Massive type IIA Massive type IIA string theory string theory

cannot be strongly cannot be strongly coupledcoupled

Daniel L. JafferisDaniel L. Jafferis

Institute for Advanced StudyInstitute for Advanced Study

16 November, 2010

Rutgers University

Based on work with Aharony, Tomasiello, and Zaffaroni

MotivationsMotivations

What is the fate of massive IIA at What is the fate of massive IIA at strong coupling?strong coupling?

What is the dual description of 3d What is the dual description of 3d CFTs at large N and fixed coupling?CFTs at large N and fixed coupling?

Explore the Explore the NN=1, 2 massive IIA AdS =1, 2 massive IIA AdS × CP× CP33 solutions and their dual CFTs. solutions and their dual CFTs.

IIA string theory at strong IIA string theory at strong couplingcoupling

The strong coupling limit of IIA string theory is M-theory, so this regime is again described by supergravity.

D0 branes have a mass 1/gs, and become light, producing the KK tower of the 11d theory.

ds211 = e¡ 2Á=3ds2

10 +e4Á=3 (dx11 +A)2

Massive IIA at strong Massive IIA at strong couplingcoupling

Would seem to be a lacuna in the web of string dualities.

The D0 branes have tadpoles, There is no free “massive” parameter in

11d supergravity.

A more fundamental question: are there any strongly coupled solutions of IIA supergravity?

RF0 AD 0

Behavior of 3d CFT at Behavior of 3d CFT at large Nlarge N

In the ‘t Hooft limit, one always finds a weakly coupled string dual.

In 3d, it is natural to consider taking N large with k fixed.

In the In the NN=6 theory, this results in light =6 theory, this results in light disorder operators corresponding to disorder operators corresponding to the light D0 branes of IIA at strong the light D0 branes of IIA at strong coupling. There is an M-theory sugra coupling. There is an M-theory sugra description with entropy Ndescription with entropy N3/23/2..

Is that the generic behavior? Is that the generic behavior?

gs » ¸=N in AdS5 and gs » ¸5=4=N in AdS4

A bound on the dilatonA bound on the dilaton

In string frame, the Einstein equations are

this is exact up to 2 derivative order

even when the coupling is large. The 00 component can be written using

frame indices as where

e¡ 2Á¡RM N + 2r M r N Á¡ 1

4HM

P Q HN P Q

¢=

Pk=0;2;4 TF k

M N

TF kM N = 1

2(k¡ 1)!FMM 2:::M k FN M 2:::M k

¡ 14k!FM 1 :::M k

F M 1:::M k gM N

14(

Pk=2;4 F 2

0;k¡ 1 +P

k=0;2;4 F 2? ;k)

Fk = e0 ^F0;k¡ 1 + F? ;k

Massive IIA solutions Massive IIA solutions cannot be strongly coupled cannot be strongly coupled

and weakly curvedand weakly curved This equation is satisfied at every

point in spacetime. All of the terms in parentheses on the left side must be small , otherwise the 2 derivative sugra action cannot be trusted.

The fluxes , on a compact a-cycle are quantized.

Thus is F0 ≠ 0, then the rhs . Therefore . Typically, the lhs is

order 1/R2, thus

(¿ ¡̀ 2s )

Z

Ca

e¡ BX

k

Fk = na(2¼̀s)a¡ 1

> 1=̀2s

eÁ ¿ 1

eÁ»< s̀=R.

In strongly curved In strongly curved backgrounds?backgrounds?

In a generic background with string scale curvature, the notion of 0-form flux is not even defined.

No signs of strong coupling in known massive IIA AdS solutions.

UV completion of Sagai-Sugimoto still unknown, but the region between the D8 branes is not both weakly curved and at large coupling.

In some special cases, one might make sense out of a strongly curved, strongly coupled region in a massive IIA solution:

If it were a part of a weakly curved solution, probably it will be small (string scale).

If there were enough supersymmetry, it might be related by duality to a better description. For example T-dualizing to a background without F0 flux. [Hull,…]

Massive IIA AdS duals of Massive IIA AdS duals of large N 3d CFTslarge N 3d CFTs

To gain further insight into this result, will look at AdS vacua of massive IIA.

This results in interesting statements about the dual field theories.

We will find that the string coupling never grows large.

At large N for fixed couplings, the behavior will be completely different than the massless case.

The N=6 CSM theory of N The N=6 CSM theory of N M2 branes in CM2 branes in C44/Z/Zkk

U(N)k x U(N)-k CSM with a pair of bifundamental hypermultiplets

Field content:

SU(2) x SU(2) global symmetry, which does not commute with SO(3)R, combining to form SU(4)R

(CI )¤;(ÃI )¤ in ( ¹N ;N ) their conjugatesCI ;ÃI in (N; ¹N ) matter ¯elds

A¹ ; ~A¹ gauge ¯elds

C I = (Aa;B¤_a):

W = 2¼k ²ab² _a_b(AaB _aAbB _b)

Dual geometryDual geometry The gauge theory coupling is 1/k. Fixing , the usual ‘t Hooft limit

is a string theory. One obtains IIA on AdS4 × CP3 with N

units of F4 and k units of F2 in CP3

For , one finds small curvature and a large dilaton. Lifts to M-theory on

¸ = N=k, N ! 1

gI I A » ¸ 1=4

kR2

str = 25=2¼p

¸

N À k5

AdS4 £ S7=Zk

Massive IIAMassive IIA Consider deforming the N=6 CSM theory

by the addition of a level a CS term for the second gauge group.

