Holographic realization of large-Nc orbifold equivalence with non-zero chemical potential

0.45 MB·English

by  Masanori Hanada

January 19, 2012
TAUP-2941/11
Holographic realization of large-N orbifold equivalence
c
with non-zero chemical potential
2
1 Masanori Hanadaa,b,1 Carlos Hoyosa,c,2 Andreas Karcha3 and Laurence G. Yaffea4
0
2
aDepartment of Physics, University of Washington, Seattle, WA 98915-1560, USA
n
a bKEK Theory Center, High Energy Accelerator Research Organization (KEK),
J
Tsukuba 305-0801, Japan

8
cRaymond and Beverly Sackler School of Physics and Astronomy, Tel-Aviv University,
1
Ramat-Aviv 69978, Israel

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h
t
-
p
e
h
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Abstract
1
v
8 Recently, it has been suggested that large-N orbifold equivalences may be applicable to cer-
c
1
tain theories with chemical potentials, including QCD, in certain portions of their phase diagram.
7
3 Whenvalid, suchanequivalenceoffersthepossibilityofrelatinglarge-N QCDatnon-zerobaryon
c
1. chemical potential, a theory with a complex fermion determinant, to a related theory whose
0 fermion determinant is real and positive. In this paper, we provide a test of this large N equiva-
c
2
lence using a holographic realization of a supersymmetric theory with baryon chemical potential
1
: and a related theory with isospin chemical potential. We show that the two strongly-coupled,
v
i large-Nc theories are equivalent in a large region of the phase diagram.
X
r
a
1E-mail address: [email protected]
2E-mail address: [email protected]
3E-mail address: [email protected]
4E-mail address: [email protected]
1
Contents
1 Introduction 2
2 Orbifold projections for large-N QCD 5
c
2.1 From SO(2N ) to QCD with a baryon chemical potential . . . . . . . . . . . . . 6
c F
2.2 From SO(2N ) to QCD with an isospin chemical potential . . . . . . . . . . . . . 7
c F
2.3 Validity of large-N equivalences and their application to the sign problem . . . . . 8
c
3 A holographic realization 9
3.1 Orientifold and orbifold projections . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 From isospin to baryon chemical potential . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 Validity of the equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.4 N /N corrections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
f c
4 Conclusion 15
A A quick introduction to the sign problem 16
B Projection of the DBI action 17
1 Introduction
Understanding QCD at non-zero baryon density is an important goal, both for its intrinsic
interest and for applications such as the structure of neutron stars and the mechanism of core-
collapse supernova. Due to the notorious sign problem,5 we lack generally effective methods for
performing numerical simulations of gauge theories with a baryon chemical potential. When a
non-zero baryon number chemical potential is present, the determinant of the Euclidean Dirac
operatorisnolongerpositiveandstandardMarkov-chainMonte-Carlomethodsarenotapplicable.
Although many schemes have been proposed to address the sign problem [1, 2, 3, 4, 5, 6, 7], it is
fair to say that no fully satisfactory solution has been found. At the same time, condensed matter
phases of several QCD-like theories which do not suffer from the sign problem have been studied
numerically, in the hope that one may extract lessons about strongly interacting finite density
systems which will also apply to QCD at finite baryon density. Examples include SU(2) Yang-
Mills (YM) with even numbers of fundamental flavors [8, 9], SU(N ) YM with adjoint fermions
c
[9], and QCD with an isospin chemical potential [10, 11]. However, there is no solid argument
delineating the extent to which these theories can reproduce properties of QCD with a baryon
chemical potential.
In recent years, it has been understood that a network of large-N equivalences relate various
c
non-Abelian gauge theories with differing gauge groups and matter content [12, 13, 14, 15, 16].
These equivalences, which are generated by appropriate orbifold projections, relate the leading
large N behavior of connected correlators of specific classes of observables. The large-N equiv-
c c
alences are valid provided certain symmetry realizations are satisfied [17]. For example, SU(N )
c
5More properly, this should be called a phase problem. See Appendix A for a brief summary.
