that the total amount of energy and mass in the Universe is zero. But as is said in different context, it is like the Bank Credits and Debits, with the individual debits and Credits being conserved, holistically, the conservation and preservation of Debits and Credits occur, and manifest in the form of General Ledger. Transformations of energy also take place individually in the same form and if all such transformations are classified and written as a Transfer Scroll, it should tally with the total, universalistic transformation. This is a very important factor to be borne in mind. Like accounts are classifiable based on rate of interest, balance standing or the age, we can classify the factors and parameters in the Universe, be it age, interaction ability, mass, energy content.
    [Show full text]
  • Week 161 1 Sigma Models
    Week 161 1 Sigma models Let us consider a target space for type II string theory, with some metric, some B field turned on, and the dilaton turned on. We want the mani- fold to have large radius of curvature, so that we can make a low energy approximation that results into a supergravity theory. How does the string couple to all of these spacetime fields? We need that the string action have the following properties 1. The worldsheet action must be supersymmetric. 2. We would expect the action to couple linearly to the spacetime fields, because we are expanding around weakly coupled flat space. 3. Consistency of worldsheet gravity requires that the theory be confor- maly invariant in order to satisfy the Virasoro constraints. Thus, in the large radius of curvature limit we expect that the theory is close to flat 10D space. Putting all of this ingredients together, we get what is called a non-linear σ-model. The action will look as follows, in order to get a classically conformally (scale) invariant theory Z 1p Z 1 Z p d2σ −ηηαβg @ Xµ@ Xν + B dXµ^dXν + −ηR Φ+ fermions 2 µν α β 2 µν η (1) The form of the action given above contains the coupling of the worldsheet metric η to the fields. R is the Ricci curvature tensor associated to η. As written, the action is classically scale invariant (we can rescale η by a constant factor and the action does not change). However, we need more than classical scale invariance. We need the worldsheet theory to be conformally invariant at the quantum level.
    [Show full text]
  • Hypersurfaces of Einstein Manifolds
    ANNALES SCIENTIFIQUES DE L’É.N.S. NORIHITO KOISO Hypersurfaces of Einstein manifolds Annales scientifiques de l’É.N.S. 4e série, tome 14, no 4 (1981), p. 433-443 <http://www.numdam.org/item?id=ASENS_1981_4_14_4_433_0> © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1981, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systé- matique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. scient. EC. Norm. Sup., 40 serie, t. 14, 1981, p. 433 a 443. HYPERSURFACES OF EINSTEIN MANIFOLDS BYNORIHITO KOISO(1) 0. Introduction and results Let (M, g) be an Einstein manifold of dimension n +1 (n ^ 2). We consider certain classes of hypersurfaces in (M, g). First, let (M, g} be a totally umbilical hypersurface in (M, g), i.e., we assume that the second fundamental form a satisfies ^=jg for some function y on M. If we know completely the curvature tensor of (M, g}, we can get much information on (M, g). For example, if(M, g) is a symmetric space, then (M, g) is also a locally symmetric space, and so the classification of such pairs [(M, g), (M, g)} reduces to Lie group theory (see Chen [4] (2), Chen and Nagano [5], Naitoh [10]).
    [Show full text]
  • Hep-Th/0505211V3 12 Dec 2005 Sebasti´An Franco Brane and Tilings Geometry Toric from Theories Gauge VERSION HYPER - Style
    Preprint typeset in JHEP style. - HYPER VERSION MIT-CTP-3646, CERN-PH-TH/2005-084, HUTP-05/A0027 Gauge Theories from Toric Geometry and Brane Tilings Sebasti´an Franco1, Amihay Hanany1, Dario Martelli2, James Sparks3, David Vegh1, and Brian Wecht1 1. Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. 2. Department of Physics, CERN Theory Division, 1211 Geneva 23, Switzerland. 3. Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02318, U.S.A. and Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, U.S.A. [email protected], [email protected], [email protected], [email protected], [email protected], [email protected] arXiv:hep-th/0505211v3 12 Dec 2005 Abstract: We provide a general set of rules for extracting the data defining a quiver gauge theory from a given toric Calabi–Yau singularity. Our method combines information from the geometry and topology of Sasaki–Einstein manifolds, AdS/CFT, dimers, and brane tilings. We explain how the field content, quantum numbers, and superpotential of a superconformal gauge theory on D3–branes probing a toric Calabi–Yau singularity can be deduced. The infinite family of toric singularities with known horizon Sasaki–Einstein manifolds La,b,c is used to illustrate these ideas. We construct the corresponding quiver gauge theories, which may be fully specified by giving a tiling of the plane by hexagons with certain gluing rules. As checks of this construction, we perform a-maximisation as well as Z-minimisation to compute the exact R-charges of an arbitrary such quiver. We also examine a number of examples in detail, including the infinite subfamily La,b,a, whose smallest member is the Suspended Pinch Point.
