Microstate counting for AdS black holes Alberto Zaffaroni - PowerPoint PPT Presentation

Microstate counting for AdS black holes Alberto Zaffaroni Milano-Bicocca PRIN Kick-off Meeting Pisa, October 2019 Alberto Zaffaroni (Milano-Bicocca) HoloBH 2019 1 / 25

Introduction Introduction A major achievement of string theory is the counting of micro-states for a class of asymptotically flat black holes [Vafa-Strominger’96] ◮ The entropy is obtained by counting states in the corresponding string/D-brane system ◮ Remarkable precision tests including higher derivatives No similar results for asymptotically AdS 4 or AdS 5 black holes until very recently. Alberto Zaffaroni (Milano-Bicocca) HoloBH 2019 2 / 25

Introduction Introduction Recent progress • initiated with static magnetically charged black holes in AdS 4 × S 7 [Benini-Hristov-AZ, 2015] • continued for electrically charged and rotating black holes in AdS 5 × S 5 with results in various overlapping limits [Choi,Kim,Kim,Naamgoong, 2018] [Cabo-Bizet,Cassani,Martelli,Murthy, 2018] [Benini-Milan, 2018] These results have been obtained through localisation and have been extended to other compactifications and dimensions. Alberto Zaffaroni (Milano-Bicocca) HoloBH 2019 3 / 25

Framework Field Theory Perspective In field theory we compute a grancanonical partition function � ( − 1) F e i ( Q I ∆ I + J i ω I ) e − β H � � e S ( q , j ) e i ( q I ∆ I + j i ω i ) Z (∆ I , ω i ) = Tr = q , j topologically twisted or superconformal index The entropy S ( q , j ) of a black hole with charge q and angular momentum j in a saddle point approximation is a Legendre Transform d ∆ = d I d I S BH ( q , j ) ≡ I (∆ , ω ) = log Z (∆ I , ω i ) − i ( q I ∆ I + j i ω i ) d ω = 0 sometimes referred as I -extremization for magnetically charged black holes Alberto Zaffaroni (Milano-Bicocca) HoloBH 2019 4 / 25

Framework Dual Field Theory Perspective The partition function is exactly computable only in the supersymmetric case � ( − 1) F e i ( Q I ∆ I + J i ω I ) e − β H p � • Z susy S d − 2 × S 1 (∆ I , ω i ) = Tr • cancellation between massive boson and fermions (Witten index) • sum over supersymmetric ground states H p = 0; What’s about ( − 1) F ? we assume no cancellation between bosonic and fermionic ground states. Seems to be true in the limit of large charges. Alberto Zaffaroni (Milano-Bicocca) HoloBH 2019 5 / 25

Framework Localization Exact quantities in supersymmetric theories with a charge Q 2 = 0 can be obtained by a saddle point approximation � � S | class × det fermions e − S = e − S + t { Q , V } = t ≫ 1 e − ¯ Z = det bosons � { Q , V } e − S + t { Q , V } = 0 ∂ t Z = Very old idea that has become very concrete recently, with the computation of partition functions on spheres and other manifolds supporting supersymmetry. Alberto Zaffaroni (Milano-Bicocca) HoloBH 2019 6 / 25

Framework Localization Localization ideas apply to path integral of Euclidean supersymmetric theories • Compact space provides IR cut-off, making path integral well defined • Localization reduces it to a finite dimensional integral, a matrix model � i < j sinh 2 u i − u j sinh 2 v i − v j N 1 N 2 � � u 2 i − � v 2 � � 4 π ( ik j ) 2 2 du i dv j e � i < j cosh 2 u i − v j i =1 j =1 2 ABJM, 3d Chern-Simon theories, [Kapustin,Willet,Yakoov;Drukker,Marino,Putrov] Alberto Zaffaroni (Milano-Bicocca) HoloBH 2019 7 / 25

Framework Localization Carried out recently in many cases • many papers on topological theories • S 2 , T 2 • S 3 , S 3 / Z k , S 2 × S 1 , Seifert manifolds • S 4 , S 4 / Z k , S 3 × S 1 , ellipsoids • S 5 , S 4 × S 1 , Sasaki-Einstein manifolds with addition of boundaries, codimension-2 operators, . . . Pestun 07; Kapustin,Willet,Yakoov; Kim; Jafferis; Hama,Hosomichi,Lee, too many to count them all · · · Alberto Zaffaroni (Milano-Bicocca) HoloBH 2019 8 / 25

Framework Localization In all cases, it reduces to a finite-dimensional matrix model on gauge variables, possibly summed over different topological sectors � � Z M ( y ) = dx Z int ( x , y ; m ) C m with different integrands and integration contours. When backgrounds for flavor symmetries are introduced, Z M ( y ) becomes an interesting and complicated function of y which can be used to test dualities • Sphere partition function, Kapustin-Willet-Yakoov; · · · • Superconformal index, Spironov-Vartanov; Gadde,Rastelli,Razamat,Yan; · · · • Topologically twisted index, Benini,AZ; Closset-Kim; · · · Alberto Zaffaroni (Milano-Bicocca) HoloBH 2019 9 / 25

Framework Entropy functional In gravity we typically define an entropy functional [Ferrara-Kallosh-Strominger 97; OSV 04; Sen 05] I ( X I , Ω i ) = E ( X I , Ω i ) − i ( q I X I + j i Ω i ) depending on the gravity scalar fields and other modes whose extremization realises the attractor mechanism : � S BH ( q , j ) ≡ I ( ¯ X I , ¯ � Ω i ) � crit X I , ¯ ¯ Ω i ≡ horizon value Alberto Zaffaroni (Milano-Bicocca) HoloBH 2019 10 / 25

