Transcript PowerPoint Presentation

Massive type IIA string
theory cannot be strongly
coupled
Daniel L. Jafferis
Institute for Advanced Study
16 November, 2010
Rutgers University
Based on work with Aharony,
Tomasiello, and Zaffaroni
Motivations

What is the fate of massive IIA at strong
coupling?

What is the dual description of 3d CFTs at large
N and fixed coupling?

Explore the N=1, 2 massive IIA AdS × CP3
solutions and their dual CFTs.
IIA string theory at strong coupling

The strong coupling limit of IIA string theory is
M-theory, so this regime is again described by
supergravity.
ds2 = e¡
11

2Á=3 ds2
10
+ e4Á=3 (dx 11 + A) 2
D0 branes have a mass 1/gs, and become light,
producing the KK tower of the 11d theory.
Massive IIA at strong coupling




Would seem to be a lacuna in the web of string
R
dualities.
F0 A D 0
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?
Behavior of 3d CFT at large N

In the ‘t Hooft limit, one always finds a weakly
coupled string dual.
gs » ¸ =N in AdS5 and gs » ¸ 5=4 =N in AdS4



In 3d, it is natural to consider taking N large
with k fixed.
In the N=6 theory, this results in light disorder
operators corresponding to the light D0 branes
of IIA at strong coupling. There is an M-theory
sugra description with entropy N3/2.
Is that the generic behavior?
A bound on the dilaton

In string
frame, the Einstein equations
¡
¢ P are
e¡
2Á
R M N + 2r
T Fk =
MN
M
r
N
Á¡
1
FM M 2 :::M k
2( k ¡ 1) !
1H PQH
N PQ
4 M
FN M
:::M k
2
¡
=
1 F
4k ! M 1 :::M k
k = 0;2;4
TFk
MN
F M 1 :::M k gM N
this is exact up to 2 derivative order even when
the coupling is large.
P
P
 The 00 component
can
be
written Fusing
frame
1(
2 )
F2
+
k = 2;4 0;k ¡ 1
k = 0;2;4 ? ;k
4
indices as
F k = e0 ^ F 0;k ¡ 1 + F ? ;k
where
Massive IIA solutions cannot be
strongly coupled and weakly curved




This equation is satisfied at every point in
spacetime. All of the terms¡ 2in parentheses on
(¿ ` )
the left side must be smalls
, otherwise the
2 derivativeZsugraXaction cannot be trusted.
e¡ B
Fk = n a (2¼` s ) a¡ 1
The fluxes C
, on a compact
k
a-cycle are quantized.
> 1=` 2
s
Thus is F0 ≠ 0, then the rhs
.
eÁ ¿ 1
Therefore eÁ < `. Typically,
the lhs is order
=R.
» s
2
1/R , thus
a
In strongly curved 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 large 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 M2
branes in C4/Zk


U(N)k x U(N)-k CSM with a pair of bifundamental
hypermultiplets
Field
A ¹ ; A~¹
gauge ¯elds
content: C ; ÃI in (N ; N¹ ) mat t er ¯elds
I
(CI ) ¤ ; (ÃI ) ¤ in ( N¹ ; N ) t heir conjugat es
W =

2¼² ² (A B A B )
a a_ b b_
k ab a_b_
C I = (A a ; B ¤ ):
SU(2) x SU(2) global symmetry, which does not
commute with SO(3)R, combining to form SU(4)R
a_
Dual geometry


The gauge theory coupling is 1/k. Fixing
¸ = N =k, N ! 1
, 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
2
R st
r

p
= 25=2 ¼ ¸
N À k5
gI I A »
¸
1=4
k
For
, one finds small curvature
a
AdS4 and
£ S7 =Z
k
large dilaton. Lifts to M-theory on
Massive IIA

Consider deforming the N=6 CSM theory by the
addition of a level a CS term for the second gauge
U(N 1 ) k £ U(N 2 ) ¡ k + n
group.
0

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]

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.
kCS(A 1 ) + (n 0 ¡ k)CS(A 2 ) + jX j 2 (A 1 ¡ A 2 ) 2

