Three Extremalization Principles in AdS/CFT with Eight Supercharges Yuji Tachikawa (Univ. of Tokyo, Hongo) partially based on [YT, hep-th/0507057] with special thanks to Y. Ookouchi March, 2006 @ KEK 0/31
⋄ type IIB on AdS 5 × X 5 ⇐ ⇒ CFT 4 8 supercharges: ⋄ • N = 1 SCFT 4 • N = 2 5d sugra • X 5 : Sasaki-Einstein R -symmetry in the superconformal algebra ⋄ = linear combination of U (1) charges which is determined by the extremalization principles ⋄ 1/31
⋄ type IIB on AdS 5 × X 5 ⇐ ⇒ CFT 4 8 supercharges: ⋄ a • N = 1 SCFT 4 ⇒ -maximization [Intriligator-Wecht] P • N = 2 5d sugra ⇒ -minimization [YT] Z • ⇒ X 5 : Sasaki-Einstein -minimization [Martelli-Sparks-Yau] R -symmetry in the superconformal algebra ⋄ = linear combination of U (1) charges which is determined by the extremalization principles ⋄ 1/31
⋄ type IIB on AdS 5 × X 5 ⇐ ⇒ CFT 4 8 supercharges: ⋄ central charge • N = 1 SCFT 4 ⇒ -maximization superpotential • N = 2 5d sugra ⇒ -minimization volume • ⇒ X 5 : Sasaki-Einstein -minimization R -symmetry in the superconformal algebra ⋄ = linear combination of U (1) charges which is determined by the extremalization principles ⋄ 1/31
CONTENTS ⇒ 1. a -maximization ⋄ 2. P -minimization ⋄ 3. Z -minimization ⋄ 4. Conclusion 2/31
1. a -maximization Intriligator-Wecht, hep-th/0304128 2/31
⋄ Studying Dynamics in 4d ⇒ Extra symmetry helps N = 1 Susy Algebra in 4d { Q α , Q † β } = σ µ β P µ ˙ α ˙ Conformal Algebra in 4d [ K µ , P ν ] ∼ ∆ δ µν N = 1 Superconformal Algebra in 4d { Q α , S α } ∼ R SC + · · · [ R SC , Q α ] = − Q α [ R SC , S α ] = S α 3/31
R SC carries a lot of info: ⋄ ∆ ≥ 3 2 R SC , • equal for chiral primaries a = 3 32 (3 tr R 3 SC − tr R SC ) • cf. On a curved mfd, � T µ µ � = a × Euler + c × Weyl 2 where Euler and Weyl are polynomials in R µνρσ 4/31
Many global symmetries Q I with [ Q I , Q α ] = − ˆ ⋄ P I Q α . R SC is a linear combination : R SC = r I Q I . ⋄ How can we find R SC ? 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 5/31
[ r I Q I , Q α ] = − Q α ⇒ r I ˆ ⋄ P I = 1 . 32 (3 tr R ( s ) 3 − tr R ( s )) where R ( s ) = s I Q I . Let a ( s ) = 3 ⋄ 9 tr Q F R SC R SC = tr Q F ⇒ r I extremizes a ( s ) under s I ˆ ⋄ P I = 1 . ⋄ Unitarity ⇒ it’s a local maximum. a -maximization ! ⋄ c IJK = tr Q I Q J Q K and ˆ ˆ c I = tr Q I : Calculable at UV using ’t Hooft’s anomaly matching. 6/31
の 図 の 図 の 図 と AdS 5 × Y p,q ⇔ Y 4 , 3 SU ( N ) 2 p gauge theory with four bifundamentals W α U i V i Y Z I p − q multiplicity 2 p p + q p q 1 − ( s 1 + s 2 ) / 2 1 + ( s 2 − s 1 ) / 2 R-sym 1 s 1 s 2 Maximize a ( s 1 , s 2 ) ⇒ � 3 � a = p 2 � 4 p 2 − 3 q 2 − 8 p 3 + 9 pq 2 + 4 q 4 7/31
CONTENTS � 1. a -maximization ⇒ 2. P -minimization ⋄ 3. Z -minimization ⋄ 4. Conclusion 8/31
2. P -minimization YT, hep-th/0507057 8/31
⋄ Any CFT phenomena ⇒ AdS counterpart via Gubser-Klebanov-Polyakov, Witten AdS CFT O φ � ˆ � φ ( x ) O ( x ) d 4 x � � e − � x 5 = ∞ = ˆ Z [ φ ( x ) φ ( x )] = J µ A I ↔ I : current for Q I µ � ˆ µ J µ A I � � e − I � SCF T Z [ A I � x 5 = ∞ = ˆ A I µ ( x ) µ ( x )] How about central charge maximization ? ⋄ ⇒ superpotential minimization ! 9/31
Q I has triangle anomalies among them : � ˆ � 1 µ J µ A I c IJK χ I F J ∧ F K δ χ ( � e − I � SCF T ) = d 4 x 24 π 2 ˆ � 1 c IJK dχ I ∧ F J ∧ F K d 5 x = 24 π 2 ˆ �� � 1 c IJK A I ∧ F J ∧ F K d 5 x = δ χ 24 π 2 ˆ ⇒ Presence of Chern-Simons terms in AdS c I A I ∧ tr R ∧ R , higher derivative effect. ⋄ c I = tr Q I ↔ ˆ ˆ 10/31
⋄ 4d N = 1 SCFT ⇒ 5d N = 2 supergravity Multiplet structure Minimal number of supercharges is 8 , called N = 2 ⋄ Gravity multiplet, Vector multiplet, Hyper multiplet ⋄ Gravity multiplet ψ i A I g µν , µ , µ ⋄ i = 1 , 2 : index for SU (2) R . ⋄ I : explained in a second 11/31
