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THE ROAD TO GRAVITATIONAL S–DUALITY H. Garcia–Compeán* Departamento de Física, Centro de Investigación y Estudios Avanzados del IPN P.O. Box 14-740, 07000, México D.F., México. O. Obregón † Instituto de Física de la Universidad de Guanajuato P.O. Box E–143, 37150, León Gto., México. C. Ramirez ‡ Facultad de Ciencias Físico Matemáticas, Universidad Autónoma de Puebla P.O. Box 1364, 72000, Puebla, México. M. Sabido § Instituto de Física de la Universidad de Guanajuato P.O. Box E–143, 37150, León Gto., México. Abstract We overview the road to defining S–duality analogues for non–supersymmetric theories of gravity. The case of pure topological gravity in four dimensions, and, MacDowell–Mansouri gauge theory of gravity are discussed. Three–dimensional dimensional reductions from the topological gravitational sector in four dimensions, enable to recuperate the 2 + 1 Chern–Simons gravity and the corresponding S–dual theory, from the notion of self–duality in the four–dimensional theory. Keywords: S–duality, MacDowell–Mansouri gauge theories, Chern–Simons gravity. *E–mail: compean@fis.cinvestav.mx † E–mail: octavio@ifug3.ugto.mx ‡ E–mail: cramirez@fcfm.buap.mx § E–mail: msabido@ifug3.ugto.mx Exact Solutions and Scalar Fields in Gravity: Recent Developments Edited by Macias et al., Kluwer Academic/Plenum Publishers, New York, 2001 111
112 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY 1. INTRODUCTION The study of symmetries in physics, has become a subject of most importance, for example in particle physics, gauge symmetry is the central idea to understanding the fundamental forces, Lorentz invariance is crucial in quantum field theory, general coordinate covariance in General Relativity, etc. In recent years a new wave of dualities has emerge, as a most important technique in the understanding of non–perturbative aspects of quantum field theory, and string theory. In superstring theory, duality shows the equivalence among different types of perturbative string theories and gives insight of a new underlying theory known as M–theory. Specially interesting are T–duality, and S– duality; the first one gives a new view of space time physics, revealing the existence of a minimal length. Duality is an old known type of symmetry which by interchanging the electric, and magnetic fields leaves invariant the vacuum Maxwell equations. It was extended by Dirac to include sources, with the well known price of the predictions of monopoles, which appear as the dual particles of the electrically charged ones, and whose existence has not been confirmed. Dirac obtained that the charges (couplings) of the magnetical and electrical charged particles are inverse to each other, so if the electrical force is “weak”, then the magnetic force between monopoles will be “strong”, then one description would be treated perturbatively, and other one not, but of course we could always use the duality to go from one theory to the other. The Strong/weak coupling duality (S–duality) in superstring and supersymmetric gauge theories in various dimensions has been, in the last five years, the major tool to study the strong coupling dynamics of these theories. Much of these results require supersymmetry through the notion of BPS state. These states describe the physical spectrum and they are protected of quantum corrections leaving the strong/coupling duality under control to extract physical information. In particular, the exact Wilson effective action of supersymmetric gauge theories has been computed, showing the duality symmetries of these effective theories. It turns out that the dual description is quite adequate to address standard non-perturbative problems of Yang–Mills theory. In the non–supersymmetric case there are no BPS states and the situation is much more involved. This latter case is an open question and it is still under current investigation. In the specific case of non–supersymmetric gauge theories in four dimensions, the subject has been explored recently in the Abelian as well
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EXACT SOLUTIONS AND SCALAR FIELDS I
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To the 65th birthday of Heinz Dehne
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CONTENTS Preface Contributing Autho
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Contents ix 5 The case of 84 Part I
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Contents xi Luis O. Pimentel, Cesar
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Contents xiii 2.2 String theory ind
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PREFACE This book is dedicated to h
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CONTRIBUTING AUTHORS Eloy Ayón-Bea
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Contributing Authors xix Jaime Klap
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Contributing Authors xxi A.P. 70-54
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Contributing Authors xxiii L. Artur
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I EXACT SOLUTIONS
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SELF-GRAVITATING STATIONARY AXI- SY
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Self-gravitating stationary axisymm
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REFERENCES 13 be shown that the ana
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NEW DIRECTIONS FOR THE NEW MILLENNI
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New Directions for the New Millenni
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New Directions for the New Millenni
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REFERENCES 21 Acknowledgments The r
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ON MAXIMALLY SYMMETRIC AND TOTALLY
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On maximally symmetric and totally
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DISCUSSION OF THE THETA FORMULA FOR
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Theta formula for the Ernst potenti
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REFERENCES 51 Rosenhain’s theta f
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THE SUPERPOSITION OF NULL DUST BEAM
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The superposition of null dustbeams
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Page 86 and 87: REFERENCES 61 geodesic with respect
