arXiv:0704.0262v3  [hep-th]  18 Jun 2007
Preprint typeset in JHEP style. - HYPER VERSION SISSA-16/2007/EP
Stringy Instantons at Orbifold Singularities
Riccardo Argurio1, Matteo Bertolini2, Gabriele Ferretti3,
Alberto Lerda4 and Christoffer Petersson3
1Physique Théorique et Mathématique and International Solvay Institutes
Université Libre de Bruxelles, CP 231, 1050 Bruxelles, Belgium
2SISSA/ISAS and INFN - Sezione di Trieste
Via Beirut 2; I-34014 Trieste, Italy
3Department of Fundamental Physics
Chalmers University of Technology, 412 96 Göteborg, Sweden
4Dipartimento di Scienze e Tecnologie Avanzate
Università del Piemonte Orientale, I-15100 Alessandria, Italy
Istituto Nazionale di Fisica Nucleare - sezione di Torino, Italy
rargurio@ulb.ac.be, bertmat@sissa.it, ferretti@chalmers.se,
lerda@to.infn.it, chrpet@chalmers.se
Abstract: We study the effects produced by D-brane instantons on the holomorphic
quantities of a D-brane gauge theory at an orbifold singularity. These effects are not
limited to reproducing the well known contributions of the gauge theory instantons
but also generate extra terms in the superpotential or the prepotential. On these
brane instantons there are some neutral fermionic zero-modes in addition to the ones
expected from broken supertranslations. They are crucial in correctly reproducing
effects which are dual to gauge theory instantons, but they may make some other
interesting contributions vanish. We analyze how orientifold projections can remove
these zero-modes and thus allow for new superpotential terms. These terms contribute
to the dynamics of the effective gauge theory, for instance in the stabilization of runaway
directions.
Keywords: Instantons, D-branes.
http://arxiv.org/abs/0704.0262v3
mailto:rargurio@ulb.ac.be, bertmat@sissa.it, ferretti@chalmers.se, lerda@to.infn.it, chrpet@chalmers.se
http://jhep.sissa.it/stdsearch?keywords=Instantons+D-branes
Contents
1. Introduction 1
2. Preliminaries 3
3. The N = 1 Z2 × Z2 orbifold 7
3.1 Instanton sector 10
3.2 Recovery of the ADS superpotential 11
3.3 Absence of exotic contributions 14
3.4 Study of the back-reaction 16
4. The N = 1 Z2 × Z2 orientifold 17
4.1 Instanton sector 19
5. An N = 2 example: the Z3 orientifold 21
5.1 Instanton sector 24
6. Conclusions 25
1. Introduction
It has long been realized that instantons in string theory are often in close correspon-
dence with instantons in gauge theories [1, 2, 3, 4, 5, 6]. Recently it was found that in
some situations stringy instantons can dynamically generate some terms which from a
low-energy effective point of view enter as ordinary external couplings in the superpo-
tential of gauge theories living on space-filling branes [7, 8, 9, 10, 11, 12, 13, 14]. By
instantons in string theory we generally mean instantons which are geometrically real-
ized as Euclidean extended objects wrapped on some non-trivial cycles of the geometry.
Thus, in a sense, a stringy instanton has a “life of its own”, not requiring an underlying
gauge theory. This opens up the possibility of having contributions originating from
instantons that do not admit a standard gauge theory realization. We shall refer to
these instantons as exotic.
There has been some debate in the recent literature about the instances where
such exotic instantons can actually contribute to the gauge theory superpotential in a
non-trivial manner. In this work we will contribute to such a debate by considering
backgrounds where a simple CFT description is possible, such as orbifolds or orientifolds
thereof.
We present various simple examples of what we believe to be a rather generic
situation. Namely, the presence of extra zero-modes for these instantons, in addition to
those required by the counting of broken symmetries, makes some of their contributions
vanish. Such extra zero-modes should not come as a surprise, since a D-brane instanton
in a CY manifold breaks a total of four out of eight supercharges, i.e. it has two extra
fermionic zero-modes from the point of view of holomorphic N = 1 gauge theory
quantities. We give some arguments as to why the backreaction of the space-filling
branes on the geometry might not help in lifting these extra zero-modes. We further
argue that only more radical changes of the background, such as the introduction of
fluxes, deformations of the CY geometry or the introduction of orientifold planes, can
remove these zero-modes. When this happens, exotic instantons do contribute to the
gauge theory superpotential and may provide qualitative changes in the low energy
effective dynamics, as for instance the stabilization of otherwise runaway directions.
We will be interested in Euclidean D-branes in type II theories. We will work
with IIB fractional branes at orbifold and orientifold singularities rather than type IIA
wrapped branes. The motivation for this choice of setting is two-fold. First, recent
advances in the gauge/gravity correspondence require the study of exotic instantons,
whose effects tend to stabilize the gauge theory rather than unstabilize it [15, 16, 9, 17],
and the gauge/gravity correspondence is more naturally defined in the context of IIB
theory. Second, similar effects are used in string phenomenology to try to understand
possible mechanisms for neutrino masses [7, 8, 13]. This latest activity is mainly done
in the type IIA scenario, but we find it easier to address some subtle issues in the IIB
orbifold case.
While working in an exact string background, our considerations will nonetheless be
only local, i.e. we will not be concerned with global issues such as tadpole cancellation
that arise in proper compactifications. This is perfectly acceptable in the context of the
gauge/gravity correspondence where the internal manifold is non-compact but, even for
string phenomenology, the results we obtain stand (locally) when properly embedded
in a consistent compactification.
The paper is organized as follows: In section 2 we set up the notation and discuss
some preliminary material. In section 3 we discuss our first case, namely the N = 1
Z2 × Z2 orbifold. After briefly recovering the usual instanton generated corrections to
the superpotential we discuss the possible presence of additional exotic contributions
and find that they are not present because of the additional zero-modes. We conclude
by giving a CFT argument on why such zero-modes are not expected to be lifted
even by taking into account the backreaction of the D-branes, unless one is willing
to move out the orbifold point in the CY moduli space. Sections 4 and 5 present
two separate instances where exotic contributions are present after having removed the
extra zero-modes by orientifolding. The first is an N = 1 orientifold, the second is an
N = 2 orientifold, displaying corrections to the superpotential and the prepotential,
respectively. We end with some conclusions and a discussion of further developments.
2. Preliminaries
In this section we briefly review the generic setup in the well understoodN = 4 situation
in order to introduce the notation for the various fields and moduli and their couplings.
The more interesting theories we will consider next will be suitable projections of the
N = 4 theory. In fact, the exotic cases can all be reduced to orbifolds/orientifolds
of this master case once the appropriate projections on the Chan-Paton factors are
performed.
Since we are interested in instanton physics (for comprehensive reviews see [18] and
the recent [19]) we will take the ten dimensional metric to be Euclidean. We consider
a system where both D3-branes and D(−1)-branes (D-instantons) are present. To be
definite, we take N D3’s and k D-instantons 1.
Quite generically we can distinguish three separate open string sectors:
• The gauge sector, made of those open strings with both ends on a D3-brane. We
assume the brane world-volumes are lying along the first four coordinates xµ and
are orthogonal to the last six xa. The massless fields in this sector form an N = 4
SYM multiplet [22]. We denote the bosonic components by Aµ and X
a. Written
in N = 1 language this multiplet is formed by a gauge superfield whose field
strength is denoted by Wα and three chiral superfields Φ
1,2,3. With a slight abuse
of notation, the bosonic components of the chiral superfields will also be denoted
by Φ, i.e. Φ1 = X4+ iX5 and so on. In N = 2 language we have instead a gauge
superfield A and a hypermultiplet H , all in the adjoint representation. The low
energy action of these fields is a four dimensional N = 4 gauge theory. All these
fields are N ×N matrices for a gauge group SU(N).
1These D3/D(−1) brane systems (and their orbifold projections) are very useful and efficient in
studying instanton effects from a stringy perspective even in the presence of non-trivial closed string
backgrounds, both of NS-NS type [20] and of R-R type [21].
• The neutral sector, which comprises the zero-modes of strings with both ends
on the D-instantons. It is usually referred to as the neutral sector because these
modes do not transform under the gauge group. The zero-modes are easily ob-
tained by dimensionally reducing the maximally supersymmetric gauge theory to
zero dimensions. We will use an ADHM [23] inspired notation [5, 6]. We denote
the bosonic fields as aµ and χ
a, where the distinction between the two is made by
the presence of the D3-branes. The fermionic zero-modes are denoted by MαA
and λα̇A, where α and α̇ denote the (positive and negative) four dimensional chi-
ralities and A is an SU(4) (fundamental or anti-fundamental) index denoting the
chirality in the transverse six dimensions. The ten dimensional chirality of both
fields is taken to be negative. In Euclidean space M and λ must be treated as
independent. When needed, we will also introduce the triplet of auxiliary fields
Dc, directly analogous to the four dimensional D, that can be used to express the
various interactions in an easier form as we will see momentarily. All these fields
are k × k matrices where k is the instanton number.
• The charged sector, comprising the zero-modes of strings stretching between a
D3-brane and a D-instanton. For each pair of such branes we have two conjugate
sectors distinguished by the orientation of the string. In the NS sector, where the
world-sheet fermions have opposite modding as the bosons, we obtain a bosonic
spinor ωα̇ in the first four directions where the GSO projection picks out the neg-
ative chirality. In the conjugate sector, we will get an independent bosonic spinor
ω̄α̇ of the same chirality. Similarly, in the R sector, after the GSO projection
we obtain a pair of independent fermions (one for each conjugate sector) both
in the fundamental of SU(4) which we denote by µA and µ̄A. These fields are
rectangular matrices N × k and k ×N .
The couplings of the fields in the gauge sector give rise to a four dimensional gauge
theory. The instanton corrections to such a theory are obtained by constructing the
Lagrangian describing the interaction of the gauge sector with the charged sector zero-
modes while performing the integral over all zero-modes, both charged and neutral. A
crucial point to notice and which will be important later is that while the neutral modes
do not transform under the gauge group, their presence affects the integral because of
their coupling to the charged sector.
The part of the interaction involving only the instanton moduli is well known from
the ADHM construction and it is essentially the reduction of the interacting gauge
Lagrangian for these modes in a specific limit where the Yukawa terms for λ and the
quadratic term for D are scaled out (see [18, 6] for details). The final form of this part
of the interaction is:
S1 = tr
− [aµ, χ
+ χaω̄α̇ω
α̇χa +
(Σ̄a)ABµ̄
AµBχa −
(Σ̄a)ABM
αA[χa,M
µ̄Aωα̇ + ω̄α̇µ
A + σ
βα̇[M
βA, aµ]
λα̇A − iD
ω̄α̇(τ c)
α̇ωβ̇ + iη̄
µν [a
µ, aν ]
(2.1)
where the sum over colors and instanton indices is understood. τ denotes the usual
Pauli matrices, η̄ (and η) the ’t Hooft symbols and Σ̄ (and Σ) are used to construct
the six-dimensional gamma-matrices
Σ̄a 0
. (2.2)
The above interactions can all be understood in terms of string diagrams on a disk with
open string vertex operators inserted at the boundary in the α′ → 0 limit.
The interaction of the charged sector with the scalars of the gauge sector can be
worked out in a similar way and yields
S2 = tr
ω̄α̇X
(Σ̄a)ABµ̄
. (2.3)
Let us rewrite the above action in a way which will be more illuminating in the following
sections. Since we will be mainly focusing on situations where we have N = 1 super-
symmetry, it is useful to write explicitly all indices in SU(4) notation, and then break
them into SU(3) representations. We thus write the six scalars Xa as the antisymmetric
representation of SU(4) as follows
XAB = −XBA ≡ (Σ̄
a)ABXa . (2.4)
The action S2 then reads
S2 = tr
ǫABCDω̄α̇XABXCDω
µ̄AXABµ
. (2.5)
Splitting now the indices A into i = 1 . . . 3 and 4, we can identify Φ
i ≡ Xi4 in the 3̄ of
SU(3) and Φi ≡ 1
ǫijkXjk in the 3 of SU(3). Thus we can rewrite the action (2.5) as
S2 = tr
ωα̇ +
ǫijkµ̄
iΦjµk
. (2.6)
In the above form, it is clear which zero-modes couple to the holomorphic superfields
and which others couple to the anti-holomorphic ones. This distinction will play an
important role later.
The main object of our investigation is the integral of e−S1−S2 over all moduli
Z = C
d{a, χ,M, λ,D, ω, ω̄, µ, µ̄} e−S1−S2 , (2.7)
where we have lumped all field independent normalization constants (including the
instanton classical action and the appropriate powers of α′ required by dimensional
analysis) into an overall coefficient C. There are, of course, other interactions involving
the fermions and the gauge bosons but, as far as the determination of the holomorphic
quantities are concerned, they can be obtained from the previous ones and supersym-
metry arguments. For example, a term in the superpotential is written as the integral
over chiral superspace
dx4dθ2 of a holomorphic function of the chiral superfields, but
such a function is completely specified by its value for bosonic arguments at θ = 0.
Thus, if we can “factor out” a term
dx4dθ2 from the moduli integral (2.7), whatever
is left will define the complex function to be used in the superpotential and similarly
for the prepotential in the N = 2 case if we succeed in factoring out an integral over
N = 2 chiral superspace
dx4dθ4.
The coordinates x and θ must of course come from the (super)translations bro-
ken by the instanton and they will be associated to the center of mass motion of the
D-instanton, namely, xµ = tr aµ and θαA = trMαA for some values of A.2 One must
pay attention however to the presence of possible additional neutral zero-modes coming
either from the traceless parts of the above moduli or from the fields λ and χ. These
modes must also be integrated over in (2.7) and their effects, as we shall see, can be
quite dramatic. In particular, the presence of λ in some instances is crucial for the
implementation of the usual ADHM fermionic constraints whereas in other circum-
stances it makes the whole contribution to the superpotential vanish. These extra λ
zero-modes are ubiquitous in orbifold theories and generically make it difficult to obtain
exotic instanton corrections for these models. As we shall see, they can however be
easily projected out by an orientifold construction making the derivation of such terms
possible.
