GRAVITATION, GAUGE THEORIES, AND DIFFERENTIAL GEOMETRY
CONTENTS HISTORICAL BACKGROUND REFERENCES GEOMETRY OF GAUGE FIELDS DIFFERENTIABLE MANIFOLDS TENSOR FIELDS AND EXTERIOR FORMS FIBER BUNDLES AS AN ENLARGED GEOMETRIC ARENA ASSOCIATED BUNDLES AND PHYSICAL FIELDS CONNECTION AND COVARIANT DERIVATIVE CURVATURE GAUGE TRANSFORMATIONS TOPOLOGICAL INVARIANTS MAXWELL AND YANG–MILLS THEORY THE LAGRANGIAN FORMALISM G-EQUIVALENCE PRINCIPLE MAXWELL’S THEORY IN DIFFERENTIAL FORMS CONSTRAINED BF SCHEME FOR MAXWELL FIELDS YANG–MILLS FIELDS INSTANTONS RELATION TO THE SEIBERG–WITTEN EQUATIONS HIGGS FIELDS TRANSLATION OF TERMINOLOGIES GRAVITATION AS A GAUGE THEORY AF?NE FRAMES AF?NE GAUGE THEORY WITH TORSION AF?NE “HIGGS” MECHANISM METRIC STRUCTURE OF THE “SPONTANEOUSLY BROKEN” POINCARÉ GAUGE THEORY GRAVITATIONAL FIELD EQUATIONS BIANCHI IDENTITIES AND THEIR CONTRACTIONS NOETHER IDENTITIES MASS AND SPIN OF THE KERR–ADS SOLUTION GRAVITATIONAL AHARONOV–BOHM EFFECT AND CARTAN CIRCUITS EINSTEIN–CARTAN THEORY INTRODUCTION DIRAC FIELDS IN RIEMANN–CARTAN SPACETIME CLASSICAL EINSTEIN–CARTAN THEORY EFFECTIVE EINSTEIN EQUATIONS ASYMPTOTIC SAFETY OF EC THEORY? THE ISSUE OF FOUR-FERMION INTERACTIONS CONSTRAINTS FROM THE WEAK EQUIVALENCE PRINCIPLE TELEPARALLELISM CHIRAL TELEPARALLELISM PARITY-VIOLATING TOPOLOGICAL INVARIANTS IN GRAVITY FROM CHERN–SIMONS TO EINSTEINIAN GRAVITY YANG–MILLS-TYPE FORMULATION OF COMPLEX GRK ENERGY–MOMENTUM COMPLEX HAMILTONIAN FORMULATION OF COMPLEX GRK POISSON BRACKETS CHERN–SIMONS SOLUTIONS OF THE CHIRAL TELEPARALLELISM CONSTRAINTS TORSION INSTANTONS WILSON LOOPS AND LINKS TOPOLOGY OF CARTAN CIRCUITS? TOPOLOGICALLY MODI?ED EINSTEIN–CARTAN THEORY DYNAMICS OF QUADRATIC POINCARÉ GAUGE THEORY CP VIOLATION IN QUANTUM GRAVITY? YANG’S THEORY OF GRAVITY DUAL CONFORMAL STRUCTURE SKY GRAVITY DOUBLE-DUAL SKY GRAVITY THE PATH INTEGRAL DOMINATED BY EINSTEIN SPACES GRAVITON SPECTRUM MASS GAP IN YANG–MIELKE THEORY OF GRAVITY? FIELD REDE?NITION SCHEME OF RENORMALIZATION LEGENDRE TRANSFORMATION VANISHING HESSIAN: GR AS A STABLE FIXED POINT APPENDIX: CLIFFORD-ALGEBRA-VALUED EXTERIOR FORMS BRST QUANTIZATION OF GRAVITY TOPOLOGICAL BRST TRANSFORMATIONS OF GAUGE FIELDS BRST QUANTIZATION OF TRANSLATIONS DIFFEOMORPHISMS TOPOLOGICAL GRAVITY ACTION EFFECTIVE SELF-DUAL SKY GRAVITY SYMMETRY BREAKING VIA DUALITY ROTATIONS GENERALIZED DOUBLE DUALITY CLASSICAL GR BY CONSTRAINING TORSION TO