New $AdS_4\times X_7$ Geometries with $\CN=6$ in M Theory

0.28 MB·English

by  Ki-Myeong Lee

KIAS-P06001,hep-th/0605214
PreprinttypesetinJHEPstyle-HYPERVERSION
New AdS X Geometries
4 7
×
7 = 6
with in M Theory
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N
0
2

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a
J

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1 Ki-Myeong Lee

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v School of Physics, Korea Institute for Advanced Study, Seoul 130-012 KOREA
4 [email protected]
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5 Ho-Ung Yee
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0 School of Physics, Korea Institute for Advanced Study, Seoul 130-012 KOREA
/
h [email protected]
t
-
p
e
h
: Abstract: We study supersymmetric AdS4 X7 solutions of 11-dim supergravity
v ×
i where the tri-Sasakian space X7 has generically U(1)2 SU(2)R isometry. The com-
X ×
pact and regular 7-dim spaces X = S(t ,t ,t ) is originated from 8-dim hyperkahler
r 7 1 2 3
a quotient of a 12-dim flat hyperkahler space by U(1) and belongs to the class of the
Eschenburg space. We calculate the volume of X and that of the supersymmetric
7
five cycle via localization. From this we discuss the 3-dim dual superconformal field
theories with = 3 supersymmetry.
N
Contents
1. Introduction and Conclusion 1
2. Hyperkahler Space (t) 5
8
M
3. Tri-Sasakian Space X (t) 10
7
3.1 The volume of tri-Sasakian space X (t) 11
7
3.2 The volumes of supersymmetric 5-cycle Σ 18
5
4. Dual Superconformal Field Theory 20
A. A Generalization of Caloron Moduli Space 23
1. Introduction and Conclusion
The AdS-CFT correspondence predicts that the type IIB-theory on the supergravity
solution AdS S5 with appropriate 5-form field strength is dual to a = 4 su-
5
× N
persymmetric 4-dim SU(N) gauge theory, which is a superconformal field theory[1].
When the space S5 is replaced by a 5-dim Sasaki Einstein space, the dual gauge the-
ory has less supersymmetry with more complicated group and matter structure[2].
These conformal theories can be regarded as a field theory on a stack of D3 branes
sitting at the the singular tip of a Ricci flat 6-dim cone whose base is the Sasaki-
Einstein space[3, 4, 5].
The AdS-CFT correspondence for the Freund-Rubin form, AdS X , of a su-
4 7
×
persymmetric solution of 11-dim supergravity implies that the M-theory on such
background is dual to a supersymmetric 3-dim superconformal field theory. Again
the field theory arises as the SCFT on a stack of M2 branes at the singular apex of
8-dim Ricci-flat space with special holonomy, whose base is 7-dim X . One has to
7
have at least one Killing spinor on X to have a special holonomy. Recently there
7
– 1 –
have been found several countable series of Sasaki-Einstein space in 5-dim and 7-
dim, and their AdS-CFT correspondence has been studied[6, 7, 8]. Especially the
corresponding 3-dim SCFT’s have = 1, or 2 supersymmetries.
N
In this work we focus on a class of countable series of the AdS X spaces,
4 7
×
which has not been studied before. The dual SCFT has = 3 and is the SCFT
N
on the M2 branes on the singular tip of the 8-dim hyperkahler cone with Sp(2)
holonomy and with its base being a tri-Sasakian space X . The 8-dim hyperkahler
7
cone is obtained by a hyperkahler quotient of R12 by a U(1) symmetry group[9]. (In
general, onecouldhavestartedfromflatR4n+8 spacewithU(1)n hyperkahler quotient
with n 1 but here we restrict to n = 1 case for simplicity.) Our tri-Sasakian
≥
space X (t) = S(t ,t ,t ) is characterized by three natural numbers t ,t ,t and has
7 1 2 3 1 2 3
SU(2) U(1)2, SU(2) SU(2) U(1) or SU(2) SU(3) isometry depending on
R R R
× × × ×
none, two, or all of t coincide, respectively. We calculate the volume of the X and
a 7
its supersymmetric 5-cycles Σ and obtain a rational expression for the ratio of the
5
volume of X and that of the unit 7-sphere and so on. We also discuss the dual 3-dim
7
superconformal field theories with = 3. During our investigation, we found these
N
type of space has appeared before in the mathematics literature [10, 11, 12] where it
is known as the Eschenburg space [13]. However, our calculation of the volumes of
the space and the super 5-cycles found in this paper seems original.
