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Localization & Exact Holography

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Localization & Exact Holography Motivation Strategy Supergravity String Theory Conclusions Localization & Exact Holography Atish Dabholkar Centre National de la Recherche Scientifique Universit´ePierre et Marie Curie Paris VI Great Lakes Conference 29 April 2011 Atish Dabholkar (Paris) Localization & Exact Holography April 2011, Chicago 1 / 40 Motivation Strategy Supergravity String Theory Conclusions 1 Motivation Quantum Entropy AdS2=CFT1 Holography 2 Strategy Microscopic Counting Localization 3 Supergravity Off-shell supergravity for Localization Localizing Instanton and its Renormalized action 4 String Theory Reducing the problem to supergravity Putting it all together 5 Conclusions Summary and Outlook Atish Dabholkar (Paris) Localization & Exact Holography April 2011, Chicago 2 / 40 Motivation Strategy Supergravity String Theory Conclusions Based on A.D. Jo~aoGomes, Sameer Murthy, \Quantum Black Holes, Localization, and the Topological String," arXiv:1012.0265 A.D. Jo~aoGomes, Sameer Murthy, \Localization and Exact Holography," arXiv:1105.nnnn A.D. Jo~aoGomes, Sameer Murthy, Ashoke Sen; \Supersymmetric Index from Black Hole Entropy," arXiv:1009.3226 Atish Dabholkar (Paris) Localization & Exact Holography April 2011, Chicago 3 / 40 Motivation Strategy Supergravity String Theory Conclusions Quantum Entropy Two Related Motivations Entropy of black holes remains one of the most important and precise clues about the microstructure of quantum gravity. Can we compute exact quantum entropy of black holes including all corrections both microscopically and macroscopically? Holography has emerged as one of the central concepts regarding the degrees of freedom of quantum gravity. Can we find simple example of AdS=CFT holgraphy where we might be able to `prove' it exactly? Atish Dabholkar (Paris) Localization & Exact Holography April 2011, Chicago 4 / 40 Motivation Strategy Supergravity String Theory Conclusions Quantum Entropy Two Related Motivations Entropy of black holes remains one of the most important and precise clues about the microstructure of quantum gravity. Can we compute exact quantum entropy of black holes including all corrections both microscopically and macroscopically? Holography has emerged as one of the central concepts regarding the degrees of freedom of quantum gravity. Can we find simple example of AdS=CFT holgraphy where we might be able to `prove' it exactly? Atish Dabholkar (Paris) Localization & Exact Holography April 2011, Chicago 4 / 40 Motivation Strategy Supergravity String Theory Conclusions Quantum Entropy Two Related Motivations Entropy of black holes remains one of the most important and precise clues about the microstructure of quantum gravity. Can we compute exact quantum entropy of black holes including all corrections both microscopically and macroscopically? Holography has emerged as one of the central concepts regarding the degrees of freedom of quantum gravity. Can we find simple example of AdS=CFT holgraphy where we might be able to `prove' it exactly? Atish Dabholkar (Paris) Localization & Exact Holography April 2011, Chicago 4 / 40 Motivation Strategy Supergravity String Theory Conclusions Quantum Entropy Two Related Motivations Entropy of black holes remains one of the most important and precise clues about the microstructure of quantum gravity. Can we compute exact quantum entropy of black holes including all corrections both microscopically and macroscopically? Holography has emerged as one of the central concepts regarding the degrees of freedom of quantum gravity. Can we find simple example of AdS=CFT holgraphy where we might be able to `prove' it exactly? Atish Dabholkar (Paris) Localization & Exact Holography April 2011, Chicago 4 / 40 Motivation Strategy Supergravity String Theory Conclusions Quantum Entropy Black Hole Entropy Bekenstein [72]; Hawking[75] For a BPS black hole with charge vector (q; p), for large charges, the leading Bekenestein- Hawking entropy precisely matches the logarithm of the degeneracy of the corresponding quantum microstates A(q; p) = log(d(q; p)) + O(1=Q) 4 Strominger & Vafa [96] This beautiful approximate agreement raises two important questions: What exact formula is this an approximation to? Can we systematically compute corrections to both sides of this formula, perturbatively and nonperturbatively in 1/Q and may be even exactly for arbitrary finite values of the charges? FiniteAtish size Dabholkar corrections (Paris) to theLocalization Bekenstein-Hawking & Exact Holography entropy.April 2011, Chicago 5 / 40 Motivation Strategy Supergravity String Theory Conclusions Quantum Entropy Black Hole Entropy Bekenstein [72]; Hawking[75] For a BPS black hole with charge vector (q; p), for large charges, the leading Bekenestein- Hawking