VANISH EFFECTIVE EINSTEINIAN “BACKGROUND” GRAVITATIONAL INSTANTONS EXACT SOLUTIONS SOLUTIONS WITH TORSION INSTANTONS WITH TORSION TOPOLOGICAL INVARIANTS ON MANIFOLDS QUANTUM MEANING OF GRAVITATIONAL INSTANTONS EULER TERM AND INDUCED WORMHOLE CON?GURATIONS THREE-DIMENSIONAL GRAVITY CHERN–SIMONS GRAVITY WITH TORSION IN 3D NOETHER THEOREM IN 3D GRAVITY TOPOLOGICAL MIELKE–BAEKLER MODEL “PROLONGATION” OF ANTI-DE SITTER TO BLACK HOLE SOLUTIONS S-DUALITY IN 3D MODI?ED S-DUALITY IN 3D TOWARD INTEGRABILITY IN 3D NONLINEAR GRAVITONS EFFECTIVE PROCA EQUATION ENERGY–MOMENTUM AND SPIN COMPLEXES CENTRAL CHARGES IN TOPOLOGICAL GRAVITY COUPLING TO THE SYMMETRIC COTTON TENSOR GRAPHENE AND EMERGENT GRAVITY DIRAC EQUATION IN 3D TOPOLOGICAL MASSIVE PHOTONS IN 3D MEMBRANES WITH TORSION DEFECTS SUPERGRAVITY RARITA–SCHWINGER LAGRANGIAN IN 3D 1010 TOPOLOGICAL SUPERSYMMETRY IN 3D SPINOR BUNDLES GLOBAL SPINOR FIELDS COVARIANT DIRAC EQUATION NONLINEAR HEISENBERG–PAULI–WEYL SPINOR EQUATION SOLITONS SOLITON-TYPE SOLUTIONS OF THE NONLINEAR DIRAC EQUATION MASS SPECTRUM QUANTUM-THEORETIC MEANING OF NONLINEAR CLASSICAL FIELD THEORIES? CHIRAL ANOMALIES ANOMALIES FOR PEDESTRIANS CLASSICAL AXIAL ANOMALY AND SPIN AXIAL CURRENT IN THE EINSTEIN–CARTAN THEORY CHIRAL ANOMALY IN QUANTUM FIELD THEORY CHIRAL ANOMALY IN SUGRA COMPARISON WITH THE HEAT KERNEL METHOD HAMILTONIAN INTERPRETATION OF ANOMALIES APPENDIX: GRAVITATIONAL CHERN–SIMONS AND PONTRYAGIN TERMS TOPOLOGICAL SL(5, RÞ GAUGE-INVARIANT ACTION MODI?ED BF SCHEME METALINEAR GROUP VERSUS DE SITTER GROUP GRADED AF?NE VERSUS CARTAN CONNECTION GRAVITY FROM SPONTANEOUS SYMMETRY-BREAKING EFFECTIVE MACDOWELL–MANSOURI THEORY INDUCED HILBERT–EINSTEIN ACTION INDUCED SPACETIME METRIC IN 4D RENORMALIZABILITY OF TOPOLOGICAL GRAVITY? THE ISSUE OF CHIRAL ANOMALIES OUTLOOK: MACH-TYPE HIGGS VACUUM? APPENDIX: TOPOLOGICAL INVARIANTS GEOMETRODYNAMICS AND ITS EXTENSIONS RAINICH GEOMETRIZATION OF ELECTROMAGNETIC FIELDS NONABELIAN KALUZA–KLEIN THEORIES STANDARD MODEL COMPACTI?CATION FERMION SPECTRUM FROM HIGHER-DIMENSIONAL MODELS KERR–NEWMAN SOLUTION NUT SOLUTION WITH DUAL MASS COLOR GEOMETRODYNAMICS TENSOR DOMINANCE OF STRONG INTERACTION PARTICLE SPECTRUM EINSTEIN–CARTAN THEORY WITH INTERNAL DEGREES OF FREEDOM OUTLOOK: GAUGE UNI?CATIONS IN FOUR DIMENSIONS? GEOMETRIC MODEL OF QUARK CON?NEMENT? BAG MODELS CON?NING POTENTIALS FROM STRONG GRAVITY BLACK SOLITON MASS FORMULA