IFT-UAM/CSIC-07-12
CERN-PH-TH/2007-061
hep-th/yymmnnn
Instanton Induced Neutrino Majorana Masses
in CFT Orientifolds with MSSM-like spectra
L.E. Ibáñez1 A.N. Schellekens2 and A. M. Uranga3
1 Departamento de F́ısica Teórica C-XI and Instituto de F́ısica Teórica C-XVI,
Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
2 NIKHEF, Kruislaan 409, 1009DB Amsterdam, The Netherlands;
IMAPP, Radboud Universiteit, Nijmegen, The Netherlands;
Instituto de Matemáticas y F́ısica Fundamental, CSIC,
Serrano 123, Madrid 28006, Spain.
3 PH-TH Division, CERN CH-1211 Geneva 23, Switzerland
(On leave from IFT, Madrid, Spain)
Abstract
Recently it has been shown that string instanton effects may give rise to neutrino Majorana masses
in certain classes of semi-realistic string compactifications. In this paper we make a systematic search
for supersymmetric MSSM-like Type II Gepner orientifold constructions admitting boundary states
associated with instantons giving rise to neutrino Majorana masses and other L- and/or B-violating
operators. We analyze the zero mode structure of D-brane instantons on general type II orientifold
compactifications, and show that only instantons with O(1) symmetry can have just the two zero
modes required to contribute to the 4d superpotential. We however discuss how the addition of
fluxes and/or possible non-perturbative extensions of the orientifold compactifications would allow
also instantons with Sp(2) and U(1) symmetries to generate such superpotentials. In the context of
Gepner orientifolds with MSSM-like spectra, we find no models with O(1) instantons with just the
required zero modes to generate a neutrino mass superpotential. On the other hand we find a number
of models in one particular orientifold of the Gepner model (2, 4, 22, 22) with Sp(2) instantons with a
few extra uncharged non-chiral zero modes which could be easily lifted by the mentioned effects. A
few more orientifold examples are also found under less stringent constraints on the zero modes. This
class of Sp(2) instantons have the interesting property that R-parity conservation is automatic and
the flavour structure of the neutrino Majorana mass matrices has a simple factorized form.
http://arxiv.org/abs/0704.1079v2
1 Introduction
String Theory, as the leading candidate for a unified theory of Particle Physics and
Gravity, should be able to describe all observed particle phenomena. One the most
valuable experimental pieces of information obtained in the last decade concerns neu-
trino masses. Indeed the evidence from solar, atmospheric, reactor and accelerator
experiments indicates that neutrinos are massive. The simplest explanation of the
smallness of neutrino masses is the see-saw mechanism [1]. The SM gauge symmetry
allows for two types of operators bilinear on the neutrinos (with dimension ≤ 4) :
Lν = MabνaRνbR + habνaRH̄Lb (1.1)
where νR is the right-handed neutrino, L is the left-handed lepton doublet and H̄ is
the Higgs field. In supersymmetric theories, this term arise from a superpotential with
the above structure, upon replacing fields by chiral superfields. If Mab is large, the
lightest neutrino eigenvalues have masses
Mν = < H̄ >
2 hTM−1h (1.2)
For M ∼ 1010 − 1013 GeV and Dirac neutrino masses of order charged lepton masses,
the eigenvalues are consistent with experimental results.
What is the structure of neutrinos and their masses in string theory? In specific
compactifications giving rise to the MSSM spectra singlet fields corresponding to right-
handed neutrinos νR generically appear. Dirac neutrino masses are also generically
present but the required Majorana νR masses are absent. This is because most MSSM-
like models constructed to date have extra U(1) symmetries, under which the right-
handed neutrinos are charged, which hence forbid such masses. In many models, such
symmetries are associated to a U(1)B−L gauge boson beyond the SM. In order to
argue for the existence of νR masses, string model builders have searched for non-
renormalizable couplings of the type (νRνRN̄RN̄R) with extra singlets NR. Once the
latter fields get a vev, U(1)B−L is broken and a Majorana mass appears for the νR.
Although indeed such couplings (or similar ones with higher dimensions) exist in some
semi-realistic compactifications, such a solution to the neutrino mass problem in string
theory has two problems: 1) The typical νR masses so generated tend to be too small
due to the higher dimension of the involved operators and 2) The vevs for the NR fields
breaks spontaneously R-parity so that dimension 4 operators potentially giving rise to
fast proton decay are generated. This is in a nutshell the neutrino problem in string
compactifications (see [2] for a recent discussion in heterotic setups).
In [3] (see also [4]) two of the present authors pointed out that there is a built-
in mechanism in string theory which may naturally give rise to Majorana masses for
right-handed neutrinos. It was pointed out that string theory instantons may generate
such masses through operators of the general form
Mstring e
−U νRνR . (1.3)
Here U is a linear combination of closed string moduli whose imaginary part gets
shifted under a U(1)B−L gauge transformation in such a way that the operator is fully
gauge invariant. The exponential factor comes from the semi-classical contribution
of a certain class of string instantons. This a pure stringy effect distinct from the
familiar gauge instanton effects which give rise to couplings violating anomalous global
symmetries like (B+L) in the SM. Here also (B−L) (which is anomaly-free) is violated.
This operator is generated due to existence of instanton fermionic zero modes which
are charged under (B − L) and couple to the νR chiral superfield. Although the effect
can take place in different constructions, the most intuitive description may be obtained
for the case of Type IIA CY orientifold compactifications with background D6-branes
wrapping 3-cycles in the CY. In the simplest configurations one has four SM stacks
of D6-branes labeled a,b, c,d which correspond to U(3), SU(2) (or U(2)), U(1)R and
U(1)L gauge interactions respectively, which contain the SM group. One can construct
compactifications with the MSSM particle spectrum in which quarks and leptons lie
at the intersections of those SM D6-branes. Then the relevant instantons correspond
to euclidean D2-branes wrapping 3-cycles in the CY (satisfying specific properties so
as to lead to the appropriate superspace interaction). The D2-D6 intersections lie the
additional fermionic zero modes which are charged under (B−L). For instantons with
the appropriate number of intersections with the appropriate D6-branes, and with open
string disk couplings among the zero modes and the νR chiral multiplet (see fig.(3.1)),
the operator in (1.3) is generated.
The fact that the complex modulus U transforms under U(1)B−L gauge transfor-
mations indicates that the U(1)B−L gauge boson gets a mass from a Stückelberg term.
So a crucial ingredient in the mechanism to generate non-perturbative masses for the
νR’s is that there should be massless U(1)B−L gauge boson which become massive by a
Stückelberg term. It turns out that not many semi-realistic models with U(1)B−L mass
from Stückelberg couplings have been constructed up to date. In the literature there
are two main classes of RR tadpole free models with massive B-L. The first class are
non-susy type IIa toroidal orientifold models first constructed in [5]. The second class
are the type II Gepner orientifold models constructed by one of the present authors and
collaborators [6, 7]. The former were already considered in [3]. In the present paper we
will concentrate on the RCFT Gepner model constructions, which lead to a large class
of MSSM like models, more representative of the general Calabi-Yau compactifications
(for a recent discussion of instanton-induced neutrino masses in a model with no RR
tadpole cancellation, see [8]).
The class of constructions in [6, 7] start with any of the 168 Type II compacti-
fications obtained by tensoring N = 2 SCFT minimal models. In addition one can
choose a number of modular invariant partition functions (MIPF), leading to a total of
5403. Then different consistent orientifold projections are performed on the different
models. This yields a total of 49304 Type II orientifolds. The open string sector of
the theory is defined in terms of the boundary states of the theory. Intuitively, they
play the same role as D-branes wrapping cycles in the geometrical settings. Thus one
associates boundary states a,b, c,d to the gauge groups giving rise to the SM. Differ-
ent choices for the SM boundary states lead to different spectra. In the present paper
we will make use of the data in [6] which contains 211634 different MSSM-like spectra
(including also different hidden sectors). Although this number is huge, most of these
models are really extensions of the MSSM, since they have either an extra U(1)B−L or
SU(2)R × U(1)B−L group factor beyond the SM group. As we said, we are actually
only interested in models in which the U(1)B−L gets a Stückelberg mass. Then we
find that the number of MSSM-like models with these characteristics is dramatically
reduced: only 0.18 percent of the models (391) have a massive U(1)B−L.
As we said, in the geometrical setting of IIA orientifolds with intersecting D6-branes
[9, 10] (see [11] for reviews and [12, 13] for the IIB counterparts), instantons are associ-
ated to D2-branes wrapping 3-cycles, like the background D6-branes do. Analogously,
in the RCFT setting the same class of boundary states appearing in the SM construc-
tions are the ones corresponding to instantons. The zero modes on the instanton is
computable from the overlaps of instanton brane boundary states (zero modes un-
charged under the 4d gauge group) or of instanton and 4d spacefilling brane boundary
states (zero modes charged under the corresponding gauge factor). We find that the
criteria for a non-perturbative superpotential to be generated [14] are only fulfilled if
the Chan-Paton (CP) symmetry of the instantons is O(1). For instantons with CP
symmetry 1 Sp(2) or U(1) we find that there are a few extra uncharged fermionic
zero modes which would preclude the formation of the searched superpotentials. On
the other hand we argue that the addition of fluxes and/or possible non-perturbative
1We adopt the convention that the fundamental representation of Sp(m) is m-dimensional.
extensions of the orientifold compactifications would allow also instantons with Sp(2)
and U(1) symmetries to generate such superpotentials. We thus include all O(1), Sp(2)
and U(1) instantons 2 in our systematic search. The computation of charged and un-
charged fermion zero modes may be easily implemented as a routine in a systematic
computer search for instanton zero modes in Gepner MSSM-like orientifolds. Results
of such a systematic computer search are presented in this article.
We find that out of the 391 models with massive U(1)B−L, there are very few ad-
mitting instantons with the required minimal O(1) CP symmetry, and in fact none
of them without additional vector-like zero modes. On the other hand we do find
32 models admitting Sp(2) symmetric instantons with just the required charged zero
mode content (and the minimal set of non-chiral fermion zero modes). In fact they
are all variations of the same orientifold Gepner model based on the tensor product
(k1, k2, k3, k4) = (2, 4, 22, 22). These models all in fact correspond to the same MIPF
and orientifold projection, they only differ on which particular boundary states corre-
sponding to the four a,b, c,d SM ‘stacks’. All models have the same chiral content,
exactly that of the MSSM , with extra vectorlike chiral fields which may in principle
become massive in different points of the CY moduli space. They have no hidden
sector, i.e., the gauge group is just that of the SM. For each of the models there are 8
instantons with Sp(2) CP symmetry with just the correct charged zero mode structure
allowing for the superpotential coupling (1.3) giving rise to νR Majorana masses. As
we said, they have extra uncharged fermion zero modes beyond the two required to
generate a superpotentials. However one would expect that these unwanted zero modes
might be lifted in more generic situations in which e.g. NS/RR fluxes are added.
We thus see that, starting with a ’large’ landscape of 211634 MSSM-like models,
and searching for instantons inducing neutrino masses, we find there are none admitting
the O(1) instantons with exactly the required zero mode structure, and only few (32)
examples with Sp(2) instantons with next-to-minimal uncharged zero mode structure
(and exactly the correct charged zero modes). Let us emphasize though that it is
the existence of massive U(1)B−L models which is rare. Starting with the subset of
models with a massive U(1)B−L, finding models with correct instanton charged zero
modes within that class is relatively frequent, 10 percent of the cases. Furthermore, we
will see that there are further models beyond those 32 which contain extra non-chiral
instanton zero modes and which may also be viable if these modes get massive by some
2We refer to the different kinds of instanton by their Chan-Paton symmetry on their volume. Since
we are not interested in gauge theory instantons, this notation should not be confusing.
effect (like e.g. the presence of RR/NS fluxes).
Instantons may generate some other interesting superpotential couplings in addition
to νR masses, some possibly beneficial and others potentially dangerous. In particular
we find that in the models which contain Sp(2) instantons which might induce νR
masses, there are also other instantons which would give rise directly to the Weinberg
operator [15]
(LHLH) (1.4)
Once the Higgs field gets a vev, this gives rise directly to left-handed neutrino masses.
Thus we find that in that class of models both the see-saw mechanism (which also gives
rise to a contribution to the Weinberg operator) and an explicit Weinberg operator
might contribute to the physical masses of neutrinos. Which effect dominates will
depend on the relative size of the corresponding instanton actions as well as on the size
of the string scale. Among potentially dangerous operators which might be generated
stand the R-parity violating operators of dimension < 5, which might give rise e.g. to
fast proton decay. We make an study of the possible generation of those, and find that
in all models in which νR masses might be generated R-parity is exactly conserved.
This is a very encouraging result.
A natural question to ask is whether one can say something about the structure
of masses and mixings for neutrinos. As argued in [3] generically large mixing angles
are expected, however to be more quantitative we also need to know the structure
of Yukawa couplings for leptons. In principle those may be computed in CFT but
in practice this type of computation has not yet been developed for CFT orientifolds.
Nevertheless we show that, in the case of instantons with Sp(2) CP symmetry, a certain
factorization of the flavor structure takes place, which could naturally give rise to a
hierarchical structure of eigenvalues for neutrino masses.
The structure of this article is as follows. In the next section we present a discussion
of instanton induced superpotentials in Type II orientifolds. This discussion will apply
both to Type IIA and Type IIB CY orientifolds as well as to more abstract CFT
orientifolds. We discuss the structure of both uncharged and charged instanton zero
modes. In particular we show that only instantons with O(1) CP symmetry have the
appropriate uncharged zero mode content to induce a superpotential contribution. We
also discuss how Sp(2) and U(1) might still generate superpotential contributions if
extra ingredients are added to the general setting. In section 3 we apply that discussion
to the specific case of the generation of νR Majorana masses, showing what is the
required zero mode structure in this case. We show how the flavor structure of the
Majorana mass term factorizes in the case of instantons with Sp(2) CP symmetry,
leading potentially to a hierarchical structure of eigenvalues. We further discuss the
generation of other B/L-violating operators including the generation of the Weinberg
operator as well as R-parity violating couplings. In section 4 we review the RCFT
Type II orientifold constructions in [6, 7]. A general discussion of zero fermion modes
for instantons in RCFT orientifolds is presented in section 5.
The results of our general search for instantons generating νR masses are presented
in section 6. We provide a list of all Gepner orientifolds which admit instanton con-
figurations potentially giving rise to νR Majorana masses. We describe the structure
of the models with Sp(2) instantons having the required charged zero modes for that
superpotential to be generated. We also describe the boundary states of the corre-
sponding instantons and provide the massless spectrum of the relevant MSSM-like
models. Other orientifolds with zero mode structure close to the minimal one are also
briefly discussed. A brief discussion about the possible generation of R-parity violating
superpotentials is included. We leave section 7 for some final comments. Some nota-
tion on the CFT orientifold constructions, and a discussion of the CFT symmetries in
the Sp(2) examples are provided in two appendices.
As this paper was ready for submission, we noticed [16, 17], whose discussion of
instanton zero modes partially overlaps with our analysis in Section 2.2.
2 Instanton induced superpotentials in Type II ori-
entifolds
In this Section we review the generation of superpotentials involving 4d charged fields
via D-brane instantons in type II compactifications. The discussion applies both to type
IIA and IIB models, and to geometrical compactification as well as to more abstract
internal CFT’s. For recent discussions on D-brane instantons, see [4, 3, 25, 8] .
Before starting, a notational remark in in order. Our notation is adapted to working
in the covering theory, namely the type II compactification, and orientifolding in a
further step. Thus we describe the brane configurations as a system of branes (described
by boundary states for abstract CFT’s), labeled k, and their orientifold images labeled
k′. Similarly, we denote M the brane / boundary state corresponding to the instanton
brane, andM ′ its orientifold image. If a brane is mapped to itself under the orientifold
action, we call it a ‘real’ brane / boundary, and ‘complex’ otherwise.
2.1 D-brane instantons, gauge invariance and effective opera-
A basic feature of type II orientifold compactifications with D-branes is the generic
presence of Stückelberg couplings between the U(1) gauge fields on the D-branes, and
certain 4d RR closed string 2-forms. These couplings are required by the Green-
Schwarz mechanism when the U(1)’s have non-zero triangle contributions to mixed
anomalies [18, 19], but can also exist for certain non-anomalous U(1)’s [5, 20]. These
couplings make the U(1) gauge bosons massive, but the symmetry remains as a global
symmetry exact in perturbation theory. Since the closed string moduli involved are
scalars (0-forms) in the RR sector, the natural candidate non-perturbative effects to
violate these U(1) symmetries are instantons arising from euclidean D-branes coupling
to these fields.
In computing the spacetime effective interaction mediated by the instanton, one
needs to integrate over the instanton zero modes. In the generic case (and in particular
for the case of our interest) there are no bosonic zero modes beyond the universal ones
(namely, the four translational bosonic zero modes associated to the position of the
instanton). On the other hand, the structure of fermion zero modes will be crucial.
Since we are interested in models with non-trivial 4d gauge group, arising from a set
of background 4d spacetime filling branes, we consider separately fermion zero modes
which are uncharged under the 4d gauge group and those which are charged. In this
paper we restrict our discussion to 4d N = 1 supersymmetric models, and this will
simplify the analysis of zero modes.
Fermion zero modes which are uncharged under the 4d gauge group determine
the kind of 4d superspace interaction which is generated by the instanton. We are
interested in generating superpotential interactions, which, as is familiar, requires the
instanton to have two fermion zero modes to saturate the d2θ superspace integration.
For this, a necessary (but not sufficient!) condition is that the D-branes are half-BPS,
so these fermion zero modes are the Goldstinos of the two broken supersymmetries.
In the string description, uncharged zero modes arise from open strings in the MM
sector (in our notation, the one leading to adjoint representations), which in particular
contain these Goldstinos, and the MM ′ sector (in our notation, the one leading to
two-index symmetric or antisymmetric tensors). Note that both are the same for ‘real’
branes. Hence we are primarily interested in D-branes whose MM sector contains
just two fermion zero modes, and whose MM ′ sector (for ‘complex’ branes) does not
contain additional fermion zero modes.
In analogy with the familiar case of gauge theory instantons [21], charged fermion
zero modes determine the violation of perturbative global symmetries by the instanton-
induced interaction. Namely, in order to saturate the integration over the charged
fermions zero modes, the spacetime interaction must contain insertions of fields charged
under the 4d gauge symmetry, and in particular under the global U(1) factors, which
are thus violated by the D-brane instanton. Notice that this holds irrespectively of the
number of uncharged fermion zero modes, namely of the kind of superspace interaction
induced by the instanton. Restricting to superpotential interactions, the resulting
operator in the 4d effective action has roughly the form
Winst = e
−U Φ1 . . .Φn (2.1)
Here the fields Φ1, . . . ,Φn are 4d N = 1 chiral multiplets charged under the 4d gauge
group, and in particular also under the U(1) symmetries. Note also that the instanton
amplitude contains a prefactor (which in general depends on closed and open string
moduli) arising from the Gaussian path integral over (massive) fluctuations of the
instanton (hence described by an open string annulus partition function, see [22, 23]
for related work), which we can ignore for our purposes in this paper.
For D-brane instantons, U is the closed string modulus to which the euclidean D-
brane couples. In the D-brane picture, instanton fermion zero modes charged under
the gauge factor carried by the kth stack of 4d space-filling branes (and its image
k′) arise from open strings in the Mk and Mk′ sectors, transforming as usual in the
( M , k) and ( M , k) representations, respectively (with both related in the case of
‘real’ branes). The (net) number of instanton fermion zero modes with such charges is
given by certain multiplicities 3 IMk, IMk′.
A D-brane instanton, irrespectively of the superspace structure of the 4d interac-
tions it may generate, violates U(1)k charge conservation by an amount IMk − IMk′ for
‘complex’ branes and IMk for ‘real’ branes. In particular, this is the total charge of the
field theory operator Φ1 . . .Φn in (2.1). From the Stückelberg couplings, it is possible
to derive [3] (see [23, 24, 25] for related work, also [26])
3In geometric type IIA compactifications with 4d spacetime-filling branes and instanton branes
given by D6- and D2-branes wrapped on Special Lagrangian 3-cycles, IMk corresponds to the inter-
section number between the 3-cycles corresponding to the kthD6- and the D2-brane M (and similarly
for IMk′ ). In geometric type IIB compactifications, it corresponds to the index of a suitable Dirac
operator. In general (even for abstract CFT’s) it can be defined as the Witten index for the 2d theory
on the open string with the boundary conditions corresponding to the two relevant branes. We will
often abuse language and refer to this quantity as intersection number, even in Section 6 where we
work in the non-geometric regime of type IIB compactifications.
that for ‘complex’ instantons, gauge transformations of the U(1)k vector multiplets
Vk → Vk + Λk, transform U as
U → U +
Nk(IMk − IMk′)Λk (2.2)
For ‘real’ brane instantons, which were not considered in [3], the shift is given by 4
U → U +
NkIMkΛk (2.3)
(this new possibility will be an important point in our instanton scan in Section 6).
The complete interaction (2.1) is invariant under the U(1) gauge symmetries. How-
ever, from the viewpoint of the 4d low-energy effective field theory viewpoint, it leads to
non-perturbative violations of the perturbative U(1) global symmetries, by the amount
mentioned above.
In the string theory construction there is a simple microscopic explanation for the
appearance of the insertions of the 4d charged fields (related to the mechanism in
[27]). The instanton brane action in general contains cubic terms αΦ γ, involving two
instanton fermions zero modes α in the ( M , k) and γ in the ( p, M) coupling to the 4d
spacetime field Φ in the ( k, p) of the 4d gauge group
5. Performing the Gaussian path
integral over the instanton fermion zero modes leads to an insertion of Φ in the effective
spacetime interaction. In general, and for a ‘complex’ instanton, there are several
fermion zero modes αi, γi in the fundamental (resp. antifundamental) of the instanton
gauge group, coupling to a 4d spacetime chiral operators Oij (which could possibly
be elementary charged fields, or composite chiral operators). Gaussian integration
over the fermion zero modes leads to an insertion of the form detO (for ‘real’ brane
instantons, detO should be interpreted as a Pfaffian). It is straightforward to derive
our above statement on the net charge violation from this microscopic mechanism.
Note that the above discussions show that instantons in different topological sec-
tors (namely with different RR charges, and different intersection numbers with the
4d spacefilling branes) contribute to different 4d spacetime operators. In particular,
multiwrapped instantons, if they exist as BPS objects, contribute to operators different
4Equivalently, one may use (2.2), but must include an additional factor of 1/2 from the reduction
of the volume for a real brane (which is invariant under the orientifold action).
5 Although there is no chirality in 0+0 dimension, the fermion zero modes α and γ are distinguished
by their chirality with respect to the SO(4) global symmetry of the system (which corresponds to
rotations in the 4d spacetime dimensions, longitudinal to the space-filling branes, and transverse to
the instanton brane). Supersymmetry of the instantons constrains the couplings on the instanton
action (such as the cubic couplings above) to have a holomorphic structure.
from the singly wrapped instanton. This implies that the instanton expansion for a
fixed operator is very convergent, and could even be finite.
Another important implication of the above discussion is that, in order to generate
a specific operator via an instanton process, a necessary condition is that the instanton
has an appropriate number and structure of charged zero modes. However, this is not
sufficient. Insertions of 4d fields appear only if the fields couple to the instanton fermion
zero modes via terms at most quadratic in the zero modes. In equivalent terms, only
zero modes appearing in the Gaussian part of the instanton action can be saturated
by insertions of 4d fields (those to which they couple). The requirement that the zero
modes have appropriate couplings to the 4d fields may be non-trivial to verify in certain
constructions. This is the case for the Gepner model orientifolds in coming sections,
whose couplings are computable in principle, but unknown in practice. In such cases
we will assume that any coupling which is not obviously forbidden by symmetries will
be non-vanishing. Unfortunately there are no arguments to estimate the actual values
of such non-vanishing couplings, hence we can argue about the existence of certain
instanton induced operators, but not about the coefficients of such terms.
2.2 Zero mode structure for D-brane instantons
In this section we describe more concretely different kinds of instantons and the struc-
ture of interesting and unwanted zero modes. Our discussion will be valid for general
D-brane models in perturbative type II orientifolds without closed string fluxes, al-
though we also make some comments on more general F-theory vacua and the effects
of fluxes. A more specific discussion is presented in Section 5.
2.2.1 Uncharged zero modes
We start discussing zero modes uncharged under the 4d gauge group. These are crucial
in determining the kind of superspace interaction induced by the instanton on the 4d
theory. In particular, we are interested in instantons contributing to the 4d superpo-
tential, namely those which contain just two fermion zero modes in this sector. We are
also interested in instantons which may contain additional fermion zero modes, and the
possible mechanisms that can be used to lift them. Let us discuss ‘real’ and ‘complex’
brane instantons in turn.
• Real brane instantons
Real brane instantons correspond to branes which are mapped to themselves by the
orientifold action, hence M = M ′. Uncharged zero modes arise from the MM open
string sector. As discussed in Section 5, there is a universal sector of zero modes, in the
sense that it is present in any BPS D-brane instanton, which we now describe. Before
the orientifold projection, we have a gauge group U(n) on the volume of n coincident
instantons. Notice that, although there are no gauge bosons in 0 + 0 dimensions, the
gauge group is still well-defined, since it acts on charged states (open string ending on
the instanton brane). There are four real bosonic zero modes and four fermion zero
modes in the adjoint representation. For the minimal U(1) case, the four bosons are the
translational Goldstones. The four fermions arise as follows. This sector is insensitive
to the extra 4d spacefilling branes, and feels an accidental 4d N = 2 supersymmetry.
The BPS D-brane instanton breaks half of this, and leads to four Goldstinos, which
are the described fermions 6.
The orientifold projection acts on this universal sector as follows (see Section 5 for
further discussion). The gauge group is projected down to orthogonal or symplectic.
Hence instanton branes with symplectic gauge group must have even multiplicity (a
related argument, in terms of the orientifold action on Chan-Paton indices, is given in
Section 5). For instantons with O(n) gauge symmetry, the orientifold projects the four
bosonic zero modes and two fermion zero modes (related by the two supercharges of 4d
N = 1 supersymmetry broken by the instanton) to the two-index symmetric represen-
tation, and the other two fermion zero modes (related by the other two supercharges of
the accidental 4d N = 2 in this sector) to the antisymmetric representation. Hence for
O(1) instantons (namely instantons with O(1) gauge group on their volume), we have
just two fermion zero modes, which are the Goldstinos of 4d N = 1 supersymmetry,
and the instanton can in principle contribute to the superpotential (if no additional
zero modes arise from other non-universal sectors). For instantons with Sp(n) gauge
symmetry, the orientifold projects the four bosonic zero modes and two fermion zero
modes to the two-index antisymmetric representation, and the other two fermion zero
modes to the symmetric representation. Hence, even for the minimal case of Sp(2)
instantons, we have just two fermion zero modes in the triplet representation, in ad-
dition to the two 4d N = 1 Goldstinos. Hence Sp(2) instantons cannot contribute to
the superpotential in the absence of additional effects which lift these zero modes (see
later) 7.
6We thank F. Marchesano for discussions on this point.
7 For D-brane instantons corresponding to gauge instantons, the additional fermion zero modes
in the universal sector couple to the boson and fermion zero modes from open strings stretched
between the instanton and the 4d spacefilling brane. They act as Lagrange multipliers which impose
In addition to this universal sector, there exist in general additional modes, whose
presence and number depends on the detailed structure of the branes. Namely, on
the geometry of the brane in the 6d compact space in geometric compactifications, or
on the boundary state of the internal CFT in more abstract setups. They lead to a
number of boson and fermion zero modes in the symmetric or antisymmetric represen-
tation. The computation of these multiplicities in terms of the precise properties of
the instanton branes is postponed to Section 5. In order to generate a superpotential,
one must require these modes to be absent, except for the case of antisymmetrics of
O(1) instantons, which are actually trivial.
An important point is that extra fermion zero modes (including the extra triplet
fermion zero modes in the universal sector of Sp(2) instantons, and any two-index
tensor fermion zero mode in the non-universal sectors) are in principle not protected
against acquiring non-zero masses once the model is slightly modified. In other words,
such fermions are non-chiral, in terms of the SO(4) chirality in footnote 5. One
such modification is motion in the closed string moduli space, which can lift the non-
universal modes if there are non-trivial couplings between them and closed string mod-
uli (unfortunately, such correlators are difficult to compute, even in cases where the
CFT is exactly solvable, like the Gepner models). Note that extra zero modes in the
universal sector of Sp(2) instantons cannot be lifted by this effect, since it does not
break the accidental 4d N = 2 in this sector. A second possible modification which
in general can lift extra zero modes is the addition of fluxes, generalizing the results
for D3-brane instantons in geometric compactifications [29] (for non-geometric CFT
compactifications, we also expect a similar effect, once fluxes are introduced following
[30]). Note that fluxes can lift extra zero modes in the universal sector as well, since
they can break the accidental 4d N = 2 susy in this sector. A last mechanism arising in
more general F-theory compactifications and discussed below for complex instantons,
is valid for real instanton branes as well.
The bottom line is that in the absence of such extra effects, only O(1) instantons can
contribute to superpotential terms. However, in modifications of the model such extra
effects can easily lift the extra fermion zero modes. Hence, this kind of extra vector-like
zero modes will not be considered catastrophic, and real instantons (including the O(1)
and Sp(2) cases) with such zero modes are considered in our scan in Section 6.
the fermionic constraints in the ADHM construction [28], and may not spoil the generation of a
superpotential.
• Complex brane instantons
Zero modes uncharged under the 4d gauge group can arise from the MM and
MM ′ open string sectors. Notice that the orientifold action maps the MM sector
to the M ′M ′, hence we simply discuss the former and impose no projection. The
discussion of the MM sector is as for real brane instantons before the orientifold
projection. The universal sector leads to a U(n) gauge symmetry, and four bosonic and
four fermionic zero modes in the adjoint representation. The bosons are translational
Goldstones, while the fermions are Goldstinos of the accidental 4d N = 2 present in
this sector. Hence, even in the minimal case of U(1) brane instantons there are two
extra fermion zero modes, beyond the two fermion zero modes corresponding to the
4d N = 1 Goldstinos. Hence U(1) instanton (except for those corresponding to gauge
instantons, see footnote 7) cannot contribute to superpotential terms in the absence of
additional effects, like closed string fluxes . However, keeping in mind the possibility
of additional effects lifting them in modifications of the model, we include them in the
discussion. Also, in what follows we will use the U(n) notation for the different fields
to keep track of the Chan-Paton index structure.
The above statement would seem in contradiction8 with computations of non-
perturbative superpotentials [14] induced by M5-branes instantons in M-theory com-
pactifications on Calabi-Yau fourfolds, which are dual to D3-brane instantons (with
world-volume U(1) gauge group) on type IIB compactifications. The resolution is that
the M5-branes that contribute are intersected by different (p, q) degenerations of the
elliptic fiber. This implies that U(1) D3-brane instanton only contribute if they are
intersected by mutually non-local (p, q) 7-branes. The two extra fermion zero modes
exist locally on the D3-brane volume, but cannot be defined globally due to the 7-brane
monodromies. Hence such effect can take place only on non-perturbative type IIB com-
pactifications including (p, q) 7-branes. Note that in perturbative compactifications,
namely IIB orientifolds, the (p, q) 7-branes are hidden inside orientifold planes [31] with
their monodromy encoding the orientifold projection; hence the only branes that can
contribute to the superpotential are real branes, for which the projection/monodromy
removes the extra fermion zero modes, as discussed above.
In addition to this universal sector, theMM sector may contain a non-universal set
of fermions and bosons, in the adjoint representation (hence uncharged under U(1)).
They depend on the specific properties of the brane instanton, and will be discussed
in Section 5. These additional zero modes should be absent in order for the instanton
8We thank S. Kachru for discussions on the ideas in this paragraph.
to induce a non-trivial superpotential. Notice however that these zero modes are un-
charged under any gauge symmetry, and hence vector-like. Thus, there could be lifted
in modifications of the model, as discussed for real instantons.
The MM ′ sector is mapped to itself under the orientifold action. Hence it leads to
a number of bosons and fermions in the two index symmetric or antisymmetric repre-
sentations. Notice that the two-index antisymmetric representation is trivial for U(1),
so these modes are actually not present. On the other hand, fermion zero modes in the
two-index symmetric representation are chiral and charged under the brane instanton
gauge symmetry. Hence they cannot be lifted by any of the familiar mechanisms, and
thus spoil the appearance of a non-perturbative superpotential, even if the model is
modified. Such fermion zero modes are considered catastrophic and we will look for
models avoiding them in our scan in Section 6.
2.2.2 Charged fermion zero modes
• Real brane instantons
Instanton zero modes charged under the 4d gauge group arise fromMk open string
sectors (and their image Mk′). In the generic case, there are no scalar zero modes in
these sectors. This is because in mixed Mk open string sectors the 4d spacetime part
leads to DN boundary conditions, which already saturate the vacuum energy in the NS
sector. Only in the special case where the internal structure of the spacetime filling
brane k and the instanton brane are the same, there may be NS ground states of the
internal CFT which do not contribute extra vacuum energy, hence leading to massless
scalars. However, this corresponds to brane instantons which can be interpreted as
instantons of the 4d gauge theory on the 4d space-filling branes. The instantons we
are interested in for the generation of neutrino Majorana mass terms are not of this
kind [3] (see e.g. [32, 33, 28, 25, 34] for discussions on gauge theory instantons from
D-brane instantons).
Hence we focus on charged fermion zero modes, which are generically present in
any mixed Mk sector. Let us define LMk, LMk′ the (positive by definition) number
of left-handed chiral fermion zero modes in the representations ( M , k), ( M , k),
respectively. The net number of chiral fermion zero modes in the ( M , k) is given
by IMk = LMk′ − LMk. This controls the violation of the U(1)a global charge by the
instanton. Namely, such fermion zero modes in the Mk, Mp sectors lead (if suitable
couplings are present) to the insertion of 4d charged fields Φkp and/or Φkp′.
