Factorizable Sheaves and Quantum Groups Book

TABLE OF CONTENTS Introduction 1 Acknowledgement 5 Part 0. OVERVIEW 1. Introduction Chapter 1. Local—???? 2. The category $ÖCC$ 11 3. Braiding local systems 15 4. Factorizable sheaves 19 5. Tensor product 20 6. Vanishing cycles 23 Chapter 2. Global (genus 0) 7. Cohesive local systems 27 8. Gluing 30 9. Semi-infinite cohomology 31 Bindestrich einfuegen, auch im Ms 10. Conformal blocks (genus 0) 33 11. Integration 35 12. Regular representation 36 13. Regular sheaf 37 Chapter 3. Modular 14. Heisenberg local system 40 15. Fusion structures on $ÖFS$ 46 16. Conformal blocks (higher genus) 48 Part I. INTERSECTION COHOMOLOGY OF REAL ARRANGEMENTS 1. Introduction 50 2. Topological preliminaries 51 3. Vanishing cycles functors 55 4. Computations for standard sheaves 63 Part II. CONFIGURATION SPACES AND QUANTUM GROUPS 1. Introduction 71 Chapter 1. Algebraic discussion 2. Free algebras and bilinear forms 73 3. Hochschild complexes 82 4. Symmetrization 84 5. Quotient algebras 87 Chapter 2. Geometric discussion 6. Diagonal stratification and related algebras 89 7. Principal stratification 94 8. Standard sheaves 100 Chapter 3. Fusion 9. Additivity theorem 110 10. Fusion and tensor products 112 Chapter 4. Category $ÖCC$ 11. Simply laced case 116 12. Non-simply laced case 119 Part III. TENSOR CATEGORIES ARISING FROM CONFIGURATION SPACES 1. Introduction 122 Chapter 1. Category $ÖFS$ 2. Space $ÖCA$ 124 3. Braiding local system $ÖCI$ 126 4. Factorizable sheaves 128 5. Finite sheaves 130 6. Standard sheaves 132 Chapter 2. Tensor structure 7. Marked disk operad 134 8. Cohesive local systems $ÖCI$ 138 9. Factorizable sheaves over $ÖCA$ 139 10. Gluing 142 11. Fusion 145 Chapter 3. Functor $ÖPhi$ 12. Functor $ÖPhi$ 150 13. Main properties of $ÖPhi$ 159 äÖbf Chapter 4. Equivalence 14. Truncation functors 162 15. Rigidity 165 16. Steinberg sheaf 167 17. Equivalence 169 18. The case of generic $Özeta$ 170 Part IV. LOCALIZATION OVER $ÖBP$ 1. Introduction 173 Chapter 1. Gluing over $ÖBP$ 2. Cohesive local system 174 3. Gluing 176 Chapter 2. Semiinifinite cohomology 4. Semi-infinite functors $Ext$ and $Tor$ in $ÖCC$ 180 5. Some calculations 184 Chapter 3. Global sections 6. Braiding and balance in $ÖCC$ and $ÖFS$ 188 7. Global sections over $ÖCA(K)$ 189 8. Global sections over $ÖCP$ 190 9. Application to conformal blocks 193 Part V. MODULAR STRUCTURE ON THE CATEGORY $ÖFS$ 1. Introduction 197 Chapter 1. Heisenberg local system 2. Notations and statement of the main result 199 3. The scheme of construction 202 4. The universal line bundle 204 5. The universal local system 207 6. Factorization isomorphisms 216 Chapter 2. The modular property of the Heisenberg system 7. Degeneration of curves: recollections and notations 219 8. Proof of Theorem 7.6(a) 221 9. Proof of Theorem 7.6(b) 224 Chapter 3. Regular representation 10. A characterization of the regular bimodule 231 11. The adjoint representation 233 Chapter 4. Quadratic degeneration in genus zero 12. $I$-sheaves 235 13. Degenerations of quadrics 236 14. The $I$-sheaf $ÖCR$ 237 15. Convolution 239 Chapter 5. Modular functor 16. Gluing over $C$ 242 17. Degeneration of factorizable sheaves 246 18. Global sections over $C$ 248 Chapter 6. Integral representations of conformal blocks 19. Conformal blocks in arbitrary genus 249 References 252 Index of Notation 266 Index of Terminology 283