jaffer is holography book

8/12/2019 Jaffer Is hologrAPHY BOOK

1/55

Recent Advances in 3dRecent Advances in 3dHolographyHolography

Daniel LouisDaniel Louis Jafferis JafferisInstitute for Advanced Study Institute for Advanced Study

Crete Conference On Gauge Theories And The Structure

Of Spacetime

14 September, 2010


2/55

OutlineOutlineReview Review

Regimes of large N gauge theoriesRegimes of large N gauge theories

SS33 partition functionspartition functions

AdS AdS44 generalizationsgeneralizations

Cascades and sCascades and s--rulerule


3/55

Understanding MUnderstanding M--theory theory

Physics of 3d quantum field theory Physics of 3d quantum field theory

CFT duals for the LandscapeCFT duals for the LandscapeNew exactly solvable theory New exactly solvable theory

Condensed matter applicationsCondensed matter applications

MotivationsMotivations


4/55

Last yearLast year s outlook s outlook Complete the analysis ofComplete the analysis of N N =2 CSM duals to M2=2 CSM duals to M2branesbraneson CY on CY 44. Explore. Explore N N =1 theories, for example in the large=1 theories, for example in the largeNNf f limit.limit.Duals for landscape IIADuals for landscape IIA vacua vacua..Learn about the strong coupling limit of massive IIALearn about the strong coupling limit of massive IIAExplain NExplain N3/23/2 -- derive the 1/derive the 1/suppression ofsuppression ofdofsdofs..More tests ofMore tests ofintegrability integrability ..Further explore the connection to 2+1 condensedFurther explore the connection to 2+1 condensedmatter systems.matter systems. Learn something new

about M-theory?


5/55

M2M2 branesbranes The conformal field theory on M2 branes is theinfrared limit of theN N = 8, 2+1 dimensional U(N) Yang-Mills theory of N D2-branes in flat space. The YM coupling in 3d is dimensionful, and the theorybecomes strongly coupled in the IR. The IR R-symmetries that control the dimensions of operatorsare not manifest in the UV.

The lack of adjustable coupling in M-theory suggestthat theIR CFT had noIR CFT had noLagrangianLagrangiandescription.description.Moreover, the black D2Moreover, the black D2branebranehas no smooth nearhas no smooth nearhorizon region in 10d SUGRA.horizon region in 10d SUGRA.


6/55

A background in which they become weaklycoupled was found, due to the presence of asmall circle.

Moreover, reducing to IIA string theory along tnatural U(1) isometry results in a background

which the black D2 brane solution has a smoo AdS near horizon limit.

Why is there an M2Why is there an M2 branebraneLagrangianLagrangian ??

= 2/k


7/55

Black D2/M2Black D2/M2 branesbranes The black D2 gravity solution in asymptotically flat The black D2 gravity solution in asymptotically flat

space does not have a smoothspace does not have a smooth AdS AdSnear horizon limit.near horizon limit. The string coupling blows up near the D2, so one lif The string coupling blows up near the D2, so one lif

to Mto M--theory. The black M2 solution in 11d has atheory. The black M2 solution in 11d has asmooth near horizon geometry with Nsmooth near horizon geometry with Nunits of flux inunits of flux in AdS AdS..

The effective The effective worldvolume worldvolumegauge theory on N D2gauge theory on N D2branesbranesis theis the N N = 8 super Yang = 8 super Yang --Mills withMills withdimensionfuldimensionfulcoupling.coupling.

AdS 4 S 7


8/55

A different IIA reduction A different IIA reduction The has a natural U(1) The has a natural U(1)isometry isometry ,,

associated to the description of Sassociated to the description of S77

as a Sas a S11

bundle overbundle overCPCP33. In the. In thett HooftHooftlimit one gets IIA onlimit one gets IIA on with k units of F with k units of F22 flux.flux.

This extends to the entire black M2 solution. This gi This extends to the entire black M2 solution. This gia background of IIA, with varyinga background of IIA, with varyingdilatondilatonand Fand F22 flux,flux,in which the black D2in which the black D2doesdoes have a smooth nearhave a smooth nearhorizon.horizon.

The string coupling is small if k is large. The string coupling is small if k is large.

