Quantum Field Theory and Manifold Invariants (Ias/Park City Mathematics, 28)

This volume contains lectures from the Graduate Summer School “Quantum Field Theory and Manifold Invariants” held at Park City Mathematics Institute 2019. The lectures span topics in topology, global analysis, and physics, and they range from introductory to cutting edge. Topics treated include mathematical gauge theory (anti-self-dual equations, Seiberg-Witten equations, Higgs bundles), classical and categorified knot invariants (Khovanov homology, Heegaard Floer homology), instanton Floer homology, invertible topological field theory, BPS states and spectral networks. This collection presents a rich blend of geometry and topology, with some theoretical physics thrown in as well, and so provides a snapshot of a vibrant and fast-moving field. Graduate students with basic preparation in topology and geometry can use this volume to learn advanced background material before being brought to the frontiers of current developments. Seasoned researchers will also benefit from the systematic presentation of exciting new advances by leaders in their fields. Background 10 Definition 11 Examples 11 TQFT’s from path integrals 12 TQFT’s from supersymmetry 13 Introduction 19 Bundles and connections 21 Vector bundles 21 Principal bundles 28 The Levi–Civita connection 39 Classification of \U(1) and \SU(2) bundles 40 The Chern–Weil theory 44 The Chern–Weil theory 44 The Chern–Simons functional 48 The modui space of flat connections 49 Dirac operators 52 Spin groups and Clifford algebras 52 Dirac operators 54 Spin and Spin^{c} structures 55 The Weitzenböck formula 59 Linear elliptic operators 61 Sobolev spaces 61 Elliptic operators 64 Elliptic complexes 68 Fredholm maps 70 The Kuranishi model and the Sard–Smale theorem 70 The \Z/2\Z degree 72 The parametric transversality 73 The determinant line bundle 75 Orientations and the \Z–valued degree 76 An equivariant setup 77 The Seiberg–Witten gauge theory 79 The Seiberg–Witten equations 79 The Seiberg–Witten invariant 89 Prelude: Knots and the Jones polynomial 95 Knots 95 Generalizations 98 New knots from old 98 The Jones polynomial 99 Connections and Further Reading 104 The Alexander Polynomial 106 The knot group 106 The infinite cyclic cover 107 The Alexander polynomial 109 Fox calculus 110 Fibred knots 112 The Seifert genus 114 The Seifert Matrix 117 Links 119 Connections and Further Reading 121 Khovanov Homology 122 Cube of resolutions 122 The Cobordism Category 124 Applying a TQFT 125 The TQFT \aA 126 Gradings 128 Invariance 129 Functoriality 130 Deformations 132 Connections and Further Reading 138 Khovanov Homology for Tangles 139 Tangles 139 Planar Tangles 141 The Kauffman Bracket 143 Category theory 144 The Krull-Schmidt property 146 Chain complexes over a category 147 The Cube of Resolutions 148 Bar-Natan’s category 149 How to Compute 152 Connections and Further Reading 156 HOMFLY-PT Homology 156 Braid Closures 157 The Hecke algebra 158 Structure of H_{n} 159 The cube of resolutions 162 The Kazhdan-Lusztig basis 164 Soergel Bimodules 164 Hochschild homology and cohomology 167 The Rouquier complex 170 HOMFLY-PT homology 171 Connections and Further Reading 173 Λ^{k} colored polynomials 173 The yoga of WRT 174 Webs 176 The MOY bracket 177 The web category 179 The Λ^{k} colored HOMFLY-PT polynomial 180 Categorification 181 Connections and Further Reading 183 Heegaard splittings and diagrams 189 Heegaard splittings 189 Heegaard diagrams 190 Doubly pointed Heegaard diagrams 193 Heegaard Floer homology 196 Overview 196 The Heegaard Floer chain complex 199 Knot Floer homology 203 Overview 203 The knot Floer complex 203 Algebraic variations 206 Computations 207 Heegaard Floer homology of knot surgery 209 Large surgery 209 Integer surgery 211 Applications 214 Introduction 219 The geometry of the moduli space of Higgs bundles 220 Higgs bundles for complex groups. 220 Real Higgs bundles. 223 Parabolic Higgs bundles. 224 Wild Higgs bundles. 226 Problem set I. 230 The geometry of the Hitchin fibration 232 The Hitchin fibration and the Teichmüller component. 232 The regular fibres of the Hitchin fibration. 233 The singular locus of the Hitchin fibration. 239 Problem set II. 240 Branes in the moduli space of Higgs bundles 242 Branes through finite group actions. 243 Branes through anti-holomorphic involutions. 246 Problem set III. 253 Higgs bundles and correspondences 255 Group homomorphisms. 255 Polygons and Hyperpolygons. 258 Langlands duality. 