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Holography, Localization, Black Holes Holography, localization, black holes Alberto Zaffaroni Milano-Bicocca Paris, June 2016 2nd Workshop on String Theory and Gender Alberto Zaffaroni (Milano-Bicocca) CFT june 2016 1 / 39 Introduction Introduction Lot of recent activity in the study of supersymmetric and superconformal theories in various dimensions in closely related contexts A deeply interconnected web of supersymmetric theories arising from branes, • most often strongly coupled, related by various types of dualities. Progresses in the evaluation of exact quantum observables. Related to • localization and the study of supersymmetry in curved space. Many results on indices and counting problems that can be also related to • BH physics. Alberto Zaffaroni (Milano-Bicocca) CFT june 2016 2 / 39 Introduction Introduction A large superconformal zoo (2,0) M5 Global symmetry enhancement 5d SCFT AGT correspondence N=4 SYM 4d SCFT N=2/N=1 non Lagrangian theories Several non conformal defs Non trivial dynamics ABJM N=4/N=2 mirror symmetry 3d SCFT 2d SCFT 1d SCFT Black Hole entropy Alberto Zaffaroni (Milano-Bicocca) CFT june 2016 3 / 39 Introduction Introduction A large superconformal zoo with holographic duals (2,0) M5 AdS7 Global symmetry enhancement 5d SCFT AdS6 AGT correspondence N=4 SYM 4d SCFT N=2/N=1 non Lagrangian theories AdS5 Several non conformal defs Non trivial dynamics ABJM N=4/N=2 mirror symmetry 3d SCFT AdS4 2d SCFT AdS3 1d SCFT Black Hole entropy AdS2 Alberto Zaffaroni (Milano-Bicocca) CFT june 2016 4 / 39 Introduction Introduction A large superconformal zoo with holographic duals (2,0) M5 AdS7 Global symmetry enhancement 5d SCFT AdS6 AGT correspondence N=4 SYM 4d SCFT N=2/N=1 non Lagrangian theories AdS5 Several non conformal defs Non trivial dynamics ABJM N=4/N=2 mirror symmetry 3d SCFT AdS4 2d SCFT AdS3 1d SCFT Black Hole entropy AdS2 Alberto Zaffaroni (Milano-Bicocca) CFT june 2016 5 / 39 Introduction Introduction 2 Supersymmetric black holes have horizon AdS2 S × CFT1 living at the horizon of black holes • density of states = BH entropy • Alberto Zaffaroni (Milano-Bicocca) CFT june 2016 6 / 39 Introduction Introduction Conformal algebra in 1d involves SL(2; R) H ; D; K with various superconformal extensions. AdS2/CFT1 still poorly understood. • Alberto Zaffaroni (Milano-Bicocca) CFT june 2016 7 / 39 Introduction Introduction Supersymmetric AdS black holes are also related by holography to states of CFT3 and CFT4 (2,0) M5 Global symmetry enhancement 5d SCFT AGT correspondence N=4 SYM 4d SCFT AdSN=2/N=15 rotating non Lagrangian BH theories Several non conformal defs Non trivial dynamics ABJM N=4/N=2 mirror symmetry 3d SCFT AdS4 static BH 2d SCFT 1d SCFT Black Hole entropy Alberto Zaffaroni (Milano-Bicocca) CFT june 2016 8 / 39 Introduction Introduction Entropy of AdS BH should be related to the counting of supersymmetric states in CFT: We still don't know where to look for AdS5: people tried with • superconformal index, 1/4-BPS partition functions, ... We have now an answer for AdS4: the index of the topologically twisted • theory Alberto Zaffaroni (Milano-Bicocca) CFT june 2016 9 / 39 In between a pletora of possibilities. TopologicalSCFT on Spheres Field Theories[Pestun 2007][Witten, 1988] A general method is to couple to supergravity and find solution of • [Festuccia-Seiberg] n n NJust= use2 theories conformal in 4d map with fromSU(2)R RtoR-symmetryS can be consistently defined • on any M4 euclidean manifoldδ (x) = + 0 µ rµ · · · ≡ The resulting theory is topological: independent of metric. • Backgrounds with twisted (conformal) Killing spinors have been classified in various dimensions and with various amount of supersymmetry. T = Q; µν f · · · g [klare,tomasiello,A.Z.;dumitruescu,festuccia,seiberg;Closser,Dumitrescu,Festuccia,Komargodski; ...] Euclidean SCFT Euclidean SCFT in finite volume To regularize IR physics we can put the theory on a Euclidean compact manifold. Supersymmetry restricts the manifold Alberto Zaffaroni (Milano-Bicocca) CFT june 2016 10 / 39 In between a pletora of possibilities. SCFT on Spheres [Pestun 2007] A general method is to couple to supergravity and find solution of • [Festuccia-Seiberg] Just use conformal map from Rn to S n • δ (x) = + 0 µ rµ · · · ≡ Backgrounds with twisted (conformal) Killing spinors have been classified in various dimensions and with various amount of supersymmetry. [klare,tomasiello,A.Z.;dumitruescu,festuccia,seiberg;Closser,Dumitrescu,Festuccia,Komargodski; ...] Euclidean SCFT Euclidean SCFT in finite volume To regularize IR physics we can put the theory on a Euclidean compact manifold. Supersymmetry restricts the manifold Topological Field Theories [Witten, 1988] N = 2 theories in 4d with SU(2)R R-symmetry can be consistently defined • on any M4 euclidean manifold The resulting theory is topological: independent of metric. • T = Q; µν f · · · g Alberto Zaffaroni (Milano-Bicocca) CFT june 2016 10 / 39 In between a pletora of possibilities. Topological Field Theories [Witten, 1988] A general method is to couple to supergravity and