Geometry and Topology of Low Dimensional Systems : Chern-Simons Theory with Applications

Preface
Acknowledgements
Contents
List of Figures
Part I
	1 Introduction
		1.1 Topology, Maps, Classes, Homotopic Groups
			1.1.1 Topological Space
			1.1.2 Topological Invariants
				Dimension, Compactness, Connectedness as Invariants
		1.2 Homotopy Groups
			1.2.1 The Fundamental Group
			1.2.2 Quotient Space
		1.3 Higher Homotopy Groups
		Appendix
		Exercises
		References
	2 Differentiable Manifolds and Geometry
		2.1 Euclidean Geometry, Metric Spaces, Smooth Manifolds
		2.2 Differentiable Manifolds
		2.3 Differential Calculus on Manifolds
			2.3.1 Closed and Exact Forms
		2.4 Homology
		2.5 Simplicial Complex and the Fundamental Group
		2.6 Cohomology
			2.6.1 Integration on a Manifold with Boundary
		Exercises
		References
	3 Riemannian and Pseudo Riemannian Geometry
		3.1 Riemannian Geometry
			3.1.1 Inner Product and the Metric
			3.1.2 Parallel Transport and Connection
		3.2 Riemann Tensor
			3.2.1 Geodesics
		3.3 Laplace Beltrami Operator
		3.4 Self Adjoint Operators
			3.4.1 Self-Adjoint Laplace Beltrami Operator on a Manifold with Boundary
		3.5 Pseudo Riemannian Geometry
			3.5.1 Einstein Hilbert Action
			3.5.2 Rindler Spacetime
		Exercises
		References
Part II
	4 Topological Understanding of Defects in Crystalline Structure
		4.1 Point Defects, Line Defects, and More
			4.1.1 Defects and Topology
			4.1.2 Group Theory and Order Parameter Space
				Brief Remarks About Continuous Groups
				Groups and Order Parameter Spaces
				Examples
				Homotopy Groups of Lie Groups and Coset Spaces
		Appendix: Exact Sequences
		Exercises
		References
	5 Configuration Space Topology and Topological Conservation Laws
		5.1 Paths and Path Connectedness, Homotopy
			5.1.1 Quantization Ambiguity and Fundamental Group
		5.2 Quantum Mechanical Systems on Different Topological Spaces
			5.2.1 A Particle Moving on a Circle S1
			5.2.2 Particle Confined Within the Interval [-1,1]
			5.2.3 Particle on R1+
			5.2.4 Particle Moving in the Space R2- D2
			5.2.5 Rigid Rotor
		5.3 System of N Identical Particles and Origin of Statistics
			5.3.1 Anyons in d=2 Dimensions
		5.4 Field Theory on Topological Spaces
			5.4.1 Kink Soliton
			5.4.2 O(3) Nonlinear σ Model
		Exercises
		References
	6 Spin-Statistics Theorem, Low Dimensional Topologyand Geometry
		6.1 History
			6.1.1 Assumptions
		6.2 Anyons and Fractional Spin
			6.2.1 O(3) Nonlinear Sigma Model
			6.2.2 Interpretation of Hopf Invariant
		6.3 Low Dimensional Topology and Geometry
			6.3.1 Genus g Riemann Spaces
				Arbitrary Genus Surfaces
		6.4 Three-Dimensional Geometry and Topology
			6.4.1 Three Manifolds
		6.5 Chern-Simons Gauge Theory on S3
		Exercises
		References
Part III
	7 Braid Group, Knots, Three Manifolds
		7.1 Braids, Braid Group
			7.1.1 Knots
			7.1.2 Some Examples
			7.1.3 Knot Invariants
		7.2 Chern Simons Theory and Knot/Link Invariants
			7.2.1 U(1) Chern-Simons Theory and Linking Invariants
				Self-linking Invariant
			7.2.2 SU(2) Chern Simons Theory on S3
			7.2.3 Canonical Quantisation
			7.2.4 Brief Introduction to the SU(2)k Kac-Moody Algebra
			7.2.5 Hilbert Space for Flat Connections
				Hilbert Space with ∂M = S2
				Partition Function Z(S3)
		7.3 Jones Polynomial from CS Theory
			7.3.1 Wess-Zumino Witten Models, Quantum Groups, Vertex Models
			7.3.2 Vertex Models and Knot Invariants
			7.3.3 Knot Invariant from Vertex Model
			7.3.4 Composite Braids and Murakami Invariants
		Exercises
		References
	8 Three-Manifold Invariants
		8.1 Surgery Procedure
		8.2 Kirby Moves
		8.3 Three-Manifold Invariants
		8.4 Conclusion
		Exercises
		References
	9 3D Gravity and BTZ Blackhole
		9.1 3D Gravity
			9.1.1 Killing Symmetries of AdS3
		9.2 Chern Simons Formulation of 3D Gravity
			9.2.1 BTZ Blackhole
			9.2.2 BTZ Blackhole and Statistical Mechanics
			9.2.3 dS3 Gravity and Chern Simons Formulation
			9.2.4 Euclidean de Sitter Space in 3D
			9.2.5 Logarithmic Correction to the Entropy
			9.2.6 Logarithmic Corrections and AdS/CFT
		Exercises
		References