- •Preface
- •Acknowledgements
- •Contents
- •2.1 Introduction and a Short History of Black Holes
- •2.2 The Kruskal Extension of Schwarzschild Space-Time
- •2.2.1 Analysis of the Rindler Space-Time
- •2.2.2 Applying the Same Procedure to the Schwarzschild Metric
- •2.2.3 A First Analysis of Kruskal Space-Time
- •2.3 Basic Concepts about Future, Past and Causality
- •2.3.1 The Light-Cone
- •2.3.2 Future and Past of Events and Regions
- •Achronal Sets
- •Time-Orientability
- •Domains of Dependence
- •Cauchy surfaces
- •2.4.1 Conformal Mapping of Minkowski Space into the Einstein Static Universe
- •2.4.2 Asymptotic Flatness
- •2.5 The Causal Boundary of Kruskal Space-Time
- •References
- •3.1 Introduction
- •3.2 The Kerr-Newman Metric
- •3.2.1 Riemann and Ricci Curvatures of the Kerr-Newman Metric
- •3.3 The Static Limit in Kerr-Newman Space-Time
- •Static Observers
- •3.4 The Horizon and the Ergosphere
- •The Horizon Area
- •3.5 Geodesics of the Kerr Metric
- •3.5.2 The Hamilton-Jacobi Equation and the Carter Constant
- •3.5.3 Reduction to First Order Equations
- •3.5.4 The Exact Solution of the Schwarzschild Orbit Equation as an Application
- •3.5.5 About Explicit Kerr Geodesics
- •3.6 The Kerr Black Hole and the Laws of Thermodynamics
- •3.6.1 The Penrose Mechanism
- •3.6.2 The Bekenstein Hawking Entropy and Hawking Radiation
- •References
- •4.1 Historical Introduction to Modern Cosmology
- •4.2 The Universe Is a Dynamical System
- •4.3 Expansion of the Universe
- •4.3.1 Why the Night is Dark and Olbers Paradox
- •4.3.2 Hubble, the Galaxies and the Great Debate
- •4.3.4 The Big Bang
- •4.4 The Cosmological Principle
- •4.5 The Cosmic Background Radiation
- •4.6 The New Scenario of the Inflationary Universe
- •4.7 The End of the Second Millennium and the Dawn of the Third Bring Great News in Cosmology
- •References
- •5.1 Introduction
- •5.2 Mathematical Interlude: Isometries and the Geometry of Coset Manifolds
- •5.2.1 Isometries and Killing Vector Fields
- •5.2.2 Coset Manifolds
- •5.2.3 The Geometry of Coset Manifolds
- •5.2.3.1 Infinitesimal Transformations and Killing Vectors
- •5.2.3.2 Vielbeins, Connections and Metrics on G/H
- •5.2.3.3 Lie Derivatives
- •5.2.3.4 Invariant Metrics on Coset Manifolds
- •5.2.3.5 For Spheres and Pseudo-Spheres
- •5.3 Homogeneity Without Isotropy: What Might Happen
- •5.3.1 Bianchi Spaces and Kasner Metrics
- •5.3.1.1 Bianchi Type I and Kasner Metrics
- •5.3.2.1 A Ricci Flat Bianchi II Metric
- •5.3.3 Einstein Equation and Matter for This Billiard
- •5.3.4 The Same Billiard with Some Matter Content
- •5.3.5 Three-Space Geometry of This Toy Model
- •5.4 The Standard Cosmological Model: Isotropic and Homogeneous Metrics
- •5.4.1 Viewing the Coset Manifolds as Group Manifolds
- •5.5 Friedman Equations for the Scale Factor and the Equation of State
- •5.5.1 Proof of the Cosmological Red-Shift
- •5.5.2 Solution of the Cosmological Differential Equations for Dust and Radiation Without a Cosmological Constant
- •5.5.3 Embedding Cosmologies into de Sitter Space
- •5.6 General Consequences of Friedman Equations
- •5.6.1 Particle Horizon
- •5.6.2 Event Horizon
- •5.6.3 Red-Shift Distances
