- •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
9.2 Black Holes Once Again |
357 |
will be our systematic choice, we can invert the above mentioned relations, expressing the derivatives of the ZM fields in terms of the charge vector QM and the inverse of the matrix M4. Upon substitution in the σ -model Lagrangian (9.2.3), we obtain the effective Lagrangian for the D = 4 scalar fields zi and the warping factor U given by (9.2.35)–(9.2.37).
The important thing is that, thanks to various identities of special geometry, the effective geodesic potential admits the following alternative representation:
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|Z|2 |
+ |Zi |2 ≡ − |
1 |
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VBH (z, |
z, Q) = − |
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ZZ + Zi gij Zj |
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(9.2.37) |
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2 |
2 |
where the symbol Z denotes the complex scalar field valued central charge of the supersymmetry algebra:
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Z ≡ V T CQ = MΣ pΣ − LΛqΛ |
(9.2.38) |
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and Zi denote its covariant derivatives: |
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Zi = i Z = Ui CQ; |
Zj = gj i Zi |
(9.2.39) |
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j = j Z = |
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j CQ; |
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i = gij |
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U |
Z |
Z |
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Equation (9.2.37) is a result in special geometry whose proof can be found in several articles and reviews of the late nineties.4
9.2.5 Critical Points of the Geodesic Potential and Attractors
The structure of the geodesic potential illustrated above allows for a detailed discussion of its critical points, which are relevant for the asymptotic behavior of the scalar fields.
By definition, critical points correspond to those values of zi for which the first derivative of the potential vanishes: ∂i VBH = 0. Utilizing the fundamental identities of special geometry and (9.2.37), the vanishing derivative condition of the potential can be reformulated as follows:
0 = 2Zi |
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+ iCij k |
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j |
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k |
(9.2.40) |
Z |
Z |
Z |
From this equation it follows that there are three possible types of critical points:
Zi = 0; |
Z = 0; |
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k = 0 |
BPS attractor |
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Zi = 0; |
Z = 0; |
iCij k |
Z |
Z |
non-BPS attractor I (9.2.41) |
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Zi = 0; |
Z = 0; |
iCij k |
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k = −2Zi |
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non-BPS attractor II |
Z |
Z |
Z |
4See for instance the lecture notes [19].
358 |
9 Supergravity: An Anthology of Solutions |
It should be noted that in the case of one-dimensional special geometries, only BPS attractors and non-BPS attractors of type II are possible. Indeed non-BPS attractors of type I are forbidden unless Czzz vanishes identically.
In order to characterize the various type of attractors, the authors of [20] and [21] introduced a certain number of special geometry invariants that obey different and characterizing relations at attractor points of different type. They are defined as follows. Let us introduce the symbols:
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3 ≡ Ci j k Zi Zj Zk |
(9.2.42) |
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N |
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and let us set: |
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i1 = ZZ |
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i3 = |
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i5 = Cij k C m n Zj Zk Zm Zn gi ;
An important identity satisfied by the above invariants, that depend both on the scalar fields zi and the charges (p, q), is the following one:
I4(p, q) = |
1 |
(i1 |
− i2)2 + i4 − |
1 |
i5 |
(9.2.44) |
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where: |
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I4(p, q) = IMN P R QM QN QP QR |
(9.2.45) |
is a quartic polynomial in the electromagnetic charges defined by a symmetric tensor IMN P R which is invariant with respect to all transformations of the isometry group UD = 4 symplectically embedded in Sp(2nv , R). This means that in the combination (9.2.44) the dependence on the fields zi cancels identically.
The generic existence in supergravity models of the quartic invariant (9.2.45) and its relation with the Black-Hole area/entropy is one of the most profound and intriguing contributions of the supergravity/superstring studies to Gravity Theory. On one hand it opens a window on the statistical interpretation of the black holes since, in the underlying superstring microscopic interpretation of supergravity, charges are related to branes and to the counting of their wrapping modes, on the other hand it is quite possible that the group-theoretical structures related to the quartic invariant might have a more general validity beyond purely supersymmetric theories. In this book we will not enter the very rich classification of black-holes in the various supergravity models. We will just confine ourselves to an ultra short illustration of the main features of such black holes by means of the simplest N = 2 supergravity model containing just one vector multiplet with non-trivial couplings. This is done in the next subsection. After this anticipation we continue with the classification of critical points.