In addition, there are PMk = min(LMk′ , LMk) vector-like pairs of fermion zero
modes. Since they are vector-like, they may be lifted by slight modifications of the
model, like moving in the closed string moduli space, or by introducing additional
ingredients, like fluxes. In addition, they may be lifted by moving in the open string
moduli space of the 4d spacefilling branes, as follows. The zero modes may lead to
insertions of 4d fields Φkk, if the kk sector contains such 4d chiral multiplets (or to
insertions of composite 4d operators in the adjoint of the kth 4d gauge factor), and if
they couple to the zero modes. Although this may not be generically not the case,
many of our models in coming section contain such adjoint fields. Hence, a non-
trivial vev for the latter can lift these extra vector-like zero modes, hence leading to
instanton generating the superpotential of interest. Given these diverse mechanisms to
lift these zero modes, their presence of such zero modes is thus unwanted, but again
not necessarily catastrophic.
One last comment, related to the concrete kind of instanton search we will be
interested in. Namely, we will be searching for instantons leading to a specific operator,
carrying non-trivial charges under a specific set of 4d gauge factors. Postponing the
detailed discussion to Sections 3.1, 4.2 , let us denote a, b, c, d the set of branes
leading to a field theory sector, denoted ‘observable’ (and which reproduces the SM
in our examples). We will require the instanton to have a prescribed number of chiral
fermion zero modes charged under these branes, namely we require the intersection
numbers of the instanton with these branes IMa, . . . , IMd to have specific values (as
mentioned above, in the most restrictive scan we forbid vector-like pairs of zero modes
under these branes). In addition, the model in general contains an additional sector of
branes, denoted ‘hidden’ (since there is zero net number of chiral multiplets charged
under both sectors) and labeled hi, required to fulfill the RR tadpole cancellation
conditions. In general there may be instanton fermion zero modes from e.g. the
Mh1, h2M sectors, which would contribute to insertions of the 4d fields in the h1h2
sector if there are suitable cubic couplings. These extra insertions could be avoided
if such 4d fields in the hidden sector acquire vevs (note that vevs for the (vector-like)
fields charged under the visible and hidden sectors would typically break hypercharge,
and should be avoided), and hence lift the zero modes. Equivalently, from the 4d
perspective, the unwanted extra h1h2 field insertion is replaced by its vev. However,
this renders the discussion very model dependent. Moreover, the possibility of hidden
brane recombination was not included in the search for SM-like models in [6, 7] (namely,
the possibility of allowing for chiral fields charged under the observable and hidden
gauge groups, which may become non-chiral and possibly massive upon hidden brane
recombination). Hence we will consider these chiral fermion zero modes as unwanted
(as usual, non-chiral modes are unwanted but not catastrophic, hence they are allowed
for in a more relaxed search).
• Complex brane instantons
The discussion of ‘complex’ brane instantons is somewhat analogous to the previous
one, with the only complication that the braneM and its imageM ′ lead to independent
modes, leading to a more involved pattern of fermion zero modes. Instanton zero modes
charged under the 4d gauge group arise from the Mk,Mk′ and related sectors. As for
‘real’ brane instantons, there are generically no scalars in these sectors (and certainly
not in our case of interest). Hence we focus on charged fermion zero modes, which are
generically present in any mixed sector.
In contrast with ‘real’ brane instantons, a net combination of fermion zero modes
in the ( M , k) + ( M , k) is not vector-like, but chiral under the instanton gauge
symmetry. Such a pair cannot therefore be lifted even by modifications of the theory.
In general, if the instanton has a mismatch in the total numbers nα, nγ of fermion zero
modes αi in the M and γj in the M , the instanton amplitude automatically vanishes.
Namely, the matrix of operators Oij coupling to the zero modes necessarily has rank
at most min(nα, nγ). That is , if nα > nγ there are linear combinations of the αi which
do not couple, and cannot lead to insertions. Moreover, they are not liftable by the
familiar mechanisms 9, thus in our instanton search in Section 6 such excess zero modes
are forbidden even in relaxed scans.
Let us thus discuss a sector of fermion zero modes with equal number nα = nγ .
Considering a given 4d space-filling brane k, let us denote LMk, LM ′k′, LMk′, LM ′k the
(positive by definition) number of left-handed chiral fermion zero modes in the repre-
sentations ( M , k), ( M , k), ( M , k), and ( M , k) respectively. The net number of
chiral fermion zero modes in the ( M , k) and ( M , k) is given by IMk = LMk′ −LM ′k′
and IMk′ = LMk−LM ′k, respectively. This net number of fermions zero modes controls
the violation of the U(1)a global charge by the instanton. Namely, such fermion zero
modes in the Mk, Mp, Mk′, Mp′ sectors lead (if suitable couplings are present) to the
insertion of 4d charged fields Φkp and/or Φkp′.
9Note that such a mismatch is always correlated with the existence of extra chiral zero modes in
the MM ′ sectors, discussed above. Denoting ~Qa, ~Qorient the vector of RR charges of the 4d space-
filling branes and orientifold planes, they satisfy the RR tadpole conditions
Na ~Qa +
Na ~Qa′ +
~Qorient. = 0. By taking the ‘intersection’ bilinear with the RR charges ~QM of the brane instanton, we
have IMa+IMa′+IM,orient = 0. This implies that the number of fundamentals minus anti-fundamentals
of the instanton gauge group is related to the number of two-index tensors.
The remaining fields in this sector are vector-like pairs, in the ( M , k) + ( M , k)
or the ( M , k) + ( M , k), which in principle lead to the vanishing of the instanton
amplitude, but which can be lifted by additional effects (motion in closed or open string
moduli space, or addition of fluxes), in a way consistent with the gauge symmetries in
4d spacetime and on the instanton.
Just like for ‘real’ brane instantons, we conclude by commenting on our concrete
instanton search in models with a set of visible branes a, b, c, d and a set of hidden
branes hi. The requirement that the instanton leads to an operator with specific
charges under the visible branes fixes the values of the quantities IMa − IMa′, etc. As
we described for real branes, one may still have fermion zero modes charged under the
hidden sector branes, but they lead to additional insertions, hence we rather focus on
instantons with IMhi − IMh′j = 0. The two kinds of conditions, on intersection numbers
with visible and hidden branes, still leave the possibility of combinations of fermion
zero modes of the kind ( M , k) + ( M , k), which do not contribute to IMk, or of the
kind ( M , k) + ( M , k), which does not contribute to IMk′. Such combinations are
automatically vector-like, and thus may be lifted in modifications of the theory. But the
condition also allow combinations like ( M , k)+( M , k), which exploit a cancellation
between IMk and IMk′ (as also does ( M , k) + ( M , k)). Such combinations are
chiral by themselves, and in general imply a mismatch of modes in the M and the
M . The total mismatch can be arranged to vanish using combinations of the kind
( M , k) + ( M , k) and ( M , p) + ( M , p) for different branes. However, the only
way to lift these pairs is by breaking the gauge symmetry on the 4d space-filling branes
k and p. This can be done without damage to the visible sector if these are hidden
branes, but this corresponds to the recombination of hidden branes that, as mentioned
already, we are not going to consider. Hence only vector-like pairs with respect to
each brane are considered to be liftable in simple modifications of the theory. In our
instanton search, these are the only additional fermion zero modes which we allow in
relaxed scans (but they are clearly not allowed for in restricted scans)
3 Instanton induced Majorana neutrino masses
In this Section we discuss the possible physical effects of D-brane instantons in string
models with SM-like spectrum. In particular we describe the conditions to generate
right-handed neutrino Majorana masses. We also comment on other possible B and/or
L violating operators that can be generated by instantons. In this section we will again
Sp(2),U(2)
U  , D 
gluon
d− Leptonic
a− Baryonic
b− Left c− Right
Figure 1: Quarks and leptons at intersecting branes
use the geometrical language of IIA intersecting D-branes but it should be clear that
our discussion equally applies to general CFT orientifolds like the ones presented in
the next section.
3.1 The MSSM on the branes
Let us now specify the discussion in the previous section to the case of the generation
of a right-handed neutrino mass term. In order to do that we need some realistic
orientifold construction with the gauge group and fermion spectrum of the Standard
Model (SM). In the context of Type II orientifolds perhaps the most economical brane
configuration leading to a SM spectrum is the one first considered in [5]. This consists
of four stacks, labelled a,b, c,d. The gauge factor on branes a is U(3), and contains
QCD and baryon number. The d factor is U(1)d, and corresponds to lepton number.
Stack b contains SU(2)Weak either embedded in U(2) or Sp(2). Finally brane c can
either provide a U(1) or an O(2) factor. In the brane intersection language, the chiral
fermions of the SM live at the intersections of these branes, as depicted in Fig. 3.1.
The U(1)Y factor of the standard model is embedded in the Chan-Paton factors of
branes a,c and d as
(QB−L −QR) (3.1)
where Qx denotes the generator of the U(1) of brane stack x (in case the Chan-Paton
factor of brane c is O(2) one should use the properly normalized O(2) generator).
Note that in this convention the Qd generator appears with sign opposite to other
conventions in the literature, e.g. in [3]. In addition to Y these models have two
additional U(1) gauge symmetries:
Qanom = 3Qa +Qd = 9QB +QL
Y ′ =
Qa +Qc −Qd = QB−L +QR (3.2)
The first is anomalous whereas the second, which we will call B − L (with a slight
abuse of language, since it is in fact a linear combination of B − L and hypercharge),
is anomaly free. In models in which the electroweak gauge group is embedded in U(2),
rather than in Sp(2), there is a second anomalous U(1)b. The charges of the SM
particles under these U(1) symmetries are given in table 1.
Intersection D = 4 fields/ zero modes Qa Qc Qd Y QM
(ab),(ab’) QL 3(3, 2) 1 0 0 1/6 0
(ca) UR 3(3̄, 1) -1 1 0 -2/3 0
(c’a) DR 3(3̄, 1) -1 -1 0 1/3 0
(db),(db’) L 3(1, 2) 0 0 1 -1/2 0
(c’d) ER 3(1, 1) 0 -1 -1 1 0
(cd) νR 3(1, 1) 0 1 -1 0 0
(Mc) αi 2(0, 0) 0 -1 0 1/2 1
(dM) γi 2(0, 0) 0 0 1 -1/2 -1
Table 1: Standard model spectrum and U(1) charges of particles and zero modes. QM stands
for the world-volume gauge symmetry in the case of U(1) complex instantons.
The U(1)k gauge symmetries have couplings with the RR 2-forms Br of the model,
as follows
SBF =
Nk(pkr − pk′r)
Br ∧ Fk (3.3)
where pkr, pk′,r are given by the RR charges of the D-branes. These imply that under
a U(1)k gauge transformation Ak → Ak + dΛk the scalar ar dual to Br transforms as
ar → ar +
Nk (pkr − pk′r) Λk (3.4)
This has two effects: 1) The linear combination of axion fields
r(pkr − pk′,r)ar is
eaten up by the U(1)k massless gauge boson, making it massive. 2) For anomalous
U(1)k, the anomalies cancel through a 4d version of the Green-Schwarz mechanism.
This works due to the existence of appropriate ar F ∧F couplings, involving the gauge
fields in the theory.
It is obvious that all anomalous U(1)’s become massive by this mechanism. However
it is important to realize [5] that gauge bosons of anomaly-free symmetries like U(1)B−L
may also become massive by combining with a linear combination of axions. This is
interesting since it provides a mechanism to reduce the gauge symmetry of the model
without needing explicit extra Higgsing. In the models in which U(1)B−L becomes
massive in this way, the gauge group left over is purely that of the SM. Moreover, we
will see that having (B-L) massive by this Stückelberg mechanism is crucial to allow
the generation of instanton-induced Majorana neutrino masses.
Note that the B∧F couplings may also be potentially dangerous, since in principle
they could also exist for hypercharge, removing U(1)Y from the low-energy spectrum.
As we will see in our RCFT examples later on, having massless U(1)Y but massive
U(1)B−L turns out to be a strong constraint in model building.
3.2 Majorana mass term generation
As discussed in the previous section, string instantons can give rise to non-perturbative
superpotentials breaking explicitly the perturbative global U(1) symmetries left-over
from U(1) gauge bosons made massive through the Stückelberg mechanism. The kind
of operator we are interested in has the form
W ≃ e−Sins νRνR (3.5)
where νR is the right-handed neutrino superfield
10. Here Sins transforms under both
U(1)B−L and U(1)R in such a way that the overall operator is gauge invariant. This
operator may be created if the mixed open string sectors lead to fermionic zero modes
αi, γi , i = 1, 2, appropriately charged under the 4d gauge factors. As we discussed in
the previous section, to generate a superpotential one needs instanton with O(1) Chan-
Paton symmetry, in order to lead to two uncharged fermion zero modes to saturate
the d2θ 4d superspace integration. On the other hand, as we argued, instantons with
Sp(2) or U(1) CP symmetries may also induce the required superpotentials if there
is some additional dynamics getting rid of the extra uncharged zero modes which in
principle appear in instantons with these symmetries. We thus consider all O(1), Sp(2)
and U(1) instantons in our discussion.
In order to to get a νR bilinear, the intersection numbers of instanton M and d, c
branes are as follows
Sp(2) case : IMc = 1 ; IMd = −1 (3.6)
(since there is an extra multiplicity from the two branes required to produce Sp(2))
O(1) case : IMc = 2 ; IMd = −2 (3.7)
10 Actually we denote by νR the left-handed ν
L field following the usual (a bit confusing) convention.
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Figure 2: Disk amplitude coupling two charged zero modes to νR in the geometrical Type
IIA intersecting brane approach.
U(1) case : IMc = 2 ; IMd = −2 or IMd′ = 2 ; IMc′ = −2 (3.8)
Furthermore there must be cubic couplings involving the right-handed neutrino super-
field νa in the ath family and the fermionic zero modes αi, γj
Lcubic ∝ dija (αi νaγj) , a = 1, 2, 3 (3.9)
In type IIA geometric compactifications, this coupling arises from open string disk
instantons, see Fig. 3.1. In general type IIA models (resp. IIB models), the coefficients
dija depend on the Kähler (resp. complex structure) moduli, and possibly on open string
moduli. In simple CFT models (like e.g. in toroidal cases) these quantities may be in
principle explicitly computed.
These trilinear couplings appear in the instanton action and after integration of the
fermionic zero modes αi, γi one gets a superpotential coupling proportional to
d2α d2γ e−d
a (αiν
aγj) =
d2θ νaνb ( ǫijǫkld
b ) (3.10)
yielding a right-handed neutrino mass term. This term is multiplied by the exponential
of the instanton euclidean action so that the final result for the right-handed neutrino
mass (up to a 1-loop prefactor) has the form
MRab = Ms( ǫijǫkld
b ) exp(−
qM,rar ) (3.11)
For geometric compactifications VΠM is roughly related to the wrapped volume. We
keep the same notation to emphasize that the effect is non-perturbative in gs. In
supersymmetric models the term in the exponential is the linear combination U of
complex structure moduli to which the instanton D-brane couples, as described in the
previous section. As explained, the gauge U(1)c, U(1)d transformation of the bilinear
piece and the e−SD2 factor nicely cancel. Note that from the viewpoint of the 4d SM
effective field theory, the instanton has generated a Majorana neutrino mass violating
B−L. Notice also that since this symmetry is non-anomalous, its violation cannot be
associated to a gauge instanton, hence this is a pure string theory instanton effect.
3.3 Flavor and the special case of Sp(2) instantons
In order to extract more specific results for the flavor structure of the obtained Majo-
rana mass operator, one needs to know more details about the quantities dija coming
from the disk correlators. However in the particular case of Sp(2) instantons, the la-
bels i, j are Sp(2) doublet indices, and the symmetry requires dija = daǫ
ij. The mass
matrix for the three neutrinos is given by MRab =2Msdadb exp (−U), so that the flavour
dependence on a, b = 1, 2, 3 factorizes. More generally, as we will see in our RCFT
search in Section 6, there are typically several different instantons contributing to the
amplitude, so that we actually have a result for the mass
MRab = 2Ms
d(r)a d
−Ur (3.12)
where the sum goes over the different contributing instantons. One thus has a structure
of the form
e−Urdiag (d
1 , d
2 , d
3 ) ·
1 1 1
1 1 1
1 1 1
· diag (d(r)1 , d
2 , d
3 ) . (3.13)
This structure is very interesting. Indeed, each instanton makes one particular (instanton-
dependent) linear combination of the neutrinos massive, leaving two linear combina-
tions massless. Hence, for three or more instantons, one generically has a matrix with
three non-zero eigenvalues. It is easy to imagine a hierarchical structure among the
three eigenvalues if e.g. the exponential suppression factors exp(−ReUr) are different
for each instanton.
3.4 Other B− and L−violating operators
Our main focus in this paper is on the generation of right-handed neutrino Majorana
masses. However instantons may induce other L- and B-violating operators which we
briefly summarize in this subsection.
3.4.1 The Weinberg operator
A right-handed neutrino Majorana mass term is not the only possible operator violating
lepton number. Instanton effects may also give rise to dimension 5 operators not
involving νR. Specifically, the Weinberg operator
(LHLH) . (3.14)
might be generated. Once Higgs fields get a vev v this operator gives rise directly to
Majorana masses for the left-handed neutrinos of order ≃ v2/M . Indeed, it is easy to
check that in this case the required instanton M must verify
Sp(2) case : IMc = −1 ; IMd = 1 (3.15)
O(1) case : IMc = −2 ; IMd = 2 (3.16)
U(1) case : IMc = −2 ; IMd = 2 or IMc′ = 2 ; IMd′ = −2 (3.17)
(here we are assuming SU(2)weak to be embedded in an Sp(2)). Note that these
intersection numbers are different to those giving rise to νR mass terms. In particular
they lead to a transformation under B − L opposite to that of νR mass operators 11.
In the present case there are altogether four fermionic zero modes αi,γi corresponding
to the intersections of the instanton M with the branes c, d. These zero modes can
have couplings involving the left-handed leptons L and the u-type Higgs multiplet H
Ldisk ∝ cija (αi(LaH)γj) . (3.18)
Upon integration over the fermionic zero modes one recovers the Weinberg operator.
In the present case the scale M of the Weinberg operator will be the string scale Ms
and the coupling λ ≃ exp(−Sins). Again, in the particular case of Sp(2) instantons the
situation simplifies (cija = c
aǫij) and one gets left-handed neutrino Majorana masses
MLab =
< H >2
2c(r)a c
−Sr (3.19)
where r runs over the different contributing instantons and Sr is their corresponding
action. The flavour structure of this left-handed neutrino mass matrix is the same as
in eq.(3.13) and again may potentially lead to a hierarchical structure of left-handed
neutrino masses, as is experimentally observed.
11Instantons with these intersection numbers will be denoted with a plus sign in the instanton search
later on
In a given model both this kind of instanton and the one giving rise to right-
handed neutrino masses (which is different) may be present. This contribution to the
left-handed neutrino Majorana mass is in principle sub-leading compared to the see-saw
contribution
MLab(see-saw) =
< H >2
d(r)a d
−Sr)−1 hD (3.20)
where is the ordinary Yukawa coupling constant habD (ν
b). In principle the former
is doubly suppressed both by 1/Ms and the exponential factor. On the other hand
if the exponential suppression is not too large this mechanism involving directly the
Weinberg operator may be the most relevant source of neutrino masses. This is because
the see-saw contribution coming from νR exchange is proportional to the square of the
ordinary Yukawa couplings habD which could be small. One could even think of having
just the Weinberg operator as the unique source of the observed left-handed neutrino
masses. Note however that in string vacua like this, in which the νR’s are present and
massless at the perturbative level, having just the Weinberg operator would not be
phenomenologically correct, and instantons of the first class are still needed so that the
νR’s get a sufficiently large mass.
3.4.2 R-parity violating operators
In the case ofN = 1 SUSYmodels like the MSSM there might be operators of dimension
3 and 4 violating lepton and/or baryon number. These are the superpotential couplings
WRp = µ
aH + λabcQ
aDbLc + λ′abcU
aDbDc + λ′′abcL
aLbEc (3.21)
in standard notation. Unlike the neutrino operators mentioned above, these operators
violate B −L in one unit (rather than 2). It is well known that the standard R-parity
of the MSSM may be identified with a Z2 subgroup of U(1)B−L, so these terms are odd
under R-parity. The simultaneous presence of all these couplings is phenomenologically
unacceptable. Indeed, the third coupling violates baryon number, and the other three
violate lepton number. Together they lead to proton decay at an unacceptably large
rate. On the other hand couplings violating either B or L are phenomenologically
allowed.
It is an interesting question whether any of these operators may be induced by
string instanton effects. A first point to note is that instantons with Sp(2) Chan-Paton
symmetry can never generate operators of this type. The reason is that all charged
zero modes will necessarily come in Sp(2) doublets and hence the charged operators
induced will always involve an even number of charged D = 4 fields and R-parity is
automatically preserved. On the other hand O(1) and U(1) instantons may generate
R-parity violating operators. In particular, the LH bilinear is essentially the square
root of the Weinberg operator, and may be induced if a U(1) or O(1) instanton M
exists with
IMc = −1 ; IMd = 1 or IMc′ = 1 ; IMd′ = −1 . (3.22)
(in the O(1) case the second option is not independent from the first). Again, if
the appropriate disk couplings are non-vanishing a term with µaL ∼ Ms exp(−Sins) is
generated. The rest of the operators in WRp may also be generated. Possible instanton
zero modes which may induce them are shown in table 2. For example, the QDL
operator may be induced if a U(1) instanton M with intersection numbers
IMb = −1 ; IMc′ = 1 ; IMd = 1 (3.23)
is present and in addition couplings
Ldisk ∝ cab (α(UaQbj)γj) + c′a(βLajγj) (3.24)
exist. Here α, β, γ are zero modes corresponding to (Mc′), (Md) and (bM) intersections
and a, b(j) are flavor(SU(2)L) indices. Analogous trilinear or quartic disk amplitudes
involving two charged zero modes should exist to generate the rest of the R-parity
violating amplitudes in table 2.
D = 4 Operator IMa IMa′ IMb IMc IMc′ IMd IMd′
νRνR 0 0 0 2 0 -2 0
LH̄LH̄ 0 0 0 -2 0 2 0
LH̄ 0 0 0 -1 0 1 0
QDL 0 0 -1 0 1 1 0
UDD -1 0 0 1 2 0 0
LLE 0 0 -1 0 1 1 0
QQQL 1 0 -2 0 0 1 0
UUDE -1 0 0 2 2 -1 0
Table 2: Zero modes required to generate Lepton/Baryon-number violating superpoten-
tial operators. Sp(2) instantons cannot give rise to R-parity violating operators whereas
O(1),U(1) instantons may in principle contribute to all of them. In the case of U(1) instantons
there are additional zero mode possibilities which are obtained by exchanging IMx ↔ −IMx′ .
3.4.3 Dimension 5 proton decay operators
There are also superpotential dimension-5 operators violating B and L which may be
constructed from the MSSM matter superfields. Indeed the dimension 5 operators
)QQQL ; (
)UUDE (3.25)
are in fact the leading source of proton decay in SUSY GUT models with R-parity.
Unlike the other operators considered here these ones preserve B−L (hence R-parity)
but not B +L. These operators do not contribute directly to a proton decay but need
to be ’dressed’ by a one loop exchange of some fermionic SUSY particle. This makes
that, even although they are suppressed only by one power of the relevant fundamental
scale, the loop factor and the corresponding couplings make the overall rate in SUSY-
GUTS (barely) consistent with present experimental bounds for M of order the GUT
scale or larger.
These dimension 5 operators may also be induced in D-brane models of the class
here considered by the presence of instantons with appropriate intersection numbers.
For instance, the first operator may be induced through O(1) or U(1) instantons M
IMb = IMb′ = −2 ; IMa = 1 ; IMd = 1 (3.26)
Again Sp(2) instantons cannot induced this operator, since the the Ma intersection
would yield 6 (rather than 3) colored fermionic zero modes. The proton decay rate
obtained from these operators depend on the ratio exp(−Sins) × 1/Ms. For Ms of
order 1016 GeV, the rate is consistent with present bounds if exp(−Sins) provides a
suppression of a few orders of magnitude. On the other hand, models with a low string
scale may be in danger unless the exponential suppression is sufficiently large (or such
particular instantons are absent).
As a general conclusion, these phenomenological aspects of instanton induced oper-
ators very much depend on the action of the instanton, e.g. the volume of the wrapped
D2-instanton in the intersecting D-brane constructions. In any event it is clear that
the instantons here considered may indeed induce proton decay at a model-dependent
rate. However in certain models R-parity will be preserved and prevent too rapid pro-
ton decay. Indeed, this is what we find in our instanton search in Gepner orientifolds.
As we said Sp(2) instantons automatically preserve R-parity. More generally, models
that violate R-parity are rare, and the corresponding instantons actually generate very
high dimensional operators, so R-parity breaking effects seems quite suppressed. In
fact in our search within MSSM-like models in Gepner model orientifolds we do not
find instantons with just the correct charged zero modes to generate the low dimen-
sional couplings discussed above. So, at least within our class of RCFT constructions,
R-parity preservation is quite a common feature.
4 CFT orientifolds
In this section we describe the 4d string models we consider, namely orientifolds of
type IIB Gepner model compactifications. This is a very large class, on which one can
carry out large scans for certain desired properties. And moreover at present the only
known class of (SUSY) models with massive B − L.
4.1 Construction of the models
In general, RCFT orientifolds are orientifold projections of closed string theories con-
structed using rational conformal field theory. Although this includes in principle
rational tori and orbifolds, the real interest lies in cases where the two-dimensional
CFT is interacting, because such theories are hard to access by other methods. A
disadvantage of the use of RCFT is that this method is algebraic, and not geometric in
nature, so that one cannot easily explore small deformations of a certain string theory.
It is best thought of as a rational scan of moduli spaces.
The most easily accessible examples are the orientifolds of tensor products of mini-
mal N = 2 conformal field theories (“Gepner models”) forming a type IIB closed string
theory. During the last decade, examples in this class have been studied by many au-
thors (see [35][36][37][38][39][40][41][42]), and searched systematically in [6] and [7].
Although the Gepner models form only a small subset of RCFT’s, they already offer
a large number of possibilities. The total number of tensor products with the required
central charge c = 9 is 168. On top of this, one can choose a large number of distinct
modular invariant partition functions on the torus. The orientifold formalism is not
available for all of them, but it has been completely worked out [43] for all simple cur-
rent invariants (based on the charge conjugation invariant). This yields a total of 5403
distinct MIPFs. On top of this, we may choose various orientifold projections. Here
the only known possibilities are a class of simple-current based choices [44][45][46][47].
This then yields a total of 49304 orientifolds.
For each orientifold choice, the full open string partition function is
a,b,i
NaNbA
abχi(
aχ̂i(
 (4.1)
Here Aiab are the annulus coefficients, M
a the Moebius coefficients, Na the Chan-Paton
multiplicities and χ(τ) are the closed string characters, and χ̂i(τ) = T
−1/2χi(τ). The
set of integers i is simply the set of primary fields of the closed string CFT, and depends
only on the tensor product. The integers a, b are the boundary labels; this set depends
on the MIPF. Our notation and labelling conventions for these CFT quantities are
explained in Appendix A. The integers Aiab and M
a depend in addition also on the
orientifold choice; in the case of Aiab the latter dependence is very simple: all distinct
annuli can be written as A
cb, where Ω is the orientifold choice (which we
usually do not specify explicitly) and CΩcb is the boundary conjugation matrix, which
acts as an involution on the set of boundaries.
Suppressing some details (which can be found in [43]) we may write these integers
m,J,K
SimRa,(m,J)g
JK Rb,(m,K)
(4.2)
MΩ,ia =
m,J,K
P imRa,(m,J)g
(m,K)
(4.3)
Here m is the label of an Ishibashi-state (the set of states that propagates in the
transverse (or closed string) channel of the the annulus or Moebius diagrams). It is
a subset of the set of closed string labels i, but in general there are degeneracies,
so that more than one distinct Ishibashi state belongs to a given closed string label.
These degeneracies are distinguished by the labels J,K (see Appendix A). The complex
numbers R and U are respectively the boundary and crosscap coefficients. Note that
the latter depend on the orientifold choice, but the former do not. The only dependence
of the annulus coefficients on the orientifold choice is through the Ishibashi metric gΩJK ,
which is a matrix on each Ishibashi degeneracy space, and which can be a sign if there
are no degeneracies. Finally, the matrix P is given by P =
TST 2S
T , where S and
T are the generators of the modular group of the torus. Similar expressions exist for
the Klein bottle multiplicities defining the unoriented closed sector, but they will not
be needed in this paper.
The boundary labels a, b, . . . refer to all boundaries that respect the bulk symmetries
of the CFT. This includes the individual N = 2 chiral algebras of the factors in the ten-
sor product, the alignment currents12 that ensure the proper definition of world-sheet
supersymmetry and the space-time supersymmetry generator that imposes a general-
ized GSO-projection on the spectrum. The latter implies that all characters χi respect
(at least) N = 1 space-time supersymmetry. By construction, the boundary states
are then supersymmetric as well. Both conditions (boundary and bulk space-time su-
persymmetry) can in principle be relaxed within the formalism, but this leads to a
much larger set of bulk and boundary states. The precise labelling of the boundaries
is explained in Appendix A and involves a subset of the closed string labels i and a
degeneracy label, distinct from the one used for the Ishibashi states. The set of bound-
ary labels is complete in the sense of [45]. This means that no additional boundary
states exist that respect all the aforementioned symmetries. It also means that the
matrices R are square matrices (although their rows and columns are defined in terms
of different index sets). It is in principle possible to write down additional boundary
states that break some of the world-sheet symmetries. This is an important possibility
to keep in mind, but we will not consider it here.
The massless spectrum is obtained by restricting the characters χi to massless
states. Since the characters are supersymmetric those massless states are either vector
multiplets or chiral multiplets. The latter can be restricted to one chirality (e.g left-
handed); the other choice merely produces the CPT conjugates. Boundaries are called
real if a = a′, where the conjugate boundary a′ is defined by CΩa,a′ = 1, and complex
otherwise. The Chan-Paton multiplicities Na give rise to gauge groups U(Na) for
complex boundaries and SO(Na) or Sp(Na) for real ones. In the latter case Na must
be even. To count bi-fundamentals we define
Lab ≡
Aiabχi(
)massless,L . (4.4)
Note that because of the factor 1
in (4.1) and the fact that Lab is symmetric, the value
of Lab is indeed precisely the number of bi-fundamentals in the representation (Na, Nb).
12These are spin-3 currents consisting of products of the world-sheet supercurrents of the factors in
the tensor product, including the NSR space-time factor.
It is convenient to introduce the intersection matrix13
Iab ≡ Lab′ − La′b , (4.5)
which is manifestly antisymmetric in a and b. Note that for a pair of complex bound-
aries a, b with conjugates a′, b′ one can define four quantities that are relevant for the
massless spectrum, two of which are chiral, namely Iab and Iab′ .
It is often convenient to associate a geometric picture to these integers. Thus we
will often refer to the boundary labels and their multiplicities as “stacks of branes”, and
view the integers Iab as brane intersection numbers. This is only done for convenience
and does not imply a concrete brane realization; indeed, it does not make sense to
say that a given boundary label corresponds to a Dp-brane for some give p. Such an
interpretation might be valid in a large radius limit, assuming such a limit exists.
In general, for a choice of Chan-Paton multiplicities Na there will be tadpoles in
the one-point closed string amplitudes on the disk and the crosscap. These have to be
cancelled in order to make the theory consistent (since we work with supersymmetric
strings we do not have the option of cancelling RR and NS-NS tadpoles separately).
This leads to a condition on the Chan-Paton multiplicities:
NaRa,(m,J) = 4ηmUm,J (4.6)
where η0 = 1 and all other η’s are −1; there is such a condition for any Ishibashi label
(m, J) that leads to a massless scalar in the transverse channel. The one for m = 0
(which is non-degenerate) is the dilaton tadpole condition. It has the special feature
that all coefficients Ra0 are real and positive. The crosscap coefficient U0 is also real
and can be chosen positive (in the CFT both signs are acceptable). If U0 6= 0 (4.6)
limits the Chan-Paton multiplicities; if U0 = 0 the only solution is Na = 0 for all a,
which rules out any realization of the Standard Model. This reduces the number of
usable orientifolds to 33012.
Tadpole cancellation condition implies cancellation of RR-charges coupling to long-
range fields, and absence of local anomalies. There is a second condition that has
13Note that Lab is a symmetric matrix giving the number of chiral multiplets in the ( a, b) bi-
fundamental. This is a natural quantity in unoriented CFT’s, where a symmetric definition for the
annulus amplitude exists. In oriented CFT the annulus is, in general, not symmetric, but on the
other hand it is possible to choose the branes in such a way that only ( , ) bi-fundamentals appear.
This has become the customary way of counting states in the intersecting brane literature, even for
orientifold models. The quantity Iab is defined in such a way that it is anti-symmetric in a and b.
This is why boundary conjugations appear in the right hand side. This has the additional advantage
of making I a more familiar quantity for readers used to the standard intersection brane conventions.
to be taken into account, which has to do with Z2 charges that do not couple to
long-range fields, usually referred to as “K-theory charges” in geometric constructions.
Uncancelled K-theory charges may lead to global anomalies in symplectic factors of the
gauge group. But even if this symptom is absent, the disease may still exist. A much
more general way to probe for uncancelled K-theory charges is to require the absence
of global anomalies not only in the Chan-Paton gauge group but also on all symplectic
brane-anti-brane pairs that can be added to it as “probe-branes” [48]. Presently this
is the most general constraint that be imposed in these models, but it is not known
if additional ones are required. This probe brane constraint leads to a large number
of mod-2 constraint and is potentially very restrictive, but almost harmless in practice
[49]. It is satisfied by all models we consider in the present paper.
4.2 Search for SM-like models
The complete set of solutions to these conditions is finite but huge, but the vast major-
ity is of no phenomenological interest. In the last few years systematic searches have
been carried out for models that contain the Standard Model. The models that were
considered have the property that the set of Chan-Paton labels can be split into two
subsets, the observable and the hidden sector. The former has been limited, for prac-
tical reasons, to at most four complex brane stacks, required to contain the Standard
Model gauge group and the right intersections to yield three families of quarks and
leptons, plus (in general) some non-chiral (vector-like) additional matter. The hidden
sector is only constrained by the requirement that there be no net number of chiral
multiplets charged under both the observable and hidden sector, and by practical com-
putational limitations. The main purpose of the hidden sector in these models is to
provide variables that can be used to satisfy the tadpole and global anomaly condi-
tions, since the multiplicities in the observable sector are already fixed. In some cases
the observable sector already satisfies the constraints by itself, and there is no hidden
sector.
