Wave Functions and Yukawa Couplings in Local String Compactifications
Abstract:
We consider local models of magnetised D7 branes in IIB string compactifications, focussing on cases where an explicit metric can be written for the local 4cycle. The presence of an explicit metric allows analytic expressions for the gauge bundle and for the chiral matter wavefunctions through solving the Dirac and Laplace equations. The triple overlap of the normalised matter wavefunctions generates the physical Yukawa couplings. Our main examples are the cases of D7 branes on and . We consider both supersymmetric and nonsupersymmetric gauge backgrounds and both Abelian and nonAbelian gauge bundles. We briefly outline potential phenomenological applications of our results.
Contents
1 Introduction
Understanding the structure of the Standard Model  the gauge groups, matter content and Yukawa couplings  represents one of the principal problems of theoretical physics. The Standard Model is not selfjustifying and does not motivate a reason for its parameter values. Any deeper explanation of the Standard Model will therefore likely involve new physical ideas and concepts, possibly of a very different nature to those used in the Standard Model itself. In this respect string theory stands out as an attractive and powerful complex of ideas.
One attractive feature of string theory is that it naturally gives rise to chiral matter and nonAbelian gauge groups, thereby reproducing the gross features of the Standard Model. One of the main tasks of string phenomenology consists in finding vacuum configurations resembling, as closely as possible, the gauge group and matter content of the Standard Model. In the heterotic string this is achieved through appropriate gauge bundles and Wilson lines, while for the type II theories this requires an appropriate model of intersecting Dbranes. Such constructions should be regarded as proofs of principle: no current construction reproduces all details of the Standard Model and, in any case, it is unclear what the desired lowenergy matter content actually is, as nothing precludes the existence of extra massive vectorlike matter. Recent reviews of string theoretic model building can be found in [1, 2, 3, 4, 5].
Such string constructions can be classified as either global or local. Global constructions rely on the topological features of the totality of a compact space, with the classic example being the weaklycoupled heterotic string where the spectrum and gauge group are determined by a bundle over the entire CalabiYau. In this case the gauge coupling is given by the volume of the entire CalabiYau. In local models, the principal avatar of which is models of branes at singularities [6], the gauge group and matter content depend only on local physics and are independent of the details of the bulk geometry. The distinction between global and local models is that in a global model the Standard Model gauge couplings always vanish in the limit that the bulk volume is taken to infinity; in local models such a limit leaves the gauge couplings finite. Recent years have seen a renewal of interest in local models, and examples of work in this direction include [7, 8, 9, 10, 11, 12, 13, 14, 15].
This is one of the most attractive features of local models. It is wellknown that for global models the two control requirements of large volume and weak coupling can never be parametrically satisfied: the known running of the Standard Model gauge couplings implies that at the compactification scale
(1) 
A weak string coupling therefore implies , making it difficult to control the expansion. However in local models eq. (1) no longer holds, allowing both large volume and weak coupling to be simultaneously realised. In some cases the use of local models can even be forced upon us by the moduli stabilisation procedure. An example is the LARGE volume scenario of [16, 17], where the volume is stabilised exponentially large, thereby generating hierarchically small supersymmetry breaking. As , a TeV gravitino mass requires a volume , which implies that any realisation of the Standard Model in this scenario is necessarily a local realisation.
A second attractive feature of local models is that they drastically simplify the geometric complexity of model building. The local geometry is noncompact and typically involves far fewer moduli than the hundreds present in typical CalabiYaus. Local models can also be constructed on very simple geometries such as singularities or their resolutions. In some cases the local CalabiYau metric is known exactly, in contrast to the cases of global compact CalabiYaus where no such explicit metrics are known. In the limit that the bulk volume is very large  which is the case for the LARGE volume models  the exact local metric is a parametrically good approximation to the true CalabiYau metric.