In the full expression for the instanton corrections there will also be a field-inde-
pendent normalization factor coming from the one-loop string diagrams and giving for
instance the proper gYM dependence in the case of the usual instanton corrections.
In this paper we will only focus on the integral over the zero-modes, which gives the
proper field-dependence, referring the reader to [10, 11] for a discussion of these other
issues.
2Obviously, for the case of an anti-instanton, the roles of M and λ are reversed.
3. The N = 1 Z2 × Z2 orbifold
In order to present a concrete example of the above discussion, let us study a simple
C3/Z2 × Z2 orbifold singularity. The resulting N = 1 theory is a non-chiral four-node
quiver gauge theory with matter in the bi-fundamental. Non-chirality implies that the
four gauge group ranks can be chosen independently [24]. This corresponds to being
able to find a basis of three independent fractional branes in the geometry (for a review
on fractional branes on orbifolds see e.g. [25]).
The field content can be conveniently summarized in a quiver diagram, see Fig. 1,
which, together with the cubic superpotential
W = Φ12Φ23Φ31 − Φ13Φ32Φ21 + Φ13Φ34Φ41 − Φ14Φ43Φ31
+Φ14Φ42Φ21 − Φ12Φ24Φ41 + Φ24Φ43Φ32 − Φ23Φ34Φ42 , (3.1)
uniquely specifies the theory.
SU(N  ) SU(N  )
SU(N  )SU(N  )
Figure 1: Quiver diagram for the Z2 × Z2 orbifold theory. Round circles correspond to
SU(Nℓ) gauge factors while the lines connecting quiver nodes represent the bi-fundamental
chiral superfields Φℓm.
A stack of N regular D3-branes amounts to having one and the same rank assign-
ment on the quiver. The gauge group is then SU(N)4 and the theory is anN = 1 SCFT.
Fractional branes correspond instead to different (but anomlay free) rank assignments.
Quite generically, fractional branes can be divided into three different classes, depend-
ing on the IR dynamics they trigger [26]. The non-chiral nature and the particularly
symmetric structure of the orbifold under consideration allows one to easily construct
any such instance of fractional brane class.
If we turn on a single node, we are left with a pure SU(N) SYM gauge theory,
with no matter fields and no superpotential. This theory is believed to confine. The
geometric dual effect is that the corresponding fractional brane leads to a geometric
transition where the branes disappear leaving behind a deformed geometry. Indeed,
there is one such deformation in the above singularity.
Turning on two nodes leads already to more varied phenomena. There are now
two bi-fundamental superfields, but still no tree level superpotential. Thus, the system
is just like two coupled massless SQCD theories or, by a slightly asymmetric point of
view, massless SQCD with a gauged diagonal flavor group. The low-energy behavior
depends on the relative ranks of the two nodes.
If the ranks are different, the node with the highest rank is in a situation where it
has less flavors than colors. Then an Affleck-Dine-Seiberg (ADS) superpotential [27, 28]
should be dynamically generated, leading eventually to a runaway behavior. This set
up of fractional branes is sometimes referred to as supersymmetry breaking fractional
branes [29, 26, 30].
If the ranks are the same we are in a situation similar to Nf = Nc SQCD for both
nodes. Hence we expect to have a moduli space of SUSY vacua, which gets deformed,
but not lifted, at the quantum level. This moduli space is roughly identified in the
geometry with the fact that the relevant fractional branes are interpreted as D5-branes
wrapped on the 2-cycle of a singularity which is locally C× (C2/Z2). Such a fractional
brane can move in the C direction. This is what is called an N = 2 fractional brane
since, at least geometrically, it resembles very much the situation of fractional branes
at N = 2 singularities.
In what follows we use the two-node example as a simple setting in which we can
analyze the subtleties involved in the integration over the neutral modes. For the gauge
theory instanton case it is known that there are extra neutral fermionic zero-modes in
addition to those required to generate the superpotential. Their integration allows to
recover the fermionic ADHM constraints on the moduli space of the usual field theory
instantons. For such instantons, we will be able to obtain the ADS superpotential
and corresponding runaway behavior in the familiar context with Nc and Nf fractional
branes at the respective nodes, for Nf = Nc−1. On the other hand, we will argue that
the presence of such extra zero-modes rules out the possibility of having exotic instanton
effects, such as terms involving baryonic operators in the Nf = Nc case. It was the
desire to study such possible contributions that constituted the original motivation for
this investigation. We will first show that such effects are absent for this theory as it
stands, and we will later discuss when and how this problem can be cured.3
3In a situation where the CFT description is less under control than in the setting discussed in
the present paper, it has been argued in [17] that such baryonic couplings do arise in the context of
fractional branes on orbifolds of the conifold, possibly at the expense of introducing O-planes. Also
in a IIA set up similar to the ones of [7, 8, 10, 11, 13] it seems reasonable that one can wrap an
ED2-brane along an O6-plane and produce such couplings on other intersecting D6-branes.
Our orbifold theory can be easily obtained as an orbifold projection of N = 4 SYM.
The orbifolding procedure and the derivation of the superpotential (3.1) are by now
standard. We briefly recall the main points in order to fix the notation and because
some of the details will be useful later in describing the instantons in such a set up.
The group Z2 × Z2 has four elements: the identity e, the generators of the two Z2
that we denote with g1 and g2 and their product, denoted by g3 = g1g2. If we introduce
complex coordinates (z1, z2, z3) ∈ C
z1 = x4 + ix5 , z2 = x6 + ix7 , z3 = x8 + ix9 (3.2)
the action of the orbifold group can be defined as in Table 1.
z1 z2 z3
e z1 z2 z3
1 −z2 −z3
g2 −z
1 z2 −z3
g3 −z
1 −z2 z3
Table 1: The action of the orbifold generators.
Let γ(g) be the regular representation of the orbifold group on the Chan-Paton
factors. If the orbifold is abelian, as always in the cases we shall be interested in, we
can always diagonalize all matrices γ(g). We will assume that the two generators have
the following matrix representation
γ(g1) = σ3 ⊗ 1 =
1 0 0 0
0 1 0 0
0 0 −1 0
0 0 0 −1
, γ(g2) = 1⊗ σ3 =
1 0 0 0
0 −1 0 0
0 0 1 0
0 0 0 −1
(3.3)
where the 1’s denote Nℓ ×Nℓ unit matrices (ℓ = 1, ..., 4). Then, the orbifold projection
amounts to enforcing the conditions
Aµ = γ(g)Aµγ(g)
−1 , Φi = ±γ(g)Φiγ(g)−1 (3.4)
where the sign ± must be chosen according to the action of the orbifold generators
g that can be read off from Table 1. With the choice (3.3), the vector superfields
are block diagonal matrices of different size (N1, N2, N3, N4), one for each node of the
quiver, while the three chiral superfields Φi have the following form [24]
0 × 0 0
× 0 0 0
0 0 0 ×
0 0 × 0
, Φ2 =
0 0 × 0
0 0 0 ×
× 0 0 0
0 × 0 0
, Φ3 =
0 0 0 ×
0 0 × 0
0 × 0 0
× 0 0 0
, (3.5)
where the crosses represent the non-zero entries Φℓm appearing in the superpotential
(3.1).
3.1 Instanton sector
Now consider D-instantons in the above set up. Such instantons preserve half of the
4 supercharges preserved by the system of D3-branes plus orbifold. In this respect
recall that the fractional branes preserve exactly the same supercharges as the regular
branes.4 Using the N = 4 construction of the previous section and the structure of the
orbifold presented in eq. (3.5), we now proceed in describing the zero-modes for such
instantons.
The neutral sector is very similar to the gauge sector. Indeed, in the (−1) su-
perghost picture, the vertex operators for such strings will be exactly the same, except
for the eip·X factor which is absent for the instanton. The Chan-Paton structure will
also be the same, so that the same pattern of fractional D-instantons will arise as for
the fractional D3-branes. In particular, the only regular D-instanton (which could be
thought of as deriving from the one of N = 4 SYM) is the one with rank (instanton
number) one at every node. All other situations can be thought of as fractional D-
instantons, which can be interpreted as Euclidean D1-branes wrapped on the two-cycles
at the singularity, ED1 for short. Generically, we can then characterize an instanton
configuration in our orbifold by (k1, k2, k3, k4).
Following the notation introduced in section 2, the bosonic modes will comprise a
4×4 block diagonal matrix aµ, and six more matrix fields χ1, . . . χ6, that can be paired
into three complex matrix fields χ1+iχ2, χ3+iχ4, χ5+iχ6, having the same structure as
(3.5) but now where each block entry is a kℓ×km matrix. On the fermionic zero-modes
MαA and λα̇A (also matrices) the orbifold projection enforces the conditions
MαA = R(g)AB γ(g)M
αBγ(g)−1 , λα̇A = γ(g)λα̇Bγ(g)
−1R(g)BA (3.6)
4There is another Euclidean brane which preserves two supercharges, namely the Euclidean (anti)
D3-branes orthogonal to the 4 dimensions of space-time. We will be considering here only the D-
instantons, leaving the complete analysis of the other effects to future work. In this context, note that
the extended brane instantons would have an infinite action (and thus a vanishing contribution) in
the strict non-compact set up we are using here.
where R(g) is the orbifold action of Table 1 in the spinor representation which can be
chosen as
R(g1) = −Γ
6789 , R(g2) = −Γ
4589 . (3.7)
It is easy to find an explicit representation of the Dirac matrices such thatMαA and λα̇A
for A = 1, 2, 3 also have the structure of (3.5) while for A = 4 they are block diagonal.
Equivalently, one could write the spinor indices in the internal space in terms of the
three SO(2) charges associated to the embedding SO(2) × SO(2) × SO(2) ⊂ SO(6) ≃
SU(4)
Mα−++ =Mα1 , Mα+−+ =Mα2 , Mα++− =Mα3 , Mα−−− =Mα4 ,
λα̇+−− = λα̇1 , λα̇−+− = λα̇2 , λα̇−−+ = λα̇3 , λα̇+++ = λα̇4 . (3.8)
The most notable difference between the neutral sector and the gauge theory on the
D3-branes is that, whereas in the four-dimensional theory the U(1) gauge factors are
rendered massive by a generalization of the Green-Schwarz mechanism and do not
appear in the low energy action, for the instanton they are in fact present and enter
crucially into the dynamics.
Let us finally turn to the charged sector, describing strings going from the instan-
tons to the D3-branes. The analysis of the spectrum and the action of the orbifold
group on the Chan-Paton factors show, in particular, that the bosonic zero-modes are
diagonal in the gauge factors. There are four block diagonal matrices of bosonic zero-
modes ωα̇, ω̄α̇ with entries Nℓ×kℓ and kℓ×Nℓ respectively and eight fermionic matrices
µA, µ̄A with entries Nℓ × km and km × Nℓ, that again display the same structure as
above – same as (3.5) for A = 1, 2, 3 and diagonal for A = 4.
3.2 Recovery of the ADS superpotential
The measure on the moduli space of the instantons and the ADHM constraints are
simply obtained by inserting the above expressions into the moduli integral (2.7). If
one chooses some of the Nℓ or kℓ to vanish one can deduce immediately from the
structure of the projection which modes will survive and which will not.
As a consistency check, one can try to reproduce the ADS correction to the super-
potential [27, 28] for the theory with two nodes. Take fractional branes corresponding
to a rank assignment (Nc, Nf , 0, 0), and consider the effect of a ED1 corresponding to
instanton numbers (1, 0, 0, 0).
The only chiral fields present are the two components of Φ1 connecting the first
and second node
0 Q 0 0
Q̃ 0 0 0
0 0 0 0
0 0 0 0
. (3.9)
Since the instanton is sitting only at one node, all off diagonal neutral modes are absent,
as they connect instantons at two distinct nodes. Thus, the only massless modes present
in the neutral sector are four bosons xµ, denoting the upper-left component of aµ, two
fermions θα denoting the upper-left component of Mα4 and two more fermions λα̇
denoting the upper-left component of λα̇4. We have identified the non zero entries of
aµ and Mα4 with the super-coordinates xµ and θα since they precisely correspond to
the Goldstone modes of the super-translation symmetries broken by the instanton and
do not appear in S1+ S2 (cfr. (2.1) and (2.3)). Their integration produces the integral
over space-time and half of Grassmann space which precedes the superpotential term
to which the instanton contributes. On the contrary, λα̇ appears in S1 and when it is
integrated it yields the fermionic ADHM constraint.
In the charged sector, we have bosonic zero-modes ωuα̇ and ω̄α̇u, with u an index
in the fundamental or anti-fundamental of SU(Nc). In addition, there are fermionic
zero-modes µu and µ̄u with indices in SU(Nc), together with additional fermionic zero-
modes µ′f and µ̄′f where the index f is now in the fundamental or anti-fundamental of
SU(Nf ).
5 Note that the µ zero-modes carry an SU(4) index 4 (being on the diagonal)
while the µ′ zero-modes carry an SU(4) index 1, since they are of the same form as Φ1.
All this can be conveniently summarized in a generalized quiver diagram as rep-
resented in Fig. 2, which accounts for both the brane configuration and the instanton
zero-modes.
For a single instanton, the action (2.1) greatly simplifies since many fields are
vanishing as well as all commutators and one gets
S1 = i (µ̄uω
α̇ + ω̄α̇uµ
u) λα̇ − iDcω̄α̇u(τ
. (3.10)
Similarly, the coupling of the charged modes to the chiral superfield can be expressed
by writing eq. (2.3) as
ω̄α̇u
v + Q̃
ωα̇v −
µ̄uQ̃
µ̄′fQ
u . (3.11)
Note that it is the anti-holomorphic superfields that enter in the couplings with the
fermionic zero-modes, as is clear by comparing with (2.6). The above action is exactly
the same which appears in the ADHM construction as reviewed in [18].