The observable sector can be realized in many different ways if one only imposes
the constraint that the standard model should be contained in it. These possibilities
were recently explored in [7]. We will focus on the realization described in Section 3.1,
first considered in [5]. There are four stacks, namely a (containing QCD and baryon
number as U(3)), b (containing electroweak SU(2) embedded as U(2) or Sp(2)), c
(providing a U(1) or an O(2) factor14, and d (providing another U(1) factor).
14In [6] also Sp(2) was considered, but this requires an additional Higgs mechanism.
The standard model hypercharge generator is , defined in (3.1):
Qd (4.7)
where Qx denotes the generator of the U(1) of brane stack x; in case the Chan-Paton
factor of brane c is O(2) one should use the properly normalized O(2) generator. In
addition to Y these models have two or three additional U(1) gauge symmetries (the
latter case if electroweak SU(2) arises from U(2)). These (except the combination
B − L) are anomalous, with anomaly cancelled by the Green-Schwarz mechanism,
implying the existence of a B ∧ F coupling making them massive. In fact, as already
mentioned, such Stückelberg couplings may be present for non-anomalous U(1)’s as
well. We are interested in models where the hypercharge gauge boson does not have
such couplings (otherwise the model would be phenomenologically unacceptable), but
where the B−L gauge boson is massive by such couplings (both in order that the gauge
group reduces to the SM one, and that neutrino Majorana masses may be induced by
string instantons, as discussed in previous sections).
The combined requirements of having a massive B − L and a massless Y turn out
to be difficult to satisfy. In fact, if the group on brane c is O(2) they are impossible
to satisfy simultaneously, because the O(2) component of the vector boson does not
couple to any axions, and hence the B−L and Y bosons have the same mass. But even
in models with a U(1) group on brane c it happens rather rarely that both constraints
are satisfied simultaneously, at least in the searches that have been done so far.
We will make use here of the data presented in [6, 7], which are available in slightly
improved form on the website www.nikhef.nl/∼t58/filtersols.php. This database con-
sist of 211634 distinct spectra. Here “distinct” means that they are physically different
for a given MIPF15 if the hidden sector is ignored. Hence the differences can be the
number of vector-like states of various kinds or the dilaton couplings of branes a, b, c,
d. Geometrically, these spectra may originate from the same moduli space, but then in
any case from different points on this moduli space. The improvements in comparison
with the data presented in [6] consist of taking into account the full global anomaly
conditions from probe branes. In some cases this required nothing more than checking
these conditions for an existing solution of the tadpole conditions, but in other cases a
new solution had to be found. As a result, a few models disappeared from the original
database, but due to improved algorithms a few new ones could be added. The net re-
15Rare cases of identical spectra and couplings originating from different MIPFs are treated as
distinct.
http://www.nikhef.nl/~t58/filtersols.php
sult is some small but inconsequential changes in the total number of models of various
kinds. The numbers we will mention below are based on the improved database.
The total number of models in that database with a Chan-Paton group U(3) ×
Sp(2)× U(1)× U(1) is 10587. Of these, 391 (about 4%) have a massive B − L vector
boson. For U(3)×U(2)×U(1)×U(1) these numbers are, respectively, 51 and 0. Hence
no examples of the latter type were found, although they were found with 1,2 and 4
families (in a limited search), in a few percent of the total number of models. It seems
therefore reasonable to expect that U(3) × U(2) × U(1) × U(1) with massive B − L
do exist, and that their absence is just a matter of statistics. Just for comparison, the
total number of U(3)× Sp(2)× O(2)× U(1) models is 56627.
5 Fermion zero modes for instantons on RCFT’s
In this section we discuss D-brane instantons for general compactifications, including
abstract CFT ones. We also provide the spectrum of zero modes on an instanton brane,
using the information about their internal structure i.e. in the compactified dimension
in geometric models, or of the internal CFT in more abstract setups like in previous
section. We will be interested in the latter case.
A first question that should be addressed is what this internal structure is. For
instance, in type IIA geometric compactifications, it corresponds to a supersymmetric
(i.e. special lagrangian) 3-cycle. Notice that these are the same kind of 3-cycles already
used to wrap the D6-branes that give rise to the 4d gauge symmetry of such models.
For general CFT’s, D-branes are described as boundary states. To describe instantons,
one can simply use the same boundary state of the internal CFT to describe the 4d
space-filling branes present in the model and the instanton branes. The only difference
is that boundaries satisfy Neumann conditions in the 4d space-filling case, and Dirichlet
in the instanton case. This exploits the fact that whenever a boundary state of the
internal CFT, and with Neumann boundary conditions in the 4d space is an acceptable
state of the full CFT, the same boundary state of the internal CFT, combined with
Dirichlet boundary conditions in the 4d space also gives an acceptable state of the full
CFT. For geometric compactifications this is related to Bott periodicity of the K-theory
classes associated to the D-brane charges, but it is possible to show it in general.
Since instanton D-branes can thus be naturally associated to the boundary states
of 4d space-filling branes, it is convenient to express the spectrum of zero modes of the
former in terms of the massless states of the latter. This is particularly useful, since
the computation of the spectra on 4d space-filling branes for Gepner model orientifolds
has already been described (although the arguments below are valid also for geometric
compactifications). Hence, let us denote by M a 4d space-filling brane associated with
the same boundary state of the internal CFT as the instanton brane M of interest.
Note that the 4d space-filling brane M is an auxiliary tool, and need not be (and, for
our instantons of interest, will not be) one of the 4d space-filling branes present in the
model.
‘Real’ brane instantons
Let us first consider the case of ‘real’ brane instantons. Consider a set ofm 4d space-
filling branes M, and focus first on the massless spectrum in the MM sector. Before
the orientifold projection, it leads to a universal 4d N = 1 U(m) vector multiplet, and
a number LMM of adjoint chiral multiplets. The orientifold operation maps this sector
to itself, acting on the Chan-Paton with a matrix γΩ,M. This matrix satisfies
γTΩ,Mγ
Ω,M = ±1m (5.1)
The two possibilities can be chosen to correspond to γΩ,M = 1m or γΩ,M = ǫm, with
−1r 0
, and m = 2r hence necessarily even in the latter case. They corre-
spond to the SO and Sp projections, respectively.
The orientifold projection on the N = 1 vector multiplet Chan-Paton matrices is
given by
λ = −γΩ,M λT γ−1Ω,M (5.2)
and leads to SO(m) or Sp(m) vector multiplets for the SO or Sp projection (hence
the name). Concerning the N = 1 chiral multiplets, they fall in two classes of p−, p+
(with p− + p+ = LMM) which suffer the projections
λ = ±γΩ,M λT γ−1Ω,M (5.3)
For the SO projection, this leads to p+, p− chiral multiplets in the , representation.
For the Sp projection, there are p+, p− chiral multiplets in the , representation.
The sectors Ma (where a is a 4d space-filling branes present in the model) are
mapped to sectors Ma′, so it is enough to focus on the former. After the orientifold
projection one gets LMa, LMa′ chiral multiplets in the ( M, a), ( M, a).
Let us now obtain the zero modes for a set of m instanton branes M in terms
of the above spectrum. The MM sector is closely related to the MM sector, by
changing the NN boundary conditions in 4d spacetime to DD boundary conditions
(which can be done in a covariant formalism, but not in the light-cone gauge). Before
the orientifold projection, one obtains the same set of states (since moddings for NN
and DD boundary conditions are identical, both in the NS and R sector), but with
different world-volume interpretation. Also, the change in boundary conditions implies
that some polarization states which are unphysical for the 4d spacefilling brane are
physical in the instanton brane. Hence, the U(m) gauge bosons on the 4d space-filling
brane M correspond to four adjoint real scalars in the instanton brane M . Similarly,
the 4d spinors in M, correspond to four fermion zero modes on M , transforming as
two spinors of opposite chiralities θα, θ̃α̇ of the SO(4) rotation group in transverse
space. The orientifold projection maps the MM sector to itself, acting on Chan-Paton
indices with a matrix γΩ,M . In close analogy with the argument in [50] for the familiar
D5-D9-brane system in type I (see [51, 52] for related derivations), one can show that
the condition (5.1) flips sign upon changing four NN boundary conditions to DD, hence
γTΩ,Mγ
Ω,M = ∓1m (5.4)
Namely, the instanton brane has Sp(m) gauge group when the 4d space-filling brane
(with same internal boundary state) has gauge group O(m), and vice-versa. We still
refer to these projections as SO and Sp, hoping no confusion arises. Note that, as
mentioned in Section 2.2, although there are no gauge bosons in 0+ 0 dimensions, the
gauge group is present on the instantons in that it acts on open string endpoints.
Let us consider the effect of the orientifold projection on the MM states, as com-
pared with the effect onMM states. Again, following arguments familiar in the D5-D9
brane system in type I, one can show that the signs in conditions like (5.2), (5.3) re-
main unchanged upon changing four NN dimensions to DD, except for bosonic modes
polarized along the directions longitudinal to these four dimensions (and for fermions
related to them by the unbroken susy of the total system). To be concrete, consider-
ing the four MM adjoint bosons, and two MM adjoint fermions θα associated to the
universal MM vector multiplets, they suffer the projection
λ = +γΩ,M λ
T γ−1M (5.5)
Hence they transform in the of Sp(m) for the SO projection, and in the of
SO(m) for the Sp projection. On the other hand, for the two fermion zero modes θ̃α̇,
the projection is
λ = −γΩ,M λT γ−1M (5.6)
and leads to two fermion zero modes in the of Sp(m) for the SO projection, and in
the of SO(m) for the Sp projection.
This implies that in order to obtain two fermion zero modes from this univer-
sal multiplet, in order to generate a superpotential, one should consider instantons
with orthogonal gauge group and multiplicity one (O(1) instantons). For instantons
with symplectic gauge group and multiplicity two (Sp(2) instantons), there are two
additional fermion zero modes in the triplet representation. As mentioned, we will
continue to consider such instantons in our relaxed scan. Multiple instantons, i.e.
boundary states with higher multiplicity, lead to a larger amount of additional fermion
zero modes (due to the larger gauge representations for the fermions), and do not
contribute to superpotentials; we will not consider such cases even in relaxed scans,
since they also very often lead to too many charged fermion zero modes and cannot
contribute to the operators of interest (except possibly for O(2) and U(2) instantons
with low intersections, which are kept in our scan as a curiosity).
Similarly, for the p± sets of MM scalars and fermions associated to the MM 4d
chiral multiplets, the projection is
λ = ±γΩ,M λT γ−1Ω,M (5.7)
with the same sign choice as in (5.3). The different structure of γΩ implies that, for
the SO projection we get p+, p− sets of scalars and fermions in the , , while for
the Sp projection there are p+, p− sets of scalars and fermions in the , .
This concludes the discussion of the MM sector. Let us not consider the Ma sec-
tors, from the information from the Ma sectors. Notice that this implies changing
four NN boundary conditions to DN, which have different moddings. Hence the states
are different in both situations, but the information on the multiplicities is preserved.
Specifically, in the NS sector the DN boundary condition introduce an additional vac-
uum energy which generically makes all states massive. Hence there are no massless
scalar zero modes in generic Ma sectors. In the R sector, the change in the mod-
dings reduces the dimension of the massless ground state, leading to a single (chiral)
fermionic degree of freedom. Since the orientifold action maps the Ma sector to Ma′
sectors, there are no subtleties in the orientifold projection. The end result is LMa,
LMa′ fermion zero modes in the ( M, a), ( M, a). The net number of chiral fermion
zero modes in the ( M, a) is given by IMa = LMa′ − LMa, i.e. the net number of
chiral multiplets in the related Ma sector.
The results for orientifold projections for real branes are shown in table 3.
Proj. Multiplet in M M (before orient.) M (after orient.) M (after orient.)
SO N = 1 vect. mult. U(m) O(m) Sp(m)
2 f + 2 f + 4 b
N = 1 ch. mult. (p+ + p−) Ad p+ + p− 2p+ ( f + b ) +
2p− ( f + b )
Sp N = 1 vect. mult. U(m) Sp(m) O(m)
2 f + 2 f + 4 b
N = 1 ch. mult. (p+ + p−) Ad p+ + p− 2p+ ( f + b ) +
2p− ( f + b )
Any N = 1 ch. mult. LMa′( M, a)+ LMa′( M, a)+ LMa′( M , a) f
LMa( M, a) LMa( M, a) LMa( M , a) f
net IMa( M, a) net IMa( M , a) f
Table 3: Orientifold projection for real branes: Massless modes of the 4d space-filling branes
M (before and after the orientifold projection) and zero modes on the instanton branes M
(denoted with sub-indices b, f for bosonic and fermionic modes)
Complex brane instantons
We now consider the case of complex brane instantons. The arguments are very
similar, hence the discussion is more sketchy. Consider m 4d spacefilling branes M,
associated to the internal boundary state of the instanton brane M of interest. The
MM leads to a 4d N = 1 U(m) vector multiplet and a number LMM′ of adjoint chiral
multiplets. The orientifold action maps it to the M′M′ sector, hence we may keep just
the former and impose no projection. The MM′ sector is mapped to itself under the
orientifold projection. Denoting by γΩ,M the action on Chan-Paton indices, the MM′
modes split into sets L±MM , L
M ′M ′, which suffer a projection
λ = ±γΩ,M λT γ−1Ω,M (5.8)
leading, for γΩ,M = 1m, to L
chiral multiplets in the , , and L+
chiral multiplets in the , . The net number of chiral multiplets in the ,
is I+
M,M − L+M′M′, I−MM′ = L−M,M − L−M′M′ . And oppositely for γΩ,M = ǫm.
Finally, the Ma, Ma′ and related sectors lead, after the orientifold projection, to
LMa′ , LMa, LM′a′ , LM′a chiral multiplets in the ( M, a), ( M, a), ( M, a), ( M, a).
In order to simplify notation, we replaceM →M in these expressions in our discussions
of instanton zero modes.
Let us now consider m brane instantons M and compute their zero mode spectrum
in terms of the above. In the MM (and its image M ′M ′) sector there are four scalar
modes and four fermions in the adjoint of the U(m) gauge symmetry group; these are
related to the 4d vector multiplet in the MM sector. In addition, there are LMM ′
sets of scalars and fermions in the adjoint, related to the LMM′ non-universal chiral
multiplets in the MM sector. The MM ′ sector is mapped to itself, and one has to
impose the orientifold projection (recalling that the matrix γΩ,M differs from γΩ,M).
For γΩ,M = 1, hence γΩ,M = ǫ, we obtain L
MM , L
MM chiral multiplets in the , ,
and L+M ′M ′ , L
M ′M ′ chiral multiplets in the , . The net number of chiral multiplets
in the , is I+MM ′ = L
MM − L+M ′M ′, I−MM ′ = L−MM − L−M ′M ′ . And oppositely for
γΩ,M = ǫ hence γΩ,M = 1.
In theMa,Ma′ and related sectors, there are generically no bosonic zero modes, and
there are LMa, LM′a′ , LMa′ , LM′a chiral fermion zero modes in the ( M , a), ( M , a),
( M , a), and ( M , a) respectively. The net number of chiral fermion zero modes in
the ( M , a) and ( M , a) is given by IMa = LMa′ −LM′a′ and IMa′ = LMa−LM′a. In
order to simplify notation, we replace M →M in these expressions in our discussions
of instanton zero modes.
The results for orientifold projections for real branes are shown in table 4.
6 Search for M instantons
In this section we perform a search of models which admit an instanton inducing a
right-handed neutrino Majorana mass operator. Namely, for each model with the
chiral content of the SM in the classification described in Section 4.2, we first scan over
boundary states, searching for all instantons with the required uncharged and charged
fermion zero mode structure to yield neutrino masses. We then relax our criteria a bit
and allow for instantons with correct charged zero mode structure but having extra
non-chiral zero modes (both charged and uncharged). The idea is that these non-chiral
zero modes could be lifted by diverse effects, as discussed.
It is important to recall that the cubic couplings between instanton zero modes and
4d chiral multiplets are difficult to compute in Gepner model orientifolds. Hence, we
will simply assume that such couplings are non-zero if there is no symmetry forbidding
them.
Proj. Multiplet in M M (before orient.) M (after orient.) M (after orient.)
Any N = 1 vect. mult. U(m)× U(m)′ U(m) U(m)
4Ad f + 4Ad b
N = 1 ch. mult. padj Ad + padjAd
′ padjAd 2padj ( Ad f + Ad b )
SO N = 1 ch.mult. LMM( M, M′) L
2L+MM b,f + 2L
MM b,f
LM′M′( M, M′) L
M′M′ M
M′M′ M
2L+M ′M ′ b,f + 2L
M ′M ′ b,f
Sp N = 1 ch.mult. LMM( M, M′) L
2L+MM b,f + 2L
MM b,f
LM′M′( M, M′) L
M′M′ M
M′M′ M
L+M ′M ′ b,f + L
M ′M ′ b,f
Any N = 1 ch. mult. LMa′( M, a)+ LMa′( M, a) LMa′( M , a) f
. . . LMa( M, a) LMa( M , a) f
. . . LM′a′( M, a) LM ′a′( M , a) f
. . . LM′a( M, a) LM ′a( M , a) f
net IMa( M, a) net IMa( M , a) f
net IMa′( M, a) net IMa′( M , a) f
Table 4: Orientifold projection for complex branes: Massless modes of the 4d space-filling
branesM (before and after the orientifold projection) and zero modes on the instanton branes
M (denoted with sub-indices b, f for bosonic and fermionic modes)
6.1 The instanton scan
Our detailed strategy will become clear along the description of the results. Given a
set of a,b,c,d standard model branes, we must look for additional boundary states M
that satisfy the requirements of a (B−L)-violating instanton. From the internal CFT
point of view this is just another boundary state, differing from 4d spacefilling branes
only in the fully localized 4d spacetime structure. The minimal requirement for such
a boundary state is B − L violation, which means explicitly
IMa − IMa′ − IMd + IMd′ 6= 0 (6.1)
It is easy to see that the existence of such an instanton implies (and hence requires) the
existence of a Stückelberg coupling making B−L massive. To see this, consider adding
to the Standard Model configuration a 4d spacefilling brane M (in fact used in Section
5) associated to the boundary stateM (RR tadpoles can be avoided by simultaneously
including M antibranes, which will not change the argument). The new sector in
the chiral spectrum charged under the branes M can be obtained by reversing the
argument in Section 5, and is controlled by the intersection numbers of M . From
the above condition it follows that the complete system has mixed U(1)B−L × (GM)2
anomalies, where GM is the Chan-Paton-factor of brane M. These anomalies are
cancelled by a Green-Schwarz mechanism involving a (B −L)-axion bilinear coupling,
which ends up giving a mass to B−L via the Stückelberg mechanism. This coupling is
in fact not sensitive to the presence of the brane M, hence it must have been present
already in the initial model (without M).
Hence the existence of a boundary labelM that satisfies (6.1) implies that B−L is
massive. Unfortunately the converse is not true: even if B−L has a Stückelberg mass,
this still does not imply the existence of suitable instantons satisfying (6.1)16 Indeed,
in several models we found not a single boundary state satisfying (6.1).
Note that, since hypercharge must be massless, one can use the reverse argument
and obtain that
IMa − IMa′ − IMc + IMc′ − IMd + IMd′ = 0 (6.2)
in all models. We verified this for all models we considered as a check on the compu-
tations.
As already discussed in Section 4.2, in the search for SM constructions in Gepner
orientifold, there are 391 models with massless hypercharge and massive B−L. In these
models we found a total of 29680 instantons with B−L violation, i.e. with intersection
numbers satisfying (6.1). Of course, in order to serve our purpose of generating a
Majorana mass superpotential, the instantons have to satisfy some more conditions.
Let us consider them in order of importance, and start with the conditions on the net
number of chiral fermion zero modes charged under the 4d observable sector. Clearly
we need IMa = IMa′ and IMb = IMb′ . The latter condition is automatically satisfied in
this case, because the b-brane is real in all 391 models. The chiral conditions on the
zero modes charged under the branes c and d are as in [3]17 and are given in equations
(3.6), (3.7) (3.8) of the present paper. These are the instantons of most interest, and
on which we mainly focus. However, as discussed in Section 3.4, other important B-
and/or L- violating operators (such as the Weinberg operator or the LH operator) can
16From intuition in geometric compactifications, one expects that there may always exist a D-brane
with the appropriate topological pairings, but there is no guarantee that there is a supersymmetric
representative in that topological sector, and even less that it would have no additional fermion zero
modes. Note also that even if such D-brane instantons exists, there is no guarantee that it will fall in
the scan over RCFT boundary states.
17Note that there is a sign change in the contribution of the U(1)d generator to Y in comparison
to [3]
be generated by instantons with similar intersection numbers, up to a factor of 2 and
a sign, see table 2. For this reason we also allow at this stage any instanton which has
the correct number of charged zero modes to generate them. Imposing these conditions
reduces the number of candidate instantons potentially contributing to neutrino masses
in any of the models to 1315.
All instantons satisfying these requirements are summarized in the table 5. In
columns 1,2 and 3 we list the tensor combination, MIPF and orientifold choice for
which the model occurred. The latter two numbers codify simple current data that
describe respectively a MIPF and an orientifold. MIPFs are in general defined by
means of a subgroup H of the simple current group G, plus a certain matrix X of
rational numbers [55]. Orientifolds are defined by a simple current and a set of signs
[43]. In previous work [6] we have enumerated these quantities (up to permutation
symmetries) and assigned integer labels to them for future reference. We only refer
to these numbers here, but further details are available upon request. Usually for
each MIPF and orientifold which contains the standard model there are several choices
a,b,c,d for which it is obtained. For a given choice of tensor combination, MIPF and
orientifold and SM branes there may be several instantons. For clarity we put all such
instantons together in the information in table 5. In column 4 we indicate which type
of instanton branes were found. Five types are distinguished: O1, O2, S2, U1 and
U2, corresponding to O(1), O(2), Sp(2), U(1) and U(2) Chan-Paton symmetry on the
instanton volume. The number indicates the instanton brane multiplicity that gives
the correct number of instanton charged zero modes from the a, b, c, d branes, to lead
to right-handed neutrino Majorana masses. The number of zero modes is in general
the product of the instanton brane multiplicity and ‘intersection number’ with the
corresponding 4d spacefilling brane. As discussed in Section 5, for symplectic branes
the smallest possible brane multiplicity is 2. As we discussed there, only O1 instantons
may have the required universal minimal set of two zero modes in the uncharged sector.
Still we look for all O(1), Sp(2) and U(1) instantons which may yield a superpotential
if the extra uncharged fermion zero modes. In this vein we also include a search for
O2 and U2 instantons. Note also that such O2 or U2 instantons imply the existence
of other instantons involving the same boundary state, but with multiplicity 1, which
may lead to the R-parity violating operator LH . We will discuss the generation of
R-parity violating operators at the end of this section. The third character (+ or −)
in the instanton in table 5 is the sign of IMc′ − IMc. For the instantons giving rise to
right-handed neutrino Majorana masses this sign should be negative, whereas it should
be positive for instantons giving rise to the Weinberg operator (or the LH operator),
see table 2.
The 1315 instantons are divided in the following way over the different types: 3 of
types O1+ and O1−, 46 of type U1+, 24 of type U1−, 550 S2+, 627 S2−, 27 of types
U2+ and U2− and four of types O2+ and O2−. Notice that the vast majority (97.5%)
of the instanton solutions are of type S2+ and S2−. This is encouraging given the
nice properties of such instantons, concerning e.g. R-parity conservation. Note also
that in almost all cases both S− and S+ are simultaneously present,18 so both sources
of physical neutrino Majorana masses (from the see-saw mechanism or the Weinberg
operator) are present. The other instanton classes possibly generating right-handed
neutrino masses are O1− and U1−, which are much less abundant. There is just one
orientifold with O1− instantons, for which one can obtain cancellation of RR tadpoles,
see below. On the other hand we have found no orientifold with U1− instantons and
cancellation of tadpoles, see below.
Table 5: Summary of instanton branes.
Tensor MIPF Orientifold Instanton Solution
(1,16,16,16) 12 0 S2+, S2− Yes
(2,4,12,82) 19 0 S2−! ?
(2,4,12,82) 19 0 U2+!, U2−! No
(2,4,12,82) 19 0 U1+, U1− No
(2,4,14,46) 10 0
(2,4,14,46) 16 0
(2,4,16,34) 15 0
(2,4,16,34) 15 1
(2,4,16,34) 16 0 S2+, S2− Yes
(2,4,16,34) 16 1
(2,4,16,34) 18 0 S2− Yes
(2,4,16,34) 18 0 U1+, U1−, U2+, U2− No
(2,4,16,34) 49 0 U2+, S2−!, U1+ Yes
Continued on next page
18In some models contributing many instantons there is an exact symmetry between S− and S+.
This explains the approximate symmetry in the full set. In some cases this symmetry can be understood
in terms of flipping the degeneracy labels of boundary states. We regard it as accidental, since it is
not found in all models.
Table 5 – continued from previous page
Tensor MIPF Orientifold Instanton Solution
(2,4,16,34) 49 0 U1− No
(2,4,18,28) 17 0
(2,4,22,22) 13 3 S2+!, S2−! Yes!
(2,4,22,22) 13 2 S2+!, S2−! Yes
(2,4,22,22) 13 1 S2+, S2− No
(2,4,22,22) 13 0 S2+, S2− Yes
(2,4,22,22) 31 1 U1+, U1− No
(2,4,22,22) 20 0
(2,4,22,22) 46 0
(2,4,22,22) 49 1 O2+, O2−, O1+, O1− Yes
(2,6,14,14) 1 1 U1+ No
(2,6,14,14) 22 2
(2,6,14,14) 60 2
(2,6,14,14) 64 0
(2,6,14,14) 65 0
(2,6,10,22) 22 2
(2,6,8,38) 16 0
(2,8,8,18) 14 2 S2+!, S2−! Yes
(2,8,8,18) 14 0 S2+!, S2−! No
(2,10,10,10) 52 0 U1+, U1− No
(4,6,6,10) 41 0
(4,4,6,22) 43 0
(6,6,6,6) 18 0
Most models have a hidden sector containing extra boundary states beyond the
SM ones. In the same spirit of imposing chiral conditions first, we should require
that IMh = IMh′, where h is a hidden sector brane. This is to guarantee that the
generated superpotential does not violate some hidden sector gauge symmetry which
would require the presence of hidden sector fields along with the νR bilinear. The latter
condition is not imposed on the previously known hidden sector (i.e. the one in [6, 7]),
but instead a new search for tadpole solutions was performed, for each M , restricting
the candidate hidden sector branes to those satisfying IMh = IMh′ (as discussed in
Section 5). This is because in general the known hidden sector in [6, 7] is just a sample
out of a huge number of possibilities.
In column 5 we indicate for which instantons it was possible to satisfy the tadpole
conditions with this additional constraint. With regard to observable-hidden matter we
use the same condition as in [6], namely that it is allowed only if it is vector-like. Such
a solution could be found for 879 of the 1315 instantons, with ten cases inconclusive
(i.e it was computationally too difficult to decide if a solution does or does not exist).
The latter are indicated with a question mark in column 5 (for most of the undecidable
cases there is a tadpole solution for a different instanton with the same characteristics;
for that reason just one question mark appears).
Independently of the RR tadpole condition (since there may be alternative sources
for its cancellation, or hidden sectors which fall beyond the reach of RCFT), we can
also consider the further constraint that the number of charged fermion zero modes is
exactly right, not just in the chiral sense. This means IMa = IMa′ = IMb = IMb′ = 0,
IMc = 2, IMc′ = 0 and IMd = −2, IMd′ = 0 or vice-versa. Furthermore we require that
there are no adjoint or rank-2 tensor zero-modes (note that the latter could be chiral
if the instanton brane is complex, and indeed they are in some of the 1315 cases).
This reduces the 1315 instantons to 263. In column 4 we indicate those cases with
an exclamation mark. It is noteworthy that the success rate for solving the tadpole
conditions is highest for these instantons: 254 of the 263 allow a solution (with 3
undecided). If an exclamation mark appears in column 4, this only indicates that
some of the instantons are free of the aforementioned zero modes, not that all of them
are. But in all cases, if there are tadpole solutions, they exist in particular for the
configurations with an exclamation mark. Finally we may impose the condition that
IMh and IMh′ are separately zero. This is indicated with and exclamation mark in
column 5. This turns out to be very restrictive. The only cases where this happens
have no hidden sector at all.
It is worth remarking that the only instantons having exactly the correct set of
charged zero modes and cancelling tadpoles are of S2± type. Also those instantons are
the only cases marked with an exclamation mark in column 4 and 5. These examples,
which will be discussed below in some detail, also have just the minimal set of fermion
zero modes, except for the universal sector (which for Sp(2) instantons contains two
extra triplets).
The main conclusion about this scan is that we did not find any instantons with
exactly the zero mode fermions to generate the neutrino mass superpotential. However
we have found a number of examples which come very close to that, with exactly the
required charged zero modes and a very reduced set of extra uncharged zero modes
from the universal sector. These extra zero modes are non-chiral and hence one expects
that e.g. RR/NS fluxes or other effects may easily lift them, as we discussed in section
2. Concerning O(1) instantons, which have just the two required fermion zero modes
in the universal sector, we have found one example, with the appropriate net structure
of charged zero modes. However, it has plenty of other extra zero modes. We discuss
examples of O(1) and Sp(2) instantons in the following subsections.
6.2 An O1 example
Let us first discuss the case of O(1) instantons. In principle they would be the more
attractive since they have no undesirable universal zero modes at all. Unfortunately
this type of instanton is rare within the set we scanned, and we found just one example
with a solution to the tadpole equations without any unwanted chiral zero-modes. The
instanton however has a very large number of uncharged and charged vector-like zero
modes.
The standard model brane configuration occurs for tensor product (2, 4, 22, 22),
MIPF 49, orientifold 1, boundaries (a,b,c,d) = (487, 1365, 576, 486). As usual we only
provide this information in order to locate this model in the database. Further details
are available on request.
The bi-fundamental fermion spectrum of this model in the (a,b,c,d) sector is fairly
close to the MSSM: there is an extra up-quark mirror pair, two mirror pairs of lepto-
quarks with down quark charges and one with up-quark charges, plus two extra right-
handed neutrinos (i.e. a total of five right-handed neutrinos). There are three MSSM
Higgs pairs. The tensor spectrum is far less appealing, in particular for brane c: this
has 25 adjoints and 7 vector-like pairs of anti-symmetric tensors.
As we said, there is just one instanton brane of type O1−. It has exactly the
right number of zero-modes with brane d, but five superfluous pairs of vector-like zero-
modes with brane c, plus one vector-like pair with brane a. In addition there are four
symmetric tensor zero-modes on the instanton brane (which of course are vector-like,
since it is a real brane): the parameter p+ in table 3 is equal to 2.
The tadpole solution that is (chirally speaking) compatible with this instanton has
a large hidden sector: O(1)×O(2)4×O(3)×U(1)2×Sp(2)2×U(3) (there are other pos-
sibilities, but no simple ones). This hidden sector introduces more undesirable features:
vector-like observable/hidden matter, vector-like instanton/hidden sector modes, plus
chiral and non-chiral matter within the hidden sector. Finally the coupling ratios are
as follows: α3/α2 = .54, sin
2θw = .094, and the instanton coupling is 3.4 times weaker
than the QCD coupling (α3/αInstanton = 3.4).
Despite these unappealing features this model does demonstrate the existence of
this kind of solution.
6.3 The S2 models
As we have mentioned, these are the examples which come closer to the minimal set of
fermion zero modes. As we see in Table 5, all such instantons satisfying the criteria on
the zero mode structure (except for the extra universal zero modes) appear for models
based on the same CFT orientifold. It is the one obtained from the (2, 4, 22, 22) Gepner
model with MIPF 13 and orientifold 3 in the table. The model is obtained as follows.
6.3.1 The closed string sector
We start with the tensor product (2,4,22,22). This yields a CFT with 12060 primary
fields, 48 of which are simple currents, forming a discrete group G = Z12×Z2×Z2. After
taking into account the permutation symmetry of the last two factors, we find that this
tensor product has 54 symmetric MIPFs, and we choose one of them to build the model
of interest. For convenience we specify all quantities in terms of a standard minimal
model notation, but also in terms of the labelling of the computer program “kac” that
generates the spectrum. This particular MIPF is nr. 13. To build it we choose a
subgroup of G, which is isomorphic to H = Z12 × Z2. The generator of the Z12 factor
is primary field nr. 1, (0, 0, 0, {24,−24, 0}, {24, 20, 0}), and the Z2 factor is generated
by primary field nr. 24, (0, 0, 0, 0, {24, 20, 2}). The representations are specified on a
basis (NSR, k = 2, k = 4, k = 22, k = 22), i.e. the boundary conditions of the NSR-
fermions and the four minimal models in the tensor product. Here 0 indicates the CFT
vacuum, and for all other states we use the familiar (l, q, s) notation for the N = 2
minimal models. The first generator has conformal weight h = 11
and has ground state
dimension 1. The second has weight h = 11
and has ground state dimension 2: the
ground state contains both (0, 0, 0, 0, {24, 20, 2}) and (0, 0, 0, {24, 20, 2}, 0). The matrix
X defining the MIPF according to the prescription given in [53][54][55] is
(6.3)
This simple current modification is applied to the charge conjugation invariant of the
tensor product. This defines a MIPF that corresponds to an automorphism of the
fusion rules, and that pairs all the primaries in the CFT off-diagonally. The number
of Ishibashi states, and hence the number of boundary states is 1080. The MIPF is
invariant under exchange of the two k = 22 factors: this maps current 24 to itself, and
current 1 to current 11, which is also in H. Hence this symmetry of the tensor product
maps H into itself, and it also preserves the matrix X .
