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Holographic Renormalization and the Holographic Cotton Tensor

Recent Advances in Topological Quantum Field Theory University of Lisbon, September 14, 2012 Sebastian de Haro (ITFA and AUC, University of Amsterdam). Holographic Renormalization and the Holographic Cotton Tensor. Outline. AdS /CFT: generalities and holographic renormalization

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Holographic Renormalization and the Holographic Cotton Tensor

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  1. Recent Advances in Topological Quantum Field Theory University of Lisbon, September 14, 2012 Sebastian de Haro (ITFA and AUC, University of Amsterdam) Holographic Renormalization and the HolographicCotton Tensor

  2. Outline • AdS/CFT: generalitiesandholographicrenormalization • Boundarygraviton: dualityand CFT coupledtogravity • Instantonsolutions Holographic Cotton Tensor. SdH, UvA

  3. AdS/CFT • Holographic duality that relates: • Gravity (string theory, M-theory) in -dimensional AdS space to: • A CFT on the (conformal) boundary of this space (-dimensional). • The duality works for both Euclidean and Lorentzian signatures. • Euclidean case much better understood. • Interested in instanton solutions. Holographic Cotton Tensor. SdH, UvA

  4. EuclideanAdS • Quadric in : • One can choose coordinates: • There is a boundary at . • We will use local coordinates (half space ): • Induces (conformally) flat metric on the boundary. • The CFT lives on this conformal boundary. Holographic Cotton Tensor. SdH, UvA

  5. Bulk Hilbert Space • Setting up QFT in AdS space, fields may contain a classical and a quantum part: • Both satisfy equation of motion in the bulk. • is part of Hilbert space normalizable, • Holographically: vev of operator • need not be: background • Holographically: source Holographic Cotton Tensor. SdH, UvA

  6. AdS/CFT • If is independent of , it is the coupling constant forthatoperator. Holographic Cotton Tensor. SdH, UvA

  7. AdS/CFT • Semi-classical approximation in the bulk: • Boundary theory is strongly coupled in that case. Typically: large and large (for ): strong ‘t Hooft coupling. • Expectation values of operators: • bulk solution for (normalizable mode ) Holographic Cotton Tensor. SdH, UvA

  8. Example: 2-point function • , conformallycoupledscalar field in bulk: • is boundarycondition at . • Regularity at imposes: Holographic Cotton Tensor. SdH, UvA

  9. Holographic Renormalization • When computing correlation functions from bulk, we encounter divergences as . • Need formalism for generic boundary metric not just flat. • Allows computation correlation functions of stress-energy tensor. • Allows computation CFT in any background. • Take into account back-reaction. • Holographic renormalization systematic method to do this. Holographic Cotton Tensor. SdH, UvA

  10. Holographic Renormalization • Bulk metric: • SolveEinstein’sequationsperturbativelyin for given boundary values of metric: Holographic Cotton Tensor. SdH, UvA

  11. Holographic Renormalization • Solve Einstein’sequationsperturbativelyin : • undetermined (= b.c.) • Higher ’s: Holographic Cotton Tensor. SdH, UvA

  12. Holographic Renormalization • Holographic recipe: • Regularize the bulk action at • Addcountertermsthat do notmodifyeom • Send, obtain finite result. = inducedmetric Holographic Cotton Tensor. SdH, UvA

  13. Holographic Renormalization • To compute, need to solve eom all the way to interior. • Only need its variation • Compute the regularized action with counterterm subtraction, varyand take limit. • It is enoughtoknow the divergent terms. Holographic Cotton Tensor. SdH, UvA

  14. Holographic Renormalization • The expectationvalue: • Result: Holographic Cotton Tensor. SdH, UvA

  15. The BoundaryGraviton • Goal: understandholography of gravitonandwhether CFT canbecoupledtodynamicalgravity. What does thisgive in bulk? • Standard normalizability analysis: • normalizable, depends on a choice of solution: vev of stress-energy. • is non-normalizable: b.c., corresponds to source on boundary. Or is it? • Ishibashi-Wald (2004): both modes are normalizable. Holographic Cotton Tensor. SdH, UvA

  16. The BoundaryGraviton • This means thateither or can be interpreted as boundary gravitons. • Correspond to different CFT’s. • Since both modes can be normalized in the bulk, they need not be a priori fixed. • We may set up dynamical equation that selects certain solutions. • May couple CFT to gravity. Holographic Cotton Tensor. SdH, UvA

  17. Instantons • Investigate the dynamics of boundarygraviton in simple case: self-dualWeyl. • Boundary graviton does not solve full Einstein equations, but more restrictive one. • Physically, self-dual solutions are ‘gravitational instantons’ that signal instability of bulk under deformation of b.c. Decay towards new vacuum. Holographic Cotton Tensor. SdH, UvA

  18. Self-Dual Solutions • In spaces without cosmological constant, naturalcondition is self-duality of Riemann. • It automaticallyimplies. • If cosmological constant non-zero, need to choose a different condition. Self-duality of Weyl tensor is compatible with cosmological constant and Euclidean signature: Holographic Cotton Tensor. SdH, UvA

  19. VanishingWeyl Tensor • Simplest case: vanishingWeyl tensor. • Bulk metric is conformally flat. • On-shell, the Weyl tensor reducesto: • TogetherwithEinstein’sequations: Holographic Cotton Tensor. SdH, UvA

  20. VanishingWeyl Tensor • Equation implies: • Series terminates at order : is last non-vanishingcoefficient. canbeintegrated: • Equationimpliesthat the Cotton tensor of vanishes: • The boundarymetric is conformally flat. • Bulk metricconformally flat iffboundarymetricis conformally flat. Holographic Cotton Tensor. SdH, UvA

  21. Self-Dual Solutions • Next case: non-zero, self-dualWeyl tensor • Solvecoupledequationsasymptotically: Holographic Cotton Tensor. SdH, UvA

  22. Self-Dual Solutions Bulk Einstein • Result: • Combine withholographicinterpretation of as 1-point function of stress-energy: • Integrate stress-tensor to obtain boundary generating functional: Holographic Cotton Tensor. SdH, UvA

  23. Self-Dual Solutions • This canbeintegratedtoyield the boundarygeneratingfunction: • We get: Holographic Cotton Tensor. SdH, UvA

  24. Self-Dual Solutions • At next level , we get a compatibility condition for the curvature: Also: • Non-linear gravity theory in 3d. Needs to be studied further. Holographic Cotton Tensor. SdH, UvA

  25. Summary • Holographic dictionary: • boundary graviton • Stress-energy tensor: • Both modes are normalizable (linearized fluctuations). • is related to boundary graviton of CFT2 via Cotton tensor. • Solutions with zero bulk Weyl tensor have zero boundary Cotton tensor. • Bulk solution with BTZ black hole on boundary. Holographic Cotton Tensor. SdH, UvA

  26. Summary • Solutions with self-dual Weyl tensor: • Generating functional can be integrated and gives the gravitational Chern-Simons action. • At next level, we get a non-linear consistency condition for the curvature. Holographic Cotton Tensor. SdH, UvA

  27. References • Maldacena (1997) • Witten (1998) • Skenderis, Solodukhin (2000) • SdH, Skenderis, Solodukhin (2000) • SdH, Petkou (2008, 2012) Holographic Cotton Tensor. SdH, UvA

  28. Thankyou! Holographic Cotton Tensor. SdH, UvA

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