Phases of Thermal N=2 Quiver Gauge Theories

0.69 MB·

by  Kasper J. Larsen

PreprinttypesetinJHEPstyle-HYPERVERSION arXiv:0708.3199 [hep-th℄
= 2
Phases of Thermal N Quiver Gauge Theories
8
0
0
2

n
a
Kasper J. Larsen and Niels A. Obers
J

0 The Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark
3
E-mail: kjlarsennbi.dk,obersnbi.dk

]
h
t N U(N)M = 2 S1 S3.
- Abstra
t: We
onsider large thermal N quiver gauge theories on ×
p
R
e We obtain a phase diagram of the theory with -symmetry
hemi
al potentials, separating
h
[ a low-temperature/high-
hemi
al potential region from a high-temperature/low-
hemi
al
= 4
potential region. In
lose analogy with the N SYM
ase, the free energy is of order
3 (1) (N2M)
v O in the low-temperature region and of order O in the high-temperature phase.
9 = 2
We
on
lude that the N theory undergoes a (cid:28)rst order Hagedorn phase transition at
9
1 the
urve in the phase diagram separating these two regions. We observe that in the region
3
of zero temperature and
riti
al
hemi
al potential the Hilbert spa
e of gauge invariant
.
8
operators trun
ates to smaller subse
tors. We
ompute a 1-loop e(cid:27)e
tive potential with
0
7 non-zero VEV's for the s
alar (cid:28)elds in a se
tor where the VEV's are homogeneous and
0
mutually
ommuting. At low temperatures the eigenvalues of these VEV's are distributed
: S5/Z S5/Z
v M M
uniformly over an whi
h we interpret as the emergen
e of the fa
tor of
i AdS S5/Z
X 5 M
the holographi
ally dual geometry × . Above the Hagedorn transition the
r
a eigenvalue distribution of the PolySa5k/oZv loopSo1pens Sa6g/aZp, resulting in the
ollapse of the
M M
joint eigenvalue distribution from × into .
1/N
Keywords: Expansion, AdS-CFT Corresponden
e, Matrix Models.
Contents
1. Introdu
tion 2
= 2 R
2. N quiver gauge theory with -symmetry
hemi
al potentials 5
= 2
2.1 Review of N quiver gauge theory 5
R
2.2 Lagrangian density with -symmetry
hemi
al potentials 8
3. Zero-
oupling limit and the matrix model 10
3.1 One-loop quantum e(cid:27)e
tive a
tion 10
3.2 The matrix model 12
4. Phase stru
ture 14
4.1 Low-temperature solution and phase transition 14
4.2 Solution above the Hagedorn temperature 18
4.3 Quantum me
hani
al se
tors 21
5. One-loop quantum e(cid:27)e
tive a
tion with s
alar VEV's 24
5.1 Quantum
orre
tions from bosoni
(cid:29)u
tuations 27
5.2 Quantum
orre
tions from fermioni
(cid:29)u
tuations 30
Z
M
5.3 Generalization to other orbifold (cid:28)eld theories 30
6. Topology transition and emergent spa
etime 31
6.1 Low-temperature eigenvalue distribution 32
6.2 Eigenvalue distribution above the Hagedorn temperature 34
7. Dis
ussion and
on
lusions 37
= 2
A. Detailed des
ription of N quiver gauge theory 39
= 4
A.1 Relation to N SYM theory 39
R
A.2 -symmetry 46
B. Bosoni
and fermioni
(cid:29)u
tuation determinants 48
B.1 Bosoni