- •5.7 Conceptual Problems of the Standard Cosmological Model
- •5.8 Cosmic Evolution with a Scalar Field: The Basis for Inflation
- •5.8.1 de Sitter Solution
- •5.8.2 Slow-Rolling Approximate Solutions
- •5.8.2.1 Number of e-Folds
- •5.9 Primordial Perturbations of the Cosmological Metric and of the Inflaton
- •5.9.1 The Conformal Frame
- •5.9.2 Deriving the Equations for the Perturbation
- •5.9.2.1 Meaning of the Propagation Equation
- •5.9.2.2 Evaluation of the Effective Mass Term in the Slow Roll Approximation
- •5.9.2.3 Derivation of the Propagation Equation
- •5.9.3 Quantization of the Scalar Degree of Freedom
- •5.9.4 Calculation of the Power Spectrum in the Two Regimes
- •5.9.4.1 Short Wave-Lengths
- •5.9.4.2 Long Wave-Lengths
- •5.9.4.3 Gluing the Long and Short Wave-Length Solutions Together
- •5.9.4.4 The Spectral Index
- •5.10 The Anisotropies of the Cosmic Microwave Background
- •5.10.1 The Sachs-Wolfe Effect
- •5.10.2 The Two-Point Temperature Correlation Function
- •5.10.3 Conclusive Remarks on CMB Anisotropies
- •References
- •6.1 Historical Outline and Introduction
- •6.1.1 Fermionic Strings and the Birth of Supersymmetry
- •6.1.2 Supersymmetry
- •6.1.3 Supergravity
- •6.2 Algebro-Geometric Structure of Supergravity
- •6.3 Free Differential Algebras
- •6.3.1 Chevalley Cohomology
- •Contraction and Lie Derivative
- •Definition of FDA
- •Classification of FDA and the Analogue of Levi Theorem: Minimal Versus Contractible Algebras
- •6.4 The Super FDA of M Theory and Its Cohomological Structure
- •6.4.1 The Minimal FDA of M-Theory and Cohomology
- •6.4.2 FDA Equivalence with Larger (Super) Lie Algebras
- •6.5 The Principle of Rheonomy
- •6.5.1 The Flow Chart for the Construction of a Supergravity Theory
- •6.6 Summary of Supergravities
- •Type IIA Super-Poicaré Algebra in the String Frame
- •The FDA Extension of the Type IIA Superalgebra in the String Frame
- •The Bianchi Identities
- •6.7.1 Rheonomic Parameterizations of the Type IIA Curvatures in the String Frame
- •Bosonic Curvatures
- •Fermionic Curvatures
- •6.7.2 Field Equations of Type IIA Supergravity in the String Frame
- •6.8 Type IIB Supergravity
- •SL(2, R) Lie Algebra
- •Coset Representative of SL(2, R)/O(2) in the Solvable Parameterization
- •The SU(1, 1)/U(1) Vielbein and Connection
- •6.8.2 The Free Differential Algebra, the Supergravity Fields and the Curvatures
- •The Curvatures of the Free Differential Algebra in the Complex Basis
- •The Curvatures of the Free Differential Algebra in the Real Basis
- •6.8.3 The Bosonic Field Equations and the Standard Form of the Bosonic Action
- •6.9 About Solutions
- •References
- •7.1 Introduction and Conceptual Outline
- •7.2 p-Branes as World Volume Gauge-Theories
- •7.4 The New First Order Formalism
- •7.4.1 An Alternative to the Polyakov Action for p-Branes
- •7.6 The D3-Brane: Summary
- •7.9 Domain Walls in Diverse Space-Time Dimensions
- •7.9.1 The Randall Sundrum Mechanism
- •7.9.2 The Conformal Gauge for Domain Walls
- •7.10 Conclusion on This Brane Bestiary
- •References
- •8.1 Introduction
- •8.2 Supergravity and Homogeneous Scalar Manifolds G/H
- •8.2.3 Scalar Manifolds of Maximal Supergravities in Diverse Dimensions
