Three Extremalization Principles in AdS/CFT with Eight Supercharges - - PowerPoint PPT Presentation

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Three Extremalization Principles in AdS/CFT with Eight Supercharges - - PowerPoint PPT Presentation

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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

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SLIDE 1

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

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SLIDE 2

⋄

type IIB on AdS5 × X5 ⇐

⇒ CFT4 ⋄

8 supercharges:

  • N = 1 SCFT4
  • N = 2 5d sugra
  • X5: Sasaki-Einstein

⋄

R-symmetry in the superconformal algebra

= linear combination of U(1) charges

⋄

which is determined by the extremalization principles

1/31

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SLIDE 3

⋄

type IIB on AdS5 × X5 ⇐

⇒ CFT4 ⋄

8 supercharges:

  • N = 1 SCFT4

⇒

a

  • maximization

[Intriligator-Wecht]

  • N = 2 5d sugra

⇒

P

  • minimization

[YT]

  • X5: Sasaki-Einstein

⇒

Z

  • 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

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SLIDE 4

⋄

type IIB on AdS5 × X5 ⇐

⇒ CFT4 ⋄

8 supercharges:

  • N = 1 SCFT4

⇒

central charge

  • maximization
  • N = 2 5d sugra

⇒

superpotential

  • minimization
  • X5: Sasaki-Einstein

⇒

volume

  • minimization

⋄

R-symmetry in the superconformal algebra

= linear combination of U(1) charges

⋄

which is determined by the extremalization principles

1/31

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SLIDE 5

CONTENTS

⇒

1.

a-maximization ⋄

2.

P -minimization ⋄

3.

Z-minimization ⋄

4. Conclusion

2/31

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SLIDE 6
  • 1. a -maximization

Intriligator-Wecht, hep-th/0304128

2/31

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SLIDE 7

⋄

Studying Dynamics in 4d ⇒ Extra symmetry helps

N = 1 Susy Algebra in 4d {Qα, Q†

˙ β} = σµ α ˙ βPµ

Conformal Algebra in 4d

[Kµ, Pν] ∼ ∆δµν N = 1 Superconformal Algebra in 4d {Qα, Sα} ∼ RSC + · · · [RSC, Qα] = −Qα [RSC, Sα] = Sα

3/31

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SLIDE 8

⋄

RSC carries a lot of info:

  • ∆ ≥ 3

2RSC,

equal for chiral primaries

  • a = 3

32(3 tr R3 SC − tr RSC)

  • cf. On a curved mfd,

T µ

µ = a × Euler + c × Weyl2

where Euler and Weyl are polynomials in Rµνρσ

4/31

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SLIDE 9

⋄

Many global symmetries QI with [QI, Qα] = − ˆ

PIQα. ⋄ RSC is a linear combination : RSC = rIQI.

How can we find RSC ?

⋄

Call QF = fIQI with [QF , Qα] = 0 a flavor symmetry.

RSC RSC QF

← →

SUSY

Tµν QF Tµν

9 tr QF RSCRSC =

tr QF

5/31

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SLIDE 10

⋄ [rIQI, Qα] = −Qα ⇒ rI ˆ PI = 1. ⋄

Let a(s) = 3

32(3 tr R(s)3 − tr R(s)) where R(s) = sIQI.

⋄ 9 tr QF RSCRSC = tr QF ⇒ rI extremizes a(s) under sI ˆ PI = 1. ⋄

Unitarity ⇒ it’s a local maximum.

a -maximization !

⋄ ˆ cIJK = tr QIQJQK and ˆ cI = tr QI :

Calculable at UV using ’t Hooft’s anomaly matching.

6/31

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SLIDE 11

AdS5 × Y p,q ⇔

Y 4,3 図 の 図 の 図 と の

SU(N)2p gauge theory with four bifundamentals W α

I

Y Z Ui V i

multiplicity

2p p + q p − q p q

R-sym 1

s1 s2 1 − (s1 + s2)/2 1 + (s2 − s1)/2

Maximize a(s1, s2) ⇒

a = p2 4q4

  • −8p3 + 9pq2 +
  • 4p2 − 3q2

3

7/31

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SLIDE 12

CONTENTS

  • 1.

a-maximization

⇒

2.

P -minimization ⋄

3.