ase 48
B.2 Fermioni

ase 49
Referen
es 51
(cid:21) 1 (cid:21)
1. Introdu
tion
N U(N)
The phase stru
ture of large gauge theories at (cid:28)nite temperature is in itself a very
ri
h and interesting subje
t that may provide qualitative insight into the phase stru
ture of
QCD. Even more so, the AdS/CFT
orresponden
e [1, 2, 3℄ has provided a general frame-
work for translating results obtained in weakly
oupled thermal gauge theory into results
about the (cid:28)nite temperature behavior of the physi
s of bla
k holes and stringy geometry
at strong
oupling. One su
h
onne
tion was suggested by Witten [4℄ who argued that the
AdS AdS
5 5
Hawking-Page phasetransition[5℄betweenthermal andthelarge S
hwarzs
hild
bla
k holeshouldhaveaholographi
dualdes
riptionasa
on(cid:28)nement/de
on(cid:28)nement tran-
S1 S3
sitioninthedualthermal(cid:28)eldtheoryde(cid:28)nedonthe
onformalboundary × ofthermal
AdS
5
.
N U(N) S3
A general framework for studying large gauge theories on at (cid:28)nite tem-
= 4 U(N)
perature was given in [6℄. In parti
ular, this
onsidered N SYM theory whi
h
= 4
was also independently studied in [7℄. Furthermore, for the N
ase the analysis was
R
extended in [8, 9℄ to in
lude
hemi
al potentials
onjugate to the -
harges. In this way a
phase diagram of the theory as a fun
tion of both temperature and
hemi
al potentials was
obtained. As one appli
ation of the phase diagram, in [9℄ the observation was made that in
= 4
regions of small temperature and
riti
al
hemi
al potential N SYM theory redu
es to
XXX
1/2 1
quantum me
hani
al subse
tors, in
luding the Heisenberg spin
hain.
= 4
Again for N SYM theory, the framework of [6, 7℄ was generalized in a di(cid:27)erent
dire
tion in [11℄ by allowing non-zero VEV's for the s
alar (cid:28)elds. There a one-loop e(cid:27)e
tive
S3
potentialforthetheoryat(cid:28)nitetemperature on atweak'tHooft
oupling was
omputed
under the assumption that the VEV's of the s
alar (cid:28)elds are
onstant and diagonal ma-
2
tri
es. The potential was used there to study the manifestation of the Gregory-La(cid:29)amme
AdS
3 5
instability of the small bla
k hole from the weakly
oupled gauge theory point of
view. The solutions to the equations of motion obtained from the e(cid:27)e
tive potential of [11℄
were given in [13℄ in terms of a joint eigenvalue distribution of the Polyakov loop and the
s
alar VEV's. Within the se
tor of
onstant and
ommuting s
alar VEV's itwas found that
the topology of the eigenvalue distribution of these VEV's undergoes a phase transition
S1 S5 S6 S5
× → at the Hagedorn temperature. The authors interpreted the eigenvalue dis-
S5
tribution of the s
alar VEV's as the emergen
e of the fa
tor of the holographi
ally dual
AdS S5
5
thermal × geometry. It should be noted that, while the trun
ation to
ommuting
matri
es is
onsistent, this se
tor will not des
ribe the absolute minima of the a
tion [14℄.
For this reasonthe observed phasetransitions in the
ommuting saddlesstudied inRef.[13℄
are not transitions in the full gauge theory.
The dis
overy that the eigenvalues of s
alar VEV's re
onstru
t the dual spa
etime ge-
ometry was originally made by Berenstein et al. in [15, 16, 17, 18℄ by setting up matrix
models for the various se
tors of BPS operators in the
hiral ring. In parti
ular the model
1
Re
entlyotherde
oupling limits have been found in near-
riti
al regions byextending this analysis to
S3
in
lude the
hemi
al potentials
onjugate to theangular momentaon [10℄.
2
This potential was
omputedearlier in [8℄ for the spe
ial
ase of zero Polyakovloop eigenvalues.
3
See[12℄ for a re
entreview of theGregory-La(cid:29)ammeinstability.
(cid:21) 2 (cid:21)
for 1/8 BPS operators was developed in [17℄ where the dynami
s was shown to redu
e to
Z,X,Y
that of the eigenvalues of three
ommuting Hermitian matri
es plus two fermioni
W
4 α
matri
es . The quantum me
hani
al Hamiltonian for the eigenvalues involves an at-
tra
tive harmoni
os
illator part and a repulsive Vandermonde type part. These for
es are
C3
balan
ed when the eigenvalues are lo
alized to a hypersurfa
e in whi
h is taken to be an
S5 SO(6) S5
due to the invarian
e of the quantum Hamiltonian. This was identi(cid:28)ed with
S5 AdS S5
5
the fa
tor of the holographi
ally dual geometry × .
= 2 U(N)M
The purposeof thispaper istoinvestigate thephase stru
ture of N quiver
5
gauge theories at (cid:28)nite temperature. De(cid:28)ned at zero temperature and on a (cid:29)at spa
etime,
= 2
thesegaugetheoriesareN supersymmetri
and
onformallyinvariant[24,25℄.We
arry
out the investigation of the phase stru
ture in two dire
tions. First, we
onsider the
ase
R
of non-zero -symmetry
hemi
al potentials. One interesting question here is whether the
high-temperature phase admits several solutions. A further point of interest is to examine
whether one
an un
over information about
losed subse
tors of the as yet not
ompletely
= 2
settled underlying spin
hain of N quiver gauge theory by studying the near-
riti
al
(T,µ)

