Transcript M2 branes, CS theories and AdS3/CFT4 correspondence

3-Sasakian geometry
from M2 branes
Daniel L. Jafferis
Rutgers University
Based on work with: A. Tomasiello;
X. Yin; D. Gaiotto.; and work in progress
Kähler and Sasakian
Geometry in Rome
19 June, 2009
Outline

Introduction

N=3 Chern-Simons-matter and 3-Sasakian 7manifolds

Quantum corrections to N=3 CSM moduli
spaces and duals to AdS4 with D6 branes

N=2 speculations
Motivation



Extend AdS/CFT correspondence between 4d
gauge theory and SE 5-manifolds to relation
between 3d Chern-Simons theories and 3Sasakian and SE 7-manifolds.
Requires extra data of a U(1) action that
commutes with Reeb vector. What happens
when this action has fixed loci?
New mathematical objects associated to 3-cycles
in 8-manifolds with flux?
M2 branes


These are 2+1 dimensional objects in 11d Mtheory. Their explicit description in physics was
mysterious since their discovery over a decade
ago.
Now we have found that they carry
(super)conformally invariant theory of ChernSimons coupled to charged scalars.
AdS/CFT
Equivalence between 4(3)d theory on the
boundary and AdS5(4) x SE5(7)
Focus on BPS sector (roughly topological sector)
Classical limit:
Moduli space of stable representations of a quiver
Moduli space of stable objects in the derived
category of coherent sheaves on the CY cone
Simplest version

The moduli space of the abelian quiver, thought
of as a quotient of the set of solutions to some
equations is the CY cone itself.
Example:

A1, A2, B1, B2. Equations dW = 0 for W =
Tr(A1 B1 A2 B2 – A1 B2 A2 B1).
Results in C4//U(1) acting by 1, 1, -1, -1.
This is the cone over T1,1


The correspondence also says that stable
representations of the path algebra
A = C[f 0 ; f 1 ]hA 1 ; A 2 ; B 1 ; B 2 i =hB 1 A i B 2 ¡ B 2 A i B 1 ; A 1 B i A 2 ¡ A 2 B i A 1 i
correspond to stable objects in the derived
category of compactly supported sheaves in the
conifold.

At the level of equations, King’s theorem says
that solving μ=θ and quotienting by U(N) is
equivalent to imposing an algebraic stability
condition and quotienting by GL(N).
¹ = A yA i ¡ B i B y
i
i
CY4 cone
SE7 with
U(1)B action
/U(1)_R
KE6
/U(1)B
//U(1)B
//U(1)B
M6
/U(1)R
CY3 cone
SE5
M2 theory for SE7 is related to D3
theory for SE5
M6 is an S2 bundle over KE4
KE4
hK2 cone
3-S with
U(1)B action
/U(1)B
///U(1)B
hK1 cone
M6
S3/Γ
M2 theory for 3-S is related to
hyperKähler quiver for hK1
Toric hyperKähler 8-manifold
ds2 = Ui j d~
x i ¢d~
x j + U i j (d'
A i = d~
x j ¢~
! j i = dx j ! a
µa j i
P
p2 hi
i
U = 1+
i
pi qi hi


i
+ A i )(d'
j
+ Aj )
[Bielawski, Dancer; Gauntlett,
Gibbons, Papadopulos, Townsend]
@j ! b ¡ @k ! a = ² abc @j Uk i
x
ji
x a k i¶
xc
b
pi qi hi
1
;
hi =
2
q hi
2j p i ~
x 1 qi ~
x2j
i
We know the associated M2 field theory in the
case all p = 1 this last summer.
New development in March: p=0 as well.
Hyperkahler singularity

In particular, the pair of U(1) isometries of the
T^2 fiber are compatible with the hyperkahler
structure, and
one
obtains
the
hypertoric
n
+
1
H
===N ,
manifold
where N is the kernel of
the map
µ
¶
¯ : U(1) n + 1 ! U(1) 2 ;

¯=
1
p1
1
p2
::: 1
: : : pn
0
m
In the
case of D6
branes in CP^3, this reduces
H 3 ===U(1);
(m,m,k)
to
[Bielawski Dancer]
D-term equation for M2

These are now cubic equations, and the analog
of King’s theorem is not known. Here a,b are
Lie algebra indices, and i,j are representation
indices.
(k ¡ 1 ) ab¹ a (TR ) i j qj = 0
b


¹ m = ³ km
One branch of solutions has all
Here k is diagonal on each U(N) factor, indexed
by m.
D6 branes in AdS_4
N = 3



We now know a large class of
quiver
CSM theories describing a stack of M2 branes at
a hypertoric singularity. In the ‘t Hooft
the
AdSlimit,
£
M6
4
w
dual geometry is a warped product
Introducing D6 branes wrapping an internal 3RP3
CP3
cycle (
in the
case) adds fundamental
hypermultiplets to the quiver.
Interestingly, conformality is preserved.
Dual CFT for

