The Many Facets of Geometry: A Tribute to Nigel Hitchin

CONTENTS......Page 10
PREFACE......Page 17
LIST OF EDITORS AND CONTRIBUTORS......Page 18
1 Geometry and physics: a personal view......Page 20
2 Mathematical work of Nigel Hitchin......Page 30
3 The Einstein–Maxwell equations, extremal Kähler metrics, and Seiberg–Witten theory......Page 36
4.1 Introduction......Page 53
4.2.1 Two types of invariant self-dual gauge fields on R[sup(4)]......Page 54
4.2.2 The Nahm transform......Page 56
4.3.1 Upstairs: twistor transform for calorons......Page 58
4.3.2 Downstairs: caloron as a KaČ–Moody monopole......Page 59
4.4.1 Flows of sheaves......Page 65
4.4.2 Boundary conditions......Page 67
4.5.1 Starting with a caloron......Page 74
4.5.2 Starting with a solution to Nahm\'s equation......Page 77
4.5.3 From Nahm to caloron to Nahm to caloron......Page 83
4.6 Moduli......Page 85
5.1 Introduction......Page 90
5.2 An infinite-dimensional Riemannian manifold......Page 93
5.3 Three equivalent problems......Page 97
5.3.1 θ equation ⇒ φ equation......Page 98
5.3.3 U equation ⇒ θ equation......Page 100
5.4.1 Monge–Ampère and the results of Chen......Page 101
5.4.2 Comparison with the free-boundary literature......Page 103
5.5 Relation with Nahm\'s equations......Page 105
6.1 Moduli of vector bundles......Page 111
6.2 Hecke correspondence......Page 113
6.3 Moduli of Higgs pairs......Page 114
6.4 Higgs pairs and the fundamental group......Page 116
6.5 Non-abelian Hodge theory......Page 117
6.6 Hitchin morphism......Page 118
6.7 Quantization......Page 120
6.8 Hecke transformation and Hitchin discriminant......Page 121
6.9 Hitchin component......Page 122
6.10 Reductive groups and principal bundles......Page 124
6.11 Reductive groups and Higgs pairs......Page 125
6.12 Real forms......Page 127
7.1 A-model and B-model......Page 132
7.2 Mirror symmetry and Hitchin\'s equations......Page 133
7.3 Hitchin fibration......Page 135
7.3.1 A few hints......Page 136
7.4 Ramification......Page 137
7.5 Wild ramification......Page 139
7.6 Four-dimensional gauge theory and stacks......Page 142
7.6.1 Stacks......Page 144
8.1 Introduction......Page 148
8.2.2 Representation variety......Page 149
8.2.4 Deformation space......Page 151
8.3.2 Multiplicative characters: G = C*......Page 152
8.3.3 Jacobi variety of a Riemann surface......Page 153
8.4 Stable vector bundles and Higgs bundles......Page 154
8.5.1 Geometric structures......Page 156
8.5.3 Examples of hyperbolic structures......Page 157
8.6.1 Deformation spaces of geometric structures......Page 161
8.6.2 Fuchsian components of Hom(Π, G)/G......Page 162
8.6.3 Characteristic classes and maximal representations......Page 163
8.6.4 Quasi-Fuchsian representations: G = PSL(2, C)......Page 164
8.6.5 Teichmüller space: marked conformal structures......Page 167
8.6.6 Holomorphic vector bundles and uniformization......Page 168
8.7.1 Harmonic metrics......Page 169
8.7.2 Higgs pairs and branched hyperbolic structures......Page 170
8.7.3 Uniformization with singularities......Page 171
8.8.1 Convex RP[sup(2)]-structures: G = SL(3, R)......Page 172
8.8.2 Higgs bundles and affine spheres......Page 173
8.8.3 Hyperconvex curves......Page 174
8.9.1 Unitary groups U(p, q)......Page 175
8.9.2 Symplectic groups Sp(n, R)......Page 176
8.9.3 Geometric structures associated to Hitchin representations......Page 177
9 Locality of holomorphic bundles, and locality in quantum field theory......Page 183
9.1 Noncommutative geometry......Page 184
9.2 Algebraic structures up to homotopy......Page 188
9.3 Quantum field theory......Page 190
10.1 Introduction......Page 196
10.1.1 Geometric construction of Z[sup((n))][sub(k)]......Page 198
10.1.2 Asymptotic faithfulness......Page 200
10.1.3 Kashdan\'s property (T) and the mapping class group......Page 201
10.1.4 Geometric construction of the curve operators......Page 203
10.1.5 The Nielsen–Thurston classification of mapping classes is determined by TQFT......Page 204
10.2 The Hitchin connection......Page 207
10.3 Toeplitz operators and Berezin–Toeplitz deformation quantization......Page 213
10.4 The formal Hitchin connection......Page 217
10.5 Asymptotics flatness of Toeplitz operators......Page 220
10.6 General asymptotic faithfulness......Page 223
11.1 Introduction......Page 229
11.1.1 Naive generalizations......Page 230
11.1.2 Non-linear analogue: the Painlevé VI equation......Page 232
11.2 What is Painlevé VI?......Page 233
11.2.1 Conceptual approach to Painlevé VI......Page 235
11.2.2 Explicit non-linear equations......Page 236
11.2.3 Monodromy of Painlevé VI......Page 237
11.3.2 (A) ↦ (C)......Page 238
11.4 Beyond Platonic Painlevé VI solutions......Page 240