hemi
al potential and low temperature regions of the phase diagram of the theory
= 4
as done for N SYM theory [9, 26℄.
S5
Another question of interest is to what extent the eigenvalue distribution of the
= 4
N SYM s
alar VEV's found in [17℄ and [13℄
an be interpreted as the emergen
e of
S5 AdS S5
5
the fa
tor of the dual string theory geometry × . To examine this question, we
= 2 Z
M
make use of the fa
t that N quiver gauge theory
an be realized as a proje
tion
= 4 = 2
of N SYM theory. The holographi
ally dual spa
etime of the N theory is thus
AdS S5/Z Z S5
5 M M
× where onlya
tsonthe fa
tor.Iftheaboveinterpretation ofemergent
S5/Z
M
spa
etime is
orre
t, we should then expe
t to (cid:28)nd an eigenvalue distribution for
= 2
the VEV's of the s
alar (cid:28)elds of N quiver gauge theory. This has been studied via

ounting of BPS operators in [27, 28, 29℄. Our approa
h to the problem is
omplementary
in that it isvalid for weak 't Hooft
oupling, andit isvalid for all temperatures inthe range
0 TR λ 1/2 T = 0
−
≤ ≪ unlike [27, 28, 29℄ whi
h is only valid for . In parallel with Ref. [13℄,
we restri
t to the se
tor of
onstant and
ommuting s
alar VEV's. Whereas this enables us
to study phase transitions in the eigenvalue distributions, revealing interesting dynami
s, it
does not ne
essarily re(cid:29)e
t the full phase stru
ture. However, we (cid:28)nd it enlightening to see
how the geometry of the dual AdS spa
etime is mirrored in the stru
ture of the quantum
e(cid:27)e
tive a
tion
omputed in this se
tor.
The outline and summary of the results in this paper are as follows. In Se
tion 2 we
= 2 S1 S3
give an introdu
tion to N quiver gauge theory on × with
hemi
al potentials
R = 2

onjugate to the -
harges. In Se
tion 3 we evaluate the quantum e(cid:27)e
tive a
tion of N
R
quiver gauge theory with non-zero -symmetry
hemi
al potentials and zero s
alar VEV's
g 0
YM
in the → limit and express it in terms of single-parti
le partition fun
tions. We use
= 2 S1 S3
the e(cid:27)e
tive a
tion to
onstru
t a matrix model for N quiver gauge theory on × .
M
The model turns out to be an -matrix model with adjoint and bifundamental potentials.
W
4 α
However,throughout theanalysis of thedynami
sin [17℄, thefermioni
matri
es are disregarded.
5
Seealso Refs. [19, 20, 21, 22, 23℄ for related work onother supersymmetri
gauge theories.
(cid:21) 3 (cid:21)
In Se
tion 4 we study the saddle points of the matrix model as fun
tions of tempera-
ture and
hemi
al potential and thereby examine the phase stru
ture of the model. In the
low-temperature phase we (cid:28)nd a saddle point
orresponding to a uniform distribution of
(1)
6
the eigenvalues of the Polyakov loop . In this phase the free energy is O with respe
t
N
to . This behavior of the free energy suggests that the model in this phase des
ribes a
non-intera
ting gas of
olor singlet states, and the phase is therefore labelled (cid:16)
on(cid:28)ning(cid:17).
This saddle point is observed to be
ome unstable when the temperature is raised above a

ertain threshold temperature (whi
h depends on the
hemi
al potential). The model then
N2M N
enters a new phase in whi
h the free energy s
ales as as → ∞. This phase is
thus interpreted as des
ribing a non-intera
ting plasma of
olor non-singlet states and is
labelled (cid:16)de
on(cid:28)ned(cid:17). The(cid:16)de
on(cid:28)nement(cid:17) transitionisof(cid:28)rstorder andisidenti(cid:28)ed witha
Hagedorn phasetransition.The
onditionofstability ofthelow-temperature saddlepointis
translated into a phase diagram of the gauge theory as a fun
tion of both temperature and

hemi
al potentials. We subsequently study the phase diagram in regions of small temper-
ature and
riti
al
hemi
al potential. We observe that the Hilbert spa
e of gauge invariant
SU(2)
operators trun
ates to the subse
tor when the
hemi
al potential
orresponding to
SU(2) R SU(2) U(1)
R R R
the fa
tor of the -symmetry group × is turned on, whereas when
both
hemi
al potentials are turned on and set equal, it trun
ates to a larger subse
tor that
SU(23) = 4