AdS4 £ N 010
.
One¤ of2 the most symmetric 3-Sasakians, its cone
T CP
is
, and it can be written as a quotient
SU ( 3)
U ( 1)
k
C =Z k
Classical moduli
space
is
,
but
quantum
2
¤
3
T CP = H ===U(1); (1; 1; 1)
corrected to
-k
4


Attempts in the 90’s where close:
[Billo` Fabbri Fre` Merlatti Zaffaroni]
Quantum correction

The cone~is modified to
M m = (M £ H)===U(1)
where U(1) acts as U(1) B on M and with the
natural charge m action on C2

Applied to C4 this results in the cone over
N010
ADHM quiver for D2 in D6


In addition to the branch of moduli space where
M2 branes probe the geometry including the lift
of the D6 branes (which is always via quantum
correction), the D2 branes may dissolve into the
D6 branes (M2 branes fractionate).
Fundamentals now get a VEV, quiver is exactly
ADHM quiver.
Fantasy


The quiver theory describing D3 branes at a
singular Calabi-Yau 3-fold can be determined by
resolving the singularity, thus blowing up the
fractional branes into wrapped D5 and D7
branes. Mathematically, the arrows in the quiver
are the Ext groups between a primitive objects
in the derived category of coherent sheaves.
It would be extremely interesting to find the
analog for M2 brane theories.

The physically natural picture would be the
resolution of fractional M2 branes into wrapped
M5 branes. There are two differences, however:
The fractional branes are typically pure torsion
There are no supersymmetric 3-cycles in hyperkahler, CY4,
Spin(7) 8-manifolds.

The resolution is probably that M-theory 4-form
flux plays a crucial role. Conjecture that there
are supersymmetric resolutions of CY4
singularities with flux. Fuzzy M5 branes would
naturally have s-rule.
Quantum correction to
hypermultiplet moduli space



In Yang-Mills theories with eight supercharges,
the moduli space of hypermultiplets is normally
not corrected, since one can promote the
coupling to a vector superfield, which decouples
from the hypers.
In CSM theories, no such argument exists for
the CS level. However, corrections to the metric
must respect the hyperkahler structure.
We will find a correction of this type.
Branches of CSM moduli space



In Chern-Simons theories, there is no Coulomb
branch, as the vector multiplets are all effectively
massive.
N=3 supersymmetry protects the dimensions of
ordinary chiral operators formed from the
matter fields; the quantum correction depends
on the existence of monopole operators.
Rich structure of branches distinguished by the
spectrum of allowed monopoles.
Chern-Simons-matter theory

We first consider the case with N=2 susy. It
consists of a vector multiplet in the adjoint of
the gauge group,Rand chiral multiplets in
i
R
representations
k
2
SN = 2 =
CS


4¼
(A ^ dA +
3
A 3 ¡ ÂÂ
¹ + 2D ¾)
The kinetic term for the chiral multiplets
¹ ¾2 Á ¡ ù ¾Ã
¡ Á
includes couplings i i i i
¹ DÁ
Á
There is the usual D term i i
¾
We integrate
out D, , and
Z
SN = 2 =
Â
k
2
¹ D ¹ Á + i ù ° ¹ D Ã
(A ^ dA + A 3 ) + D ¹ Á
i
i
i
¹ i
4¼
3
16¼2 ¹
4¼ ¹
¹
¹
a
b
a
b
¡
( Ái T Ái )( Áj T Áj )( Ák T T Ák ) ¡
( Ái T a Ái )( ù j T a Ãj )
Ri
Rj
Rk Rk
Ri
Rj
k2
k
8¼ ¹
¹ T a à ):
¡
( Ãi T a Ái )( Á
j R
j
Ri
k
j
Note that this action has classically marginal
couplings. It is has been argued that it does not
renormalize, up to shift of k, and so is a CFT.
[Gaiotto Yin]
N=3 CS-matter



To obtain a more supersymmetric theory, begin
with N=4 YM-matter. Then add the CS term,
breaking to N=3.
'
Thus we add a chiral multiplet, ,with no
kintetic term in the adjoint, and the matter chiral
©i ; ©~i
multiplets,
must come in pairs.
W = ¡ k T r (' 2 )
8¼
There is a superpotential,
from the
CS term.
'

Integrating out
one obtains the same action
as before, but with a superpotential:
W =

~ Ta
4¼ ( ©
i R
k
i
~ Ta © )
©i )( ©
j
j
Rj
These N=3 theories are completely rigid, and
hence superconformal. It is impossible to have
more supersymmetry in a YM-CS-matter theory,
but we shall see that for particular choices of
gauge groups and matter representations, the
pure CSM can have enhanced supersymmetry.
[Schwarz; Gaiotto Yin]
Simple example



