Quantum Field Theory and Manifold Invariants

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^{?} 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 ?_{?}
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^{?} colored polynomials
The yoga of WRT
Webs
The MOY bracket
The web category
The Λ^{?} 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 ?(?) and ??(?)-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 ?⁴ 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 ?
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 ?-series
?-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
?-cohomology
Richer examples
Field theory
Supersymmetric field theory
Lectures 2-3: 2d theories, ??* 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
??* geometry
Lecture 4: 4d theories and spectral networks
Class ? 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