Black holes with baryonic charge and I -extremization Hyojoong Kim Kyung Hee University Based on 1904.05344 with Nakwoo Kim See also 1904.04269 (HZ) and 1904.04282 (GMS) Strings and Fields 2019, YITP Aug. 23, 2019 Hyojoong Kim (KHU) Black holes with baryonic charge and I -extremization Aug. 23, 2019 1 / 25
Recently, there has been progress on the microscopic understanding of the AdS black hole entropy such as the magnetically charged static AdS black holes and the topologically twisted index the electrically charged rotating AdS black holes and the superconformal index with complex chemical potentials (See Morteza’s overview talk) In this talk, I will discuss the magnetically charged static AdS black holes with baryonic flux from the viewpoint of the extremization principle and toric geometry. Hyojoong Kim (KHU) Black holes with baryonic charge and I -extremization Aug. 23, 2019 2 / 25
Four Extremizations It is known that there are four extremization principles in supersymmetric gauge theories. a-maximization F-maximization [Intrilligator, Wecht 03] [Jafferis 10] 4 d, N =1 3 d, N =2 S 3 free energy F (∆ a ) central charge a trial (∆ a ) Hyojoong Kim (KHU) Black holes with baryonic charge and I -extremization Aug. 23, 2019 3 / 25
Four Extremizations It is known that there are four extremization principles in supersymmetric gauge theories. a-maximization F-maximization [Intrilligator, Wecht 03] [Jafferis 10] 4 d, N =1 3 d, N =2 S 3 free energy F (∆ a ) central charge a trial (∆ a ) � gravity dual 1 √ 1 AdS 5 × Y 5 , a trial ∼ AdS 4 × Y 7 , F ∼ vol(Y 5 ) vol(Y 7 ) Geometric dual of a- & F-maximization : volume minimization [Martelli, Sparks, Yau 05] Hyojoong Kim (KHU) Black holes with baryonic charge and I -extremization Aug. 23, 2019 3 / 25
Compactify theories on Σ g with a topological twist c-extremization I -extremization [Benini, Bobev 12] [Benini, Hristov, Zaffaroni 15] 2 d, N =(0,2) 1 d, N =2 central charge c r (∆ a , n a ) topologically twisted index I (∆ a , n a ) Hyojoong Kim (KHU) Black holes with baryonic charge and I -extremization Aug. 23, 2019 4 / 25
Compactify theories on Σ g with a topological twist c-extremization I -extremization [Benini, Bobev 12] [Benini, Hristov, Zaffaroni 15] 2 d, N =(0,2) 1 d, N =2 central charge c r (∆ a , n a ) topologically twisted index I (∆ a , n a ) � gravity dual AdS 3 × Σ g , AdS 2 × Σ g the entropy of the magnetically charged static AdS black hole Aim : Finding a geometric dual of c- and I -extremization. The geometric dual of c-extremization was studied. [Couzens, Gauntlett, Martelli, Sparks 1810; GMS 1812 ; Hosseini, Zaffaroni 1901] In this talk, I will focus on the I -extremization. [HZ 1901; HZ; GMS; KK 1904] Hyojoong Kim (KHU) Black holes with baryonic charge and I -extremization Aug. 23, 2019 4 / 25
