Getting N=1 super Yang-Mills from strings

0.23 MB·English

by  P. Merlatti

= 1
7
0 Getting N super Yang-Mills
0
2
from strings

n
a
J

2 Paolo Merlatti
1

1
v
0 Departamento de Fisi
a de Parti
ulas, Universidade de Santiago de Compostela
2 E-15782, Santiago de Compostela, Spain
1
merlattifpaxp1.us
.es
1
0
7
0
/ Abstra
t
h
t
- In the
ontext of the gauge-string
orresponden
e, we dis
uss the
p
e spontaneousS5partial breaking of supersymmetry. Starting from the
h orbifold of , supersymmetry breaking leads us to
onsider the (re-
v: solved)
onifold ba
kground and some of the gauge dynami
s en
oded
Xi in that geometry. Using this gravity=d1ual, we
ompute the low energy
e(cid:27)e
tive superpotential for su
h N theories. We are naturally led
r
a to extend the Veneziano-Yankielowi
z one: glueball (cid:28)elds appear.
Contents
1 Introdu
tion 1
2 Partial supersymmetry breaking and a new path to the
oni-
fold 3
2.1 Probe analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Away from the orbifold limit . . . . . . . . . . . . . . . . . . . 7
3 E(cid:27)e
tive superpotentials from supergravity 11
3.1 An extension of the Veneziano-Yankielowi
z superpotential . . 12
4 Con
lusions 14
1 Introdu
tion
Even if the most
elebrated example of the gauge/string
orresponden
e is
= 4
theN AdS-CFTduality[1℄(see[2℄forareview), manyinterestingresults
have been also found for less (super)-symmetri
theories (see [3, 4℄ for some
extensive reviews).
= 1
For the
ase of N super Yang-Mills theories in four dimensions (i.e.
4 real super
harges) the relevant ba
kground geometry is a warped prod-
u
t of four-dimensional Minkowski spa
e times the
onifold, in presen
e of
ba
kground (cid:29)uxes. There are two known expli
it supergravity solutions
or-
responding to this
ase, the MN solution [5℄ and the KS one [6℄. Both the
solutions preserve the same amount of supersymmetry but they
orrespond
to two di(cid:27)erent dual gauge theories, at least in the ultraviolet. It seems also
that, despite des
ribing similar infrared dynami
s, they
orrespond to two
di(cid:27)erent universality
lasses of
on(cid:28)ning theories. It is then important to
understand how duality works
ase by
ase.
In thistalkwe make
onta
t witha ni
e andfruitfulway oflookingatthis
duality. Su
h perspe
tive hasbeenintrodu
edin[7℄forthetopologi
alstring.
There, it has been shownNthat in topologi
al striSng3 theory the deformSed3

onifold ba
kground, with 3-brane wrapped on (equivalent to the
Chern-Simon theory [8℄), is dual to the same topologi
al string but without
string modes and on a di(cid:27)erent ba
kground. The new ba
kground is the
N
resolved
onifold,whoseparametersdependnowon . Su
hgauge/geometry