orresponds to an orbifolded version of the | se
tor found in N SYM theory.
= 2 S1 S3
In Se
tion 5 we develop a matrix model for N quiver gauge theory on × with
R
non-zero VEV's for the s
alar (cid:28)elds and zero -symmetry
hemi
al potentials. We
arry
out this
omputation in the spe
ial
ase where the ba
kground (cid:28)elds are assumed to be
(cid:16)
ommuting(cid:17) in a sense that
onforms to the quiver stru
ture. Furthermore the ba
kground
SO(4)
(cid:28)elds will be taken to be stati
and spatially homogeneous in order to preserve the
S3
isometry of the spatial manifold. The method employed for
omputing the e(cid:27)e
tive
potential will be the standard ba
kground (cid:28)eld formalism. That is,we expand the quantum
(cid:28)elds about
lassi
al ba
kground (cid:28)elds and path integrate over the (cid:29)u
tuations, dis
arding
terms of
ubi
or higher order in the (cid:29)u
tuations. The resulting (cid:29)u
tuation operators turn
out to have a parti
ular tridiagonal stru
ture in their quiver indi
es. By exploiting the
va
uum stru
ture of the theory we (cid:28)nd that the determinants fa
torize, leading to an
= 2 U(N)M
expression for the quantum e(cid:27)e
tive a
tion of N quiver gauge theory that
Z
M
expli
itly displays the stru
ture of the theory. Finally we generalize our results to a
Z = 4
M
spe
i(cid:28)

lassof(cid:28)eld theoriesthat
anbeobtained as proje
tions ofN SYMtheory,
= 2
of whi
h N quiver gauge theory is a spe
ial
ase.
N
In Se
tion 6we(cid:28)nd theminima of thematrix model of Se
tion 5inthe large limit in
a
oarse grained approximation. We
onsider the joint eigenvalue distribution of the s
alar
VEV's and the Polyakov loop and (cid:28)nd that the topology of the eigenvalue distribution is
tied to the Hagedorn phase transition. Below the Hagedorn temperature the eigenvalues
S5/Z
M
of the s
alar VEV's are distributed uniformly over an and the eigenvalues of the
S1
Polyakov loop are distributed uniformly over an . Thus, the joint eigenvalue distribution
6
We are using a somewhat sloppy terminology here: by `Polyakov loop' we really mean the holonomy
matrixofa
losed
urvewindingaboutthethermal
ir
le andnotjustitstra
e.Throughoutthis paperwe
will usetheword to des
ribe both and leave thepre
ise meaningto be determinedfrom the
ontext.
(cid:21) 4 (cid:21)
S5/Z S1 S5/Z
M M
is an (cid:28)bered trivially over . We interpret this as the emergen
e of the
S5/Z AdS S5/Z
M 5 M
fa
tor of the holographi
ally dual × geometry. Above the Hagedorn
temperature the eigenvalue distribution of the Polyakov loop be
omes gapped and is thus
S5/Z
M
an interval. The s
alar VEV's are now distributed uniformly over an (cid:28)bered over
S5/Z
M
this interval, with the radius of the at any point in the interval proportional to
TR S5/Z
M
the density of Polyakov loop eigenvalues at that point (for (cid:28)xed ). The thus
shrinks to zero radius at the endpoints of the interval: the topology of the joint eigenvalue
S6/Z Z S5
M M
distribution is an where the is understood to a
t on the transverse to an
S1 Z
M
diameter. Finally we generalize our results to the orbifold (cid:28)eld theories dis
ussed at
the end of Se
tion 5. In parti
ular we (cid:28)nd that the geometry of the dual AdS spa
etime is
mirrored in the stru
ture of the quantum e(cid:27)e
tive a
tion in a pre
ise way within this
lass
of orbifold (cid:28)eld theories.
InSe
tion7wedis
usstheresultswehaveobtainedinthispaperandsuggestdire
tions
= 2 U(N)M
for future study. In Appendix A further details about N quiver gauge theory
aregiven,someofwhi
htheauthorsofthispaperhavenotfoundelsewhereintheliterature.
SU(2) U(1)
R R
In parti
ular, we write the full Lagrangian density in terms of × invariants.
InAppendix Bwegivefurther te
hni
al detailsofthe
omputation ofthequantume(cid:27)e
tive
a
tion obtained in Se
tion 5.
= 2 R
2. N quiver gauge theory with -symmetry
hemi
al potentials
= 2U(N)M S1 S3 R
In thisse
tion we review N quivergauge theories on × with -symmetry
= 2 S1 S3