AdS 4 CP 3

(Nk )3/ 2

/k = N 2

(N/k ) 1 / 2

AdS 4 S 7 /Z k

[Nilsson Pope, VolkovSorokin Tkach]


9/55

ChernChern --SimonsSimons --matter theorymatter theory We first consider the case with N=2 We first consider the case with N=2susy susy . It consists of. It consists ofa vectora vectormultipletmultipletin thein theadjointadjointof the gauge group, andof the gauge group, andchiralchiralmultipletsmultipletsin representationsin representations

The kinetic term for the The kinetic term for thechiralchiralmultipletsmultipletsincludesincludescouplingscouplings

There is the usual D term There is the usual D term

R i

i 2i i i

i D i

S N =2CS = k4

(A

dA + 23 A3

+ 2 D )


10/55

We integrate out D, , andS N =2 = k4 (AdA + 23A3 ) + D i D i + ii D i

162

k2 (i T aR i i )( j T

bR j j )( k T

aR k T

bR k k )

4k

(i T aR i i )( j T aR j j )

8k

( i T aR i i )( j T aR j j ).

Note that this action has classically marginal

couplings. It is has been argued that it does notrenormalize, up to shift of k, and so is a CFT.


11/55

N=3 CSN=3 CS --mattermatter To obtain a more To obtain a moresupersymmetricsupersymmetrictheory, begin withtheory, begin withN=4 YMN=4 YM--matter. Then add the CS term, breaking tomatter. Then add the CS term, breaking toN=3.N=3.

Thus we add an Thus we add anadjointadjointchiralchiralmultipletmultiplet, ,with no, ,with nokinetic term in the CS limit, and the matterkinetic term in the CS limit, and the matterchiralchiralmultipletsmultiplets, , which must come in pairs., , which must come in pairs.

There is a superpotential, , needed tosupersymmetrize the CS term.

i , i

W = k8 T r (2)


12/55

Integrating out one obtains the same action asIntegrating out one obtains the same action asbefore, but with abefore, but with asuperpotentialsuperpotential::

These N=3 theories are completely rigid, and hence These N=3 theories are completely rigid, and hence

superconformalsuperconformal. It is impossible to have more. It is impossible to have moresupersymmetry in a YMsupersymmetry in a YM--CSCS--matter theory, but formatter theory, but forparticular choices of gauge groups and matterparticular choices of gauge groups and matter

representations, the pure CSM can have enhancedrepresentations, the pure CSM can have enhancedsupersymmetry.supersymmetry.

[ [ Zupnik Zupnik ,, KhetseliusKhetselius, Kao, Lee, Lee, Schwarz,, Kao, Lee, Lee, Schwarz,GaiottoGaiotto, Yin], Yin]

W = 4k

(i T aR i

i )( j T aR j

j )


13/55

The N=6 CSM theory of N M2The N=6 CSM theory of N M2branesbranes in Cin C 44/Z/Z k k

U(N)U(N)k k x U(N)x U(N)--k k CSM with a pair ofCSM with a pair ofbifundamentalbifundamental

hypermultipletshypermultipletsField content:Field content:

SU(2) x SU(2) global symmetry, which doesSU(2) x SU(2) global symmetry, which doesnotnotcommute with SO(3)commute with SO(3)R R , combining to form SU(4), combining to form SU(4)R R

(C I ), (I ) in ( N , N ) their conjugatesC I ,

I

in (N, N ) matter elds

A , A gauge elds

C I = ( Aa , Ba ).W = 2k ab a b(Aa B a AbB b)

[Aharony, Bergman, Maldacena,DLJ]

[ABJM, Benna, Klebanov, Klose, Smedback, Bandres,Lipstein, Schwarz, Schanbl, Tachikawa]


14/55

tt HooftHooft LimitLimit The gauge theory coupling is 1/k. Fix The gauge theory coupling is 1/k. Fix

Perhaps disappointingly, but unsurprisingly, thPerhaps disappointingly, but unsurprisingly, th

usualusualtt HooftHooftlimit is a string theory.limit is a string theory.One obtains IIA on AdSOne obtains IIA on AdS44 CPCP33 with N units of with N units ofFF44 and k units of Fand k units of F22 in CPin CP33

= N/k , N

R 2str = 2 5/ 2 + 124 1 1k 2 + 22k 2Higher order curvature correction[Bergman Hirano]gIIA

1 / 4

k


15/55

Why is CSM a theory of M2Why is CSM a theory of M2 branesbranes ?? There is an extra circle which emerges only at strong There is an extra circle which emerges only at strongcoupling, , due to the monopole operators.coupling, , due to the monopole operators.