- •8.3 Duality Symmetries in Even Dimensions
- •8.3.1 The Kinetic Matrix N and Symplectic Embeddings
- •8.3.2 Symplectic Embeddings in General
- •8.5 Summary of Special Kähler Geometry
- •8.5.1 Hodge-Kähler Manifolds
- •8.5.2 Connection on the Line Bundle
- •8.5.3 Special Kähler Manifolds
- •8.6 Supergravities in Five Dimension and More Scalar Geometries
- •8.6.1 Very Special Geometry
- •8.6.3 Quaternionic Geometry
- •8.6.4 Quaternionic, Versus HyperKähler Manifolds
- •References
- •9.1 Introduction
- •9.2 Black Holes Once Again
- •9.2.2 The Oxidation Rules
- •Orbit of Solutions
- •The Schwarzschild Case
- •The Extremal Reissner Nordström Case
- •Curvature of the Extremal Spaces
- •9.2.4 Attractor Mechanism, the Entropy and Other Special Geometry Invariants
- •9.2.5 Critical Points of the Geodesic Potential and Attractors
- •At BPS Attractor Points
- •At Non-BPS Attractor Points of Type I
- •At Non-BPS Attractor Points of Type II
- •9.2.6.2 The Quartic Invariant
- •9.2.7.1 An Explicit Example of Exact Regular BPS Solution
- •The Metric
- •The Scalar Field
- •The Electromagnetic Fields
- •The Fixed Scalars at Horizon and the Entropy
- •The Metric
- •The Scalar Field
- •The Electromagnetic Fields
- •The Fixed Scalars at Horizon and the Entropy
- •9.2.9 Resuming the Discussion of Critical Points
- •Non-BPS Case
- •BPS Case
- •9.2.10 An Example of a Small Black Hole
- •The Metric
- •The Complex Scalar Field
- •The Electromagnetic Fields
- •The Charges
- •Structure of the Charges and Attractor Mechanism
- •9.2.11 Behavior of the Riemann Tensor in Regular Solutions
- •9.3.4 The SO(8) Spinor Bundle and the Holonomy Tensor
- •9.3.5 The Well Adapted Basis of Gamma Matrices
- •9.3.6 The so(8)-Connection and the Holonomy Tensor
- •9.3.7 The Holonomy Tensor and Superspace
- •9.3.8 Gauged Maurer Cartan 1-Forms of OSp(8|4)
- •9.3.9 Killing Spinors of the AdS4 Manifold
- •9.3.10 Supergauge Completion in Mini Superspace
- •9.3.11 The 3-Form
- •9.4.1 Maurer Cartan Forms of OSp(6|4)
- •9.4.2 Explicit Construction of the P3 Geometry
- •9.4.3 The Compactification Ansatz
- •9.4.4 Killing Spinors on P3
- •9.4.5 Gauge Completion in Mini Superspace
- •9.4.6 Gauge Completion of the B[2] Form
- •9.5 Conclusions
- •References
- •10.1 The Legacy of Volume 1
- •10.2 The Story Told in Volume 2
- •Appendix A: Spinors and Gamma Matrix Algebra
- •A.2 The Clifford Algebra
- •A.2.1 Even Dimensions
- •A.2.2 Odd Dimensions
- •A.3 The Charge Conjugation Matrix
- •A.4 Majorana, Weyl and Majorana-Weyl Spinors
- •Appendix B: Auxiliary Tools for p-Brane Actions
- •B.1 Notations and Conventions
- •Appendix C: Auxiliary Information About Some Superalgebras
- •C.1.1 The Superalgebra
- •C.2 The Relevant Supercosets and Their Relation
- •C.2.1 Finite Supergroup Elements
- •C.4 An so(6) Inversion Formula
- •Appendix D: MATHEMATICA Package NOVAMANIFOLDA
- •Coset Manifolds (Euclidian Signature)
- •Instructions for the Use
- •Description of the Main Commands of RUNCOSET
- •Structure Constants for CP2
- •Spheres
- •N010 Coset
- •RUNCOSET Package (Euclidian Signature)
- •Main
- •Spin Connection and Curvature Routines
- •Routine Curvapack
- •Routine Curvapackgen
- •Contorsion Routine for Mixed Vielbeins