Indeed in [20] it was proposed that the three types of critical points can be characterized by the following relations among the above invariants holding at the attractor point:
9.2 Black Holes Once Again |
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359 |
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At BPS Attractor Points We have: |
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i1 = 0; |
i2 = i3 = i4 = i5 = 0 |
(9.2.46) |
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At Non-BPS Attractor Points of Type I |
We have: |
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i2 = 0; |
i1 = i3 = i4 = i5 = 0 |
(9.2.47) |
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At Non-BPS Attractor Points of Type II |
We have: |
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i2 = 3i1; |
i3 = 0; |
i4 = −2i12; |
i5 = 12i12 |
(9.2.48) |
9.2.6 The N = 2 Supergravity S3-Model
The pedagogical example we consider in this book is the simplest possible case of vector multiplet coupling in N = 2 supergravity: we just introduce one vector multiplet. This means that we have two vector fields in the theory and one complex scalar field z. This scalar field parameterizes a one-dimensional special Kähler manifold which, in our choice, will be the complex lower half-plane endowed with the standard Poincaré metric. In other words:5
gzz |
∂μz∂μ |
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1 |
∂μz∂μ |
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(9.2.49) |
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(Im z)2 |
is the σ -model part of the Lagrangian (8.4.1). From the point of view of geometry the lower half-plane is the symmetric coset manifold SL(2, R)/SO(2) SU(1, 1)/U(1) which admits a standard solvable parameterization as it follows. Let:
L0 = |
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L+ = |
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L− = |
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−
be the standard three generators of the sl(2, R) Lie algebra satisfying the commu-
tation relations [L0, L±] = ±L± and [L+, L−] = 2L0. The coset manifold SL(2,R)
SO(2)
is metrically equivalent with the solvable group manifold generated by L0 and L+. Correspondingly we can introduce the coset representative:
L4(φ, y) = exp[yL1] exp[ϕL0] = |
eϕ/2 |
e−ϕ/2y |
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(9.2.51) |
0 |
e−ϕ/2 |
Generic group elements of SL(2, R) are just 2 × 2 real matrices with determinant one:
SL(2, R) # A = |
a |
b |
; ad − bc = 1 |
(9.2.52) |
c |
d |
5The special overall normalization of the Poincaré metric is chosen in order to match the general definitions of special geometry applied to the present case.
360 |
9 Supergravity: An Anthology of Solutions |
and their action on the lower half-plane is defined by usual fractional linear transformations:
A z |
az + b |
(9.2.53) |
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+ |
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The correspondence between the lower complex half-plane C− and the solvableparameterized coset (9.2.51) is easily established observing that the entire set of Im z < 0 complex numbers is just the orbit of the number i under the action of
L(φ, y):
L |
(φ, y) |
: |
i |
→ |
−eϕ/2i + e−ϕ/2y |
= |
y |
− |
ieϕ |
(9.2.54) |
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e−ϕ/2 |
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This simple argument shows that we can rewrite the coset representative terms of the complex scalar field z as follows:
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√ |
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Re z |
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Im z |
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L(φ, y) in
(9.2.55)
The issue of special Kähler geometry becomes clear at this stage. If we did not have vectors in the game, the choice of the coset metric would be sufficient and nothing more would have to be said. The point is that we still have to define the kinetic matrix of the vector and for that the symplectic bundle is necessary. On the same base manifold SL(2, R)/SO(2) we have different special structures which lead to different physical models and to different σ -model groups Uσ . The special structure is determined by the choice of the symplectic embedding SL(2, R) → Sp(4, R). The symplectic embedding that defines our pedagogical model and which eventually leads to the σ -model group Uσ = G2(2) is cubic and it is described in the following subsection.