Vector Multiplet A I λ x φ x µ , i , ⋄ I : 0 , . . . , n V and x : 1 , . . . , n V cf. I labels integral basis; graviphoton is a mixture of A I µ . φ x parametrize a hypersurface ⋄ F = c IJK h I h J h K = 1 in ( n V + 1) dim space of { h I } ; c IJK constants h I = h I ( φ x ) : special coordinates ⋄ 12/31
⋄ Kinetic terms determined by c IJK : − 1 2 g xy ( φ ) ∂ µ φ x ∂ µ φ y − 1 4 a IJ ( φ ) F I µν F J µν Presence of the Chern-Simons term : ⋄ 1 c IJK ǫ µνρστ A I µ F J νρ F K √ στ 6 6 √ 6 ⇒ c IJK = 16 π 2 tr Q I Q J Q K . 13/31
Potential P r I : triplet generalization of the Fayet-Iliopoulos term ⋄ where r = 1 , 2 , 3 label the triplets in SU (2) R . ⋄ The potential V is given by V = 3 g xy ∂ x P r ∂ y P r − 4 P r P r where P r = P r I h I , Gukov-Vafa-Witten type . 14/31
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 where P i j = P r σ ri j 15/31
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 16/31
SUSY condition for sugra δλ = ǫ j ∂ x P ij = 0 ⇒ � h I ,x � P ij ⋄ = 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 ⇒ 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 � ∝ P r =3 : Attractor Eq. I 17/31
⋄ Recall AdS CFT Q α , S α ↔ ψ µ A I ↔ Q I µ jI ψ j D ν ψ i µ = ∂ ν ψ i µ + A I µ P i [ Q I , Q α ] = − ˆ ↔ P I Q α . I ⇒ P r =1 , 2 = 0 ⇒ charge of Q α , S α under Q I is ± P r =3 . I ⇒ P r =3 = ˆ P I . I 18/31
SUSY tr. of hypers ⇒ R SC = r I Q I ∝ � h I � Q I . ⋄ � h I � Recall r I P I = 1 . ⇒ r I = ⋄ . � h I � P I h I Suppose s I = R trial = s I Q 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 19/31
CONTENTS � 1. a -maximization � 2. P -minimization ⇒ 3. Z -minimization ⋄ 4. Conclusion 20/31
3. Z -minimization Martelli-Sparks-Yau, hep-th/0503183 hep-th/0603021 20/31
⋄ 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 (1+3) d 1d 5d 21/31
Low energy theory on the D3 Near Horizon Limit of the D3 × × AdS 5 × X 5 Some quiver gauge theory Describe the same physics. AdS/CFT correspondence [Maldacena] 22/31
Field theory on M 3 , 1 Gravity on AdS 5 × X 5 Conformal symmetry Isometry of AdS = Isometry of X Global symmetry = R SC symmetry Reeb vector = central charge a 1 / Vol( X ) = ⋄ a is from a -maximization. How about Vol( X ) ? Many explicit metrics for X 5 : S 5 , T 1 , 1 , Y p,q and L p,q,r ⋄ [Gauntlett et al.], [Cvetiˇ c et al.] ⇒ Central charge a = (Vol) − 1 23/31
Y p,q : explicit 5d Sasaki-Einstein metric found in 2004 ⋄ X 2 2 S S A B S 1 fibered over S 2 A × S 2 B ; S 2 ⋄ B squashed S 1 ‘winds’ p times on S 2 A and q times on S 2 ⋄ B 4 p 2 − 3 q 23 � − 1 � Vol( Y p,q ) = π 3 q 4 � − 8 p 3 + 9 pq 2 + p 2 24/31
What to do without an explicit metric ? ⋄ ⋄ X 5 Sasaki-Einstein ⇒ C ( X ) : Calabi-Yau ⋄ X 5 Sasaki ⇒ C ( X ) : K¨ ahler Dilation along the cone r ∂ ⋄ and the complex structure J ∂r ⇒ the Reeb vector R = Jr ∂ : isometry of X 5 ∂r 25/31
⋄ Given C ( X ) as an complex manifold, find Einstein metric on X ⇒ HARD . • find the volume of Einstein metric ⇒ Tractable . • ⋄ The cone has isometries k 1 , 2 ,... : Isometry in X ⇒ gauge field in AdS ⇒ Global sym. in CFT Reeb vector R = s i k i ↔ R-sym. R = s I Q I ⋄ 26/31
� √ g ( R − 12) ⋄ Let Z = X ⋄ Z = 8Vol( X ) if g is Sasaki Z : depends only on the Reeb vector , because ⋄ � � e − r 2 / 2 vol = e − H ω 3 Vol( X ) ∝ cone cone where H : the Hamiltonian for the Reeb vector ⇒ Apply Duistermaat-Heckman ! or, one can show explicitly show the integrand = total derivative Einstein metric extremizes Z ⇒ True s i extremizes Z ( s ) ! ⋄ 27/31
a and Z Z ( s 1 , s 2 , s 3 ) Toric Sasaki-Einstein: Z = a ( s 1 , s 2 , s 3 , s 1 B , s 2 Field Theory: a = B , . . . ) a ( s ) cubic, Z ( s ) rational. ⋄ s 1 , 2 , 3 and s 1 , 2 ,... ⋄ correspond to B the mesonic and the baryonic , resp. a ( s ) is cubic in s 1 , 2 , 3 and quadratic in s 1 , 2 ,... ⋄ B Maximization in s 1 , 2 ,... is Linear ⋄ B ⇒ It can be done easily ⇒ a = a ( s 1 , s 2 , s 3 ) 28/31
Recommend
More recommend