- •Calculation of the Contorsion for General Manifolds
- •Calculation for Cartan Maurer Equations and Vielbein Differentials (Euclidian Signature)
- •Routine Thoft
- •AdS Space in Four Dimensions (Minkowski Signature)
- •Lie Algebra of SO(2, 3) and Killing Metric
- •Solvable Subalgebra Generating the Coset and Construction of the Vielbein
- •Killing Vectors
- •Trigonometric Coordinates
- •Test of Killing Vectors
- •MANIFOLDPROVA
- •The 4-Dimensional Coset CP2
- •Calculation of the (Pseudo-)Riemannian Geometry of a Kasner Metric in Vielbein Formalism
- •References
- •Index
7.10 Conclusion on This Brane Bestiary |
299 |
7.10 Conclusion on This Brane Bestiary
The snapshot survey of the p-brane bestiary we have presented in this chapter was meant to illustrate the wealth of classical solutions of supergravity and their profound interpretration in connection with gauge-theories and many other aspects of quantum field theory. The main message we would like to convey to the student is that supergravity is just a natural extension of General Relativity (the main topic of this book) which stands on its feet independently from string theory. Furthermore the rich park of solutions and mechanisms contributed by supergravity requires attentive consideration and certainly is part of the general theory of gravity.
References
1.Green, M.B., Schwarz, J.H.: Supersymmetrical dual string theory. Nucl. Phys. B 181, 502 (1981)
2.Green, M.B., Schwarz, J.H.: Supersymmetrical dual string theory (II). Phys. Lett. B 109, 444 (1982)
3.Duff, M.J., Khuri, R.R., Lu, J.X.: String solitons. Phys. Rep. 259, 213–326 (1995). hep-th/9412184
4.Tonin, M.: Consistency condition for kappa anomalies and superspace constraints in quantum heterotic superstrings. Int. J. Mod. Phys. A 4, 1983 (1989)
5.Grisaru, M.T., Howe, P., Mezincescu, L., Nilsson, B., Townsend, P.K.: N = 2 superstrings in a supergravity background. Phys. Lett. B 162, 116 (1985)
6.Townsend, P.K.: Spacetime supersymmetric particles and strings in background fields. In: D’Auria, R., Frè, P. (eds.) Superunification and Extra Dimensions, p. 376. World Scientific, Singapore (1986)
7.Castellani, L., D’Auria, R., Frè, P.: Supergravity and Superstring Theory: A Geometric Perspective. World Scientific, Singapore (1990)
8.Dall’Agata, G., Fabbri, D., Fraser, C., Frè, P., Termonia, P., Trigiante, M.: The Osp(8|4) singleton action from the supermembrane. Nucl. Phys. B 542, 157 (1999). hep-th/9807115
9.Billó, M., Cacciatori, S., Denef, F., Frè, P., Van Proeyen, A., Zanon, D.: The 0-brane action in a general D = 4 supergravity background. Class. Quantum Gravity 16, 2335–2358 (1999). hep-th/9902100
10.Castellani, L., Pesando, I.: The complete superspace action of chiral N = 2 D = 10 supergravity. Nucl. Phys. B 226, 269 (1983)
11.Pesando, I.: A kappa fixed type IIB superstring action on AdS5 × S5. J. High Energy Phys. 11, 002 (1998). hep-th/9808020
12.Polyakov, A.M.: Quantum geometry of bosonic strings. Phys. Lett. B 103, 207–211 (1981)
13.Nambu, Y.: Lectures at Copenhagen Symposium (1970)
14.Goto, T.: Relativistic quantum mechanics of one-dimensional mechanical continuum and subsidiary condition of dual resonance model. Prog. Theor. Phys. 46, 1560 (1971)