AdS solutions from wrapped D3- and M2-branes AdS 3 solutions in type IIB [Nakwoo Kim 05] ds 2 10 = L 2 e − B/ 2 � ds 2 ( AdS 3 ) + ds 2 ( Y 7 ) � , F 5 = − L 4 (vol AdS 3 ∧ F + ∗ 7 F ) . AdS 2 solutions in d=11 supergravity [N. Kim, Jong-Dae Park 06] ds 2 11 = L 2 e − 2 B/ 3 � ds 2 ( AdS 2 ) + ds 2 ( Y 9 ) � , F 5 = L 3 vol AdS 2 ∧ F. ∗ SUSY requires a Killing vector ξ in Y 2 n +1 and the foliation Y 2 n to be a K¨ ahler manifold. ∗∗ The 2n-dimensional K¨ ahler metrics satisfy the gauge field equation of motion � 2 n R − 1 2 R 2 + R ij R ij = 0 , where n = 3 for IIB and n = 4 for d=11. Hyojoong Kim (KHU) Black holes with baryonic charge and I -extremization Aug. 23, 2019 5 / 25
CGMS-Extremization Imposing the supersymmetry condition ( ∗ ) and relaxing the equation of motion ( ∗∗ ), the supersymmetric solution can be obtained by extremizing (2n+1)-dimensional action [Couzens, Gauntlett, Martelli, Sparks 1810] J n − 1 � η ∧ ρ ∧ S SUSY = ( n − 1)! . Y 2 n +1 For n=3, the central charge 3 L 8 c sugra = 3 L = S SUSY | on-shell . (2 π ) 6 g 2 s ℓ 8 2 G 3 s For n=4, the Bekenstein-Hawking entropy 4 πL 9 1 S BH = = S SUSY | on-shell . (2 π ) 8 ℓ 9 4 G 2 p Hyojoong Kim (KHU) Black holes with baryonic charge and I -extremization Aug. 23, 2019 6 / 25
OLD & NEW extremizations � AdS 5 × SE 5 and AdS 4 × SE 7 solutions C ( X 2 n − 1 ) is K¨ ahler : X 2 n − 1 is Sasakian. volume minimization : relax Einstein conditions and extremize the Sasakian volume V ( b i ) . [Martelli, Sparks, Yau 05] � b is a Killing vector, called Reeb vector, which is dual to a U(1) R-symmetry in the field theory. It corresponds to a geometric dual of a- and F-maximization Hyojoong Kim (KHU) Black holes with baryonic charge and I -extremization Aug. 23, 2019 7 / 25
OLD & NEW extremizations � AdS 5 × SE 5 and AdS 4 × SE 7 solutions C ( X 2 n − 1 ) is K¨ ahler : X 2 n − 1 is Sasakian. volume minimization : relax Einstein conditions and extremize the Sasakian volume V ( b i ) . [Martelli, Sparks, Yau 05] � b is a Killing vector, called Reeb vector, which is dual to a U(1) R-symmetry in the field theory. It corresponds to a geometric dual of a- and F-maximization � AdS 3 × Y 7 and AdS 2 × Y 9 solutions C ( Y 2 n +1 ) is not K¨ ahler : Y 2 n +1 is no longer Sasakian. Focus on a special case where Y 2 n − 1 → Y 2 n +1 → Σ g and C ( Y 2 n − 1 ) is toric. [Gauntlett, Martelli, Sparks 1812] For a given toric data of C ( Y 2 n − 1 ) , we can calculate a master volume V ( b i ; { λ a } ) where λ a is the transverse K¨ ahler class. Extremizing 2 n + 1 -dimensional action S SUSY corresponds to a geometric dual of c- and I -extremization. Hyojoong Kim (KHU) Black holes with baryonic charge and I -extremization Aug. 23, 2019 7 / 25
GMS-Extremization with the master volume Step 1. Construct the master volume V ( b i ; { λ a } ) for a given toric data. Step 2. Solve the constraint equation and the flux quantization conditions for λ a , A d d 4 d ∂ 2 V ∂ 2 V ∂ V � ∂λ a ∂λ b = 2 πn 1 � � n i � A ∂λ a − 2 πb 1 ∂λ a ∂b i , a,b =1 a =1 i =1 a =1 d d 4 ∂ 2 V ∂ 2 V ∂ V n a N = − A � � � n i N = − ∂λ a , ∂λ a ∂λ b − b 1 ∂λ a ∂b i . 2 π a =1 b =1 i =1 Step 3. Obtain the entropy functional and the R-charges of baryonic operators � d 4 �� ∂ V n i ∂ V � S ( b i , n a ) = − 8 π 2 � � A ∂λ a + 2 πb 1 , � ∂b i � � a =1 i =1 λ a ,A � R a ( b i , n a ) = − 2 ∂ V ˜ � . � N ∂λ a � λ a ,A Step 4. Extremize the entropy functional S ( b i , n a ) with respect to b 2 , b 3 and b 4 after setting b 1 = 1 . Hyojoong Kim (KHU) Black holes with baryonic charge and I -extremization Aug. 23, 2019 8 / 25