find solution of • [Festuccia-Seiberg] N = 2 theories in 4d with SU(2)R R-symmetry can be consistently defined • on any M4 euclidean manifoldδ (x) = + 0 µ rµ · · · ≡ The resulting theory is topological: independent of metric. • Backgrounds with twisted (conformal) Killing spinors have been classified in various dimensions and with various amount of supersymmetry. T = Q; µν f · · · g [klare,tomasiello,A.Z.;dumitruescu,festuccia,seiberg;Closser,Dumitrescu,Festuccia,Komargodski; ...] Euclidean SCFT Euclidean SCFT in finite volume To regularize IR physics we can put the theory on a Euclidean compact manifold. Supersymmetry restricts the manifold SCFT on Spheres [Pestun 2007] Just use conformal map from Rn to S n • Alberto Zaffaroni (Milano-Bicocca) CFT june 2016 10 / 39 TopologicalSCFT on Spheres Field Theories[Pestun 2007][Witten, 1988] n n NJust= use2 theories conformal in 4d map with fromSU(2)R RtoR-symmetryS can be consistently defined • on any M4 euclidean manifold The resulting theory is topological: independent of metric. • T = Q; µν f · · · g Euclidean SCFT Euclidean SCFT in finite volume To regularize IR physics we can put the theory on a Euclidean compact manifold. Supersymmetry restricts the manifold In between a pletora of possibilities. A general method is to couple to supergravity and find solution of • [Festuccia-Seiberg] δ (x) = + 0 µ rµ · · · ≡ Backgrounds with twisted (conformal) Killing spinors have been classified in various dimensions and with various amount of supersymmetry. [klare,tomasiello,A.Z.;dumitruescu,festuccia,seiberg;Closser,Dumitrescu,Festuccia,Komargodski; ...] Alberto Zaffaroni (Milano-Bicocca) CFT june 2016 10 / 39 R for theories with more supercharges and in different dimensions, Aa is • promoted to a non-abelian gauge field, there can be other backgrounds tensor fields and extra conditions. Euclidean SCFT Superconformal theories on curved spaces For SCFT, the variation of the gravitino can be written as a (twisted) conformal Killing equation R A 1 A ( a iA )+ + γa− = 0 = + = γa + r − a ) ra d r Alberto Zaffaroni (Milano-Bicocca) CFT june 2016 11 / 39 Euclidean SCFT Superconformal theories on curved spaces For SCFT, the variation of the gravitino can be written as a (twisted) conformal Killing equation R A 1 A ( a iA )+ + γa− = 0 = + = γa + r − a ) ra d r R for theories with more supercharges and in different dimensions, Aa is • promoted to a non-abelian gauge field, there can be other backgrounds tensor fields and extra conditions. Alberto Zaffaroni (Milano-Bicocca) CFT june 2016 11 / 39 Euclidean SCFT Superconformal theories on curved spaces A possible solution is to have covariantly constant spinors A 1 A A + = γa + solved by + = 0 ra d r ra Topological Field Theories can be realized this way by canceling the spin connection with a background R-symmetry A 1 αβ αβ R + = @a+ + ! Γ + + A + = @a+ ra 4 a a which can be solved by + = constant on any manifold Alberto Zaffaroni (Milano-Bicocca) CFT june 2016 12 / 39 Euclidean SCFT Superconformal theories on curved spaces The well know examples of Topological Field Theories back in the eighties A 1 αβ αβ R + = @a+ + ! Γ + + A + = @a+ ra 4 a a N = 2 theories on any M4: SU(2)R background gauge field • + transform in the (2; 0) representation of the local Euclidean group SO(4) = SU(2) × SU(2) A twist in 2d on any Riemann surface Σg : U(1)R background gauge field • Alberto Zaffaroni (Milano-Bicocca) CFT june 2016 13 / 39 Euclidean SCFT Superconformal theories on curved spaces A possible solution is to have Killing spinors A 1 A + = γa + solved by a+ = γa+ ra d r r Realized for Theories on Spheres, with zero background field AR = 0. Alberto Zaffaroni (Milano-Bicocca) CFT june 2016 14 / 39 The path integral on S d S 1 can be written as a trace over a Hilbert space × P F −βfQ;Sg+ ∆aJa Z d 1 = Tr( 1) e S ×S − This is the superconformal index: it counts BPS states graded by dimension and charges [Romelsberger; Kinney,Maldacena,Minwalla,Rahu] computed for large class of 4d SCFT [Gadde,Rastelli,Razamat,Yan] Euclidean SCFT Superconformal theories on curved spaces Or combinations: Killing spinors on S d and R- and flavor holonomies on S 1 A 1 A + = γa + solved by a+ = γa+ ; At = i ra d r r Alberto Zaffaroni (Milano-Bicocca) CFT june 2016 15 / 39 Euclidean SCFT Superconformal theories on curved spaces Or combinations: Killing spinors on S d and R- and flavor holonomies on S 1 A 1 A + = γa + solved by a+ = γa+ ; At = i ra d r r The path integral on S d S 1 can be written as a trace over a Hilbert space × P F −βfQ;Sg+ ∆aJa Z d 1 = Tr( 1) e S ×S − This is the superconformal index: it counts BPS states graded by dimension and charges [Romelsberger; Kinney,Maldacena,Minwalla,Rahu] computed for large class of 4d SCFT [Gadde,Rastelli,Razamat,Yan] Alberto Zaffaroni (Milano-Bicocca) CFT june 2016 15 / 39 1 The path integral on Σg S can be written as a Witten index (elliptic genus) × F F iJF A −βH Z 1 = TrH ( 1) e e Σg ×S − 2 F Q = H − σ JF holomorphic in AF + iσF of the dimensional reduced theory on Σg and counts ground states graded by charges.
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