2
O(2N )
c
N fundamentals
f
chemical potential µ
F
(cid:0) (cid:64)
(cid:0) (cid:64)
(cid:0) (cid:64)
(cid:0)(cid:9) (cid:64)(cid:82)
U(N ) U(N )
c c
N fundamentals N fundamentals
f f
baryon chemical potential µ isospin chemical potential µ
B I
Figure1: AnorbifoldprojectionactingonaparentO(2N )Yang-MillstheorywithN fundamental
c f
representation Dirac fermions and a flavor-singlet chemical potential µ may generate a U(N )
F c
daughter theory with N fermions and a baryon chemical potential µ (left) or, provided N is
f B f
even, the same U(N ) theory with an isospin chemical potential µ (right) [21]. In the parent
c I
O(2N )theorywithevenN ,thereisnodistinctionbetweenabaryonorisospinchemicalpotential.
c f
and SO(2N ) Yang-Mills theories have coinciding large N limits of all Wilson loop expectation
c c
values(aswellasconnectedcorrelatorsofrealpartsofWilsonloops), providedchargeconjugation
symmetry is not spontaneously broken in the SU(N ) theory [18].
c
We will be concerned with QCD-like theories containing fundamental representation fermions
and non-zero chemical potentials. Specifically, we will discuss:
1. SO(2N ) Yang-Mills with N Dirac fundamental representation fermions and a non-zero
c f
fermion chemical potential µ , under which all N flavors have charge +1. For brevity, we
F f
will denote this theory as SO(2N ) .
c F
2. SU(N ) Yang-Mills with N fundamental representation fermions and a non-zero baryon
c f
chemical potential µ , under which all N fermion flavors have charge +1. For brevity, we
B f
will denote this theory as QCD .
B
3. SU(N ) Yang-Mills with N fundamental representation fermions, with N even, and a non-
c f f
zero isospin chemical potential µ , under which half the fermion flavors have charge +1 and
I
half have charge −1. For brevity, we will denote this theory as QCD .
I
Although QCD suffers from a sign problem, this is not the case for either QCD or SO(2N ) ,
B I c F
as both of these theories have a real and positive fermion determinant [10, 21].
As figure 1 schematically depicts, starting from the SO(2N ) theory one choice of orbifold
c F
projection yields QCD , while a different choice yields QCD .6 Based on this observation, it has
B I
recently been suggested that large N equivalences may relate suitable observables in the parent
c
6Strictlyspeaking,theorbifoldprojectionmapsaparenttheorywithSO(2N )gaugegrouptoadaughterU(N )
c c
gauge theory. But the difference between U(N ) and SU(N ) theories is sub-dominant in the large N limit. Note
c c c
that in a U(N ) theory with chemical potential, the U(1) part of the gauge field is to be fixed at infinity. In finite
c
volume,thetheoryshouldbedefinedwithDirichletboundaryconditionsontheU(1)gaugefield,notperiodic. (In
practice, for lattice simulations, it is more convenient to simply use the SU(N ) theory.)
c
3
SO(2N ) theory to corresponding observables in either QCD or QCD [21, 22, 23]. In portions
c F B I
of the phase diagram where both equivalences are valid (if such regions exist), this implies that
one may obtain quantitative information about large-N QCD with a baryon chemical potential
c
from studies of the same theory with an isospin chemical potential, thereby circumventing the
sign problem.7
When N → ∞ with N fixed, a comparison of planar Feynman diagrams in the parent
c f
SO(2N ) and daughter SU(N ) theories shows that they coincide [12, 13, 21, 22]. (For other ap-
c c
proachessee[15].) Thisanalysisisunaffectedbythepresenceofanon-zerochemicalpotential. Co-
inciding perturbative expansions does not, however, necessarily imply a genuine non-perturbative
equivalence. Necessary conditions for a valid equivalence include a requirement that the symme-
tries used to define the orbifold projection not be spontaneously broken in the parent theory [15].
