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 : M2dCFT − → R such that the RG flow is a gradient line of c-function: βi(g) = −G(g)ij ∂c(g) ∂gj . 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

• rdering

energy scale radial coordinate flow equation RG flow BPS equation fixed point CFT horizon geometry insensitive to the microscopic detail

• f the system

the asymptotic values

• f 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

2R(O).

◮ Conjecture : a-function defined by a =

3 32

( 3 tr R3 − tr R ) decreases along RG flow: aUV > aIR. ◮ 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 Rφ = R0 +

n

∑

i=1

φiFi, the correct U(1)R current is given by the φ which attains the maximum of the “trial” a-function a(φ) = 3 32 ( 3 tr R3

φ − tr Rφ

) .

a-functions from toric diagrams

For a toric diagram P with vertices v1, · · · , vn, the a function of the corresponding quiver gauge theory is given by a(φ) = 9 32 N 2 2

n

∑

i,j,k=1

cijkφiφjφk, cijk = | det(vi, vj, vk)|, where φ1 + · · · + φn = 2, φi > 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

• f 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 v1, · · · , vn. ◮ The trial a-function ˆ FP : Rn → R, ˆ FP(φ) = ∑

1≤i<j<k≤n

| det(vi, vj, vk)| φiφjφk. ◮ Physical range of R-charges Γn := { (φ1, · · · , φn) ∈ Rn : φi ≥ 0, ∑n

i=1 φi = r

} ⊂ Rn. ◮ Extremize the function FP : Γn → R. ◮ Modulus := normalized maximum value of a-function M(P) := (3 r )3 max

φ∈Γn

ˆ FP(φ).

Some Examples

• ˆ

F(φ) = 2φ1φ2φ3 + 4φ1φ2φ4 + 3φ1φ3φ4 + φ2φ3φ4 ∃ unique extremal (maximal) point in relint(Γ4). Toric gauge theory.

• ˆ

F(φ) = 2φ1φ2φ3 + 8φ1φ2φ4 + 3φ1φ3φ4 + φ2φ3φ4 No critical points in relint(Γ4) ; maximized at a point on ∂(Γ4) but this is not a critical point

• ˆ

F(φ) = 4φ1φ2φ3 + 2φ1φ2φ4 + 9φ1φ3φ4 + 7φ2φ3φ4 + 4φ1φ3φ5 + φ2φ3φ5 + φ1φ4φ5 + 10φ2φ4φ5 + 4φ3φ4φ5 ∃ two critical points in relint(Γ5); 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 v1, · · · , vn. Then the function FP : Γn → R has a unique critical point φ∗ in the relative interior of Γn and φ∗ is also the unique global maximum of FP. ◮ Universal Upper Bound The critical point φ∗ satisfies the universal bound 0 < φi

∗ ≤ r/3

for all i. The equality φi

∗ = 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)⋉Z2, 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 ⊂ Rd conv(S) := {λx + (1 − λ)y ∈ Rd : x, y ∈ S, 0 ≤ λ ≤ 1}. ◮ Minkowski sum of A, B ⊂ Rd A + B := {x + y : x ∈ A, y ∈ B}. ◮ Dilatation rA := {rx : x ∈ A}.

Zonotopes

For a vector configuration X = {x1, · · · , xn} ⊂ Rd, the zonotope is given by Z(X) = {x ∈ Rd : x = λ1x1 + · · · + λnxn, 0 ≤ λi ≤ 1}

• Theorem. [Shephard, McMullen]

vold(Z(X)) = ∑

1≤i1<···<id≤n

| det(xi1, · · · , xid)|. Corollary. a-function is proportional to the volume of a zonotope: ˆ FP(φ) ∝ vol3(ZP(φ)), ZP(φ) := Z(φ1v1, φ2v2, · · · , φnvn)

Zonotope generators

zone cube

Brunn-Minkowski inequality

Theorem. (vold(−))1/d is concave on the set of d-dimensional

• bodies. Namely, if 0 ≤ λ ≤ 1 and A, B ⊂ Rd are convex bodies,

then (vold((1−λ)A + λB))1/d ≥ (1−λ) (vold(A))1/d + λ (vold(B))1/d . Equality ⇐ ⇒ A and B are homothetic Corollary. (FP)1/3 : Γn → R is strictly 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∈Γn−1 FQ(φ) < max ψ∈Γn FP(ψ).

FP : Γn → R cannot attain its maximum on the boundary ∂Γn. · · · + continuity + strict concavity of (FP)1/3 = ⇒ Existence & uniqueness of local maximum = ⇒ Monotinicity of M(P).