To define an orientifold, we must specify a “Klein bottle current” plus two signs
defined on the basis of the simple current group. For the current K we use the
generator of the second Z2 in G, primary field nr. 12. This is the representation
(0, 0, {4,−4, 0}, {(24, 16, 2)}, {(24,−12, 2)}) which is degenerate with nine other states,
all of dimension 1 and conformal weight 7. The crosscap signs are chosen, on the afore-
mentioned basis of H as (+,−). This results in a crosscap coefficient of 0.0464731, and
it is orientifold nr. 3 of a total of 8. The orientifold is also invariant under permutation
of the identical factors.
The closed string spectrum contains 14 vector multiplets and 60 chiral multiplets.
6.3.2 The standard model branes
To build a standard model configuration we have to specify the boundary state labels.
It turns out that we have four choices for label a and b, one for c and two for d.
This leads to a total of 32 possibilities. Among these 32 there are 22 have distinct
spectra (distinguished by the number of vector-like states), but for all 32 choices one
obtains the same set of dilaton couplings. It seems plausible that these choices simply
correspond to putting the a, b and d branes in slightly different positions, so that we
move the configuration in brane moduli space. The choices are as follows (these are
boundary labels assigned by the computer program, and can be decomposed in terms
of minimal model representations; this will be explained in table 6 below)
a : 10, 22, 130, 142
b : 210, 282, 290, 291
c : 629
d : 712, 797
There are additional possibilities, but they do not give rise to additional distinct spec-
Table 6: Branes appearing in standard model configurations
Label Orbit/Deg. Reps Weight Dimension
10 240 (0, 0, 0, 0, {10, 0, 0}) 5/4 1
130 2760 (0, 0, 0, {10, 0, 0}, 0) 5/4 1
22 [528,0] (0, 0, 0, {1,−1, 0}, {11, 1, 0}) 3/2 1
(0, 0, 0, {1, 1, 0}, {11,−1, 0}) 3/2 1
142 [3048,0] (0, 0, 0, {11,−1, 0}, {1, 1, 0}) 3/2 1
(0, 0, 0, {11, 1, 0}, {1,−1, 0}) 3/2 1
210 4248 (0, 0, {3, 3, 0}, {3,−3, 0}, {9,−9, 0}) 1/2 1
282 5760 (0, 0, {3, 3, 0}{9,−9, 0}{3,−3, 0}) 1/2 1
290 [5952,0] (0, 0, {1, 1, 0}{9, 7, 0}{11,−11, 0}) 5/6 1
291 [5952,24] (0, 0, {1, 1, 0}{9, 7, 0}{11,−11, 0}) 5/6 1
629 [9348,30] (0, (1,−1, 0), 0, {9, 9, 0}{5,−3, 0} 7/12 1
712 [9852,0] (0, {1, 1, 0}{3,−3, 0}{1, 1, 0}{5, 5, 0}) 1/2 2
(0, {1, 1, 0}{1,−1, 0}{1, 1, 0}{5,−3, 0}) 1/2 2
797 [10356,30] (0, {1, 1, 0}{3,−3, 0}{5, 5, 0}{1, 1, 0}) 1/2 2
(0, {1, 1, 0}{1,−1, 0}{5,−3, 0}{1, 1, 0}) 1/2 2
The second column gives the boundary labels in terms of a primary field label and
a degeneracy label (boundaries not indicated by square brackets are not degenerate).
The labels appearing in columns 1 and 2 are assigned by the computer program, and
are listed here only for the purpose of reproducing the results using that program.
In column 2, the boundary labels are expressed in terms of primary field labels, as
in formula (A.4). If a single number appears, this is a representative of an H-orbit
corresponding to the boundary. If square brackets are used, this means that the H-
orbit has fixed points, and that it corresponds to more than one boundary label. The
second entry in the square brackets is the degeneracy label, and refers to a character of
the “Central Stabilizer” defined in [43]; the details of the definition and the labelling
will not be important here. In this case the first entry within the square brackets refers
to an orbit representative.
These orbit representatives can also be expressed in a standard form for minimal
model tensor products. This is done in column 3. This is basically the same expansion
shown in (A.4), except that the degeneracy label ΨI turns out to be trivial in all cases,
both for the standard model and for the instanton branes shown below (although the
theory does contain primaries with non-trivial Ψ’s). In columns 4 and 5 we specify the
weight and ground state dimension of the corresponding highest weight representation.
These data are not directly relevant for the boundary state, but helps in identifying it.
Since boundaries are specified by orbit representatives, it is not straightforward to
compare them, since the standard choice (the one listed in column 2) is arbitrary. For
this reason we have used another representative in columns 3, 4 and 5, selected by an
objective criterion: we choose the one of minimal dimension and minimal conformal
weight (in that order). If there is more than one representative satisfying these criteria
we list all.
6.3.3 The open string spectrum
In Table 7 we summarize the spectra of the 32 models. The first four columns list the
a,b,c,d brane labels. The last eight columns specify the total number of multiplets of
types Q (quark doublet), U (up quark singlet), D (down quark singlet), L (lepton dou-
blet), E (charged lepton singlet), N (neutrino singlet), Y (lepto-quark) and H (Higgs).
The numbers given are for the total number of lefthanded fermions in the represen-
tation, plus their complex conjugates. So for example a 7 in column “Q” means that
there are 5 quark doublets in the usual representation (3, 2, 1
), plus two in the complex
conjugate representation (3∗, 2,−1
This yields the required three families of quark doublets, plus two mirror pairs.
Hence the smallest number that can occur in the six columns QUDLEN is three, if
there are no mirrors (note that cubic anomaly cancellation requires three right-handed
neutrinos in this class of models). The lepto-quarks Y are all in the same representation
as the down-quarks (D), or the conjugate thereof, and they occur only as vector-like
mirror pairs. They differ from D-type mirror quarks because they carry lepton number,
because they come from open strings ending on the d-brane instead of the c-brane.
In general, there can also exist U-type lepto-quarks, but in these models they do not
occur. Finally the numbers 10, 18 and 26 in column ’H’ mean that there are 5, 9
or 13 MSSM Higgs pairs H + H̄. It is worth noticing that right-handed quarks U,D
and neutrinos N = νR do not have vectorlike copies. On the other hand right-handed
leptons E always have one and the left-handed fields Q,L may have up to 3 vector-like
copies.
Table 7: Spectrum all 32 configurations.
U(3) Sp(2) U(1) U(1) Q U D L E N Y H
10 210 629 712 7 3 3 9 5 3 6 10
22 210 629 712 7 3 3 9 5 3 6 10
130 210 629 712 3 3 3 9 5 3 2 10
142 210 629 712 3 3 3 9 5 3 2 10
10 282 629 712 3 3 3 5 5 3 6 26
22 282 629 712 3 3 3 5 5 3 6 26
130 282 629 712 7 3 3 5 5 3 2 26
142 282 629 712 7 3 3 5 5 3 2 26
10 290 629 712 3 3 3 3 5 3 6 18
22 290 629 712 3 3 3 3 5 3 6 18
130 290 629 712 3 3 3 3 5 3 2 18
142 290 629 712 3 3 3 3 5 3 2 18
10 291 629 712 3 3 3 3 5 3 6 18
22 291 629 712 5 3 3 3 5 3 6 18
130 291 629 712 3 3 3 3 5 3 2 18
142 291 629 712 3 3 3 3 5 3 2 18
10 210 629 797 7 3 3 5 5 3 2 10
22 210 629 797 7 3 3 5 5 3 2 10
130 210 629 797 3 3 3 5 5 3 6 10
142 210 629 797 3 3 3 5 5 3 6 10
10 282 629 797 3 3 3 9 5 3 2 26
22 282 629 797 3 3 3 9 5 3 2 26
130 282 629 797 7 3 3 9 5 3 6 26
142 282 629 797 7 3 3 9 5 3 6 26
10 290 629 797 3 3 3 3 5 3 2 18
22 290 629 797 3 3 3 3 5 3 2 18
130 290 629 797 3 3 3 3 5 3 6 18
142 290 629 797 3 3 3 3 5 3 6 18
10 291 629 797 3 3 3 3 5 3 2 18
22 291 629 797 5 3 3 3 5 3 2 18
130 291 629 797 3 3 3 3 5 3 6 18
Continued on next page
Table 7 – continued from previous page
U(3) Sp(2) U(1) U(1) Q U D L E N Y H
142 291 629 797 3 3 3 3 5 3 6 18
In the following table we list the multiplicities Laa and Laa′ of the branes that occur
in these models, leading to vector-like sets of adjoints and rank-2 tensors. Since brane
b is symplectic, the number of adjoints is equal to the number of symmetric tensors.
Table 8: 4d matter from the aa and aa′ sectors.
Boundary Adjoints Anti-symm. Symm.
a(10) 2 2 6
a(22) 2 2 2
a(130) 2 2 6
a(142) 2 2 2
b(210) - 14 10
b(282) - 14 10
b(290) - 14 6
b(291) - 14 6
c(629) 9 - 14
d(712) 3 - 6
d(797) 3 - 6
It should be emphasized that CFT constructions generically correspond to par-
ticular points in moduli space of CY orientifolds. Due to this, they usually have an
‘enhanced’ massless particle content with extra vector-like matter and closed string
gauge interactions. Thus one would expect that many of the massless vector-like chiral
fields present in this class of models could gain masses while moving to a nearby point
in moduli space.
6.3.4 The instantons
Each of these 32 Standard Model compactifications admits 8 instantons. The instanton
labels are identical for all the 32 models. They are listed in Table 9. The first five
columns use the same notation as for the standard model boundary labels. In column
6 we list the numerical value of the dilaton coupling to the instanton brane. This
quantity is proportional to 1
. It is instructive to compare these couplings to the gauge
couplings, in order to gain intuition on the suppression factor for our instantons. In
these models the U(3) dilaton couplings are 0.00622, so that the instantons are more
strongly coupled than QCD19 On the other hand in this particular model the ratio
α3/α2 at the string scale is 3.23 (the value of sin
2θw at the string scale is 0.527). All
of these couplings are subject to renormalization group running, and there are plenty
of vector-like states to contribute to this, if one assumes that they acquire masses at
a sufficiently low scale. One should perform a detailed renormalization group analysis
to check whether one may obtain consistency with the gauge couplings measured at
low-energies. Let us emphasize however that one expects that moving in moduli space
many of these vector-like states will gain masses and also the values of the different
gauge couplings will also generically vary.
Since the value of the Type II dilaton is a free parameter at this level, one can get the
appropriate (intermediate) mass scale for the right-handed neutrino Majorana masses
by choosing an appropriate value for the dilaton. In this context, it is satisfactory to
verify that the instanton couplings are unrelated to the gauge couplings, as expected
since they do not correspond to gauge instantons [3], and are in fact less suppressed
than the latter.
Note that the 8 instantons fall into two distinct classes (evidently not related by
any discrete symmetry, since the conformal weight on the boundary orbit is distinct,
and the coupling is different as well). Within each class, the orbits of the four instan-
ton boundaries appear to be related by the Z2 symmetries of interchange of the last
two tensor factors, and simultaneous inversion of the charge q of the minimal model.
However, one has to be very careful in reading off symmetries directly from the labels
in columns 3 of Tables (6) and (9) for a number of reasons. First of all the entries in
column 3 are representatives of boundary orbits, and these representatives themselves
are merely representatives of extension orbits. Secondly the action of any discrete
19 Note that the Type II dilaton in this compactifications is an arbitrary parameter which can
always be chosen so that we consistently work at weak coupling. It is the relative value of gauge
couplings which we are comparing here.
Table 9: Instantons for all 32 configurations
Lbl. Orbit/Deg. Reps Weight Dim. coupling
414 [8064,0] (0, {1, 1, 0}, 0, {22,−22, 0}, {20, 16, 0}) 5/2 1 0.0016993
417 [8076,30] (0, {1,−1, 0}, 0, {22, 22, 0}, {20,−16, 0}) 5/2 1 0.0016993
456 [8316,0] (0, {1, 1, 0}, 0, {20, 16, 0}, {22,−22, 0}) 5/2 1 0.0016993
459 [8328,30] (0, {1,−1, 0}, 0, {20,−16, 0}, {22, 22, 0}) 5/2 1 0.0016993
418 [8088,0] (0, {1, 1, 0}, 0, {22,−22, 0}, {18, 16, 0}) 5/3 1 0.0027033
420 [8100,0] (0, {1,−1, 0}, 0, {22, 22, 0}, {18,−16, 0}) 5/3 1 0.0027033
502 [8592,0] (0, {1, 1, 0}, 0, {18, 16, 0}, {22,−22, 0}) 5/3 1 0.0027033
505 [8604,30] (0, {1,−1, 0}, 0, {18,−16, 0}, {22, 22, 0}) 5/3 1 0.0027033
symmetry on the degeneracy labels can be non-trivial. In appendix B we discuss these
symmetries in more detail.
6.4 Other examples
The Sp(2) instanton examples just discussed are the ones which get closer to the
required minimal set of fermion zero modes. Under slightly weaker conditions, we
find many more solutions. In all these cases some additional mechanism beyond exact
RCFT will be needed to lift some undesirable zero modes.
The simplest such case is the following. The tensor product is (2, 8, 8, 18), MIPF
nr. 14, orientifold 2 (the precise spectra may be found using this information in the
database www.nikhef.nl/∼t58/filtersols.php). There are three distinct brane configu-
rations for which almost perfect instantons exist, namely (a,b, c,d) = (64, 562, 389, 67)
and (64, 577, 389, 67) and (65, 560, 189, 66). Each has six instantons, three of type S2+
and three of type S2−. As in the foregoing example, the six instantons are identical
for the three standard model configurations. In this example, they have three differ-
ent dilaton coupling strengths: .00254, .00665 and .0108 (each value occurs once for
S2+ and once for S2−). By comparison, the U(3)-brane dilaton coupling strength is
0.0119338, so that the instanton brane coupling is quite a bit stronger than the QCD
coupling. This is again an interesting point if we want that νR masses are not too much
http://www.nikhef.nl/~t58/filtersols.php
suppressed. Furthermore in this example there are three distinct instanton couplings,
so that one may expect three non-zero eigenvalues (with a hierarchy) in the mass ma-
trix. As in the previous examples there is not gauge coupling unification, one rather
has α3/α2 = .4813 and sin
2(θw) = .183 at the string scale. Again a full renormalization
group analysis should be performed in order to check consistency with the measured
low-energy gauge coupling values.
These models all have a hidden sector consisting of a single Sp(2) factor. They
have respectively 3, 1 and 3 susy Higgs pairs, and a spectrum of bi-fundamentals that
is closer to that of the standard model than the previously discussed Sp(2) examples:
quarks and leptons do not have vector-like copies (there are only some vector-like
leptoquarks), and even one of the three models have the minimal set of Higgs fields
of the MSSM. The rest of the spectrum is purely vector-like, and contains a number
of rank-2 tensors, including eight or six adjoints of U(3). Furthermore there is vector-
like observable-hidden matter. The only undesirable instanton zero-mode is a single
bi-fundamental between the hidden sector Sp(2) brane and the instanton brane. Still,
these SM brane configurations without the hidden sector, provide interesting and very
simple local models of D-brane sectors admitting instantons generating neutrino masses
(with the additional ingredients required to eliminate the extra universal triplets of
fermion zero modes).
6.5 R-parity violation
We now turn to the generation of other possible superpotentials violating B − L. An
instanton violates R-parity if the amount of B − L violation,
IMa − IMa′ − IMd + IMd′ (6.4)
is odd. Examples of instantons with that property were found in the following ten-
sor product/MIPF/orientifold combinations: [(1, 16, 16, 16), 12, 0], [(2, 4, 16, 34), 49, 0],
[(2, 4, 12, 82), 19, 0] [(2, 4, 22, 22), 49, 0] and [(2, 4, 16, 34), 18, 0]. Note that all cases for
which O2 or U2 instantons were found necessarily have R-parity violating instantons as
well: the corresponding O1 and U1 instantons have IMd or IMd′ equal to ±1, whereas
the intersection with the a is non-chiral. In principle, there are many more ways
to obtain R-parity violating instantons (either due to non-vanishing contributions to
IMa − IMa′ or higher values of IMd − IMd′), and indeed, many such instantons turn
out to exist. But the number of tensor product/MIPF/orientifold combinations where
they occur hardly increases: only in the case [(1, 16, 16, 16), 12, 0] we found R-parity
violating instantons, but no U1 or O1 instantons. This suggests that in the other cases
R-parity is a true symmetry of the model. Unfortunately we have no way of rigorously
ruling out any other non-perturbative effects, but at least the set we can examine re-
spects R-parity. This includes in particular the models without hidden sector (found
for [(2, 4, 22, 22), 13, 3] ) discussed above.
The following table list the total number of instantons with the chiral intersections
listed in table 2. The total number of instantons (boundaries violating the sum rule,
as defined in (6.1)) is 29680, for all standard model configurations combined. The
last four columns indicate how many unitary instantons satisfy the sum rule exactly
as listed in table (6.1), how many satisfy it with IMx ↔ −IMx′ (the column U’), and
how many O-type and S-type instantons there are. Here ‘S’ refers to boundaries with
a symplectic Chan-Paton group if the boundary is used as an instanton brane. All
intersection numbers for type S have been multiplied by 2 before comparing with table
2. For real branes, the relevant quantities used in the comparison are IMa − IMa′,
IMc−IMc′ and IMd−IMd′ , while IMb = 0. There are fewer unitary instantons possibly
generating Majorana masses then the numbers mentioned above because the conditions
we use here are stricter: we require here that IMx and IMx′ match exactly, not just their
difference. Note however that this still allows additional vector-like zero-modes. If we
only wish to consider cases without any spurious zero-modes, we may limit ourselves
to the O-type instantons in the last column. There are very few to inspect, and all of
them turn out to have a few non-universal zero modes.
D = 4 Operator U U’ S O
νRνR 1 2 627 3
LH̄LH̄ 0 5 550 3
LH̄ 3 3 0 4
QDL 8 4 0 4
UDD 0 0 0 4
LLE 8 4 0 4
QQQL 0 4 0 3892
UUDE 4 0 0 3880
Table 10: Number of instantons in our search which may induce neutrino masses (first 2
rows), R-parity violation (next 4 rows) or proton decay operators (last 2 rows).
The last two cases are B − L preserving dimension five operators, and obviously
do not come from the set of 29680 B −L violating instantons. They were searched for
separately, but the search was limited to the same 391 models we used in the rest of the
paper. Obviously, one could equally well look for such instantons in the full database,
since their existence does not require a massive B − L.
It is interesting to note that in the classes of MSSM-like models discussed earlier in
this section with the closest to minimal zero mode structure, there are no instantons
al all generating either R-parity violating or the B − L dim=5 operators in the table.
This makes them particularly attractive.
Note that all numbers in table 10 refer to the occurrence of instantons in the set
of 391 tadpole-free models with massive B-L, but without checking the presence of
zero-modes between the hidden sector and the instanton. It makes little sense to use
the hidden sector in the database for such a check, since this is just one sample from a
(usually) large number of possibilities. A meaningful question would be: can one find a
hidden sector that has no zero-modes with the instanton. We have done such a search
for the B − L violating instantons (see the exclamation marks in the last column of
table (5)), but not for the B − L preserving instantons.
7 Conclusions and outlook
In this paper we have presented a systematic search for MSSM-like Type II Gepner
orientifold models allowing for boundary states associated to instantons giving rise to
neutrino Majorana masses. This search is very well motivated since neutrino masses
are not easily accommodated in the semi-realistic compactifications constructed up to
now. String instanton induced Majorana masses provides a novel and promising way
to understand the origin of neutrino masses in the string theory context.
The string instantons under discussion are not gauge instantons. Thus, for example,
they not only break B + L symmetry (like ’t Hooft instantons do) but also B − L,
allowing for Majorana neutrino mass generation. The obtained mass terms are of
order Ms exp(−V/g2) but this suppression is unrelated to the exponential suppression
of e.g. electroweak instantons and may be mild. In fact we find in our most interesting
examples that the instanton action is typically substantially smaller than that of QCD
or electroweak instantons, and hence these effects are much less suppressed than those
coming from gauge theory instantons.
To perform our instanton search we have analyzed the structure of the zero modes
that these instantons must have in order to induce the required superpotential. This
analysis goes beyond the particular context of Gepner orientifolds and has general
validity for Type II CY orientifolds. We have found that instantons with O(1) CP
symmetry have the required universal sector of just two fermionic zero modes for the
superpotential to be generated. Instantons with Sp(2) and U(1) CP symmetries have
extra unwanted universal fermionic zero modes, which however may be lifted in a va-
riety of ways in more general setups, as we discuss in the text. In fact we find in
our search that around 98 % of the instantons with the correct structure of charged
zero modes have Sp(2) CP symmetry. Indeed, from a number of viewpoints the Sp(2)
instantons are specially interesting. The instantons we find with the simplest structure
of fermionic zero modes are Sp(2) instantons which are also the ones which are present
more frequently in the MSSM-like class of Gepner constructions considered. They have
also some interesting features from the phenomenological point of view. Indeed, due
to the non-Abelian structure of the CP symmetry, the structure in flavor space of the
neutrino Majorana masses factorizes. This makes that, irrespective of what particu-
lar compactification is considered, Sp(2) instantons may easily lead to a hierarchical
structure of neutrino masses. It would be important to further study the possible
phenomenological applications of the present neutrino mass generating mechanism.
String instanton effects can also give rise to other B- or L-violating operators. Of
particular interest is the dimension 5 Weinberg operator giving direct Majorana masses
to the left-handed neutrinos. We find that in the most interesting cases, different
instantons giving rise to the Weinberg operator and to νR Majorana masses are both
simultaneously present. Which effect is the dominant one in the generation of the
physical light neutrino masses depends on the values of the instanton actions and
amplitudes as well as on the value of the string scale. Instantons may also generate
dim< 5 operators violating R-parity. We find however that instantons inducing such
operators are extremely rare, and in fact are completely absent in the Gepner models
with the simplest Sp(2) instantons inducing neutrino masses.
There are many avenues yet to be explored. It would be important to understand
better the possible sources (moving in moduli space, addition of RR/NS backgrounds
etc.) of uplifting for the extra uncharged fermionic zero modes in the most favoured
Sp(2) instantons. A second important question is that we have concentrated on check-
ing the existence of instanton zero modes appropriate to generate neutrino masses; one
should further check that the required couplings among the fermionic zero modes and
the relevant 4d superfields (i.e. νR or LH̄) are indeed present in each particular case.
This is in principle possible in models with a known CFT description but could be
difficult in practice for the Gepner models here described.
Instantons can also generate other superpotentials with interesting physical appli-
cations. One important example is the generation of a Higgs bilinear (i.e. a µ-term)
in MSSM-like models [4, 3]. Thus, e.g., one could perform a systematic search for
instantons (boundary states) generating a µ-term in the class of CFT Gepner orien-
tifolds considered in the present article. Other possible application is the search for
instantons inducing superpotential couplings involving only closed string moduli. The
latter may be useful for the moduli-fixing problem, or for non-perturbative corrections
to perturbatively allowed couplings [56].
Finally, it would be important to search for analogous instanton effects inducing
neutrino masses in other string constructions (heterotic, M-theory etc.). A necessary
condition is that the anomaly free U(1)B−L gauge boson should become massive due
to a Stückelberg term.
The importance of neutrino masses in physics beyond the Standard Model is un-
questionable. We have shown that string theory instantons provide an elegant and
simple mechanism to implement them in semi-realistic MSSM-like string vacua, and a
powerful constraint in model building. In our opinion, the conditions of the existence
of appropriate instantons to generate neutrino masses should be an important guide in
a search for a string description of the Standard Model.
Acknowledgements
We thank M. Bertolini, R. Blumenhagen, S. Franco, M. Frau, S. Kachru, E. Kiritsis,
A. Lerda, D. Lüst, F. Marchesano, T. Weigand for useful discussions. A.M.U. thanks
M. González for encouragement and support. The research of A.N. Schellekens was
funded in part by program FP 57 of the Foundation for Fundamental Research of
Matter (FOM), and Research Project FPA2005-05046 of de Ministerio de Educacion y
Ciencia, Spain. The research by L.E. Ibáñez and A.M. Uranga has been supported by
the European Commission under RTN European Programs MRTN-CT-2004-503369,
MRTN-CT-2004-005105, by the CICYT (Spain), and the Comunidad de Madrid under
project HEPHACOS P-ESP-00346.
Appendix
A CFT Notation
Here we summarize the labelling conventions for various CFT quantities. Further
details and explanations can be found in [43].
It is important to keep in mind that there are four steps in the construction, each
involving choices of some quantities. The steps are
• A CFT tensor product
• An extension of the chiral algebra of this tensor product
• The choice of a MIPF
• The choice of an orientifold
The second and third step are easily confused. A MIPF can itself be of extension
type (although it can also be of automorphism or mixed type), meaning that it implies
an extension of the chiral algebra. The crucial difference between step two and three in
that case is that in step 2 all fields that are non-local with respect to the extension are
projected out, and the symmetry of the extension is imposed on all states of the CFT,
i.e in particular on all boundary states. The extension in step three acts as a bulk
invariant, but the boundary states are not required to respect the symmetry implied
by the extension.
Primary fields of N = 2 minimal models are labeled in the usual way by three
integers (l, q, s). In addition to these minimal models, one building block of our CFT’s
is of course a set of NSR fermions in four dimensions. They can be represented by the
four conjugacy classes (0), (v), (s), (c) analogous to those of a root lattice of type D.
Primary fields in a tensor product of M factors are therefore labelled as
I = ((x), (l1, q1, s1), . . . , (lM , qM , sM)) (A.1)
where x = 0, v, s or c.
This tensor product is extended by the alignment currents and the spin-1 field
corresponding to the space-time supersymmetry generator. This organizes the tensor
product fields into orbits, which can be labelled by one of the elements of the orbit.
We always choose the field of minimal conformal weight (or one of them, in case there
are more) as the orbit representative labelling the orbit. The supersymmetry generator
may have fixed points, leading to orbits appearing more than once as primary fields
of the extended theory. In those cases we need an additional degeneracy label to
distinguish them. It is convenient to choose for this label a character of the discrete
group that is causing the degeneracy, the “untwisted stabilizer”, which depends on I.
Denoting this character as ΨI we get then the following set of labels for the primaries
of the extended CFT
i = [I,ΨI ] (A.2)
where I has the form (A.1). If there are no degeneracies we will leave out the square
brackets and the ΨI .
In boundary CFT’s two new labels appear: the labels of Ishibashi-states that propa-
gate in the transverse channel of the annulus, and the boundary labels. In the simplest,
“Cardy” case both sets of labels are in one-to-one correspondence with the extended
CFT labels i. But if we consider non-trivial MIPFs Zij both sets of labels are different.
The Ishibashi states are in one-to-one correspondence with the fields i with Ziic 6= 0.
Degeneracies can occur here if Ziic 6> 0. This requires the introduction of a degeneracy
label. Such degeneracies may occur if the stabilizer of i (the set of simple currents that
fix i) is non-trivial. It is convenient to use elements J of the stabilizer as degeneracy
labels, so that the Ishibashi labels get the following form
m = (i, J) , (A.3)
where i is an extended CFT label, as defined above (to be precise, in some cases a
non-trivial degeneracy label is introduced even if Ziic = 1. The details will not matter
here).
Boundary states correspond to orbits of the simple current group H that defines a
MIPF. To label such orbits we choose a representative. There is no obvious canonical
representative (one could use one of minimal conformal weight, but the conformal
weight of orbit members of a boundary state does not play any rôle in the formalism,
unlike the conformal weight of a primary). So in this case we just make an arbitrary
choice. Once again there can be degeneracies. In this case they are due to a subgroup
of the stabilizer called the “Central Stabilizer”. It is convenient to label the boundary
states by an orbit representative i and a character ψi of the central stabilizer. If we
expand the boundary state label in all of its components we get
a = [i, ψi] = [[I,ΨI ], ψi] = [((x), (l1, q1, s1), . . . , (lM , qM , sM)),ΨI ], ψi] . (A.4)
Note that i is just a representative of a boundary orbit, and that I is just a represen-
tative of an orbit of the extension of the CFT.
B Instanton boundary symmetries
In the hidden-sector free example discussed in some detail in section 6 we have en-
countered Sp(2)-type instantons, the most common kind in our scan. This particular
model is the one that comes closest to the required zero mode count, although the only
superfluous zero modes are rather awkward. Let us assume that the effect of these su-
perfluous universal zero-modes instantons can be avoided. Then there is still another
problem we have to face, namely that the two zero modes αi and γi are related by
an Sp(2) transformation of the label i. Then we we need at least three independent
instantons (with unrelated couplings) to generate three non-zero neutrino masses, as
discussed in Section 3.3. Since the technology to compute the couplings is not yet
available, we cannot be completely sure that the relevant couplings are distinct, or
indeed that they are non-vanishing, but at least we can inspect if there are obvious
symmetries relating them.
The unextended tensor product (2, 4, 22, 22) has 64 discrete symmetries: five sep-
arate charge conjugations of the factors (including the NSR space-time factor) and
the interchange of the two identical k = 22 minimal models. To get space-time su-
persymmetry this tensor product is extended with the product of the simple current
Ramond ground states of each factor. These Ramond ground states are not invariant
under charge conjugation. Therefore this extension breaks the discrete symmetries to
Z2 × Z2, the combined charge conjugation of all five factors and the permutation of
the two identical factors. The combined charge conjugation also acts non-trivially on
the Ramond ground states in each factor, but the result is the charge conjugate of the
space-time supersymmetry generator, which is in the chiral algebra. The combined
conjugation is in fact the charge conjugation symmetry of the extended CFT. It turns
out that only a Z2 subgroup of Z2 × Z2 acts non-trivially on the simple currents of
the extended CFT: the permutation of the k = 22 factors acts in the same way as
charge conjugation. The action of these symmetries on the complete set of primary
fields is more complicated. It is easy to see that the permutation acts differently than
charge conjugation. In general the primary fields of the extended CFT are labelled
as i = [((x), (i1), . . . , (i4)),Ψ] (see Appendix A). The action of the permutation is to
interchange i3 and i4, but in cases with a non-trivial degeneracy it is not a priori
clear which of the degenerate states is the image of the map. This can be resolved by
examining the fusion rules, which should be invariant under the permutation:
Nijk = Nπ(i)π(j)π(k) , (B.1)
where N is a fusion coefficient and π a permutation or other automorphism. In general
there may be more than one way to resolve these ambiguities, resulting in additional
automorphism of the CFT. The standard example of this situation is the extension of
the affine algebra A1 level 4 by the simple current. The resulting CFT has an outer
automorphism, non-trivial charge conjugation, that has no counterpart in A1 level 4.
As mentioned above, the Z2 permutation symmetry is respected by the MIPF and
the orientifold, and since charge conjugation acts in the same way on the simple currents
as the permutation, charge conjugation is respected as well.
In this way we end up with (at least) a surviving Z2×Z2 discrete symmetry acting
on the boundary labels, or a larger discrete symmetry if that symmetry is extended
by the action on the degeneracy labels of the extension. The foregoing story repeats
itself for the action on the boundary labels. The boundary labels are given in terms
of the CFT labels plus a second degeneracy label, the one indicated by the second
entry in the square brackets in column 2 of tables (6) and (9). Once again one has to
determine not only how a symmetry acts on the first entry (this is just the action of
the symmetry in the extended CFT, respecting its fusion rules), but also how it acts
on the degeneracy labels. In this case the precise action can be determined from the
invariance of the annulus coefficients
Aiab = A
π̂(a)π̂(b) , (B.2)
where π is the action on the primaries of the extended CFT (as determined above) and
π̂ is the action on the boundary labels induced by π.
Since the orientifold choice is non-trivial, boundary charge conjugation does not
coincide with CFT charge conjugation. Indeed, the eight instanton boundary states
are invariant under boundary charge conjugation (which they must be in order to
produce a “real” Sp(2)-type instanton). However, just as permutations, CFT charge
conjugation may induce a non-trivial discrete symmetry on the boundary states.
In addition to these “outer automorphisms” there is the notion of boundary simple
currents, introduced in the appendix of [6]. These may be thought of as remnants of
the original simple currents, and imply relations between annulus amplitudes of the
Aiab′ = A
Ja(Jb)′ (B.3)
All of the aforementioned symmetries might relate instanton couplings, and hence
threaten their numerical independence. However, in order to do that they have to be
symmetries of the full standard model/instanton configuration, not just relate some of
the eight instantons to each other. It is easy to see that the permutation of the k = 22
factors changes the standard model brane configuration. Consider brane c: it turns out
that under permutation boundary state 629 it is mapped to boundary state 544 or 545
(depending on the action on the degeneracy label), which in any case is distinct. Hence
even if the instanton boundaries 414 and 456 resp. 418 and 502 are mapped to each
other by boundary permutation, at the same time the standard model configuration is
mapped to a distinct one.
This means that we may expect at least four distinct couplings, which should be
sufficient. It is of course possible to work out the discrete symmetries exactly, but in
view of this argument this would not yield any additional insight.
We do know the exact boundary orbits. The orbit of instanton label 414 is
(414, 415, 416, 417), so that instantons 414 and 417 are related. But the orbit of brane
c under the same action is (629, 628, 626, 627). Hence the action that relates 414 and
417 maps 629 to 627. In fact all four standard model boundaries a,b,c,d are mapped
to different ones. This implies that instantons 414 and 417 may produce different
couplings as well, so that all eight instantons may contribute in a different way.
These distinctions concern the disk correlators d(r)a in (3.12). The factors exp(−Re Ur)
will be related by discrete symmetries, and it seems reasonable to expect them to be
identical for instantons 414, 417, 456 and 459, which is indeed correct. However there
is no reason to expect the other four instantons to have the same suppression factor,
and indeed they do not.
Note that these symmetries imply the existence of a much larger set of standard
model configurations than the 32 discussed here. However, as mentioned before, the
32 models considered here display all possible distinct spectra.
