Extremization problems in AdS/CFT a -maximization and attractor - PowerPoint PPT Presentation

Extremization problems in AdS/CFT a -maximization and attractor mechanism Akishi KATO (Math. Sci., Univ. Tokyo) March 13, 2007, KEK ref. hep-th/0610266 “Zonotopes and four-dimensional superconformal field theories”; A.K. & Y. Tachikawa

Some Extremization problems ◮ Zamoldchikov c -theorem There exists a real-valued function c : M 2dCFT − → R such β i ( g ) = that the RG flow is a gradient line of c -function: − G ( g ) ij ∂c ( g ) ∂g j . The critical value of c is the Virasoro central charge of the corresponding CFT. ◮ Attractor mechanism (Ferrara, Kallosh, Strominger, ....) The radial evolution of moduli fields of N =2 extremal black hole is given by a gradient flow line of the central charge function. The critical value gives the area (entropy) of the horizon, and depends on the electric and magnetic charges of the black hole but not the asymptotic values of the moduli fields.

Extremization principle ◮ “Extremization principle” natually explains the universality — “critical values” are insensitive the small change of initial values. ◮ Universality in QFT v.s. attractor mechanism in black holes universality attractor ordering energy scale radial coordinate flow equation RG flow BPS equation fixed point CFT horizon geometry insensitive to the microscopic detail the asymptotic values of the system of the moduli scalar fields ◮ may shed new lights on AdS/CFT correspondence.

Plan Meta-question : Does extremization principle really work? Solution exists? Is it unique ? Monotonicity? ◮ a -maximization of toric quiver gauge theory – specified by combinatorial data (toric diagrams). – height function is given by a simple cubic polynomial. – exact results for 4D superconformal FT. – will serve as a test case to see the power/limts of “extremization principle” ◮ Black hole attractors in 5D N = 2 gauged supergravity – very special geometry – dual problem to a -maximization – uniqueness of attractor solution

4d Superconformal Field Theories and U (1) R ◮ Global symmetry contains SU (2 , 2 | 1) ⊃ SO (4 , 2) × U (1) R ◮ Scaling dimension of chiral operators are protected from quantum corrections: ∆( O ) = 3 2 R ( O ) . 3 tr R 3 − tr R 3 ( ) ◮ Conjecture : a -function defined by a = 32 decreases along RG flow: a UV > a IR . ◮ U (1) R is thus extremely useful provided that it can be correctly identified. In general, however, abelian part of non- R global flavor symmetry G can mix with U (1) R .

a-maximization Theorem. [Intriligator-Wecht 2003] Exact U (1) R charges maximize a : Among all possible combination of abelian currents n ∑ φ i F i , R φ = R 0 + i =1 the correct U (1) R current is given by the φ which attains the maximum of the “trial” a -function a ( φ ) = 3 3 tr R 3 ( ) φ − tr R φ . 32

a-functions from toric diagrams For a toric diagram P with vertices v 1 , · · · , v n , the a function of the corresponding quiver gauge theory is given by n N 2 a ( φ ) = 9 ∑ c ijk φ i φ j φ k , c ijk = | det( v i , v j , v k ) | , 32 2 i,j,k =1 where φ 1 + · · · + φ n = 2 , φ i > 0 . Hanany-Iqbal, Benvenuti-Franco-Hanany-Martelli-Sparks, Butti-Zaffaroni, Franco- Hanany-Kennaway-Vegh-Wecht, Benvenuti-Kruczenski, ... Benvenuti-Zayas-Tachikawa, Lee-Rey

Basic Questions ◮ Does a -maximization always have a solution? ◮ Is it unique? No saddle points? ◮ Do non-extremal points in toric diagrams play their role in a - maximization? ◮ How does the change of toric diagrams influence the maxima of trial a -functions? Does a -function decrease whenever a toric diagram shrinks? ◮ Don’t want to build conjectures upon other conjectures.... Can we answer these questions without assuming AdS/CFT correspondence?