orresponden
e has been su

essfully embedded in superstring theory in [9℄.
1
The
entral idea is the
on
ept of geometri
transition. The (cid:28)eld theory
D
has to be engineered via -branes wrapped over
ertain
y
les of a non-
trivial Calabi-Yau geometry. The low energy dual arises from a geometri
transition of the Calabi-Yau, where the branes have disappeared and have
= 1
been repla
ed by (cid:29)uxes. The simplest example [9, 10℄ is pure N super
N
DYa5ng-Mills theory. Before the transitionS2the (cid:28)eld theory is engineered on
branes wrapped on the blown-Nup of a resolved
onifold. AfSte3r the
transition we are instead left with units of R-R (cid:29)ux through the of a
deformed
onifold [6℄.
= 1
Another way to make
onta
t with the N
onifold theory is to start
= 2
from a moresupersymmetri
theory, namely a quiver NS5/Z (cid:28)eld theory [11℄.
2
In that
ase one starts from the orbifold ba
kground , that is dual to
= 2 U(N) U(N)
the N super-
onforma(lNY,aN¯ng)-Mi(lNl¯s,N) × the=ory1 with hyper-
multiplets transforming in ⊕ . From an N perspe
tive,
the hypermultiplets
orrespond to
hiral multiplets and other
hiral multi-
U(N)
plets in the adjoint representations of the two 's are present. There
is also a non-trivial superpotential for the multiplets in the bi-fundamental
representations of the gauge group. One
an add now a relevant term to this
superpotential. To integrate outthe adjoint multipletsinpresen
e of thSis5/rZel-
2
evant perturbation
orresponds to blow up the orbifold singularity of
[11℄. Remarkably, in [11℄ it has also been shown that su
h blown-up spa
e is
T1,1
topologi
ally equivalent to the
oset spa
e , the base of the
onifold.
We will des
ribe the (cid:29)ow to the
onifold theory in a di(cid:27)erent way [12℄,
hoping to
ontribute to
larify some of its intri
ate aspe
ts. We start from
= 2
the N super
onformal Yang-Mills theory. When
onformal invarian
e
is broken introdu
ing fra
tional branes, we (cid:28)nd how partial supersymme-
try breaking
an be des
ribed on the supergravity side. Within this purely
ten dimensional superstring
ontext, we see then how the resolved
onifold
= 1
ba
kground emerges naturally as the proper one to engineer N super-
Yang-Mills theory, with no matter and just one gauge group. We follow all
the steps in terms of expli
it supergravity solutions.
Having eventually at our disposal a proper gravitational dual des
ription
= 1
of pure N super-Yang-Millstheory, many results
an be obtained for the
latter. Some of them are
he
ks of previously known (cid:28)eld theoreti
al
ompu-
tations. Othersarenewpredi
tions. Theinterplaybetween gaugetheory and
gravity has indeed a quite ri
h stru
ture. Among many, we will des
ribe one
of the possible appli
ations. We will see how the dual gravitational des
rip-
tion
an be used to determine the low energy e(cid:27)e
tive superpotential of the
2
(cid:28)eld theory under
onsideration, namely the Veneziano-Yankielowi
z [13℄ su-
1
perpotential . The same gravitational des
ription seems also to suggest that
other degrees of freedom, with the same quantum number of a glueball, are
relevant at low energy. We (cid:28)nally determine the form of the generalized
Veneziano-Yankielowi
z superpotential, with the in
lusion of su
h glueball
states [16℄. Some physi
al impli
ations are dis
ussed.
2 Partial supersymmetry breaking and a new
path to the
onifold
WestartfromthesimplestgeneralizationoftheoriginalAdS/CFT
orrespon-
den
e, nSa5mely we will
onsideSr5t/hZe
ase in whi
h the internal manifold is not
2
simply , but the orbifold [17℄. As dis
ussed in the introdu
tion,
= 2 U(N) U(N)
the dual gauge theory is N super-
onform(Nal,NY¯a)ng-M(N¯il,lsN) ×
theory with hypermultiplets transforming in ⊕ . As for the
= 4
original N
ase, on the string side we know the expli
it supergravity
solution
onne
ting the UV geometry (the relevant one for the engineering
of the gauge theory) and the IR one.SI5f/wZe parameterize the six dimensional
2
transverse spa
e, (i.e. the
one over ), in terms of the
oordinates
X = rsinξcosτ
4
(1)
X = rsinξsinτ
5
(2)
θ β +θ
X = rcosξcos cos
6
2 2 (3)
θ β +θ
X = rcosξcos sin
7
2 2 (4)
θ β θ
X = rcosξsin cos −
8
2 2 (5)
θ β θ
X = rcosξsin sin − ,
9
2 2 (6)
0 < β < 2π 0 < ξ < π 0 < θ < π
the ri0gh<t pτres<
ri2pπtion for the orbifoldingis Z , ,
2
and . It is easy to see the a
ts just on the
oordinates
1
Inpresen
eof(cid:29)avors,alsotheA(cid:31)e
k-Dine-Seiberg[14℄superpotential
anbederived
in an analogous way [15℄
3
X , X , X , X
6 7 8 9
. The metri
of the solitoni
solution we are dis
ussing is:
3
ds2 = H(r)−1/2dx2 +H(r)1/2 dr2 +r2 cos2ξ σ2 +sin2ξdτ2 +dξ2 ,
1,3 i
!
(cid:16) Xi=1 (cid:17)
(7)
σ
i
where the are de(cid:28)ned as
2σ = sinβdθ+cosβsinθdφ,
1
− (8)
2σ = cosβdθ+sinβsinθdφ,
2
(9)
2σ = dβ +cosθdφ
3
(10)
dσ = ε σ σ H(r)
i ijk j k
and they satisfy ∧ . The expli
it expression for is:
N
H(r) = 1+ .
r4 (11)
r M
1,3
We sSee5/tZhat for → ∞ wer re
over thAedSUV gSe5o/mZetry times the
one
2 5 2
over , while for small the dual × geometry is found. The
AdS
fa
tor implies the dual gauge theory is
onformal. It is well known that
in orbifold spa
es a way to break
onformal invarian
e is to add fra
tional
branes. Su
h branes are
harged under the twisted (cid:28)elds that appear in the
orbifold spe
trum. As we want to break
onformal invarian
e, we try now
to swit
h on the proper twisted (cid:28)elds (the ones that
ouple to fra
tional
D3-branes in su
h ba
kground). We start perturbing the same UV geometry
and we hope to be able to (cid:28)nd a new solution.
Before doing this, we make the following
hange of
oordinates in the six
dimensional transverse spa
e:
ρ
r = ρ2 +R2 , tanξ = .
R (12)
p
S5/Z
2
We see that the metri
on the
one over be
omes
3
ds2 = dR2 +R2 σ2 + dρ2 +ρ2dτ2.
6 i (13)
i=1
X
It is thus easy to see thCat2/tZhis spa
e fa
torizes inS3t/hZe produ
t of a four-
2 2
Rdi,mβen,sθional AφLE spa
e ( , i.e. the
one ovRe2r , parameterizρed by
and ) and a two dimensional plane ( , parameterized by and
4
τ S5/Z
2
). In this way we re
Rov2er Cth2e/Zisomorphism between the
one over
2
and the orbifold spa
e × . This isomorphism makes all the pi
ture
D3