SincetheprojectionleadingtoQCD isgeneratedbyacombinationofagaugetransformationand
B
a U(1) phase rotation, this projection can only lead to a valid large-N equivalence in portions
F c
of the phase diagram where the U(1) global symmetry associated with net fermion number is
F
unbroken. In other words, a large N equivalence relating QCD with baryon and isospin chemical
c
potentials can only apply to portions of the phase diagram in which fermions do not condense to
form a superfluid. In simpler examples, analogous conditions on symmetry realizations are both
necessary and sufficient conditions for the validity of large N equivalences [15]; whether this is
c
the case in the present example is not yet clear.
In this paper, we use gauge/gravity duality to test the validity of analogous possible large N
c
equivalences relating supersymmetric generalizations of the above theories. Although the trio of
theories we consider will not include QCD itself, the arguments of refs. [21, 22, 23] are equally
applicable to the supersymmetric theories we consider. By considering supersymmetric theories,
and using holographic methods, it will be possible to examine relations between theories with
different types of chemical potential directly in the limit of strong coupling (and large N ) using
c
simple analytic methods. We will find that, in a large region of the phase diagram with no
spontaneous breaking of flavor symmetries, large N equivalences between our theories are valid.
c
Our holographic construction involves an orbifold and an orientifold projection of the D3/D7-
system, with N D3 branes, N D7 branes, and always N (cid:28) N . At large-N and strong ’t Hooft
c f f c c
coupling, the low energy theory on the D3 branes is described by classical type IIB supergravity
in AdS ×S5, with probe D7 branes wrapping an S3 in the S5 [24]. The projections act on the
5
geometry, changing it to AdS × RP5. An isospin chemical potential in the original theory is
5
described by a particular configuration of the gauge field on the D7 brane. We show that after the
projectiontheisospinchemicalpotentialbecomesabaryonchemicalpotential. Figure2illustrates
the connections between the corresponding field theories. We prove that, provided the projection
symmetries are not broken, the equations of motion of the D7 branes coincide in both theories.
As we discuss, this implies that the conjectured large N equivalences are valid in these theories
c
in those regions of the phase diagram where flavor symmetries are unbroken and no additional
fields become active.
The paper is organized as follows: in §2 we review in more detail the proposed equivalences
involvinglarge-N QCD.In§3weprovideaholographicrealizationforoursupersymmetricgener-
c
alization. Finally, in §4 we discuss the regime of validity of the equivalence, and the consequences
for the phase diagram.
7Readers should refer to section 2, and refs. [21, 22, 23], for more details and more nuanced discussion.
4
SU(2N ), 2N fundamentals, 3 adjoints
c f
isospin chemical potential µ
I
(cid:63)
SO(2N ), N fundamentals, 1 adjoint, 2 symmetrics
c f
chemical potential µ
F
(cid:63)
SU(N ), N fundamentals, 1 adjoint, 2 symmetrics
c f
baryon chemical potential µ
B
Figure2: N = 1supersymmetrictheoriesrelatedbyorbifoldprojections. IntheSO(2N )theory,
c
there is no distinction between a baryon or isospin chemical potential.
2 Orbifold projections for large-N QCD
c
Consider an SO(2N ) Yang-Mills theory coupled to N fundamental representation Dirac
c f
fermions. The Lagrange density is
N
L = 1 trF2 +(cid:88)f ψ¯ (cid:0)γµD +m +µ γ0(cid:1)ψ , (1)
SO 4g2 µν a µ q F a
SO a=1
whereF isthefieldstrengthoftheSO(2N )gaugefieldA , D ≡ ∂ +A , ψ isaDiracfermion
µν c µ µ µ µ a
in the vector representation of SO(2N ), and m and µ are the quark mass and fermion chemical
c q F
potential,respectively. Becausethegaugefieldisreal,theDiracoperatorD ≡ (γµD +m +µ γ0)
µ q F
satisfies (Cγ )D(Cγ )−1 = D∗, where C is the charge conjugation matrix defined by Cγ C−1 =
5 5 µ
−γT = −γ∗.8 If v is an eigenvector of the Dirac operator D with an eigenvalue λ, Dv = λv, then
µ µ
(Cγ )−1v∗ is another eigenvector of D with eigenvalue λ∗, and is linearly independent of v even
5
when λ is real. (See §2.3 of ref. [20].) Therefore the determinant of D is always real and positive,
implying that standard Markov chain Monte-Carlo simulation techniques may be used [21].