    [Show full text]
  • Momentum Sections in Hamiltonian Mechanics and Sigma Models
    Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 076, 16 pages Momentum Sections in Hamiltonian Mechanics and Sigma Models Noriaki IKEDA Department of Mathematical Sciences, Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan E-mail: [email protected] Received May 24, 2019, in final form September 29, 2019; Published online October 03, 2019 https://doi.org/10.3842/SIGMA.2019.076 Abstract. We show a constrained Hamiltonian system and a gauged sigma model have a structure of a momentum section and a Hamiltonian Lie algebroid theory recently intro- duced by Blohmann and Weinstein. We propose a generalization of a momentum section on a pre-multisymplectic manifold by considering gauged sigma models on higher-dimensional manifolds. Key words: symplectic geometry; Lie algebroid; Hamiltonian mechanics; nonlinear sigma model 2010 Mathematics Subject Classification: 53D20; 70H33; 70S05 1 Introduction Recently, relations of physical systems with a Lie algebroid structure and its generalizations have been found and analyzed in many contexts. For instance, a Lie algebroid [23] and a generalization such as a Courant algebroid appear in topological sigma models [18], T-duality [9], quantizations, etc. Blohmann and Weinstein [2] have proposed a generalization of a momentum map and a Hamil- tonian G-space on a Lie algebra (a Lie group) to Lie algebroid setting, based on analysis of the general relativity [1]. It is called a momentum section and a Hamiltonian Lie algebroid. This structure is also regarded as reinterpretation of compatibility conditions of geometric quantities such as a metric g and a closed differential form H with a Lie algebroid structure, which was analyzed by Kotov and Strobl [21].
    [Show full text]
  • Introduction to Effective Field Theory
    IV Introduction to effective field theory The purpose of an effective field theory is to represent in a simple way the dynamical content of a theory in the low energy limit. One uses only those light degrees of freedom that are active at low energy, and treats their interactions in a full field theoretic framework. The effective field theory is often technically non-renormalizable, yet loop diagrams are included and renormalization of the physical parameters is readily accomplished. Effective field theory is used in all aspects of the Standard Model and beyond, from QED to superstrings. Perhaps the best setting for learning about the topic is that of chiral symmetry. Besides being historically important in the development of effective field theory techniques, chiral symmetry is a rather subtle subject which can be used to illustrate all aspects of the method, viz. the low energy expansion, non-leading be- havior, loops, renormalization and symmetry breaking. In addition, the results can be tested directly by experiment since the chiral effective field theory provides a framework for understanding the very low energy limit of QCD. In this chapter we introduce effective field theory by a study of the linear sigma model, and discuss the generalization of these techniques to other settings. IV{1 Effective lagrangians and the sigma model The linear sigma model, introduced in Sects. I{4, I{6, provides a `user friendly' introduction to effective field theory because all the relevant ma- nipulations can be explicitly demonstrated. The Goldstone boson fields, the pions, are present at all stages of the calculation.
    [Show full text]
  • Lecture 23 and CY Geometry Notes
    Lecture 23 and CY Geometry Notes McGreevy November 28, 2007 Reading: GSW ch. 12, 14, 15, 16. 1 Geometry and topology summary I’ve stolen this summary from some notes by Shamit Kachru. 1.1 Cohomology and Homology A p-form is a completely antisymmetric p-index tensor, 1 A ≡ A dxm1 ∧ ... ∧ dxmp . p! m1...mp The wedge product of a p-form A and a q-form B is a p + q form with components (p + q)! (A ∧ B)m ...m = A B 1 p+q p!q! [m1...mp mp+1...mp+q] where [..] means sum over permutations with a −1 for odd permutations. The space of p-forms on a manifold X is sometimes denoted Ωp(X). The exterior derivative d acts on forms as d :Ωp(X) → Ωp+1 by (dAp)m1...mp+1 = (p + 1)∂[m1 Am2...mp+1]. You can check that d2 = 0. 1 Because of this property d2 = 0, it is possible to define cohomology – the image of one d :Ωp → Ωp+1 is in the kernel of the next d :Ωp+1 → Ωp+2 (i.e. the Ωps form a chain complex). A form ωp is closed if it is killed by d: dωp = 0. A form ωp is exact if it is d of something: ωp = dαp−1. The p-th de Rham cohomology is defined to be closed p−forms on X Hp(X) ≡ . exact p−forms on X bp = dimHp(X) is the pth betti number. bp are topological invariants of X, and the euler character χ can be written in terms of them d=dimX χ(X)= (−1)pbp(X).
    [Show full text]