5d Supergravity Dual of a-maximization Yuji Tachikawa (Univ. of - PowerPoint PPT Presentation

5d Supergravity Dual of a-maximization Yuji Tachikawa (Univ. of Tokyo, Hongo) based on [YT, hep-th/0507057] October, 2005 @ Caltech 0/31

1. Introduction ⋄ Consider a D3 brane probing the tip of the CY cone Y Y = dr 2 + r 2 ds 2 ds 2 X D3 at the tip X :Sasaki-Einstein Y :Calabi-Yau ⋄ Two way of analysis • Field theory on the D3 ⇒ Some quiver gauge theory • Near Horizon Limit of the D3 ⇒ AdS 5 × X 5 1/31

Do the properties match ? ⋄ Both give a theory with SU (2 , 2 | 1) symmetry. • 4d N = 1 SCFT from the field theory p.o.v. • Isometry of spacetime from the gravity p.o.v. Basic quantity : Central charge a and c ⋄ a − c : 1 /N correction / higher-derivative correction How can we calculate a ? ⋄ 2/31

Field Theory Side CY cone as toric singularity ⇒ quiver gauge theory [Hanany et al.] ⇒ a -maximization [Intriligator-Wecht] Sasaki-Einstein Side CY cone as toric singularity ⇒ Need to find Einstein metric on Sasaki-manifold ⇒ Z -minimization [Martelli-Sparks-Yau] 3/31

They match ! ⋄ A great check of AdS/CFT, ⋄ Why do they match ? cf. [Butti-Zaffaroni 0506232] showed by brute force that Z = 1 /a after maximizing for baryonic symmetry, but no clear physical explanation yet. cf. [Barnes-Gorbatov-Intriligator-Wright 0507146] showed the equiva- lence Z = 1 /a on the vacua by considering two-point current correla- tors. 4/31

Divide and Conquer ... type IIB on AdS 5 × X 5 ↓ ← − in a slow progress 5d gauged sugra on AdS 5 � ← − my work 4d SCFT 5d gauged sugra is very constrained , ⋄ just as the Seiberg-Witten theory. ⋄ Some insight might be expected from this point of view ? 5/31

⋄ My message today : a -max imization in 4d is P -min imization in 5d. More common name for P -maximization is the Attractor Eq. ⋄ Historical Curiosity In the penultimate paragraph in [Ferrara-Zaffaroni ‘N=1,2 SCFT and Supergravity in 5d’ 9803060]: The presence of a scalar potential for supergravities in AdS5 allows to study critical points for different possible vacua in the bulk theory. It is natural to conjecture that these critical points should have a dual interpretation in the boundary superconformal field theory side. They could have discovered a -maximization before [Intriligator-Wecht, 0304128] ! 6/31

CONTENTS � 1. Introduction ⇒ 2. a -maximization ⋄ 3. 5d Gauged supergravity ⋄ 4. How they match ⋄ 5. Conclusion & Outloook 7/31

2. a -maximization N = 1 Superconformal Algebra in 4d { Q α , S α } ∼ R SC + · · · [ R SC , Q α ] = − Q α R SC carries a lot of info: ⋄ ∆ ≥ 3 • 2 R SC a = 3 32 (3 tr R 3 • SC − tr R SC ) cf. � T µ µ � = a × Euler + c × Weyl 2 8/31

s I Q I . ⋄ Let Q I be integral U (1) charges. ⇒ R SC = ˜ s I ? How can we find ˜ Let [ Q I , Q α ] = − ˆ ⋄ P I Q α . Call Q F = f I Q I with [ Q F , Q α ] = 0 a flavor symmetry. ⋄ R SC T µ ν Q F Q F ← → SUSY R SC T µ ν 9 tr Q F R SC R SC = tr Q F 9/31

s I ˆ s I Q I , Q α ] = − Q α ⇒ ˜ ⋄ [˜ P I = 1 . 32 (3 tr R ( s ) 3 − tr R ( s )) where R ( s ) = s I Q I . Let a ( s ) = 3 ⋄ s I extremizes a ( s ) under s I ˆ ⋄ 9 tr Q F R SC R SC = tr Q F ⇒ ˜ P I = 1 . ⋄ Unitarity ⇒ it’s a local maximum. a -maximization ! ⋄ Let ˆ c IJK = tr Q I Q J Q K and ˆ c I = tr Q I . cf. Both calculable at UV using ’t Hooft’s anomaly matching. 10/31