onsistent. The (cid:28)eld theory on the -branes is indeed engineered on the
latter orbifold [18℄.
D3
We
an now disregard the bulk -branes originating the metri
(7) and
try to (cid:28)nd an analogous solution just for tRhe2 frCa
2t/iZonal branes. This has
22
been done expli
itely in [19℄ for the orbifold × and we
an borrow
N
their results. They dRes1
,5ribeCa/Zsta
k of fra
tional D3-branes in the =IIB2
2 2
orbifold ba
kground ( × ) [19, 21℄. On their world volume a N
SU(N)
supersymmetri
Yang-Mills theory lives.
The solution
an be written in term˜s of the metri
, a R-R four form
C b c
4
potential ( ), a NS-NS twisted s
alar ( ) and a R-R twisted one ( ):
ds2 = H(r,ρ)−1/2η dxαdxβ + H(r,ρ)1/2δ dxidxj,
αβ ij
C = H(r,ρ)−1 1 dx0 ... dx3,
4
− ∧ ∧ (14)
ρ
˜
c = (cid:0)NK τ ,(cid:1) b = NKlog ,
ǫ
α,β = 0,...,3; i,j = 4,...,9; r = δ xixj; K = 4πg α′; ǫ
ij s
where is a
ρ τ x4, x5
regulator; and are polar
oordinates in thepplane . The self-duality

