11

Marginal Deformations

and

Penrose limits with continuous spectrum

Toni Mateos

Imperial College London

Universitat de Barcelona, December 22, 2005

22

Introduction

• In general, CFTs are isolated points in space of couplings

( gi ) = 0 ! fixes all gi

- deformation, breaks SU(3) flavor ! U(1) £ U(1)

• other SCFTs, like Klebanov-Witten (T1,1) also admit - defomations,

SU(2) £ SU(2) flavor ! U(1) £ U(1)

• Susy ) -functions » anomalous

dimensions,

# anomalous dimensions < # marginal

couplings ) continuous families of CFTs

• e.g. N=4 , # exactly marginal deformations = 3C

33

Continuous families of CFTs=

Continuous familes of AdS5 £ X5 solutions

• Lunin and Maldacena: simple way out for – deformations.

• If SCFT has U(1) £ U(1)

8d solution, with SL(2,R) £ SL(3,R) duality group

• How to construct them? General case not known.

SL(2,R) acts on

• Original solution regular ) final solution regular, if

U

Introduction

AdS5 £ X5, isom( X5 ) ¾ U(1) £ U(1)

44

• Applicable to any FT with U(1) £ U(1) global symmetry (even non CFT !) (even non SUSY ! )

• Deformation of Lagrangian is simple to obtain: · ! *

• Sugra side very simple, and for finite

e.g. N=4 :

Introduction

stringy SL(2,Z) X

• If N 0 ) final N depends on number of Killing spinors

invariant under U(1) £ U(1).

55

Introduction

NN = 4 = 4 - deformed: - deformed:

66

Contents

Part II : Penrose Limit of -deformation of 4d N = 4 SYM

T. M.hep-th/0505243

Part I : Marginal deformations of 3d FTs with AdS4 duals

J. Gauntlett, S. Lee, T. M., D. Waldramhep-th/0505207

Part 0 : Exactly Marginal Deformations

• see also C. Ahn, J.F. Vazquez-Portitz, hep-th/0505168

• see also R. Mello Koch, J. Murugan, S. Smolic, M. Smolic, hep-th/0505227

Lunin, Maldacena, hep-th/0502086

77

Part I: AdS4 and 3d Field Theories

• At least at the level of supergravity, method generalises to AdS solutions of D=11.

AdS4 £ Y7 , Isom(Y7) ¾ U(1)3

8d solution, with SL(2,R) £ SL(3,R) duality group

SL(2,R) acts on

• Are there similar exactly marginal deformations of 3d CFTs ?

• FT on M2 branes much less understood (strongly coupled IR)

• Solid proposals have been made for some cases

Part I: Supergravity

Field theory

U

88

I.a. Supergravity

M2

C(Y7)

Y7

ds2 = dr2 + r2 ds2(Y7)

• Susy after deformation: L = 0

Part I: AdS4 and 3d Field Theories

• Sasaki-Einstein: N = 2

N = 0, if U(1)3

N = 2, if U(1)4

• Tri-Sasaki: N = 3 ! N = 1

( very non-trivial !)• Weak G2 : N = 1 ! N= 1

99

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1010

I.a. Supergravity

Part I: AdS4 and 3d Field Theories

• Deformation procedure simplified. Pick 3 U(1)'s and...

1111

I.a. Supergravity

Part I: AdS4 and 3d Field Theories

• Deformation of AdS4 £ Yp,q (Sasaki-Einstein, 2 ! 2)

1212

I.a. Supergravity

Part I: AdS4 and 3d Field Theories

• Deformation of AdS4 £ S7squashed (weak G2, 1 ! 1)

1313

Q(1,1,1) M(3,2) N(1,1)

• Moduli space = C(Y7)

X

X

X• Spectrum of chiral operators = KK spectrum Y7

• of baryons = energy of M5 wrapping 5-cycles

Part I: AdS4 and 3d Field Theories

• Like in QCD !: empirical data in IR ) UV lagrangian

I.b. Field Theory

probe particles / M-branes in

AdS5 £ { Q(1,1,1), M(3,2), N(1,1) }

1414

I.b. Field Theory

How to identify -deformation without NC open string intuition?