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 Rsymmetry.
[Tomasiello; Gaiotto Tomasiello]
Review of massive AdS4 solutions

The dual geometries are topologically the same as
the N=6 solution, but are now warped.
ds2
N = 1;2;3
= ds2
war ped A dS
4
+ ds2
C P 3 ; N = 1;2;3
Metric on CP3 has SO(5), SO(4), SO(3) isometry in the N
= 1,2,3 cases. Last solution only known
perturbatively.
R
n4 =
F4 = N 2 ¡ N 1
n 0 = F 0 = k1 ¡ k2
CP2
R
R
n6 =
F6 = N 1
n2 =
F2 =
k2
CP3

CP1
Large 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.
N=1 detailed analysis

The SO(5) invariant metric on CP3 is given by
³
ds2
C P 3; N = 1

= R2
´
1 (dx i
8
+ ²i j k A j x k )2 +
1 ds2
2¾ S 4
where the space is regarded as an S2 bundle over
S4.
¾2 [ 2 ; 2]
5
The parameter
, where 2 is the FubiniStudy metric.
p
q
R A dS =
R
2
5
( 2¾+ 1)
B = ¡
( 2¡ ¾) ( ¾¡ 2=5)
¾+ 2
J+ ¯
Parameters and fluxes

The four parameters of the sugra solution are
related to R
the quantized fluxes,
e¡
0
B
B
B
B
B
B
B
@
1
l gs
f 0 (¾)
1
0
B
Fk = n k (2¼` s ) k ¡
nb
0
1
0
1
C B
C B
B
B b
l f (¾) C
C
b C
n
2
B
C
gs
C B 2 C B
´ B
C = B
C
B
l 3 f (¾) C
b
B
C
B 1 b2
n
C
4
4
@
A
@ 2
gs
A
1 b3
nb
l 5 f (¾)
6
6
6
gs
0
1
b
1 b2
2
` ´ R A dS=(2¼` s )
1
0 0
1 0
C
0 0 C
C
C
C
1 0 C
A
b 1
n0
1
B
C
B n C
B 2 C
B
C
B
C
B n4 C
@
A
n6
A new regime


These relations can be
inverted explicitly.
Take n4 = 0, n2 = k, n6=N
When N ¿
k3
n2
When N À
0
0
¾! 2, t he Fubini-St udy met ric,
and ` » N 1 = 4 , gs » N 1 = 4
k 1=4
k 5=4
deformat ion of t he N = 6 solut ion.
k3
n2
¾! 1, t he nearly-Kahler met ric,
1
and ` » N 1 = 6 , gs »
n 1=6
N 1=6 n 5=6
0
a new weakly coupled regime!
0
Particle-like probe branes

R
In the massive IIA solutions, D0 branes have a 1tadpole.
F2 ^ A D 2
2¼`
Just as in the massless case, so do D2 branes,
R state has a total worldvolume tadpole
A D2/D0 bound
nD 2 = n0, nD 0 = ¡ n2.
(n D 2 n 2 + n D 0 n 0 )
A
R
.
. Take
Consider n0=1,Lnp2 =2 k for
simplicity. Then the mass of
k + L 4.
the D-brane is g
» k2
In the first phase, the D0s dominate the mass » while
N 2=3
in the second phase, the D2 dominates the mass
L
D4 branes always exist, and have mass g in AdS units,
which is order N in both phases, as expected for a
baryon.
s


s

5

s
Field theory interpretation

Define the ‘t Hooft couplings,
¸1=


N
k1
; ¸2=
N
¡ k2
; ¸§ = ¸1§ ¸2
where n4=0 for simplicity.
In these variables,
the transition occurs for
n3
N »
2
n2
0
)
¸¡ » ¸2
+
To have better control over the CFT, we turn to
the N=2 case.
Light disorder operators in 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 operators


There are monopole operators in YM-CS-matter
theories, which we follow to the IR CSM.
R
In radial quantization,
is na classical background with
F a =it2¼
magnetic flux S 2
, and constant scalar,
¾a = k ¡ 1 ¹
¾= n=2
. Of course, in the CSM limit,

It is crucial that the fields in μ are not charged under a.