hemi
al potentials. An introdu
tory review of N quiver gauge theories on ×
is given in Se
tion 2.1. Details, some of whi
h the authors have not found elsewhere in
the literature, are deferred to Appendix A. In Se
tion 2.2 we then write up the
omplete
R
Lagrangian density in
luding -symmetry
hemi
al potentials.
= 2
2.1 Review of N quiver gauge theory
= 2 U(N)M
N quiver gauge theory with gauge group arises as the world-volume theory
N C3/Z
M
of open strings ending on a sta
k of D3-branes pla
ed on the orbifold . The
Z
M
gauge theory is thus super
onformal [25℄ with 16 super
harges. It
an be obtained as a
= 4 U(NM)
proje
tion ofN SYMtheory asexplained indetail inAppendix A.The result-
U(N)M U(N)
ing gauge group is where all the fa
tors of the gauge group have the same
g i = 1,...,M
YM
gauge
oupling
onstant asso
iated with them. Letting and identifying
i i+M M
7
≃ , the (cid:28)eld
ontent
an be summarized as follows. There are ve
tor multiplets
(A ,Φ ,ψ ,ψ ) A ψ Φ
µi i Φ,i i µi i i
where isthegauge(cid:28)eld, isthegaugino, isa
omplex s
alar(cid:28)eld,
ψ Φ ψ ψ
Φ,i i i Φ,i
and is the superpartner of . We take and to be 2-
omponent Weyl spinors.
M (A ,B ,χ ,χ ) A
i,(i+1) (i+1),i A,i B,i i,(i+1)
Furthermore there are hypermultiplets where and
B χ χ
(i+1),i A,i B,i
are
omplex s
alar (cid:28)elds and and are their respe
tive superpartners whi
h
i
we will take as 2-
omponent Weyl spinors. The (cid:28)elds in the 'th ve
tor multiplet all trans-
i U(N)
form in the adjoint representation of the 'th fa
tor of the gauge group. The (cid:28)elds
N =1
7
We will use an notation throughoutsin
e this proves
onvenient.
(cid:21) 5 (cid:21)
i i
in the 'th hypermultiplet transform in a bifundamental representation of the 'th and
(i+1) N
i
'th fa
tors. More spe
i(cid:28)
ally, letting denote the fundamental representation of the
i U(N) N A
i i,(i+1)
'th fa
tor and the
orresponding antifundamental representation, and
χ N N B
A,i i i+1 (i+1),i
its superpartner transform in the ⊗ representation, whereas and its
χ N N
B,i i i+1
superpartner transform in the ⊗ representation.
The (cid:28)eld
ontent is
onveniently summarized in the quiver diagram in Figure 1. The
M i = 1,...,M i i+M
diagram
onsists of nodes, labelled by with the identi(cid:28)
ation ≃ .
i i U(N) i
The 'th node represents the 'th gauge group fa
tor. Fields belonging to the 'th
i i
ve
tor multiplet are drawn as arrows that start and end on the 'th node. For the 'th
N N
i i+1
hypermultiplet, the(cid:28)eldstransforming inthe ⊗ representation aredrawn asarrows
i (i+1)
that start at the 'th node and end at the 'th node; the (cid:28)elds transforming in the
N N (i+1) i
i i+1
⊗ are depi
ted as arrows going from the 'th to the 'th node.
F
1
FM A A F2
M,1 1,2
B B
1,M 2,1
A A
(M-1),M B B 2,3
M,(M-1) 3,2
FM-1 FM-1
B
A (M-1),(M-2)
(M-2),(M-1)
FM-2
= 2U(N)M
Figure 1: Quiver diagram summarizing the (cid:28)eld
ontent of N quiver gauge theory.
U(N)
Ea
h of the bla
k dots (
alled nodes) represents a gauge group fa
tor. The nodes are labelled
i=1,...,M i i+M
by with the identi(cid:28)
ation ≃ . Arrows go from fundamental to antifundamental
A ,B Φ
i,(i+1) (i+1),i i
representations of the
orresponding gauge group fa
tors. The s
alar (cid:28)elds and
are shown in the (cid:28)gure, whereas the gauge (cid:28)elds and all the superpartners have been left impli
it.
= 2
The holographi
dual of N quiver gauge theory was found in [25℄ to be Type
AdS S5/Z S5/Z S5
5 M M
IIB string theory on × . The quotient is obtained by embedding
C3 Z AdS
M 5
in where the a
tion of is as de(cid:28)ned in (A.1). The spa
e has a radius given
R2 = 4πg (α)2NM g NM
by AdS s ′ where s is the Type IIB string