When m = µ = 0, the Lagrangian (1) has a manifest SU(N ) × SU(N ) × U(1) ×
q F f L f R F
U(1) flavor symmetry, just like SU(N ) QCD. However, the flavor symmetry of the theory
A c
is actually larger than this due to the fact that SO(2N ) is a real gauge group; classically it
c
extends to U(2N ) [25, 26]. The axial U(1) ⊂ U(2N ) is anomalous, and at the quantum
f A f
level the (continuous part of the) flavor symmetry is SU(2N ), which spontaneously breaks to
f
SO(2N ) ⊇ SU(N ) due to the formation of a chiral condensate (cid:104)ψ¯ψ(cid:105). The resulting massless
f f V
Nambu-Goldstone bosons span the SU(2N )/SO(2N ) coset space. In contrast to QCD, some
f f
of these Nambu-Goldstone bosons, which we will refer to as baryonic pions, are charged under
U(1) . Ordinary pions are created by operators that look like ψ¯ γ ψ , while baryonic pions are
F a 5 b
created by color-singlet operators of the form ψTCγ ψ and ψ¯ Cγ ψ¯T.
a 5 b a 5 b
8We use (++++) metric signature and Hermitian gamma matrices.
5
2.1 From SO(2N ) to QCD with a baryon chemical potential
c F
To perform an orbifold projection, one identifies a discrete subgroup of the symmetry group
of the parent theory, which for us is the SO(2N ) theory, and then removes all of the degrees of
c F
freedom in the parent theory which are not invariant under the chosen discrete symmetry. This
yields a daughter theory, which will turn out to be large-N QCD.
c
The required orbifold projection is a Z subgroup of the SO(2N ) gauge×U(1) flavor sym-
2 c F
metryoftheSO(2N ) theory. Todefinetheorbifoldprojection, whichwewilldenoteasP , take
c F B
J ∈ SO(2N ) to be given by J = iσ ⊗1 , where 1 denotes an N ×N identity matrix.
2Nc c 2Nc 2 Nc N
The group element J generates a Z subgroup of SO(2N ). Next, let ω = eiπ/2 ∈ U(1) denote
2Nc 4 c F
the phase which generates a Z subgroup of U(1) . The discrete symmetry which will define the
4 F
orbifold projection acts on on the fields A , ψ as
µ a
A → J A J−1 , ψ → ωJ ψ . (2)
µ 2Nc µ 2Nc a 2Nc a
Since J2 = −1 , and (ωJ )2 = +1 , this symmetry transformation generates a Z sub-
2Nc 2Nc 2Nc 2Nc 2
group of SO(2N )×U(1) .
c F
The action of the orbifold projection on the basic fields is
(cid:16) (cid:17)
P A = 1 (cid:0)A +J A J−1 (cid:1), P ψ = 1 ψ+J ψK−1 (3)
B µ 2 µ 2Nc µ 2Nc B 2 2Nc Nf
where, for later convenience, we have defined a matrix K−1 ≡ i1 acting on flavor indices. To
Nf Nf
display the action of the projection more explicitly, it is convenient to block-decompose the gauge
and fermion fields. The gauge field A may be written in terms of four N ×N blocks as
µ c c
(cid:18) AA+BA CA−DS (cid:19)
A ≡ µ µ µ µ , (4)
µ CA+DS AA−BA
µ µ µ µ
where fields marked with an ‘A’ or ‘S’ superscript are anti-symmetric or symmetric matrices,
respectively. Under the Z symmetry transformation (3), AA, and DS are even while BA, and CA
2 µ µ µ µ
are odd, so the orbifold projection sets BA = CA = 0. Hence
µ µ
(cid:18) AA −DS (cid:19)
P A = µ µ . (5)
B µ DS AA
µ µ
If one defines the unitary matrix
(cid:18) (cid:19)
1 1 i1
P = √ Nc Nc , (6)
2 1Nc −i1Nc
then
(cid:18) (cid:19)
A 0
P P A P−1 = µ , (7)
B µ 0 −AT
µ
where A ≡ AA+iDS is a U(N ) gauge field. At large N , we can neglect the difference between
µ µ µ c c
U(N ) and SU(N ) up to 1/N2 corrections.