If one gives anIf one gives aneigenvalueeigenvaluea VEV, , soa VEV, , soone of the M2one of the M2branesbranesis at distance then theis at distance then themass of the off mass of the off --diagonal modes scales likediagonal modes scales like

This is the area of a cone, rather than a length, as This is the area of a cone, rather than a length, asexpected from a wrapped M2.expected from a wrapped M2.

R = 3/ 2P v,

[Mukhi Papageorgakis, Lambert Tong, Distler MukhiPapageorgakis van Raamsdonk, Berenstein Trancanelli]

C I = v 00 0

1k v

2 = Rk 3P

N k5


16/55

Two natural generalizationsTwo natural generalizationsFind other 7d conicalFind other 7d conical

backgrounds in IIA withbackgrounds in IIA with vanishing vanishingdilatondilatonat theat theorigin, in which the blackorigin, in which the black

D2D2branebrane will have a will have asmooth near horizon.smooth near horizon.

Most natural method forMost natural method forM2M2branesbranesononsusy susy 88--manifolds.manifolds.

Marginally deform theMarginally deform the

CFT, or follow a relevantCFT, or follow a relevantoperator to a new fixedoperator to a new fixedpoint, and identify thepoint, and identify the

dual geometry.dual geometry.

Typically gives Typically gives vacua vacua with fluxes on the with fluxes on theinternal manifold.internal manifold.


17/55

Fractional M2Fractional M2 branesbranesOne can also consider theOne can also consider theU(N+U(N+ ) )k k U(N)U(N)--k k CSM theory. This retainsCSM theory. This retains N N =6 supersymmetry.=6 supersymmetry.In the DIn the D--branebraneconstruction, it corresponds toconstruction, it corresponds tounequal numbers of stretched D3unequal numbers of stretched D3branesbranes..In MIn M--theory, one obtains N M2theory, one obtains N M2branesbranes, together, together

with with fractional M2 at the singularity fractional M2 at the singularity

5(1,k)

D3N

D3N+l

NS5

[Hosomichi Lee Lee Lee Park, Aharony Bergman DLJ]

[Aharony Bergman DLJ]


18/55

BB--field in IIA field in IIA In the IIA near horizon limit this corresponds In the IIA near horizon limit this corresponds turning on the B field in the internal space.turning on the B field in the internal space.Roughly speaking, this is the reduction of theRoughly speaking, this is the reduction of theflat Cflat C--field. It doesnfield. It doesnt affect the equations oft affect the equations ofmotion, but does shift the quantizationmotion, but does shift the quantizationcondition for Fcondition for F44..Since supersymmetry requires that FSince supersymmetry requires that F44 vanishes, vanishes,B is dynamically quantized in units of 1/k J.B is dynamically quantized in units of 1/k J. Appears to be a mysterious shift by Appears to be a mysterious shift by..

[Aharony Hashimoto Hirano Ouyang]


19/55

Massive IIA Massive IIA Consider deforming the N=6 CSM theory by theConsider deforming the N=6 CSM theory by the

addition of a level a CS term for the second gaugeaddition of a level a CS term for the second gaugegroup.group.

In this theory the monopole operators correspondingIn this theory the monopole operators correspondingto D0to D0branesbranesdevelop a tadpole, since the induceddevelop a tadpole, since the inducedelectric charge (k, aelectric charge (k, a--k) cannot be cancelled with thek) cannot be cancelled with thematter fields.matter fields.

This motivates the idea that the total CS level should This motivates the idea that the total CS level shouldrelated to the Frelated to the F00 flux.flux.

[Gaiotto Tomasiello, Fujita Li Ryu Takayanagi, PetriniZaffaroni]

U (N )k U (N )k+ a


20/55

The light U(1) on the The light U(1) on themodulimodulispace has a levelspace has a levela a ChernChern--Simons term, matching the coupling ofSimons term, matching the coupling of

the D2the D2 worldvolume worldvolumeto the Romans mass.to the Romans mass.

For such deformations ofFor such deformations of N N =6 CSM, there are=6 CSM, there arefield theories withfield theories with N N = 3,2,1,0 differing by the= 3,2,1,0 differing by the

breaking of the SU(4) into flavor and R breaking of the SU(4) into flavor and R --symmetry. Still have the topologysymmetry. Still have the topology

kCS (A1) + ( a k)CS (A2) + |X |2(A1 A2)2

[Tomasiello; Gaiotto Tomasiello]

AdS4 CP 3


21/55


22/55


23/55

tt HooftHooft limitlimit The string coupling always becomes weak. The string coupling always becomes weak.