5Recall that the bosonic zero-modes are diagonal in the gauge factors; therefore there are no ω
and ω̄α̇f zero-modes.
SU(N  ) SU(N  )f
Figure 2: Quiver diagram describing an ordinary instanton in a SU(Nc) × SU(Nf ) theory.
Gauge theory nodes are represented by round circles, instanton nodes by squares. The ED1
is wrapped on the same cycle as the color branes. All zero-modes are included except the θ’s
and the xµ’s, which only contribute to the measure for the integral over chiral superspace.
We are now ready to perform the integral (2.7) over all the existing zero-modes.
Writing
dx4dθ2W , (3.12)
we see that the instanton induced superpotential is
W = C
d{λ,D, ω, ω̄, µ, µ̄} e−S1−S2 . (3.13)
The integrals over D and λ enforce the bosonic and fermionic ADHM constraints,
respectively. Thus
W = C
d{ω, ω̄, µ, µ̄} δ(µ̄uω
α̇ + ω̄α̇uµ
u) δ(ω̄α̇u(τ
) e−S2 . (3.14)
We essentially arrive at the point of having to evaluate an integral over a set of zero-
modes which is exactly the same as the one discussed in detail in the literature, e.g. [18].
We thus quickly go to the result referring the reader to the above review for further
details. First of all, it is easy to see that, due to the presence of extra µ modes in
the integrand from the fermionic delta function, only when Nf = Nc − 1 we obtain
a non-vanishing result. After having integrated over the µ and µ′, we are left with
a (constrained) gaussian integration that can be performed e.g. by going to a region
of the moduli space where the chiral fields are diagonal, up to a row/column of ze-
roes. Furthermore, the D-terms in the gauge sector constrain the quark superfields
to obey QQ† = Q̃†Q̃, so that the bosonic integration brings the square of a simple
determinant in the denominator. The last fermionic integration conspires to cancel the
anti-holomorphic contributions and gives
WADS =
Λ2Nc+1
det(Q̃Q)
, (3.15)
which is just the expected ADS superpotential for Nf = Nc − 1, the only case where
such non-perturbative contribution is generated by a genuine one-instanton effect and
not by gaugino condensation. In (3.15) Λ is the SQCD strong coupling scale that is
reconstructed by the combination of e−8π
2/g2 coming from the instanton action with
various dimensional factors coming from the normalization of the instanton measure
[18].
3.3 Absence of exotic contributions
Until now, we have reproduced from stringy considerations the effect that is supposed
to be generated also by instantons in the gauge theory. Considering a slightly different
set up, we would like to study the possibility of generating other terms.
Let us consider a system with rank assignment (Nc, Nf , 0, 0), as before, but frac-
tional instanton numbers (0, 0, 1, 0). In other words, we study the effect of a single
fractional instanton sitting on an unoccupied node of the gauge theory. The quiver
diagram, with the relevant zero-modes structure, is given in Fig. 3.
The neutral zero-modes of the instanton sector are the same as before. This is
because the quantization of this sector does not know the whereabouts of the D3-
branes and thus all nodes are equivalent, in this respect. In the mixed sector, we have
no bosonic zero-modes now, since the ω and ω̄ are diagonal. Note that, although we
always have four mixed (ND) boundary conditions, due to the quiver structure induced
by the orbifold, here we effectively realize the same situation one has when there are
eight ND directions, namely that the bosonic sector of the charged moduli is empty.
On the other hand, there are fermionic zero-modes µu, µ̄u, µ
′f and µ̄′f , as in the
previous case. Note that despite having the same name, these zero-modes correspond
actually to different Chan-Paton matrix elements with respect to the previous ones,
the difference being in the instanton index that is not written explicitly. In particular
we can think of µ and µ′ as carrying an SU(4) index 2 and 3 respectively.
Because of the absence of bosonic charged modes, the action (2.1) is identically
SU(N  )fSU(N  )
Figure 3: Quiver diagram describing an exotic instanton in a SU(Nc) × SU(Nf ) theory.
Gauge theory nodes are represented by round circles, instanton nodes by squares. The ED1
is wrapped on a different cycle with respect to both sets of quiver branes.
zero and the action (2.3) contains only the last term:
S1 = 0
µ̄′fQ̃
u. (3.16)
Note that in this case it is the holomorphic superfields which appear above, as is clear
from (2.6) and from noticing that the diagonal fermionic zero-mode µ4 is not present.
We are thus led to consider
W = C
d{λ,D, µ, µ̄} e−S2 . (3.17)
One notices right away that the integral over the charged modes is non vanishing
(only) for the case Nf = Nc and gives a tantalizing contribution proportional to BB̃,
where B = detQ and B̃ = det Q̃ are the baryon fields of the theory. However, we
must carefully analyze the integration over the remaining zero-modes of the neutral
sector. Now neither D nor λ appear in the integrand. The integral over D does not
raise any concern: it is, after all, an auxiliary field and its disappearance from the
integrand is due to the peculiarities of the ADHM limit. Before taking this limit, D
appeared quadratically in the action and could be integrated out, leaving an overall
normalization constant. The integral over λ is another issue. In this case, λ is absent
from the integrand even before taking the ADHM limit and its integration multiplies
the above result by zero, making the overall contribution of such instantons to the
superpotential vanishing. Of course, the presence of such extra zero-modes should not
come as a surprise since they correspond to the two extra broken supersymmetries of
an instanton on a CY.
Therefore we see that the neutral zero-modes contribution, in the exotic instanton
case, plays a dramatic role and conspires to make everything vanishing (as opposite
to the ADS case analyzed before). A natural question is to see whether these zero-
modes get lifted by some effect we have not taken into account, yet. For one thing,
supersymmetry arguments would make one think that taking into account the back-
reaction of the D3-branes might change things. However, in the following subsection
we show that this seems not to be the case.
3.4 Study of the back-reaction
Let us stick to the case Nf = Nc, which is the only one where the integral (3.17)
might give a non-vanishing contribution. In this case the fractional brane system
is nothing but a stack of (Nc) N = 2 fractional branes. These branes couple to
only one of the 3 closed string twisted sectors [24]. More specifically, they source
the metric hµν , the R-R four-form potential Cµνρσ and two twisted scalars b and c from
the NS-NS and R-R sector respectively. This means that the disk one-point function
of their vertex operators [31, 32] is non vanishing when the disk boundary is attached
to such D3-branes. (Indeed in this way or, equivalently, by using the boundary-state
formalism [33, 34], one can derive the profile for these fields.)
If the back-reaction of these fields on the instanton lifted the extra zero-modes
λ’s, this should be visible when computing the one point function of the corresponding
closed string vertex operators on a disk with insertions on this boundary of the vertex
operators for such moduli. To see whether such coupling is there, we first need to write
down the vertex operators for the λ’s in the (±1/2) superghost pictures. The vertex in
the (−1/2) picture is found e.g. in [6] and reads
λ (z) = λα̇AS
α̇(z)SA(z)e−φ(z)/2 , (3.18)
where Sα̇(z) and SA(z) are the spin-fields in the first four and last six directions re-
spectively. For our argument we need to focus on the SA(z) dependence. Since the
modulus that survives the orbifold projection is, with our conventions, λα̇4 = λα̇+++,
we write the corresponding spin-field as
S+++(z) = eiH1(z)/2eiH2(z)/2eiH3(z)/2, (3.19)
where Hi(z) is the free boson used to bosonize the fermionic sector in the i-th complex
direction: ψi(z) = eiHi(z). The vertex operator in the +1/2 picture can be obtained by
applying the picture-changing operator to (3.18)
λ (z) = [QBRST, ξV
λ (z)] . (3.20)
The crucial part in QBRST is [31]
QBRST =
ψµ∂Xµ + ψ̄i∂Z i + ψi∂Z̄ i
+ . . . (3.21)
Because of the nature of the supercurrent, we see that (3.21) flips at most one sign in
(3.19), hence the product V
λ will always carry an unbalanced charge in some
of the three internal SO(2) groups. On the other hand, the vertex operators for the
fields sourced by the fractional D3’s cannot compensate such an unbalance. Hence, their
correlation function on the D-instanton with the insertion of V
λ carries a charge
unbalance and therefore vanishes. Therefore, at least within the above perturbative
approach, the neutral zero-modes seem not to get lifted by the back-reaction of the
D3-branes.
One might consider some additional ingredients which could provide the lifting. A
natural guess would be moving in the CY moduli space or adding suitable background
fluxes [35, 36]. There are indeed non-vanishing background fields at the orbifold point,
i.e. the b fields of the twisted sectors which the N = 2 fractional branes do not
couple to. These fields, however, being not associated to geometric deformations of
the internal space should be described by a CFT vertex operator uncharged under the
SO(2)’s, simply because of Lorentz invariance in the internal space. Therefore, the
only way to get an effective mass term for the zero-modes λ would be to move out of
the orbifold point in the CY moduli space. Indeed, the other moduli of the NS-NS
twisted sector, being associated to geometric blow-ups of the singularity, are charged
under (some of) the internal SO(2)’s and can have a non vanishing coupling with the
λ’s. More generically, complicated closed string background fluxes might be suitable.
This is an interesting option which however we do not pursue here, since we want to
stick to situations where a CFT description is available.
A more radical thing to do is to remove the zero-modes from the very start, for
instance by means of an orientifold projection [37, 38]. This is the option we are going
to consider in the remainder of this work.
4. The N = 1 Z2 × Z2 orientifold
In this section we supplement our orbifold background by an O3 orientifold and show
that in this case exotic instanton contributions do arise and provide new terms in the
superpotential. We refer to e.g. [39, 40, 41] for a comprehensive discussion of N = 1
and N = 2 orientifolds.
The first ingredient we need is the action of the O3-plane on the various fields.
Denote by Ω the generator of the orientifold. The action of Ω on the vertex operators
for the various fields (ignoring for the time being the Chan-Paton factors) is well known.
The vertex operators for the bosonic fields on the D3-brane contain, in the 0 picture,
the following terms: Aµ ∼ ∂τx
µ and Φi ∼ ∂σz̄
i. They both change sign under Ω,
the first because of the derivative ∂τ and the second because the orientifold action for
the O3-plane is always accompanied by a simultaneous reflection of all the transverse
coordinates zi.
The action of the orientifold on the Chan-Paton factors is realized by means of
a matrix γ(Ω) which in presence of an orbifold must satisfy the following consistency
condition [39]
γ(g)γ(Ω)γ(g)T = + γ(Ω) (4.1)
for all orbifold generators g. This amounts to require that the orientifold projection
commutes with the orbifold projection. The matrix γ(Ω) can be either symmetric or
anti-symmetric. We choose to perform an anti-symmetric orientifold projection on the
D3 branes and denote the corresponding matrix by γ−(Ω). This requires having an
even number Nℓ of D3 branes on each node of the quiver so that we can write
γ−(Ω) =
ǫ1 0 0 0
0 ǫ2 0 0
0 0 ǫ3 0
0 0 0 ǫ4
(4.2)
where the ǫℓ’s are Nℓ × Nℓ antisymmetric matrices obeying ǫ
ℓ = −1. Using (3.3) and
(4.2) it is straightforward to verify that the consistency condition (4.1) is verified.
The field content of the stacks of fractional D3-branes in this orientifold model is
obtained by supplementing the orbifold conditions (3.4) with the orientifold ones
Aµ = −γ−(Ω)A
µγ−(Ω)
−1 , Φl = −γ−(Ω)Φ
lTγ−(Ω)
−1. (4.3)
This implies that Aµ = diag (A
µ) with A
µ = ǫℓA
µ ǫℓ. Thus, the result-
ing gauge theory is a USp(N1) × USp(N2) × USp(N3) × USp(N4) theory. The chiral
superfields, which after the orbifold have the structure (3.5), are such that the Φℓm
component joining the nodes ℓ and m of the quiver, must obey the orientifold condi-
tion Φℓm = ǫℓΦ
mℓǫm. In the following, we will take N3 = N4 = 0 so that we are left
with only two gauge groups and no tree level superpotential.
4.1 Instanton sector
Let us now consider the instanton sector, starting by analyzing the zero-mode content
in the neutral sector. There are two basic changes to the previous story. The first is
that the vertex operator for aµ is now proportional to ∂σx
µ, not to ∂τx
µ and it remains
invariant under Ω (the vertex operator for χa still changes sign). The second is that
the crucial consistency condition discussed in [38] requires that we now represent the
action of Ω on the Chan-Paton factors of the neutral modes by a symmetric matrix
which can be taken to be
γ+(Ω) =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
, (4.4)
where the 1’s are kℓ × kℓ unit matrices. The matrix aµ will be 4 × 4 block diagonal,
e.g. aµ = diag (a
µ), but now a
µ = a
µ . The most generic situation is to have
a configuration with instanton numbers (k1, k2, k3, k4). By considering a configuration
with k3 = 1 and k1 = k2 = k4 = 0, we can project out all bosonic zero-modes except
for the four components a3µ that we denote by xµ. The scalars χ
4 . . . χ9 are off-diagonal
and we shall not consider them further.
The nice surprise comes when considering the orientifold action on the fermionic
neutral zero-modes MαA and λα̇A. The orbifold part of the group acts on the spinor
indices as in (3.7), while the orientifold projection acts as the reflection in the transverse
space, namely
R(Ω) = −iΓ456789 (4.5)
Putting together the orbifold projections (3.6) with the orientifold ones
MαA = RAB(Ω)γ+(Ω)(M
αB)Tγ+(Ω)
−1 , λα̇A = γ+(Ω)(λα̇B)
Tγ+(Ω)
−1RBA(Ω) (4.6)
we can find the spectrum of surviving fermionic zero-modes. Using (4.4) and (4.5), it
is easy to see that (4.6) implies
MαA = (MαA)T , λα̇A = −(λα̇A)
T . (4.7)
Thus, for the simple case where k3 = 1 and k1 = k2 = k4 = 0, all λ’s are projected out
and only two chiral M zero-modes remain: Mα−−−, to be identified with the N = 1
chiral superspace coordinates θα.