The simplest case with X = S(1,1,1)is known as N(1,1) and its cone is the rel-
7
ative moduli space of a single instanton in SU(3) gauge group, which is hyperkahler
8-dim space with one scale and the coset space SU(3)/U(1). There has been consid-
erable work on the AdS-CFT correspondence on the AdS N(1,1) space[14, 15, 16].
4
×
Especially as N(1,1) is homogeneous, one can study the Kaluza-Klein modes of the
theorytocomparethemwiththedualSCFT.Morerecent investigation onAdS X
4 7
×
space with known X and its marginal deformation can be found in Ref.[17].
7
Our work has been motivated in part by the effort to understand the mysterious
3-dim = 16 supersymmetric conformal field theory which is the low energy theory
N
ofN parallelM2branes. Theycanberegardedasthestrongcouplinglimite2 of
→ ∞
= 16 supersymmetric Yang-Mills theories, as one can easily see in the M-theory
N
limit of D2 branes. One may deform the = 16 supersymmetric Yang-Mills by
N
adding Chern-Simons terms, so that the resulting theory has a less supersymmetry
= 3 [18][19]. In the infrared limit or strong coupling limit e2 , the theory
N → ∞
becomes purely Chern-Simons Higgs, which is superconformal. While the Chern-
Simons level k is quantized to be integer, the small k limit is the strong-coupling
| |
limit. Unfortunately its physics is not well understood. One may still hope that the
physics near k = 0 is similar to that of N = 16 superconformal theory. (See for a
similar idea in Ref.[20].)
– 2 –
Another motivation was to try to understand further the old result on the su-
perconformal field theory dual for the AdS geometry with tri-Sasakian space X =
7
N(1,1) whose 8-dim cone is the relative moduli space of a single instanton in SU(3)
gauge theory. While the corresponding field theory may have some component of
Chern-Simons theory, the t’Hooft coupling from the geometry seems to be related
to the parameter N of the corresponding gauge group SU(N) SU(N), instead of
×
the generic ’tHooft coupling N/k of the Chern-Simons-Higgs theory where k is the
integer quantized Chern-Simons level. Our models would provide more examples
along this line.
Final motivation was to try to construct new tri-Sasakian geometry similar to
instanton moduli space by generalizing the moduli space of three distinct magnetic
monopoles, which could be constituent of a single instanton of SU(3) theory in
R3 S1 geometry[21]. Thus one wants to generalize the moduli space of N distinct
×
magnetic monopoles in SU(N) theory broken toU(1)N−1 inR3 S1. There would be
×
magnetic monopoles for each simple root of the extended Dynkin diagram of SU(N)
gauge group. But we generalize the interaction strength between nearest neighbor
by arbitrary magnitude. Only requirement is that the geometry is smooth when-
ever magnetic monopoles are coming together except when all of them are coming
together. In appendix this is shown to lead to the interaction strength between each
link to be some natural number instead of the unity as in the SU(N) case. In the
limit where N 1 monopoles become massless, the relative geometry has only one
−
scale parameter which controls the overall size of the system, and so a cone-like ge-
ometry with a singularity at the apex of the cone. Our geometry X can also obtain
7
from this approach.