Framework Comparison The two pictures are expected to match: • dyonic static AdS 4 × S 7 black holes: QFT computation = attractor mechanism in N = 2 gauged supergravity [Ferrara-Kallosh-Strominger 96; Dall’Agata-Gnecchi 10] Not always the attractor mechanism is known: entropy functional can be written combining field theory and gravity intuition • Kerr-Newman AdS 5 × S 5 : entropy functional found empirically [Hristov-Hosseini-AZ, 17]   3 2 I (∆ a , ω i ) = i π N 2 ∆ 1 ∆ 2 ∆ 3 � � ∆ 1 + ∆ 2 + ∆ 3 + ω 1 + ω 2 = 1 ∆ a Q a −   , + 2 π i ω i J i ω 1 ω 2 a =1 i =1 – Reproduced in QFT in various overlapping limits using the superconformal index [Choi,Hwang,Kim,Nahmgoong;Cabo-Bizet,Cassani,Martelli,Murthy; Benini-Milan;Cabo-Bizet-Murthy] – Similar functionals proposed in higher dimensions and also computed via on-shell actions [Hristov-Hosseini-AZ;Choi,Hwang,Kim,Nahmgoong;Cabo-Bizet,Cassani,Martelli,Murthy; Cassani-Papini] Alberto Zaffaroni (Milano-Bicocca) HoloBH 2019 11 / 25

AdS 4 black holes Example I: Static black holes in AdS 4 × S 7 Black holes in M theory on AdS 4 × S 7 : [Cacciatori, Klemm 08; Dall’Agata, Gnecchi; Hristov, Vandoren 10; Katmadas; Halmagyi 14; Hristov, Katmadas, Toldo 18] • preserves two real supercharges (1 / 16 BPS) and horizon AdS 2 × Σ g • four electric q a and magnetic p a charges under U (1) 4 ⊂ SO (8); only six independent parameters • supersymmetry preserved with a topological twist • entropy goes like O ( N 3 / 2 ) and is a complicated function � � I 4 (Γ , Γ , G , G ) 2 − 64 I 4 (Γ) I 4 ( G ) S BH ( p a , q a ) ∼ I 4 (Γ , Γ , G , G ) ± I 4 symplectic quartic invariant Γ = ( p 1 , p 2 , p 3 , p 4 , q 1 , q 2 , q 3 , q 4 ) [Halmagyi 13] G = (0 , 0 , 0 , 0 , g , g , g , g ) Alberto Zaffaroni (Milano-Bicocca) HoloBH 2019 12 / 25

AdS 4 black holes The relevant index Topologically twisted index = QM Witten index � ( − 1) F e i � 4 a =1 Q a ∆ a e − β H p � Z Σ g × S 1 (∆ I , p a ) = Tr H � �� � � 4 a =1 ∆ a ∈ 2 π Z • magnetic charges p a enter in the Hamiltonian H g , electric charges q a introduced through chemical potentials ∆ a • number of fugacities equal to the number of conserved charges Alberto Zaffaroni (Milano-Bicocca) HoloBH 2019 13 / 25

AdS 4 black holes ABJM twisted index Luckily enough, the topologically twisted partition function for ABJM can be evaluated using localization � N N 1 dx i d ˜ x i � 1 − x i � � 1 − ˜ x i � � � � Z susy x − k � x k m i m i × × S 2 × S 1 = ˜ i i ( N !) 2 2 π ix i 2 π i ˜ ˜ x i x j x j m , � m ∈ Z N i =1 i � = j � � m j − p 1 +1 � � x i x i x j y 1 x j y 2 N � m i − � � m i − � m j − p 2 +1 � ˜ ˜ × 1 − x i 1 − x i x j y 1 x j y 2 ˜ ˜ i , j =1 � � m j − m i − p 3 +1 � � x j ˜ x j ˜ � � � � x i y 3 x i y 4 m j − m i − p 4 +1 1 − ˜ x j 1 − ˜ x j x i y 3 x i y 4 � � p a = 2 a y a = 1 , and solved in the large N limit. There is no cancellation between bosons and fermions and log Z = O ( N 3 / 2 ). [Benini-AZ; Benini-Hristov-AZ] Alberto Zaffaroni (Milano-Bicocca) HoloBH 2019 14 / 25

AdS 4 black holes QFT/Gravity comparison • dyonic static AdS 4 × S 7 black holes: QFT computation = attractor mechanism in N = 2 gauged supergravity Entropy functional: [Ferrara-Kallosh-Strominger 96; Dall’Agata-Gnecchi 10] � � � � a i p a ∂ F � � S BH ( p a , q a ) = log Z ( X a , p a ) − iX a q a � crit = ∂ X a − iX a q a � crit a � gauged supergravity prepotential F ∼ X 1 X 2 X 3 X 4 � X a = 2 π horizon scalar fields Localization (topologically twisted index): [Benini,Hristov,AZ 05] � � � � a i p a ∂ W � � S ( p a , q a ) = log Z (∆ a , p a ) − i ∆ a q a � crit = ∂ ∆ a − i ∆ a q a � crit a 3 iN 3 / 2 � twisted superpotential W on − shell = 2 2∆ 1 ∆ 2 ∆ 3 ∆ 4 � 4 a =1 ∆ a = 2 π Re ∆ a ∈ [0 , 2 π ] Alberto Zaffaroni (Milano-Bicocca) HoloBH 2019 15 / 25