oupling. There are also
AdS
units of 5-foprm RR-(cid:29)ux through the 5. Due to the orbifold a
tion the volume of the
S5/Z S5 M
M
quotient equals the volume of the
overing spa
e divided by a fa
tor where
S5 AdS N
5
the has the same radius as . Similarly, there are units of 5-form RR-(cid:29)uxthrough
S5/Z NM
M
the fa
tor whi
h originate from units of (cid:29)ux in the
overing spa
e. Finally,
(cid:21) 6 (cid:21)
U(N) g
YM
we note that the Yang-Mills
oupling for ea
h gauge group fa
tor is related to
g2 = 4πg M
the Type IIB
oupling by YM s . This means that the 't Hooft
oupling relevant
λ = g2 N = 4πg NM
for ea
h fa
tor is YM s . This is the same as the 't Hooft
oupling on the
NM
original D3-branes before orbifolding, for whi
h the Yang-Mills
oupling was equal to
4πg g
s
. In the following we will often denote the Yang-Mills
oupling simply by .
= 2 U(N)M S1 S3
Thea
tion ofN quivergauge theory de(cid:28)ned on × isgiven asfollows.
F = ∂ A ∂ A +ig[A ,A ] D = ∂ +ig[A , ].
µν µ ν ν µ µ ν µ µ µ
To (cid:28)x our
onventions, we set − and · We
S1 β
will denote the
ir
umferen
e of the thermal
ir
le with and the radius of the spatial
S3 R = 2 S1 S3
with . The Eu
lidean a
tion of N quiver gauge theory on × is then
S = d4x g + +
gauge scalar ferm
| | L L L (2.1)
ZS1×S3 p (cid:0) (cid:1)
where the gauge boson, s
alar (cid:28)eld and spinor (cid:28)eld Lagrangian densities are given by,
8
respe
tively
1
= TrF F
gauge µν µν
L 4 (2.2)
= Tr D AD A+D BD B+D ΦD Φ
scalar µ µ µ µ µ µ
L
h(cid:16) 1 (cid:17) 2
+R 2 AA+BB+ΦΦ + g2 [A,A]+[B,B]+[Φ,Φ]
−
2
2g2 (cid:16)[A,B] 2+ [A,Φ(cid:17)] 2+ [B(cid:16),Φ] 2 (cid:17)
− (2.3)
ferm = iTr χAτµD↔(cid:16)(cid:12)(cid:12)µχA+(cid:12)(cid:12)χBτ(cid:12)(cid:12)µD↔µχB(cid:12)(cid:12) +ψ(cid:12)(cid:12) τµD↔µ(cid:12)(cid:12)ψ(cid:17)+iψΦτµD↔µψΦ
L
g(cid:16) (cid:17)
+ Tr χ [A,ψ ] [B,ψ] + χ [A,ψ]+[B,ψ ]
A Φ B Φ
√2 −
(cid:16)
(cid:0) (cid:1) (cid:0) (cid:1)
ψ [A,χ ] [B,χ ] ψ [A,χ ]+[B,χ ]
B A Φ A B
− − −
+χA(cid:0) [A,ψΦ] [B,ψ](cid:1)+ χB(cid:0)[A,ψ]+[B,ψΦ] (cid:1)
−
ψ (cid:0)[A,χ ] [B,χ ](cid:1) ψ (cid:0)[A,χ ]+[B,χ(cid:1)]
B A Φ A B
− − −
+χ(cid:0)[Φ,χ ] χ [Φ,(cid:1)χ ] + (cid:0)ψ[Φ,ψ ] ψ [Φ,(cid:1)ψ]
A B B A Φ Φ
− −
+χ [Φ,χ ] χ [Φ,χ ] + ψ[Φ,ψ ] ψ [Φ,ψ] .
A B B A Φ Φ
− − (2.4)
(cid:17)
NM NM χ ,χ ,ψ ,ψ
A B Φ
The tra
es are taken over the × matri
es. The spinor (cid:28)elds are
τµ = (1,iσ) D↔µ
undotted 2-
omponent Weyl spinors. We de(cid:28)ne . The operator is de(cid:28)ned
by ψ1D↔µψ2 ≡ 21 ψ1Dµψ2 −(Dµψ1)ψ2 . It is implied that the (cid:28)elds A,B,Φ,Aµ et
. take
the orbifold proje(cid:0)
tion invariant forms g(cid:1)iven in Eqs. (A.15)-(A.16) and (A.31)-(A.32). Note
S3
that the s
alar (cid:28)elds are
onformally
oupled to the
urvature of the spatial manifold
R 2Tr AA+BB+ΦΦ
−
through the term in (2.3). This e(cid:27)e
tively indu
es a mass for the
s
alar (cid:28)elds. (cid:0) (cid:1)
χ ,χ ,ψ ,ψ
8 A B Φ
Note that for all (cid:28)elds, in
luding the Weyl spinors , the bars denote the Hermitian
on-
jugate, notthe
omplexor Weyl
onjugate. E.g., (χA)αβ =(χA)∗βα whereα,β are gaugegroupindi
esand
∗
the denotes
omplex
onjugation. Furthermore, in the third line of Eq. (2.3), the notation means, e.g.,
|[A,B]|2 ≡[A,B][A,B]
.
(cid:21) 7 (cid:21)
R SU(4) = 4
The orbifolding breaks the -symmetry group of N SYM theory into
SU(2) U(1) Φ z C3
R R 1
× . As des
ribed in Appendix A, is asso
iated with the dire
tion of
Z A B
M
whi
h isinert under thea
tion ofthe orbifold group ,while and areasso
iatedwith
z z U(1) z eiζz
2 3 R 1 1
and respe
tively. The fa
tor
orresponds to the transformation → and
Φ A B
therefore a
ts on the (cid:28)elds by multiplying phase rotations. The and (cid:28)elds have zero
U(1) SU(2) A B
R R