9.2.6.1 The Cubic Special Kähler Structure on SL(2, R)/SO(2)
The group SL(2, R) is also locally isomorphic to SO(1, 2) and the fundamental representation of the first corresponds to the spin J = 12 of the latter. The spin J = 32 representation is obviously four-dimensional and, in the SL(2, R) language, it corresponds to a symmetric three-index tensor tabc . Let us explicitly construct the 4 × 4 matrices of such a representation. This is easily done by choosing an order for the four independent components of the symmetric tensor tabc . For instance we can identify the four axes of the representation with t111, t112, t122, t222. So doing, the image of the group element A in the cubic symmetric tensor product representation is the following 4 × 4 matrix:
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a3 |
3a2b |
ab2 |
b3 |
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D3(A) = |
a2c |
da2 |
2bca |
cb23 |
2adb |
b2d |
(9.2.56) |
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ac2 |
bc2 |
+ 2adc |
ad2 + |
2bcd |
bd2 |
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+ |
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3c2 d |
3cd2 |
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9.2 Black Holes Once Again |
361 |
By explicit evaluation we can easily check that:
T |
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(A)C4D3 |
(A) = C4 |
where C4 |
0 |
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7 |
7 |
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Since 7C4 is antisymmetric, (9.2.57) is already a clear indication that the triple symmetric representation defines a symplectic embedding. To make this manifest it suffices to change basis. Consider the matrix:
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3 |
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1 |
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and define:
Λ(A) = S−1D3(A)S
We can easily check that: |
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(9.2.58)
(9.2.59)
1 0
01
(9.2.60)
00
0 0
So we have indeed constructed a standard symplectic embedding SL(2, R) → Sp(4, R) whose explicit form is the following:
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c d |
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(9.2.61) |
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The 2× 2 blocks A, B, C, D of the 4× 4 symplectic matrix Λ(A) are easily readable from (9.2.61) so that, assuming now that the matrix A(z) is the coset representative of the manifold SU(1, 1)/U(1), we can apply the Gaillard-Zumino formula (8.3.67) and obtain the explicit form of the kinetic matrix NΛΣ :
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(c+id)(ac+bd) |
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ibc iad 2bd |
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N |
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id)(ac |
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bd) |
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id)2(2ac |
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ibc |
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iad |
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2bd) |
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Inserting the specific values of the entries a, b, c, d corresponding to the coset representative (9.2.55), we get the explicit dependence of the kinetic period matrix
362 9 Supergravity: An Anthology of Solutions
on the complex scalar field z:
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z) |
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N ΛΣ (z) |
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This might conclude the determination of the Lagrangian of our master example, yet we have not yet seen the special Kähler structure induced by the cubic embedding. Let us present it.
The key point is the construction of the required holomorphic symplectic section Ω(z). As usual the transformation properties of a geometrical object indicate the way to build it explicitly. For consistency we should have that:
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(9.2.64) |
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where Λ(A) is the symplectic representation (9.2.61) of the considered SL(2, R)
matrix a b and f (z) is the associated transition function for that line-bundle
c d
whose Chern-class is the Kähler class of the base-manifold. The identification of the symplectic fibres with the cubic symmetric representation provide the construction mechanism of Ω. Consider a vector vv12 that transforms in the fundamental doublet representation of SL(2, R). On one hand we can identify the complex coordinate z on the lower half-plane as z = v1/v2, on the other we can construct a symmetric three-index tensor taking the tensor products of three vi , namely: tij k = vi vj vk . Dividing the resulting tensor by (v2)3 we obtain a four vector:
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Next, recalling the change of basis (9.2.58), (9.2.59) required to put the cubic representation into a standard symplectic form we set:
−√3z2
Ω(z) = SΩ(z) = |
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and we can easily verify that this object transforms in the appropriate way. Indeed we obtain:
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(9.2.67) |
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The pre-factor (cz + d)−3 is the correct one for the prescribed line-bundle. To see this let us first calculate the Kähler potential and the Kähler form. Inserting (9.2.66)
9.2 Black Holes Once Again |
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363 |
into (8.5.18) we get: |
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K = − log i Ω | |
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This shows that the constructed symplectic bundle leads indeed to the standard Poincaré metric and the exponential of the Kähler potential transforms with the prefactor (cz + d)3 whose inverse appears in (9.2.67).
To conclude let us show that the special geometry definition of the period matrix N agrees with the Gaillard-Zumino definition holding true for all symplectically embedded cosets. To this effect we calculate the necessary ingredients:
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Then according to (8.5.29) we obtain:
Λ |
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and applying definition (8.5.30) we exactly retrieve the same form of NΛΣ as given in (9.2.63).
For completeness and also for later use we calculate the remaining items pertaining to special geometry, in particular the symmetric C-tensor. From the general definition (8.5.23) applied to the present one-dimensional case we get:
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Czzz = − |
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zUz = iCzzzhzz |
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As for the standard Levi-Civita connection we have: |
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Γzzz |
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Γzz z = − |
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all other components vanish |
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z − z |
z − z |
This concludes our illustration of the cubic special Kähler structure on SL(2,R) .
SO(2)