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	Introduction
	Instanton induced superpotentials in Type II orientifolds
	D-brane instantons, gauge invariance and effective operators
	Zero mode structure for D-brane instantons
	Uncharged zero modes
	Charged fermion zero modes
	Instanton induced Majorana neutrino masses
	The MSSM on the branes
	Majorana mass term generation
	Flavor and the special case of Sp(2) instantons
	Other B- and L-violating operators
	The Weinberg operator 
	R-parity violating operators 
	Dimension 5 proton decay operators 
	CFT orientifolds
	Construction of the models
	Search for SM-like models
	Fermion zero modes for instantons on RCFT's
	Search for M instantons
	The instanton scan
	An O1 example
	The S2 models
	The closed string sector
	The standard model branes
	The open string spectrum
	The instantons
	Other examples
	R-parity violation
	Conclusions and outlook
	Appendix
	CFT Notation
	Instanton boundary symmetries

  Recently it has been shown that string instanton effects may give rise to
neutrino Majorana masses in certain classes of semi-realistic string
compactifications. In this paper we make a systematic search for supersymmetric
MSSM-like Type II Gepner orientifold constructions admitting boundary states
associated with instantons giving rise to neutrino Majorana masses and other L-
and/or B-violating operators. We analyze the zero mode structure of D-brane
instantons on general type II orientifold compactifications, and show that only
instantons with O(1) symmetry can have just the two zero modes required to
contribute to the 4d superpotential. We however discuss how the addition of
fluxes and/or possible non-perturbative extensions of the orientifold
compactifications would allow also instantons with $Sp(2)$ and U(1) symmetries
to generate such superpotentials. In the context of Gepner orientifolds with
MSSM-like spectra, we find no models with O(1) instantons with just the
required zero modes to generate a neutrino mass superpotential. On the other
hand we find a number of models in one particular orientifold of the Gepner
model $(2,4,22,22)$ with $Sp(2)$ instantons with a few extra uncharged
non-chiral zero modes which could be easily lifted by the mentioned effects. A
few more orientifold examples are also found under less stringent constraints
on the zero modes. This class of $Sp(2)$ instantons have the interesting
property that R-parity conservation is automatic and the flavour structure of
the neutrino Majorana mass matrices has a simple factorized form.

Introduction
String Theory, as the leading candidate for a unified theory of Particle Physics and
Gravity, should be able to describe all observed particle phenomena. One the most
valuable experimental pieces of information obtained in the last decade concerns neu-
trino masses. Indeed the evidence from solar, atmospheric, reactor and accelerator
experiments indicates that neutrinos are massive. The simplest explanation of the
smallness of neutrino masses is the see-saw mechanism [1]. The SM gauge symmetry
allows for two types of operators bilinear on the neutrinos (with dimension ≤ 4) :
Lν = MabνaRνbR + habνaRH̄Lb (1.1)
where νR is the right-handed neutrino, L is the left-handed lepton doublet and H̄ is
the Higgs field. In supersymmetric theories, this term arise from a superpotential with
the above structure, upon replacing fields by chiral superfields. If Mab is large, the
lightest neutrino eigenvalues have masses
Mν = < H̄ >
2 hTM−1h (1.2)
For M ∼ 1010 − 1013 GeV and Dirac neutrino masses of order charged lepton masses,
the eigenvalues are consistent with experimental results.
What is the structure of neutrinos and their masses in string theory? In specific
compactifications giving rise to the MSSM spectra singlet fields corresponding to right-
handed neutrinos νR generically appear. Dirac neutrino masses are also generically
present but the required Majorana νR masses are absent. This is because most MSSM-
like models constructed to date have extra U(1) symmetries, under which the right-
handed neutrinos are charged, which hence forbid such masses. In many models, such
symmetries are associated to a U(1)B−L gauge boson beyond the SM. In order to
argue for the existence of νR masses, string model builders have searched for non-
renormalizable couplings of the type (νRνRN̄RN̄R) with extra singlets NR. Once the
latter fields get a vev, U(1)B−L is broken and a Majorana mass appears for the νR.
Although indeed such couplings (or similar ones with higher dimensions) exist in some
semi-realistic compactifications, such a solution to the neutrino mass problem in string
theory has two problems: 1) The typical νR masses so generated tend to be too small
due to the higher dimension of the involved operators and 2) The vevs for the NR fields
breaks spontaneously R-parity so that dimension 4 operators potentially giving rise to
fast proton decay are generated. This is in a nutshell the neutrino problem in string
compactifications (see [2] for a recent discussion in heterotic setups).
In [3] (see also [4]) two of the present authors pointed out that there is a built-
in mechanism in string theory which may naturally give rise to Majorana masses for
right-handed neutrinos. It was pointed out that string theory instantons may generate
such masses through operators of the general form
Mstring e
−U νRνR . (1.3)
Here U is a linear combination of closed string moduli whose imaginary part gets
shifted under a U(1)B−L gauge transformation in such a way that the operator is fully
gauge invariant. The exponential factor comes from the semi-classical contribution
of a certain class of string instantons. This a pure stringy effect distinct from the
familiar gauge instanton effects which give rise to couplings violating anomalous global
symmetries like (B+L) in the SM. Here also (B−L) (which is anomaly-free) is violated.
This operator is generated due to existence of instanton fermionic zero modes which
are charged under (B − L) and couple to the νR chiral superfield. Although the effect
can take place in different constructions, the most intuitive description may be obtained
for the case of Type IIA CY orientifold compactifications with background D6-branes
wrapping 3-cycles in the CY. In the simplest configurations one has four SM stacks
of D6-branes labeled a,b, c,d which correspond to U(3), SU(2) (or U(2)), U(1)R and
U(1)L gauge interactions respectively, which contain the SM group. One can construct
compactifications with the MSSM particle spectrum in which quarks and leptons lie
at the intersections of those SM D6-branes. Then the relevant instantons correspond
to euclidean D2-branes wrapping 3-cycles in the CY (satisfying specific properties so
as to lead to the appropriate superspace interaction). The D2-D6 intersections lie the
additional fermionic zero modes which are charged under (B−L). For instantons with
the appropriate number of intersections with the appropriate D6-branes, and with open
string disk couplings among the zero modes and the νR chiral multiplet (see fig.(3.1)),
the operator in (1.3) is generated.
The fact that the complex modulus U transforms under U(1)B−L gauge transfor-
mations indicates that the U(1)B−L gauge boson gets a mass from a Stückelberg term.
So a crucial ingredient in the mechanism to generate non-perturbative masses for the
νR’s is that there should be massless U(1)B−L gauge boson which become massive by a
Stückelberg term. It turns out that not many semi-realistic models with U(1)B−L mass
from Stückelberg couplings have been constructed up to date. In the literature there
are two main classes of RR tadpole free models with massive B-L. The first class are
non-susy type IIa toroidal orientifold models first constructed in [5]. The second class
are the type II Gepner orientifold models constructed by one of the present authors and
collaborators [6, 7]. The former were already considered in [3]. In the present paper we
will concentrate on the RCFT Gepner model constructions, which lead to a large class
of MSSM like models, more representative of the general Calabi-Yau compactifications
(for a recent discussion of instanton-induced neutrino masses in a model with no RR
tadpole cancellation, see [8]).
The class of constructions in [6, 7] start with any of the 168 Type II compacti-
fications obtained by tensoring N = 2 SCFT minimal models. In addition one can
choose a number of modular invariant partition functions (MIPF), leading to a total of
5403. Then different consistent orientifold projections are performed on the different
models. This yields a total of 49304 Type II orientifolds. The open string sector of
the theory is defined in terms of the boundary states of the theory. Intuitively, they
play the same role as D-branes wrapping cycles in the geometrical settings. Thus one
associates boundary states a,b, c,d to the gauge groups giving rise to the SM. Differ-
ent choices for the SM boundary states lead to different spectra. In the present paper
we will make use of the data in [6] which contains 211634 different MSSM-like spectra
(including also different hidden sectors). Although this number is huge, most of these
models are really extensions of the MSSM, since they have either an extra U(1)B−L or
SU(2)R × U(1)B−L group factor beyond the SM group. As we said, we are actually
only interested in models in which the U(1)B−L gets a Stückelberg mass. Then we
find that the number of MSSM-like models with these characteristics is dramatically
reduced: only 0.18 percent of the models (391) have a massive U(1)B−L.
As we said, in the geometrical setting of IIA orientifolds with intersecting D6-branes
[9, 10] (see [11] for reviews and [12, 13] for the IIB counterparts), instantons are associ-
ated to D2-branes wrapping 3-cycles, like the background D6-branes do. Analogously,
in the RCFT setting the same class of boundary states appearing in the SM construc-
tions are the ones corresponding to instantons. The zero modes on the instanton is
computable from the overlaps of instanton brane boundary states (zero modes un-
charged under the 4d gauge group) or of instanton and 4d spacefilling brane boundary
states (zero modes charged under the corresponding gauge factor). We find that the
criteria for a non-perturbative superpotential to be generated [14] are only fulfilled if
the Chan-Paton (CP) symmetry of the instantons is O(1). For instantons with CP
symmetry 1 Sp(2) or U(1) we find that there are a few extra uncharged fermionic
zero modes which would preclude the formation of the searched superpotentials. On
the other hand we argue that the addition of fluxes and/or possible non-perturbative
1We adopt the convention that the fundamental representation of Sp(m) is m-dimensional.
extensions of the orientifold compactifications would allow also instantons with Sp(2)
and U(1) symmetries to generate such superpotentials. We thus include all O(1), Sp(2)
and U(1) instantons 2 in our systematic search. The computation of charged and un-
charged fermion zero modes may be easily implemented as a routine in a systematic
computer search for instanton zero modes in Gepner MSSM-like orientifolds. Results
of such a systematic computer search are presented in this article.
We find that out of the 391 models with massive U(1)B−L, there are very few ad-
mitting instantons with the required minimal O(1) CP symmetry, and in fact none
of them without additional vector-like zero modes. On the other hand we do find
32 models admitting Sp(2) symmetric instantons with just the required charged zero
mode content (and the minimal set of non-chiral fermion zero modes). In fact they
are all variations of the same orientifold Gepner model based on the tensor product
(k1, k2, k3, k4) = (2, 4, 22, 22). These models all in fact correspond to the same MIPF
and orientifold projection, they only differ on which particular boundary states corre-
sponding to the four a,b, c,d SM ‘stacks’. All models have the same chiral content,
exactly that of the MSSM , with extra vectorlike chiral fields which may in principle
become massive in different points of the CY moduli space. They have no hidden
sector, i.e., the gauge group is just that of the SM. For each of the models there are 8
instantons with Sp(2) CP symmetry with just the correct charged zero mode structure
allowing for the superpotential coupling (1.3) giving rise to νR Majorana masses. As
we said, they have extra uncharged fermion zero modes beyond the two required to
generate a superpotentials. However one would expect that these unwanted zero modes
might be lifted in more generic situations in which e.g. NS/RR fluxes are added.
We thus see that, starting with a ’large’ landscape of 211634 MSSM-like models,
and searching for instantons inducing neutrino masses, we find there are none admitting
the O(1) instantons with exactly the required zero mode structure, and only few (32)
examples with Sp(2) instantons with next-to-minimal uncharged zero mode structure
(and exactly the correct charged zero modes). Let us emphasize though that it is
the existence of massive U(1)B−L models which is rare. Starting with the subset of
models with a massive U(1)B−L, finding models with correct instanton charged zero
modes within that class is relatively frequent, 10 percent of the cases. Furthermore, we
will see that there are further models beyond those 32 which contain extra non-chiral
instanton zero modes and which may also be viable if these modes get massive by some
2We refer to the different kinds of instanton by their Chan-Paton symmetry on their volume. Since
we are not interested in gauge theory instantons, this notation should not be confusing.
effect (like e.g. the presence of RR/NS fluxes).
Instantons may generate some other interesting superpotential couplings in addition
to νR masses, some possibly beneficial and others potentially dangerous. In particular
we find that in the models which contain Sp(2) instantons which might induce νR
masses, there are also other instantons which would give rise directly to the Weinberg
operator [15]
(LHLH) (1.4)
Once the Higgs field gets a vev, this gives rise directly to left-handed neutrino masses.
Thus we find that in that class of models both the see-saw mechanism (which also gives
rise to a contribution to the Weinberg operator) and an explicit Weinberg operator
might contribute to the physical masses of neutrinos. Which effect dominates will
depend on the relative size of the corresponding instanton actions as well as on the size
of the string scale. Among potentially dangerous operators which might be generated
stand the R-parity violating operators of dimension < 5, which might give rise e.g. to
fast proton decay. We make an study of the possible generation of those, and find that
in all models in which νR masses might be generated R-parity is exactly conserved.
This is a very encouraging result.
A natural question to ask is whether one can say something about the structure
of masses and mixings for neutrinos. As argued in [3] generically large mixing angles
are expected, however to be more quantitative we also need to know the structure
of Yukawa couplings for leptons. In principle those may be computed in CFT but
in practice this type of computation has not yet been developed for CFT orientifolds.
Nevertheless we show that, in the case of instantons with Sp(2) CP symmetry, a certain
factorization of the flavor structure takes place, which could naturally give rise to a
hierarchical structure of eigenvalues for neutrino masses.
The structure of this article is as follows. In the next section we present a discussion
of instanton induced superpotentials in Type II orientifolds. This discussion will apply
both to Type IIA and Type IIB CY orientifolds as well as to more abstract CFT
orientifolds. We discuss the structure of both uncharged and charged instanton zero
modes. In particular we show that only instantons with O(1) CP symmetry have the
appropriate uncharged zero mode content to induce a superpotential contribution. We
also discuss how Sp(2) and U(1) might still generate superpotential contributions if
extra ingredients are added to the general setting. In section 3 we apply that discussion
to the specific case of the generation of νR Majorana masses, showing what is the
required zero mode structure in this case. We show how the flavor structure of the
Majorana mass term factorizes in the case of instantons with Sp(2) CP symmetry,
leading potentially to a hierarchical structure of eigenvalues. We further discuss the
generation of other B/L-violating operators including the generation of the Weinberg
operator as well as R-parity violating couplings. In section 4 we review the RCFT
Type II orientifold constructions in [6, 7]. A general discussion of zero fermion modes
for instantons in RCFT orientifolds is presented in section 5.
The results of our general search for instantons generating νR masses are presented
in section 6. We provide a list of all Gepner orientifolds which admit instanton con-
figurations potentially giving rise to νR Majorana masses. We describe the structure
of the models with Sp(2) instantons having the required charged zero modes for that
superpotential to be generated. We also describe the boundary states of the corre-
sponding instantons and provide the massless spectrum of the relevant MSSM-like
models. Other orientifolds with zero mode structure close to the minimal one are also
briefly discussed. A brief discussion about the possible generation of R-parity violating
superpotentials is included. We leave section 7 for some final comments. Some nota-
tion on the CFT orientifold constructions, and a discussion of the CFT symmetries in
the Sp(2) examples are provided in two appendices.
As this paper was ready for submission, we noticed [16, 17], whose discussion of
instanton zero modes partially overlaps with our analysis in Section 2.2.
2 Instanton induced superpotentials in Type II ori-
entifolds
In this Section we review the generation of superpotentials involving 4d charged fields
via D-brane instantons in type II compactifications. The discussion applies both to type
IIA and IIB models, and to geometrical compactification as well as to more abstract
internal CFT’s. For recent discussions on D-brane instantons, see [4, 3, 25, 8] .
Before starting, a notational remark in in order. Our notation is adapted to working
in the covering theory, namely the type II compactification, and orientifolding in a
further step. Thus we describe the brane configurations as a system of branes (described
by boundary states for abstract CFT’s), labeled k, and their orientifold images labeled
k′. Similarly, we denote M the brane / boundary state corresponding to the instanton
brane, andM ′ its orientifold image. If a brane is mapped to itself under the orientifold
action, we call it a ‘real’ brane / boundary, and ‘complex’ otherwise.
2.1 D-brane instantons, gauge invariance and effective opera-
A basic feature of type II orientifold compactifications with D-branes is the generic
presence of Stückelberg couplings between the U(1) gauge fields on the D-branes, and
certain 4d RR closed string 2-forms. These couplings are required by the Green-
Schwarz mechanism when the U(1)’s have non-zero triangle contributions to mixed
anomalies [18, 19], but can also exist for certain non-anomalous U(1)’s [5, 20]. These
couplings make the U(1) gauge bosons massive, but the symmetry remains as a global
symmetry exact in perturbation theory. Since the closed string moduli involved are
scalars (0-forms) in the RR sector, the natural candidate non-perturbative effects to
violate these U(1) symmetries are instantons arising from euclidean D-branes coupling
to these fields.
In computing the spacetime effective interaction mediated by the instanton, one
needs to integrate over the instanton zero modes. In the generic case (and in particular
for the case of our interest) there are no bosonic zero modes beyond the universal ones
(namely, the four translational bosonic zero modes associated to the position of the
instanton). On the other hand, the structure of fermion zero modes will be crucial.
Since we are interested in models with non-trivial 4d gauge group, arising from a set
of background 4d spacetime filling branes, we consider separately fermion zero modes
which are uncharged under the 4d gauge group and those which are charged. In this
paper we restrict our discussion to 4d N = 1 supersymmetric models, and this will
simplify the analysis of zero modes.
Fermion zero modes which are uncharged under the 4d gauge group determine
the kind of 4d superspace interaction which is generated by the instanton. We are
interested in generating superpotential interactions, which, as is familiar, requires the
instanton to have two fermion zero modes to saturate the d2θ superspace integration.
For this, a necessary (but not sufficient!) condition is that the D-branes are half-BPS,
so these fermion zero modes are the Goldstinos of the two broken supersymmetries.
In the string description, uncharged zero modes arise from open strings in the MM
sector (in our notation, the one leading to adjoint representations), which in particular
contain these Goldstinos, and the MM ′ sector (in our notation, the one leading to
two-index symmetric or antisymmetric tensors). Note that both are the same for ‘real’
branes. Hence we are primarily interested in D-branes whose MM sector contains
just two fermion zero modes, and whose MM ′ sector (for ‘complex’ branes) does not
contain additional fermion zero modes.
In analogy with the familiar case of gauge theory instantons [21], charged fermion
zero modes determine the violation of perturbative global symmetries by the instanton-
induced interaction. Namely, in order to saturate the integration over the charged
fermions zero modes, the spacetime interaction must contain insertions of fields charged
under the 4d gauge symmetry, and in particular under the global U(1) factors, which
are thus violated by the D-brane instanton. Notice that this holds irrespectively of the
number of uncharged fermion zero modes, namely of the kind of superspace interaction
induced by the instanton. Restricting to superpotential interactions, the resulting
operator in the 4d effective action has roughly the form
Winst = e
−U Φ1 . . .Φn (2.1)
Here the fields Φ1, . . . ,Φn are 4d N = 1 chiral multiplets charged under the 4d gauge
group, and in particular also under the U(1) symmetries. Note also that the instanton
amplitude contains a prefactor (which in general depends on closed and open string
moduli) arising from the Gaussian path integral over (massive) fluctuations of the
instanton (hence described by an open string annulus partition function, see [22, 23]
for related work), which we can ignore for our purposes in this paper.
For D-brane instantons, U is the closed string modulus to which the euclidean D-
brane couples. In the D-brane picture, instanton fermion zero modes charged under
the gauge factor carried by the kth stack of 4d space-filling branes (and its image
k′) arise from open strings in the Mk and Mk′ sectors, transforming as usual in the
( M , k) and ( M , k) representations, respectively (with both related in the case of
‘real’ branes). The (net) number of instanton fermion zero modes with such charges is
given by certain multiplicities 3 IMk, IMk′.
A D-brane instanton, irrespectively of the superspace structure of the 4d interac-
tions it may generate, violates U(1)k charge conservation by an amount IMk − IMk′ for
‘complex’ branes and IMk for ‘real’ branes. In particular, this is the total charge of the
field theory operator Φ1 . . .Φn in (2.1). From the Stückelberg couplings, it is possible
to derive [3] (see [23, 24, 25] for related work, also [26])
3In geometric type IIA compactifications with 4d spacetime-filling branes and instanton branes
given by D6- and D2-branes wrapped on Special Lagrangian 3-cycles, IMk corresponds to the inter-
section number between the 3-cycles corresponding to the kthD6- and the D2-brane M (and similarly
for IMk′ ). In geometric type IIB compactifications, it corresponds to the index of a suitable Dirac
operator. In general (even for abstract CFT’s) it can be defined as the Witten index for the 2d theory
on the open string with the boundary conditions corresponding to the two relevant branes. We will
often abuse language and refer to this quantity as intersection number, even in Section 6 where we
work in the non-geometric regime of type IIB compactifications.
that for ‘complex’ instantons, gauge transformations of the U(1)k vector multiplets
Vk → Vk + Λk, transform U as
U → U +
Nk(IMk − IMk′)Λk (2.2)
For ‘real’ brane instantons, which were not considered in [3], the shift is given by 4
U → U +
NkIMkΛk (2.3)
(this new possibility will be an important point in our instanton scan in Section 6).
The complete interaction (2.1) is invariant under the U(1) gauge symmetries. How-
ever, from the viewpoint of the 4d low-energy effective field theory viewpoint, it leads to
non-perturbative violations of the perturbative U(1) global symmetries, by the amount
mentioned above.
In the string theory construction there is a simple microscopic explanation for the
appearance of the insertions of the 4d charged fields (related to the mechanism in
[27]). The instanton brane action in general contains cubic terms αΦ γ, involving two
instanton fermions zero modes α in the ( M , k) and γ in the ( p, M) coupling to the 4d
spacetime field Φ in the ( k, p) of the 4d gauge group
5. Performing the Gaussian path
integral over the instanton fermion zero modes leads to an insertion of Φ in the effective
spacetime interaction. In general, and for a ‘complex’ instanton, there are several
fermion zero modes αi, γi in the fundamental (resp. antifundamental) of the instanton
gauge group, coupling to a 4d spacetime chiral operators Oij (which could possibly
be elementary charged fields, or composite chiral operators). Gaussian integration
over the fermion zero modes leads to an insertion of the form detO (for ‘real’ brane
instantons, detO should be interpreted as a Pfaffian). It is straightforward to derive
our above statement on the net charge violation from this microscopic mechanism.
Note that the above discussions show that instantons in different topological sec-
tors (namely with different RR charges, and different intersection numbers with the
4d spacefilling branes) contribute to different 4d spacetime operators. In particular,
multiwrapped instantons, if they exist as BPS objects, contribute to operators different
4Equivalently, one may use (2.2), but must include an additional factor of 1/2 from the reduction
of the volume for a real brane (which is invariant under the orientifold action).
5 Although there is no chirality in 0+0 dimension, the fermion zero modes α and γ are distinguished
by their chirality with respect to the SO(4) global symmetry of the system (which corresponds to
rotations in the 4d spacetime dimensions, longitudinal to the space-filling branes, and transverse to
the instanton brane). Supersymmetry of the instantons constrains the couplings on the instanton
action (such as the cubic couplings above) to have a holomorphic structure.
from the singly wrapped instanton. This implies that the instanton expansion for a
fixed operator is very convergent, and could even be finite.
Another important implication of the above discussion is that, in order to generate
a specific operator via an instanton process, a necessary condition is that the instanton
has an appropriate number and structure of charged zero modes. However, this is not
sufficient. Insertions of 4d fields appear only if the fields couple to the instanton fermion
zero modes via terms at most quadratic in the zero modes. In equivalent terms, only
zero modes appearing in the Gaussian part of the instanton action can be saturated
by insertions of 4d fields (those to which they couple). The requirement that the zero
modes have appropriate couplings to the 4d fields may be non-trivial to verify in certain
constructions. This is the case for the Gepner model orientifolds in coming sections,
whose couplings are computable in principle, but unknown in practice. In such cases
we will assume that any coupling which is not obviously forbidden by symmetries will
be non-vanishing. Unfortunately there are no arguments to estimate the actual values
of such non-vanishing couplings, hence we can argue about the existence of certain
instanton induced operators, but not about the coefficients of such terms.
2.2 Zero mode structure for D-brane instantons
In this section we describe more concretely different kinds of instantons and the struc-
ture of interesting and unwanted zero modes. Our discussion will be valid for general
D-brane models in perturbative type II orientifolds without closed string fluxes, al-
though we also make some comments on more general F-theory vacua and the effects
of fluxes. A more specific discussion is presented in Section 5.
2.2.1 Uncharged zero modes
We start discussing zero modes uncharged under the 4d gauge group. These are crucial
in determining the kind of superspace interaction induced by the instanton on the 4d
theory. In particular, we are interested in instantons contributing to the 4d superpo-
tential, namely those which contain just two fermion zero modes in this sector. We are
also interested in instantons which may contain additional fermion zero modes, and the
possible mechanisms that can be used to lift them. Let us discuss ‘real’ and ‘complex’
brane instantons in turn.
• Real brane instantons
Real brane instantons correspond to branes which are mapped to themselves by the
orientifold action, hence M = M ′. Uncharged zero modes arise from the MM open
string sector. As discussed in Section 5, there is a universal sector of zero modes, in the
sense that it is present in any BPS D-brane instanton, which we now describe. Before
the orientifold projection, we have a gauge group U(n) on the volume of n coincident
instantons. Notice that, although there are no gauge bosons in 0 + 0 dimensions, the
gauge group is still well-defined, since it acts on charged states (open string ending on
the instanton brane). There are four real bosonic zero modes and four fermion zero
modes in the adjoint representation. For the minimal U(1) case, the four bosons are the
translational Goldstones. The four fermions arise as follows. This sector is insensitive
to the extra 4d spacefilling branes, and feels an accidental 4d N = 2 supersymmetry.
The BPS D-brane instanton breaks half of this, and leads to four Goldstinos, which
are the described fermions 6.
The orientifold projection acts on this universal sector as follows (see Section 5 for
further discussion). The gauge group is projected down to orthogonal or symplectic.
Hence instanton branes with symplectic gauge group must have even multiplicity (a
related argument, in terms of the orientifold action on Chan-Paton indices, is given in
Section 5). For instantons with O(n) gauge symmetry, the orientifold projects the four
bosonic zero modes and two fermion zero modes (related by the two supercharges of 4d
N = 1 supersymmetry broken by the instanton) to the two-index symmetric represen-
tation, and the other two fermion zero modes (related by the other two supercharges of
the accidental 4d N = 2 in this sector) to the antisymmetric representation. Hence for
O(1) instantons (namely instantons with O(1) gauge group on their volume), we have
just two fermion zero modes, which are the Goldstinos of 4d N = 1 supersymmetry,
and the instanton can in principle contribute to the superpotential (if no additional
zero modes arise from other non-universal sectors). For instantons with Sp(n) gauge
symmetry, the orientifold projects the four bosonic zero modes and two fermion zero
modes to the two-index antisymmetric representation, and the other two fermion zero
modes to the symmetric representation. Hence, even for the minimal case of Sp(2)
instantons, we have just two fermion zero modes in the triplet representation, in ad-
dition to the two 4d N = 1 Goldstinos. Hence Sp(2) instantons cannot contribute to
the superpotential in the absence of additional effects which lift these zero modes (see
later) 7.
6We thank F. Marchesano for discussions on this point.
7 For D-brane instantons corresponding to gauge instantons, the additional fermion zero modes
in the universal sector couple to the boson and fermion zero modes from open strings stretched
between the instanton and the 4d spacefilling brane. They act as Lagrange multipliers which impose
In addition to this universal sector, there exist in general additional modes, whose
presence and number depends on the detailed structure of the branes. Namely, on
the geometry of the brane in the 6d compact space in geometric compactifications, or
on the boundary state of the internal CFT in more abstract setups. They lead to a
number of boson and fermion zero modes in the symmetric or antisymmetric represen-
tation. The computation of these multiplicities in terms of the precise properties of
the instanton branes is postponed to Section 5. In order to generate a superpotential,
one must require these modes to be absent, except for the case of antisymmetrics of
O(1) instantons, which are actually trivial.
An important point is that extra fermion zero modes (including the extra triplet
fermion zero modes in the universal sector of Sp(2) instantons, and any two-index
tensor fermion zero mode in the non-universal sectors) are in principle not protected
against acquiring non-zero masses once the model is slightly modified. In other words,
such fermions are non-chiral, in terms of the SO(4) chirality in footnote 5. One
such modification is motion in the closed string moduli space, which can lift the non-
universal modes if there are non-trivial couplings between them and closed string mod-
uli (unfortunately, such correlators are difficult to compute, even in cases where the
CFT is exactly solvable, like the Gepner models). Note that extra zero modes in the
universal sector of Sp(2) instantons cannot be lifted by this effect, since it does not
break the accidental 4d N = 2 in this sector. A second possible modification which
in general can lift extra zero modes is the addition of fluxes, generalizing the results
for D3-brane instantons in geometric compactifications [29] (for non-geometric CFT
compactifications, we also expect a similar effect, once fluxes are introduced following
[30]). Note that fluxes can lift extra zero modes in the universal sector as well, since
they can break the accidental 4d N = 2 susy in this sector. A last mechanism arising in
more general F-theory compactifications and discussed below for complex instantons,
is valid for real instanton branes as well.
The bottom line is that in the absence of such extra effects, only O(1) instantons can
contribute to superpotential terms. However, in modifications of the model such extra
effects can easily lift the extra fermion zero modes. Hence, this kind of extra vector-like
zero modes will not be considered catastrophic, and real instantons (including the O(1)
and Sp(2) cases) with such zero modes are considered in our scan in Section 6.
the fermionic constraints in the ADHM construction [28], and may not spoil the generation of a
superpotential.
• Complex brane instantons
Zero modes uncharged under the 4d gauge group can arise from the MM and
MM ′ open string sectors. Notice that the orientifold action maps the MM sector
to the M ′M ′, hence we simply discuss the former and impose no projection. The
discussion of the MM sector is as for real brane instantons before the orientifold
projection. The universal sector leads to a U(n) gauge symmetry, and four bosonic and
four fermionic zero modes in the adjoint representation. The bosons are translational
Goldstones, while the fermions are Goldstinos of the accidental 4d N = 2 present in
this sector. Hence, even in the minimal case of U(1) brane instantons there are two
extra fermion zero modes, beyond the two fermion zero modes corresponding to the
4d N = 1 Goldstinos. Hence U(1) instanton (except for those corresponding to gauge
instantons, see footnote 7) cannot contribute to superpotential terms in the absence of
additional effects, like closed string fluxes . However, keeping in mind the possibility
of additional effects lifting them in modifications of the model, we include them in the
discussion. Also, in what follows we will use the U(n) notation for the different fields
to keep track of the Chan-Paton index structure.
The above statement would seem in contradiction8 with computations of non-
perturbative superpotentials [14] induced by M5-branes instantons in M-theory com-
pactifications on Calabi-Yau fourfolds, which are dual to D3-brane instantons (with
world-volume U(1) gauge group) on type IIB compactifications. The resolution is that
the M5-branes that contribute are intersected by different (p, q) degenerations of the
elliptic fiber. This implies that U(1) D3-brane instanton only contribute if they are
intersected by mutually non-local (p, q) 7-branes. The two extra fermion zero modes
exist locally on the D3-brane volume, but cannot be defined globally due to the 7-brane
monodromies. Hence such effect can take place only on non-perturbative type IIB com-
pactifications including (p, q) 7-branes. Note that in perturbative compactifications,
namely IIB orientifolds, the (p, q) 7-branes are hidden inside orientifold planes [31] with
their monodromy encoding the orientifold projection; hence the only branes that can
contribute to the superpotential are real branes, for which the projection/monodromy
removes the extra fermion zero modes, as discussed above.
In addition to this universal sector, theMM sector may contain a non-universal set
of fermions and bosons, in the adjoint representation (hence uncharged under U(1)).
They depend on the specific properties of the brane instanton, and will be discussed
in Section 5. These additional zero modes should be absent in order for the instanton
8We thank S. Kachru for discussions on the ideas in this paragraph.
to induce a non-trivial superpotential. Notice however that these zero modes are un-
charged under any gauge symmetry, and hence vector-like. Thus, there could be lifted
in modifications of the model, as discussed for real instantons.
The MM ′ sector is mapped to itself under the orientifold action. Hence it leads to
a number of bosons and fermions in the two index symmetric or antisymmetric repre-
sentations. Notice that the two-index antisymmetric representation is trivial for U(1),
so these modes are actually not present. On the other hand, fermion zero modes in the
two-index symmetric representation are chiral and charged under the brane instanton
gauge symmetry. Hence they cannot be lifted by any of the familiar mechanisms, and
thus spoil the appearance of a non-perturbative superpotential, even if the model is
modified. Such fermion zero modes are considered catastrophic and we will look for
models avoiding them in our scan in Section 6.
2.2.2 Charged fermion zero modes
• Real brane instantons
Instanton zero modes charged under the 4d gauge group arise fromMk open string
sectors (and their image Mk′). In the generic case, there are no scalar zero modes in
these sectors. This is because in mixed Mk open string sectors the 4d spacetime part
leads to DN boundary conditions, which already saturate the vacuum energy in the NS
sector. Only in the special case where the internal structure of the spacetime filling
brane k and the instanton brane are the same, there may be NS ground states of the
internal CFT which do not contribute extra vacuum energy, hence leading to massless
scalars. However, this corresponds to brane instantons which can be interpreted as
instantons of the 4d gauge theory on the 4d space-filling branes. The instantons we
are interested in for the generation of neutrino Majorana mass terms are not of this
kind [3] (see e.g. [32, 33, 28, 25, 34] for discussions on gauge theory instantons from
D-brane instantons).
Hence we focus on charged fermion zero modes, which are generically present in
any mixed Mk sector. Let us define LMk, LMk′ the (positive by definition) number
of left-handed chiral fermion zero modes in the representations ( M , k), ( M , k),
respectively. The net number of chiral fermion zero modes in the ( M , k) is given
by IMk = LMk′ − LMk. This controls the violation of the U(1)a global charge by the
instanton. Namely, such fermion zero modes in the Mk, Mp sectors lead (if suitable
couplings are present) to the insertion of 4d charged fields Φkp and/or Φkp′.
In addition, there are PMk = min(LMk′ , LMk) vector-like pairs of fermion zero
modes. Since they are vector-like, they may be lifted by slight modifications of the
model, like moving in the closed string moduli space, or by introducing additional
ingredients, like fluxes. In addition, they may be lifted by moving in the open string
moduli space of the 4d spacefilling branes, as follows. The zero modes may lead to
insertions of 4d fields Φkk, if the kk sector contains such 4d chiral multiplets (or to
insertions of composite 4d operators in the adjoint of the kth 4d gauge factor), and if
they couple to the zero modes. Although this may not be generically not the case,
many of our models in coming section contain such adjoint fields. Hence, a non-
trivial vev for the latter can lift these extra vector-like zero modes, hence leading to
instanton generating the superpotential of interest. Given these diverse mechanisms to
lift these zero modes, their presence of such zero modes is thus unwanted, but again
not necessarily catastrophic.