R
Consider a U(1) x U(1) CSM 2k
theory,
a ^with
db a BFlike Chern-Simons coupling 4¼ ~
~)
(X ; X ); (Y; Y
Take a pair of matter hypers,
in
the fundamental of the first and second U(1).
In this theory the supersymmetry is enhanced to
N=4; one can check that the boson-fermion
coupling is invariant under a separate SU(2)
acting of each fundamental hyper.
Classical moduli space




The superpotential is dictated 4¼
by N=3
supersymmetry to be W = k (X X~)(Y Y~ )
M
M
b
Thus there are two branches, a and
, on
which the respective U(1) is unbroken.
Naively, one would quotient by the nontrivially
R U(1), but would leave 3d, so can’t be.
acting
2k
a ^ db
4¼
is only invariant under a Z_k .
M
a
= M
2 =Z
=
C
b
k
Extra massless fields at origin

The two branches intersect at the origin, where
there
extra massless fields. In particular,
~)
Mare
(Y; Y
a
on
which is parameterized by
the X
fields have µa mass
¶
4¼ ¹
4¼ 1
¹~
~ j 2 ); Y Y
~; Y
¹Y
Y( a Yb) =
(jY j 2 ¡ j Y
k
k 2

´ m
~X
We will see that integrating out these fields
changes the singularity at the origin.
Mukhi-Papageorgakis effect
(X ; X~ )


Forget about the
mutliplet for a moment.
~)
(Y; Y
Going onto the moduli space by turning on
gives a mass to the broken gauge field, b.
Integrating out b gives Yang-Mills kinetics to the
unbroken gauge
' Y field, a! It can then be dualized
to a scalar,
which transforms under the U(1)
in the same way as the phase of the hyper Y, but
with charge k. The Z_k arises by gauge fixing.
Correction to the hyperkahler metric

As familiar from the Coulomb branch of N=4
2+1 gauge theories, integrating out a charged
massive hypermultiplet at 1 loop gives rise to a
Z
term
1
1
(@¹ m
~ ¢@¹ m
~ ¡ jdaj 2 ) + ² ¹ º ½² i j k @i (
8¼j mj
~
8¼j mj
~
Z
1
=
(@¹ m
~ ¢@¹ m
~ ¡ jdaj 2 ) + (¤da) ¹ ! i @¹ m i
8¼j mj
~

)a¹ @º m j @½m k
Note that this already introduces a Yang-Mills
term for the gauge field a.

Before integrating out the broken gauge field b,
we dualize a, treating F_a as the fundamental
Z
Z
1
k
variable
.
jD Y j 2 +
(@ m
~ ¢@¹ m
~ ¡ j F~ j 2 ) +
F~ ^ (d' + ! dm +
b)
¹
8¼j mj
~
ZIntegrating
1
jD ¹
¹
a
a
Y
i
i
out F_a leads to
k
+
@¹ m
~
+ 2¼j mj(@
~ ¹ ' Y + ! i @¹ m i +
b¹ ) 2
8¼j mj
~
2¼
k
Y ! ei ¤ Y; b¹ ! b¹ + @¹ ¤ ; ' Y ! ' Y ¡
¤
2¼
Y j2
¢@¹ m
~
~ , and '
Y; Y

2¼
Y
The U(1)_b acts on the space of
.
The metric is nontrivial due to the quantum
correction as seen.
M
a
= C2 =Z k + 1
.
Monopoles in the chiral ring


There are monopole operators in YM-CS-matter
theories, which we follow to the IR CSM.
R it is a classical background
In radial quantization,
F a = 2¼n
S2
with magnetic
flux
,
and
constant
¾= n=2
scalar,
. Of course, in the CSM limit,
~j2)
¾a = 1=k (jY j 2 ¡ j Y


It is crucial that Y is not charged under a.
Call this monopole operator T.
[Borokhov Kapustin Wu]
CS induced charge of T


The Chern-Simons term induces a charge for
RT we have just
R defined.
the operator
2k
a ^ db = kn
b
r adi u s
Writing 4¼
in the
monopole background, it is a particle of charge
n k under U(1)_b
Equivalently, in radial quantization, the Gauss’
law constraint is modified, and some matter field
zero modes must be turned on.
Anomalous dimension

The dimension of the monopole operator will
X the two
X
be theQ sum
contributions
1 of
= (
¡
)jq j
0
i
2
i 2 h y per
i 2 v ect or
and the dimension of the scalar fields used in
the dressing.
 This was calculated in Borokhov-Kapustin-Wu
by quantizing the matter fields
in the monopole
¾= n=2
background with constant