When When

is small, always have large curvature inis small, always have large curvature in

string units. Whenstring units. When is large, it dependsis large, it depends forforexampleexample N N =2=2 U(N)U(N)k k with g > 2 with g > 2adjointsadjointshas ahas astringy dual.stringy dual.

Open question: how can one tell given a CFTOpen question: how can one tell given a CFT

gs /N in AdS5 and gs 5/ 4 /N in AdS4


24/55

Weakly coupled planar limitWeakly coupled planar limitShould be dual to the very weakly interactingShould be dual to the very weakly interactinglimit of string theory in a highly curvedlimit of string theory in a highly curvedbackground.background.

Extreme example is the singlet sector of a freeExtreme example is the singlet sector of a freetheory of N fields. This is dual to thetheory of N fields. This is dual to the Vasiliev Vasiliev sshigher spin gauge theory. Thehigher spin gauge theory. Themasslessmasslesshigherhigherspin fields correspond to the higher spinspin fields correspond to the higher spincurrents of the (trivially)currents of the (trivially)integrableintegrablefree theory.free theory.

[Klebanov Polyakov; Sezgin Sundell; Giombi Yin; ]


25/55

Vasiliev Vasiliev theorytheoryIn the large N limit, it seems that the ( In the large N limit, it seems that the ( MM AdS AdS/M/Mplpl ) )perturbativeperturbativeexpansion of the nonlocalexpansion of the nonlocal Vasiliev Vasiliev theory should be a change of variables from ththeory should be a change of variables from th1/N expansion in the boundary theory, with a1/N expansion in the boundary theory, with a

huge gauge redundancy. This is furtherhuge gauge redundancy. This is furthersuggested by the existence of a gauge in whicsuggested by the existence of a gauge in whicthe dependence onthe dependence on AdS AdSdrops out.drops out.

On the other hand, can one engineer it in strinOn the other hand, can one engineer it in strintheory?theory?


26/55

Strict large N limitStrict large N limitIn theIn the N N =6 theory, this limit results in light=6 theory, this limit results in lightmonopole operators corresponding to the lighmonopole operators corresponding to the lighD0D0branesbranesof IIA at strong coupling. There is anof IIA at strong coupling. There is anMM--theorytheorysugrasugradescriptiondescription which does not which does notallow us to calculate much beyondallow us to calculate much beyondsupergravity supergravity ..

Not the only possible behavior: one can insteaNot the only possible behavior: one can insteaobtain weakly interacting strings again!obtain weakly interacting strings again!


27/55

A new weakly coupled string regime A new weakly coupled string regimeConsider the massive IIA solution dual toConsider the massive IIA solution dual toU(N)U(N)k k U(N)U(N)--k+ak+a

This is in spite of the fact that the This is in spite of the fact that the N N =2 theory=2 theoryhas light monopole operators.has light monopole operators.It would be interesting to understand the geneIt would be interesting to understand the genebehavior.behavior.

Rstr N a 1/ 6gs 1(Na 5 ) 1 / 6 1G N

N 5/ 3a1/ 3

[Aharony DJ Tomasiello Zaffaroni]


28/55

Partition functions on SPartition functions on S33

This partition function of the Euclidean theory This partition function of the Euclidean theoryis given in classicalis given in classicalsupergravity supergravity by minus theby minus theEuclidean Einstein action of theEuclidean Einstein action of the AdS AdS..

The surface term is an integral of the extrinsic The surface term is an integral of the extrinsiccurvature of the boundary, the last term is acurvature of the boundary, the last term is ahigher derivative boundaryhigher derivative boundarycountertermcounterterm..Bulk CS terms would lead to an imaginary paBulk CS terms would lead to an imaginary pa

S =

1

16G N d4x g(R

2) + S surf + S ct =

2GN [Balasubramanian, Emparan, Johnson, Kraus, Larsen, Myers, Siebelink,Taylor,]


29/55


30/55

Matrix integralMatrix integral All fields are set to zero, except All fields are set to zero, except== -- D,D,constant on the sphere. Results in a matrixconstant on the sphere. Results in a matrixintegral after going to theintegral after going to theeigenvalueseigenvalues.. The gauge field 1 The gauge field 1--loop determinant togetherloop determinant together with the with the Vandermonde Vandermondegives a measure factor of gives a measure factor of

The matter sector localization requires that th The matter sector localization requires that thfields have canonical dimensions ( fields have canonical dimensions ( ieie.. N N = 3).= 3).

i


31/55

NN3/23/2

from the field theoryfrom the field theory

Solved bySolved byDrukkerDrukkerMarinoMarinoPutrov Putrov ! The leading free! The leading freeenergy when N = M isenergy when N = M is

This even captures the This even captures thesubleading subleading corrections to thecorrections to theM2 charge. When BM2 charge. When B0, there is a nontrivial phase,0, there is a nontrivial phase, which remains to be understood in the gravity dual. which remains to be understood in the gravity dual.Note that NNote that N3/23/2 is special to this matrix model.is special to this matrix model.