entropy precisely matches the logarithm of the degeneracy of the corresponding quantum microstates A(q; p) = log(d(q; p)) + O(1=Q) 4 Strominger & Vafa [96] This beautiful approximate agreement raises two important questions: What exact formula is this an approximation to? Can we systematically compute corrections to both sides of this formula, perturbatively and nonperturbatively in 1/Q and may be even exactly for arbitrary finite values of the charges? FiniteAtish size Dabholkar corrections (Paris) to theLocalization Bekenstein-Hawking & Exact Holography entropy.April 2011, Chicago 5 / 40 Motivation Strategy Supergravity String Theory Conclusions AdS2=CFT1 Holography Quantum Entropy and AdS2=CFT1 Sen [08] 2 Near horizon geometry of a BPS black hole is AdS2 × S . Quantum entropy can be defined as as partition function of the AdS2. Z 2π finite i W (q; p) = exp − i qi A dθ : 0 AdS2 Functional integral over all string fields. The Wilson line insertion is necessary so that classical variational problem is well defined. The black hole is made up of a complicated brane configuration. The worldvolume theory typically has a gap. Focusing on low energy states gives CFT1 with a degenerate, finite dimensional Hilbert space. Partition function d(q; p) is simply the dimension of this Hilbert space. Atish Dabholkar (Paris) Localization & Exact Holography April 2011, Chicago 6 / 40 Motivation Strategy Supergravity String Theory Conclusions AdS2=CFT1 Holography Functional Integral Boundary Conditions For a theory with some vector fields Ai and scalar fields φa, we have the fall-off conditions dr 2 ds2 = v r 2 + O(1) dθ2 + ; 0 r 2 + O(1) φa = ua + O(1=r) ; Ai = −i ei (r − O(1))dθ ; (1) Magnetic charges are fluxes on the S2. Constants v; ei ; ua fixed to i a attractor values v∗; e∗; u∗ determined purely in terms of the charges, and set the boundary condition for the functional integral. Quantum entropy is purely a function of the charges (q; p). The functional integral is infrared divergent due to infinite volume of the AdS2. Holographic renormalization to define the finite part. Atish Dabholkar (Paris) Localization & Exact Holography April 2011, Chicago 7 / 40 Motivation Strategy Supergravity String Theory Conclusions AdS2=CFT1 Holography Renormalized functional integral Put a cutoff at a large r = r0. Lagrangian Lbulk is a local functional, hence the action has the form −1 Sbulk = C0r0 + C1 + O(r0 ) ; with C0; C1 independent of r0. C0 can be removed by a boundary counter-term (boundary cosmological constant). C1 is field dependent to be integrated over. Quantum Entropy gives a proper generalization of Wald entropy to include not only higher-derivative local terms but also the effect of integrating over massless fields. This is essential for duality invariance and for a systematic comparison since nonlocal effects can contribute to same order. Atish Dabholkar (Paris) Localization & Exact Holography April 2011, Chicago 8 / 40 Motivation Strategy Supergravity String Theory Conclusions AdS2=CFT1 Holography Renormalized functional integral Put a cutoff at a large r = r0. Lagrangian Lbulk is a local functional, hence the action has the form −1 Sbulk = C0r0 + C1 + O(r0 ) ; with C0; C1 independent of r0. C0 can be removed by a boundary counter-term (boundary cosmological constant). C1 is field dependent to be integrated over. Quantum Entropy gives a proper generalization of Wald entropy to include not only higher-derivative local terms but also the effect of integrating over massless fields. This is essential for duality invariance and for a systematic comparison since nonlocal effects can contribute to same order. Atish Dabholkar (Paris) Localization & Exact Holography April 2011, Chicago 8 / 40 Motivation Strategy Supergravity String Theory Conclusions AdS2=CFT1 Holography Renormalized functional integral Put a cutoff at a large r = r0. Lagrangian Lbulk is a local functional, hence the action has the form −1 Sbulk = C0r0 + C1 + O(r0 ) ; with C0; C1 independent of r0. C0 can be removed by a boundary counter-term (boundary cosmological constant). C1 is field dependent to be integrated over. Quantum Entropy gives a proper generalization of Wald entropy to include not only higher-derivative local terms but also the effect of integrating over massless fields. This is essential for duality invariance and for a systematic comparison since nonlocal effects can contribute to same order. Atish Dabholkar (Paris) Localization & Exact Holography April 2011, Chicago 8 / 40 Motivation Strategy Supergravity String Theory Conclusions AdS2=CFT1 Holography Renormalized functional integral Put a cutoff at a large r = r0. Lagrangian Lbulk is a local functional, hence the action has the form −1 Sbulk = C0r0 + C1 + O(r0 ) ; with C0; C1 independent of r0. C0 can be removed by a boundary counter-term (boundary cosmological constant).
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