The geometry we are interested in is the AdS X type solutions of D = 11
4 7
×
supergravity where X is an Einstein manifold and the four-form field strength F
7 4
∼
vol . We normalize the metric on X so that R (X ) = 6g (X ). To preserve
AdS4 7 µν 7 µν 7
some of 32 supersymmetry of 11-dim supergravity, the eight dimensional cone
8
M
over X with the metric
7
ds2 = dr2 +r2ds2(X ) (1.1)
M8 7
should be Ricci-flat and have special holonomy. (For example see Ref. [22].) When
the cone has Sp(2) = SO(5) holonomy, and so hyper-K¨ahler, the space X is tri-
7
Sasakian and the dual theory is a = 3 SCFT.
N
Such a geometry arises as the near horizon limit of M2 branes lying at the
singular apex of the Ricci-flat cone . The dual SCFT lives on the M2 branes.
8
M
The flux of F on X is proportional to the number of M2 branes. Baryonic states
4 7
of SCFT are dual to five-branes wrapping five-cycles Σ in the manifold X . For
5 7
supersymmetirc states the five-cycles must lift to supersymmetric 6-cycles in the
cone . A supersymmetric 6-cycle will be holomorphic with respect to one of the
8
M
– 3 –
three complex structures, breaking two of six supersymmetries. The dimension of
the baryonic operators are given by the geometric formula [15]
πN Vol(Σ )
5
∆ = . (1.2)
6 Vol(X )
7
In SCFT, they can often be predicted from their R-charges. Comparing the two
predictions is then a non-trivial check for the gauge/gravity correspondence.
This Eschenburg space X = S(t ,t ,t ) [10] can be regarded as a left-quotient
7 1 2 3
spaceofSU(3)groupmanifoldbyU(1)groupwhoseelementsarediag(eit1ψ,eit2ψ,eit3ψ).
Their homological properties seem to be known. Here we provide an explicit metric
and calculation of the volume of the space and supersymmetric 5-cycles. The space
X = S(t ,t ,t ) depends only on the t ’s up to overall common factor. It is homo-
7 1 2 3 a
geneous with SU(3) SU(2) symmetry when t = t = t . When t = t = t , the
R 1 2 3 1 2 3
× 6
space has co-homogeneity one with SU(2) U(1) SU(2) symmetry. When all t
R a
× ×
are different from each other, the space has co-homogeneity two with U(1)2 SU(2)
R
×
symmetry. We find the two kind of expressions for the metric for X (t). The first one
7
is explicit but not useful. The second one is more implicit but shows the symmetric
and cone structures clearly.
Instead of using the metric, we use the equivarient cohomology and localization
technique [23] to calculate the volume X (t). This approach is somewhat esoteric
7
and so the detail is provided here. We find that the ratio of the volume of the tri-
Sasakian space X and that for any supersymmetric 5-dim cycle Σ (t) is independent
7 5
of t = (t ,t ,t ) and identical to the ratio for the volume of unit 7-sphere and that
1 2 3
of unit 5-sphere. The explicit form for the volumes are
vol(X (t)) vol(Σ (t)) t t t (t t +t t +t t )
7 5 1 2 3 1 2 2 3 3 1
= = , (1.3)
vol(S ) vol(S ) l.c.m.(t t ,t t ,t t )(t +t )(t +t )(t +t )
7 5 1 2 2 3 3 1 1 2 2 3 3 1
where l.c.m. means the least common multiple and vol(S ) = π4/3 for the unit 7
7
sphere and vol(S ) = π3 for the unit 5 sphere. The maximum of the above ratio for
5
any t appears when t = t = t = 1 for the well-known space N(1,1).
1 2 3
The dual superconformal field theory in three dimension is SU(N) SU(N)
1 2
×
gauge theory with = 3 supersymmetry. The matter fields U = (u , v∗) are
N a a − a
hypermultiplets in = 4 language belonging to the symmetrized product represen-
N
tation Symta(N) of the fundamental representation of SU(N) and that Symta(N¯)
1
of the anti-fundamental representation of SU(N) . The internal global symmetry
2
is again SU(2) U(1)2 for distinct t . The chiral primary operators and baryonic
R a
×
operators show that one can assign a chiral dimension 1/2 for the U field and its
a
complex conjugates.