The result was that the spectrum of fermions
from the hypermultiplets became asymmetric,
E + = jnj=2 + p;
p



E ¡ = ¡ jnj=2 ¡ p;
p
E 0 = + jnj=2
The spectrum of scalars was found to be
symmetric. Thus only the fermions contributed
to the vacuum energy, which is exactly the
anomalous dimension of the operator.
We will include the CS terms simply noting this
operator is charged under the gauge group.
This is sensible since the matter fields needed to
“dress” the operator are neutral under the
magnetic U(1). Needed for it to be in chiral ring
Our example




Ta ; T~a ; Tb; T~b
We have monopoles
, the firstMtwo
Ma
b
on the branch
, and the latter pair on
Each has one hypermultiplet charged under the
associated U(1), so it gets a dimension ½
The CS induced charge of T is (0,k) under the
U(1) x U(1) gauge group.
Ma
~, T Y
~ k , T~Y k
YY
The chiral operators on
are
M a = C2 =Z k + 1
exactly as expected for
D6 branes in AdS_4 x CP^3



We consider introducing D6 branes wrapping
the AdS. This should be similar to adding D7
branes in AdS5.RP3
They wrap an
cycle in the internal
manifold. Thus there is a Z_2 Wilson line,
distinguishing two types of D6 branes.
One can also add D6 branes to more general
S3 =Z n
N=3 AdS4 backgrounds, where they wrap
[DJ Tomasiello] CSM quiver with n nodes
IIB engineering


Consider N D3 branes wrapping a circle and
intersecting an NS5 and (1,k) 5 brane. This
engineers the ABJM theory.
Add some D5 branes, some intersecting each
half of the stack of D3. Breaks supersymmetry
to N=3, and adds fundamentals to the quiver.
D5
NS5
(1,k) 5
D5
M-theory lift



T-dualize the circle: NS5 branes turn into TaubNUT, D5 charge become D6, D3 becomes D2.
Near the D2 horizon, lift to M-theory:
Gibbons-Gauntless-Papadopolus-Townsend
showed this is purely geometry
U = U1
1
+
2j~
x1j
µ
1 0
0 0
¶
1
+
2j~
x 1 + k~
x2j
µ
1 k
k k2
¶
m
+
2j~
x 2j
µ
0 0
0 1
¶
Hyperkahler singularity

In particular, the pair of U(1) isometries of the
T^2 fiber are compatible with the hyperkahler
structure, and
one
obtains
the
hypertoric
n
+
1
H
===N ,
manifold
where N is the kernel of
the map
µ
¶
¯ : U(1) n + 1 ! U(1) 2 ;

¯=
1
p1
1
p2
::: 1
: : : pn
0
m
In the
case of D6
branes in CP^3, this reduces
H 3 ===U(1);
(m,m,k)
to
[Bielawski Dancer]
Quantum corrected geometric
branch

There are ordinary
chiral
operators
of
the
form
T r (A B A B )
i



j
k
l
On the moduli space of diagonal matrices, the
diagonal U(1)^N is unbroken, and there are
monopoles operators with such magnetic fluxes.
They have CS induced charge k, and anomalous
dimension m/2.
For m=1, k=1, at dimension 1, one Hhas
8 gauge
3 ====U(1)
111
invariant operators as expected
for
~
T r (A i B j ); T B i ; T A i
Completely Higgsed branch



If the number of fundamentals is at least twice
the rank of the gauge groups, there is a branch
in which the entire gauge symmetry is Higgsed.
This branch must have all moments set to zero,
resulting exactly the ordinary Kahler quotient
for the ADHM quiver of N instantons of rank
m on C^2/Z_n.
FI parameters resolve the singularity, each node
is a fractional brane Zthat blows up into a D4.
ki =
F2
S2
(i )
Supergravity limit

The volume of these 3-Sasakians is known:

This implies that the radius of curvature in M³
´1
theory is given by R M 7 = 26 ¼2 N ( m + k ) 2 6
V ol(M 7 ) = V ol(S7 )
`p

m + 2k
2( m + k ) 2
m + 2k
It is a warped compactification, but using the
inverse ofRM3the lightest
8¼N 1=2 D0 mass, a typical value
R2 »
= p
st r
m+ k
m + 2k
of
7
Stuffing fundamentals with dof

It is simplest to determine the number of
degrees of freedom at high temperature from
the M-theory supergravity limit. It is dominated
by the large AdS4 black hole, and the internal
manifold only enters via the four dimension
Planck scale. (m + k) p 2
N2
3mN p
¯F = ¡ 27=2 3¡ 2 ¼2 N 3=2 p

m + 2k
V2 T 2 » p
¸
Note the enhancement of N m!
+
4
¸ + :::