One last comment, related to the concrete kind of instanton search we will be
interested in. Namely, we will be searching for instantons leading to a specific operator,
carrying non-trivial charges under a specific set of 4d gauge factors. Postponing the
detailed discussion to Sections 3.1, 4.2 , let us denote a, b, c, d the set of branes
leading to a field theory sector, denoted ‘observable’ (and which reproduces the SM
in our examples). We will require the instanton to have a prescribed number of chiral
fermion zero modes charged under these branes, namely we require the intersection
numbers of the instanton with these branes IMa, . . . , IMd to have specific values (as
mentioned above, in the most restrictive scan we forbid vector-like pairs of zero modes
under these branes). In addition, the model in general contains an additional sector of
branes, denoted ‘hidden’ (since there is zero net number of chiral multiplets charged
under both sectors) and labeled hi, required to fulfill the RR tadpole cancellation
conditions. In general there may be instanton fermion zero modes from e.g. the
Mh1, h2M sectors, which would contribute to insertions of the 4d fields in the h1h2
sector if there are suitable cubic couplings. These extra insertions could be avoided
if such 4d fields in the hidden sector acquire vevs (note that vevs for the (vector-like)
fields charged under the visible and hidden sectors would typically break hypercharge,
and should be avoided), and hence lift the zero modes. Equivalently, from the 4d
perspective, the unwanted extra h1h2 field insertion is replaced by its vev. However,
this renders the discussion very model dependent. Moreover, the possibility of hidden
brane recombination was not included in the search for SM-like models in [6, 7] (namely,
the possibility of allowing for chiral fields charged under the observable and hidden
gauge groups, which may become non-chiral and possibly massive upon hidden brane
recombination). Hence we will consider these chiral fermion zero modes as unwanted
(as usual, non-chiral modes are unwanted but not catastrophic, hence they are allowed
for in a more relaxed search).
• Complex brane instantons
The discussion of ‘complex’ brane instantons is somewhat analogous to the previous
one, with the only complication that the braneM and its imageM ′ lead to independent
modes, leading to a more involved pattern of fermion zero modes. Instanton zero modes
charged under the 4d gauge group arise from the Mk,Mk′ and related sectors. As for
‘real’ brane instantons, there are generically no scalars in these sectors (and certainly
not in our case of interest). Hence we focus on charged fermion zero modes, which are
generically present in any mixed sector.
In contrast with ‘real’ brane instantons, a net combination of fermion zero modes
in the ( M , k) + ( M , k) is not vector-like, but chiral under the instanton gauge
symmetry. Such a pair cannot therefore be lifted even by modifications of the theory.
In general, if the instanton has a mismatch in the total numbers nα, nγ of fermion zero
modes αi in the M and γj in the M , the instanton amplitude automatically vanishes.
Namely, the matrix of operators Oij coupling to the zero modes necessarily has rank
at most min(nα, nγ). That is , if nα > nγ there are linear combinations of the αi which
do not couple, and cannot lead to insertions. Moreover, they are not liftable by the
familiar mechanisms 9, thus in our instanton search in Section 6 such excess zero modes
are forbidden even in relaxed scans.
Let us thus discuss a sector of fermion zero modes with equal number nα = nγ .
Considering a given 4d space-filling brane k, let us denote LMk, LM ′k′, LMk′, LM ′k the
(positive by definition) number of left-handed chiral fermion zero modes in the repre-
sentations ( M , k), ( M , k), ( M , k), and ( M , k) respectively. The net number of
chiral fermion zero modes in the ( M , k) and ( M , k) is given by IMk = LMk′ −LM ′k′
and IMk′ = LMk−LM ′k, respectively. This net number of fermions zero modes controls
the violation of the U(1)a global charge by the instanton. Namely, such fermion zero
modes in the Mk, Mp, Mk′, Mp′ sectors lead (if suitable couplings are present) to the
insertion of 4d charged fields Φkp and/or Φkp′.
9Note that such a mismatch is always correlated with the existence of extra chiral zero modes in
the MM ′ sectors, discussed above. Denoting ~Qa, ~Qorient the vector of RR charges of the 4d space-
filling branes and orientifold planes, they satisfy the RR tadpole conditions
Na ~Qa +
Na ~Qa′ +
~Qorient. = 0. By taking the ‘intersection’ bilinear with the RR charges ~QM of the brane instanton, we
have IMa+IMa′+IM,orient = 0. This implies that the number of fundamentals minus anti-fundamentals
of the instanton gauge group is related to the number of two-index tensors.
The remaining fields in this sector are vector-like pairs, in the ( M , k) + ( M , k)
or the ( M , k) + ( M , k), which in principle lead to the vanishing of the instanton
amplitude, but which can be lifted by additional effects (motion in closed or open string
moduli space, or addition of fluxes), in a way consistent with the gauge symmetries in
4d spacetime and on the instanton.
Just like for ‘real’ brane instantons, we conclude by commenting on our concrete
instanton search in models with a set of visible branes a, b, c, d and a set of hidden
branes hi. The requirement that the instanton leads to an operator with specific
charges under the visible branes fixes the values of the quantities IMa − IMa′, etc. As
we described for real branes, one may still have fermion zero modes charged under the
hidden sector branes, but they lead to additional insertions, hence we rather focus on
instantons with IMhi − IMh′j = 0. The two kinds of conditions, on intersection numbers
with visible and hidden branes, still leave the possibility of combinations of fermion
zero modes of the kind ( M , k) + ( M , k), which do not contribute to IMk, or of the
kind ( M , k) + ( M , k), which does not contribute to IMk′. Such combinations are
automatically vector-like, and thus may be lifted in modifications of the theory. But the
condition also allow combinations like ( M , k)+( M , k), which exploit a cancellation
between IMk and IMk′ (as also does ( M , k) + ( M , k)). Such combinations are
chiral by themselves, and in general imply a mismatch of modes in the M and the
M . The total mismatch can be arranged to vanish using combinations of the kind
( M , k) + ( M , k) and ( M , p) + ( M , p) for different branes. However, the only
way to lift these pairs is by breaking the gauge symmetry on the 4d space-filling branes
k and p. This can be done without damage to the visible sector if these are hidden
branes, but this corresponds to the recombination of hidden branes that, as mentioned
already, we are not going to consider. Hence only vector-like pairs with respect to
each brane are considered to be liftable in simple modifications of the theory. In our
instanton search, these are the only additional fermion zero modes which we allow in
relaxed scans (but they are clearly not allowed for in restricted scans)
3 Instanton induced Majorana neutrino masses
In this Section we discuss the possible physical effects of D-brane instantons in string
models with SM-like spectrum. In particular we describe the conditions to generate
right-handed neutrino Majorana masses. We also comment on other possible B and/or
L violating operators that can be generated by instantons. In this section we will again
Sp(2),U(2)
U  , D 
gluon
d− Leptonic
a− Baryonic
b− Left c− Right
Figure 1: Quarks and leptons at intersecting branes
use the geometrical language of IIA intersecting D-branes but it should be clear that
our discussion equally applies to general CFT orientifolds like the ones presented in
the next section.
3.1 The MSSM on the branes
Let us now specify the discussion in the previous section to the case of the generation
of a right-handed neutrino mass term. In order to do that we need some realistic
orientifold construction with the gauge group and fermion spectrum of the Standard
Model (SM). In the context of Type II orientifolds perhaps the most economical brane
configuration leading to a SM spectrum is the one first considered in [5]. This consists
of four stacks, labelled a,b, c,d. The gauge factor on branes a is U(3), and contains
QCD and baryon number. The d factor is U(1)d, and corresponds to lepton number.
Stack b contains SU(2)Weak either embedded in U(2) or Sp(2). Finally brane c can
either provide a U(1) or an O(2) factor. In the brane intersection language, the chiral
fermions of the SM live at the intersections of these branes, as depicted in Fig. 3.1.
The U(1)Y factor of the standard model is embedded in the Chan-Paton factors of
branes a,c and d as
(QB−L −QR) (3.1)
where Qx denotes the generator of the U(1) of brane stack x (in case the Chan-Paton
factor of brane c is O(2) one should use the properly normalized O(2) generator).
Note that in this convention the Qd generator appears with sign opposite to other
conventions in the literature, e.g. in [3]. In addition to Y these models have two
additional U(1) gauge symmetries:
Qanom = 3Qa +Qd = 9QB +QL
Y ′ =
Qa +Qc −Qd = QB−L +QR (3.2)
The first is anomalous whereas the second, which we will call B − L (with a slight
abuse of language, since it is in fact a linear combination of B − L and hypercharge),
is anomaly free. In models in which the electroweak gauge group is embedded in U(2),
rather than in Sp(2), there is a second anomalous U(1)b. The charges of the SM
particles under these U(1) symmetries are given in table 1.
Intersection D = 4 fields/ zero modes Qa Qc Qd Y QM
(ab),(ab’) QL 3(3, 2) 1 0 0 1/6 0
(ca) UR 3(3̄, 1) -1 1 0 -2/3 0
(c’a) DR 3(3̄, 1) -1 -1 0 1/3 0
(db),(db’) L 3(1, 2) 0 0 1 -1/2 0
(c’d) ER 3(1, 1) 0 -1 -1 1 0
(cd) νR 3(1, 1) 0 1 -1 0 0
(Mc) αi 2(0, 0) 0 -1 0 1/2 1
(dM) γi 2(0, 0) 0 0 1 -1/2 -1
Table 1: Standard model spectrum and U(1) charges of particles and zero modes. QM stands
for the world-volume gauge symmetry in the case of U(1) complex instantons.
The U(1)k gauge symmetries have couplings with the RR 2-forms Br of the model,
as follows
SBF =
Nk(pkr − pk′r)
Br ∧ Fk (3.3)
where pkr, pk′,r are given by the RR charges of the D-branes. These imply that under
a U(1)k gauge transformation Ak → Ak + dΛk the scalar ar dual to Br transforms as
ar → ar +
Nk (pkr − pk′r) Λk (3.4)
This has two effects: 1) The linear combination of axion fields
r(pkr − pk′,r)ar is
eaten up by the U(1)k massless gauge boson, making it massive. 2) For anomalous
U(1)k, the anomalies cancel through a 4d version of the Green-Schwarz mechanism.
This works due to the existence of appropriate ar F ∧F couplings, involving the gauge
fields in the theory.
It is obvious that all anomalous U(1)’s become massive by this mechanism. However
it is important to realize [5] that gauge bosons of anomaly-free symmetries like U(1)B−L
may also become massive by combining with a linear combination of axions. This is
interesting since it provides a mechanism to reduce the gauge symmetry of the model
without needing explicit extra Higgsing. In the models in which U(1)B−L becomes
massive in this way, the gauge group left over is purely that of the SM. Moreover, we
will see that having (B-L) massive by this Stückelberg mechanism is crucial to allow
the generation of instanton-induced Majorana neutrino masses.
Note that the B∧F couplings may also be potentially dangerous, since in principle
they could also exist for hypercharge, removing U(1)Y from the low-energy spectrum.
As we will see in our RCFT examples later on, having massless U(1)Y but massive
U(1)B−L turns out to be a strong constraint in model building.
3.2 Majorana mass term generation
As discussed in the previous section, string instantons can give rise to non-perturbative
superpotentials breaking explicitly the perturbative global U(1) symmetries left-over
from U(1) gauge bosons made massive through the Stückelberg mechanism. The kind
of operator we are interested in has the form
W ≃ e−Sins νRνR (3.5)
where νR is the right-handed neutrino superfield
10. Here Sins transforms under both
U(1)B−L and U(1)R in such a way that the overall operator is gauge invariant. This
operator may be created if the mixed open string sectors lead to fermionic zero modes
αi, γi , i = 1, 2, appropriately charged under the 4d gauge factors. As we discussed in
the previous section, to generate a superpotential one needs instanton with O(1) Chan-
Paton symmetry, in order to lead to two uncharged fermion zero modes to saturate
the d2θ 4d superspace integration. On the other hand, as we argued, instantons with
Sp(2) or U(1) CP symmetries may also induce the required superpotentials if there
is some additional dynamics getting rid of the extra uncharged zero modes which in
principle appear in instantons with these symmetries. We thus consider all O(1), Sp(2)
and U(1) instantons in our discussion.
In order to to get a νR bilinear, the intersection numbers of instanton M and d, c
branes are as follows
Sp(2) case : IMc = 1 ; IMd = −1 (3.6)
(since there is an extra multiplicity from the two branes required to produce Sp(2))
O(1) case : IMc = 2 ; IMd = −2 (3.7)
10 Actually we denote by νR the left-handed ν
L field following the usual (a bit confusing) convention.
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Figure 2: Disk amplitude coupling two charged zero modes to νR in the geometrical Type
IIA intersecting brane approach.
U(1) case : IMc = 2 ; IMd = −2 or IMd′ = 2 ; IMc′ = −2 (3.8)
Furthermore there must be cubic couplings involving the right-handed neutrino super-
field νa in the ath family and the fermionic zero modes αi, γj
Lcubic ∝ dija (αi νaγj) , a = 1, 2, 3 (3.9)
In type IIA geometric compactifications, this coupling arises from open string disk
instantons, see Fig. 3.1. In general type IIA models (resp. IIB models), the coefficients
dija depend on the Kähler (resp. complex structure) moduli, and possibly on open string
moduli. In simple CFT models (like e.g. in toroidal cases) these quantities may be in
principle explicitly computed.
These trilinear couplings appear in the instanton action and after integration of the
fermionic zero modes αi, γi one gets a superpotential coupling proportional to
d2α d2γ e−d
a (αiν
aγj) =
d2θ νaνb ( ǫijǫkld
b ) (3.10)
yielding a right-handed neutrino mass term. This term is multiplied by the exponential
of the instanton euclidean action so that the final result for the right-handed neutrino
mass (up to a 1-loop prefactor) has the form
MRab = Ms( ǫijǫkld
b ) exp(−
qM,rar ) (3.11)
For geometric compactifications VΠM is roughly related to the wrapped volume. We
keep the same notation to emphasize that the effect is non-perturbative in gs. In
supersymmetric models the term in the exponential is the linear combination U of
complex structure moduli to which the instanton D-brane couples, as described in the
previous section. As explained, the gauge U(1)c, U(1)d transformation of the bilinear
piece and the e−SD2 factor nicely cancel. Note that from the viewpoint of the 4d SM
effective field theory, the instanton has generated a Majorana neutrino mass violating
B−L. Notice also that since this symmetry is non-anomalous, its violation cannot be
associated to a gauge instanton, hence this is a pure string theory instanton effect.
3.3 Flavor and the special case of Sp(2) instantons
In order to extract more specific results for the flavor structure of the obtained Majo-
rana mass operator, one needs to know more details about the quantities dija coming
from the disk correlators. However in the particular case of Sp(2) instantons, the la-
bels i, j are Sp(2) doublet indices, and the symmetry requires dija = daǫ
ij. The mass
matrix for the three neutrinos is given by MRab =2Msdadb exp (−U), so that the flavour
dependence on a, b = 1, 2, 3 factorizes. More generally, as we will see in our RCFT
search in Section 6, there are typically several different instantons contributing to the
amplitude, so that we actually have a result for the mass
MRab = 2Ms
d(r)a d
−Ur (3.12)
where the sum goes over the different contributing instantons. One thus has a structure
of the form
e−Urdiag (d
1 , d
2 , d
3 ) ·
1 1 1
1 1 1
1 1 1
· diag (d(r)1 , d
2 , d
3 ) . (3.13)
This structure is very interesting. Indeed, each instanton makes one particular (instanton-
dependent) linear combination of the neutrinos massive, leaving two linear combina-
tions massless. Hence, for three or more instantons, one generically has a matrix with
three non-zero eigenvalues. It is easy to imagine a hierarchical structure among the
three eigenvalues if e.g. the exponential suppression factors exp(−ReUr) are different
for each instanton.
3.4 Other B− and L−violating operators
Our main focus in this paper is on the generation of right-handed neutrino Majorana
masses. However instantons may induce other L- and B-violating operators which we
briefly summarize in this subsection.
3.4.1 The Weinberg operator
A right-handed neutrino Majorana mass term is not the only possible operator violating
lepton number. Instanton effects may also give rise to dimension 5 operators not
involving νR. Specifically, the Weinberg operator
(LHLH) . (3.14)
might be generated. Once Higgs fields get a vev v this operator gives rise directly to
Majorana masses for the left-handed neutrinos of order ≃ v2/M . Indeed, it is easy to
check that in this case the required instanton M must verify
Sp(2) case : IMc = −1 ; IMd = 1 (3.15)
O(1) case : IMc = −2 ; IMd = 2 (3.16)
U(1) case : IMc = −2 ; IMd = 2 or IMc′ = 2 ; IMd′ = −2 (3.17)
(here we are assuming SU(2)weak to be embedded in an Sp(2)). Note that these
intersection numbers are different to those giving rise to νR mass terms. In particular
they lead to a transformation under B − L opposite to that of νR mass operators 11.
In the present case there are altogether four fermionic zero modes αi,γi corresponding
to the intersections of the instanton M with the branes c, d. These zero modes can
have couplings involving the left-handed leptons L and the u-type Higgs multiplet H
Ldisk ∝ cija (αi(LaH)γj) . (3.18)
Upon integration over the fermionic zero modes one recovers the Weinberg operator.
In the present case the scale M of the Weinberg operator will be the string scale Ms
and the coupling λ ≃ exp(−Sins). Again, in the particular case of Sp(2) instantons the
situation simplifies (cija = c
aǫij) and one gets left-handed neutrino Majorana masses
MLab =
< H >2
2c(r)a c
−Sr (3.19)
where r runs over the different contributing instantons and Sr is their corresponding
action. The flavour structure of this left-handed neutrino mass matrix is the same as
in eq.(3.13) and again may potentially lead to a hierarchical structure of left-handed
neutrino masses, as is experimentally observed.
11Instantons with these intersection numbers will be denoted with a plus sign in the instanton search
later on
In a given model both this kind of instanton and the one giving rise to right-
handed neutrino masses (which is different) may be present. This contribution to the
left-handed neutrino Majorana mass is in principle sub-leading compared to the see-saw
contribution
MLab(see-saw) =
< H >2
d(r)a d
−Sr)−1 hD (3.20)
where is the ordinary Yukawa coupling constant habD (ν
b). In principle the former
is doubly suppressed both by 1/Ms and the exponential factor. On the other hand
if the exponential suppression is not too large this mechanism involving directly the
Weinberg operator may be the most relevant source of neutrino masses. This is because
the see-saw contribution coming from νR exchange is proportional to the square of the
ordinary Yukawa couplings habD which could be small. One could even think of having
just the Weinberg operator as the unique source of the observed left-handed neutrino
masses. Note however that in string vacua like this, in which the νR’s are present and
massless at the perturbative level, having just the Weinberg operator would not be
phenomenologically correct, and instantons of the first class are still needed so that the
νR’s get a sufficiently large mass.
3.4.2 R-parity violating operators
In the case ofN = 1 SUSYmodels like the MSSM there might be operators of dimension
3 and 4 violating lepton and/or baryon number. These are the superpotential couplings
WRp = µ
aH + λabcQ
aDbLc + λ′abcU
aDbDc + λ′′abcL
aLbEc (3.21)
in standard notation. Unlike the neutrino operators mentioned above, these operators
violate B −L in one unit (rather than 2). It is well known that the standard R-parity
of the MSSM may be identified with a Z2 subgroup of U(1)B−L, so these terms are odd
under R-parity. The simultaneous presence of all these couplings is phenomenologically
unacceptable. Indeed, the third coupling violates baryon number, and the other three
violate lepton number. Together they lead to proton decay at an unacceptably large
rate. On the other hand couplings violating either B or L are phenomenologically
allowed.
It is an interesting question whether any of these operators may be induced by
string instanton effects. A first point to note is that instantons with Sp(2) Chan-Paton
symmetry can never generate operators of this type. The reason is that all charged
zero modes will necessarily come in Sp(2) doublets and hence the charged operators
induced will always involve an even number of charged D = 4 fields and R-parity is
automatically preserved. On the other hand O(1) and U(1) instantons may generate
R-parity violating operators. In particular, the LH bilinear is essentially the square
root of the Weinberg operator, and may be induced if a U(1) or O(1) instanton M
exists with
IMc = −1 ; IMd = 1 or IMc′ = 1 ; IMd′ = −1 . (3.22)
(in the O(1) case the second option is not independent from the first). Again, if
the appropriate disk couplings are non-vanishing a term with µaL ∼ Ms exp(−Sins) is
generated. The rest of the operators in WRp may also be generated. Possible instanton
zero modes which may induce them are shown in table 2. For example, the QDL
operator may be induced if a U(1) instanton M with intersection numbers
IMb = −1 ; IMc′ = 1 ; IMd = 1 (3.23)
is present and in addition couplings
Ldisk ∝ cab (α(UaQbj)γj) + c′a(βLajγj) (3.24)
exist. Here α, β, γ are zero modes corresponding to (Mc′), (Md) and (bM) intersections
and a, b(j) are flavor(SU(2)L) indices. Analogous trilinear or quartic disk amplitudes
involving two charged zero modes should exist to generate the rest of the R-parity
violating amplitudes in table 2.
D = 4 Operator IMa IMa′ IMb IMc IMc′ IMd IMd′
νRνR 0 0 0 2 0 -2 0
LH̄LH̄ 0 0 0 -2 0 2 0
LH̄ 0 0 0 -1 0 1 0
QDL 0 0 -1 0 1 1 0
UDD -1 0 0 1 2 0 0
LLE 0 0 -1 0 1 1 0
QQQL 1 0 -2 0 0 1 0
UUDE -1 0 0 2 2 -1 0
Table 2: Zero modes required to generate Lepton/Baryon-number violating superpoten-
tial operators. Sp(2) instantons cannot give rise to R-parity violating operators whereas
O(1),U(1) instantons may in principle contribute to all of them. In the case of U(1) instantons
there are additional zero mode possibilities which are obtained by exchanging IMx ↔ −IMx′ .
3.4.3 Dimension 5 proton decay operators
There are also superpotential dimension-5 operators violating B and L which may be
constructed from the MSSM matter superfields. Indeed the dimension 5 operators
)QQQL ; (
)UUDE (3.25)
are in fact the leading source of proton decay in SUSY GUT models with R-parity.
Unlike the other operators considered here these ones preserve B−L (hence R-parity)
but not B +L. These operators do not contribute directly to a proton decay but need
to be ’dressed’ by a one loop exchange of some fermionic SUSY particle. This makes
that, even although they are suppressed only by one power of the relevant fundamental
scale, the loop factor and the corresponding couplings make the overall rate in SUSY-
GUTS (barely) consistent with present experimental bounds for M of order the GUT
scale or larger.
These dimension 5 operators may also be induced in D-brane models of the class
here considered by the presence of instantons with appropriate intersection numbers.
For instance, the first operator may be induced through O(1) or U(1) instantons M
IMb = IMb′ = −2 ; IMa = 1 ; IMd = 1 (3.26)
Again Sp(2) instantons cannot induced this operator, since the the Ma intersection
would yield 6 (rather than 3) colored fermionic zero modes. The proton decay rate
obtained from these operators depend on the ratio exp(−Sins) × 1/Ms. For Ms of
order 1016 GeV, the rate is consistent with present bounds if exp(−Sins) provides a
suppression of a few orders of magnitude. On the other hand, models with a low string
scale may be in danger unless the exponential suppression is sufficiently large (or such
particular instantons are absent).
As a general conclusion, these phenomenological aspects of instanton induced oper-
ators very much depend on the action of the instanton, e.g. the volume of the wrapped
D2-instanton in the intersecting D-brane constructions. In any event it is clear that
the instantons here considered may indeed induce proton decay at a model-dependent
rate. However in certain models R-parity will be preserved and prevent too rapid pro-
ton decay. Indeed, this is what we find in our instanton search in Gepner orientifolds.
As we said Sp(2) instantons automatically preserve R-parity. More generally, models
that violate R-parity are rare, and the corresponding instantons actually generate very
high dimensional operators, so R-parity breaking effects seems quite suppressed. In
fact in our search within MSSM-like models in Gepner model orientifolds we do not
find instantons with just the correct charged zero modes to generate the low dimen-
sional couplings discussed above. So, at least within our class of RCFT constructions,
R-parity preservation is quite a common feature.
4 CFT orientifolds
In this section we describe the 4d string models we consider, namely orientifolds of
type IIB Gepner model compactifications. This is a very large class, on which one can
carry out large scans for certain desired properties. And moreover at present the only
known class of (SUSY) models with massive B − L.
4.1 Construction of the models
In general, RCFT orientifolds are orientifold projections of closed string theories con-
structed using rational conformal field theory. Although this includes in principle
rational tori and orbifolds, the real interest lies in cases where the two-dimensional
CFT is interacting, because such theories are hard to access by other methods. A
disadvantage of the use of RCFT is that this method is algebraic, and not geometric in
nature, so that one cannot easily explore small deformations of a certain string theory.
It is best thought of as a rational scan of moduli spaces.
The most easily accessible examples are the orientifolds of tensor products of mini-
mal N = 2 conformal field theories (“Gepner models”) forming a type IIB closed string
theory. During the last decade, examples in this class have been studied by many au-
thors (see [35][36][37][38][39][40][41][42]), and searched systematically in [6] and [7].
Although the Gepner models form only a small subset of RCFT’s, they already offer
a large number of possibilities. The total number of tensor products with the required
central charge c = 9 is 168. On top of this, one can choose a large number of distinct
modular invariant partition functions on the torus. The orientifold formalism is not
available for all of them, but it has been completely worked out [43] for all simple cur-
rent invariants (based on the charge conjugation invariant). This yields a total of 5403
distinct MIPFs. On top of this, we may choose various orientifold projections. Here
the only known possibilities are a class of simple-current based choices [44][45][46][47].
This then yields a total of 49304 orientifolds.
For each orientifold choice, the full open string partition function is
a,b,i
NaNbA
abχi(
aχ̂i(
 (4.1)
Here Aiab are the annulus coefficients, M
a the Moebius coefficients, Na the Chan-Paton
multiplicities and χ(τ) are the closed string characters, and χ̂i(τ) = T
−1/2χi(τ). The
set of integers i is simply the set of primary fields of the closed string CFT, and depends
only on the tensor product. The integers a, b are the boundary labels; this set depends
on the MIPF. Our notation and labelling conventions for these CFT quantities are
explained in Appendix A. The integers Aiab and M
a depend in addition also on the
orientifold choice; in the case of Aiab the latter dependence is very simple: all distinct
annuli can be written as A
cb, where Ω is the orientifold choice (which we
usually do not specify explicitly) and CΩcb is the boundary conjugation matrix, which
acts as an involution on the set of boundaries.
Suppressing some details (which can be found in [43]) we may write these integers
m,J,K
SimRa,(m,J)g
JK Rb,(m,K)
(4.2)
MΩ,ia =
m,J,K
P imRa,(m,J)g
(m,K)
(4.3)
Here m is the label of an Ishibashi-state (the set of states that propagates in the
transverse (or closed string) channel of the the annulus or Moebius diagrams). It is
a subset of the set of closed string labels i, but in general there are degeneracies,
so that more than one distinct Ishibashi state belongs to a given closed string label.
These degeneracies are distinguished by the labels J,K (see Appendix A). The complex
numbers R and U are respectively the boundary and crosscap coefficients. Note that
the latter depend on the orientifold choice, but the former do not. The only dependence
of the annulus coefficients on the orientifold choice is through the Ishibashi metric gΩJK ,
which is a matrix on each Ishibashi degeneracy space, and which can be a sign if there
are no degeneracies. Finally, the matrix P is given by P =
TST 2S
T , where S and
T are the generators of the modular group of the torus. Similar expressions exist for
the Klein bottle multiplicities defining the unoriented closed sector, but they will not
be needed in this paper.
The boundary labels a, b, . . . refer to all boundaries that respect the bulk symmetries
of the CFT. This includes the individual N = 2 chiral algebras of the factors in the ten-
sor product, the alignment currents12 that ensure the proper definition of world-sheet
supersymmetry and the space-time supersymmetry generator that imposes a general-
ized GSO-projection on the spectrum. The latter implies that all characters χi respect
(at least) N = 1 space-time supersymmetry. By construction, the boundary states
are then supersymmetric as well. Both conditions (boundary and bulk space-time su-
persymmetry) can in principle be relaxed within the formalism, but this leads to a
much larger set of bulk and boundary states. The precise labelling of the boundaries
is explained in Appendix A and involves a subset of the closed string labels i and a
degeneracy label, distinct from the one used for the Ishibashi states. The set of bound-
ary labels is complete in the sense of [45]. This means that no additional boundary
states exist that respect all the aforementioned symmetries. It also means that the
matrices R are square matrices (although their rows and columns are defined in terms
of different index sets). It is in principle possible to write down additional boundary
states that break some of the world-sheet symmetries. This is an important possibility
to keep in mind, but we will not consider it here.
The massless spectrum is obtained by restricting the characters χi to massless
states. Since the characters are supersymmetric those massless states are either vector
multiplets or chiral multiplets. The latter can be restricted to one chirality (e.g left-
handed); the other choice merely produces the CPT conjugates. Boundaries are called
real if a = a′, where the conjugate boundary a′ is defined by CΩa,a′ = 1, and complex
otherwise. The Chan-Paton multiplicities Na give rise to gauge groups U(Na) for
complex boundaries and SO(Na) or Sp(Na) for real ones. In the latter case Na must
be even. To count bi-fundamentals we define
Lab ≡
Aiabχi(
)massless,L . (4.4)
Note that because of the factor 1
in (4.1) and the fact that Lab is symmetric, the value
of Lab is indeed precisely the number of bi-fundamentals in the representation (Na, Nb).
12These are spin-3 currents consisting of products of the world-sheet supercurrents of the factors in
the tensor product, including the NSR space-time factor.
It is convenient to introduce the intersection matrix13
Iab ≡ Lab′ − La′b , (4.5)
which is manifestly antisymmetric in a and b. Note that for a pair of complex bound-
aries a, b with conjugates a′, b′ one can define four quantities that are relevant for the
massless spectrum, two of which are chiral, namely Iab and Iab′ .
It is often convenient to associate a geometric picture to these integers. Thus we
will often refer to the boundary labels and their multiplicities as “stacks of branes”, and
view the integers Iab as brane intersection numbers. This is only done for convenience
and does not imply a concrete brane realization; indeed, it does not make sense to
say that a given boundary label corresponds to a Dp-brane for some give p. Such an
interpretation might be valid in a large radius limit, assuming such a limit exists.
In general, for a choice of Chan-Paton multiplicities Na there will be tadpoles in
the one-point closed string amplitudes on the disk and the crosscap. These have to be
cancelled in order to make the theory consistent (since we work with supersymmetric
strings we do not have the option of cancelling RR and NS-NS tadpoles separately).
This leads to a condition on the Chan-Paton multiplicities:
NaRa,(m,J) = 4ηmUm,J (4.6)
where η0 = 1 and all other η’s are −1; there is such a condition for any Ishibashi label
(m, J) that leads to a massless scalar in the transverse channel. The one for m = 0
(which is non-degenerate) is the dilaton tadpole condition. It has the special feature
that all coefficients Ra0 are real and positive. The crosscap coefficient U0 is also real
and can be chosen positive (in the CFT both signs are acceptable). If U0 6= 0 (4.6)
limits the Chan-Paton multiplicities; if U0 = 0 the only solution is Na = 0 for all a,
which rules out any realization of the Standard Model. This reduces the number of
usable orientifolds to 33012.
Tadpole cancellation condition implies cancellation of RR-charges coupling to long-
range fields, and absence of local anomalies. There is a second condition that has
13Note that Lab is a symmetric matrix giving the number of chiral multiplets in the ( a, b) bi-
fundamental. This is a natural quantity in unoriented CFT’s, where a symmetric definition for the
annulus amplitude exists. In oriented CFT the annulus is, in general, not symmetric, but on the
other hand it is possible to choose the branes in such a way that only ( , ) bi-fundamentals appear.
This has become the customary way of counting states in the intersecting brane literature, even for
orientifold models. The quantity Iab is defined in such a way that it is anti-symmetric in a and b.
This is why boundary conjugations appear in the right hand side. This has the additional advantage
of making I a more familiar quantity for readers used to the standard intersection brane conventions.
to be taken into account, which has to do with Z2 charges that do not couple to
long-range fields, usually referred to as “K-theory charges” in geometric constructions.
Uncancelled K-theory charges may lead to global anomalies in symplectic factors of the
gauge group. But even if this symptom is absent, the disease may still exist. A much
more general way to probe for uncancelled K-theory charges is to require the absence
of global anomalies not only in the Chan-Paton gauge group but also on all symplectic
brane-anti-brane pairs that can be added to it as “probe-branes” [48]. Presently this
is the most general constraint that be imposed in these models, but it is not known
if additional ones are required. This probe brane constraint leads to a large number
of mod-2 constraint and is potentially very restrictive, but almost harmless in practice
[49]. It is satisfied by all models we consider in the present paper.
4.2 Search for SM-like models
The complete set of solutions to these conditions is finite but huge, but the vast major-
ity is of no phenomenological interest. In the last few years systematic searches have
been carried out for models that contain the Standard Model. The models that were
considered have the property that the set of Chan-Paton labels can be split into two
subsets, the observable and the hidden sector. The former has been limited, for prac-
tical reasons, to at most four complex brane stacks, required to contain the Standard
Model gauge group and the right intersections to yield three families of quarks and
leptons, plus (in general) some non-chiral (vector-like) additional matter. The hidden
sector is only constrained by the requirement that there be no net number of chiral
multiplets charged under both the observable and hidden sector, and by practical com-
putational limitations. The main purpose of the hidden sector in these models is to
provide variables that can be used to satisfy the tadpole and global anomaly condi-
tions, since the multiplicities in the observable sector are already fixed. In some cases
the observable sector already satisfies the constraints by itself, and there is no hidden
sector.