Thereareseveraldirectionstopursue. OurSCFTisagainmysterious asthe =
N
8 superconformal field theory in 3-dim as it is not quite the Chern-Simons theory.
– 4 –
They may be the strong-coupling limit of the supersymmetric Yang-Mills Chern-
Simons theory with κ 0. One curious aspect of our AdS-CFT correspondence is
→
that there is no obvious geometry for the dual theory when t = t = t = 1 as the
1 2 3
6
X (t) is defined only up to common factors of t .
7 a
The plan of the paper is as follows. In Sec.2 we define the 8-dim hyperkahler
space (t) which is a singular cone and is obtained from a hyperkahler quotient of
8
M
3-dim quaternion space H3 = R12 by using a single U(1) group. We find its metric
explicitly and also show the space has the cone geometry. We identify its isometry.
In Sec.3 we review the homology property of tri-Sasakian space X (t) first. Then
7
we calculate the volumes of X (t) and supersymmetric 5-cycles Σ in X in the
7 5 7
language of equivariant cohomology. In Sec.4, we identify the dual SCFT and study
its properties. In Appendix, we generalize the caloron moduli space.
2. Hyperkahler Space (t)
8
M
Let us start from 12-dim flat hyperkahler space H3 defined by the three quaternions
q ,q ,q . (See for example Ref. [24] for an introduction.) Each quaternion is defined
1 2 3
as
q = q4 +iσ q , q¯ = q4 iσ q , (2.1)
a a · a a a − · a
with four real numbers qµ,µ = 1,2,3,4 and three Pauli matrices σ1,σ2,σ3. The
a
Euclidean flat metric on 12-dim is
1
ds2 = tr(dq dq¯ ) = dqµdqµ, (2.2)
2 a ⊗ a a a
a a
X X
and the three K¨ahler forms are
1
ω σ = dq dq¯. (2.3)
· 2 ∧
Sometimes we use complex coordinates for quaternions as
u v
a a
q = , (2.4)
a
v¯ u¯
(cid:18)− a a(cid:19)
in which the metric becomes
1
ds2 = (du du¯ +dv dv¯ +c.c.). (2.5)
a a a a
2 ⊗ ⊗
a
X
Another useful coordinate for quaternions is
q = p eiσ3ψa , (2.6)
a a
– 5 –
where p is pure imaginary, or p¯ = p . In terms of the 3-dimCartesian coordinates
a a a
−
r such that
a
ir σ = qiσ q¯= ip σ3p (2.7)
a 3 a a
· −
and angle variable ψ , the flat metric on 12-dim becomes
a
1 dr2
ds2 = a +r (dψ +w dr )2 , (2.8)
a a a a
4 r ·
a (cid:18) a (cid:19)
X
where r = r and w (r ) = (1/r ).
a a a a a
| | ∇× ∇
For each triple natural numbers t ,t ,t , we consider a corresponding abelian
1 2 3
symmetry Ut(1), under which
q q eiσ3taχ, a = 1,2,3 . (2.9)
a a
→
The Ut(1) is unique up to a common factor on triples. The corresponding moment
map µ is
µ σ = t q σ q¯ = t r
a a 3 a a a
·
a a
X X
u 2 v 2 2u v
a a a a
= t | | −| | − . (2.10)
a 2u¯ v¯ u 2 + v 2
a (cid:18) − a a −| a| | a| (cid:19)
X
The space which satisfies the constraint µ = 0 becomes 9-dimensional. Once we
mod outU(1) on this space, the resulting quotient space becomes 8-dim hyperkahler
t
space,
8(t) = µ−1(0)/Ut(1). (2.11)
M
This process of hyperkahler quotient is defined by three natural numbers t ,t ,t .
1 2 3
As (t) is hyperkahler, it is Ricci-flat automatically.