This operator creates a vortex.
[Borokhov Kapustin Wu]
Anomalous dimension
N=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
P finds
gauged) U(1). One
¡

1
2
f er m i on s
jqe jQF
This is an addition to the usual, mesonic charge
of the operator.
Monopoles in the massive duals
¾i =
1 diag(w1 ; : : : ; wN i
i
i
2
).
ni =
P
wa
i
Take
Then
(ki w1 ; : : : ; ki wN )
i
i
 Sits in an irrep with weight
.
P
 Gauge invariant combinations
ki n i = 0 with the matter
fields require that
w1 = (1; : : :k : : : ; 1; 0; : : : ) and
2
 In our case, take
w = (1; : : : : : : ; 1; 0; : : : )

i
2

k1
There are solutions to the equations:
.
AA y ¡ B y B = k 1 w1 ,
2¼
w1 A = Aw2 ;
B B y ¡ A yA = ¡
w2 B = B w1
k2 w
2¼ 2
Dimensions


There are two adjoint fermions with R-charge
+1 in the vector multiplets, and four bifundamental fermions with R-charge -1/2.
This results in a quantum correction to the Rcharge
of the monopole
(n ¡ n ) 2 ¡ (N ¡ N )(n ¡ n )
1

2
1
2
1
2
Combines with the matter dimension to give
k1 k2
2
+ (k 2 ¡ k1 ) 2 ¡ (k2 ¡ k1 )(N 1 ¡ N 2 )
N=2 solution

ds2
6



The internal metric is SO(4) invariant.
e2B 1 ( t )
e2B 2 ( t )
1
1
2
2
2
2
=
ds +
ds + ² (t)dt +
¡ 2 (t)(da + A 2 ¡ A 1 ) 2
2
2
S
S
4
4
8
64
1
2
It has the form of T1,1 fibered over an interval.
One S2 shrinks at each end.
p
Ã1 2 [0; 3]
Depends on 4 parameters, L, gs, b,
,
where 0 is the undeformed solution.
Related to the four quantized fluxes.
N=2 solution

The solution can be reduced to three first order
differential equations.
Ã0 =
sin(4Ã) Ct ;Ã (w1 + w2 ) + 2 cos2 (2t)w1 w2
sin(4t) Ct ;Ã (w1 + w2 ) cos2 (2Ã) + 2w1 w2
w0
1
4w1 Ct ;Ã (w1 w2 ¡ 2 w2 ¡ 2 sin2 (2Ã)w1 )
=
sin(4t) Ct ;Ã (w1 + w2 ) cos2 (2Ã) + 2w1 w2
w0
2
4w2 Ct ;Ã (w1 w2 ¡ 2 w1 ¡ 2 sin2 (2Ã)w2 )
=
sin(4t) Ct ;Ã (w1 + w2 ) cos2 (2Ã) + 2w1 w2
Where wi = 4e2B i ¡ 2A , Ct ;Ã = cos2 (2t) cos2 (2Ã) ¡ 1, e2A is t he warp fact or
of t he AdS met ric, and à appears in t he spinors.
The 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 again
Take n4 = 0,
n2 = k, n6=N
When N ¿
k3
n2
, Ã1 ! 0,
When N À
0
n2
, Ã1 !
p
3,
0
get Fubini-St udy met ric,
and ` » N 1 = 4 , gs » N 1 = 4 .
k 1= 4
k3
k 5=4
a conifold singularity appears,
1
and ` » N 1 = 6 , gs »
.
n 1=6
0
N 1=6 n 5=6
0
Match 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
mD 2 L =
n0L
(2¼) 2 gs ` 3
s


Rp
det (g + F ¡ B ) =
¡
n2
2
2
¢
¡ n 0 n 4 F (Ã1 )
Remarkably, computing numerically, F is a
constant, equal to 1.
Precisely matches the field theory.
A new weakly coupled string regime

In the massive IIA solution dual to U(N)k ×
U(N)³-k+n0´, we found
Rst r »


N
n0
1=6
gs »
1
(N n 5 )1=6
0
1
GN
» N 5=3 n 1=3
0
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.
Conclusions



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.