GEOMETRY AND PHYSICS

By T. HISAS

CONTENTS PART AN ELEMENTARY INTRODUCTION TO BRANES IN STRING THEORY INTRODUCTION BRANES IN STRING THEORY THE SUPERSTRING EFFECTIVE ACTIONS OF TYPE II TYPE IIA TYPE IIB GENERAL CONSTRUCTION EXPLICIT SOLUTIONS FUNDAMENTAL STRING NS 5-BRANE DP-BRANES THE GEOMETRY OF THE D3-BRANE OF TYPE IIB THE BOUNDARY STATE DESCRIPTION OF D-BRANES THE BOUNDARY STATE WITH AN EXTERNAL ?ELD THE EFFECTIVE ACTION OF D-BRANES CLASSICAL D-BRANES FROM THE BOUNDARY STATE REFERENCES PART PHYSICAL ASPECTS TWO-DIMENSIONAL CONFORMAL ?ELD THEORY ON OPEN AND UNORIENTED SURFACES GENERAL PROPERTIES OF TWO-DIMENSIONAL CFT THE STRESS–ENERGY TENSOR IN TWO DIMENSIONS RATIONAL CONFORMAL ?ELD THEORIES NON-ABELIAN CONFORMAL CURRENT ALGEBRAS PARTITION FUNCTION, MODULAR INVARIANCE CORRELATION FUNCTIONS IN CURRENT ALGEBRA MODELS PROPERTIES OF THE CHIRAL CONFORMAL BLOCKS REGULAR BASIS OF 4-POINT FUNCTIONS IN THE SU(2) MODEL MATRIX REPRESENTATION OF THE EXCHANGE ALGEBRA TWO-DIMENSIONAL BRAID INVARIANT GREEN FUNCTIONS CFT ON SURFACES WITH HOLES AND CROSSCAPS OPEN SECTOR, SEWING CONSTRAINTS CLOSED UNORIENTED SECTOR, CROSSCAP CONSTRAINT PARTITION FUNCTIONS KLEIN BOTTLE PROJECTION ANNULUS PARTITION FUNCTION M¨OBIUS STRIP PROJECTION SOLUTIONS FOR THE PARTITION FUNCTIONS ACKNOWLEDGMENTS TOPICS IN STRING TACHYON DYNAMICS WHY TACHYONS? TACHYONS IN ADS: THE C = 1 BARRIER TACHYON ?-MODEL BETA-FUNCTIONS OPEN STRINGS AND COSMOLOGICAL CONSTANT: THE FISCHLER–SUSSKIND MECHANISM FISCHLER–SUSSKIND MECHANISM: CLOSED-STRING CASE OPEN-STRING CONTRIBUTION TO THE COSMOLOGICAL CONSTANT: THE ?LLING BRANE THE EFFECTIVE ACTION A WARMING-UP EXERCISE NON-CRITICAL DIMENSION AND TACHYON CONDENSATION D-BRANES, TACHYON CONDENSATION AND K-THEORY EXTENDED OBJECTS AND TOPOLOGICAL STABILITY A GAUGE THEORY ANALOGUE FOR D-BRANES IN TYPE II STRINGS K-THEORY VERSION OF SEN’S CONJECTURE TYPE IIA STRINGS SOME ?NAL COMMENTS ON GAUGE THEORIES MATHEMATICAL DEVELOPMENTS DEFORMATION THEORY, HOMOLOGICAL ALGEBRA AND MIRROR SYMMETRY CLASSICAL DEFORMATION THEORY HOLOMORPHIC STRUCTURE ON VECTOR BUNDLES FAMILIES OF HOLOMORPHIC STRUCTURES ON VECTOR BUNDLES COHOMOLOGY AND DEFORMATIONS BUNDLE VALUED HARMONIC FORMS CONSTRUCTION OF A VERSAL FAMILY AND FEYNMAN DIAGRAMS THE KURANISHI FAMILY FORMAL DEFORMATIONS HOMOLOGICAL ALGEBRA AND DEFORMATION THEORY HOMOTOPY THEORY OF A?AND L?ALGEBRAS MAURER–CARTAN EQUATION AND MODULI FUNCTORS CANONICAL MODEL, KURANISHI MAP AND MODULI SPACE SUPERSPACE AND ODD VECTOR ?ELDS—AN ALTERNATIVE FORMU- LATION OF L?ALGEBRAS APPLICATION TO MIRROR SYMMETRY NOVIKOV RINGS AND ?LTERED A?, L?ALGEBRAS REVIEW OF A PART OF GLOBAL SYMPLECTIC GEOMETRY FROM LAGRANGIAN SUBMANIFOLD TO A?ALGEBRA MAURER–CARTAN EQUATION FOR ?LTERED A?ALGEBRAS HOMOLOGICAL MIRROR SYMMETRY LARGE N DUALITIES AND TRANSITIONS IN GEOMETRY GEOMETRY AND TOPOLOGY OF TRANSITIONS THE LOCAL TOPOLOGY OF A CONIFOLD TRANSITION TRANSITIONS OF CALABI–YAU THREEFOLDS TRANSITIONS AND MIRROR SYMMETRY TRANSITIONS, BLACK HOLES ETC CHERN–SIMONS THEORY CHERN–SIMONS’ FORM AND ACTION THE HAMILTONIAN FORMULATION OF THE CHERN–SIMONS QFT (FOLLOWING WITTEN’S CANONICAL QUANTIZATION) COMPUTABILITY AND LINK INVARIANTS THE GOPAKUMAR–VAFA CONJECTURE MATCHING THE FREE ENERGIES THE MATCHING OF EXPECTATION VALUES LIFTING TO M-THEORY RIEMANNIAN HOLONOMY, G2 MANIFOLDS AND CALABI–YAU, REVISITED THE GEOMETRY BRANES AND M-THEORY LIFTS M-THEORY LIFT AND M-THEORY ?OP APPENDIX: SOME NOTATION ON SINGULARITIES AND THEIR RESOLUTIONS APPENDIX: MORE ON THE GREENE–PLESSER CONSTRUCTION APPENDIX: MORE ON TRANSITIONS IN SUPERSTRING THEORY APPENDIX: PRINCIPAL BUNDLES, CONNECTIONS ETC APPENDIX: MORE ON WITTEN’S OPEN-STRING THEORY INTERPRETATION OF QFT