8
M
Let us consider in detail the symmetry of the hyperkahler space (t). The
8
M
first one is the SU(2) symmetry which rotates three complex structures in 12-dim
R
space,
i
q exp( ǫ σ)q , a = 1,2,3, (2.12)
a a
→ −2 ·
where ǫ are the SU(2) parameters. Under this SU(2) transformation, r for each
a
a transforms as a vector. This is commuting with the hyperK¨ahler quotient and so
the resulting space has SU(2) symmetry. The additional symmetry arises from the
R
transformation
q q expiTσ3 (2.13)
a → b ba
where T is the U(3) generator which co(cid:0)mmutes w(cid:1)ith the U(1)t generator
t = diag(t ,t ,t ), (2.14)
1 2 3
– 6 –
and leaves Ut(1) invariant subspace invariant. Thus when t1 = t2 = t3, the resulting
symmetry is SU(3). When t = t = t , the resulting symmetry is SU(2) U(1).
1 2 3
6 ×
When t ,t ,t are all different, the resulting symmetry would be U(1)2.
1 2 3
For t = t = t , the resulting 8-dim space is the moduli space of a single
1 2 3
SU(3) instanton in the center of mass frame. The 8 parameters denote a single scale
parameter and 7 coordinates for the coset space SU(3)/U(1), and so the space is
cone-like. For generic t , the metric is more complicated. A simple way to write the
a
metric is to start from the flat metric (2.8) and express the r in terms of r and r
3 1 2
by using the moment map (2.10) so that with A = 1,2
4ds2 = C dr dr +CAB(dψ +w dr )(dψ +w dr ), (2.15)
AB A B A AC C B BD D
· ·
where
1 t2
C = + 1 ,
11 r t t r +t r
1 3 1 1 2 2
| |
1 t2
C = + 2 ,
22 r t t r +t r
2 3 1 1 2 2
| |
t t
1 2
C = C = , (2.16)
12 21 t3 t1r1 +t2r2
| |
and the vector potential satisfies w = C . This metric is hyperkahler
C AB C AB
∇ × ∇
and regular unless r = r = 0 simultaneously[25, 26].
1 2
Toexpressthemetricsothatthecone-structureismanifestneedsmorework. Let
us focus on the generic case where all t are different. The moment map vanishes and
a
so t1r1+t2r2+t3r3 = 0, which defines a triangle whose side length are t1r1,t2r2,t3r3.
Using the SU(2) transformation, we can put this triangle on the 1-2 plane. In this
R
case, the complex coordinates u and v satisfy
a a
3
r
a
u = v = , t u v = 0. (2.17)
a a a a a
| | | | 2
r a=1
X
The general configuration would be made of the rotation of the triangle in space and
also the phase rotation of u and v in opposite way. Triangle has three independent
a a
parameters. The spatial rotation has three independent parameters. The relative
phase of u ,v variables has three independent parameters, one of which is the global
a a
U(1) which should be mode out. Thus there are eight independent parameters.
To specify the moduli parameter for the triangle, we choose the parameters to
be u = v = r /2 = u eiϕa/2, which implies the moduli metric of the triangle to
a a a a
| |
be
p
3
1 1
ds2 = dr2 +r dϕ2 , (2.18)
∆ 4 r a a a
a=1(cid:18) a (cid:19)
X
– 7 –
with the condition (2.17) being
t r eiϕa = 0. (2.19)
a a
a
X
This is the condition on for three complex vectors t r eiϕa to form a triangle. This
a a
constraint depends only on the relative angles θ of vectors as
a
θ = ϕ ϕ , θ = 2π +ϕ ϕ , θ = ϕ ϕ , (2.20)
1 3 2 2 1 3 3 2 1
− − −
where 0 ϕ < 2π. Only two of the relative angles are independent as θ +θ +θ =
a 1 2 3
≤
2π. The overall orientation angle ϕ = ϕ1+ϕ2+ϕ3 of the triangle are a part of the
3
rotational degrees from SU(2) . The above triangle condition (2.19) implies the
R
three following conditions on the length and relative angles as given in elementary
geometry:
t2r2 = t2r2 +t2r2 +2t r t r cosθ ,
1 1 2 2 3 3 2 2 3 3 1
t2r2 = t2r2 +t2r2 +2t r t r cosθ ,
2 2 3 3 1 1 3 3 1 1 2
t r2 = t2r2 +t2r2 +2t r t r cosθ , (2.21)
3 3 1 1 2 2 1 1 2 2 3
of which only two are independent. Thus these conditions reduce the independent
variables to three, which we choose as one length variable, and two relative angle
variables.