1N !M !

dui2

dvj2

i


32/55

CSM from geometry and anCSM from geometry and an isometryisometry The theory of M2 The theory of M2branesbranesprobing a geometry with aprobing a geometry with a

U(1)U(1)isometry isometry is part of a family made by takingis part of a family made by takingZZk k quotients with weakly coupled gauge groups in the lquotients with weakly coupled gauge groups in the lk limit.k limit.

Moreover, all of the theories now known to describeMoreover, all of the theories now known to describeM2M2branesbraneshave a baryonic U(1) associated to have a baryonic U(1) associated to . This includes those with pure Yang This includes those with pure Yang --Mills, where weakMills, where weakcoupling is obtained in the largecoupling is obtained in the largeNNf f limit, giving exactlylimit, giving exactlyaa

i T r(F i )

Z N f U (1)B quotient.


33/55

Varieties of U(1) Varieties of U(1)BB

actionaction1) There are no fixed points away from the origin.1) There are no fixed points away from the origin.

2) If the U(1) shrinks away from the origin, then the2) If the U(1) shrinks away from the origin, then the will be explicit D6 will be explicit D6branesbranesin the IIA reduction. Ex:in the IIA reduction. Ex:QQ111111, which is a circle bundle over S, which is a circle bundle over S22 x Sx S22 x Sx S22, with, withU(1) action generated by rotation of two of the spherU(1) action generated by rotation of two of the spher

3) It is also possible that Y has non3) It is also possible that Y has non--isolatedisolatedorbifoldorbifoldsingularities. For example, Csingularities. For example, C44 with a with a

acting with charges 1,acting with charges 1,--1, p,1, p,--p, which is related to NS5p, which is related to NS5 (p,q)5 configurations, has a(p,q)5 configurations, has anonisolatednonisolatedZZpp singularitysingularityin the IIA reduction.in the IIA reduction.

Z q U (1)

The weakly coupled manifestly N=4 theory is mysterious in IIB. Inprinciple determined by Gaiotto Witten


34/55

Corresponding field theoriesCorresponding field theoriesIn the first case, theIn the first case, the worldvolume worldvolumetheory will betheory will bejust that of D2just that of D2branesbranesin the 7d cone. Within the 7d cone. With N N ==2, they are D2 on CY 32, they are D2 on CY 3--fold plus CS terms.fold plus CS terms. The second case results in theories with flavo The second case results in theories with flavo The generic case 3 is probably described by The generic case 3 is probably described bystrongly coupled CFT interacting via CS gaugstrongly coupled CFT interacting via CS gaug

theories.theories.Changing the U(1) used gives a dual theory,Changing the U(1) used gives a dual theory,generalizing 3d mirror symmetry.generalizing 3d mirror symmetry.


35/55

ModuliModuli space of N=2 CSMspace of N=2 CSM There are F There are F--term equations of the usual typeterm equations of the usual type

We have a D3 quiver on Y with n nodes, all fields ar We have a D3 quiver on Y with n nodes, all fields aradjointsadjointsoror bifundamentalsbifundamentals..

The D The D--term equations are replaced by cubic equationterm equations are replaced by cubic equation where whereaa are the usual moment maps.are the usual moment maps.

The M The M--theory geometry is the solutiontheory geometry is the solutionaa = k = k aa r, andr, andthe CS term implies that one only gauges the kernel the CS term implies that one only gauges the kernel This is precisely the geometry X. This is precisely the geometry X.

: (u1 , , u n )U (1)n uk 11 ...u k nn

1k a a T ib q i = 0,

[Tomasiello DLJ, Martelli Sparks, Hanany Zaffaroni]

W = 0.


36/55

Monopoles in theMonopoles in the chiralchiral ringring There are monopole operators in YM There are monopole operators in YM--CSCS--mattermattertheories, which we follow to the IR CSM.theories, which we follow to the IR CSM.

