Quantum Field Theory and Manifold Invariants

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. Contents Preface Introduction Background Definition Examples TQFT’s from path integrals TQFT’s from supersymmetry Introduction to Gauge Theory Introduction Bundles and connections Vector bundles Principal bundles The Levi–Civita connection Classification of \U(1) and \SU(2) bundles The Chern–Weil theory The Chern–Weil theory The Chern–Simons functional The modui space of flat connections Dirac operators Spin groups and Clifford algebras Dirac operators Spin and Spin^{c} structures The Weitzenböck formula Linear elliptic operators Sobolev spaces Elliptic operators Elliptic complexes Fredholm maps The Kuranishi model and the Sard–Smale theorem The \Z/2\Z degree The parametric transversality The determinant line bundle Orientations and the \Z–valued degree An equivariant setup The Seiberg–Witten gauge theory The Seiberg–Witten equations The Seiberg–Witten invariant Knots, Polynomials, and Categorification Prelude: Knots and the Jones polynomial Knots Generalizations New knots from old The Jones polynomial Connections and Further Reading The Alexander Polynomial The knot group The infinite cyclic cover The Alexander polynomial Fox calculus Fibred knots The Seifert genus The Seifert Matrix Links Connections and Further Reading Khovanov Homology Cube of resolutions The Cobordism Category Applying a TQFT The TQFT \aA Gradings Invariance Functoriality Deformations Connections and Further Reading Khovanov Homology for Tangles Tangles Planar Tangles The Kauffman Bracket Category theory The Krull-Schmidt property Chain complexes over a category The Cube of Resolutions Bar-Natan’s category How to Compute Connections and Further Reading HOMFLY-PT Homology Braid Closures The Hecke algebra Structure of H_{n} The cube of resolutions The Kazhdan-Lusztig basis Soergel Bimodules Hochschild homology and cohomology The Rouquier complex HOMFLY-PT homology Connections and Further Reading Λ^{k} colored polynomials The yoga of WRT Webs The MOY bracket The web category The Λ^{k} colored HOMFLY-PT polynomial Categorification Connections and Further Reading Lecture notes on Heegaard Floer homology Introduction Heegaard splittings and diagrams Heegaard splittings Heegaard diagrams Doubly pointed Heegaard diagrams Heegaard Floer homology Overview The Heegaard Floer chain complex Knot Floer homology Overview The knot Floer complex Algebraic variations Computations Heegaard Floer homology of knot surgery Large surgery Integer surgery Applications Advanced topics in gauge theory:mathematics and physics of Higgs bundles Introduction The geometry of the moduli space of Higgs bundles Higgs bundles for complex groups. Real Higgs bundles. Parabolic Higgs bundles. Wild Higgs bundles. Problem set I. The geometry of the Hitchin fibration The Hitchin fibration and the Teichmüller component. The regular fibres of the Hitchin fibration. The singular locus of the Hitchin fibration. Problem set II. Branes in the moduli space of Higgs bundles Branes through finite group actions. Branes through anti-holomorphic involutions. Problem set III. Higgs bundles and correspondences Group homomorphisms. Polygons and Hyperpolygons. Langlands duality. Problem set IV. Gauge theory and a few applications to knot theory Preamble: Some background material not included in the lectures A brief introduction to Morse homology The negative gradient flow and (un)stable manifolds Palais-Smale condition C Towards a compactification of the space of unparameterized trajectories The Morse Homology Orientations Homology with Local Coefficients Novikov-Morse Homology Principal Bundles, connections, Chern-Weil theory and the Chern-Simons invariant Principal bundles The big and little adjoint bundles Connections The action of gauge transformations on connections The Curvature The holonomy representation of flat connections Deformations of connections Classifications of U(n) and SU(n)-bundles on 3-manifolds Characteristic classes and the Chern-Simons invariant Floer Homology overview The Chern-Simons Functional Formal Gradients Hessians Novikov Rings? Exercises for the preamble Some examples of representation spaces and computations of the Chern-Simons functional of three manifolds and knots The first variation of Chern-Simons and its periods Examples of representation spaces and computations of the Chern-Simons invariant Seifert Fibered Spaces Representation spaces of knot groups The ASD Equation: Examples and Basic Properties The gradient of Chern-Simons and the ASD equation The linear case The basic instanton More instantons on S4 by conformal transformations Gauge Transformations Sobolev Completions Local Charts The Construction of Slices The local structure of the moduli space on a closed 4-manifold Uhlenbeck’s Fundamental Lemma The Curvature Map Is Proper The Uhlenbeck Compactness Theorem. Lecture 3: The Construction of Floer Homology The ASD Equation and Gauge Fixing on A Cylinder The Extended Hessian The Spectral Flow and Fredholm Index The Spectral Flow in the Instanton Floer Homology Compactification of the Moduli Space A Criterion to Avoid Reducible Flat Connections Instanton Floer Homology Floer Homology for links in three manifolds, exact triangles and Khovanov Homology Representation spaces of knot complements. Khovanov Homology Exact Triangles Some Non-Orientable Surfaces in 4-Manifolds Floer’s Exact Triangle The Second Step The Final Step A Spectral Sequence Lecture on Invertible Field Theories Introduction Lecture 1: Cobordisms Classical definitions Cobordism categories, first attempt Categories, groupoids, and spaces Invertible field theories (poor man’s version) Lecture 2: Topologically enriched (cobordism) categories Topologically enriched cobordism categories Manifold bundles and bundles of cobordisms Infinite dimensional ambient space Categories versus enriched categories The main theorem Lecture 3: More structure Symmetric monoidal structures Symmetric monoidal structure on the universal groupoid under C Little disks Structure on embedded cobordism categories Topological categories Conclusion Lecture 4: Cobordism classes and characteristic classes Stable homology of moduli spaces of surfaces Cohomology of morphism spaces Cobordisms with connectivity restrictions Cohomology of morphism spaces Exercises (by A. Debray, S. Galatius, M. Palmer) Exercise set 1 Exercise set 2 Exercise set 3 Solutions to selected exercises (by A. Debray, S. Galatius, M. Palmer) Solution to Exercise 5.2.2(b) Solution to Exercise 5.3.1(e) Topological Quantum Field Theories, Knots and BPS states Lecture I Motivation Dehn surgery and Kirby calculus Witten-Reshetikhin-Turaev invariant and Chern-Simons TQFT Lecture II 2d TQFTs 3d TQFTs Lecture III Asymptotic expansion conjecture Analytically continued Chern-Simons Lecture IV Problems in categorifing WRT invariant Abelian flat connections and linking pairing WRT invariant and q-series q-series invariant for plumbed 3-manifolds Exercises Lecture I Lecture II Lecture III Lecture IV Solutions Lecture I Lecture II Lecture III Lecture IV Lectures on BPS states and spectral networks Lecture 1: What is a BPS state? Quantum mechanics The superparticle Q-cohomology Richer examples Field theory Supersymmetric field theory Lectures 2-3: 2d theories, tt* geometry and Stokes phenomenon Landau-Ginzburg models BPS solitons Wall-crossing formula and spectral networks for 2d \N=(2,2) theories Chiral rings and vacua The topological connection Contour integrals Spectral network as jumping locus for covariantly constant sections Wall-crossing formula via the spectral network tt* geometry Lecture 4: 4d theories and spectral networks Class S and surface defects Chiral rings BPS solitons Spectral networks Adding punctures BPS indices in the \fsl2 case BPS particles in higher rank cases Families of flat connections Abelianization Presents 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.