Page 88 and 89: SOLVING EQUILIBRIUM PROBLEM FOR THE
Page 90 and 91: Solving equilibrium problem for the
Page 92 and 93: REFERENCES 67 where the subindices
Page 94 and 95: ROTATING EQUILIBRIUM CONFIGURATIONS
Page 96 and 97: Rotating equilibrium configurations
Page 98 and 99: Rotating equilibrium configurations
Page 100 and 101: REFERENCES 75 5. DISCUSSION The ana
Page 102 and 103: INTEGRABILITY OF SDYM EQUATIONS FOR
Page 104 and 105: Integrability of SDYM Equations for
Page 112 and 113: REFERENCES 87 [18] [19] [20] [21] [
Page 114 and 115: II ALTERNATIVE THEORIES AND SCALAR
Page 116 and 117: THE FRW UNIVERSES WITH BAROTROPIC F
Page 118 and 119: The FRW Universes with Barotropic F
Page 124 and 125: REFERENCES 99 (1979) 515; H. Dehnen
Page 126 and 127: HIGGS-FIELD AND GRAVITY Heinz Dehne
Page 128 and 129: Higgs-Field and Gravity 103 is defi
Page 130 and 131: Higgs-Field and Gravity 105 Consequ
Page 132 and 133: Higgs-Field and Gravity 107 From th
Page 134 and 135: REFERENCES 109 Evidently by inserti
Page 138 and 139: The road to Gravitational S-duality
Page 146 and 147: REFERENCES 121 The partition functi
Page 148 and 149: EXACT SOLUTIONS IN MULTIDIMENSIONAL
Page 150 and 151: Exact solutions in multidimensional
Page 156 and 157: REFERENCES 131 For possible physica
Page 158 and 159: EFFECTIVE FOUR-DIMENSIONAL DILATON
Page 160 and 161: Dilaton gravity from Chern-Simons g
Page 166 and 167: A PLANE-FRONTED WAVE SOLUTION IN ME
Page 168 and 169: A plane-fronted wave solution in me
Page 176 and 177: REFERENCES 6. SUMMARY 151 We invest
Page 178 and 179: III COSMOLOGY AND INFLATION
Page 180 and 181: NEW SOLUTIONS OF BIANCHI MODELS IN
Page 182 and 183: New solutions of Bianchi models in
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New solutions of Bianchi models in
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REFERENCES 163 Further solutions of
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SCALAR FIELD DARK MATTER Tonatiuh M
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Scalar field dark matter 167 tional
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Scalar field dark matter 169 being
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Scalar field dark matter 171 of dar
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Scalar field dark matter 173 growin
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Scalar field dark matter 175 This s
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Scalar field dark matter 177 rotati
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Scalar field dark matter 179 where
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Scalar field dark matter 181 while
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REFERENCES 183 The hypothesis of th
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INFLATION WITH A BLUE EIGENVALUE SP
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Inflation with a blue eigenvalue sp
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REFERENCES 193 problem” for nonli
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CLASSICAL AND QUANTUM COSMOLOGY WIT
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Quantum cosmology with self-interac
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REFERENCES 203 [13] has considered
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THE BIG BANG IN GOWDY COSMOLOGICAL
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The Big Bang in Gowdy Cosmological
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THE INFLUENCE OF SCALAR FIELDS IN P
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The influence of scalar fields in p
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The influence of scalar fields in p
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REFERENCES 221 [3] by R.C. Kennicut
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ADAPTIVE CALCULATION OF A COLLAPSIN
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Adaptive Calculation of a Collapsin
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REFERENCES 233 [5] [6] [7] [8] [9]
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REVISITING THE CALCULATION OF INFLA
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Revisiting the calculation of infla
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CONFORMAL SYMMETRY AND DEFLATIONARY
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Conformal symmetry and deflationary
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REFERENCES 259 [16] R. Maartens, D.
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IV EXPERIMENTS AND OTHER TOPICS
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STATICITY THEOREM FOR NON-ROTATING
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Staticity Theorem for Non-Rotating
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Staticity Theorem for Non-Rotating
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REFERENCES 269 The boundary is cons
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QUANTUM NONDEMOLITION MEASUREMENTS
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Quantum nondemolition measurements
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REFERENCES 279 [11] M. Kasevich and
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ON ELECTROMAGNETIC THIRRING PROBLEM
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On Electromagnetic Thirring Problem
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REFERENCES 293 [8] [9] D. Brill and
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ON THE EXPERIMENTAL FOUNDATION OF M
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On the experimental foundation of M
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REFERENCES 309 [12] [13] [14] [15]
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LORENTZ FORCE FREE CHARGED FLUIDS I
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Lorentz force free charged fluids i
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REFERENCES 319 force can be foresee
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Index Axisymmetries and configurati
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INDEX 323 Theorem (cont.) staticity
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Prof. Heinz Dehnen: Brief Biography
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Prof. Dietrich Kramer: Brief Biogra
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