Also the charged zero-modes are easy to discuss in this simple scenario. There are
no bosonic modes since the D-instanton and the D3-branes sit at different nodes while
the bosonic modes are necessarily diagonal. Most of the fermionic zero-modes µA and
µ̄A are also projected out by the orbifold condition
µA = R(g)ABγ(g)µ
Bγ(g)−1 , µ̄A = R(g)ABγ(g)µ̄
Bγ(g)−1 . (4.8)
Finally, the orientifold condition relates this time the fields in the conjugate sectors,
allowing one to express µ̄ as a linear combination of the µ
µ̄A = R(Ω)ABγ+(Ω)(µ
B)Tγ−(Ω)
−1 . (4.9)
The only charged modes surviving these projections can be expressed, in block 4 × 4
notation, as
0 0 µ13 0
0 0 0 0
0 0 0 0
0 0 0 0
, µ̄2 =
0 0 0 0
0 0 0 0
µ̄31 0 0 0
0 0 0 0
0 0 0 0
0 0 µ23 0
0 0 0 0
0 0 0 0
, µ̄3 =
0 0 0 0
0 0 0 0
0 µ̄32 0 0
0 0 0 0
, (4.10)
where the entries, to be thought of as column/row vectors in the fundamental/anti-
fundamental of SU(Nℓ) depending on their position, are such that µ̄31 = −µ
13ǫ1 and
µ̄32 = −µ
23ǫ2.
Thus, in the case where we have fractional D3 branes (N1, N2, 0, 0) and an exotic
instanton (0, 0, 1, 0), the only surviving chiral field is Φ12 ≡ ǫ1Φ
21ǫ2, the orientifold
projection eliminates the offending λ’s and we are left with just the neutral zero-modes
xµ and θ
α and the charged ones µ13 and µ23. This is summarized in the generalized
quiver of Fig. 4.
In this case the instanton partition function is
dx4dθ2 W (4.11)
where the superpotential W is
W = C
dµ e−S1−S2 = C
dµ13dµ23 e
iµT13ǫ1Φ12µ23 . (4.12)
This integral clearly vanishes unless N1 = N2, in which case we have
W ∝ det(Φ12) (4.13)
USP(N  )USP(N  )1 2
Φ Φ1221
2313µ
Figure 4: The generalized Z2 ×Z2 orientifold quiver and the exotic instanton contribution.
We thus see that exotic instanton corrections are possible in this simple model.6
It is interesting to note that the above correction is present in the same case
(N1 = N2 ≡ N) where the usual ADS superpotential for USp(N) is generated [42]
WADS =
Λ2N+3
det(Φ12)
(4.14)
and its presence stabilizes the runaway behavior and gives a theory with a non-trivial
moduli space of supersymmetric vacua given by det(Φ12) = const. Of course, the ADS
superpotential for this case can also be constructed along the same lines as section 3.2,
see e.g. [18]. In fact, this derivation is somewhat simpler than the one for the SU(N)
gauge group since there are no ADHM constraints at all in the one instanton case.
We think the above situation is not specific to the background we have been consid-
ering, but is in fact quite generic. As soon as the λ zero-modes are consistently lifted,
we expect the exotic instantons to contribute new superpotential terms. As a further
example, in the next section we will consider a N = 2 model, where exotic instantons
will turn out to contribute to the prepotential.
5. An N = 2 example: the Z3 orientifold
Let us now consider the quiver gauge theory obtained by placing an orientifold O3-plane
at a C × C2/Z3 orbifold singularity. In what follows we will use N = 1 superspace
notation. We first briefly repeat the steps that led to the constructions of such a quiver
6The gauge invariant quantity above can be rewritten as the Pfaffian of a suitably defined mesonic
matrix.
theory in the seminal paper [39]. Define ξ = e2πi/3 and let the generator of the orbifold
group act on the first two complex coordinates as
0 ξ−1
, (5.1)
while leaving the third one invariant. This preserves N = 2 SUSY. The action of the
generator g on the Chan-Paton factors is given by the matrix
γ(g) =
1 0 0
0 ξ 0
0 0 ξ2
 . (5.2)
The N = 2 theory obtained this way, summarized in Fig. 5, is a three node quiver
gauge theory with gauge groups SU(N1)× SU(N2)× SU(N3), supplemented by a cubic
superpotential which is nothing but the orbifold projection of the N = 4 superpotential
(its precise form is not relevant for the present purposes).
SU(N  ) SU(N  )
SU(N  )1
Figure 5: The Z3 (un-orientifolded) theory. The lines with both ends on a single node
represent adjoint chiral multiplets which, together with the vector multiplets at each node
constitute the N = 2 vector multiplets. Similarly, lines between nodes represent chiral mul-
tiplets which pair up into hyper-multiplets, in N = 2 language.
As for the action of Ω on the Chan-Paton factors, we choose again to perform the
symplectic projection on the D3-branes. To do so, we must take N1 to be even and
N2 = N3, so that we can write
γ−(Ω) =
ǫ 0 0
0 0 1
0 −1 0
 , (5.3)
where ǫ is a N1×N1 antisymmetric matrix obeying ǫ
2 = −1 and the 1’s denote N2×N2
identity matrices. The matrices γ(g) and γ−(Ω) satisfy the usual consistency condi-
tion [38, 39] as in (4.1).
The field content on the fractional D3-branes at the singularity will be given by
implementing the conditions
Aµ = γ(g)Aµγ(g)
−1 , Φi = ξ−iγ(g)Φiγ(g)−1 ,
Aµ = −γ−(Ω)A
µγ−(Ω)
−1 , Φi = −γ−(Ω)Φ
iTγ−(Ω)
−1 . (5.4)
The orbifold part of these conditions forces Aµ and Φ
3 to be 3 × 3 block diagonal
matrices, e.g. Aµ = diag (A
µ), while the orientifold imposes that A
µ = ǫA
and A2µ = −A
µ . The resulting gauge theory is thus a USp(N1)× SU(N2) theory. It is
convenient, however, to still denote A2µ and A
µ diagramatically as belonging to different
nodes with the understanding that these should be identified in the above sense.
The projection on the chiral fields can be done similarly and we obtain, denoting
by Φℓm the non-zero entries of the fields Φ
1 and Φ2 (only one can be non-zero for each
pair ℓm)
Φ12 = −ǫΦ
31, Φ13 = +ǫΦ
21, Φ23 = Φ
23, Φ32 = Φ
32 . (5.5)
The field content is summarized in Table 2.
USp(N1) SU(N2)
Φ12 � �
Φ21 � �
Φ13 � �
Φ31 � �
Φ23 · ��
Φ32 · ��
Table 2: Chiral fields making up the quiver gauge theory.
The theory we want to focus on in the following has rank assignment (N1, N2) =
(0, N). This yields an N = 2 SU(N) gauge theory with an hyper-multiplet in the sym-
metric/(conjugate)symmetric representation. We denote the N = 2 vector multiplet
by A whose field content in the block 3× 3 notation is thus
0 0 0
0 A 0
0 0 −AT
 . (5.6)
In what follows we will be interested in studying corrections to the prepotential F
coming from exotic instantons associated to the first node (the one that is not populated
by D3-branes). Let us then analyze the structure of the stringy instanton sector of the
present model, first.
5.1 Instanton sector
The most generic situation is to have a configuration with instanton numbers (k1, k2)
(later we will be mainly concerned with a configuration with instanton numbers (1, 0)).
Let us start analyzing the zero-modes content in neutral sector. The story is
pretty similar to the one discussed in the previous section. The vertex operator for aµ
is proportional to ∂σx
µ and so it remains invariant under Ω. The action on the Chan-
Paton factors of these D-instantons must now be represented by a symmetric matrix
which we take to be
γ+(Ω) =
1′ 0 0
0 0 1
0 1 0
 (5.7)
where 1′ is a k1 × k1 unit matrix and the 1’s are k2 × k2 unit matrices.
Because of the different orientifold projection, the matrices of bosonic zero-modes
behave slightly differently. The matrices aµ, χ
8 and χ9 will still be 3×3 block diagonal,
e.g. aµ = diag (a
µ), but now a
µ = a
µ and a
µ = a
µ whereas the same relations
for χ8 and χ9 will have an additional minus sign. The remaining fields χ4...7 are off
diagonal and we shall not consider them further since we will consider only the case of
one type of instanton. By considering a configuration with k1 = 1 and k2 = 0, we can
project out all bosonic zero-modes except for the four components a1µ that we denote
by xµ.
Let us now consider the orientifold action on the fermionic neutral zero-modesMαA
and λα̇A. The orbifold part of the group acts on the internal spinor indices as a rotation
R(g) = e
Γ45e−
Γ67 , (5.8)
while the orientifold acts through the matrix R(Ω) given in (4.5). The orbifold and
orientifold projections thus require
MαA = R(g)ABγ(g)M
αBγ(g)−1 , λα̇A = γ(g)λα̇Bγ(g)
−1R(g)BA , (5.9)
MαA = R(Ω)ABγ+(Ω)(M
αB)Tγ+(Ω)
−1 , λα̇A = γ+(Ω)(λα̇B)
Tγ+(Ω)
−1R(Ω)BA .
Using the explicit expressions for the various matrices, we see that, for the simple
case where k1 = 1 and k2 = 0, all λ’s are projected out and only four chiral M zero-
modes remain: Mα−−− and Mα++− to be identified with the N = 2 chiral superspace
coordinates θ1α and θ
α. Hence, also in this case the orientifold projection has cured the
problem encountered in section 3 (albeit in a N = 2 context now) and we can rest
assured that the integration over the charged modes will yield a contribution to the
prepotential.
Let us now move to the charged zero-modes sector. Just as in the previous model,
there are no bosonic modes since the D-instanton and the D3-branes sit at different
nodes while the bosonic modes are necessarily diagonal. Most of the fermionic zero-
modes µA and µ̄A are projected out by the orbifold condition which is formally the same
as in (4.8), while the orientifold condition relates the fields in the conjugate sectors,
giving µ̄ as a linear combination of the µ’s according to
µ̄A = R(Ω)ABγ+(Ω)(µ
B)Tγ−(Ω)
−1 . (5.10)
To summarize, the only charged modes surviving the projection can be expressed, in
block 3× 3 notation as
0 0 0
0 0 0
µ 0 0
 , µ̄1 =
0 µT 0
0 0 0
0 0 0
0 0 0
µ′ 0 0
0 0 0
 , µ̄2 =
0 0 −µ′T
0 0 0
0 0 0
 (5.11)
where the entries are to be thought of as column/row vectors in the fundamental/anti-
fundamental of SU(N) depending on their position.
As anticipated, the configuration we want to consider is a (0, N) fractional D3-
branes system together with an exotic (1, 0) instanton. The quiver structure, including
the relevant moduli, is depicted in Fig. 6. It is now easy to see that inserting the
expressions (5.6) and (5.11) into Eqs. (2.1), (2.3) and (2.7) we finally obtain
dx4dθ4F with F = C
dµdµ′ eiµ
TAµ′ ∝ detA . (5.12)
It would be interesting to study the potential implications of this result in the gauge
theory. There are many other simple models that could be analyzed along these lines.
6. Conclusions
In this paper we have presented some simple examples of what seem to be rather generic
phenomena in the context of string instanton physics. We paid particular attention to
Φ Φ3223
U(N) U(N)
Figure 6: The extended Z3 orientifold theory with (0, N) fractional D3-branes and (1, 0)
instanton number. The upper node (which would represent the USp(N1) gauge group and
disappears when we set N1 = 0 as in the case under consideration) is where the instanton
sits. The lower nodes denote only one gauge group. The charged fermionic zero-modes follow
Eq. (5.11). For simplicity we have not drawn the lines denoting the adjoint.
the study of the fermionic zero-modes and their effects on the holomorphic quantities
of the theory. We have seen both examples where the instanton contributions vanish
due to the presence of extra zero-modes and where they do not. In the second case, as
explicitly shown in a N = 1 example, exotic instantons can have a stabilizing effect on
the theory.
Although we have only considered some simple examples, we would like to stress
that these results are quite generic and can be carried over to all orbifold gauge theories.
A future direction would be to try to be more systematic and analyze the various
possibilities encountered in more complex N = 2 and N = 1 models. In a similar
spirit, one should analyze the multi-instanton contributions as well, since the total
correction to the holomorphic quantities will be the sum of all such terms. The study
of the zero-modes is expected to be even more relevant in this case as it will probably
make many contributions vanish. With an eye to string phenomenology, one should
also incorporate these models into globally consistent compactifications and study the
effects of these terms there.
Lastly, it would be interesting to study the dynamical implications of some of the
terms generated. We briefly touched upon this at the end of section 4 when we men-
tioned the stabilizing effect of the exotic instanton on the USp(N) theory. Although
from the strict field theory point of view these terms are thought of as ordinary polyno-
mial terms in the holomorphic quantities,7 they are “special” when seen from the point
of view of string theory and they might therefore induce a particular type of dynamics.