11.4.1 Construction......Page 241
11.4.2 Relating connections (A) and (B)......Page 242
11.4.3 New solutions......Page 245
11.5 Pullbacks......Page 247
11.6.1 Up to degree 24......Page 250
11.6.2 Quadratic/Landen/folding transformations......Page 251
11.7 Conclusion......Page 252
12 An introduction to bundle gerbes......Page 256
12.1 Introduction......Page 257
12.2 Background......Page 261
12.3 Bundle gerbes......Page 262
12.3.3 Characteristic class......Page 264
12.3.4 Connective structure......Page 265
12.4 Triviality......Page 266
12.4.1 Holonomy......Page 269
12.5 Examples of bundle gerbes......Page 270
12.5.1 Lifting bundle gerbe......Page 271
12.5.2 Projective bundles......Page 272
12.5.3 Bundle gerbes on Lie groups......Page 273
12.6.1 Wess–Zumino–Witten term......Page 274
12.6.2 Faddeev–Mickelsson anomaly......Page 275
12.7 Other matters......Page 276
13.1 Introduction......Page 280
13.2 Projective hulls......Page 283
13.3 Projective linking and projective winding numbers......Page 286
13.4 Quasi-plurisubharmonic functions......Page 288
13.5 Boundaries of positive holomorphic chains......Page 289
13.6 Projective Alexander–Wermer theorem for curves......Page 290
13.7 Theorems for general projective manifolds......Page 294
13.8 Relative holomorphic cycles......Page 296
14.1 Skyrmions......Page 300
14.2 Rational map ansatz......Page 302
14.3.1 B = 8......Page 305
14.3.2 B = 12......Page 306
14.3.3 B = 16......Page 308
14.3.4 B = 24......Page 309
14.3.6 B = 32......Page 310
14.4 Quantization and calibration......Page 311
14.5 Conclusions......Page 313
Appendix: Double rational map ansatz......Page 314
15.1 Introduction......Page 318
15.2.2 Mirror symmetry conjectures......Page 319
15.2.3 SYZ transform......Page 320
15.2.4 Mirror of elliptic fibrations......Page 321
15.3.2 Elliptic fibrations and their duals......Page 323
15.3.3 FM transform and spectral cover construction......Page 324
15.4.1 Symplectic Fourier–Mukai transform......Page 325
15.4.2 Twin Lagrangian fibrations......Page 326
15.4.3 Dual twin Lagrangian fibration and its FM transform......Page 331
15.4.4 Examples of twin Lagrangian fibrations......Page 332
15.5.1 SYZ transform of dual elliptic fibrations......Page 334
15.5.2 SYZ transform of the universal Poincaré sheaf......Page 336
15.5.3 SYZ transforms commute with FM transforms......Page 339
15.6 Conclusions and discussions......Page 340
16.1 Introduction......Page 343
16.2.1 Moduli of Yang–Mills instantons on R[sup(4)]......Page 345
16.2.2 Moduli space of magnetic monopoles on R[sup(3)]......Page 346
16.3.1 L[sup(2)] harmonic forms on complete manifolds......Page 347
16.3.2 Results on L[sup(2)] harmonic forms......Page 348
16.4.1 Mixed Hodge structure of Deligne......Page 350
16.4.2 Arithmetic and topological content of the E-polynomial......Page 351
16.5.1 Nakajima quiver varieties......Page 352
16.5.2 Spaces diifeomorphic to the Hitchin moduli space M(C, PU( n))......Page 355
16.5.3 Topological mirror test......Page 356
16.5.4 Mirror symmetry for finite groups of Lie type......Page 357
16.5.5 Conjectural answer......Page 358
17.1 Introduction......Page 365
17.2.1 Holonomy......Page 368
17.2.2 A non-trivial case......Page 369
17.2.3 Hyper-Kähler viewpoint......Page 370
17.3.2 A sharper result......Page 372
17.4 G[sub(2)]-Manifolds......Page 377
17.4.1 Hypersurfaces......Page 378
17.5 Spin(7)-manifolds......Page 381
17.5.1 Hypersurfaces......Page 383
17.5.2 Flow interpretation......Page 384
18.1 Introduction......Page 387
18.2 Gerbe trivializations......Page 388
18.3 Generalized connections......Page 389
18.4 Generalized holomorphic bundles and branes......Page 393
18.4.1 Generalized holomorphic bundles......Page 394
18.4.2 Generalized complex branes......Page 396
18.5 Multiple branes and holomorphic Poisson varieties......Page 399
18.6 Relation to generalized Kähler geometry......Page 404
18.7 Construction of generalized Kähler metrics......Page 407
18.8 Relation to non-commutative algebraic geometry......Page 410
19 Consistent orientation of moduli spaces......Page 414
19.1.1 Quantum field theory......Page 416
19.1.2 Topological construction......Page 417
19.1.3 Remarks......Page 419
19.2.1 Ordinary cohomology......Page 420
19.2.2 K-theory......Page 422
19.3.2 Closed surfaces......Page 424
19.3.3 Surfaces with boundary......Page 427
19.3.4 The level......Page 430
19.3.5 Pushforward maps......Page 432
19.4 Families of surfaces, twistings, and anomalies......Page 434
B......Page 440
C......Page 441
D......Page 442
E......Page 443
G......Page 444
H......Page 445
K......Page 446
M......Page 447
O......Page 448
Q......Page 449
S......Page 450
T......Page 452
Z......Page 453