The observable sector can be realized in many different ways if one only imposes
the constraint that the standard model should be contained in it. These possibilities
were recently explored in [7]. We will focus on the realization described in Section 3.1,
first considered in [5]. There are four stacks, namely a (containing QCD and baryon
number as U(3)), b (containing electroweak SU(2) embedded as U(2) or Sp(2)), c
(providing a U(1) or an O(2) factor14, and d (providing another U(1) factor).
14In [6] also Sp(2) was considered, but this requires an additional Higgs mechanism.
The standard model hypercharge generator is , defined in (3.1):
Qd (4.7)
where Qx denotes the generator of the U(1) of brane stack x; in case the Chan-Paton
factor of brane c is O(2) one should use the properly normalized O(2) generator. In
addition to Y these models have two or three additional U(1) gauge symmetries (the
latter case if electroweak SU(2) arises from U(2)). These (except the combination
B − L) are anomalous, with anomaly cancelled by the Green-Schwarz mechanism,
implying the existence of a B ∧ F coupling making them massive. In fact, as already
mentioned, such Stückelberg couplings may be present for non-anomalous U(1)’s as
well. We are interested in models where the hypercharge gauge boson does not have
such couplings (otherwise the model would be phenomenologically unacceptable), but
where the B−L gauge boson is massive by such couplings (both in order that the gauge
group reduces to the SM one, and that neutrino Majorana masses may be induced by
string instantons, as discussed in previous sections).
The combined requirements of having a massive B − L and a massless Y turn out
to be difficult to satisfy. In fact, if the group on brane c is O(2) they are impossible
to satisfy simultaneously, because the O(2) component of the vector boson does not
couple to any axions, and hence the B−L and Y bosons have the same mass. But even
in models with a U(1) group on brane c it happens rather rarely that both constraints
are satisfied simultaneously, at least in the searches that have been done so far.
We will make use here of the data presented in [6, 7], which are available in slightly
improved form on the website www.nikhef.nl/∼t58/filtersols.php. This database con-
sist of 211634 distinct spectra. Here “distinct” means that they are physically different
for a given MIPF15 if the hidden sector is ignored. Hence the differences can be the
number of vector-like states of various kinds or the dilaton couplings of branes a, b, c,
d. Geometrically, these spectra may originate from the same moduli space, but then in
any case from different points on this moduli space. The improvements in comparison
with the data presented in [6] consist of taking into account the full global anomaly
conditions from probe branes. In some cases this required nothing more than checking
these conditions for an existing solution of the tadpole conditions, but in other cases a
new solution had to be found. As a result, a few models disappeared from the original
database, but due to improved algorithms a few new ones could be added. The net re-
15Rare cases of identical spectra and couplings originating from different MIPFs are treated as
distinct.
http://www.nikhef.nl/~t58/filtersols.php
sult is some small but inconsequential changes in the total number of models of various
kinds. The numbers we will mention below are based on the improved database.
The total number of models in that database with a Chan-Paton group U(3) ×
Sp(2)× U(1)× U(1) is 10587. Of these, 391 (about 4%) have a massive B − L vector
boson. For U(3)×U(2)×U(1)×U(1) these numbers are, respectively, 51 and 0. Hence
no examples of the latter type were found, although they were found with 1,2 and 4
families (in a limited search), in a few percent of the total number of models. It seems
therefore reasonable to expect that U(3) × U(2) × U(1) × U(1) with massive B − L
do exist, and that their absence is just a matter of statistics. Just for comparison, the
total number of U(3)× Sp(2)× O(2)× U(1) models is 56627.
5 Fermion zero modes for instantons on RCFT’s
In this section we discuss D-brane instantons for general compactifications, including
abstract CFT ones. We also provide the spectrum of zero modes on an instanton brane,
using the information about their internal structure i.e. in the compactified dimension
in geometric models, or of the internal CFT in more abstract setups like in previous
section. We will be interested in the latter case.
A first question that should be addressed is what this internal structure is. For
instance, in type IIA geometric compactifications, it corresponds to a supersymmetric
(i.e. special lagrangian) 3-cycle. Notice that these are the same kind of 3-cycles already
used to wrap the D6-branes that give rise to the 4d gauge symmetry of such models.
For general CFT’s, D-branes are described as boundary states. To describe instantons,
one can simply use the same boundary state of the internal CFT to describe the 4d
space-filling branes present in the model and the instanton branes. The only difference
is that boundaries satisfy Neumann conditions in the 4d space-filling case, and Dirichlet
in the instanton case. This exploits the fact that whenever a boundary state of the
internal CFT, and with Neumann boundary conditions in the 4d space is an acceptable
state of the full CFT, the same boundary state of the internal CFT, combined with
Dirichlet boundary conditions in the 4d space also gives an acceptable state of the full
CFT. For geometric compactifications this is related to Bott periodicity of the K-theory
classes associated to the D-brane charges, but it is possible to show it in general.
Since instanton D-branes can thus be naturally associated to the boundary states
of 4d space-filling branes, it is convenient to express the spectrum of zero modes of the
former in terms of the massless states of the latter. This is particularly useful, since
the computation of the spectra on 4d space-filling branes for Gepner model orientifolds
has already been described (although the arguments below are valid also for geometric
compactifications). Hence, let us denote by M a 4d space-filling brane associated with
the same boundary state of the internal CFT as the instanton brane M of interest.
Note that the 4d space-filling brane M is an auxiliary tool, and need not be (and, for
our instantons of interest, will not be) one of the 4d space-filling branes present in the
model.
‘Real’ brane instantons
Let us first consider the case of ‘real’ brane instantons. Consider a set ofm 4d space-
filling branes M, and focus first on the massless spectrum in the MM sector. Before
the orientifold projection, it leads to a universal 4d N = 1 U(m) vector multiplet, and
a number LMM of adjoint chiral multiplets. The orientifold operation maps this sector
to itself, acting on the Chan-Paton with a matrix γΩ,M. This matrix satisfies
γTΩ,Mγ
Ω,M = ±1m (5.1)
The two possibilities can be chosen to correspond to γΩ,M = 1m or γΩ,M = ǫm, with
−1r 0
, and m = 2r hence necessarily even in the latter case. They corre-
spond to the SO and Sp projections, respectively.
The orientifold projection on the N = 1 vector multiplet Chan-Paton matrices is
given by
λ = −γΩ,M λT γ−1Ω,M (5.2)
and leads to SO(m) or Sp(m) vector multiplets for the SO or Sp projection (hence
the name). Concerning the N = 1 chiral multiplets, they fall in two classes of p−, p+
(with p− + p+ = LMM) which suffer the projections
λ = ±γΩ,M λT γ−1Ω,M (5.3)
For the SO projection, this leads to p+, p− chiral multiplets in the , representation.
For the Sp projection, there are p+, p− chiral multiplets in the , representation.
The sectors Ma (where a is a 4d space-filling branes present in the model) are
mapped to sectors Ma′, so it is enough to focus on the former. After the orientifold
projection one gets LMa, LMa′ chiral multiplets in the ( M, a), ( M, a).
Let us now obtain the zero modes for a set of m instanton branes M in terms
of the above spectrum. The MM sector is closely related to the MM sector, by
changing the NN boundary conditions in 4d spacetime to DD boundary conditions
(which can be done in a covariant formalism, but not in the light-cone gauge). Before
the orientifold projection, one obtains the same set of states (since moddings for NN
and DD boundary conditions are identical, both in the NS and R sector), but with
different world-volume interpretation. Also, the change in boundary conditions implies
that some polarization states which are unphysical for the 4d spacefilling brane are
physical in the instanton brane. Hence, the U(m) gauge bosons on the 4d space-filling
brane M correspond to four adjoint real scalars in the instanton brane M . Similarly,
the 4d spinors in M, correspond to four fermion zero modes on M , transforming as
two spinors of opposite chiralities θα, θ̃α̇ of the SO(4) rotation group in transverse
space. The orientifold projection maps the MM sector to itself, acting on Chan-Paton
indices with a matrix γΩ,M . In close analogy with the argument in [50] for the familiar
D5-D9-brane system in type I (see [51, 52] for related derivations), one can show that
the condition (5.1) flips sign upon changing four NN boundary conditions to DD, hence
γTΩ,Mγ
Ω,M = ∓1m (5.4)
Namely, the instanton brane has Sp(m) gauge group when the 4d space-filling brane
(with same internal boundary state) has gauge group O(m), and vice-versa. We still
refer to these projections as SO and Sp, hoping no confusion arises. Note that, as
mentioned in Section 2.2, although there are no gauge bosons in 0+ 0 dimensions, the
gauge group is present on the instantons in that it acts on open string endpoints.
Let us consider the effect of the orientifold projection on the MM states, as com-
pared with the effect onMM states. Again, following arguments familiar in the D5-D9
brane system in type I, one can show that the signs in conditions like (5.2), (5.3) re-
main unchanged upon changing four NN dimensions to DD, except for bosonic modes
polarized along the directions longitudinal to these four dimensions (and for fermions
related to them by the unbroken susy of the total system). To be concrete, consider-
ing the four MM adjoint bosons, and two MM adjoint fermions θα associated to the
universal MM vector multiplets, they suffer the projection
λ = +γΩ,M λ
T γ−1M (5.5)
Hence they transform in the of Sp(m) for the SO projection, and in the of
SO(m) for the Sp projection. On the other hand, for the two fermion zero modes θ̃α̇,
the projection is
λ = −γΩ,M λT γ−1M (5.6)
and leads to two fermion zero modes in the of Sp(m) for the SO projection, and in
the of SO(m) for the Sp projection.
This implies that in order to obtain two fermion zero modes from this univer-
sal multiplet, in order to generate a superpotential, one should consider instantons
with orthogonal gauge group and multiplicity one (O(1) instantons). For instantons
with symplectic gauge group and multiplicity two (Sp(2) instantons), there are two
additional fermion zero modes in the triplet representation. As mentioned, we will
continue to consider such instantons in our relaxed scan. Multiple instantons, i.e.
boundary states with higher multiplicity, lead to a larger amount of additional fermion
zero modes (due to the larger gauge representations for the fermions), and do not
contribute to superpotentials; we will not consider such cases even in relaxed scans,
since they also very often lead to too many charged fermion zero modes and cannot
contribute to the operators of interest (except possibly for O(2) and U(2) instantons
with low intersections, which are kept in our scan as a curiosity).
Similarly, for the p± sets of MM scalars and fermions associated to the MM 4d
chiral multiplets, the projection is
λ = ±γΩ,M λT γ−1Ω,M (5.7)
with the same sign choice as in (5.3). The different structure of γΩ implies that, for
the SO projection we get p+, p− sets of scalars and fermions in the , , while for
the Sp projection there are p+, p− sets of scalars and fermions in the , .
This concludes the discussion of the MM sector. Let us not consider the Ma sec-
tors, from the information from the Ma sectors. Notice that this implies changing
four NN boundary conditions to DN, which have different moddings. Hence the states
are different in both situations, but the information on the multiplicities is preserved.
Specifically, in the NS sector the DN boundary condition introduce an additional vac-
uum energy which generically makes all states massive. Hence there are no massless
scalar zero modes in generic Ma sectors. In the R sector, the change in the mod-
dings reduces the dimension of the massless ground state, leading to a single (chiral)
fermionic degree of freedom. Since the orientifold action maps the Ma sector to Ma′
sectors, there are no subtleties in the orientifold projection. The end result is LMa,
LMa′ fermion zero modes in the ( M, a), ( M, a). The net number of chiral fermion
zero modes in the ( M, a) is given by IMa = LMa′ − LMa, i.e. the net number of
chiral multiplets in the related Ma sector.
The results for orientifold projections for real branes are shown in table 3.
Proj. Multiplet in M M (before orient.) M (after orient.) M (after orient.)
SO N = 1 vect. mult. U(m) O(m) Sp(m)
2 f + 2 f + 4 b
N = 1 ch. mult. (p+ + p−) Ad p+ + p− 2p+ ( f + b ) +
2p− ( f + b )
Sp N = 1 vect. mult. U(m) Sp(m) O(m)
2 f + 2 f + 4 b
N = 1 ch. mult. (p+ + p−) Ad p+ + p− 2p+ ( f + b ) +
2p− ( f + b )
Any N = 1 ch. mult. LMa′( M, a)+ LMa′( M, a)+ LMa′( M , a) f
LMa( M, a) LMa( M, a) LMa( M , a) f
net IMa( M, a) net IMa( M , a) f
Table 3: Orientifold projection for real branes: Massless modes of the 4d space-filling branes
M (before and after the orientifold projection) and zero modes on the instanton branes M
(denoted with sub-indices b, f for bosonic and fermionic modes)
Complex brane instantons
We now consider the case of complex brane instantons. The arguments are very
similar, hence the discussion is more sketchy. Consider m 4d spacefilling branes M,
associated to the internal boundary state of the instanton brane M of interest. The
MM leads to a 4d N = 1 U(m) vector multiplet and a number LMM′ of adjoint chiral
multiplets. The orientifold action maps it to the M′M′ sector, hence we may keep just
the former and impose no projection. The MM′ sector is mapped to itself under the
orientifold projection. Denoting by γΩ,M the action on Chan-Paton indices, the MM′
modes split into sets L±MM , L
M ′M ′, which suffer a projection
λ = ±γΩ,M λT γ−1Ω,M (5.8)
leading, for γΩ,M = 1m, to L
chiral multiplets in the , , and L+
chiral multiplets in the , . The net number of chiral multiplets in the ,
is I+
M,M − L+M′M′, I−MM′ = L−M,M − L−M′M′ . And oppositely for γΩ,M = ǫm.
Finally, the Ma, Ma′ and related sectors lead, after the orientifold projection, to
LMa′ , LMa, LM′a′ , LM′a chiral multiplets in the ( M, a), ( M, a), ( M, a), ( M, a).
In order to simplify notation, we replaceM →M in these expressions in our discussions
of instanton zero modes.
Let us now consider m brane instantons M and compute their zero mode spectrum
in terms of the above. In the MM (and its image M ′M ′) sector there are four scalar
modes and four fermions in the adjoint of the U(m) gauge symmetry group; these are
related to the 4d vector multiplet in the MM sector. In addition, there are LMM ′
sets of scalars and fermions in the adjoint, related to the LMM′ non-universal chiral
multiplets in the MM sector. The MM ′ sector is mapped to itself, and one has to
impose the orientifold projection (recalling that the matrix γΩ,M differs from γΩ,M).
For γΩ,M = 1, hence γΩ,M = ǫ, we obtain L
MM , L
MM chiral multiplets in the , ,
and L+M ′M ′ , L
M ′M ′ chiral multiplets in the , . The net number of chiral multiplets
in the , is I+MM ′ = L
MM − L+M ′M ′, I−MM ′ = L−MM − L−M ′M ′ . And oppositely for
γΩ,M = ǫ hence γΩ,M = 1.
In theMa,Ma′ and related sectors, there are generically no bosonic zero modes, and
there are LMa, LM′a′ , LMa′ , LM′a chiral fermion zero modes in the ( M , a), ( M , a),
( M , a), and ( M , a) respectively. The net number of chiral fermion zero modes in
the ( M , a) and ( M , a) is given by IMa = LMa′ −LM′a′ and IMa′ = LMa−LM′a. In
order to simplify notation, we replace M →M in these expressions in our discussions
of instanton zero modes.
The results for orientifold projections for real branes are shown in table 4.
6 Search for M instantons
In this section we perform a search of models which admit an instanton inducing a
right-handed neutrino Majorana mass operator. Namely, for each model with the
chiral content of the SM in the classification described in Section 4.2, we first scan over
boundary states, searching for all instantons with the required uncharged and charged
fermion zero mode structure to yield neutrino masses. We then relax our criteria a bit
and allow for instantons with correct charged zero mode structure but having extra
non-chiral zero modes (both charged and uncharged). The idea is that these non-chiral
zero modes could be lifted by diverse effects, as discussed.
It is important to recall that the cubic couplings between instanton zero modes and
4d chiral multiplets are difficult to compute in Gepner model orientifolds. Hence, we
will simply assume that such couplings are non-zero if there is no symmetry forbidding
them.
Proj. Multiplet in M M (before orient.) M (after orient.) M (after orient.)
Any N = 1 vect. mult. U(m)× U(m)′ U(m) U(m)
4Ad f + 4Ad b
N = 1 ch. mult. padj Ad + padjAd
′ padjAd 2padj ( Ad f + Ad b )
SO N = 1 ch.mult. LMM( M, M′) L
2L+MM b,f + 2L
MM b,f
LM′M′( M, M′) L
M′M′ M
M′M′ M
2L+M ′M ′ b,f + 2L
M ′M ′ b,f
Sp N = 1 ch.mult. LMM( M, M′) L
2L+MM b,f + 2L
MM b,f
LM′M′( M, M′) L
M′M′ M
M′M′ M
L+M ′M ′ b,f + L
M ′M ′ b,f
Any N = 1 ch. mult. LMa′( M, a)+ LMa′( M, a) LMa′( M , a) f
. . . LMa( M, a) LMa( M , a) f
. . . LM′a′( M, a) LM ′a′( M , a) f
. . . LM′a( M, a) LM ′a( M , a) f
net IMa( M, a) net IMa( M , a) f
net IMa′( M, a) net IMa′( M , a) f
Table 4: Orientifold projection for complex branes: Massless modes of the 4d space-filling
branesM (before and after the orientifold projection) and zero modes on the instanton branes
M (denoted with sub-indices b, f for bosonic and fermionic modes)
6.1 The instanton scan
Our detailed strategy will become clear along the description of the results. Given a
set of a,b,c,d standard model branes, we must look for additional boundary states M
that satisfy the requirements of a (B−L)-violating instanton. From the internal CFT
point of view this is just another boundary state, differing from 4d spacefilling branes
only in the fully localized 4d spacetime structure. The minimal requirement for such
a boundary state is B − L violation, which means explicitly
IMa − IMa′ − IMd + IMd′ 6= 0 (6.1)
It is easy to see that the existence of such an instanton implies (and hence requires) the
existence of a Stückelberg coupling making B−L massive. To see this, consider adding
to the Standard Model configuration a 4d spacefilling brane M (in fact used in Section
5) associated to the boundary stateM (RR tadpoles can be avoided by simultaneously
including M antibranes, which will not change the argument). The new sector in
the chiral spectrum charged under the branes M can be obtained by reversing the
argument in Section 5, and is controlled by the intersection numbers of M . From
the above condition it follows that the complete system has mixed U(1)B−L × (GM)2
anomalies, where GM is the Chan-Paton-factor of brane M. These anomalies are
cancelled by a Green-Schwarz mechanism involving a (B −L)-axion bilinear coupling,
which ends up giving a mass to B−L via the Stückelberg mechanism. This coupling is
in fact not sensitive to the presence of the brane M, hence it must have been present
already in the initial model (without M).
Hence the existence of a boundary labelM that satisfies (6.1) implies that B−L is
massive. Unfortunately the converse is not true: even if B−L has a Stückelberg mass,
this still does not imply the existence of suitable instantons satisfying (6.1)16 Indeed,
in several models we found not a single boundary state satisfying (6.1).
Note that, since hypercharge must be massless, one can use the reverse argument
and obtain that
IMa − IMa′ − IMc + IMc′ − IMd + IMd′ = 0 (6.2)
in all models. We verified this for all models we considered as a check on the compu-
tations.
As already discussed in Section 4.2, in the search for SM constructions in Gepner
orientifold, there are 391 models with massless hypercharge and massive B−L. In these
models we found a total of 29680 instantons with B−L violation, i.e. with intersection
numbers satisfying (6.1). Of course, in order to serve our purpose of generating a
Majorana mass superpotential, the instantons have to satisfy some more conditions.
Let us consider them in order of importance, and start with the conditions on the net
number of chiral fermion zero modes charged under the 4d observable sector. Clearly
we need IMa = IMa′ and IMb = IMb′ . The latter condition is automatically satisfied in
this case, because the b-brane is real in all 391 models. The chiral conditions on the
zero modes charged under the branes c and d are as in [3]17 and are given in equations
(3.6), (3.7) (3.8) of the present paper. These are the instantons of most interest, and
on which we mainly focus. However, as discussed in Section 3.4, other important B-
and/or L- violating operators (such as the Weinberg operator or the LH operator) can
16From intuition in geometric compactifications, one expects that there may always exist a D-brane
with the appropriate topological pairings, but there is no guarantee that there is a supersymmetric
representative in that topological sector, and even less that it would have no additional fermion zero
modes. Note also that even if such D-brane instantons exists, there is no guarantee that it will fall in
the scan over RCFT boundary states.
17Note that there is a sign change in the contribution of the U(1)d generator to Y in comparison
to [3]
be generated by instantons with similar intersection numbers, up to a factor of 2 and
a sign, see table 2. For this reason we also allow at this stage any instanton which has
the correct number of charged zero modes to generate them. Imposing these conditions
reduces the number of candidate instantons potentially contributing to neutrino masses
in any of the models to 1315.
All instantons satisfying these requirements are summarized in the table 5. In
columns 1,2 and 3 we list the tensor combination, MIPF and orientifold choice for
which the model occurred. The latter two numbers codify simple current data that
describe respectively a MIPF and an orientifold. MIPFs are in general defined by
means of a subgroup H of the simple current group G, plus a certain matrix X of
rational numbers [55]. Orientifolds are defined by a simple current and a set of signs
[43]. In previous work [6] we have enumerated these quantities (up to permutation
symmetries) and assigned integer labels to them for future reference. We only refer
to these numbers here, but further details are available upon request. Usually for
each MIPF and orientifold which contains the standard model there are several choices
a,b,c,d for which it is obtained. For a given choice of tensor combination, MIPF and
orientifold and SM branes there may be several instantons. For clarity we put all such
instantons together in the information in table 5. In column 4 we indicate which type
of instanton branes were found. Five types are distinguished: O1, O2, S2, U1 and
U2, corresponding to O(1), O(2), Sp(2), U(1) and U(2) Chan-Paton symmetry on the
instanton volume. The number indicates the instanton brane multiplicity that gives
the correct number of instanton charged zero modes from the a, b, c, d branes, to lead
to right-handed neutrino Majorana masses. The number of zero modes is in general
the product of the instanton brane multiplicity and ‘intersection number’ with the
corresponding 4d spacefilling brane. As discussed in Section 5, for symplectic branes
the smallest possible brane multiplicity is 2. As we discussed there, only O1 instantons
may have the required universal minimal set of two zero modes in the uncharged sector.
Still we look for all O(1), Sp(2) and U(1) instantons which may yield a superpotential
if the extra uncharged fermion zero modes. In this vein we also include a search for
O2 and U2 instantons. Note also that such O2 or U2 instantons imply the existence
of other instantons involving the same boundary state, but with multiplicity 1, which
may lead to the R-parity violating operator LH . We will discuss the generation of
R-parity violating operators at the end of this section. The third character (+ or −)
in the instanton in table 5 is the sign of IMc′ − IMc. For the instantons giving rise to
right-handed neutrino Majorana masses this sign should be negative, whereas it should
be positive for instantons giving rise to the Weinberg operator (or the LH operator),
see table 2.
The 1315 instantons are divided in the following way over the different types: 3 of
types O1+ and O1−, 46 of type U1+, 24 of type U1−, 550 S2+, 627 S2−, 27 of types
U2+ and U2− and four of types O2+ and O2−. Notice that the vast majority (97.5%)
of the instanton solutions are of type S2+ and S2−. This is encouraging given the
nice properties of such instantons, concerning e.g. R-parity conservation. Note also
that in almost all cases both S− and S+ are simultaneously present,18 so both sources
of physical neutrino Majorana masses (from the see-saw mechanism or the Weinberg
operator) are present. The other instanton classes possibly generating right-handed
neutrino masses are O1− and U1−, which are much less abundant. There is just one
orientifold with O1− instantons, for which one can obtain cancellation of RR tadpoles,
see below. On the other hand we have found no orientifold with U1− instantons and
cancellation of tadpoles, see below.
Table 5: Summary of instanton branes.
Tensor MIPF Orientifold Instanton Solution
(1,16,16,16) 12 0 S2+, S2− Yes
(2,4,12,82) 19 0 S2−! ?
(2,4,12,82) 19 0 U2+!, U2−! No
(2,4,12,82) 19 0 U1+, U1− No
(2,4,14,46) 10 0
(2,4,14,46) 16 0
(2,4,16,34) 15 0
(2,4,16,34) 15 1
(2,4,16,34) 16 0 S2+, S2− Yes
(2,4,16,34) 16 1
(2,4,16,34) 18 0 S2− Yes
(2,4,16,34) 18 0 U1+, U1−, U2+, U2− No
(2,4,16,34) 49 0 U2+, S2−!, U1+ Yes
Continued on next page
18In some models contributing many instantons there is an exact symmetry between S− and S+.
This explains the approximate symmetry in the full set. In some cases this symmetry can be understood
in terms of flipping the degeneracy labels of boundary states. We regard it as accidental, since it is
not found in all models.
Table 5 – continued from previous page
Tensor MIPF Orientifold Instanton Solution
(2,4,16,34) 49 0 U1− No
(2,4,18,28) 17 0
(2,4,22,22) 13 3 S2+!, S2−! Yes!
(2,4,22,22) 13 2 S2+!, S2−! Yes
(2,4,22,22) 13 1 S2+, S2− No
(2,4,22,22) 13 0 S2+, S2− Yes
(2,4,22,22) 31 1 U1+, U1− No
(2,4,22,22) 20 0
(2,4,22,22) 46 0
(2,4,22,22) 49 1 O2+, O2−, O1+, O1− Yes
(2,6,14,14) 1 1 U1+ No
(2,6,14,14) 22 2
(2,6,14,14) 60 2
(2,6,14,14) 64 0
(2,6,14,14) 65 0
(2,6,10,22) 22 2
(2,6,8,38) 16 0
(2,8,8,18) 14 2 S2+!, S2−! Yes
(2,8,8,18) 14 0 S2+!, S2−! No
(2,10,10,10) 52 0 U1+, U1− No
(4,6,6,10) 41 0
(4,4,6,22) 43 0
(6,6,6,6) 18 0
Most models have a hidden sector containing extra boundary states beyond the
SM ones. In the same spirit of imposing chiral conditions first, we should require
that IMh = IMh′, where h is a hidden sector brane. This is to guarantee that the
generated superpotential does not violate some hidden sector gauge symmetry which
would require the presence of hidden sector fields along with the νR bilinear. The latter
condition is not imposed on the previously known hidden sector (i.e. the one in [6, 7]),
but instead a new search for tadpole solutions was performed, for each M , restricting
the candidate hidden sector branes to those satisfying IMh = IMh′ (as discussed in
Section 5). This is because in general the known hidden sector in [6, 7] is just a sample
out of a huge number of possibilities.
In column 5 we indicate for which instantons it was possible to satisfy the tadpole
conditions with this additional constraint. With regard to observable-hidden matter we
use the same condition as in [6], namely that it is allowed only if it is vector-like. Such
a solution could be found for 879 of the 1315 instantons, with ten cases inconclusive
(i.e it was computationally too difficult to decide if a solution does or does not exist).
The latter are indicated with a question mark in column 5 (for most of the undecidable
cases there is a tadpole solution for a different instanton with the same characteristics;
for that reason just one question mark appears).
Independently of the RR tadpole condition (since there may be alternative sources
for its cancellation, or hidden sectors which fall beyond the reach of RCFT), we can
also consider the further constraint that the number of charged fermion zero modes is
exactly right, not just in the chiral sense. This means IMa = IMa′ = IMb = IMb′ = 0,
IMc = 2, IMc′ = 0 and IMd = −2, IMd′ = 0 or vice-versa. Furthermore we require that
there are no adjoint or rank-2 tensor zero-modes (note that the latter could be chiral
if the instanton brane is complex, and indeed they are in some of the 1315 cases).
This reduces the 1315 instantons to 263. In column 4 we indicate those cases with
an exclamation mark. It is noteworthy that the success rate for solving the tadpole
conditions is highest for these instantons: 254 of the 263 allow a solution (with 3
undecided). If an exclamation mark appears in column 4, this only indicates that
some of the instantons are free of the aforementioned zero modes, not that all of them
are. But in all cases, if there are tadpole solutions, they exist in particular for the
configurations with an exclamation mark. Finally we may impose the condition that
IMh and IMh′ are separately zero. This is indicated with and exclamation mark in
column 5. This turns out to be very restrictive. The only cases where this happens
have no hidden sector at all.
It is worth remarking that the only instantons having exactly the correct set of
charged zero modes and cancelling tadpoles are of S2± type. Also those instantons are
the only cases marked with an exclamation mark in column 4 and 5. These examples,
which will be discussed below in some detail, also have just the minimal set of fermion
zero modes, except for the universal sector (which for Sp(2) instantons contains two
extra triplets).
The main conclusion about this scan is that we did not find any instantons with
exactly the zero mode fermions to generate the neutrino mass superpotential. However
we have found a number of examples which come very close to that, with exactly the
required charged zero modes and a very reduced set of extra uncharged zero modes
from the universal sector. These extra zero modes are non-chiral and hence one expects
that e.g. RR/NS fluxes or other effects may easily lift them, as we discussed in section
2. Concerning O(1) instantons, which have just the two required fermion zero modes
in the universal sector, we have found one example, with the appropriate net structure
of charged zero modes. However, it has plenty of other extra zero modes. We discuss
examples of O(1) and Sp(2) instantons in the following subsections.
6.2 An O1 example
Let us first discuss the case of O(1) instantons. In principle they would be the more
attractive since they have no undesirable universal zero modes at all. Unfortunately
this type of instanton is rare within the set we scanned, and we found just one example
with a solution to the tadpole equations without any unwanted chiral zero-modes. The
instanton however has a very large number of uncharged and charged vector-like zero
modes.
The standard model brane configuration occurs for tensor product (2, 4, 22, 22),
MIPF 49, orientifold 1, boundaries (a,b,c,d) = (487, 1365, 576, 486). As usual we only
provide this information in order to locate this model in the database. Further details
are available on request.
The bi-fundamental fermion spectrum of this model in the (a,b,c,d) sector is fairly
close to the MSSM: there is an extra up-quark mirror pair, two mirror pairs of lepto-
quarks with down quark charges and one with up-quark charges, plus two extra right-
handed neutrinos (i.e. a total of five right-handed neutrinos). There are three MSSM
Higgs pairs. The tensor spectrum is far less appealing, in particular for brane c: this
has 25 adjoints and 7 vector-like pairs of anti-symmetric tensors.
As we said, there is just one instanton brane of type O1−. It has exactly the
right number of zero-modes with brane d, but five superfluous pairs of vector-like zero-
modes with brane c, plus one vector-like pair with brane a. In addition there are four
symmetric tensor zero-modes on the instanton brane (which of course are vector-like,
since it is a real brane): the parameter p+ in table 3 is equal to 2.
The tadpole solution that is (chirally speaking) compatible with this instanton has
a large hidden sector: O(1)×O(2)4×O(3)×U(1)2×Sp(2)2×U(3) (there are other pos-
sibilities, but no simple ones). This hidden sector introduces more undesirable features:
vector-like observable/hidden matter, vector-like instanton/hidden sector modes, plus
chiral and non-chiral matter within the hidden sector. Finally the coupling ratios are
as follows: α3/α2 = .54, sin
2θw = .094, and the instanton coupling is 3.4 times weaker
than the QCD coupling (α3/αInstanton = 3.4).
Despite these unappealing features this model does demonstrate the existence of
this kind of solution.
6.3 The S2 models
As we have mentioned, these are the examples which come closer to the minimal set of
fermion zero modes. As we see in Table 5, all such instantons satisfying the criteria on
the zero mode structure (except for the extra universal zero modes) appear for models
based on the same CFT orientifold. It is the one obtained from the (2, 4, 22, 22) Gepner
model with MIPF 13 and orientifold 3 in the table. The model is obtained as follows.
6.3.1 The closed string sector
We start with the tensor product (2,4,22,22). This yields a CFT with 12060 primary
fields, 48 of which are simple currents, forming a discrete group G = Z12×Z2×Z2. After
taking into account the permutation symmetry of the last two factors, we find that this
tensor product has 54 symmetric MIPFs, and we choose one of them to build the model
of interest. For convenience we specify all quantities in terms of a standard minimal
model notation, but also in terms of the labelling of the computer program “kac” that
generates the spectrum. This particular MIPF is nr. 13. To build it we choose a
subgroup of G, which is isomorphic to H = Z12 × Z2. The generator of the Z12 factor
is primary field nr. 1, (0, 0, 0, {24,−24, 0}, {24, 20, 0}), and the Z2 factor is generated
by primary field nr. 24, (0, 0, 0, 0, {24, 20, 2}). The representations are specified on a
basis (NSR, k = 2, k = 4, k = 22, k = 22), i.e. the boundary conditions of the NSR-
fermions and the four minimal models in the tensor product. Here 0 indicates the CFT
vacuum, and for all other states we use the familiar (l, q, s) notation for the N = 2
minimal models. The first generator has conformal weight h = 11
and has ground state
dimension 1. The second has weight h = 11
and has ground state dimension 2: the
ground state contains both (0, 0, 0, 0, {24, 20, 2}) and (0, 0, 0, {24, 20, 2}, 0). The matrix
X defining the MIPF according to the prescription given in [53][54][55] is
(6.3)
This simple current modification is applied to the charge conjugation invariant of the
tensor product. This defines a MIPF that corresponds to an automorphism of the
fusion rules, and that pairs all the primaries in the CFT off-diagonally. The number
of Ishibashi states, and hence the number of boundary states is 1080. The MIPF is
invariant under exchange of the two k = 22 factors: this maps current 24 to itself, and
current 1 to current 11, which is also in H. Hence this symmetry of the tensor product
maps H into itself, and it also preserves the matrix X .
To define an orientifold, we must specify a “Klein bottle current” plus two signs
defined on the basis of the simple current group. For the current K we use the
generator of the second Z2 in G, primary field nr. 12. This is the representation
(0, 0, {4,−4, 0}, {(24, 16, 2)}, {(24,−12, 2)}) which is degenerate with nine other states,
all of dimension 1 and conformal weight 7. The crosscap signs are chosen, on the afore-
mentioned basis of H as (+,−). This results in a crosscap coefficient of 0.0464731, and
it is orientifold nr. 3 of a total of 8. The orientifold is also invariant under permutation
of the identical factors.
