The Squashed, Stretched The Squashed, Stretched and Warped Gets and Warped Gets Perturbed Perturbed

Igor KlebanovIgor Klebanov

PCTS and Department of PhysicsPCTS and Department of Physics

Talk at Rutgers Talk at Rutgers April 14, 2009April 14, 2009

IntroductionIntroduction• The gauge theory on coincident M2 branes has been a The gauge theory on coincident M2 branes has been a hot hot

topic topic over the past year.over the past year.

• This is a long-standing problem: how to find the world This is a long-standing problem: how to find the world volume theory on coincident super-membranes in 11-volume theory on coincident super-membranes in 11-dimensional M-theory. This is harder than the description dimensional M-theory. This is harder than the description of D-branes in string theory that is known explicitly at of D-branes in string theory that is known explicitly at small string coupling. small string coupling.

• But M-theory is inherently strongly coupled: one can think But M-theory is inherently strongly coupled: one can think of it as the strong coupling limit of a 10-dimensional of it as the strong coupling limit of a 10-dimensional superstring theory. What to do?superstring theory. What to do?

• The research on AdSThe research on AdS55/CFT/CFT44 has has rekindled interest in the rekindled interest in the maximally super-symmetric 4-d maximally super-symmetric 4-d gauge theory and provided a gauge theory and provided a host of information about its host of information about its strongly coupled limit. See the strongly coupled limit. See the January 2009 Physics Today January 2009 Physics Today article by I.K., J.Maldacena.article by I.K., J.Maldacena.

• This conformal gauge theory is This conformal gauge theory is becoming becoming `The Harmonic `The Harmonic Oscillator of 4-d Gauge Theory’ Oscillator of 4-d Gauge Theory’ in in that it may be exactly solvable. that it may be exactly solvable.

• It is has provided a It is has provided a `hyperbolic `hyperbolic cow’ cow’ approximation to various approximation to various phenomena at strong coupling. phenomena at strong coupling.

AdSAdS44/CFT/CFT33

• Besides describing all of Besides describing all of known particle physics, known particle physics, Quantum Field Theory is Quantum Field Theory is important for understanding important for understanding the vicinity of certain phase the vicinity of certain phase transitions, such as the all-transitions, such as the all-important water/vapor important water/vapor transition. transition.

• Here we are interested in a Here we are interested in a 3-d (Euclidean) QFT. 3-d (Euclidean) QFT.

• This transition is in the 3-d Ising This transition is in the 3-d Ising Model Universality Class. Model Universality Class.

• Other common transitions are Other common transitions are described by 3-d QFT with O(N) described by 3-d QFT with O(N) symmetry. symmetry.

• 3-d theories are also very important 3-d theories are also very important in describing 2-d quantum systems, in describing 2-d quantum systems, such as those in the Quantum Hall such as those in the Quantum Hall effect, high-Tc superconductors, etc. effect, high-Tc superconductors, etc.

• Can we find a Can we find a `Harmonic Oscillator’ `Harmonic Oscillator’ of 3-d Conformal Field Theory ? of 3-d Conformal Field Theory ?

O(N) Sigma ModelO(N) Sigma Model• Describes 2Describes 2ndnd order phase transitions in order phase transitions in

statistical systems with O(N) symmetry.statistical systems with O(N) symmetry.

• IR fixed point can be studied using the IR fixed point can be studied using the Wilson-Fisher expansion in Wilson-Fisher expansion in =4-d. =4-d.

• The model simplifies in the large N limit The model simplifies in the large N limit since it possesses conserved currentssince it possesses conserved currents

Higher Spin Gauge TheoryHigher Spin Gauge Theory

• An AdSAn AdS44 dual of the large N sigma model dual of the large N sigma model was proposed. was proposed. IK, Polyakov (2002)IK, Polyakov (2002)

• It is the Fradkin-Vasiliev gauge theory of an It is the Fradkin-Vasiliev gauge theory of an infinite number of interacting massless infinite number of interacting massless higher-spin gauge fields. higher-spin gauge fields.