CONTENTS � 1. Introduction � 2. a -maximization ⇒ 3. 5d Gauged supergravity ⋄ 4. How they match ⋄ 5. Conclusion & Outloook 11/31

3. 5d gauged supergravity Multiplet structure Minimal number of supercharges is 8 , called N = 2 ⋄ ⋄ Gravity multiplet, Vector multiplet, Hypermultiplet Gravity multiplet ψ i A I g µν , µ , µ ⋄ i = 1 , 2 : index for SU (2) R . ⋄ I : explained in a second 12/31

Vector Multiplet A I λ x φ x µ , i , ⋄ I : 0 , . . . , n V and x : 1 , . . . , n V cf. A I µ constitutes integral basis; graviphoton is a mixture. φ x parametrize a hypersurface ⋄ F = c IJK h I h J h K = 1 in ( n V + 1) dim space { h I } ⇒ h I = h I ( φ x ) : sp. coordinates 13/31

⋄ Kinetic terms are − 1 2 g xy ∂ µ φ x ∂ µ φ y − 1 4 a IJ F I µν F J µν where g xy = − 3 IJK h I ,x h J ,y h K a IJ = h I h J + 3 2 g xy h I,x h J,y h I ≡ c IJK h J h K (dual sp. coordinates) ⋄ There is the Chern-Simons term 1 c IJK ǫ µνρστ A I µ F J νρ F K √ στ 6 6 14/31

Hypermultiplet q X , ζ A ⋄ X : 1 , . . . , 4 n H , holonomy SO (4 n H ) ⊃ Sp ( n H ) ⊗ Sp (1) R ⋄ A : 1 , . . . , 2 n H labels the fundamental of Sp ( n H ) Vierbein f X ⋄ iA , i = 1 , 2 ⋄ Sp (1) R part of the curvature is fixed: R XY ij = − ( f XiA f A Y j − f Y iA f A Xj ) We use { ij } ← → r = 1 , 2 , 3 15/31

Gauging ⋄ Potential is associated with the gauging of the hypers: ∂ µ q X − → D µ q X = ∂ µ q X + A I µ K X I where K X is triholomorphic: I K X I R r XY = D Y P r I cf. P r I is a triplet generalization of the D -term. ⋄ The potential V is given by V = 3 g xy ∂ x P r ∂ y P r + g XY D X P r D Y P r − 4 P r P r where P r = h I P r I cf. Gukov-Vafa-Witten type superpotential. 16/31

P r appears everywhere: ⋄ • Covariant derivative of the gravitino jI ψ j D ν ψ i µ = ∂ ν ψ i µ + A I µ P i I • SUSY transformation law δ ǫ φ x = i ǫ i λ x 2¯ i � 2 3 ∂ x P ij + · · · δ ǫ λ i x = − ǫ j δ ǫ q X = − i ¯ ǫ i f XiA ζ A √ 6 I h I + · · · ǫ i f XiA K X δ ǫ ζ A = 4 ¯ 17/31

IY q Y + · · · Suppose K X = Q X ⋄ I where Q X IY is the charge matrix of the hypers. I R ij XY = D Y P ij Recall K X ⋄ I ⇒ some calculation ⇒ Q IX ∈ so (4 n H ) Y ↓ ∪ projection P ij ∈ sp (1) R I at q X = 0 . 18/31

CONTENTS � 1. Introduction � 2. a -maximization � 3. 5d Gauged supergravity ⇒ 4. How they match ⋄ 5. Conclusion & Outloook 19/31

4. How do they match? Recap. Field theory Supergravity ⇔ c IJK h I h J h K = 1 c IJK = tr Q I Q J Q K ˆ [ Q I , Q α ] = − ˆ P I Q α D µ ψ i ν = ( ∂ µ + P i jI A I µ ) ψ ν ??? a -max 20/31

AdS/CFT correspondence AdS CFT ˆ O φ � � φ ( x ) O ( x ) d 4 x � � e − Z [ ˆ φ ( x ) � x 5 = ∞ = φ ( x )] = J µ A I ↔ I : current for Q I µ � µ J µ A I Thus, we need to introduce � e − I � SCF T . 21/31