c c c
6
We can split the 2N -component fundamental fermions of the SO(2N ) theory into two N -
c c c
component fields,
(cid:18) (cid:19)
ψ
ψ = 1 , (8)
ψ
2
and then we use the matrix (6) to change basis. This yields
(cid:18) (cid:19)
ψ
Pψ = + , (9)
ψ
−
√
where ψ ≡ (ψ ±iψ )/ 2. From eq. (7), one sees that ψ and ψ transform as fundamental
± 1 2 + −
and antifundamental representations under SU(N ), respectively. After the projection, only ψ
c +
survives.
If we take the Lagrangian of the parent theory and apply the orbifold projection, it becomes
N
L = 1 TrF2 +(cid:88)f λ¯ (cid:0)γµD +m +µ γ4(cid:1)λa, (10)
4g2 µν a µ q B
SU a=1
where F is the field strength of the SU(N ) gauge field A = AA + iDS, D = ∂ + A ,
√ µν c µ µ µ µ µ µ
λa = 2ψa, and the gauge coupling is given by g2 = g2 .
+ SU SO
(p)
In the large-N limit for fixed N , connected correlation functions of operators O in the
c f i
(d)
parent SO theory which are invariant under the projection symmetry, and their counterparts O
i
in the daughter SU theory which are formed from the projected fields, coincide to all orders in
perturbation theory [12],
(p) (p) (d) (d)
(cid:104)O O ···(cid:105) = (cid:104)O O ···(cid:105) . (11)
1 2 p 1 2 d
The baryonic pion fields do not survive the projection, so there is no equivalent to them in the
daughter theory.
2.2 From SO(2N ) to QCD with an isospin chemical potential
c F
WhenthenumberofflavorsintheparentSO(2N )theoryiseven,N = 2k,itisalsopossibleto
c f
define a projection which yields large-N QCD with an isospin chemical potential. The projection
c
for the gauge field is the same as in eq. (3), but we now choose a different orbifold action on the
flavor indices of the fermions. Let us write the fermions using N ×N -component fields as
c f
(cid:32) (cid:33)
(1) (2)
ψ ψ
ψ = + + . (12)
(1) (2)
ψ ψ
− −
In this basis, the orbifold action is
ψ → J ψJ−1. (13)
2Nc 2k
This transformation also generates a Z group. The action of the orbifold projection P is
2 I
P A = 1 (cid:0)A +J A J−1 (cid:1), P ψ = 1 (cid:0)ψ+J ψJ−1(cid:1). (14)
I µ 2 µ 2Nc µ 2Nc I 2 2Nc 2k
7
√ √
(1) (2) (1) (2)
Defining ϕ = (ψ ∓iψ )/ 2 and ξ = (ψ ± iψ )/ 2, one sees that ϕ survive while ξ
± ± ± ± ± ± ± ±
is eliminated by the projection (14). Since ϕ and ϕ couple to A and AC, respectively, the
+ − µ µ
fermionic part of the action of the daughter theory can be written as
k
(cid:88)(cid:88) λ¯(f)(cid:0)γµD +m±µγ4(cid:1)λ(f), (15)
± µ ±
f=1 ±
√ √
where λ(f) = 2ϕ(f), λ(f) = 2(ϕ(f))C, and we have now written the flavor index (f) = 1,··· ,k
+ + − −
explicitly. This theory has an isospin chemical potential µ ≡ 2µ.
I
2.3 Validity of large-N equivalences and their application to the sign problem
c
The perturbative proof of the parent-daughter equivalence with isospin chemical potential
is valid also when quark loops are included in planar diagrams, so it is possible to extend the
analysis to include N /N corrections. However, this is not possible for the projection to a theory
f c
with baryon chemical potential. The difference stems from the properties of the projection in
the flavor sector, while the projection to isospin chemical potential is performed using a regular
representation [12, 13]
trJ = 0, J2 = ±1 , (16)
2k 2k 2k
theseconditionsarenotsatisfiedfortherepresentationusedtodotheprojectiontothetheorywith
a baryon chemical potential, where we have used K instead of J . Only diagrams containing a
2k 2k
single quark loop produce the same result in parent and daughter theories.