• There is no small AdS curvature limit. This There is no small AdS curvature limit. This makes the theory difficult to study in the makes the theory difficult to study in the dual AdS formulation. This is an interesting dual AdS formulation. This is an interesting problem for the future.problem for the future.

M2 Brane TheoryM2 Brane Theory• The theory on N coincident M2-branes The theory on N coincident M2-branes

has has NN=8, the maximum possible =8, the maximum possible supersymmetry in 3 dimensions. supersymmetry in 3 dimensions.

• When N is large, its dual description is When N is large, its dual description is provided by the weakly curved AdSprovided by the weakly curved AdS44 x S x S7 7

background in 11-dimensional M-theory. background in 11-dimensional M-theory.

• This dual description is tractable and This dual description is tractable and makes many non-trivial predictions. makes many non-trivial predictions.

• A general prediction of the AdS/CFT A general prediction of the AdS/CFT duality is that the number of degrees duality is that the number of degrees of freedom on a large number N of of freedom on a large number N of coincident M2-branes scales as Ncoincident M2-branes scales as N3/23/2

I.K., A. Tseytlin (1996)I.K., A. Tseytlin (1996)

• This is much smaller than the NThis is much smaller than the N22 scaling found in the 4-d SYM theory scaling found in the 4-d SYM theory on N coincident D3-branes (as on N coincident D3-branes (as described by the dual gravity). described by the dual gravity). Gubser, I.K., Peet (1996) Gubser, I.K., Peet (1996)

What is the M2 Brane What is the M2 Brane Theory?Theory?• It is the Infrared limit of the D2-brane It is the Infrared limit of the D2-brane

theory, the theory, the NN=8 supersymmetric Yang-=8 supersymmetric Yang-Mills theory in 2+1 dimensions, i.e. it Mills theory in 2+1 dimensions, i.e. it describes the degrees of freedom at describes the degrees of freedom at energy much lower than (genergy much lower than (gYMYM))2 2

• The number of such degrees of freedom The number of such degrees of freedom ~ N~ N3/23/2 is much lower than the number of is much lower than the number of UV degrees of freedom ~ NUV degrees of freedom ~ N22. .

• Is there a more direct way to characterize Is there a more direct way to characterize the Infrared Scale-Invariant Theory?the Infrared Scale-Invariant Theory?

The BLG TheoryThe BLG Theory• In a remarkable development, Bagger In a remarkable development, Bagger

and Lambert, and Gustavsson and Lambert, and Gustavsson formulated an SO(4) Chern-Simons formulated an SO(4) Chern-Simons Gauge Theory with manifest Gauge Theory with manifest NN=8 =8 superconformal gauge theory. In Van superconformal gauge theory. In Van Raamsdonk’s SU(2)xSU(2) formulation,Raamsdonk’s SU(2)xSU(2) formulation,

• XXI I are the 8 fields transforming in (2,2), are the 8 fields transforming in (2,2), which is the 4 of SO(4) which is the 4 of SO(4)

NN=2 Superspace =2 Superspace FormulationFormulation

• Define bi-fundamental superfields Define bi-fundamental superfields rotated by SU(4)rotated by SU(4)flavorflavor symmetry symmetry

• The superpotential is The superpotential is Benna, IK, Klose, Smedback,Benna, IK, Klose, Smedback,

• Using SO(4) gauge group notation, Using SO(4) gauge group notation,

The ABJM TheoryThe ABJM Theory• Aharony, Bergman, Jafferis and Maldacena Aharony, Bergman, Jafferis and Maldacena

argued that the correct description of a argued that the correct description of a pair of M2-branes is slightly different. It pair of M2-branes is slightly different. It involves U(2) x U(2) gauge theory. involves U(2) x U(2) gauge theory.