Topologically twisted indices, black hole entropy and entropy functional : ABJM case [Benini, Hristov, Zaffaroni 15] [Hosseini, Zaffaroni 1901] Hyojoong Kim (KHU) Black holes with baryonic charge and I -extremization Aug. 23, 2019 9 / 25
ABJM theory ABJM theory is a 3-dimensional U ( N ) k × U ( N ) − k Chern-Simons theory 4 bi-fundamental chiral multiplets 1 2 the quartic superpotential W ∝ tr( ǫ ab ǫ cd Z a W c Z b W d ) The dual gravity theory is the AdS 4 × S 7 solution of D=11 supergravity the SO(8)-invariant vacuum of D=4, N = 8 SO(8) gauged supergravity Hyojoong Kim (KHU) Black holes with baryonic charge and I -extremization Aug. 23, 2019 10 / 25
Topological twisted index and black hole entropy The topologically twisted index is the partition function on Σ g × S 1 with magnetic fluxes n a on Σ g . In the large N -limit, it reduce to � 4 � I (∆ a , n a ) = − π n a � 3 N 3 / 2 � 2∆ 1 ∆ 2 ∆ 3 ∆ 4 ∆ a a =1 where 4 4 � � ∆ a = 2 , n a = 2 − 2 g . a =1 a =1 The entropy of magnetically charged static D=4 AdS black holes solution is � 4 � S BH = − πL 2 n a � � X 1 X 2 X 3 X 4 . G 4 X a a =1 [Benini, Hristov, Zaffaroni 15] Hyojoong Kim (KHU) Black holes with baryonic charge and I -extremization Aug. 23, 2019 11 / 25
I -extremization Extremizing the twisted index and the black hole entropy w.r.t ∆ a and X a , respectively, leads to I| ∆ a = ¯ ∆ a ( n a ) = S BH | X = X ( r h ) ( n a ) . Comments The entropy is a function of magnetic flux. The topologically twisted index successfully reproduces the entropy of the black hole. The extremization procedure on the field theory side is called I -extremization. on the gravity side corresponds to the attractor mechanism. They agree even before extremization! (off-shell) Hyojoong Kim (KHU) Black holes with baryonic charge and I -extremization Aug. 23, 2019 12 / 25
Entropy functional Using the toric data of C 4 , MV-M111 v 1 = (1 , 0 , 0 , 0) , v 2 = (1 , 1 , 0 , 0) , v 3 = (1 , 0 , 1 , 0) , v 4 = (1 , 0 , 0 , 1) , the master volume for S 7 is easily obtained as [Hosseini, Zaffaroni 1901] V ( b i , λ a ) = 8 π 4 ( λ 1 ( b 2 + b 3 + b 4 − b 1 ) − λ 2 b 2 − λ 3 b 3 − λ 4 b 4 ) 3 . 3 b 2 b 3 b 4 ( b 1 − b 2 − b 3 − b 4 ) The entropy functional and R-charges are √ 2 N 3 / 2 � S ( b i , n a ) = − 2 π b 2 b 3 b 4 ( b 1 − b 2 − b 3 − b 4) 3 b 1 � � b 1 − b 2 − b 3 − b 4 + n 2 n 1 b 2 + n 3 b 3 + n 4 × , b 4 ∆ 1 ( b i ) = 2 ( b 1 − b 2 − b 3 − b 4 ) ∆ 2 = 2 b 2 ∆ 3 = 2 b 3 ∆ 4 = 2 b 4 , b 1 , b 1 , b 1 . b 1 Hyojoong Kim (KHU) Black holes with baryonic charge and I -extremization Aug. 23, 2019 13 / 25
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