Z-minimization ⋄

4. Conclusion

8/31

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SLIDE 13
  • 2. P -minimization

YT, hep-th/0507057

8/31

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SLIDE 14

⋄

Any CFT phenomena

⇒ AdS counterpart via Gubser-Klebanov-Polyakov, Witten

AdS CFT

φ O Z[φ(x)

  • x5=∞ = ˆ

φ(x)] = e−

ˆ φ(x)O(x)d4x

AI

µ

↔ Jµ

I : current for QI

Z[AI

µ(x)

  • x5=∞ = ˆ

AI

µ(x)]

e−

ˆ AI

µJµ I SCF T

⋄

How about central charge maximization ?

⇒ superpotential minimization !

9/31

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SLIDE 15

QI has triangle anomalies among them : δχ(e−

ˆ AI

µJµ I SCF T ) =

  • d4x

1 24π2ˆ cIJK χI F J ∧ F K =

  • d5x

1 24π2ˆ cIJK dχI ∧ F J ∧ F K = δχ

  • d5x

1 24π2ˆ cIJK AI ∧ F J ∧ F K

  • ⇒ Presence of Chern-Simons terms in AdS

⋄ ˆ cI = tr QI ↔ ˆ cIAI ∧ tr R ∧ R , higher derivative effect.

10/31

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SLIDE 16

⋄

4d N = 1 SCFT ⇒ 5d N = 2 supergravity Multiplet structure

⋄

Minimal number of supercharges is 8, called N = 2

⋄

Gravity multiplet, Vector multiplet, Hypermultiplet

Gravity multiplet

gµν, ψi

µ,

AI

µ

⋄ i = 1, 2: index for SU(2)R. ⋄ I: explained in a second

11/31

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SLIDE 17

Vector Multiplet

AI

µ,

λx

i ,

φx ⋄ I : 0, . . . , nV and x : 1, . . . , nV

  • cf. I labels integral basis; graviphoton is a mixture of AI

µ.

⋄

φx parametrize a hypersurface

F = cIJKhIhJhK = 1

in (nV + 1) dim space of {hI} ; cIJK constants

⋄ hI = hI(φx): special coordinates

12/31

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SLIDE 18

⋄

Kinetic terms determined by cIJK :

−1 2gxy(φ) ∂µφx∂µφy − 1 4aIJ(φ) F I

µνF J µν

⋄

Presence of the Chern-Simons term :

1 6 √ 6 cIJKǫµνρστAI

µF J νρF K στ

⇒ cIJK = √ 6 16π2 tr QIQJQK.

13/31

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SLIDE 19

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 = 3gxy∂xP r∂yP r − 4P rP r

where P r = P r

I hI,

Gukov-Vafa-Witten type .

14/31

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SLIDE 20

⋄

P r appears everywhere:

  • Covariant derivative of the gravitino

Dνψi

µ = ∂νψi µ + AI µP i jIψj I

  • SUSY transformation law

δǫφx = i 2¯ ǫiλx

i

δǫλi

x = −ǫj

  • 2

3∂xP ij + · · ·

where P i

j = P rσri j

15/31

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SLIDE 21

Recap.

Field theory

⇔

Supergravity

ˆ cIJK = tr QIQJQK cIJKhIhJhK = 1 [QI, Qα] = − ˆ P IQα Dµψi

ν = (∂µ + P i jIAI µ)ψν

a-max

???

16/31

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SLIDE 22

SUSY condition for sugra

⋄ δλ = ǫj∂xP ij = 0 ⇒ hI

,xP ij I

= 0 ⋄

h∗

,x and P r ∗ is perpendicular as (nV + 1) dim’l vectors. ⋄ x = 1, . . . , nV ⇒ P r=1,2,3 are parallel ⇒ P r=1,2 = 0, P r=3 = 0. ⋄ cIJKhIhJhK = 1 ⇒ cIJKhIhJ

  • hK

hK

,x = 0

hI ∝ P r=3

I

: Attractor Eq.

17/31

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SLIDE 23

⋄

Recall

AdS CFT

ψµ ↔ Qα, Sα AI

µ

↔ QI Dνψi

µ = ∂νψi µ + AI µ P i jI ψj I

↔ [QI, Qα] = − ˆ PIQα. ⇒ P r=1,2 = 0 ⇒ charge of Qα, Sα under QI is ±P r=3

I

.

⇒ P r=3 I

= ˆ PI.

18/31

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SLIDE 24

⋄

SUSY tr. of hypers ⇒ RSC = rIQI ∝ hIQI.

⋄

Recall rIPI = 1. ⇒ rI =

hI hIPI

.

⋄ Rtrial = sIQI.

Suppose sI =

hI hIPI ⇒ a(s) ∝ tr(sIQI)3 = ˆ cIJKsIsJsK = cIJKhIhJhK (hIPI)3 ∝ (hIPI)−3. ⋄

a-max = P -min !