In this theory the monopole operators corresponding to D0 branes develop a tadpole, since the induced electric charge (k, n0-k) cannot be cancelled with the matter fields.

This motivates the idea that the total CS level should be related to the F0 flux. [Gaiotto Tomasiello, Fujita Li Ryu Takayanagii]

U(N1)k £ U(N2)¡ k+n0

The light U(1) on the moduli space has a level n0 Chern-Simons term, matching the coupling of the D2 worldvolume to the Romans mass.

For such deformations of N=6 CSM, there are field theories with N = 3,2,1,0 differing by the breaking of the SU(4) into flavor and R-symmetry.

[Tomasiello; Gaiotto Tomasiello]

kCS(A1) + (n0 ¡ k)CS(A2) + jX j2(A1 ¡ A2)2

Review of massive AdSReview of massive AdS44 solutionssolutions

The dual geometries are topologically the same as the N=6 solution, but are now warped.

Metric on CP3 has SO(5), SO(4), SO(3) isometry in the N = 1,2,3 cases. Last solution only known perturbatively.

ds2N =1;2;3 = ds2

warped AdS4+ds2

C P 3; N =1;2;3

n0 = F0 = k1 ¡ k2

n2 =R

C P 1 F2 = k2

n4 =R

C P 2 F4 = N2 ¡ N1

n6 =R

C P 3 F6 = N1

Large N limitLarge N limit

In the ‘t Hooft limit, these solutions are small deformations of the AdS4 x CP3 N=6 IIA supergravity solution.

What about the large N limit for fixed levels? When n0 = 0, this results in strong coupling, and a lift to M-theory.

We now know that this is impossible for n0 ≠ 0.

NN=1 detailed analysis=1 detailed analysis

The SO(5) invariant metric on CP3 is given by

where the space is regarded as an where the space is regarded as an SS22 bundle over S bundle over S44..

The parameter , where 2 is the Fubini-Study metric.

ds2C P 3; N =1 = R2

³18(dxi +²i j kAj xk)2 + 1

2¾ds2

S4

´

¾2 [25;2]

RA dS = R2

q5

(2¾+1) B = ¡p

(2¡ ¾)(¾¡ 2=5)¾+2 J + ¯

Parameters and fluxesParameters and fluxes

The four parameters of the sugra solution are related to the quantized fluxes,

Re¡ B Fk = nk(2¼̀s)k¡ 1

0

BBBBBBB@

1lgs

f 0(¾)

lgs

f 2(¾)

l3

gsf 4(¾)

l5

gsf 6(¾)

1

CCCCCCCA

=

0

BBBBBB@

nb0

nb2

nb4

nb6

1

CCCCCCA

´

0

BBBBBB@

1 0 0 0

b 1 0 0

12b2 b 1 0

16b3 1

2b2 b 1

1

CCCCCCA

0

BBBBBB@

n0

n2

n4

n6

1

CCCCCCA

` ´ RAdS=(2¼̀s)

A new regimeA new regime

These relations can be inverted explicitly.

Take n4 = 0, n2 = k, n6=N

When N ¿ k3

n20

¾! 2, theFubini-Study metric,and ` » N 1=4

k1=4, gs » N 1=4

k5=4

deformation of theN = 6 solution.

When N À k3

n20

¾! 1, thenearly-Kahler metric,and ` » N 1=6

n1=60

, gs » 1N 1=6n5=6

0a new weakly coupled regime!

Particle-like probe Particle-like probe branesbranes

In the massive IIA solutions, D0 branes have a tadpole. Just as in the massless case, so do D2 branes,

A D2/D0 bound state has a total worldvolume tadpole . . Take

Consider n0=1, n2 = k for simplicity. Then the mass of the D-brane is

In the first phase, the D0s dominate the mass while in the second phase, the D2 dominates the mass

D4 branes always exist, and have mass in AdS units, which is order N in both phases, as expected for a baryon.

12¼̀ s

RF2 ^AD 2

(nD 2n2 +nD 0n0)R

RA

» k2

» N 2=3

L 5

gs

nD 2 = n0, nD 0 = ¡ n2.

Lgs

pk2 +L4.

Field theory Field theory interpretationinterpretation

Define the ‘t Hooft couplings,

where n4=0 for simplicity. In these variables, the transition

occurs for

To have better control over the CFT, we turn to the N=2 case.

¸1 = Nk1

; ¸2 = N¡ k2

; ¸§ = ¸1 § ¸2

N »n3

2n2

0) ¸¡ » ¸2

+

Light disorder operators in Light disorder operators in the CFT dual?the CFT dual?