• The SU(4) flavor symmetry is not manifest The SU(4) flavor symmetry is not manifest because of the choice of complex because of the choice of complex combinationscombinations

• The manifest flavor symmetry is The manifest flavor symmetry is SU(2)xSU(2) SU(2)xSU(2)

• For N M2-branes ABJM theory easily For N M2-branes ABJM theory easily generalizes to U(N) x U(N). The generalizes to U(N) x U(N). The theory with Chern-Simons coefficient theory with Chern-Simons coefficient k is then conjectured to be dual to k is then conjectured to be dual to AdSAdS44 x S x S77/Z/Zkk supported by N units of supported by N units of flux. flux.

• For k>2 this theory has For k>2 this theory has NN=6 =6 supersymmetry, in agreement with supersymmetry, in agreement with this conjecture. In particular, the this conjecture. In particular, the theory has manifest SU(4) R-theory has manifest SU(4) R-symmetry.symmetry.

SU(4)SU(4)RR Symmetry Symmetry• The global symmetry rotating the 6 The global symmetry rotating the 6

supercharges is SO(6)~SU(4). The supercharges is SO(6)~SU(4). The classical action of this theory indeed classical action of this theory indeed has this symmetry. has this symmetry. Benna, IK, Klose, SmedbackBenna, IK, Klose, Smedback

YYAA, A=1,…4, are complex N x N matrices., A=1,…4, are complex N x N matrices.

Enhanced SymmetryEnhanced Symmetry• For k=1 or 2 the global symmetry should For k=1 or 2 the global symmetry should

enhance to SO(8) according to the ABJM enhance to SO(8) according to the ABJM conjecture. This is not seen in the classical conjecture. This is not seen in the classical lagrangian but should appear in the quantum lagrangian but should appear in the quantum theory. In the superspace formulation, SU(4) theory. In the superspace formulation, SU(4) flavor symmetry should be present.flavor symmetry should be present.

• To find it, need to study the `monopole’ (or To find it, need to study the `monopole’ (or disorder) operators that create singular disorder) operators that create singular monopole field configurations at a point. monopole field configurations at a point.

Monopole OperatorsMonopole Operators

• These operators create magnetic flux These operators create magnetic flux in the diagonal U(1) subgroup. They in the diagonal U(1) subgroup. They are charged under the remaining are charged under the remaining gauge groups.gauge groups.

• For k=1 the singly-charged operator is For k=1 the singly-charged operator is

and the doubly-charged oneand the doubly-charged one• With their help, form a fundamental of With their help, form a fundamental of

SU(4)SU(4)flavorflavor IK, Klose, MuruganIK, Klose, Murugan

Proposal for U(2) x U(2)Proposal for U(2) x U(2)

• The explicit form of monopole operators The explicit form of monopole operators isis

• The operator T carries charge 1 under the The operator T carries charge 1 under the baryonic U(1) gauge group and is needed baryonic U(1) gauge group and is needed for maintaining the full gauge invariance. for maintaining the full gauge invariance.

• The superpotential generalizes that in the The superpotential generalizes that in the BLG theoryBLG theory

Relevant DeformationsRelevant Deformations

• The M2-brane theory may be The M2-brane theory may be perturbed by relevant operators that perturbed by relevant operators that cause it to flow to new fixed points cause it to flow to new fixed points with reduced supersymmetry. with reduced supersymmetry. Benna, IK, Benna, IK, Klose, Smedback; IK, Klose, Murugan; AhnKlose, Smedback; IK, Klose, Murugan; Ahn

• For example, a quadratic For example, a quadratic superpotential deformation, allowed superpotential deformation, allowed for k=1, 2, may preserve SU(3) flavor for k=1, 2, may preserve SU(3) flavor symmetrysymmetry

• For U(2) x U(2) ABJM theory, this For U(2) x U(2) ABJM theory, this becomesbecomes

• To understand the IR SCFT, we integrate To understand the IR SCFT, we integrate outout

to obtainto obtain

where A=1,2,3 where A=1,2,3

• The R-charge and dimension of these 3 The R-charge and dimension of these 3 bi-fundamental superfields is 1/3bi-fundamental superfields is 1/3