Togobeyondtheperturbativeproofoftheequivalenceoneneedstodoacarefulanalysisofthe
necessary and sufficient conditions that must be obeyed for the equivalence to hold. A necessary
condition is that the projection symmetry not be spontaneously broken in the parent [15]. The
U(1) symmetry,whichisusedfortheprojectionfromtheSO(2N )theorytoQCDwithabaryon
B c
chemical potential, breaks to Z when the baryonic pion condenses (e.g., when µ > m /2 at zero
2 π
temperature). Therefore, the parent-daughter equivalence can hold only at smaller values of the
chemical potential.9 On the other hand, the projection symmetry to obtain QCD with isospin
chemical potential should not be spontaneously broken for any µ; in this case condensation of
baryonic pions in the parent theory is mapped to pion condensation in the daughter theory.
Clearly, if it were possible to show that these equivalences hold nonperturbatively, they would
be very useful because one would be able to derive properties of a large-N QCD theory with
c
baryonic chemical potential from a SO(2N ) theory or from large-N QCD with isospin chemical
c c
potential, both of which are free of the sign problem. This could also explain why the phase
quenching approximation in QCD is quite good — for a certain class of operators (e.g., the chiral
condensate),thephasequenchingapproximationbecomesexactinthelarge-N limit.10 Thephase
c
quenching approximation for the chiral condensate is exact in the chiral random matrix model
[39, 22]. The orbifold equivalence, if true, would ensure that the phase quenching approximation
in QCD is exact for a large class of observables in the large-N limit, even beyond the parameter
c
region where the chiral random matrix model is valid (the “(cid:15)-regime”).
9Note that the chemical potential at which baryonic pions condense is temperature dependent, and should
increase with increasing temperature.
10Notethat(forN even)droppingthephaseofthefermiondeterminantturnsthefunctionalintegralforQCD
f B
into that for QCD .
I
8
ToprovideanonperturbativeproofoftheorbifoldequivalenceinQCDwithchemicalpotentials
is beyond the scope of this paper. However, in the following section we will show that analogous
equivalences hold in a class of supersymmetric cousins of QCD which have gravity duals.
3 A holographic realization
It is possible to build a simple supersymmetric model where an isospin chemical potential
is projected into a baryon chemical potential. The model is one of the examples mentioned in
ref. [27], based on the description of N = 2 theories from D4 branes suspended between NS5
branes [28]. Flavor can be introduced by adding D6 branes. We will start with a configuration
whose low energy limit on the T-dual D3 branes is N = 4 U(2N ) super Yang-Mills plus 2N
c f
hypermultiplets in the fundamental representation, so the flavor group is U(2N ). In the T-dual
f
configuration the flavor branes are D7’s, and we will work in the ’t Hooft limit of N /N (cid:28) 1 so
f c
we can neglect their backreaction just as in the D3/D7 system of ref. [24]. We then introduce an
orientifold plane to produce an SO(2N ) theory with USp(2N ) flavor group and then finally do
c f
a Z orbifold projection that reduces it to U(N ) with U(N ) flavor group. We will show that
2 c f
an isospin chemical potential in the original U(2N ) theory is projected to a baryon chemical
c
potential in the U(N ) theory and discuss when the two theories are equivalent.
c
3.1 Orientifold and orbifold projections
The construction in type IIA theory consists on a set of 2N D4 branes wrapping a circle in
c
the x6 direction and intersecting two O6+ planes at opposite sides of the circle. In addition, there
is a NS5 brane at each orientifold point and 2N D6 branes parallel to the O6 planes:
f
0 1 2 3 4 5 6 7 8 9
D4 × × × × · · × · · ·
O6/D6 × × × × · · · × × ×
NS5 × × × × × × · · · ·
Since the O6 planes are positively charged, Ramond-Ramond (RR) tadpoles do not cancel and
the β function for the ’t Hooft coupling is positive. However, in the ’t Hooft limit N (cid:28) N , the
f c
β function is suppressed by N /N at large N . So to leading order in N /N we can neglect the
f c c f c
tadpoles and consider the D6’s and O6’s as probes.