onstraint on the R-R (cid:28)ve-form (cid:28)eld strength has to be imposed by hand.
H(r,ρ)
The pre
ise fun
tional form of the fun
tion has also been deter-
mined [19℄ but it is not relevant for our purposes. It is enough for us to
note that the spa
etime (14) has a naked singularity. We are thus not able
to follow our program
ompletely and we are prevented to (cid:28)nd the dual ge-
ometry in the deep IR. This is a quite general feature of those gravitational
ba
kgrounds that are dualtonon-
onformalgauge theories. Lu
kily, inmany

ases the singularities are not a
tually present but they are removed by var-
= 2
ious me
hanisms. In theories N supersymmetri
, like this one or the
similar (cid:16)wrapped branes(cid:17) example [22℄, the relevant me
hanism is the en-
hançon [23℄. It removes a family of time-like singularities by forming a shell
of (massless) branes on whi
h the exterior geometry terminates. As a result,
the interior singularitiesare (cid:16)ex
ised(cid:17) and the spa
etime be
omes a

eptable,
even if the geometry inside the shell has still to be determined. For the
ase
at hand, the appearan
e of su
h shell, its (cid:28)eld-theoreti
interpretation and
its
onsisten
y at the supergravity level have been dis
ussed in [19, 21, 24℄.
2
see [20℄ for more general orbifolds
5
(ρ,τ)
It turns out that the enhançon shell is simply a ring in the -plane. Its
radius is ρ = ǫ e−π/(2Ngs).
e
(15)
ρ ρ
e
The geometry (14) is thus valid outside the enhançon shell, i. e. for ≥ .
On the gauge theory side the enhançon lo
us
orresponds to the lo
us where
the Yang-Mills
oupling
onstant diverges [19, 21℄.
2.1 Probe analysis
3
To understand better what is going on we
an perform a probe analysis
in terms of the (cid:28)elds appearing in (14) [12℄. We should thus
onsider the
following boundary a
tion
T 1
S = 3 d4x det G +2πα′F 1+ ˜b
bdy −√2k − αβ αβ 2π2α′ (16)
orb Z (cid:18) (cid:19)
T p 1 T α′
3 ˜ 3
+ C 1+ b + c F F.
√2k 4 2π2α′ √2k 2 ∧
orb Z (cid:18) (cid:19) orb Z
T = √π k = (2π)7/2g α′2
3 orb s
where , and we have also in
lxuad=ed2tπhαe′Φwaorld-
volume ve
tor (cid:28)elds (cid:29)u
tuations. As usual, the
oordinates are
identi(cid:28)ed with the
hiral s
alar (cid:28)elds of the Yang-Mills theory living on the
world-volume of the brane. Writing now the boundary a
tion (16) in terms
of the
lassi
al solution (14), we get:
T (2πα′)2 1 1
S = 3 d4x δ ∂αΦa∂ Φb + FαβFa
probe −√2k 2 2 ab α 4 a αβ · (17)
orb(cid:26)Z (cid:18) (cid:19)
′
NK ρ NK ρ α
1+ log +V 1+ log KNτ F F .
· 2π2α′ ǫ 4 2π2α′ ǫ − 2 ∧
(cid:18) (cid:19) (cid:18) (cid:19) Z (cid:27)
We note that a non trivial potential is generated, namely:
T NK ρ 1
3
V(Φ) = 1+ log = τ
√2k 2π2α′ ǫ (2π)3α′2 Y2 (18)
orb (cid:18) (cid:19)
τ τ +iτ 4πi + θY
where we have de(cid:28)ned Y ≡ Y1 Y2 ≡ gY2 2π as the holographi