Squashed, stretched and Squashed, stretched and warpedwarped

• The dual AdSThe dual AdS44 background of M-theory background of M-theory should also preserve should also preserve NN=2 SUSY and =2 SUSY and SU(3) flavor symmetry. Such an SU(3) flavor symmetry. Such an extremum of gauge SUGRA was found extremum of gauge SUGRA was found 25 years ago by Warner. Upon uplifting 25 years ago by Warner. Upon uplifting to 11-d Corrado, Pilch and Warner to 11-d Corrado, Pilch and Warner found a warped product of AdSfound a warped product of AdS44 and of and of a `stretched and squashed’ 7-sphere: a `stretched and squashed’ 7-sphere:

• The squashing parameter is The squashing parameter is ; the ; the stretching is stretching is

• The four complex coordinates The four complex coordinates

may be expressed in terms of the 7 angles.may be expressed in terms of the 7 angles.

• The equations of motion are satisfied The equations of motion are satisfied withwith

• The internal components break parity The internal components break parity (Englert). They preserve a flavor (Englert). They preserve a flavor SU(3), and a U(1) R-symmetrySU(3), and a U(1) R-symmetry

The Spectrum via Group The Spectrum via Group TheoryTheory

• There are only two ways of breaking There are only two ways of breaking the SO(8) R-symmetry consistent the SO(8) R-symmetry consistent with the with the

Osp(2|4) symmetry in the IR:Osp(2|4) symmetry in the IR:

• We find that Scenario I gives We find that Scenario I gives SU(3)xU(1)SU(3)xU(1)RR quantum numbers in quantum numbers in agreement with the proposed gauge agreement with the proposed gauge theory, where they are schematically theory, where they are schematically given by given by

HypermultipletsHypermultiplets

• For even n, the operators areFor even n, the operators are

• Their explicit form for U(2) x U(2) isTheir explicit form for U(2) x U(2) is

• For odd n, we need the single monopole For odd n, we need the single monopole operator operator

• The quantum numbers are in agreement The quantum numbers are in agreement with SUGRA: with SUGRA:

Spin-2 PerturbationsSpin-2 Perturbations

• Consider graviton perturbations in Consider graviton perturbations in AdS withAdS with

satisfy the minimal scalar satisfy the minimal scalar equationequation

For the (p,q) irrep of SU(3), we find the For the (p,q) irrep of SU(3), we find the angular dependence angular dependence IK, Pufu, RochaIK, Pufu, Rocha

• The R-charge isThe R-charge is

• For the j-th KK mode the mass-squared isFor the j-th KK mode the mass-squared is

• The operator dimension is determined byThe operator dimension is determined by

• For operators in the MGRAV and SGRAV For operators in the MGRAV and SGRAV multipletsmultiplets

•

• Here are the low lying operatorsHere are the low lying operators

Further DirectionsFurther Directions• Other examples of AdSOther examples of AdS44/CFT/CFT33 dualities with dualities with

NN=1,2,3,… supersymmetry are being =1,2,3,… supersymmetry are being studied by many groups.studied by many groups.

• Various famous quivers assume new Various famous quivers assume new identitites: M111, Q222, etc.identitites: M111, Q222, etc.

• Ultimate Hope: to find a `simple’ dual of a Ultimate Hope: to find a `simple’ dual of a 3-d fixed point realized in Nature.3-d fixed point realized in Nature.

• Remarkably, the `M2 Mini-revolution’ Remarkably, the `M2 Mini-revolution’ may have percolated even to the may have percolated even to the study of recently observed study of recently observed Charmonium decays:Charmonium decays:

M2 signatures in psi(2S) M2 signatures in psi(2S) radiative decays.radiative decays.Jonathan L. Rosner . EFI-08-24, Sep . EFI-08-24, Sep 2008. 8pp. 2008. 8pp. Temporary entry e-Print: e-Print: arXiv:0809.0471arXiv:0809.0471 [hep-ph]  [hep-ph]

• But this is an entirely different But this is an entirely different story…story…