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