15.Bertolini, M., Ferretti, G., Frè, P., Trigiante, M., Campos, L., Salomonson, P.: Supersymmetric 3-branes on smooth ALE manifolds with flux. Nucl. Phys. B 617, 3–42 (2001). hep-th/0106186
16.Bertolini, M., Di Vecchia, P., Frau, M., Lerda, A., Marotta, R., Pesando, I.: Fractional D-branes and their gauge duals. J. High Energy Phys. 0102, 014 (2001). hep-th/0011077
17.Bertolini, M., Di Vecchia, P., Frau, M., Lerda, A., Marotta, R.: N = 2 gauge theories on systems of fractional D3/D7 branes. Nucl. Phys. B 621, 157 (2002). hep-th/0107057
18.Di Vecchia, P., Lerda, A., Merlatti, P.: N = 1 and N = 2 super Yang-Mills theories from wrapped branes. hep-th/0205204
300 |
7 The Branes: Three Viewpoints |
19.Billó, M., Gallot, L., Liccardo, A.: Classical geometry and gauge duals for fractional branes on ALE orbifolds. Nucl. Phys. B 614, 254 (2001). hep-th/0105258
20.Howe, P.S., Tucker, R.W.: A locally supersymmetric and reparameterization invariant action for a spinning membrane. J. Phys. A 10, L155–L158 (1977)
21.Howe, P.S., Sezgin, E.: Superbranes. Phys. Lett. B 390, 133–142 (1997). hep-th/9607227
22.Howe, P.S., Sezgin, E.: D = 11, p = 5. Phys. Lett. B 394, 62–66 (1997). hep-th/9611008
23.Howe, P.S., Raetzel, O., Sezgin, E.: On brane actions and superembeddings. J. High Energy Phys. 9808, 011 (1998). hep-th/9804051
24.Polchinski, J.: String Theory, vol. 1. Cambridge University Press, Cambridge (2005)
25.Polchinski, J.: String Theory, vol. 2. Cambridge University Press, Cambridge (2005)
26.Born, M., Infeld, L.: Foundation of the new field theory. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 144, 425–451 (1934)
27.Frè, P., Modesto, L.: A new first order formalism for k-supersymmetric Born-Infeld actions: The D3 brane example. Class. Quantum Gravity 19, 5591 (2002). arXiv:hep-th/0206144
28.Fradkin, E.S., Tseytlin, A.A.: Phys. Lett. B 163, 425 (1985)
29.Tseytlin, A.A.: Self-duality of Born-Infeld action and Dirichlet 3-brane of type IIB superstring. Nucl. Phys. B 469, 51 (1996). hep-th/9602064
30.Cederwall, M., von Gussich, A., Nilsson, B.E.W., Westwrberg, A.: The Dirichlet super- three-brane in ten-dimensional type II B supergravity. Nucl. Phys. B 490, 163–178 (1997). hep-th/9610148
31.Pasti, P., Sorokin, D., Tonin, M.: Covariant action for a D = 11 five-brane with the chiral field. Phys. Lett. B 398, 41–46 (1997). hep-th/9701037
32.Bandos, I., Pasti, P., Sorokin, D., Tonin, M.: Superbrane actions and geometrical approach. DFPD 97/TH/19, ICTP IC/97/44. hep-th/9705064
33.Bandos, I., Pasti, P., Sorokin, D., Tonin, M., Volkov, D.: Superstrings and supermembranes in the doubly supersymmetric geometrical approach. Nucl. Phys. B 446, 79–118 (1995). hep-th/9501113
34.Bandos, I., Sorokin, D., Volkov, D.: On the generalized action principle for superstrings and supermembranes. Phys. Lett. B 352, 269–275 (1995). hep-th/9502141
35.Bandos, I., Sorokin, D., Tonin, M.: Generalized action principle and superfield equations of motion for d = 10 (D − p)-branes. Nucl. Phys. B 497, 275–296 (1997). hep-th/9701127
36.Bandos, I., Lechner, K., Nurmagambetov, A., Pasti, P., Sorokin, D., Tonin, M.: Covariant action for the super-five-brane of M-theory. Phys. Rev. Lett. 78, 4332–4334 (1997). hep-th/9701149