Acknowledgements
We would like to thank many people for discussions and email exchanges at various
stages of this work that helped us sharpen the focus of the presentation: M. Bianchi,
M. Billò, P. Di Vecchia, S. Franco, M. Frau, F. Fucito, S. Kachru, R. Marotta, L. Mar-
tucci, F. Morales, B. E. W. Nilsson, D. Persson, I. Pesando, D. Robles-Llana, R. Russo,
A. Tanzini, A. Tomasiello, A. Uranga, T. Weigand and N. Wyllard.
R.A., M.B. and A.L. are partially supported by the European Commission FP6
Programme MRTN-CT-2004-005104, in which R.A is associated to V.U. Brussel, M.B.
to University of Padova and A.L. to University of Torino. R.A. is a Research Associate
of the Fonds National de la Recherche Scientifique (Belgium). The research of R.A.
is also supported by IISN - Belgium (convention 4.4505.86) and by the “Interuniver-
sity Attraction Poles Programme –Belgian Science Policy”. M.B. is also supported by
Italian MIUR under contract PRIN-2005023102 and by a MIUR fellowship within the
program “Rientro dei Cervelli”. The research of G.F. is supported by the Swedish
Research Council (Vetenskapsr̊adet) contracts 622-2003-1124 and 621-2002-3884. A.L.
thanks the Galileo Galilei Institute for the hospitality and support during the comple-
tion of this work.
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  We study the effects produced by D-brane instantons on the holomorphic
quantities of a D-brane gauge theory at an orbifold singularity. These effects
are not limited to reproducing the well known contributions of the gauge theory
instantons but also generate extra terms in the superpotential or the
prepotential. On these brane instantons there are some neutral fermionic
zero-modes in addition to the ones expected from broken supertranslations. They
are crucial in correctly reproducing effects which are dual to gauge theory
instantons, but they may make some other interesting contributions vanish. We
analyze how orientifold projections can remove these zero-modes and thus allow
for new superpotential terms. These terms contribute to the dynamics of the
effective gauge theory, for instance in the stabilization of runaway
directions.

Introduction 1
2. Preliminaries 3
3. The N = 1 Z2 × Z2 orbifold 7
3.1 Instanton sector 10
3.2 Recovery of the ADS superpotential 11
3.3 Absence of exotic contributions 14
3.4 Study of the back-reaction 16
4. The N = 1 Z2 × Z2 orientifold 17
4.1 Instanton sector 19
5. An N = 2 example: the Z3 orientifold 21
5.1 Instanton sector 24
6. Conclusions 25
1. Introduction
It has long been realized that instantons in string theory are often in close correspon-
dence with instantons in gauge theories [1, 2, 3, 4, 5, 6]. Recently it was found that in
some situations stringy instantons can dynamically generate some terms which from a
low-energy effective point of view enter as ordinary external couplings in the superpo-
tential of gauge theories living on space-filling branes [7, 8, 9, 10, 11, 12, 13, 14]. By
instantons in string theory we generally mean instantons which are geometrically real-
ized as Euclidean extended objects wrapped on some non-trivial cycles of the geometry.
Thus, in a sense, a stringy instanton has a “life of its own”, not requiring an underlying
gauge theory. This opens up the possibility of having contributions originating from
instantons that do not admit a standard gauge theory realization. We shall refer to
these instantons as exotic.
There has been some debate in the recent literature about the instances where
such exotic instantons can actually contribute to the gauge theory superpotential in a
non-trivial manner. In this work we will contribute to such a debate by considering
backgrounds where a simple CFT description is possible, such as orbifolds or orientifolds
thereof.
We present various simple examples of what we believe to be a rather generic
situation. Namely, the presence of extra zero-modes for these instantons, in addition to
those required by the counting of broken symmetries, makes some of their contributions
vanish. Such extra zero-modes should not come as a surprise, since a D-brane instanton
in a CY manifold breaks a total of four out of eight supercharges, i.e. it has two extra
fermionic zero-modes from the point of view of holomorphic N = 1 gauge theory
quantities. We give some arguments as to why the backreaction of the space-filling
branes on the geometry might not help in lifting these extra zero-modes. We further
argue that only more radical changes of the background, such as the introduction of
fluxes, deformations of the CY geometry or the introduction of orientifold planes, can
remove these zero-modes. When this happens, exotic instantons do contribute to the
gauge theory superpotential and may provide qualitative changes in the low energy
effective dynamics, as for instance the stabilization of otherwise runaway directions.
We will be interested in Euclidean D-branes in type II theories. We will work
with IIB fractional branes at orbifold and orientifold singularities rather than type IIA
wrapped branes. The motivation for this choice of setting is two-fold. First, recent
advances in the gauge/gravity correspondence require the study of exotic instantons,
whose effects tend to stabilize the gauge theory rather than unstabilize it [15, 16, 9, 17],
and the gauge/gravity correspondence is more naturally defined in the context of IIB
theory. Second, similar effects are used in string phenomenology to try to understand
possible mechanisms for neutrino masses [7, 8, 13]. This latest activity is mainly done
in the type IIA scenario, but we find it easier to address some subtle issues in the IIB
orbifold case.
While working in an exact string background, our considerations will nonetheless be
only local, i.e. we will not be concerned with global issues such as tadpole cancellation
that arise in proper compactifications. This is perfectly acceptable in the context of the
gauge/gravity correspondence where the internal manifold is non-compact but, even for
string phenomenology, the results we obtain stand (locally) when properly embedded
in a consistent compactification.
The paper is organized as follows: In section 2 we set up the notation and discuss
some preliminary material. In section 3 we discuss our first case, namely the N = 1
Z2 × Z2 orbifold. After briefly recovering the usual instanton generated corrections to
the superpotential we discuss the possible presence of additional exotic contributions
and find that they are not present because of the additional zero-modes. We conclude
by giving a CFT argument on why such zero-modes are not expected to be lifted
even by taking into account the backreaction of the D-branes, unless one is willing
to move out the orbifold point in the CY moduli space. Sections 4 and 5 present
two separate instances where exotic contributions are present after having removed the
extra zero-modes by orientifolding. The first is an N = 1 orientifold, the second is an
N = 2 orientifold, displaying corrections to the superpotential and the prepotential,
respectively. We end with some conclusions and a discussion of further developments.
2. Preliminaries
In this section we briefly review the generic setup in the well understoodN = 4 situation
in order to introduce the notation for the various fields and moduli and their couplings.
The more interesting theories we will consider next will be suitable projections of the
N = 4 theory. In fact, the exotic cases can all be reduced to orbifolds/orientifolds
of this master case once the appropriate projections on the Chan-Paton factors are
performed.
Since we are interested in instanton physics (for comprehensive reviews see [18] and
the recent [19]) we will take the ten dimensional metric to be Euclidean. We consider
a system where both D3-branes and D(−1)-branes (D-instantons) are present. To be
definite, we take N D3’s and k D-instantons 1.
Quite generically we can distinguish three separate open string sectors:
• The gauge sector, made of those open strings with both ends on a D3-brane. We
assume the brane world-volumes are lying along the first four coordinates xµ and
are orthogonal to the last six xa. The massless fields in this sector form an N = 4
SYM multiplet [22]. We denote the bosonic components by Aµ and X
a. Written
in N = 1 language this multiplet is formed by a gauge superfield whose field
strength is denoted by Wα and three chiral superfields Φ
1,2,3. With a slight abuse
of notation, the bosonic components of the chiral superfields will also be denoted
by Φ, i.e. Φ1 = X4+ iX5 and so on. In N = 2 language we have instead a gauge
superfield A and a hypermultiplet H , all in the adjoint representation. The low
energy action of these fields is a four dimensional N = 4 gauge theory. All these
fields are N ×N matrices for a gauge group SU(N).
1These D3/D(−1) brane systems (and their orbifold projections) are very useful and efficient in
studying instanton effects from a stringy perspective even in the presence of non-trivial closed string
backgrounds, both of NS-NS type [20] and of R-R type [21].
• The neutral sector, which comprises the zero-modes of strings with both ends
on the D-instantons. It is usually referred to as the neutral sector because these
modes do not transform under the gauge group. The zero-modes are easily ob-
tained by dimensionally reducing the maximally supersymmetric gauge theory to
zero dimensions. We will use an ADHM [23] inspired notation [5, 6]. We denote
the bosonic fields as aµ and χ
a, where the distinction between the two is made by
the presence of the D3-branes. The fermionic zero-modes are denoted by MαA
and λα̇A, where α and α̇ denote the (positive and negative) four dimensional chi-
ralities and A is an SU(4) (fundamental or anti-fundamental) index denoting the
chirality in the transverse six dimensions. The ten dimensional chirality of both
fields is taken to be negative. In Euclidean space M and λ must be treated as
independent. When needed, we will also introduce the triplet of auxiliary fields
Dc, directly analogous to the four dimensional D, that can be used to express the
various interactions in an easier form as we will see momentarily. All these fields
are k × k matrices where k is the instanton number.
• The charged sector, comprising the zero-modes of strings stretching between a
D3-brane and a D-instanton. For each pair of such branes we have two conjugate
sectors distinguished by the orientation of the string. In the NS sector, where the
world-sheet fermions have opposite modding as the bosons, we obtain a bosonic
spinor ωα̇ in the first four directions where the GSO projection picks out the neg-
ative chirality. In the conjugate sector, we will get an independent bosonic spinor
ω̄α̇ of the same chirality. Similarly, in the R sector, after the GSO projection
we obtain a pair of independent fermions (one for each conjugate sector) both
in the fundamental of SU(4) which we denote by µA and µ̄A. These fields are
rectangular matrices N × k and k ×N .
The couplings of the fields in the gauge sector give rise to a four dimensional gauge
theory. The instanton corrections to such a theory are obtained by constructing the
Lagrangian describing the interaction of the gauge sector with the charged sector zero-
modes while performing the integral over all zero-modes, both charged and neutral. A
crucial point to notice and which will be important later is that while the neutral modes
do not transform under the gauge group, their presence affects the integral because of
their coupling to the charged sector.
The part of the interaction involving only the instanton moduli is well known from
the ADHM construction and it is essentially the reduction of the interacting gauge
Lagrangian for these modes in a specific limit where the Yukawa terms for λ and the
quadratic term for D are scaled out (see [18, 6] for details). The final form of this part
of the interaction is:
S1 = tr
− [aµ, χ
+ χaω̄α̇ω
α̇χa +
(Σ̄a)ABµ̄
AµBχa −
(Σ̄a)ABM
αA[χa,M
µ̄Aωα̇ + ω̄α̇µ
A + σ
βα̇[M
βA, aµ]
λα̇A − iD
ω̄α̇(τ c)
α̇ωβ̇ + iη̄
µν [a
µ, aν ]
(2.1)
where the sum over colors and instanton indices is understood. τ denotes the usual
Pauli matrices, η̄ (and η) the ’t Hooft symbols and Σ̄ (and Σ) are used to construct
the six-dimensional gamma-matrices
Σ̄a 0
. (2.2)
The above interactions can all be understood in terms of string diagrams on a disk with
open string vertex operators inserted at the boundary in the α′ → 0 limit.
The interaction of the charged sector with the scalars of the gauge sector can be
worked out in a similar way and yields
S2 = tr
ω̄α̇X
(Σ̄a)ABµ̄
. (2.3)
Let us rewrite the above action in a way which will be more illuminating in the following
sections. Since we will be mainly focusing on situations where we have N = 1 super-
symmetry, it is useful to write explicitly all indices in SU(4) notation, and then break
them into SU(3) representations. We thus write the six scalars Xa as the antisymmetric
representation of SU(4) as follows
XAB = −XBA ≡ (Σ̄
a)ABXa . (2.4)
The action S2 then reads
S2 = tr
ǫABCDω̄α̇XABXCDω
µ̄AXABµ
. (2.5)
Splitting now the indices A into i = 1 . . . 3 and 4, we can identify Φ
i ≡ Xi4 in the 3̄ of
SU(3) and Φi ≡ 1
ǫijkXjk in the 3 of SU(3). Thus we can rewrite the action (2.5) as
S2 = tr
ωα̇ +
ǫijkµ̄
iΦjµk
. (2.6)
In the above form, it is clear which zero-modes couple to the holomorphic superfields
and which others couple to the anti-holomorphic ones. This distinction will play an
important role later.
The main object of our investigation is the integral of e−S1−S2 over all moduli
Z = C
d{a, χ,M, λ,D, ω, ω̄, µ, µ̄} e−S1−S2 , (2.7)
where we have lumped all field independent normalization constants (including the
instanton classical action and the appropriate powers of α′ required by dimensional
analysis) into an overall coefficient C. There are, of course, other interactions involving
the fermions and the gauge bosons but, as far as the determination of the holomorphic
quantities are concerned, they can be obtained from the previous ones and supersym-
metry arguments. For example, a term in the superpotential is written as the integral
over chiral superspace
dx4dθ2 of a holomorphic function of the chiral superfields, but
such a function is completely specified by its value for bosonic arguments at θ = 0.
Thus, if we can “factor out” a term
dx4dθ2 from the moduli integral (2.7), whatever
is left will define the complex function to be used in the superpotential and similarly
for the prepotential in the N = 2 case if we succeed in factoring out an integral over
N = 2 chiral superspace
dx4dθ4.
The coordinates x and θ must of course come from the (super)translations bro-
ken by the instanton and they will be associated to the center of mass motion of the
D-instanton, namely, xµ = tr aµ and θαA = trMαA for some values of A.2 One must
pay attention however to the presence of possible additional neutral zero-modes coming
either from the traceless parts of the above moduli or from the fields λ and χ. These
modes must also be integrated over in (2.7) and their effects, as we shall see, can be
quite dramatic. In particular, the presence of λ in some instances is crucial for the
implementation of the usual ADHM fermionic constraints whereas in other circum-
stances it makes the whole contribution to the superpotential vanish. These extra λ
zero-modes are ubiquitous in orbifold theories and generically make it difficult to obtain
exotic instanton corrections for these models. As we shall see, they can however be
easily projected out by an orientifold construction making the derivation of such terms
possible.
In the full expression for the instanton corrections there will also be a field-inde-
pendent normalization factor coming from the one-loop string diagrams and giving for
instance the proper gYM dependence in the case of the usual instanton corrections.
In this paper we will only focus on the integral over the zero-modes, which gives the
proper field-dependence, referring the reader to [10, 11] for a discussion of these other
issues.
2Obviously, for the case of an anti-instanton, the roles of M and λ are reversed.