δλ = hI

,xPI = (hIPI),x = P,x = 0

19/31

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SLIDE 25

CONTENTS

  • 1.

a-maximization

  • 2.

P -minimization

⇒

3.

Z-minimization ⋄

4. Conclusion

20/31

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SLIDE 26
  • 3. Z -minimization

Martelli-Sparks-Yau, hep-th/0503183 hep-th/0603021

20/31

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SLIDE 27

⋄

Consider a D3 brane probing the tip of the CY cone Y

ds2

Y = dr2 + r2ds2 X

X :Sasaki-Einstein Y :Calabi-Yau D3 at the tip × (1+3) d 1d 5d

21/31

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SLIDE 28

Low energy theory on the D3 Near Horizon Limit of the D3

× ×

Some quiver gauge theory AdS5 × X5 Describe the same physics. AdS/CFT correspondence [Maldacena]

22/31

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SLIDE 29

Field theory on M3,1 Gravity on AdS5 × X5

Conformal symmetry

=

Isometry of AdS Global symmetry

=

Isometry of X RSC symmetry

=

Reeb vector

central charge a

=

1/Vol(X)

⋄ a is from a-maximization. How about Vol(X) ? ⋄

Many explicit metrics for X5 : S5, T 1,1, Y p,q and Lp,q,r [Gauntlett et al.], [Cvetiˇ c et al.]

⇒ Central charge a = (Vol)−1

23/31

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SLIDE 30

⋄ Y p,q: explicit 5d Sasaki-Einstein metric found in 2004

X

S S

A B 2 2

⋄ S1 fibered over S2

A × S2 B ; S2 B squashed

⋄ S1 ‘winds’ p times on S2

A and q times on S2 B

Vol(Y p,q) = π3q4

p2

  • −8p3 + 9pq2 +
  • 4p2 − 3q23−1

24/31

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SLIDE 31

⋄

What to do without an explicit metric ?

⋄ X5 Sasaki-Einstein ⇒ C(X) : Calabi-Yau ⋄ X5 Sasaki ⇒ C(X) : K¨

ahler

⋄

Dilation along the cone r ∂

∂r

and the complex structure J

⇒ the Reeb vector R = Jr ∂ ∂r

: isometry of X5

25/31

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SLIDE 32

⋄

Given C(X) as an complex manifold,

  • find Einstein metric on X ⇒ HARD.
  • find the volume of Einstein metric ⇒ Tractable .

⋄

The cone has isometries k1,2,... : Isometry in X ⇒ gauge field in AdS ⇒ Global sym. in CFT

⋄

Reeb vector R = siki ↔ R-sym. R = sIQI

26/31

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SLIDE 33

⋄

Let Z =

  • X

√g(R − 12) ⋄ Z = 8Vol(X) if g is Sasaki ⋄ Z: depends only on the Reeb vector , because Vol(X) ∝

  • cone

e−r2/2 vol =

  • cone

e−Hω3

where H : the Hamiltonian for the Reeb vector

⇒ Apply Duistermaat-Heckman !

  • r, one can show explicitly show the integrand = total derivative

⋄

Einstein metric extremizes Z ⇒ True si extremizes Z(s) !

27/31

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SLIDE 34

a and Z

Toric Sasaki-Einstein:

Z = Z(s1, s2, s3)

Field Theory:

a = a(s1, s2, s3, s1

B, s2 B, . . .)

⋄

a(s) cubic, Z(s) rational.

⋄

s1,2,3 and s1,2,...

B

correspond to the mesonic and the baryonic , resp.

⋄ a(s) is cubic in s1,2,3 and quadratic in s1,2,... B ⋄

Maximization in s1,2,...

B

is Linear

⇒ It can be done easily ⇒ a = a(s1, s2, s3)

28/31

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SLIDE 35

⋄

[Butti-Zaffaroni 0506232] showed that the equality

Z(s1, s2, s3) = a(s1, s2, s3)−1

holds before maximization.

⋄

Done by a brute force calculation.

⋄

Physical interpretation unclear.

29/31

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SLIDE 36

CONTENTS

  • 1.

a-maximization

  • 2.

P -minimization

  • 3.

Z-minimization

⇒

4. Conclusion

30/31

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SLIDE 37
  • 4. Conclusion

30/31

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SLIDE 38

DONE

Reviewed 3 kinds of extremalization principle

  • in 4d SCFT,
  • in AdS5,
  • and in the Sasaki-Einstein manifolds,

and their interrelations.

TO DO

⋄

Understand the interrelations in physical terms.

31/31