Mathematical Setup ◮ Input data : a toric diagram P with vertices v 1 , · · · , v n . F P : R n → R , ◮ The trial a -function ˆ ˆ ∑ | det( v i , v j , v k ) | φ i φ j φ k . F P ( φ ) = 1 ≤ i<j<k ≤ n ◮ Physical range of R -charges ( φ 1 , · · · , φ n ) ∈ R n : φ i ≥ 0 , ∑ n i =1 φ i = r ⊂ R n . { } Γ n := Extremize the function F P : Γ n → R . ◮ ◮ Modulus := normalized maximum value of a -function ) 3 ( 3 ˆ M ( P ) := max F P ( φ ) . r φ ∈ Γ n

Some Examples F ( φ ) = 2 φ 1 φ 2 φ 3 + 4 φ 1 φ 2 φ 4 + 3 φ 1 φ 3 φ 4 + φ 2 φ 3 φ 4 • ˆ ∃ unique extremal (maximal) point in relint(Γ 4 ) . Toric gauge theory. F ( φ ) = 2 φ 1 φ 2 φ 3 + 8 φ 1 φ 2 φ 4 + 3 φ 1 φ 3 φ 4 + φ 2 φ 3 φ 4 • ˆ No critical points in relint(Γ 4 ) ; maximized at a point on ∂ (Γ 4 ) but this is not a critical point F ( φ ) = 4 φ 1 φ 2 φ 3 + 2 φ 1 φ 2 φ 4 + 9 φ 1 φ 3 φ 4 + 7 φ 2 φ 3 φ 4 + 4 φ 1 φ 3 φ 5 + • ˆ φ 2 φ 3 φ 5 + φ 1 φ 4 φ 5 + 10 φ 2 φ 4 φ 5 + 4 φ 3 φ 4 φ 5 ∃ two critical points in relint(Γ 5 ) ; one is a local maximum and the other is a saddle point.

Main Theorems ◮ Existence and Uniqueness Let P be a toric diagram with vertices v 1 , · · · , v n . Then the function F P : Γ n → R has a unique critical point φ ∗ in the relative interior of Γ n and φ ∗ is also the unique global maximum of F P . ◮ Universal Upper Bound The critical point φ ∗ satisfies the universal bound 0 < φ i ∗ ≤ r/ 3 for all i . The equality φ i ∗ = r/ 3 holds for some i if and only if n = 3 . ◮ Monotonicity The maximum value M ( P ) depends on P only through its convex hull. M ( P ) is monotone in the sense that if P ⊂ P ′ up to integral affine transformations G := GL (2 , Z ) ⋉Z 2 , then M ( P ) ≤ M ( P ′ ) . The equality holds if and only if P = P ′ up to G -action.

Polytopes and Minkowski sums ◮ Convex hull of S ⊂ R d conv( S ) := { λ x + (1 − λ ) y ∈ R d : x , y ∈ S, 0 ≤ λ ≤ 1 } . ◮ Minkowski sum of A, B ⊂ R d A + B := { x + y : x ∈ A, y ∈ B } . ◮ Dilatation rA := { r x : x ∈ A } .

Zonotopes For a vector configuration X = { x 1 , · · · , x n } ⊂ R d , the zonotope is given by Z ( X ) = { x ∈ R d : x = λ 1 x 1 + · · · + λ n x n , 0 ≤ λ i ≤ 1 } Theorem. [Shephard, McMullen] ∑ vol d ( Z ( X )) = | det( x i 1 , · · · , x i d ) | . 1 ≤ i 1 < ··· <i d ≤ n Corollary. a -function is proportional to the volume of a zonotope: ˆ F P ( φ ) ∝ vol 3 ( Z P ( φ )) , Z P ( φ ) := Z ( φ 1 v 1 , φ 2 v 2 , · · · , φ n v n )