om-
plexi(cid:28)ed Yang-Mills
oupling [26℄:
1
τ = c+i 2π2α′ +˜b .
Y (2π√α′)2g (19)
s
(cid:16) (cid:16) (cid:17)(cid:17)
3
for a review of this method see, for example, [25℄
6
This
omputation shows that this system violates the no-for
e
ondition.
= 2
This
ould seem quite surprising as the
on(cid:28)guration is still N super-
symmetri
and usually the no-for
e
ondition is respe
ted with su
h amount
of supersymmetry (on the (cid:28)eld theory side su
h
onditionre(cid:29)e
ts the (cid:29)atness
of the Coulomb bran
h). Nevertheless, at leastfrom a purely (cid:28)eld theoreti
al
pointofview, we know thatsu
h (cid:29)atness
anbe violatedbyele
tri
-magneti
= 2
Fayet-Iliopoulosterms[27℄, thatpreserve N supersymmetrybutgenerate
a non-trivial potential:
e + mτ 2 + ξ2
V = | | ,
τ (20)
2
e, m ξ
where and are free parameters. Lu
kily, it turns out the potential
(18) is just a spe
i(cid:28)

ase of (20), namely with
mθ 1
e = Y , ξ = 0 m2 = .
− 2π and (2π)3α′2 (21)
We see in this way that the violation of the no-for
e
ondition does not

ontradi
t supersymmetry, but it
orresponds to the generation of a purely
magneti
FI term. We
all it purely magneti
be
ause a dyon with quantum
(e,m) e+ mθ
numbers has an e(cid:27)e
tive ele
tri

harge equal to 2π [28℄ and with
the values in (21) it vanishes identi
ally. Su
h values ensure also that the
τˆ
probe-brane is in a proper va
uum in the dire
tion, but for generi
values
ρ ρ = ρ
e
of it is not and it moves toward . This situation holographi
ally

orresponds to va
ua preserving all the supersymmetries but signaling a sin-
gularity. This
ontinues to be in perfe
t agreement with the gravity pi
ture
we just got: the probe-brane is pushed to the enhançon and there stays.
From the gauge theory it is moreover possible to argue [27℄ at that point of
the moduli spa
e extra massless states appear (i.e. monopoles) and to give a
physi
al interpretation of the singularity. On the gravity side it
orresponds
to the well known fa
t that at the enhançon extra massless obje
ts appear
[23℄ and it enfor
es the interpretation of the enhançon lo
us as the (cid:28)eld the-
oreti

urve of marginal stability [12℄ (we mean the
urve in moduli spa
e
where extra massless obje
ts, typi
ally dyons, appear [29℄).
2.2 Away from the orbifold limit
We are now going tosoften the orbifoldsingularityand
onsider the resulting
(smooth) Egu
hi-Hanson spa
e [30℄. We generalize the probe analysis that
7
we have done in the orbifold and (cid:28)nd partial supersymmetry breaking. We
= 1
are then led to
onsider N supersymmetri
Yang-Mills theory and the
4
resolved
onifold geometry [31℄ .
The dual (cid:28)eld theory analysis of the previous se
tion has been entirely
performed in an orbifold ba
kground and we have found purely magneti
FI
terms. It is well known [18℄ that a way to introdu
e standard (i.e. ele
tri
)
FI terms is via the blow up of the orbifold singularity. In su
h
ase the
resulting spa
e is no longer an orbifold but it is a (smooth) Egu
hi-Hanson
EH
( ) spa
e [30℄. This is an ALE spa
e. It is
hara
terized by three moduli
that
orrespond to three di(cid:27)erent ways of blowing-up the relevant two
y
le
when going away from the orbifold limit (via the triplet of massless NS-NS
= 2
twisted s
alars). Analogously to the (cid:28)eld theory des
ription of N FI
SU(2)
R
terms, it is possible to use the global symmetry and to redu
e to the