261 Problem set IV. 263 Preamble: Some background material not included in the lectures 272 A brief introduction to Morse homology 272 The negative gradient flow and (un)stable manifolds 274 Palais-Smale condition C 280 Towards a compactification of the space of unparameterized trajectories 280 The Morse Homology 282 Orientations 282 Homology with Local Coefficients 283 Novikov-Morse Homology 284 Principal Bundles, connections, Chern-Weil theory and the Chern-Simons invariant 286 Principal bundles 286 The big and little adjoint bundles 287 Connections 287 The action of gauge transformations on connections 289 The Curvature 290 The holonomy representation of flat connections 291 Deformations of connections 292 Classifications of U(n) and SU(n)-bundles on 3-manifolds 292 Characteristic classes and the Chern-Simons invariant 293 Floer Homology overview 295 The Chern-Simons Functional 295 Formal Gradients 296 Hessians 297 Novikov Rings? 299 Exercises for the preamble 299 Some examples of representation spaces and computations of the Chern-Simons functional of three manifolds and knots 300 The first variation of Chern-Simons and its periods 301 Examples of representation spaces and computations of the Chern-Simons invariant 302 Seifert Fibered Spaces 305 Representation spaces of knot groups 309 The ASD Equation: Examples and Basic Properties 310 The gradient of Chern-Simons and the ASD equation 310 The linear case 313 The basic instanton 315 More instantons on S4 by conformal transformations 318 Gauge Transformations 319 Sobolev Completions 322 Local Charts 323 The Construction of Slices 325 The local structure of the moduli space on a closed 4-manifold 328 Uhlenbeck’s Fundamental Lemma 329 The Curvature Map Is Proper 330 The Uhlenbeck Compactness Theorem. 331 Lecture 3: The Construction of Floer Homology 333 The ASD Equation and Gauge Fixing on A Cylinder 333 The Extended Hessian 334 The Spectral Flow and Fredholm Index 336 The Spectral Flow in the Instanton Floer Homology 337 Compactification of the Moduli Space 338 A Criterion to Avoid Reducible Flat Connections 340 Instanton Floer Homology 342 Floer Homology for links in three manifolds, exact triangles and Khovanov Homology 343 Representation spaces of knot complements. 343 Khovanov Homology 345 Exact Triangles 348 Some Non-Orientable Surfaces in 4-Manifolds 349 Floer’s Exact Triangle 350 The Second Step 353 The Final Step 358 A Spectral Sequence 358 Introduction 365 Lecture 1: Cobordisms 365 Classical definitions 365 Cobordism categories, first attempt 367 Categories, groupoids, and spaces 368 Invertible field theories (poor man’s version) 372 Lecture 2: Topologically enriched (cobordism) categories 373 Topologically enriched cobordism categories 373 Manifold bundles and bundles of cobordisms 374 Infinite dimensional ambient space 376 Categories versus enriched categories 377 The main theorem 380 Lecture 3: More structure 382 Symmetric monoidal structures 382 Symmetric monoidal structure on the universal groupoid under C 383 Little disks 384 Structure on embedded cobordism categories 387 Topological categories 388 Conclusion 389 Lecture 4: Cobordism classes and characteristic classes 391 Stable homology of moduli spaces of surfaces 391 Cohomology of morphism spaces 392 Cobordisms with connectivity restrictions 395 Cohomology of morphism spaces 396 Exercises (by A. Debray, S. Galatius, M. Palmer) 397 Exercise set 1 398 Exercise set 2 399 Exercise set 3 403 Solutions to selected exercises (by A. Debray, S. Galatius, M. Palmer) 405 Solution to Exercise 5.2.2(b) 405 Solution to Exercise 5.3.1(e) 409 Lecture I 421 Motivation 421 Dehn surgery and Kirby calculus 421 Witten-Reshetikhin-Turaev invariant and Chern-Simons TQFT 423 Lecture II 425 2d TQFTs 425 3d TQFTs 426 Lecture III 430 Asymptotic expansion conjecture 430 Analytically continued Chern-Simons 432 Lecture IV 437 Problems in categorifing WRT invariant 437 Abelian flat connections and linking pairing 437 WRT invariant and q-series 438 q-series invariant for plumbed 3-manifolds 440 Exercises 442 Lecture I 442 Lecture II 442 Lecture III 443 Lecture IV 444 Solutions 445 Lecture I 445 Lecture II 448 Lecture III 450 Lecture IV 451 Lecture 1: What is a BPS state? 457 Quantum mechanics 457 The superparticle 459 Q-cohomology 461 Richer examples 462 Field theory 463 Supersymmetric field theory 464 Lectures 2-3: 2d theories, tt* geometry and Stokes phenomenon 469 Landau-Ginzburg models 469 BPS solitons 469 Wall-crossing formula and spectral networks for 2d \N=(2,2) theories 472 Chiral rings and vacua 474 The topological connection 475 Contour integrals 476 Spectral network as jumping locus for covariantly constant sections 477 Wall-crossing formula via the spectral network 478 tt* geometry 479 Lecture 4: 4d theories and spectral networks 480 Class S and surface defects 481 Chiral rings 481 BPS solitons 482 Spectral networks 482 Adding punctures 484 BPS indices in the \fsl2 case 485 BPS particles in higher rank cases 488 Families of flat connections 489 Abelianization 489