Bounds on critical points

Proposition. If φ∗ ∈ Γn is the critical point of FP, then φp

∗ = r

3 · vol(Z[p]

P (φ))

vol(ZP(φ)) , (p = 1, . . . , n). where Z[p]

P (φ) denotes the union of those cubes which has at least

• ne face belonging to p-th zone.

Baryonic & flavor symmetry

Maximization in two steps Rn

≥0 πP

− → C(P) − → R ∪ ∪ ∪ Γn

πP

− → rP − → {r} b = πP(φ) =

n

∑

i=1

φivi max

φ∈Rn

≥0

F(φ) = max

b∈C(P )

( max

φ∈π−1(b) F(φ)

) .

Relation with volume minimization

Theorem. Suppose b ∈ rP i.e. b = (∗, ∗, r). Then

• ˆ

FP is a quadratic polynomial along the fiber π−1

P (b).

• In each fiber, there is a unique critical & maximum point σP(b),

determined by σi

P(b) =

r VP(b)ℓi

P(b). Here,

ℓi

P(b) :=

vi−1, vi, vi+1 b, vi−1, vi b, vi, vi+1 ∝ vol(calibrated 3-cycle) VP(b) :=

n

∑

i=1

ℓi

P(b)

∝ vol(Sasaki-Einstein mfd.)

• max

φ∈π−1

P (b)

ˆ FP(φ) = ˆ FP(σP(b)) = r VP(b). Martelli-Sparks-Yau, Butti-Zaffaroni

5D N=2 gauged supergravity

Very special geometry Gunaydin, Sierra, Townsend S = ∫ 1 2 √−gR−1 2gabdφa∧∗dφb−1 4GijF i∧∗F j− 1 12cijkF i∧F j∧Ak

• The dynamics of vector multiplets (including graviphoton) is

completely governed by the prepotential F(φ) = 1

6cijkφiφjφk.

• The

vector multiplet moduli space MV is given by the hypersurface F(φ) = 1.

• The gauge coupling matrix Gij = − 1

2 ∂2 ∂φi∂φj log F(φ).

• The moduli space metric gab is given by the restriction of Gij 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 5D BH solution (electric charges Qi)

ds2 = −e−4U(r)dt2 + e2U(r)(dr2 + r2dΩ2

3)

GijF i

tr = 1

4Kj(r), Ki(r) = ki + Qi r2

• ∫
• ut the dynamics in non-radial directions

= ⇒ effective (0 + 1)-dim theory ( SUSY quantum mechanics )

• For τ = 1/r2, BPS equation reads

dU dτ = 1 6e2U, gab dφb dτ = 1 2e2U ∂Z ∂φa Z = ∑

i

Qiφi central charge

• One can choose U as a new “time” variable

∂φa ∂U = 3gab ∂ ∂φb log Z gradient flow line of log Z

• Attractor (fixed point) ↔ supersymmetric vacuum ↔ critical

point of Z

Positivity of metric & uniqueness of attractor

Proposition. Assume

• the Chern-Simons couplings are given by cijk = | det(vi, vj, vk)|.

(e.g. IIB string theory compactified on Y5 × AdS5 with toric Sasaki-Einstein manifold Y5.)

• MV = {φ ∈ Rn

≥0 : F(φ) = 1}

(e.g. φi = volume of SUSY 3-cycle in Y5 > 0) Then,

• The gauge coupling matrix Gij and the vector multiplet moduli

space metric gab is positive definite on MV .

• The attractor point in MV is unique; it is always a local maximum.

a-maximization and attractor equation are dual

F(φ) = 1

6

∑ cijkφiφjφk (FP(φ))1/3 : concave = ⇒ log FP(φ) : concave = ⇒ − log FP(φ) : convex = ⇒ Gij = − 1

2 ∂2 ∂φi∂φj log FP(φ) : positive definite

a-maximization 5D attractor maximize F(φ) Z(φ) = ∑

i Qiφi

constraint R(φ) = ∑

i φi = 2

F(φ) = 1 Lagrange’s method F(φ) + λR(φ) →max µF(φ) + Z(φ) →max

Future problems

◮ Global structure of 4d SCFT moduli. RG flows. Seiberg dualities. ◮ Inverse problem: Does the critical value of a-function characterize the toric diagram? ◮ Relation with dimers and (co)amoebas ◮ Why black holes and the RG flows are related? ◮ Scope of extremization principle