Holographic Weyl Anomaly

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Table of contents

1 Introduction 
2 Fundamentals of Holography and its Applications 
2.1 Basic Ingredients of Holographic Duality
2.1.1 A Glimpse of the Anti-de Sitter Space
2.1.2 CFT Correlation functions via the Duality
2.2 Holography and Two-Dimensional Defect CFT
2.2.1 Holographic Weyl Anomaly
2.2.2 2D Defect CFT and Energy Transport Coefficients
2.3 Quantum Complexity and the Holographic Conjectures
2.3.1 The Two Holographic Conjectures
2.3.2 Field Theory Approach for Complexity: Nielsen Method
2.4 Modular Flow and Bulk Reconstruction
2.4.1 Entanglement Entropy and Modular Hamiltonian
2.4.2 Modular Hamiltonians in Holography: JLMS Relation
2.4.3 Code Subspace and Entanglement Wedge Reconstruction
3 Holographic complexity: “CA” or “CV” ? 
3.1 Warm up with the Defect Toy Model
3.1.1 Two-Dimensional Branes in AdS3
3.1.2 Fefferman-Graham Expansion and the Cutoff Surface
3.1.3 Wheeler-DeWitt Patch in Defect AdS3
3.2 Holographic Complexity with a Defect
3.2.1 CV Conjecture
3.2.2 CA Conjecture
3.3 Holographic Complexity for Subregions
3.3.1 Subregion CV Conjecture
3.3.2 Subregion CA Conjecture
3.4 Complexity in QFT
3.5 Discussion
4 A Careful Consideration of Holographic 2D dCFT 
4.1 Holographic Scattering States and Matching
4.2 Summary and Outlook
5 Revisiting Circuit Complexity in 2d Bosonisation 
5.1 2D Bosonisation
5.1.1 Basic Ingredients
5.1.2 Fermionic Fock space
5.1.3 Correspondence between states
5.2 Fubini-Study Method for Bosonic Coherent States
5.3 Application of Nielsen Method on Bosonic Coherent States
5.3.1 Complexity between Bosonic Ground States
5.3.2 Bosonic Coherent States with One Excited Mode
5.3.3 Complexity for Bosonic Coherent States with Shifts in More Modes
5.4 A class of Fermionic and Bosonic Bi-Gaussian States
5.4.1 A Bosonisation Identity
5.4.2 An Example with One Mode
5.4.3 An Example with Two Modes
5.5 Conclusion
6 Berry Curvature as A Probe of Bulk Curvature 
6.1 Entanglement as a Connection
6.2 Modular Berry Connection
6.2.1 A toy example
6.2.2 Gauging the modular zero modes
6.2.3 Comment on two-sided modular Hamiltonians
6.2.4 Modular Berry holonomy examples
6.3 Entanglement Wedge Connection
6.3.1 Modular zero modes in the bulk
6.3.2 Relative edge-mode frame as a connection
6.3.3 Bulk modular curvature and parallel transport
6.3.4 Example: Pure AdS3
6.4 The Proposal and Implications
7 Conclusions