There are clearly no light D-branes in this limit of the N=1 solution.

One expects that the monopole operators of the CFT will get large quantum corrections to their dimensions.

However, in the N=2 case, they are protected.

Monopoles operatorsMonopoles operators There are monopole operators in YM-CS-matter

theories, which we follow to the IR CSM.

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

. Of course, in the CSM limit,

It is crucial that the fields in μ are not charged under a.

This operator creates a vortex.

RS2 Fa = 2¼n

¾= n=2

[Borokhov Kapustin Wu]

¾a = k¡ 1¹

Anomalous dimensionAnomalous dimensionNN=2 case=2 case

We work in the UV to compute the 1-loop correction to the charge of a monopole operator under some flavor (or R–symmetry, or gauged) U(1). One finds

This is an addition to the usual, mesonic charge of the operator.

¡ 12

Pf ermions jqejQF

Monopoles in the Monopoles in the massive dualsmassive duals

Take Then Sits in an irrep with weight

. Gauge invariant combinations with

the matter fields require that In our case, take

There are solutions to the equations:

¾i = 12diag(w1

i ; :: :;wN ii ). ni =

Pwa

i

(ki w1i ; : : : ;ki w

N ii )

Pki ni = 0

w2 = (1;:: :k1: : : ;1;0;:: :)

AAy ¡ ByB = k12¼w1, BBy ¡ AyA = ¡ k2

2¼w2. w1A = Aw2; w2B = Bw1

w1 = (1;:: :k2: : : ;1;0;:: :) and

DimensionsDimensions

There are two adjoint fermions with R-charge +1 in the vector multiplets, and four bi-fundamental fermions with R-charge -1/2.

This results in a quantum correction to the R-charge of the monopole

Combines with the matter dimension to give

(n1 ¡ n2)2 ¡ (N1 ¡ N2)(n1 ¡ n2)

k1k22 + (k2 ¡ k1)2 ¡ (k2 ¡ k1)(N1 ¡ N2)

NN=2 solution=2 solution

The internal metric is SO(4) invariant.

It has the form of T1,1 fibered over an interval. One S2 shrinks at each end.

Depends on 4 parameters, L, gs, b, , where 0 is the undeformed solution.

Related to the four quantized fluxes.

ds26 =

e2B 1(t)

4ds2

S21

+e2B 2(t)

4ds2

S22

+18

²2(t)dt2 +164

¡ 2(t)(da+ A2 ¡ A1)2

Ã1 2 [0;p

3]

NN=2 solution=2 solution

The solution can be reduced to three first order differential equations.

Ã0=sin(4Ã)sin(4t)

Ct;Ã (w1 +w2) + 2cos2(2t)w1w2

Ct;Ã (w1 +w2) cos2(2Ã) +2w1w2

w01 =

4w1

sin(4t)Ct;Ã (w1w2 ¡ 2w2 ¡ 2sin2(2Ã)w1)Ct;Ã (w1 +w2) cos2(2Ã) +2w1w2

w02 =

4w2

sin(4t)Ct;Ã (w1w2 ¡ 2w1 ¡ 2sin2(2Ã)w2)Ct;Ã (w1 +w2) cos2(2Ã) +2w1w2

Where wi = 4e2B i ¡ 2A , Ct;à = cos2(2t) cos2(2Ã) ¡ 1, e2A is the warp factorof the AdS metric, and à appears in the spinors.

The shape of the solutionThe shape of the solution At each end of the interval one sphere shrinks.

The size of the other at that point is plotted versus the deformation parameter.

Develops a conifold singularity!

Two phases againTwo phases againTake nTake n44 = 0, = 0,

nn22 = k, n = k, n66=N=N

When N À k3

n20, Ã1 !

p3,

a conifold singularity appears,and ` » N 1=6

n1=60

, gs » 1N 1=6n5=6

0

.

When N ¿ k3

n20, Ã1 ! 0,

get Fubini-Study metric,and ` » N 1=4

k1=4, gs » N 1=4

k5=4.

Match of light D2 branesMatch of light D2 branes

A D2 brane wrapping the diagonal S2 can be supersymmetric. The tadpole is cancelled by appropriate worldvolume flux.

The mass can be calculated to give

Remarkably, computing numerically, F is a constant, equal to 1.

Precisely matches the field theory.

mD 2L = n0L(2¼)2gs `3

s

Rpdet (g+F ¡ B) =

¡ n22

2 ¡ n0n4¢F (Ã1)

A new weakly coupled A new weakly coupled string regimestring regime

In the massive IIA solution dual to U(N)k × U(N)-k+n0

, we found

This is in spite of the fact that the N=2 theory has light monopole operators.

It would be interesting to understand the general behavior.

Rstr »³

Nn0

´1=6gs » 1

(N n50)1=6

1GN

» N 5=3n1=30

ConclusionsConclusions

There are no strongly coupled solutions of massive IIA supergravity. Regions of strong curvature still need to be fully understood.

Conifold singularities seen to arise in AdS backgrounds.

The emergence of weakly coupled strings in a new regime of field theories.