3. The N = 1 Z2 × Z2 orbifold
In order to present a concrete example of the above discussion, let us study a simple
C3/Z2 × Z2 orbifold singularity. The resulting N = 1 theory is a non-chiral four-node
quiver gauge theory with matter in the bi-fundamental. Non-chirality implies that the
four gauge group ranks can be chosen independently [24]. This corresponds to being
able to find a basis of three independent fractional branes in the geometry (for a review
on fractional branes on orbifolds see e.g. [25]).
The field content can be conveniently summarized in a quiver diagram, see Fig. 1,
which, together with the cubic superpotential
W = Φ12Φ23Φ31 − Φ13Φ32Φ21 + Φ13Φ34Φ41 − Φ14Φ43Φ31
+Φ14Φ42Φ21 − Φ12Φ24Φ41 + Φ24Φ43Φ32 − Φ23Φ34Φ42 , (3.1)
uniquely specifies the theory.
SU(N  ) SU(N  )
SU(N  )SU(N  )
Figure 1: Quiver diagram for the Z2 × Z2 orbifold theory. Round circles correspond to
SU(Nℓ) gauge factors while the lines connecting quiver nodes represent the bi-fundamental
chiral superfields Φℓm.
A stack of N regular D3-branes amounts to having one and the same rank assign-
ment on the quiver. The gauge group is then SU(N)4 and the theory is anN = 1 SCFT.
Fractional branes correspond instead to different (but anomlay free) rank assignments.
Quite generically, fractional branes can be divided into three different classes, depend-
ing on the IR dynamics they trigger [26]. The non-chiral nature and the particularly
symmetric structure of the orbifold under consideration allows one to easily construct
any such instance of fractional brane class.
If we turn on a single node, we are left with a pure SU(N) SYM gauge theory,
with no matter fields and no superpotential. This theory is believed to confine. The
geometric dual effect is that the corresponding fractional brane leads to a geometric
transition where the branes disappear leaving behind a deformed geometry. Indeed,
there is one such deformation in the above singularity.
Turning on two nodes leads already to more varied phenomena. There are now
two bi-fundamental superfields, but still no tree level superpotential. Thus, the system
is just like two coupled massless SQCD theories or, by a slightly asymmetric point of
view, massless SQCD with a gauged diagonal flavor group. The low-energy behavior
depends on the relative ranks of the two nodes.
If the ranks are different, the node with the highest rank is in a situation where it
has less flavors than colors. Then an Affleck-Dine-Seiberg (ADS) superpotential [27, 28]
should be dynamically generated, leading eventually to a runaway behavior. This set
up of fractional branes is sometimes referred to as supersymmetry breaking fractional
branes [29, 26, 30].
If the ranks are the same we are in a situation similar to Nf = Nc SQCD for both
nodes. Hence we expect to have a moduli space of SUSY vacua, which gets deformed,
but not lifted, at the quantum level. This moduli space is roughly identified in the
geometry with the fact that the relevant fractional branes are interpreted as D5-branes
wrapped on the 2-cycle of a singularity which is locally C× (C2/Z2). Such a fractional
brane can move in the C direction. This is what is called an N = 2 fractional brane
since, at least geometrically, it resembles very much the situation of fractional branes
at N = 2 singularities.
In what follows we use the two-node example as a simple setting in which we can
analyze the subtleties involved in the integration over the neutral modes. For the gauge
theory instanton case it is known that there are extra neutral fermionic zero-modes in
addition to those required to generate the superpotential. Their integration allows to
recover the fermionic ADHM constraints on the moduli space of the usual field theory
instantons. For such instantons, we will be able to obtain the ADS superpotential
and corresponding runaway behavior in the familiar context with Nc and Nf fractional
branes at the respective nodes, for Nf = Nc−1. On the other hand, we will argue that
the presence of such extra zero-modes rules out the possibility of having exotic instanton
effects, such as terms involving baryonic operators in the Nf = Nc case. It was the
desire to study such possible contributions that constituted the original motivation for
this investigation. We will first show that such effects are absent for this theory as it
stands, and we will later discuss when and how this problem can be cured.3
3In a situation where the CFT description is less under control than in the setting discussed in
the present paper, it has been argued in [17] that such baryonic couplings do arise in the context of
fractional branes on orbifolds of the conifold, possibly at the expense of introducing O-planes. Also
in a IIA set up similar to the ones of [7, 8, 10, 11, 13] it seems reasonable that one can wrap an
ED2-brane along an O6-plane and produce such couplings on other intersecting D6-branes.
Our orbifold theory can be easily obtained as an orbifold projection of N = 4 SYM.
The orbifolding procedure and the derivation of the superpotential (3.1) are by now
standard. We briefly recall the main points in order to fix the notation and because
some of the details will be useful later in describing the instantons in such a set up.
The group Z2 × Z2 has four elements: the identity e, the generators of the two Z2
that we denote with g1 and g2 and their product, denoted by g3 = g1g2. If we introduce
complex coordinates (z1, z2, z3) ∈ C
z1 = x4 + ix5 , z2 = x6 + ix7 , z3 = x8 + ix9 (3.2)
the action of the orbifold group can be defined as in Table 1.
z1 z2 z3
e z1 z2 z3
1 −z2 −z3
g2 −z
1 z2 −z3
g3 −z
1 −z2 z3
Table 1: The action of the orbifold generators.
Let γ(g) be the regular representation of the orbifold group on the Chan-Paton
factors. If the orbifold is abelian, as always in the cases we shall be interested in, we
can always diagonalize all matrices γ(g). We will assume that the two generators have
the following matrix representation
γ(g1) = σ3 ⊗ 1 =
1 0 0 0
0 1 0 0
0 0 −1 0
0 0 0 −1
, γ(g2) = 1⊗ σ3 =
1 0 0 0
0 −1 0 0
0 0 1 0
0 0 0 −1
(3.3)
where the 1’s denote Nℓ ×Nℓ unit matrices (ℓ = 1, ..., 4). Then, the orbifold projection
amounts to enforcing the conditions
Aµ = γ(g)Aµγ(g)
−1 , Φi = ±γ(g)Φiγ(g)−1 (3.4)
where the sign ± must be chosen according to the action of the orbifold generators
g that can be read off from Table 1. With the choice (3.3), the vector superfields
are block diagonal matrices of different size (N1, N2, N3, N4), one for each node of the
quiver, while the three chiral superfields Φi have the following form [24]
0 × 0 0
× 0 0 0
0 0 0 ×
0 0 × 0
, Φ2 =
0 0 × 0
0 0 0 ×
× 0 0 0
0 × 0 0
, Φ3 =
0 0 0 ×
0 0 × 0
0 × 0 0
× 0 0 0
, (3.5)
where the crosses represent the non-zero entries Φℓm appearing in the superpotential
(3.1).
3.1 Instanton sector
Now consider D-instantons in the above set up. Such instantons preserve half of the
4 supercharges preserved by the system of D3-branes plus orbifold. In this respect
recall that the fractional branes preserve exactly the same supercharges as the regular
branes.4 Using the N = 4 construction of the previous section and the structure of the
orbifold presented in eq. (3.5), we now proceed in describing the zero-modes for such
instantons.
The neutral sector is very similar to the gauge sector. Indeed, in the (−1) su-
perghost picture, the vertex operators for such strings will be exactly the same, except
for the eip·X factor which is absent for the instanton. The Chan-Paton structure will
also be the same, so that the same pattern of fractional D-instantons will arise as for
the fractional D3-branes. In particular, the only regular D-instanton (which could be
thought of as deriving from the one of N = 4 SYM) is the one with rank (instanton
number) one at every node. All other situations can be thought of as fractional D-
instantons, which can be interpreted as Euclidean D1-branes wrapped on the two-cycles
at the singularity, ED1 for short. Generically, we can then characterize an instanton
configuration in our orbifold by (k1, k2, k3, k4).
Following the notation introduced in section 2, the bosonic modes will comprise a
4×4 block diagonal matrix aµ, and six more matrix fields χ1, . . . χ6, that can be paired
into three complex matrix fields χ1+iχ2, χ3+iχ4, χ5+iχ6, having the same structure as
(3.5) but now where each block entry is a kℓ×km matrix. On the fermionic zero-modes
MαA and λα̇A (also matrices) the orbifold projection enforces the conditions
MαA = R(g)AB γ(g)M
αBγ(g)−1 , λα̇A = γ(g)λα̇Bγ(g)
−1R(g)BA (3.6)
4There is another Euclidean brane which preserves two supercharges, namely the Euclidean (anti)
D3-branes orthogonal to the 4 dimensions of space-time. We will be considering here only the D-
instantons, leaving the complete analysis of the other effects to future work. In this context, note that
the extended brane instantons would have an infinite action (and thus a vanishing contribution) in
the strict non-compact set up we are using here.
where R(g) is the orbifold action of Table 1 in the spinor representation which can be
chosen as
R(g1) = −Γ
6789 , R(g2) = −Γ
4589 . (3.7)
It is easy to find an explicit representation of the Dirac matrices such thatMαA and λα̇A
for A = 1, 2, 3 also have the structure of (3.5) while for A = 4 they are block diagonal.
Equivalently, one could write the spinor indices in the internal space in terms of the
three SO(2) charges associated to the embedding SO(2) × SO(2) × SO(2) ⊂ SO(6) ≃
SU(4)
Mα−++ =Mα1 , Mα+−+ =Mα2 , Mα++− =Mα3 , Mα−−− =Mα4 ,
λα̇+−− = λα̇1 , λα̇−+− = λα̇2 , λα̇−−+ = λα̇3 , λα̇+++ = λα̇4 . (3.8)
The most notable difference between the neutral sector and the gauge theory on the
D3-branes is that, whereas in the four-dimensional theory the U(1) gauge factors are
rendered massive by a generalization of the Green-Schwarz mechanism and do not
appear in the low energy action, for the instanton they are in fact present and enter
crucially into the dynamics.
Let us finally turn to the charged sector, describing strings going from the instan-
tons to the D3-branes. The analysis of the spectrum and the action of the orbifold
group on the Chan-Paton factors show, in particular, that the bosonic zero-modes are
diagonal in the gauge factors. There are four block diagonal matrices of bosonic zero-
modes ωα̇, ω̄α̇ with entries Nℓ×kℓ and kℓ×Nℓ respectively and eight fermionic matrices
µA, µ̄A with entries Nℓ × km and km × Nℓ, that again display the same structure as
above – same as (3.5) for A = 1, 2, 3 and diagonal for A = 4.
3.2 Recovery of the ADS superpotential
The measure on the moduli space of the instantons and the ADHM constraints are
simply obtained by inserting the above expressions into the moduli integral (2.7). If
one chooses some of the Nℓ or kℓ to vanish one can deduce immediately from the
structure of the projection which modes will survive and which will not.
As a consistency check, one can try to reproduce the ADS correction to the super-
potential [27, 28] for the theory with two nodes. Take fractional branes corresponding
to a rank assignment (Nc, Nf , 0, 0), and consider the effect of a ED1 corresponding to
instanton numbers (1, 0, 0, 0).
The only chiral fields present are the two components of Φ1 connecting the first
and second node
0 Q 0 0
Q̃ 0 0 0
0 0 0 0
0 0 0 0
. (3.9)
Since the instanton is sitting only at one node, all off diagonal neutral modes are absent,
as they connect instantons at two distinct nodes. Thus, the only massless modes present
in the neutral sector are four bosons xµ, denoting the upper-left component of aµ, two
fermions θα denoting the upper-left component of Mα4 and two more fermions λα̇
denoting the upper-left component of λα̇4. We have identified the non zero entries of
aµ and Mα4 with the super-coordinates xµ and θα since they precisely correspond to
the Goldstone modes of the super-translation symmetries broken by the instanton and
do not appear in S1+ S2 (cfr. (2.1) and (2.3)). Their integration produces the integral
over space-time and half of Grassmann space which precedes the superpotential term
to which the instanton contributes. On the contrary, λα̇ appears in S1 and when it is
integrated it yields the fermionic ADHM constraint.
In the charged sector, we have bosonic zero-modes ωuα̇ and ω̄α̇u, with u an index
in the fundamental or anti-fundamental of SU(Nc). In addition, there are fermionic
zero-modes µu and µ̄u with indices in SU(Nc), together with additional fermionic zero-
modes µ′f and µ̄′f where the index f is now in the fundamental or anti-fundamental of
SU(Nf ).
5 Note that the µ zero-modes carry an SU(4) index 4 (being on the diagonal)
while the µ′ zero-modes carry an SU(4) index 1, since they are of the same form as Φ1.
All this can be conveniently summarized in a generalized quiver diagram as rep-
resented in Fig. 2, which accounts for both the brane configuration and the instanton
zero-modes.
For a single instanton, the action (2.1) greatly simplifies since many fields are
vanishing as well as all commutators and one gets
S1 = i (µ̄uω
α̇ + ω̄α̇uµ
u) λα̇ − iDcω̄α̇u(τ
. (3.10)
Similarly, the coupling of the charged modes to the chiral superfield can be expressed
by writing eq. (2.3) as
ω̄α̇u
v + Q̃
ωα̇v −
µ̄uQ̃
µ̄′fQ
u . (3.11)
Note that it is the anti-holomorphic superfields that enter in the couplings with the
fermionic zero-modes, as is clear by comparing with (2.6). The above action is exactly
the same which appears in the ADHM construction as reviewed in [18].
5Recall that the bosonic zero-modes are diagonal in the gauge factors; therefore there are no ω
and ω̄α̇f zero-modes.
SU(N  ) SU(N  )f
Figure 2: Quiver diagram describing an ordinary instanton in a SU(Nc) × SU(Nf ) theory.
Gauge theory nodes are represented by round circles, instanton nodes by squares. The ED1
is wrapped on the same cycle as the color branes. All zero-modes are included except the θ’s
and the xµ’s, which only contribute to the measure for the integral over chiral superspace.
We are now ready to perform the integral (2.7) over all the existing zero-modes.