Zonotope generators

cube zone

Brunn-Minkowski inequality (vol d ( − )) 1 /d is concave on the set of d -dimensional Theorem. bodies. Namely, if 0 ≤ λ ≤ 1 and A, B ⊂ R d are convex bodies, then (vol d ((1 − λ ) A + λB )) 1 /d ≥ (1 − λ ) (vol d ( A )) 1 /d + λ (vol d ( B )) 1 /d . Equality ⇐ ⇒ A and B are homothetic ( F P ) 1 / 3 : Γ n → R is strictly Corollary. concave. Concavity + existence of critical point = ⇒ uniqueness & global maximum

Changing Toric Diagrams Proposition. Suppose toric diagrams P , Q are related as follows: Then, max φ ∈ Γ n − 1 F Q ( φ ) < max ψ ∈ Γ n F P ( ψ ) . F P : Γ n → R cannot attain its maximum on the boundary ∂ Γ n . · · · + continuity + strict concavity of ( F P ) 1 / 3 = ⇒ Existence & uniqueness of local maximum = ⇒ Monotinicity of M ( P ) .

Bounds on critical points Proposition. If φ ∗ ∈ Γ n is the critical point of F P , then 3 · vol( Z [ p ] ∗ = r P ( φ )) φ p vol( Z P ( φ )) , ( p = 1 , . . . , n ) . where Z [ p ] P ( φ ) denotes the union of those cubes which has at least one face belonging to p -th zone.

Baryonic & flavor symmetry Maximization in two steps π P R n − → C ( P ) − → R n ≥ 0 ∑ φ i v i b = π P ( φ ) = ∪ ∪ ∪ π P Γ n − → rP − → { r } i =1 ( ) max F ( φ ) = max φ ∈ π − 1 ( b ) F ( φ ) max . φ ∈ R n b ∈ C ( P ) ≥ 0

Relation with volume minimization Theorem. Suppose b ∈ rP i.e. b = ( ∗ , ∗ , r ) . Then • ˆ F P is a quadratic polynomial along the fiber π − 1 P ( b ) . • In each fiber, there is a unique critical & maximum point σ P ( b ) , r determined by σ i V P ( b ) ℓ i P ( b ) = P ( b ) . Here, � v i − 1 , v i , v i +1 � ℓ i P ( b ) := � b , v i − 1 , v i � � b , v i , v i +1 � ∝ vol( calibrated 3-cycle ) n ∑ ℓ i V P ( b ) := P ( b ) ∝ vol( Sasaki-Einstein mfd. ) i =1 r F P ( φ ) = ˆ ˆ • max F P ( σ P ( b )) = V P ( b ) . φ ∈ π − 1 P ( b ) Martelli-Sparks-Yau, Butti-Zaffaroni

5D N =2 gauged supergravity Very special geometry Gunaydin, Sierra, Townsend ∫ 1 √− gR − 1 2 g ab dφ a ∧∗ dφ b − 1 4 G ij F i ∧∗ F j − 1 12 c ijk F i ∧ F j ∧ A k S = 2 • The dynamics of vector multiplets (including graviphoton) is completely governed by the prepotential F ( φ ) = 1 6 c ijk φ i φ j φ k . • The vector multiplet moduli space M V is given by the hypersurface F ( φ ) = 1 . ∂ 2 • The gauge coupling matrix G ij = − 1 ∂φ i ∂φ j log F ( φ ) . 2 • The moduli space metric g ab is given by the restriction of G ij to the hypersurface F ( φ ) = 1 .

Attractor equation • According to AdS/CFT : holographic dual Black hole ← → RG flow radial coordinate ← → RG time • Ansatz for the 5 D BH solution (electric charges Q i ) ds 2 = − e − 4 U ( r ) dt 2 + e 2 U ( r ) ( dr 2 + r 2 d Ω 2 3 ) tr = 1 K i ( r ) = k i + Q i G ij F i 4 K j ( r ) , r 2 ∫ • out the dynamics in non-radial directions = ⇒ effective (0 + 1) -dim theory ( SUSY quantum mechanics )