The closed string spectrum contains 14 vector multiplets and 60 chiral multiplets.
6.3.2 The standard model branes
To build a standard model configuration we have to specify the boundary state labels.
It turns out that we have four choices for label a and b, one for c and two for d.
This leads to a total of 32 possibilities. Among these 32 there are 22 have distinct
spectra (distinguished by the number of vector-like states), but for all 32 choices one
obtains the same set of dilaton couplings. It seems plausible that these choices simply
correspond to putting the a, b and d branes in slightly different positions, so that we
move the configuration in brane moduli space. The choices are as follows (these are
boundary labels assigned by the computer program, and can be decomposed in terms
of minimal model representations; this will be explained in table 6 below)
a : 10, 22, 130, 142
b : 210, 282, 290, 291
c : 629
d : 712, 797
There are additional possibilities, but they do not give rise to additional distinct spec-
Table 6: Branes appearing in standard model configurations
Label Orbit/Deg. Reps Weight Dimension
10 240 (0, 0, 0, 0, {10, 0, 0}) 5/4 1
130 2760 (0, 0, 0, {10, 0, 0}, 0) 5/4 1
22 [528,0] (0, 0, 0, {1,−1, 0}, {11, 1, 0}) 3/2 1
(0, 0, 0, {1, 1, 0}, {11,−1, 0}) 3/2 1
142 [3048,0] (0, 0, 0, {11,−1, 0}, {1, 1, 0}) 3/2 1
(0, 0, 0, {11, 1, 0}, {1,−1, 0}) 3/2 1
210 4248 (0, 0, {3, 3, 0}, {3,−3, 0}, {9,−9, 0}) 1/2 1
282 5760 (0, 0, {3, 3, 0}{9,−9, 0}{3,−3, 0}) 1/2 1
290 [5952,0] (0, 0, {1, 1, 0}{9, 7, 0}{11,−11, 0}) 5/6 1
291 [5952,24] (0, 0, {1, 1, 0}{9, 7, 0}{11,−11, 0}) 5/6 1
629 [9348,30] (0, (1,−1, 0), 0, {9, 9, 0}{5,−3, 0} 7/12 1
712 [9852,0] (0, {1, 1, 0}{3,−3, 0}{1, 1, 0}{5, 5, 0}) 1/2 2
(0, {1, 1, 0}{1,−1, 0}{1, 1, 0}{5,−3, 0}) 1/2 2
797 [10356,30] (0, {1, 1, 0}{3,−3, 0}{5, 5, 0}{1, 1, 0}) 1/2 2
(0, {1, 1, 0}{1,−1, 0}{5,−3, 0}{1, 1, 0}) 1/2 2
The second column gives the boundary labels in terms of a primary field label and
a degeneracy label (boundaries not indicated by square brackets are not degenerate).
The labels appearing in columns 1 and 2 are assigned by the computer program, and
are listed here only for the purpose of reproducing the results using that program.
In column 2, the boundary labels are expressed in terms of primary field labels, as
in formula (A.4). If a single number appears, this is a representative of an H-orbit
corresponding to the boundary. If square brackets are used, this means that the H-
orbit has fixed points, and that it corresponds to more than one boundary label. The
second entry in the square brackets is the degeneracy label, and refers to a character of
the “Central Stabilizer” defined in [43]; the details of the definition and the labelling
will not be important here. In this case the first entry within the square brackets refers
to an orbit representative.
These orbit representatives can also be expressed in a standard form for minimal
model tensor products. This is done in column 3. This is basically the same expansion
shown in (A.4), except that the degeneracy label ΨI turns out to be trivial in all cases,
both for the standard model and for the instanton branes shown below (although the
theory does contain primaries with non-trivial Ψ’s). In columns 4 and 5 we specify the
weight and ground state dimension of the corresponding highest weight representation.
These data are not directly relevant for the boundary state, but helps in identifying it.
Since boundaries are specified by orbit representatives, it is not straightforward to
compare them, since the standard choice (the one listed in column 2) is arbitrary. For
this reason we have used another representative in columns 3, 4 and 5, selected by an
objective criterion: we choose the one of minimal dimension and minimal conformal
weight (in that order). If there is more than one representative satisfying these criteria
we list all.
6.3.3 The open string spectrum
In Table 7 we summarize the spectra of the 32 models. The first four columns list the
a,b,c,d brane labels. The last eight columns specify the total number of multiplets of
types Q (quark doublet), U (up quark singlet), D (down quark singlet), L (lepton dou-
blet), E (charged lepton singlet), N (neutrino singlet), Y (lepto-quark) and H (Higgs).
The numbers given are for the total number of lefthanded fermions in the represen-
tation, plus their complex conjugates. So for example a 7 in column “Q” means that
there are 5 quark doublets in the usual representation (3, 2, 1
), plus two in the complex
conjugate representation (3∗, 2,−1
This yields the required three families of quark doublets, plus two mirror pairs.
Hence the smallest number that can occur in the six columns QUDLEN is three, if
there are no mirrors (note that cubic anomaly cancellation requires three right-handed
neutrinos in this class of models). The lepto-quarks Y are all in the same representation
as the down-quarks (D), or the conjugate thereof, and they occur only as vector-like
mirror pairs. They differ from D-type mirror quarks because they carry lepton number,
because they come from open strings ending on the d-brane instead of the c-brane.
In general, there can also exist U-type lepto-quarks, but in these models they do not
occur. Finally the numbers 10, 18 and 26 in column ’H’ mean that there are 5, 9
or 13 MSSM Higgs pairs H + H̄. It is worth noticing that right-handed quarks U,D
and neutrinos N = νR do not have vectorlike copies. On the other hand right-handed
leptons E always have one and the left-handed fields Q,L may have up to 3 vector-like
copies.
Table 7: Spectrum all 32 configurations.
U(3) Sp(2) U(1) U(1) Q U D L E N Y H
10 210 629 712 7 3 3 9 5 3 6 10
22 210 629 712 7 3 3 9 5 3 6 10
130 210 629 712 3 3 3 9 5 3 2 10
142 210 629 712 3 3 3 9 5 3 2 10
10 282 629 712 3 3 3 5 5 3 6 26
22 282 629 712 3 3 3 5 5 3 6 26
130 282 629 712 7 3 3 5 5 3 2 26
142 282 629 712 7 3 3 5 5 3 2 26
10 290 629 712 3 3 3 3 5 3 6 18
22 290 629 712 3 3 3 3 5 3 6 18
130 290 629 712 3 3 3 3 5 3 2 18
142 290 629 712 3 3 3 3 5 3 2 18
10 291 629 712 3 3 3 3 5 3 6 18
22 291 629 712 5 3 3 3 5 3 6 18
130 291 629 712 3 3 3 3 5 3 2 18
142 291 629 712 3 3 3 3 5 3 2 18
10 210 629 797 7 3 3 5 5 3 2 10
22 210 629 797 7 3 3 5 5 3 2 10
130 210 629 797 3 3 3 5 5 3 6 10
142 210 629 797 3 3 3 5 5 3 6 10
10 282 629 797 3 3 3 9 5 3 2 26
22 282 629 797 3 3 3 9 5 3 2 26
130 282 629 797 7 3 3 9 5 3 6 26
142 282 629 797 7 3 3 9 5 3 6 26
10 290 629 797 3 3 3 3 5 3 2 18
22 290 629 797 3 3 3 3 5 3 2 18
130 290 629 797 3 3 3 3 5 3 6 18
142 290 629 797 3 3 3 3 5 3 6 18
10 291 629 797 3 3 3 3 5 3 2 18
22 291 629 797 5 3 3 3 5 3 2 18
130 291 629 797 3 3 3 3 5 3 6 18
Continued on next page
Table 7 – continued from previous page
U(3) Sp(2) U(1) U(1) Q U D L E N Y H
142 291 629 797 3 3 3 3 5 3 6 18
In the following table we list the multiplicities Laa and Laa′ of the branes that occur
in these models, leading to vector-like sets of adjoints and rank-2 tensors. Since brane
b is symplectic, the number of adjoints is equal to the number of symmetric tensors.
Table 8: 4d matter from the aa and aa′ sectors.
Boundary Adjoints Anti-symm. Symm.
a(10) 2 2 6
a(22) 2 2 2
a(130) 2 2 6
a(142) 2 2 2
b(210) - 14 10
b(282) - 14 10
b(290) - 14 6
b(291) - 14 6
c(629) 9 - 14
d(712) 3 - 6
d(797) 3 - 6
It should be emphasized that CFT constructions generically correspond to par-
ticular points in moduli space of CY orientifolds. Due to this, they usually have an
‘enhanced’ massless particle content with extra vector-like matter and closed string
gauge interactions. Thus one would expect that many of the massless vector-like chiral
fields present in this class of models could gain masses while moving to a nearby point
in moduli space.
6.3.4 The instantons
Each of these 32 Standard Model compactifications admits 8 instantons. The instanton
labels are identical for all the 32 models. They are listed in Table 9. The first five
columns use the same notation as for the standard model boundary labels. In column
6 we list the numerical value of the dilaton coupling to the instanton brane. This
quantity is proportional to 1
. It is instructive to compare these couplings to the gauge
couplings, in order to gain intuition on the suppression factor for our instantons. In
these models the U(3) dilaton couplings are 0.00622, so that the instantons are more
strongly coupled than QCD19 On the other hand in this particular model the ratio
α3/α2 at the string scale is 3.23 (the value of sin
2θw at the string scale is 0.527). All
of these couplings are subject to renormalization group running, and there are plenty
of vector-like states to contribute to this, if one assumes that they acquire masses at
a sufficiently low scale. One should perform a detailed renormalization group analysis
to check whether one may obtain consistency with the gauge couplings measured at
low-energies. Let us emphasize however that one expects that moving in moduli space
many of these vector-like states will gain masses and also the values of the different
gauge couplings will also generically vary.
Since the value of the Type II dilaton is a free parameter at this level, one can get the
appropriate (intermediate) mass scale for the right-handed neutrino Majorana masses
by choosing an appropriate value for the dilaton. In this context, it is satisfactory to
verify that the instanton couplings are unrelated to the gauge couplings, as expected
since they do not correspond to gauge instantons [3], and are in fact less suppressed
than the latter.
Note that the 8 instantons fall into two distinct classes (evidently not related by
any discrete symmetry, since the conformal weight on the boundary orbit is distinct,
and the coupling is different as well). Within each class, the orbits of the four instan-
ton boundaries appear to be related by the Z2 symmetries of interchange of the last
two tensor factors, and simultaneous inversion of the charge q of the minimal model.
However, one has to be very careful in reading off symmetries directly from the labels
in columns 3 of Tables (6) and (9) for a number of reasons. First of all the entries in
column 3 are representatives of boundary orbits, and these representatives themselves
are merely representatives of extension orbits. Secondly the action of any discrete
19 Note that the Type II dilaton in this compactifications is an arbitrary parameter which can
always be chosen so that we consistently work at weak coupling. It is the relative value of gauge
couplings which we are comparing here.
Table 9: Instantons for all 32 configurations
Lbl. Orbit/Deg. Reps Weight Dim. coupling
414 [8064,0] (0, {1, 1, 0}, 0, {22,−22, 0}, {20, 16, 0}) 5/2 1 0.0016993
417 [8076,30] (0, {1,−1, 0}, 0, {22, 22, 0}, {20,−16, 0}) 5/2 1 0.0016993
456 [8316,0] (0, {1, 1, 0}, 0, {20, 16, 0}, {22,−22, 0}) 5/2 1 0.0016993
459 [8328,30] (0, {1,−1, 0}, 0, {20,−16, 0}, {22, 22, 0}) 5/2 1 0.0016993
418 [8088,0] (0, {1, 1, 0}, 0, {22,−22, 0}, {18, 16, 0}) 5/3 1 0.0027033
420 [8100,0] (0, {1,−1, 0}, 0, {22, 22, 0}, {18,−16, 0}) 5/3 1 0.0027033
502 [8592,0] (0, {1, 1, 0}, 0, {18, 16, 0}, {22,−22, 0}) 5/3 1 0.0027033
505 [8604,30] (0, {1,−1, 0}, 0, {18,−16, 0}, {22, 22, 0}) 5/3 1 0.0027033
symmetry on the degeneracy labels can be non-trivial. In appendix B we discuss these
symmetries in more detail.
6.4 Other examples
The Sp(2) instanton examples just discussed are the ones which get closer to the
required minimal set of fermion zero modes. Under slightly weaker conditions, we
find many more solutions. In all these cases some additional mechanism beyond exact
RCFT will be needed to lift some undesirable zero modes.
The simplest such case is the following. The tensor product is (2, 8, 8, 18), MIPF
nr. 14, orientifold 2 (the precise spectra may be found using this information in the
database www.nikhef.nl/∼t58/filtersols.php). There are three distinct brane configu-
rations for which almost perfect instantons exist, namely (a,b, c,d) = (64, 562, 389, 67)
and (64, 577, 389, 67) and (65, 560, 189, 66). Each has six instantons, three of type S2+
and three of type S2−. As in the foregoing example, the six instantons are identical
for the three standard model configurations. In this example, they have three differ-
ent dilaton coupling strengths: .00254, .00665 and .0108 (each value occurs once for
S2+ and once for S2−). By comparison, the U(3)-brane dilaton coupling strength is
0.0119338, so that the instanton brane coupling is quite a bit stronger than the QCD
coupling. This is again an interesting point if we want that νR masses are not too much
http://www.nikhef.nl/~t58/filtersols.php
suppressed. Furthermore in this example there are three distinct instanton couplings,
so that one may expect three non-zero eigenvalues (with a hierarchy) in the mass ma-
trix. As in the previous examples there is not gauge coupling unification, one rather
has α3/α2 = .4813 and sin
2(θw) = .183 at the string scale. Again a full renormalization
group analysis should be performed in order to check consistency with the measured
low-energy gauge coupling values.
These models all have a hidden sector consisting of a single Sp(2) factor. They
have respectively 3, 1 and 3 susy Higgs pairs, and a spectrum of bi-fundamentals that
is closer to that of the standard model than the previously discussed Sp(2) examples:
quarks and leptons do not have vector-like copies (there are only some vector-like
leptoquarks), and even one of the three models have the minimal set of Higgs fields
of the MSSM. The rest of the spectrum is purely vector-like, and contains a number
of rank-2 tensors, including eight or six adjoints of U(3). Furthermore there is vector-
like observable-hidden matter. The only undesirable instanton zero-mode is a single
bi-fundamental between the hidden sector Sp(2) brane and the instanton brane. Still,
these SM brane configurations without the hidden sector, provide interesting and very
simple local models of D-brane sectors admitting instantons generating neutrino masses
(with the additional ingredients required to eliminate the extra universal triplets of
fermion zero modes).
6.5 R-parity violation
We now turn to the generation of other possible superpotentials violating B − L. An
instanton violates R-parity if the amount of B − L violation,
IMa − IMa′ − IMd + IMd′ (6.4)
is odd. Examples of instantons with that property were found in the following ten-
sor product/MIPF/orientifold combinations: [(1, 16, 16, 16), 12, 0], [(2, 4, 16, 34), 49, 0],
[(2, 4, 12, 82), 19, 0] [(2, 4, 22, 22), 49, 0] and [(2, 4, 16, 34), 18, 0]. Note that all cases for
which O2 or U2 instantons were found necessarily have R-parity violating instantons as
well: the corresponding O1 and U1 instantons have IMd or IMd′ equal to ±1, whereas
the intersection with the a is non-chiral. In principle, there are many more ways
to obtain R-parity violating instantons (either due to non-vanishing contributions to
IMa − IMa′ or higher values of IMd − IMd′), and indeed, many such instantons turn
out to exist. But the number of tensor product/MIPF/orientifold combinations where
they occur hardly increases: only in the case [(1, 16, 16, 16), 12, 0] we found R-parity
violating instantons, but no U1 or O1 instantons. This suggests that in the other cases
R-parity is a true symmetry of the model. Unfortunately we have no way of rigorously
ruling out any other non-perturbative effects, but at least the set we can examine re-
spects R-parity. This includes in particular the models without hidden sector (found
for [(2, 4, 22, 22), 13, 3] ) discussed above.
The following table list the total number of instantons with the chiral intersections
listed in table 2. The total number of instantons (boundaries violating the sum rule,
as defined in (6.1)) is 29680, for all standard model configurations combined. The
last four columns indicate how many unitary instantons satisfy the sum rule exactly
as listed in table (6.1), how many satisfy it with IMx ↔ −IMx′ (the column U’), and
how many O-type and S-type instantons there are. Here ‘S’ refers to boundaries with
a symplectic Chan-Paton group if the boundary is used as an instanton brane. All
intersection numbers for type S have been multiplied by 2 before comparing with table
2. For real branes, the relevant quantities used in the comparison are IMa − IMa′,
IMc−IMc′ and IMd−IMd′ , while IMb = 0. There are fewer unitary instantons possibly
generating Majorana masses then the numbers mentioned above because the conditions
we use here are stricter: we require here that IMx and IMx′ match exactly, not just their
difference. Note however that this still allows additional vector-like zero-modes. If we
only wish to consider cases without any spurious zero-modes, we may limit ourselves
to the O-type instantons in the last column. There are very few to inspect, and all of
them turn out to have a few non-universal zero modes.
D = 4 Operator U U’ S O
νRνR 1 2 627 3
LH̄LH̄ 0 5 550 3
LH̄ 3 3 0 4
QDL 8 4 0 4
UDD 0 0 0 4
LLE 8 4 0 4
QQQL 0 4 0 3892
UUDE 4 0 0 3880
Table 10: Number of instantons in our search which may induce neutrino masses (first 2
rows), R-parity violation (next 4 rows) or proton decay operators (last 2 rows).
The last two cases are B − L preserving dimension five operators, and obviously
do not come from the set of 29680 B −L violating instantons. They were searched for
separately, but the search was limited to the same 391 models we used in the rest of the
paper. Obviously, one could equally well look for such instantons in the full database,
since their existence does not require a massive B − L.
It is interesting to note that in the classes of MSSM-like models discussed earlier in
this section with the closest to minimal zero mode structure, there are no instantons
al all generating either R-parity violating or the B − L dim=5 operators in the table.
This makes them particularly attractive.
Note that all numbers in table 10 refer to the occurrence of instantons in the set
of 391 tadpole-free models with massive B-L, but without checking the presence of
zero-modes between the hidden sector and the instanton. It makes little sense to use
the hidden sector in the database for such a check, since this is just one sample from a
(usually) large number of possibilities. A meaningful question would be: can one find a
hidden sector that has no zero-modes with the instanton. We have done such a search
for the B − L violating instantons (see the exclamation marks in the last column of
table (5)), but not for the B − L preserving instantons.
7 Conclusions and outlook
In this paper we have presented a systematic search for MSSM-like Type II Gepner
orientifold models allowing for boundary states associated to instantons giving rise to
neutrino Majorana masses. This search is very well motivated since neutrino masses
are not easily accommodated in the semi-realistic compactifications constructed up to
now. String instanton induced Majorana masses provides a novel and promising way
to understand the origin of neutrino masses in the string theory context.
The string instantons under discussion are not gauge instantons. Thus, for example,
they not only break B + L symmetry (like ’t Hooft instantons do) but also B − L,
allowing for Majorana neutrino mass generation. The obtained mass terms are of
order Ms exp(−V/g2) but this suppression is unrelated to the exponential suppression
of e.g. electroweak instantons and may be mild. In fact we find in our most interesting
examples that the instanton action is typically substantially smaller than that of QCD
or electroweak instantons, and hence these effects are much less suppressed than those
coming from gauge theory instantons.
To perform our instanton search we have analyzed the structure of the zero modes
that these instantons must have in order to induce the required superpotential. This
analysis goes beyond the particular context of Gepner orientifolds and has general
validity for Type II CY orientifolds. We have found that instantons with O(1) CP
symmetry have the required universal sector of just two fermionic zero modes for the
superpotential to be generated. Instantons with Sp(2) and U(1) CP symmetries have
extra unwanted universal fermionic zero modes, which however may be lifted in a va-
riety of ways in more general setups, as we discuss in the text. In fact we find in
our search that around 98 % of the instantons with the correct structure of charged
zero modes have Sp(2) CP symmetry. Indeed, from a number of viewpoints the Sp(2)
instantons are specially interesting. The instantons we find with the simplest structure
of fermionic zero modes are Sp(2) instantons which are also the ones which are present
more frequently in the MSSM-like class of Gepner constructions considered. They have
also some interesting features from the phenomenological point of view. Indeed, due
to the non-Abelian structure of the CP symmetry, the structure in flavor space of the
neutrino Majorana masses factorizes. This makes that, irrespective of what particu-
lar compactification is considered, Sp(2) instantons may easily lead to a hierarchical
structure of neutrino masses. It would be important to further study the possible
phenomenological applications of the present neutrino mass generating mechanism.
String instanton effects can also give rise to other B- or L-violating operators. Of
particular interest is the dimension 5 Weinberg operator giving direct Majorana masses
to the left-handed neutrinos. We find that in the most interesting cases, different
instantons giving rise to the Weinberg operator and to νR Majorana masses are both
simultaneously present. Which effect is the dominant one in the generation of the
physical light neutrino masses depends on the values of the instanton actions and
amplitudes as well as on the value of the string scale. Instantons may also generate
dim< 5 operators violating R-parity. We find however that instantons inducing such
operators are extremely rare, and in fact are completely absent in the Gepner models
with the simplest Sp(2) instantons inducing neutrino masses.
There are many avenues yet to be explored. It would be important to understand
better the possible sources (moving in moduli space, addition of RR/NS backgrounds
etc.) of uplifting for the extra uncharged fermionic zero modes in the most favoured
Sp(2) instantons. A second important question is that we have concentrated on check-
ing the existence of instanton zero modes appropriate to generate neutrino masses; one
should further check that the required couplings among the fermionic zero modes and
the relevant 4d superfields (i.e. νR or LH̄) are indeed present in each particular case.
This is in principle possible in models with a known CFT description but could be
difficult in practice for the Gepner models here described.
Instantons can also generate other superpotentials with interesting physical appli-
cations. One important example is the generation of a Higgs bilinear (i.e. a µ-term)
in MSSM-like models [4, 3]. Thus, e.g., one could perform a systematic search for
instantons (boundary states) generating a µ-term in the class of CFT Gepner orien-
tifolds considered in the present article. Other possible application is the search for
instantons inducing superpotential couplings involving only closed string moduli. The
latter may be useful for the moduli-fixing problem, or for non-perturbative corrections
to perturbatively allowed couplings [56].
Finally, it would be important to search for analogous instanton effects inducing
neutrino masses in other string constructions (heterotic, M-theory etc.). A necessary
condition is that the anomaly free U(1)B−L gauge boson should become massive due
to a Stückelberg term.
The importance of neutrino masses in physics beyond the Standard Model is un-
questionable. We have shown that string theory instantons provide an elegant and
simple mechanism to implement them in semi-realistic MSSM-like string vacua, and a
powerful constraint in model building. In our opinion, the conditions of the existence
of appropriate instantons to generate neutrino masses should be an important guide in
a search for a string description of the Standard Model.
Acknowledgements
We thank M. Bertolini, R. Blumenhagen, S. Franco, M. Frau, S. Kachru, E. Kiritsis,
A. Lerda, D. Lüst, F. Marchesano, T. Weigand for useful discussions. A.M.U. thanks
M. González for encouragement and support. The research of A.N. Schellekens was
funded in part by program FP 57 of the Foundation for Fundamental Research of
Matter (FOM), and Research Project FPA2005-05046 of de Ministerio de Educacion y
Ciencia, Spain. The research by L.E. Ibáñez and A.M. Uranga has been supported by
the European Commission under RTN European Programs MRTN-CT-2004-503369,
MRTN-CT-2004-005105, by the CICYT (Spain), and the Comunidad de Madrid under
project HEPHACOS P-ESP-00346.
Appendix
A CFT Notation
Here we summarize the labelling conventions for various CFT quantities. Further
details and explanations can be found in [43].
It is important to keep in mind that there are four steps in the construction, each
involving choices of some quantities. The steps are
• A CFT tensor product
• An extension of the chiral algebra of this tensor product
• The choice of a MIPF
• The choice of an orientifold
The second and third step are easily confused. A MIPF can itself be of extension
type (although it can also be of automorphism or mixed type), meaning that it implies
an extension of the chiral algebra. The crucial difference between step two and three in
that case is that in step 2 all fields that are non-local with respect to the extension are
projected out, and the symmetry of the extension is imposed on all states of the CFT,
i.e in particular on all boundary states. The extension in step three acts as a bulk
invariant, but the boundary states are not required to respect the symmetry implied
by the extension.
Primary fields of N = 2 minimal models are labeled in the usual way by three
integers (l, q, s). In addition to these minimal models, one building block of our CFT’s
is of course a set of NSR fermions in four dimensions. They can be represented by the
four conjugacy classes (0), (v), (s), (c) analogous to those of a root lattice of type D.
Primary fields in a tensor product of M factors are therefore labelled as
I = ((x), (l1, q1, s1), . . . , (lM , qM , sM)) (A.1)
where x = 0, v, s or c.
This tensor product is extended by the alignment currents and the spin-1 field
corresponding to the space-time supersymmetry generator. This organizes the tensor
product fields into orbits, which can be labelled by one of the elements of the orbit.
We always choose the field of minimal conformal weight (or one of them, in case there
are more) as the orbit representative labelling the orbit. The supersymmetry generator
may have fixed points, leading to orbits appearing more than once as primary fields
of the extended theory. In those cases we need an additional degeneracy label to
distinguish them. It is convenient to choose for this label a character of the discrete
group that is causing the degeneracy, the “untwisted stabilizer”, which depends on I.
Denoting this character as ΨI we get then the following set of labels for the primaries
of the extended CFT
i = [I,ΨI ] (A.2)
where I has the form (A.1). If there are no degeneracies we will leave out the square
brackets and the ΨI .
In boundary CFT’s two new labels appear: the labels of Ishibashi-states that propa-
gate in the transverse channel of the annulus, and the boundary labels. In the simplest,
“Cardy” case both sets of labels are in one-to-one correspondence with the extended
CFT labels i. But if we consider non-trivial MIPFs Zij both sets of labels are different.
The Ishibashi states are in one-to-one correspondence with the fields i with Ziic 6= 0.
Degeneracies can occur here if Ziic 6> 0. This requires the introduction of a degeneracy
label. Such degeneracies may occur if the stabilizer of i (the set of simple currents that
fix i) is non-trivial. It is convenient to use elements J of the stabilizer as degeneracy
labels, so that the Ishibashi labels get the following form
m = (i, J) , (A.3)
where i is an extended CFT label, as defined above (to be precise, in some cases a
non-trivial degeneracy label is introduced even if Ziic = 1. The details will not matter
here).
Boundary states correspond to orbits of the simple current group H that defines a
MIPF. To label such orbits we choose a representative. There is no obvious canonical
representative (one could use one of minimal conformal weight, but the conformal
weight of orbit members of a boundary state does not play any rôle in the formalism,
unlike the conformal weight of a primary). So in this case we just make an arbitrary
choice. Once again there can be degeneracies. In this case they are due to a subgroup
of the stabilizer called the “Central Stabilizer”. It is convenient to label the boundary
states by an orbit representative i and a character ψi of the central stabilizer. If we
expand the boundary state label in all of its components we get
a = [i, ψi] = [[I,ΨI ], ψi] = [((x), (l1, q1, s1), . . . , (lM , qM , sM)),ΨI ], ψi] . (A.4)
Note that i is just a representative of a boundary orbit, and that I is just a represen-
tative of an orbit of the extension of the CFT.
B Instanton boundary symmetries
In the hidden-sector free example discussed in some detail in section 6 we have en-
countered Sp(2)-type instantons, the most common kind in our scan. This particular
model is the one that comes closest to the required zero mode count, although the only
superfluous zero modes are rather awkward. Let us assume that the effect of these su-
perfluous universal zero-modes instantons can be avoided. Then there is still another
problem we have to face, namely that the two zero modes αi and γi are related by
an Sp(2) transformation of the label i. Then we we need at least three independent
instantons (with unrelated couplings) to generate three non-zero neutrino masses, as
discussed in Section 3.3. Since the technology to compute the couplings is not yet
available, we cannot be completely sure that the relevant couplings are distinct, or
indeed that they are non-vanishing, but at least we can inspect if there are obvious
symmetries relating them.
The unextended tensor product (2, 4, 22, 22) has 64 discrete symmetries: five sep-
arate charge conjugations of the factors (including the NSR space-time factor) and
the interchange of the two identical k = 22 minimal models. To get space-time su-
persymmetry this tensor product is extended with the product of the simple current
Ramond ground states of each factor. These Ramond ground states are not invariant
under charge conjugation. Therefore this extension breaks the discrete symmetries to
Z2 × Z2, the combined charge conjugation of all five factors and the permutation of
the two identical factors. The combined charge conjugation also acts non-trivially on
the Ramond ground states in each factor, but the result is the charge conjugate of the
space-time supersymmetry generator, which is in the chiral algebra. The combined
conjugation is in fact the charge conjugation symmetry of the extended CFT. It turns
out that only a Z2 subgroup of Z2 × Z2 acts non-trivially on the simple currents of
the extended CFT: the permutation of the k = 22 factors acts in the same way as
charge conjugation. The action of these symmetries on the complete set of primary
fields is more complicated. It is easy to see that the permutation acts differently than
charge conjugation. In general the primary fields of the extended CFT are labelled
as i = [((x), (i1), . . . , (i4)),Ψ] (see Appendix A). The action of the permutation is to
interchange i3 and i4, but in cases with a non-trivial degeneracy it is not a priori
clear which of the degenerate states is the image of the map. This can be resolved by
examining the fusion rules, which should be invariant under the permutation:
Nijk = Nπ(i)π(j)π(k) , (B.1)
where N is a fusion coefficient and π a permutation or other automorphism. In general
there may be more than one way to resolve these ambiguities, resulting in additional
automorphism of the CFT. The standard example of this situation is the extension of
the affine algebra A1 level 4 by the simple current. The resulting CFT has an outer
automorphism, non-trivial charge conjugation, that has no counterpart in A1 level 4.
As mentioned above, the Z2 permutation symmetry is respected by the MIPF and
the orientifold, and since charge conjugation acts in the same way on the simple currents
as the permutation, charge conjugation is respected as well.
In this way we end up with (at least) a surviving Z2×Z2 discrete symmetry acting
on the boundary labels, or a larger discrete symmetry if that symmetry is extended
by the action on the degeneracy labels of the extension. The foregoing story repeats
itself for the action on the boundary labels. The boundary labels are given in terms
of the CFT labels plus a second degeneracy label, the one indicated by the second
entry in the square brackets in column 2 of tables (6) and (9). Once again one has to
determine not only how a symmetry acts on the first entry (this is just the action of
the symmetry in the extended CFT, respecting its fusion rules), but also how it acts
on the degeneracy labels. In this case the precise action can be determined from the
invariance of the annulus coefficients
Aiab = A
π̂(a)π̂(b) , (B.2)
where π is the action on the primaries of the extended CFT (as determined above) and
π̂ is the action on the boundary labels induced by π.
Since the orientifold choice is non-trivial, boundary charge conjugation does not
coincide with CFT charge conjugation. Indeed, the eight instanton boundary states
are invariant under boundary charge conjugation (which they must be in order to
produce a “real” Sp(2)-type instanton). However, just as permutations, CFT charge
conjugation may induce a non-trivial discrete symmetry on the boundary states.
In addition to these “outer automorphisms” there is the notion of boundary simple
currents, introduced in the appendix of [6]. These may be thought of as remnants of
the original simple currents, and imply relations between annulus amplitudes of the
Aiab′ = A
Ja(Jb)′ (B.3)
All of the aforementioned symmetries might relate instanton couplings, and hence
threaten their numerical independence. However, in order to do that they have to be
symmetries of the full standard model/instanton configuration, not just relate some of
the eight instantons to each other. It is easy to see that the permutation of the k = 22
factors changes the standard model brane configuration. Consider brane c: it turns out
that under permutation boundary state 629 it is mapped to boundary state 544 or 545
(depending on the action on the degeneracy label), which in any case is distinct. Hence
even if the instanton boundaries 414 and 456 resp. 418 and 502 are mapped to each
other by boundary permutation, at the same time the standard model configuration is
mapped to a distinct one.
This means that we may expect at least four distinct couplings, which should be
sufficient. It is of course possible to work out the discrete symmetries exactly, but in
view of this argument this would not yield any additional insight.
We do know the exact boundary orbits. The orbit of instanton label 414 is
(414, 415, 416, 417), so that instantons 414 and 417 are related. But the orbit of brane
c under the same action is (629, 628, 626, 627). Hence the action that relates 414 and
417 maps 629 to 627. In fact all four standard model boundaries a,b,c,d are mapped
to different ones. This implies that instantons 414 and 417 may produce different
couplings as well, so that all eight instantons may contribute in a different way.
These distinctions concern the disk correlators d(r)a in (3.12). The factors exp(−Re Ur)
will be related by discrete symmetries, and it seems reasonable to expect them to be
identical for instantons 414, 417, 456 and 459, which is indeed correct. However there
is no reason to expect the other four instantons to have the same suppression factor,
and indeed they do not.
Note that these symmetries imply the existence of a much larger set of standard
model configurations than the 32 discussed here. However, as mentioned before, the
32 models considered here display all possible distinct spectra.
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	Introduction
	Instanton induced superpotentials in Type II orientifolds
	D-brane instantons, gauge invariance and effective operators
	Zero mode structure for D-brane instantons
	Uncharged zero modes
	Charged fermion zero modes
	Instanton induced Majorana neutrino masses
	The MSSM on the branes
	Majorana mass term generation
	Flavor and the special case of Sp(2) instantons
	Other B- and L-violating operators
	The Weinberg operator 
	R-parity violating operators 
	Dimension 5 proton decay operators 
	CFT orientifolds
	Construction of the models
	Search for SM-like models
	Fermion zero modes for instantons on RCFT's
	Search for M instantons
	The instanton scan
	An O1 example
	The S2 models
	The closed string sector
	The standard model branes
	The open string spectrum
	The instantons
	Other examples
	R-parity violation
	Conclusions and outlook
	Appendix
	CFT Notation
	Instanton boundary symmetries