37.Bergshoeff, E., Kallosh, R., Ortin, T., Papadopoulos, G.: Kappa-symmetry, supersymmetry and intersecting branes. Nucl. Phys. B 502, 149–169 (1997). hep-th/9705040
38.Frè, P.: Gaugings and other supergravity tools of p-brane physics. In: Lectures given at the RTN School Recent Advances in M-theory, Paris, 1–8 February 2001, IHP. hep-th/0102114
39.Trigiante, M.: Dualities in supergravity and solvable lie algebras. PhD thesis. hep-th/9801144
40.D’Auria, R., Frè, P.: BPS black holes in supergravity: duality groups, p-branes, central charges and the entropy. In: Frè, P. et al. (eds.) Classical and Quantum Black Holes. Lecture Notes for the 8th Graduate School in Contemporary Relativity and Gravitational Physics: The Physics of Black Holes, SIGRAV 98, Villa Olmo, Italy, 20–25 Apr. 1998, pp. 137–272, 1999. hep-th/9812160
41.The literature on this topic is quite extended. As a general review, see the lecture notes: Stelle, K.: Lectures on supergravity p-branes. Lectures presented at 1996 ICTP Summer School, Trieste. hep-th/9701088
42.For a recent comprehensive updating on M-brane solutions see also Townsend, P.K.: M-theory from its superalgebra. Talk given at the NATO Advanced Study Institute on Strings, Branes and Dualities, Cargese, France, 26 May–14 June, 1997. hep-th/9712004
43.Castellani, L., Ceresole, A., D’Auria, R., Ferrara, S., Frè, P., Trigiante, M.: G/H M-branes and AdSp+2 geometries. Nucl. Phys. B 527, 142 (1998). hep-th/9803039
44.Maldacena, J.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231 (1998). hep-th/9711200
References |
301 |
45.Claus, P., Kallosh, R., Van Proeyen, A.: Nucl. Phys. B 518, 117 (1998)
46.Claus, P., Kallosh, R., Kumar, J., Townsend, P., Van Proeyen, A.: hep-th/9801206
47.Fabbri, D., Frè, P., Gualtieri, L., Termonia, P.: M-theory on AdS4 × M111: The complete Osp(2|4) × SU(3) × SU(2) spectrum from harmonic analysis. hep-th/9903036
48.Fabbri, D., Frè, P., Gualtieri, L., Reina, C., Tomasiello, A., Zaffaroni, A., Zampa, A.: 3D superconformal theories from Sasakian seven-manifolds: new non-trivial evidence for AdS4/CFT3. Nucl. Phys. B 577, 547 (2000). hep-th/9907219
49.Klebanov, I., Witten, E.: Superconformal field theory on threebranes at a Calabi Yau singularity. Nucl. Phys. B 536, 199 (1998). hep-th/9807080
50.Billó, M., Fabbri, D., Frè, P., Merlatti, P., Zaffaroni, A.: Shadow multiplets in AdS4/CFT3 and the superHiggs mechanism. hep-th/0005220
51.Gubser, S.S.: Einstein manifolds and conformal field theories. Phys. Rev. D 59, 025006 (1999). hep-th/9807164
52.Gubser, S.S., Klebanov, I.: Baryons and domain walls in an N = 1 superconformal gauge theory. Phys. Rev. D 58, 125025 (1998). hep-th/9808075
53.Ceresole, A., Dall’Agata, G., D’Auria, R., Ferrara, S.: M-theory on the Stiefel manifold and 3d conformal field theories. J. High Energy Phys. 0003, 011 (2000). hep-th/9912107
54.Ceresole, A., Dall’Agata, G., D’Auria, R., Ferrara, S.: Spectrum of type IIB supergravity on AdS5 × T 11: Predictions on N = 1 SCFT’s. hep-th/9905226
55.Ceresole, A., Dall’Agata, G., D’Auria, R.: KK spectroscopy of type IIB supergravity on AdS5 × T 11. hep-th/9907216