In radial quantization, it is a classical background wIn radial quantization, it is a classical background wimagnetic flux , and constant scalar,magnetic flux , and constant scalar,

. Of course, in the CSM limit,. Of course, in the CSM limit,

It is crucial that the fields inIt is crucial that the fields inare not charged under a.are not charged under a.

This operator creates a vortex. This operator creates a vortex.

S 2 F a = 2n

= n/ 2

[Borokhov Kapustin Wu]

a = k1


37/55

M2M2 branesbranes onon CalabiCalabi -- Yau Yau 44--foldsfoldsConsider a conicalConsider a conicalCalabiCalabi-- Yau Yau44--fold, X, with a U(1)fold, X, with a U(1)isometry isometry that leaves thethat leaves theholomorphicholomorphic44--form invariant.form invariant.For example, aFor example, atorictoricCY CY 44 will have 3 such U(1) will have 3 such U(1)s.s.

Then the Then theK K hlerhlerquotient X//U(1) will be a CY quotient X//U(1) will be a CY 33, Y., Y.Reducing MReducing M--theory on X to IIA on this U(1) gives a 7dtheory on X to IIA on this U(1) gives a 7dcone which is Y fibered over a real line, with Fcone which is Y fibered over a real line, with F22 flux,flux, varying varyingdilatondilaton, and, and

The F The F22 flux scales with k if one starts with X/flux scales with k if one starts with X/ZZk k ..

F I a = ka r.

[DLJ Tomasiello, Martelli Sparks, Hanany Zaffaroni, Aganagic]


38/55

N=2 CSM from D3 quiversN=2 CSM from D3 quivers We want to know the theory on N M2 We want to know the theory on N M2branesbraneson X. Iton X. Itis the IR limit of the theory on N D2is the IR limit of the theory on N D2branesbraneson the 7don the 7dcone X/U(1), with Fcone X/U(1), with F22 flux.flux.

Take the reduction to 2+1 of the quiver theory Take the reduction to 2+1 of the quiver theorydescribing N D3describing N D3branesbraneson Y. In the resolved geometryon Y. In the resolved geometry

we can image this describes the fractional we can image this describes the fractionalbranesbranesas D4as D4and D6 onand D6 onholomorphicholomorphic2 and 4 cycles in Y. The CS2 and 4 cycles in Y. The CSterms arise from the D4terms arise from the D4 worldvolume worldvolumecouplingcoupling

The coupling must arise from the The coupling must arise from thefibrationfibrationofof Y over R Y over R 11..

F 2S CS (a)k2 D

[Aganagic]


39/55

D6D6 branesbranes in AdSin AdS 44Given this large class of quiver CSMGiven this large class of quiver CSMtheories describing a stack of M2theories describing a stack of M2branesbranesat a (hyper)at a (hyper)torictoricsingularity. In thesingularity. In thett HooftHooftlimit, the dual geometrylimit, the dual geometryis a warped productis a warped product

Introducing D6Introducing D6branesbranes wrapping an internal, wrapping an internal,homologically trivial, 3homologically trivial, 3--cycle ( in the case) addscycle ( in the case) addsfundamentalfundamentalhypermultipletshypermultipletsto the quiver.to the quiver.CChoice ofhoice ofZZ22 Wilson line corresponds to which node the Wilson line corresponds to which node thefundamental is attached.fundamental is attached.Interestingly,Interestingly,conformality conformality is preserved.is preserved.

AdS4 w M 6

RP 3 CP 3

[Hohenegger Kirsch, Gaiotto DLJ, Hikida Li Takayanagi, Fujita Tai]

N = 2 , 3


40/55

Adding flavors Adding flavors The addition of fundamental flavors will alter The addition of fundamental flavors will alterthethechiralchiralring, corresponding to the quantumring, corresponding to the quantumcorrection of thecorrection of themodulimodulispace.space.MesonicMesonicoperators are unaffected, but BPSoperators are unaffected, but BPSmonopole operators that appear in themonopole operators that appear in thechiralchiralreceive a quantum contribution to theirreceive a quantum contribution to their

dimension, and a corresponding change of thedimension, and a corresponding change of theOPE.OPE.