• Look for a superpotential with:

• Unique answer for N=4 and T1,1 in 4d.

W ( · ! * ) = cos WN=4 + i sin Tr (123+132)

• = 2 chiral primary

• Global symmetry ! U(1)3

• N ! N

Part I: AdS4 and 3d Field Theories

• Unique answer for the 3d known susy cases.XX X

X

1515

I.b. Field Theory

Part I: AdS4 and 3d Field Theories

• AdS4 £ Q(1,1,1), N=2

• Global Symmetry: SU(2)3 £ U(1)R

• Gauge Symmetry: SU(N)3

• W = Tr (ABC ABC) in the (3,3,3) of SU(2)3

• Chiral primaries: Tr (ABC )k ,

= k, in (k+1,k+1,k+1) of SU(2)3

• U(1)3 preserving: uniqueX

• Baryons: det A, det B, det C ! A = B = C = 1/3

1616

Summary

• Extension of -deformations for 3d CFTs via AdS4 duals.

• Prediction N = (3, 2, 1) ! N = (1, 2 / 0, 1)

• Operators identified without open string theory.

Possibility of studying modification of chiral ring,

new branches…

• Discussion in paper about non-susy deformations.

( see paper! )

Part I: AdS4 and 3d Field Theories

I.b. Field Theory

1717

Part II: Penrose Limit of deformed N=4

New exact results SCFT $ string theory ?

• Spectrum of chiral operators

S5S5

SO(6)

· ! * add complicated phases ei

[Berenstein, Leigh, Jejjala]

N = 4 :

N = 1 :

1818

• Focus on

huge discrete degeneracy

• Expectations:

N = 4 :

N = 1 :

• All states with zero charge under U(1) £ U(1)

! not affected

• Vacuum should be unique :

• Other exchanges:

Part II: Penrose Limit of deformed N=4

· ! * add complicated phases ei

1919

• Covariantly constant null Killing v + null potentials

) generalised super-GS action ( quantisable! )

• U(1) £ U(1) ½ S5 of -deformation still isometry (y2 , y4)

Part II: Penrose Limit of deformed N=4

• Penrose Limit ! IIB configuration:

NS-NS: G , B ,

R-R: F5 , F3

• Number of supersymmetries = 16 + 4 = 20

2020

• For n 0 (stringy modes) :

Quantisation of Bosonic Sector

• For n 0 (particle-like modes):

Part II: Penrose Limit of deformed N=4

2121

• n = 0, decoupling of planes y1y2 and y3y4

Quantisation of Bosonic Sector

Vacuum with 1 discrete degeneracy

= 0 : Landau problem

Part II: Penrose Limit of deformed N=4

X

2222

If 0 : Landau + spring

y2

y1

2323

If 0 : Landau + spring

Vacuum unique!

Spectrum continuous!

2424

y2

y1

Part II: Penrose Limit of deformed N=4

2525

Part II: Penrose Limit of deformed N=4

y2

y1

v2 » y1

• it takes energy to speed up / climb the wall

• constraint system (2nd class) ! Dirac bracket quantisation

2626

Field Theory Interpretation

Charge under py , Charge under U(1) £ U(1)

if uncharged ) f() X

if charged ) f() ) more energy ( )

X

Part II: Penrose Limit of deformed N=4

, Departure from

2727

Field Theory Interpretation

Part II: Penrose Limit of deformed N=4

(J,0,0)

(J,J,J)

2828

Summary

• Exploration of new phenomena of SCFTs via AdS

(chiral ring, spectrum of anomalous dimensions)

• New exactly solvable (and physically motivated)

string theory backgrounds ( F3 , F5 , H3 )

• Half-way between flat-space and pp-wave ( Modified Landau Problem )

• Predictions for of dual operators

Part II: Penrose Limit of deformed N=4

- End -

• String Theory analysis directly applicable

to other cases; e.g. Penrose limit of AdS5 £ T1,1