harge under .The symmetrya
tsonthe and (cid:28)eldsandtheir Hermitian
(A,B) ( B,A) SU(2) (ψ,ψ )
R Φ

onjugates. In fa
t, and − form doublets. Furthermore and
( ψ ,ψ) SU(2) χ χ SU(2)
Φ R A B R
− are doublets whereas and have zero
harge under . The
SU(2) U(1) R
R R
gauge (cid:28)eld is not
harged under × . We summarize the -
harges in Table
A.
R
2.2 Lagrangian density with -symmetry
hemi
al potentials
G
Givenanynon-Abeliansymmetrygroup ,one
anintrodu
e
hemi
alpotentials
onjugate
G
to the generators of a maximal torus of . In this se
tion we will
onsider the
ase where
G R SU(2) U(1) = 2
R R
is the -symmetry group × of N quiver gauge theory. The maximal
U(1) U(1) U(1) SU(2) Q
R R 1
torus is × . We will denote the Cartan generators of and by
Q µ µ
2 1 2
and , respe
tively, and the
orresponding
hemi
al potentials by and . For the
U(1) U(1)
R
fa
tor of the maximal torus that
orresponds to the eigenvalues of the Cartan
U(1) SU(2)
R
generators
an dire
tly be read o(cid:27) from Table A. For the ⊂ we
hoose as
σ SU(2)
z R
a basis for the Cartan subalgebra the diagonal generator so that the doublets
U(1) σ 1σ
will have well-de(cid:28)ned
harges under . (We
hoose z rather than 2 z as the generator
Q eiQ2θ θ θ +2π Q σ
2 2 z
be
ause we require to be invariant under → . Setting ≡ we have
eiQ2θ = diag(eiθ,e iθ)
−
whi
h is
learly invariant.) Therefore the
harges under the maximal
U(1) SU(2) SU(2)
R R
torus of will be 2 times the
harges.
Thus for the bosoni
(cid:28)elds,
(µ Q )A = µ A
a a i,(i+1) 2 i,(i+1)
(2.5)
(µ Q )B = µ B
a a (i+1),i 2 (i+1),i
(2.6)
(µ Q )Φ = µ Φ
a a i 1 i
(2.7)
(µ Q )A = 0,
a a µi
(2.8)
and for the fermioni
(cid:28)elds,
(µ Q )χ = 1µ χ
a a A,i −2 1 A,i (2.9)
(µ Q )χ = 1µ χ
a a B,i −2 1 B,i (2.10)
(µ Q )ψ = 1µ µ ψ
a a i 2 1− 2 i (2.11)
(µaQa)ψΦ,i = (cid:0)−12µ1−µ(cid:1)2 ψΦ,i. (2.12)
(cid:0) (cid:1)
The
orresponding expressions for the Hermitian
onjugate (cid:28)elds are obtained by simply