41/55

Anomalous dimension Anomalous dimensionN=2 caseN=2 case

We work in the UV to compute the 1 We work in the UV to compute the 1--looploopcorrection to the charge of a monopole operatcorrection to the charge of a monopole operatunder some flavor U(1). Followingunder some flavor U(1). FollowingBorokhov Borokhov --KapustinKapustin-- Wu, Wu, we find we find

This is an addition to the usual, This is an addition to the usual,mesonicmesonicchargechargeof the operator.of the operator.

12 fermions |q e |QF


42/55

CancellationCancellationDoing the calculation in the UV YMDoing the calculation in the UV YM--CSM theory,CSM theory,

, where R is th, where R is the exacte exactR R --symmetry in the IR CSM. Note that in thesymmetry in the IR CSM. Note that in thetorictoriccase,case,eacheachchiralchiralfield appears exactly twice in thefield appears exactly twice in thesuperpotentialsuperpotential. Therefore any flavor symmetry (which. Therefore any flavor symmetry (which

must leave W invariant) that might mix with the R must leave W invariant) that might mix with the R --symmetry cancels in the monopole dimension.symmetry cancels in the monopole dimension.

Therefore equivalent to anomaly cancellation in the Therefore equivalent to anomaly cancellation in the YM theory with the same quiver. YM theory with the same quiver.More generally, anomalies of the 4d theory becomeMore generally, anomalies of the 4d theory become

quantum corrections to this monopole in the 3d theoquantum corrections to this monopole in the 3d theo[DLJ, Niarchos, Benna Klebanov Klose]

dim = R =

12 fermions R-charge


43/55


44/55

Correction to theCorrection to the modulimoduli spacespaceSuppose we have addedSuppose we have addedNNf f flavor pairs. The flavorflavor pairs. The flavorsymmetries of the original quiver act trivially on thesymmetries of the original quiver act trivially on theand their own flavor symmetries are alwaysand their own flavor symmetries are alwaysnonabeliannonabeliangroups, which cannot mix with the R groups, which cannot mix with the R --symmetry. Thussymmetry. Thus we are justified in computing the quantum correction we are justified in computing the quantum correctionthe naive UV R the naive UV R --charge of our monopole.charge of our monopole.

This results in This results in since thesince thetotal dimension of thetotal dimension of thesuperpotentialsuperpotentialq F p must be 2.q F p must be 2. The only consistent structure for the OPE is then The only consistent structure for the OPE is then

T T

F N f .

2N f 2 (dfund 1) = N f 2 dimension( F ),


45/55

MM--theory geometrytheory geometry The modified geometry is The modified geometry is

where the U(1) acts as where the U(1) acts asNNf f times U(1)times U(1)BB on X, andon X, and with weights k, with weights k,--k on Ck on C22..

Note that this U(1) has a fixed locus whereNote that this U(1) has a fixed locus whereand are zero. This is exactly the location of tand are zero. This is exactly the location of t

D6D6branesbranes, where the fundamentals are, where the fundamentals aremasslessmassless..

X f = ( X C 2)//U (1) , tt = F N f

tt


46/55

ChiralChiral flavorsflavorsF is now in the rather than theF is now in the rather than theadjointadjoint..

Using the formula for 1Using the formula for 1--loop corrections to theloop corrections to thecharges of monopoles, we now find that therecharges of monopoles, we now find that therea quantum correction to thea quantum correction to the gauge gauge charges, sincecharges, sincethe number of fields entered and leaving eachthe number of fields entered and leaving eachnode are not equal.node are not equal.

Taking two such D6 Taking two such D6branesbranes, one can produce, one can produceQQ111111!!k k

- k

- k

( N i , N j )

W f l = p1A1q 1 + p2A2q 2


47/55

ChiralChiral exampleexample The cone over Q The cone over Q111111 is theis thetorictoricCY 4CY 4--fold, Cfold, C66//U(1)//U(1)22

The 3d The 3dKahlerKahlerquotient on which the quiver is based is then justquotient on which the quiver is based is then justthetheconifoldconifold.. There are two fixed loci defined by the equations There are two fixed loci defined by the equationsFlavored CFT gives:Flavored CFT gives:

Assuming the 4 node quiver is quantum mechanically consis Assuming the 4 node quiver is quantum mechanically consistperhaps it describes the theory with a flat Cperhaps it describes the theory with a flat C--field turned on.field turned on.

z1 z2 z3 z4 z5 z61 1

1

1 0 0

0 0 1 1 1 1U (1) B 1 0 1 0 0 0

a1 = z1z3 , a2 = z2z4 , b1 = z5 , b2 = z6a1 = 0, a2 = 0

A1 A2 B1 B2 t t1 1

1

1 1 1 tt = A1A2


48/55

GenericGeneric orbiorbi --bundle casebundle case The M The M--theory horizon manifold is smooth, buttheory horizon manifold is smooth, but

the reduction to IIA has singularities that arethe reduction to IIA has singularities that arelocallylocallyorbifoldsorbifolds. These can support fractional F. These can support fractional F22flux.flux.