This brane setup has as a T-dual a configuration involving D3 and D7 branes. The two O6
planes map to a single O7 plane and the NS5 brane to a Z singularity localized at x6 = x7 =
2
x8 = x9 = 0:
0 1 2 3 4 5 6 7 8 9
D3 × × × × · · · · · ·
O7/D7 × × × × · · × × × ×
Z × × × × × × · · · ·
2
The geometric effect of the Z action is a reflection in the transverse directions. The orientifold
2
projection Ω(cid:48) = ΩR45(−1)FL involves worldsheet parity reversal Ω, a reflection R45 in the x4 and
x5 coordinates, and (−1)FL acts as −1 in the Ramond sector of left movers. The effect on Chan-
Paton factors of open strings on D3 branes is given by the matrices γ = iJ for the orbifold
3 2Nc
9
action and ω = 1 for the orientifold action. The corresponding matrices for the D7 branes are
3 2Nc
γ = iJ and ω = iJ .
7 2N 7 2N
f f
The massless spectrum of D3 branes involves a vector multiplet on the worldvolume A
0123
and three complex scalar multiplets describing the transverse motion X , X , X . Before the
45 67 89
projection those describe the field content of N = 4 U(2N ) super Yang-Mills, that in N = 2
c
language involves a vector multiplet and a hypermultiplet in the adjoint representation. The
orientifold action is
A → −ω AT ω−1,
0123 3 0123 3
X → −ω XT ω−1, (17)
45 3 45 3
X → ω XT ω−1.
67,89 3 67,89 3
Therefore, the orientifold projection for the gauge field is
P A = 1 (cid:0)A −AT(cid:1), (18)
ω µ 2 µ µ
sotheprojectedgaugefieldisantisymmetricandspansanSO(2N )algebra. ThefieldX isinan
c 45
antisymmetric (adjoint) representation, while for the fields X the orientifold action projects
67,89
them to a symmetric representation.
The Z action of the orbifold is
2
P A → γ P A γ−1,
ω 0123 3 ω 0123 3
P X → γ P X γ−1, (19)
ω 45 3 ω 45 3
P X → −γ P X γ−1.
ω 67,89 3 ω 67,89 3
ThetransformationsofA andX areidenticalandproducefieldsintheadjointrepresentation
0123 45
ofU(N ). TheprojectiononX producesfieldsinatwo-indexsymmetricrepresentation. More
c 67,89
explicitly, for the gauge field the projection is
P P A = 1 (cid:0)P A +J P A J−1 (cid:1). (20)
γ ω µ 2 ω µ 2Nc ω µ 2Nc
The resulting theory is a N = 2 U(N ) theory with a symmetric hypermultiplet. If one considers
c
the orientifold projection alone, the theory is projected to N = 2 SO(2N ) super Yang-Mills with
c
a hypermultiplet in the two-index representation, we can think of this theory as the analog of the
SO(2N ) gauge theory of the QCD case.
c
The D3/D7 spectrum is initially described by two 2N ×2N chiral multiplets HA describing
c f
stringsfromD3toD7branesandthereversedstringsH(cid:101)A = (cid:15)ABHB†. Theorientifoldandorbifold
actions are as follows
HA → −i(cid:15) (cid:0)ω HBω−1(cid:1)∗, P HA → γ P HAγ−1. (21)
AB 3 7 ω 3 ω 7
Therefore, the projections acting on flavor fields are
(cid:16) (cid:17) (cid:16) (cid:17)
P HA = 1 HA+(cid:15) (cid:0)HB(cid:1)∗J−1 , P P HA = 1 P HA+J P HAJ−1 . (22)
ω 2 AB 2Nf γ ω 2 ω 2Nc ω 2Nf
TheresultingmasslessfieldisaN = 2hypermultipletinthe(N ,N )representation, orN flavors
c f f
in the fundamental representation of the U(N ) gauge group. In the theory obtained from the
c
orientifold projection alone there are N hypermultiplets in the fundamental representation of the
f
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