Writing
dx4dθ2W , (3.12)
we see that the instanton induced superpotential is
W = C
d{λ,D, ω, ω̄, µ, µ̄} e−S1−S2 . (3.13)
The integrals over D and λ enforce the bosonic and fermionic ADHM constraints,
respectively. Thus
W = C
d{ω, ω̄, µ, µ̄} δ(µ̄uω
α̇ + ω̄α̇uµ
u) δ(ω̄α̇u(τ
) e−S2 . (3.14)
We essentially arrive at the point of having to evaluate an integral over a set of zero-
modes which is exactly the same as the one discussed in detail in the literature, e.g. [18].
We thus quickly go to the result referring the reader to the above review for further
details. First of all, it is easy to see that, due to the presence of extra µ modes in
the integrand from the fermionic delta function, only when Nf = Nc − 1 we obtain
a non-vanishing result. After having integrated over the µ and µ′, we are left with
a (constrained) gaussian integration that can be performed e.g. by going to a region
of the moduli space where the chiral fields are diagonal, up to a row/column of ze-
roes. Furthermore, the D-terms in the gauge sector constrain the quark superfields
to obey QQ† = Q̃†Q̃, so that the bosonic integration brings the square of a simple
determinant in the denominator. The last fermionic integration conspires to cancel the
anti-holomorphic contributions and gives
WADS =
Λ2Nc+1
det(Q̃Q)
, (3.15)
which is just the expected ADS superpotential for Nf = Nc − 1, the only case where
such non-perturbative contribution is generated by a genuine one-instanton effect and
not by gaugino condensation. In (3.15) Λ is the SQCD strong coupling scale that is
reconstructed by the combination of e−8π
2/g2 coming from the instanton action with
various dimensional factors coming from the normalization of the instanton measure
[18].
3.3 Absence of exotic contributions
Until now, we have reproduced from stringy considerations the effect that is supposed
to be generated also by instantons in the gauge theory. Considering a slightly different
set up, we would like to study the possibility of generating other terms.
Let us consider a system with rank assignment (Nc, Nf , 0, 0), as before, but frac-
tional instanton numbers (0, 0, 1, 0). In other words, we study the effect of a single
fractional instanton sitting on an unoccupied node of the gauge theory. The quiver
diagram, with the relevant zero-modes structure, is given in Fig. 3.
The neutral zero-modes of the instanton sector are the same as before. This is
because the quantization of this sector does not know the whereabouts of the D3-
branes and thus all nodes are equivalent, in this respect. In the mixed sector, we have
no bosonic zero-modes now, since the ω and ω̄ are diagonal. Note that, although we
always have four mixed (ND) boundary conditions, due to the quiver structure induced
by the orbifold, here we effectively realize the same situation one has when there are
eight ND directions, namely that the bosonic sector of the charged moduli is empty.
On the other hand, there are fermionic zero-modes µu, µ̄u, µ
′f and µ̄′f , as in the
previous case. Note that despite having the same name, these zero-modes correspond
actually to different Chan-Paton matrix elements with respect to the previous ones,
the difference being in the instanton index that is not written explicitly. In particular
we can think of µ and µ′ as carrying an SU(4) index 2 and 3 respectively.
Because of the absence of bosonic charged modes, the action (2.1) is identically
SU(N  )fSU(N  )
Figure 3: Quiver diagram describing an exotic instanton in a SU(Nc) × SU(Nf ) theory.
Gauge theory nodes are represented by round circles, instanton nodes by squares. The ED1
is wrapped on a different cycle with respect to both sets of quiver branes.
zero and the action (2.3) contains only the last term:
S1 = 0
µ̄′fQ̃
u. (3.16)
Note that in this case it is the holomorphic superfields which appear above, as is clear
from (2.6) and from noticing that the diagonal fermionic zero-mode µ4 is not present.
We are thus led to consider
W = C
d{λ,D, µ, µ̄} e−S2 . (3.17)
One notices right away that the integral over the charged modes is non vanishing
(only) for the case Nf = Nc and gives a tantalizing contribution proportional to BB̃,
where B = detQ and B̃ = det Q̃ are the baryon fields of the theory. However, we
must carefully analyze the integration over the remaining zero-modes of the neutral
sector. Now neither D nor λ appear in the integrand. The integral over D does not
raise any concern: it is, after all, an auxiliary field and its disappearance from the
integrand is due to the peculiarities of the ADHM limit. Before taking this limit, D
appeared quadratically in the action and could be integrated out, leaving an overall
normalization constant. The integral over λ is another issue. In this case, λ is absent
from the integrand even before taking the ADHM limit and its integration multiplies
the above result by zero, making the overall contribution of such instantons to the
superpotential vanishing. Of course, the presence of such extra zero-modes should not
come as a surprise since they correspond to the two extra broken supersymmetries of
an instanton on a CY.
Therefore we see that the neutral zero-modes contribution, in the exotic instanton
case, plays a dramatic role and conspires to make everything vanishing (as opposite
to the ADS case analyzed before). A natural question is to see whether these zero-
modes get lifted by some effect we have not taken into account, yet. For one thing,
supersymmetry arguments would make one think that taking into account the back-
reaction of the D3-branes might change things. However, in the following subsection
we show that this seems not to be the case.
3.4 Study of the back-reaction
Let us stick to the case Nf = Nc, which is the only one where the integral (3.17)
might give a non-vanishing contribution. In this case the fractional brane system
is nothing but a stack of (Nc) N = 2 fractional branes. These branes couple to
only one of the 3 closed string twisted sectors [24]. More specifically, they source
the metric hµν , the R-R four-form potential Cµνρσ and two twisted scalars b and c from
the NS-NS and R-R sector respectively. This means that the disk one-point function
of their vertex operators [31, 32] is non vanishing when the disk boundary is attached
to such D3-branes. (Indeed in this way or, equivalently, by using the boundary-state
formalism [33, 34], one can derive the profile for these fields.)
If the back-reaction of these fields on the instanton lifted the extra zero-modes
λ’s, this should be visible when computing the one point function of the corresponding
closed string vertex operators on a disk with insertions on this boundary of the vertex
operators for such moduli. To see whether such coupling is there, we first need to write
down the vertex operators for the λ’s in the (±1/2) superghost pictures. The vertex in
the (−1/2) picture is found e.g. in [6] and reads
λ (z) = λα̇AS
α̇(z)SA(z)e−φ(z)/2 , (3.18)
where Sα̇(z) and SA(z) are the spin-fields in the first four and last six directions re-
spectively. For our argument we need to focus on the SA(z) dependence. Since the
modulus that survives the orbifold projection is, with our conventions, λα̇4 = λα̇+++,
we write the corresponding spin-field as
S+++(z) = eiH1(z)/2eiH2(z)/2eiH3(z)/2, (3.19)
where Hi(z) is the free boson used to bosonize the fermionic sector in the i-th complex
direction: ψi(z) = eiHi(z). The vertex operator in the +1/2 picture can be obtained by
applying the picture-changing operator to (3.18)
λ (z) = [QBRST, ξV
λ (z)] . (3.20)
The crucial part in QBRST is [31]
QBRST =
ψµ∂Xµ + ψ̄i∂Z i + ψi∂Z̄ i
+ . . . (3.21)
Because of the nature of the supercurrent, we see that (3.21) flips at most one sign in
(3.19), hence the product V
λ will always carry an unbalanced charge in some
of the three internal SO(2) groups. On the other hand, the vertex operators for the
fields sourced by the fractional D3’s cannot compensate such an unbalance. Hence, their
correlation function on the D-instanton with the insertion of V
λ carries a charge
unbalance and therefore vanishes. Therefore, at least within the above perturbative
approach, the neutral zero-modes seem not to get lifted by the back-reaction of the
D3-branes.
One might consider some additional ingredients which could provide the lifting. A
natural guess would be moving in the CY moduli space or adding suitable background
fluxes [35, 36]. There are indeed non-vanishing background fields at the orbifold point,
i.e. the b fields of the twisted sectors which the N = 2 fractional branes do not
couple to. These fields, however, being not associated to geometric deformations of
the internal space should be described by a CFT vertex operator uncharged under the
SO(2)’s, simply because of Lorentz invariance in the internal space. Therefore, the
only way to get an effective mass term for the zero-modes λ would be to move out of
the orbifold point in the CY moduli space. Indeed, the other moduli of the NS-NS
twisted sector, being associated to geometric blow-ups of the singularity, are charged
under (some of) the internal SO(2)’s and can have a non vanishing coupling with the
λ’s. More generically, complicated closed string background fluxes might be suitable.
This is an interesting option which however we do not pursue here, since we want to
stick to situations where a CFT description is available.
A more radical thing to do is to remove the zero-modes from the very start, for
instance by means of an orientifold projection [37, 38]. This is the option we are going
to consider in the remainder of this work.
4. The N = 1 Z2 × Z2 orientifold
In this section we supplement our orbifold background by an O3 orientifold and show
that in this case exotic instanton contributions do arise and provide new terms in the
superpotential. We refer to e.g. [39, 40, 41] for a comprehensive discussion of N = 1
and N = 2 orientifolds.
The first ingredient we need is the action of the O3-plane on the various fields.
Denote by Ω the generator of the orientifold. The action of Ω on the vertex operators
for the various fields (ignoring for the time being the Chan-Paton factors) is well known.
The vertex operators for the bosonic fields on the D3-brane contain, in the 0 picture,
the following terms: Aµ ∼ ∂τx
µ and Φi ∼ ∂σz̄
i. They both change sign under Ω,
the first because of the derivative ∂τ and the second because the orientifold action for
the O3-plane is always accompanied by a simultaneous reflection of all the transverse
coordinates zi.
The action of the orientifold on the Chan-Paton factors is realized by means of
a matrix γ(Ω) which in presence of an orbifold must satisfy the following consistency
condition [39]
γ(g)γ(Ω)γ(g)T = + γ(Ω) (4.1)
for all orbifold generators g. This amounts to require that the orientifold projection
commutes with the orbifold projection. The matrix γ(Ω) can be either symmetric or
anti-symmetric. We choose to perform an anti-symmetric orientifold projection on the
D3 branes and denote the corresponding matrix by γ−(Ω). This requires having an
even number Nℓ of D3 branes on each node of the quiver so that we can write
γ−(Ω) =
ǫ1 0 0 0
0 ǫ2 0 0
0 0 ǫ3 0
0 0 0 ǫ4
(4.2)
where the ǫℓ’s are Nℓ × Nℓ antisymmetric matrices obeying ǫ
ℓ = −1. Using (3.3) and
(4.2) it is straightforward to verify that the consistency condition (4.1) is verified.
The field content of the stacks of fractional D3-branes in this orientifold model is
obtained by supplementing the orbifold conditions (3.4) with the orientifold ones
Aµ = −γ−(Ω)A
µγ−(Ω)
−1 , Φl = −γ−(Ω)Φ
lTγ−(Ω)
−1. (4.3)
This implies that Aµ = diag (A
µ) with A
µ = ǫℓA
µ ǫℓ. Thus, the result-
ing gauge theory is a USp(N1) × USp(N2) × USp(N3) × USp(N4) theory. The chiral
superfields, which after the orbifold have the structure (3.5), are such that the Φℓm
component joining the nodes ℓ and m of the quiver, must obey the orientifold condi-
tion Φℓm = ǫℓΦ
mℓǫm. In the following, we will take N3 = N4 = 0 so that we are left
with only two gauge groups and no tree level superpotential.
4.1 Instanton sector
Let us now consider the instanton sector, starting by analyzing the zero-mode content
in the neutral sector. There are two basic changes to the previous story. The first is
that the vertex operator for aµ is now proportional to ∂σx
µ, not to ∂τx
µ and it remains
invariant under Ω (the vertex operator for χa still changes sign). The second is that
the crucial consistency condition discussed in [38] requires that we now represent the
action of Ω on the Chan-Paton factors of the neutral modes by a symmetric matrix
which can be taken to be
γ+(Ω) =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
, (4.4)
where the 1’s are kℓ × kℓ unit matrices. The matrix aµ will be 4 × 4 block diagonal,
e.g. aµ = diag (a
µ), but now a
µ = a
µ . The most generic situation is to have
a configuration with instanton numbers (k1, k2, k3, k4). By considering a configuration
with k3 = 1 and k1 = k2 = k4 = 0, we can project out all bosonic zero-modes except
for the four components a3µ that we denote by xµ. The scalars χ
4 . . . χ9 are off-diagonal
and we shall not consider them further.
The nice surprise comes when considering the orientifold action on the fermionic
neutral zero-modes MαA and λα̇A. The orbifold part of the group acts on the spinor
indices as in (3.7), while the orientifold projection acts as the reflection in the transverse
space, namely
R(Ω) = −iΓ456789 (4.5)
Putting together the orbifold projections (3.6) with the orientifold ones
MαA = RAB(Ω)γ+(Ω)(M
αB)Tγ+(Ω)
−1 , λα̇A = γ+(Ω)(λα̇B)
Tγ+(Ω)
−1RBA(Ω) (4.6)
we can find the spectrum of surviving fermionic zero-modes. Using (4.4) and (4.5), it
is easy to see that (4.6) implies
MαA = (MαA)T , λα̇A = −(λα̇A)
T . (4.7)
Thus, for the simple case where k3 = 1 and k1 = k2 = k4 = 0, all λ’s are projected out
and only two chiral M zero-modes remain: Mα−−−, to be identified with the N = 1
chiral superspace coordinates θα.