The knowledge of an explicit local metric has several consequences. The Laplace and Dirac equations can be solved directly, allowing the exact normalised wavefunctions of the chiral matter fields to be determined. Such wavefunctions give the extradimensional profile of the fields. In local models of magnetised D7 branes, the Yukawa couplings schematically descend from the triple overlap integral
(2) 
Using the form of the wavefunctions the triple overlap integrals of eq. (2) can be computed and the physical Yukawa couplings, including the nonholomorphic parts, can be evaluated. In contrast, algebrogeometric techniques, while very powerful in determining the holomorphic superpotential, are unable to determine the physical couplings which require knowledge of nonholomorphic functions such as the Kähler metrics. Furthermore, the presence of explicit metrics in principle also allows the spectrum and wavefunctions of KaluzaKlein modes to be computed. While these will not contribute to the renormalisable Lagrangian, integrating out these modes will generate highly suppressed nonrenormalisable operators. KaluzaKlein modes are generically gauge singlets and thus in the Rparity MSSM count as righthanded neutrinos. Knowledge of the explicit form of such KK wavefunctions may then have important consequences for string theory models of neutrino masses and mixing matrices [18].
The use of explicit metrics to study wavefunctions and Yukawa couplings through direct dimensional reduction has been carried out in detail for toroidal models of magnetised D9 branes [19] (for a recent discussion for orbifolds of toroidal models see [20]). The purpose of this paper is to perform a similar analysis for models of magnetised D7 branes. We shall study in detail local models of magnetised D7 branes wrapping curved spaces. We aim to compute the chiral massless spectrum and wavefunctions, and use these to analyse the structure of the resulting overlap integrals and Yukawa couplings. Our two principal examples will be and . The Dirac and Laplace equations on such geometries have also been studied in [21, 22, 23, 24, 25]. The local CalabiYau geometries these correspond to are and . While the former admits chiral supersymmetric D7 brane configurations in the geometric regime, the latter does not. The geometry is however of great interest as the resolution of the singularity, which has played a central role in phenomenologically attractive models of branes at singularities [6].
This paper is organised as follows. In section 2 we describe the dimensional reduction of the D7 brane action, the classification of the fourdimensional fields that arise, the structure of the Yukawa couplings and the equations that need to be solved to determine the wavefunctions and Yukawa couplings. In section 3 we review the solutions of the Dirac and Laplace equations on , while in sections 4 and 5 we study the cases of and respectively. In these sections we also discuss the twisting of the Dirac and Laplace equations that is necessary to account for the curved nature of the Dbrane embedding.
2 Dimensional Reduction and the Low Energy Action
In this section we aim to collect the equations of motion that are satisfied by the various fields, and to describe how the solutions of these equations can be used to compute the Yukawa couplings. We focus first on deriving the equations of motion that need to be satisfied to obtain scalar, fermion and vector zero modes, and subsequently describe the origin of the Yukawa couplings. The presence of a nontrivial brane embedding will cause the equations determining the zero modes to be twisted.
We start with the tendimensional superYang Mills action, which will be dimensionally reduced to eight dimensions. The tendimensional action is
(3) 
where is a ten dimensional WeylMajorana spinor and . On reduction to eight dimensions the field content becomes one eightdimensional gauge boson, (), one complex scalar () and a single physical fermion . The action is
(4) 
Here we retain the indices for the 8d quantities and use the index for the directions. There are also additional quartic scalar interactions in eq. (4) that we will neglect as not relevant to our purposes. There is a symmetry transverse to the brane, under which a field with charge is multiplied by under rotation by an angle in the 89 plane. The charges of the fields are
where in writing we have treated as a Weyl fermion of positive chirality.
We want to decompose the eight dimensional fields into fourdimensional ones. For the bosonic degrees of freedom this is straightforward. The transverse scalar becomes
where refers to the directions and to the directions. The eight dimensional vector decomposes into a 4 dimensional vector and 4 real scalars associated to the internal vector degrees of freedom , .
To provide an explicit decomposition of the tendimensional MajoranaWeyl fermion, we start by taking the ten dimensional gamma matrices to be in a product representation,
(5) 
where , , and . are the four dimensional Minkowski gamma matrices
(6) 
while are four dimensional Euclidean gamma matrices
(7) 
The are Pauli matrices with and . and denote the chirality matrices
(8) 
The ten dimensional chirality matrix is , and a Weyl fermion is defined by . We can also define a Majorana matrix
(9) 
which satisfies . The Majorana condition is , and we will require that be both Majorana and Weyl.