To solve the above constraints (2.21), let us introduce an angle variable A, a
length square variable L, and an area variable S such that
A = (cotθ +cotθ +cotθ ) ,
1 2 3
−
L = t2r2 ,
a a
a
X
S = t r t r sinθ = t r t r sinθ = t r t r sinθ . (2.22)
1 1 2 2 3 3 3 1 1 2 2 2 3 3 1
Note that S is twice the area of the triangle, and the triangle condition implies that
L cotθ
S = , t2r2 = L 1+ a for each a (2.23)
2A a a A
(cid:18) (cid:19)
with A 0. Let us now introduce the radial variable in 12-dim flat space as the
≥
length variable,
r = q 2 + q 2 + q 2 = √r +r +r . (2.24)
1 2 3 1 2 3
| | | | | |
Defining a function B ofpangles θ as
a
1 cotθ
a
B = 1 , (2.25)
a tas(cid:18) − cotθ1 +cotθ2 +cotθ3(cid:19)
X
– 8 –
we see the length variable L is given in terms of three independent variables r,θ as
a
follows,
r4
L = . (2.26)
B2
So the variables r can be written in terms of scale variable r and angles θ asfollows,
a a
1 cotθ 1 cotθ
r = r2ρ , ρ 1 a 1 c . (2.27)
a a a
≡ tas(cid:18) − bcotθb(cid:19)(cid:30) c tcs(cid:18) − dcotθd(cid:19)
X
P P
Note that three functions ρ of angles θ satisfies the condition ρ +ρ +ρ = 1.
a a 1 2 3
The moduli space of the triangle on the plane would be then
1 1 r2 dρ2
ds2 = dr2 +r dϕ2 = dr2 + a +dϕ2 , (2.28)
∆ 4 r a a a 4 ρ a
a (cid:18) a (cid:19) a (cid:18) a (cid:19)
X X
as ρ = 1. Note the ϕ can be written as the relative angles θ and the overall
a a a
orientation of the triangle on the plane. The metric for the can be now obtained
P M8
by parameterizing quaternion as follows
u u u
Q 1 2 3 = RQ T , (2.29)
0
≡ v¯ v¯ v¯
(cid:18)− 1 − 2 − 3(cid:19)
where the R is an SU(2) element parameterized by Euler angle, which includes
the orientation of the triangle on the plane, T is a diagonal U(3) element, say,
T = diag(eiψ1,eiψ2,eiψ3) and Q is the value of Q when the triangle is on 1-2 plane
0
and so
√r eiϕ1/2 √r eiϕ2/2 √r eiϕ3/2
1 2 3
Q = . (2.30)
0 √r e−iϕ1/2 √r e−iϕ2/2 √r e−iϕ3/2
(cid:18)− 1 − 2 − 3 (cid:19)
The metric of the triangle is ds2 = dQ dQ¯ of the metric (2.28) and so the metric
∆ 0 0
on the 9-dim space is
1
ds2 = (dQ dQ¯ +dQ¯ dQ) . (2.31)
µ−1(0) 2 ⊗ ⊗
It is trivial to mode out Ut(1) to get the 8-dim hyperkahler space 8(t). The 3
M
kahler forms are again given by
ω σ = dQ dQ¯. (2.32)
· ∧
The isometry of (t) can be easily read. First of all the SU(2) transformation
8
M
by R matrix leads to SU(2) symmetry which mixes three complex structure. In
R
addition there are U(1) U(1) isometries from the tranformations given by T matrix
×
modulo Ut(1), which are tri-holomorphic as they leave three kahler structures invari-
ant. When some of t ,t ,t become identical, these tri-holomoprhic isometries get
1 2 3
– 9 –