ase of just one modulus. The one-parameter family of metri
s we get in this
way
an be written as:
−1
a 4 a 4
ds2 = 1 dr2 + r2 1 σ2 + r2(σ2 +σ2),
EH − r − r 3 1 2 (22)
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:16) (cid:17) (cid:16) (cid:17)
a r a2
wσhere ≤ ≤ ∞, is propoSr3tional to the sizθe, oφf,tβhe blown-up
y
le, the
i
are de(cid:28)ned in terms of the Euler angles ( ) as in (9).
The solution
orresponding to fra
tional branes in the Egu
hi-Hanson
ba
kground has been found in [33℄ and it is very similar to the orbifold one.
It
an be written in terms of a metri
ds2 = H(r,ρ)−1/2η dxαdxβ + H(r,ρ)1/2 dρ2 +ρ2dτ2 + H(r,ρ)1/2ds2 ,
αβ EH
(cid:0) (cid:1) (23)
a self dual R-R (cid:28)ve-form
F = d H(r,ρ)−1 dx0 ... dx3 + ⋆d H(r,ρ)−1 dx0 ... dx3 ,
5
∧ ∧ ∧ ∧ (24)
(cid:0) (cid:1) (cid:0) (cid:1)
and a
omplex three form (a linear
ombination of the NS-NS one and the
R-R one) valued only in the transverse spa
e:
FNS +iFR = dγ(ρ,τ) ω,
H ≡ 3 3 ∧ (25)
γ(ρ,τ) R2 ω
where is an analyti
fun
tion on the transverse , is the har-
r
moni
anti-self-dualformon the Egu
hi-Hanson spa
e and isnow the radial
Egu
hi-Hanson
oordinate.
4
see also [32℄ for a way to embed the non-supersymmetri
solution of [31℄ in a super-
symmetri
set-up
8
Evenifthemi
ros
opi
interpretationofthissolutionisquiteobs
ure[33℄,
we assume that it
orresponds to the standard geometri
al interpretation of
D3 D5
fra
tional -branes as -branes wrapped on the non-trivial
y
le that
shrinks to zero size in the orbifold limit [34℄. The form of the solution is
indeed
onsistent with the interpretation of twisted (cid:28)elds as (cid:28)elds
oming
from wrapping of higher degree forms[34℄, as it was already
on(cid:28)rmed in [35℄
byexpli
it
omputationsofstrings
atteringamplitudes. Themaindi(cid:27)eren
e
with respe
t to the orbifold solution is the expli
it fun
tional form of the
H(r,ρ)
warp fa
tor , whose physi
al impli
ation is that there is still a naked
singularitys
reened by an enhançon me
hanism, but
ontrary to the orbifold

ase, in the internal dire
tions (those along the ALE spa
e) the singularity
is
ured by the blowing-up of the
y
le and there is no enhançon there [33℄.
Unlu
kily, the Egu
hi-Hanson ba
kground la
ks of a
onformal (cid:28)eld the-
ory des
ription. This makes it di(cid:30)
ult to determine the
orre
t form of the
boundary a
tions one would need to get gauge theory information. It looks
nevertheless quite reasonable to assume the proper boundary a
tion is the
same as the orbifoldone with the in
lusion of the new FI term
orresponding
to having blown up the
y
le. This
an be motivated by the knowledge of
the expli
it solution in the Egu
hi-Hanson
ase, that makes it
lear the
lose
similarity to its orbifold limit. Besides, this is very analogous in spirit to the
su

essful assumption made in [36℄. The potential the brane (cid:28)lls will be thus
the same we
omputed in the orbifold (we refer to the magneti

ase (18))
ξ = 0
but now with an extra term
orresponding to have 6 and proportional
a2
to the size of the blown-up
y
le ( ):
ξ2
V(Φ) = m2τ +
2
τ (26)
2
We see again that the no-for
e
ondition is violated but now there is an
interesting minimum where the probe-brane
an sit, i.e. at
π ξ
ρ∗ = ρeeNm,
(27)
ρe ρ∗ > ρe ρ = ρ∗
where the enhançon radius has been de(cid:28)ned in (15). As ,
de(cid:28)nesaregularspa
e-timelo
us. Fromthe(cid:28)eldtheoryanalysis[27℄weknow
this kind of va
uum preserves just half of the supersymmetry. This implies
= 1
that [12℄ the gauge theory living on the probe is N supersymmetri
with
M
a massless ve
tor multiplet and a massive
hiral one, with mass given by:
m2
′
M = τ .
2ξ h i (28)
9