For example CFor example C44/Z/Z q q acting by acting by

maps NS5 to (p, mp+1)5maps NS5 to (p, mp+1)5T(SU(N))

T(SU(N))

U(N) p

U(N) -pU(N) m

U(N) -mq = mp + 1

(z1 , z2 , z3 , z4) (z1 , 1z2 , pz3 , pz4 )

T pST m SL(2,Z )

T (SU (N )) = U (1) U (2) ... U (N 1) U (N )F


49/55

M5M5 branesbranes on 3on 3 --manifoldsmanifoldsGives rise to interacting 3dGives rise to interacting 3dSCFTsSCFTsin the IR.in the IR.

On hyperbolic 3On hyperbolic 3--manifolds, probably lack amanifolds, probably lack aLagrangianLagrangiandescription. Special case: mappingdescription. Special case: mappingtoritoriof largeof large

diffeomorphismsdiffeomorphismsof Riemann surfaces. Very specialof Riemann surfaces. Very specialcase: circle bundles over Riemann surfaces.case: circle bundles over Riemann surfaces.

Can have large duals in MCan have large duals in M--theory with internal fluxes intheory with internal fluxes inthe hyperbolic case.the hyperbolic case.[Gauntlett Conamhna Mateos Waldram]


50/55


51/55

SS--rulerule The discrete torsion fluxes live in The discrete torsion fluxes live in

Naively,Naively, can be any integer.can be any integer.But theBut theHanany Hanany -- Witten s Witten s--rule implies that therule implies that themodulimodulispace is lifted ifspace is lifted if > k. This can be seen> k. This can be seenas the nonexistence ofas the nonexistence ofSU(N)SU(N)k k N N = 3 CS theory= 3 CS theoryfor N > k.for N > k.

This is a strong coupling effect. This is a strong coupling effect.

H 4(S 7 /Z k ) = Z k


52/55

ParityParity The TheU(N)U(N)k k U(N+U(N+ ) )--k k theory is equivalent totheory is equivalent to

U(N)U(N)--k k U(N+k U(N+k -- ) )k k related by parity torelated by parity toU(N)U(N)k k U(N+k U(N+k -- ) )--k k ..

Perhaps there is a cascade rather thanPerhaps there is a cascade rather than

supersymmetry breaking in certainsupersymmetry breaking in certainconfigurations withconfigurations with > k.> k.

(1,k) 5

D3N

(N+l) D3

NS5 (1,k) 5

D3N

D3(N+kl)

NS5


53/55

CascadesCascadesEvidence for a cascade when 2Nk M(M-k)

found by Aharony Hashimoto Hirano Ouyang;Evslin Kuperstein; Hashimoto Hirano Ouyang.Extra difference in the ranks is carried by D3branes that wind several times. The supergravity dual has yet to be constructe

for the flow from the YM-CS-matter throughthe cascade.


54/55

Cascades IICascades II A cascading solution was found for the A cascading solution was found for the N N =4=4

configuration. This is a cascade starting with the 4dconfiguration. This is a cascade starting with the 4d N N =4 YM on a circle with domain walls.=4 YM on a circle with domain walls.

In the YMIn the YM--CSM case, one needs to check whenCSM case, one needs to check whenflowing upflowing up whether the cascade occurs before the whether the cascade occurs before thefield theory enters the weakly coupled regime.field theory enters the weakly coupled regime.

The s The s--rule remains to be fully understood. What is therule remains to be fully understood. What is thepotential generated on thepotential generated on themodulimodulispace?space?


55/55

For the futureFor the futureEmbedEmbed Vasiliev Vasiliev theory into string theory.theory into string theory.

Understand when weakly coupled strings, smUnderstand when weakly coupled strings, smacurvature, and/or weak gravity emerge.curvature, and/or weak gravity emerge.

More applications of localization on SMore applications of localization on S33..

A new window into the string landscape? A new window into the string landscape?

Connect explicit duals with condensed matterConnect explicit duals with condensed matter

jaffer is holography book

Documents