Also the charged zero-modes are easy to discuss in this simple scenario. There are
no bosonic modes since the D-instanton and the D3-branes sit at different nodes while
the bosonic modes are necessarily diagonal. Most of the fermionic zero-modes µA and
µ̄A are also projected out by the orbifold condition
µA = R(g)ABγ(g)µ
Bγ(g)−1 , µ̄A = R(g)ABγ(g)µ̄
Bγ(g)−1 . (4.8)
Finally, the orientifold condition relates this time the fields in the conjugate sectors,
allowing one to express µ̄ as a linear combination of the µ
µ̄A = R(Ω)ABγ+(Ω)(µ
B)Tγ−(Ω)
−1 . (4.9)
The only charged modes surviving these projections can be expressed, in block 4 × 4
notation, as
0 0 µ13 0
0 0 0 0
0 0 0 0
0 0 0 0
, µ̄2 =
0 0 0 0
0 0 0 0
µ̄31 0 0 0
0 0 0 0
0 0 0 0
0 0 µ23 0
0 0 0 0
0 0 0 0
, µ̄3 =
0 0 0 0
0 0 0 0
0 µ̄32 0 0
0 0 0 0
, (4.10)
where the entries, to be thought of as column/row vectors in the fundamental/anti-
fundamental of SU(Nℓ) depending on their position, are such that µ̄31 = −µ
13ǫ1 and
µ̄32 = −µ
23ǫ2.
Thus, in the case where we have fractional D3 branes (N1, N2, 0, 0) and an exotic
instanton (0, 0, 1, 0), the only surviving chiral field is Φ12 ≡ ǫ1Φ
21ǫ2, the orientifold
projection eliminates the offending λ’s and we are left with just the neutral zero-modes
xµ and θ
α and the charged ones µ13 and µ23. This is summarized in the generalized
quiver of Fig. 4.
In this case the instanton partition function is
dx4dθ2 W (4.11)
where the superpotential W is
W = C
dµ e−S1−S2 = C
dµ13dµ23 e
iµT13ǫ1Φ12µ23 . (4.12)
This integral clearly vanishes unless N1 = N2, in which case we have
W ∝ det(Φ12) (4.13)
USP(N  )USP(N  )1 2
Φ Φ1221
2313µ
Figure 4: The generalized Z2 ×Z2 orientifold quiver and the exotic instanton contribution.
We thus see that exotic instanton corrections are possible in this simple model.6
It is interesting to note that the above correction is present in the same case
(N1 = N2 ≡ N) where the usual ADS superpotential for USp(N) is generated [42]
WADS =
Λ2N+3
det(Φ12)
(4.14)
and its presence stabilizes the runaway behavior and gives a theory with a non-trivial
moduli space of supersymmetric vacua given by det(Φ12) = const. Of course, the ADS
superpotential for this case can also be constructed along the same lines as section 3.2,
see e.g. [18]. In fact, this derivation is somewhat simpler than the one for the SU(N)
gauge group since there are no ADHM constraints at all in the one instanton case.
We think the above situation is not specific to the background we have been consid-
ering, but is in fact quite generic. As soon as the λ zero-modes are consistently lifted,
we expect the exotic instantons to contribute new superpotential terms. As a further
example, in the next section we will consider a N = 2 model, where exotic instantons
will turn out to contribute to the prepotential.
5. An N = 2 example: the Z3 orientifold
Let us now consider the quiver gauge theory obtained by placing an orientifold O3-plane
at a C × C2/Z3 orbifold singularity. In what follows we will use N = 1 superspace
notation. We first briefly repeat the steps that led to the constructions of such a quiver
6The gauge invariant quantity above can be rewritten as the Pfaffian of a suitably defined mesonic
matrix.
theory in the seminal paper [39]. Define ξ = e2πi/3 and let the generator of the orbifold
group act on the first two complex coordinates as
0 ξ−1
, (5.1)
while leaving the third one invariant. This preserves N = 2 SUSY. The action of the
generator g on the Chan-Paton factors is given by the matrix
γ(g) =
1 0 0
0 ξ 0
0 0 ξ2
 . (5.2)
The N = 2 theory obtained this way, summarized in Fig. 5, is a three node quiver
gauge theory with gauge groups SU(N1)× SU(N2)× SU(N3), supplemented by a cubic
superpotential which is nothing but the orbifold projection of the N = 4 superpotential
(its precise form is not relevant for the present purposes).
SU(N  ) SU(N  )
SU(N  )1
Figure 5: The Z3 (un-orientifolded) theory. The lines with both ends on a single node
represent adjoint chiral multiplets which, together with the vector multiplets at each node
constitute the N = 2 vector multiplets. Similarly, lines between nodes represent chiral mul-
tiplets which pair up into hyper-multiplets, in N = 2 language.
As for the action of Ω on the Chan-Paton factors, we choose again to perform the
symplectic projection on the D3-branes. To do so, we must take N1 to be even and
N2 = N3, so that we can write
γ−(Ω) =
ǫ 0 0
0 0 1
0 −1 0
 , (5.3)
where ǫ is a N1×N1 antisymmetric matrix obeying ǫ
2 = −1 and the 1’s denote N2×N2
identity matrices. The matrices γ(g) and γ−(Ω) satisfy the usual consistency condi-
tion [38, 39] as in (4.1).
The field content on the fractional D3-branes at the singularity will be given by
implementing the conditions
Aµ = γ(g)Aµγ(g)
−1 , Φi = ξ−iγ(g)Φiγ(g)−1 ,
Aµ = −γ−(Ω)A
µγ−(Ω)
−1 , Φi = −γ−(Ω)Φ
iTγ−(Ω)
−1 . (5.4)
The orbifold part of these conditions forces Aµ and Φ
3 to be 3 × 3 block diagonal
matrices, e.g. Aµ = diag (A
µ), while the orientifold imposes that A
µ = ǫA
and A2µ = −A
µ . The resulting gauge theory is thus a USp(N1)× SU(N2) theory. It is
convenient, however, to still denote A2µ and A
µ diagramatically as belonging to different
nodes with the understanding that these should be identified in the above sense.
The projection on the chiral fields can be done similarly and we obtain, denoting
by Φℓm the non-zero entries of the fields Φ
1 and Φ2 (only one can be non-zero for each
pair ℓm)
Φ12 = −ǫΦ
31, Φ13 = +ǫΦ
21, Φ23 = Φ
23, Φ32 = Φ
32 . (5.5)
The field content is summarized in Table 2.
USp(N1) SU(N2)
Φ12 � �
Φ21 � �
Φ13 � �
Φ31 � �
Φ23 · ��
Φ32 · ��
Table 2: Chiral fields making up the quiver gauge theory.
The theory we want to focus on in the following has rank assignment (N1, N2) =
(0, N). This yields an N = 2 SU(N) gauge theory with an hyper-multiplet in the sym-
metric/(conjugate)symmetric representation. We denote the N = 2 vector multiplet
by A whose field content in the block 3× 3 notation is thus
0 0 0
0 A 0
0 0 −AT
 . (5.6)
In what follows we will be interested in studying corrections to the prepotential F
coming from exotic instantons associated to the first node (the one that is not populated
by D3-branes). Let us then analyze the structure of the stringy instanton sector of the
present model, first.
5.1 Instanton sector
The most generic situation is to have a configuration with instanton numbers (k1, k2)
(later we will be mainly concerned with a configuration with instanton numbers (1, 0)).
Let us start analyzing the zero-modes content in neutral sector. The story is
pretty similar to the one discussed in the previous section. The vertex operator for aµ
is proportional to ∂σx
µ and so it remains invariant under Ω. The action on the Chan-
Paton factors of these D-instantons must now be represented by a symmetric matrix
which we take to be
γ+(Ω) =
1′ 0 0
0 0 1
0 1 0
 (5.7)
where 1′ is a k1 × k1 unit matrix and the 1’s are k2 × k2 unit matrices.
Because of the different orientifold projection, the matrices of bosonic zero-modes
behave slightly differently. The matrices aµ, χ
8 and χ9 will still be 3×3 block diagonal,
e.g. aµ = diag (a
µ), but now a
µ = a
µ and a
µ = a
µ whereas the same relations
for χ8 and χ9 will have an additional minus sign. The remaining fields χ4...7 are off
diagonal and we shall not consider them further since we will consider only the case of
one type of instanton. By considering a configuration with k1 = 1 and k2 = 0, we can
project out all bosonic zero-modes except for the four components a1µ that we denote
by xµ.
Let us now consider the orientifold action on the fermionic neutral zero-modesMαA
and λα̇A. The orbifold part of the group acts on the internal spinor indices as a rotation
R(g) = e
Γ45e−
Γ67 , (5.8)
while the orientifold acts through the matrix R(Ω) given in (4.5). The orbifold and
orientifold projections thus require
MαA = R(g)ABγ(g)M
αBγ(g)−1 , λα̇A = γ(g)λα̇Bγ(g)
−1R(g)BA , (5.9)
MαA = R(Ω)ABγ+(Ω)(M
αB)Tγ+(Ω)
−1 , λα̇A = γ+(Ω)(λα̇B)
Tγ+(Ω)
−1R(Ω)BA .
Using the explicit expressions for the various matrices, we see that, for the simple
case where k1 = 1 and k2 = 0, all λ’s are projected out and only four chiral M zero-
modes remain: Mα−−− and Mα++− to be identified with the N = 2 chiral superspace
coordinates θ1α and θ
α. Hence, also in this case the orientifold projection has cured the
problem encountered in section 3 (albeit in a N = 2 context now) and we can rest
assured that the integration over the charged modes will yield a contribution to the
prepotential.
Let us now move to the charged zero-modes sector. Just as in the previous model,
there are no bosonic modes since the D-instanton and the D3-branes sit at different
nodes while the bosonic modes are necessarily diagonal. Most of the fermionic zero-
modes µA and µ̄A are projected out by the orbifold condition which is formally the same
as in (4.8), while the orientifold condition relates the fields in the conjugate sectors,
giving µ̄ as a linear combination of the µ’s according to
µ̄A = R(Ω)ABγ+(Ω)(µ
B)Tγ−(Ω)
−1 . (5.10)
To summarize, the only charged modes surviving the projection can be expressed, in
block 3× 3 notation as
0 0 0
0 0 0
µ 0 0
 , µ̄1 =
0 µT 0
0 0 0
0 0 0
0 0 0
µ′ 0 0
0 0 0
 , µ̄2 =
0 0 −µ′T
0 0 0
0 0 0
 (5.11)
where the entries are to be thought of as column/row vectors in the fundamental/anti-
fundamental of SU(N) depending on their position.
As anticipated, the configuration we want to consider is a (0, N) fractional D3-
branes system together with an exotic (1, 0) instanton. The quiver structure, including
the relevant moduli, is depicted in Fig. 6. It is now easy to see that inserting the
expressions (5.6) and (5.11) into Eqs. (2.1), (2.3) and (2.7) we finally obtain
dx4dθ4F with F = C
dµdµ′ eiµ
TAµ′ ∝ detA . (5.12)
It would be interesting to study the potential implications of this result in the gauge
theory. There are many other simple models that could be analyzed along these lines.
6. Conclusions
In this paper we have presented some simple examples of what seem to be rather generic
phenomena in the context of string instanton physics. We paid particular attention to
Φ Φ3223
U(N) U(N)
Figure 6: The extended Z3 orientifold theory with (0, N) fractional D3-branes and (1, 0)
instanton number. The upper node (which would represent the USp(N1) gauge group and
disappears when we set N1 = 0 as in the case under consideration) is where the instanton
sits. The lower nodes denote only one gauge group. The charged fermionic zero-modes follow
Eq. (5.11). For simplicity we have not drawn the lines denoting the adjoint.
the study of the fermionic zero-modes and their effects on the holomorphic quantities
of the theory. We have seen both examples where the instanton contributions vanish
due to the presence of extra zero-modes and where they do not. In the second case, as
explicitly shown in a N = 1 example, exotic instantons can have a stabilizing effect on
the theory.
Although we have only considered some simple examples, we would like to stress
that these results are quite generic and can be carried over to all orbifold gauge theories.
A future direction would be to try to be more systematic and analyze the various
possibilities encountered in more complex N = 2 and N = 1 models. In a similar
spirit, one should analyze the multi-instanton contributions as well, since the total
correction to the holomorphic quantities will be the sum of all such terms. The study
of the zero-modes is expected to be even more relevant in this case as it will probably
make many contributions vanish. With an eye to string phenomenology, one should
also incorporate these models into globally consistent compactifications and study the
effects of these terms there.
Lastly, it would be interesting to study the dynamical implications of some of the
terms generated. We briefly touched upon this at the end of section 4 when we men-
tioned the stabilizing effect of the exotic instanton on the USp(N) theory. Although
from the strict field theory point of view these terms are thought of as ordinary polyno-
mial terms in the holomorphic quantities,7 they are “special” when seen from the point
of view of string theory and they might therefore induce a particular type of dynamics.
Acknowledgements
We would like to thank many people for discussions and email exchanges at various
stages of this work that helped us sharpen the focus of the presentation: M. Bianchi,
M. Billò, P. Di Vecchia, S. Franco, M. Frau, F. Fucito, S. Kachru, R. Marotta, L. Mar-
tucci, F. Morales, B. E. W. Nilsson, D. Persson, I. Pesando, D. Robles-Llana, R. Russo,
A. Tanzini, A. Tomasiello, A. Uranga, T. Weigand and N. Wyllard.
R.A., M.B. and A.L. are partially supported by the European Commission FP6
Programme MRTN-CT-2004-005104, in which R.A is associated to V.U. Brussel, M.B.
to University of Padova and A.L. to University of Torino. R.A. is a Research Associate
of the Fonds National de la Recherche Scientifique (Belgium). The research of R.A.
is also supported by IISN - Belgium (convention 4.4505.86) and by the “Interuniver-
sity Attraction Poles Programme –Belgian Science Policy”. M.B. is also supported by
Italian MIUR under contract PRIN-2005023102 and by a MIUR fellowship within the
program “Rientro dei Cervelli”. The research of G.F. is supported by the Swedish
Research Council (Vetenskapsr̊adet) contracts 622-2003-1124 and 621-2002-3884. A.L.
thanks the Galileo Galilei Institute for the hospitality and support during the comple-
tion of this work.
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