The Topological Compact Model

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Table of contents

Introduction 
1 Basics 
1.1 Chiral Ring
1.2 Topological Conformal Field Theories
1.3 Summary
2 Twisted Non-Compact Models 
2.1 The Topological Conformal Field Theory
2.1.1 The Topological Compact Model
2.1.2 The Topological Non-Compact Model
2.1.3 Small Deformations
2.2 The Topological Massive Model
2.2.1 The Compact Topological Field Theory
2.2.2 The Non-Compact Topological Field Theory
2.3 The Conformal Metric
2.4 The Topological-Anti-Topological Massive Metric
2.4.1 The tt Equation
2.4.2 A Family of Theories
3 AdS3/Liouville Duality 
3.1 Conformal Field Theory and Metrics
3.1.1 The Energy-momentum Tensor and the Metric
3.1.2 Energy-momentum Tensor Expectation Value
3.1.3 Energy-momentum Tensor on the Cylinder
3.2 Universal Metrics
3.2.1 The BTZ Black Hole Metric
3.2.2 Properties of the Metrics
3.2.3 On Monodromies
3.3 Liouville Quantum Gravity
3.3.1 Properties of Liouville Theory
3.3.2 The Holographic Interpretation
3.3.3 Black Hole Correlators
3.3.4 One Loop Correction
3.4 Supersymmetric AdS3/Liouville
3.4.1 Bulk and Boundary Actions
3.4.2 The Exact Solutions
3.4.3 Supersymmetric Liouville Quantum Gravity
3.5 A Twisted Holography
3.5.1 The Twisted Supergravity Theory
3.5.2 The Topological Conformal Field Theory on the Boundary
3.5.3 The Gravitational Chiral Primaries
3.5.4 The Bulk Chiral Ring
4 Twisted String Theory in Anti-de Sitter Space 
4.1 The Generating Function
4.2 The Three-Dimensional Stringy Geometries
4.3 The twisted background
4.3.1 Degrees of Freedom
4.3.2 Flat Generalizations
4.3.3 The Explicit Boundary Gauge Field Dependence
4.3.4 The Explicit Boundary Metric Dependence
4.3.5 The Asymptotic Twisted Generalization
4.4 The Space-Time Energy-Momentum Operator
4.4.1 The Energy-Momentum
4.4.2 The R-Symmetry Current
4.4.3 The Topological Energy-Momentum Tensor
5 Topological Orbifold Theory 
5.1 The Complex Plane
5.1.1 The Hilbert Scheme of Points on the Complex Plane
5.1.2 The Topological Conformal Field Theory
5.1.3 A Plethora of Results on the Cohomology Ring
5.1.4 The Interaction
5.1.5 The Structure Constants are Hurwitz Numbers
5.1.6 The Broader Context and a Proof of a Conjecture
5.1.7 Summary and Lessons
5.2 Compact Surfaces
5.2.1 The Orbifold Frobenius Algebra
5.2.2 Cup Products of Single Cycle Elements
5.2.3 Cup Products at Low and High Order
5.2.4 Genus One and an Overlap of Three
5.2.5 Further Examples
5.2.6 Remarks
Conclusion