hanging the signs of the
hemi
al potentials.
= 2
ToobtaintheLagrangiandensityofN quivergaugetheorywith
hemi
alpotentials
µ SU(2) U(1)
a R R
for the × Cartan generators, one makes the following substitution in the
Lagrangian density
D D µ Q δ .
µ µ a a µ0
−→ − (2.13)
(cid:21) 8 (cid:21)
Below we have written the Lagrangian densities for the fundamental s
alar and spinor
= 2
(cid:28)elds of N quiver gauge theory. This will be important for the analysis in the following
se
tions in order to distinguish the adjoint from the bifundamental stru
tures.
R
The Lagrangian density for the s
alar (cid:28)elds with -symmetry
hemi
al potentials is
M
= Tr ∂ A +igA A igA A µ δ A
scalar µ i,(i+1) µi i,(i+1) i,(i+1) µ(i+1) 2 µ0 i,(i+1)
L − −
(
Xi=1 h(cid:16) (cid:17)
∂ A +igA A igA A +µ δ A
µ i,(i+1) µ(i+1) i,(i+1) i,(i+1) µi 2 µ0 i,(i+1)
× −
(cid:16) (cid:17)i
+ Tr ∂ B +igA B igB A µ δ B
µ (i+1),i µ(i+1) (i+1),i (i+1),i µi 2 µ0 (i+1),i
− −
h(cid:16) (cid:17)
∂ B +igA B igB A +µ δ B
µ (i+1),i µi (i+1),i (i+1),i µ(i+1) 2 µ0 (i+1),i
× −
(cid:16) (cid:17)i
+ Tr ∂ Φ +ig[A ,Φ ] µ δ Φ ∂ Φ +ig[A ,Φ ]+µ δ Φ
µ i µi i 1 µ0 i µ i µi i 1 µ0 i
−
h(cid:16) (cid:17)(cid:16) (cid:17)i
+R 2Tr A A +B B +Φ Φ
− i,(i+1) i,(i+1) (i+1),i (i+1),i i i
1 (cid:16) (cid:17)
+ g2Tr A A A A
i,(i+1) i,(i+1) (i 1),i (i 1),i
2 − − −
h(cid:16) 2
+B B B B +[Φ ,Φ ]
i,(i 1) i,(i 1) (i+1),i (i+1),i i i
− − −
(cid:17) i
2g2Tr A B B A
i,(i+1) (i+1),i i,(i 1) (i 1),i
− − − −
h(cid:16) (cid:17)
A B B A
(i 1),i i,(i 1) (i+1),i i,(i+1)
× − − −
(cid:16) (cid:17)i
2g2Tr A Φ Φ A A Φ Φ A
i,(i+1) i+1 i i,(i+1) i,(i+1) i i+1 i,(i+1)
− − −
h(cid:16) (cid:17)(cid:16) (cid:17)i
2g2Tr B Φ Φ B B Φ Φ B .
(i+1),i i i+1 (i+1),i (i+1),i i+1 i (i+1),i
− − − ) (2.14)
h(cid:16) (cid:17)(cid:16) (cid:17)i
N N
Here thetra
esarealways takenoverthegaugeindi
esofthe × matri
es.Observethat
µ µ Φ A ,B
1 2 i i,(i+1) (i+1),i
the
hemi
al potentials and a
tlikenegative masssquaresfor and .
S3
On a
ompa
t spatial manifold su
h as , these terms are balan
ed by the positive mass
squaretermsindu
edbythe
onformal
oupling to
urvature.Weimmediately observefrom
= 2 S1 S3 µ ,µ R 1
1 2 −
(2.14) that N quiver gauge theory on × iswell-de(cid:28)ned as long as ≤ . If
the
hemi
al potentials ex
eed this bound, the theory develops ta
hyoni
modes and there
exists no stable ground state.
R
The Lagrangian density for the spinor (cid:28)elds with -symmetry
hemi
al potentials is
M
i
= Tr χ τ ∂ χ +igA χ igχ A + 1µ δ χ
Lferm 2 A,i µ µ A,i µi A,i− A,i µ(i+1) 2 1 µ0 A,i
(
Xi=1 (cid:16) (cid:0) (cid:1)(cid:17)
i
Tr ∂ χ +igA χ igχ A 1µ δ χ τ χ
−2 µ A,i µ(i+1) A,i− A,i µi− 2 1 µ0 A,i µ A,i
i (cid:16)(cid:0) (cid:1) (cid:17)
+ Tr χ τ ∂ χ +igA χ igχ A + 1µ δ χ
2 B,i µ µ B,i µ(i+1) B,i− B,i µi 2 1 µ0 B,i
i(cid:16) (cid:0) (cid:1)(cid:17)
Tr ∂ χ +igA χ igχ A 1µ δ χ τ χ
−2 µ B,i µi B,i− B,i µ(i+1) − 2 1 µ0 B,i µ B,i
(cid:16) (cid:17)
(cid:0) (cid:1)
(cid:21) 9 (cid:21)