Q I has triangle anomalies among them � µ J µ A I ⇒ � e − I � SCF T depends on the gauge ! � 1 c IJK g I F J ∧ F K d 4 x δ g ( · · · ) = 24 π 2 ˆ � 1 c IJK δ g ( A I ) ∧ F J ∧ F K d 5 x = 24 π 2 ˆ √ 6 c IJK = 16 π 2 ˆ c IJK . c I A I ∧ tr R ∧ R in the same way. ⋄ c I = tr Q I is related to ˆ ˆ ⋄ c I ≪ ˆ In the following, we set ˆ c IJK . 22/31

SUSY condition for sugra Assume q X = 0 and δζ A = 0 . ⋄ ,x P ij Recall δλ i x ∝ ǫ j ∂ x P ij . ⇒ δλ = 0 ⇒ ˜ h I ⋄ = 0 I h ∗ ,x and P r ∗ is perpendicular as ( n V + 1) dim’l vectors. ⋄ x = 1 , . . . , n V ⇒ P r =1 , 2 , 3 are parallel . ⋄ Rotate so that P r =1 , 2 = 0 , P r =3 � = 0 . ⋄ c IJK h I h J h K = 1 ⇒ c IJK h I h J h K ⋄ ,x = 0 � �� � h K ˜ h I = kP r =3 : Attractor Eq. I 23/31

⋄ Recall AdS CFT Q α , S α ↔ ψ µ A I ↔ Q I µ and jI ψ j D ν ψ i µ = ∂ ν ψ i µ + A I µ P i I , ⇒ The charge of Q α , S α under Q I is ± P r =3 . I Recall [ Q I , Q α ] = − ˆ ⋄ P I Q α ⇒ P r =3 = ˆ P I . I 24/31

⋄ From the susy tr, { δ ǫ , δ ǫ ′ } q X ∝ (¯ ǫǫ ′ ) h I K X I s I Q I ∝ ˜ h I Q I . ⇒ R SC = ˜ s I = ˜ h I / (˜ s I P I = 1 . ⇒ ˜ h I P I ) . ⋄ Recall ˜ Extend the relation to s I = h I / ( h I P I ) ⇒ ⋄ c IJK s I s J s K = c IJK h I h J h K a ( s ) ∝ tr ( s I Q I ) 3 = ˆ ∝ ( h I P I ) − 3 . ( h I P I ) 3 ⋄ a -max = P -min ! δλ = h I ,x P I = ( h I P I ) ,x = P ,x = 0 25/31

E.g. Exactly marginal deformation AdS CFT ↔ M c ‘Conformal manifolds’ ⋄ A puzzle M c should have K¨ ahler structure from CFT p.o.v. • M c must corresponds to hyper scalars, • since vectors are fixed by a -max = P -min. Hypers are quaternionic , which is not K¨ ahler. ??? • 26/31

δζ A = 0 ⇒ K X = 0 where K X ≡ ˜ h I K X ⋄ I . Superconformal deformation ↔ M c = { q X ∈ M hyper | K X = 0 } ⋄ From D X P r = R r XY K X , P r is covariantly constant on M c . ⋄ Then M c is K¨ ahler , because ⋄ J X Y ≡ R r Y Z g XZ P r / | P r | is a covariantly constant matrix with J 2 = − 1 . 27/31

a -max with Lagrange multipliers ⋄ SCFT also has anomalous symmetries � I tr F a ∧ F a ∂ µ J µ m a I = a ⋄ [Kutasov] extended the a-max to include them, a ( s I , λ a ) = a ( s ) + λ a m a I s I where λ a are Lagrange multipliers enforcing the anomaly-free condi- tion for R SC . λ a behaves like coupling constant for F a ⋄ µν . 28/31

⋄ Recall AdS CFT J µ A µ X φ ↔ Higgs mechanism anomalous symmetry ⋄ It’s because � A µ J µ + φ X ) � : invariant under � exp( δA µ = ∂ µ ǫ, δφ = ǫ ⇒ � ǫ ( ∂ µ J µ − X )) � = 0 . � exp( 29/31