56.Lü, H., Pope, C.N., Townsend, P.K.: Domain walls form anti de sitter space. Phys. Lett. B 391, 39 (1997). hep-th/9607164
57.Bergshoeff, E., van der Schaar, J.P.: On M-9-branes. Class. Quantum Gravity 16, 23 (1999). hep-th/9806069
58.Cvetic,ˇ M., Soleng, H.H.: Naked singularities in dilatonic domain wall space-time. Phys. Rev. D 51, 5768 (1995). hep-th/9411170
59.Cvetic,ˇ M., Soleng, H.H.: Supergravity domain walls. Phys. Rep. 282, 159 (1997). hep-th/9604090
60.Cvetic,ˇ M., Lü, H., Pope, C.N.: Domain walls with localised gravity and domain wall/QFT correspondence. hep-th/0007209
61.Randall, L., Sundrum, R.: An alternative to compactification. Phys. Rev. Lett. 83, 23 (1999). hep-th/9906064
62.Lykken, J., Randall, L.: The shape of gravity. hep-th/9908076
63.Randall, L., Sundrum, R.: A large mass hierarchy from a small extra dimension. Phys. Rev. Lett. 83, 3370 (1999). hep-th/9905221
Chapter 8
Supergravity: A Bestiary in Diverse Dimensions
Incipit liber de natura quorundam animalium, et lapidum et quid significetur per eam
from a Medieval Latin Bestiary
8.1 Introduction
In the previous chapter we discussed p-branes as classical solutions of supergravity theories in diverse dimensions, while in Chap. 5 w appreciated the relevance of scalar fields in cosmology. From this viewpoint the basic information one would like to master is the following:
•The scalar field dependence of the kinetic terms of p-forms NΛΣ (φ)F Λ F Σ since this latter eventually decides the values of the coefficients a in the exponential factors of the p-brane actions (7.7.1).
•The scalar field potential V (φ) which eventually decides the form of the cosmological term in the domain wall actions (7.9.5) and plays a fundamental role in the inflationary scenarios.
•The metric gij (φ) appearing in the kinetic term gij (φ)∂μφi ∂μφj of the scalar
fields since it is needed as much as the matrix NΛΣ (φ) to determine the values of a and eventually of .
It turns out that each of the above items involves a wealth of surprisingly sophisticated geometric structures that are skillfully utilized by supergravity, first to stand on its feet at the ungauged level and, secondly, to be gauged producing non-Abelian symmetries and the scalar potential. In the present chapter we survey all these structures and we try to illustrate their meaning in relation with the parent string theory. Obviously the cause that imposes on the theory all such structures is supersymmetry and the presence of the fermions. Yet since the fermions are ugly objects to deal with while their yield, namely the geometric structure of the theory is beautiful, we will only stick to the latter and mention the fermions as seldom as possible. In Chap. 6 we illustrated the general principles that underlie the construction of supergravity theories and we hope that our reader got enough information to understand its logic. In the rest of the present chapter we confine ourselves to a mostly descriptive presentation. Nowhere we pretend to give the proof that the various supergravities are
P.G. Frè, Gravity, a Geometrical Course, DOI 10.1007/978-94-007-5443-0_8, |
303 |
© Springer Science+Business Media Dordrecht 2013 |
|