In writing the spinor it will be convenient to use the following notation. Spinors can be labelled by their chirality in each of the , and directions. We will use superscript to indicate positive chirality and subscript to indicate negative chirality, so the spinor has positive chirality () in each of the , and directions. The Weyl condition restricts a general 10dimensional spinor to
From the form of the Majorana matrix (9), we see that its action corresponds to a chirality flip in both and directions. The Majorana condition therefore imposes relations between and , and and . A general MajoranaWeyl spinor can be schematically written as
To be more explicit, we write the as
(10) 
The Majorana condition then imposes the constraints
(11) 
From a four dimensional viewpoint there are two distinct types of lefthanded spinors, distinguished by their extradimensional chirality. These can be written as
(12) 
In terms of representation content under the of 4D Minkowski space, the of the internal 4D space and the we can write
(13) 
The subscript denotes the charge. We will tend to use lefthanded spinors in Minkowski space and will treat and as the two independent dynamical degrees of freedom, with and determined as above.
The geometric background we visualise is a stack of D7 branes wrapped on a 4cycle, with a magnetic flux background turned on of the branes. The flux background breaks the original gauge group. In the case that a bundle is turned on, the low energy gauge group is . All fields start off valued in , but decompose and give bifundamentals under the flux background. An arbitrary field can be written as
(14) 
Chiral bifundamental matter arises from zero modes descending from either the or sectors. The sector gives modes and the sector modes.
We suppose that units of flux have been turned on in the sector. When writing the Dirac or Laplace equations for the or , the covariant derivative term will imply that modes in the () sector are effectively charged under a field with () units of flux. The number of fields in fundamental and antifundamental representations is determined by the number of zero modes with () units of flux, and the net chirality is given by .
2.1 Equations of Motion
In describing the equations of motion for the fields, we shall go into considerable detail for the vector modes, and then be more concise in our description of the scalar and fermion equations of motion.
An important general feature here is the fact that the brane is wrapping a cycle with a nontrivial normal bundle. This implies that many of the equations of motion will need to be twisted [26], reflecting the fact that the field is not scalarvalued but rather bundlevalued over the cycle. This does not however hold for the internal degrees of freedom valued in the tangent bundle, such as the (internal) vector modes. We first describe the equations of motion for scalar fields that come from vector degrees of freedom in the internal space.
2.1.1 Vectors
The equations of motion for vectors start from the YangMills action
(15) 
with
is valued in the adjoint representation of and the action is invariant under the gauge transformation
On dimensional reduction, the vector degrees of freedom give rise to a 4dimensional vector () and 4dimensional scalars arising from vectors in the internal space. As the vectors are all valued in the tangent bundle, they are uncharged under the Rsymmetry transverse to the brane. Their equations of motion are therefore not subject to twisting and can be found by a direct dimensional reduction of the action on the surface in the presence of a background magnetic field.
To establish notation and conventions, we start with the Lie algebra of . Our discussion will follow that in the appendix of [19]. The elements of the Lie algebra can be taken to be^{1}^{1}1Although strictly is defined for , for convenience of notation we will allow ourselves to write even when potentially , an example being the last expression of eq. (19).
The gauge field is expanded as
(16) 
Requiring that implies that is real and . Let us state some useful relations for the and :
(17) 
(18) 
(19) 
Also, , , where if and only if , and are all identical.
If we now write , we can expand
(20)  
Here and . Further expanding we obtain
(21)  
In backgrounds with Abelian magnetic flux, has a nonzero vev but does not.^{2}^{2}2For a careful study of vector modes in backgrounds with nonAbelian magnetic fluxes, such as used in section (5), we would need to construct analogous equations in which . The fluctuations of will correspond to zero modes of the low energy theory. We focus on the 2point interactions, and neglect the 3 and 4point interactions present in equation (21). Expanding the terms of (21) we obtain
(22) 
We similarly obtain
(23)  
(24) 
where . We write .
We now expand about the background fields, writing
(25)  
(26) 
Then
The action becomes^{3}^{3}3This differs in the first term by a factor of from the expression in (A.18) of [19].
There are extra interactions not included in (2.1.1), for example 3 and 4point interactions, but these are less relevant for our purposes. Note that the covariant derivatives reduce to ordinary derivatives in the absence of flux,  in this case the gauge connection generates only 3point (or higher) interactions.
We want to examine this action and work out the mass eigenstates. We will do this term by term to work out the contributing parts.

First,
where we have used . We denote by . represents the flux difference seen by the and sectors. In this case we can then write
(28) This generates a quadratic fluxdependent mass term.

The next term we consider is the term
On integration by parts, these will both give rise to terms of the form
As we will impose the gaugefixing condition , these will vanish and will not generate mass terms.

The next nontrivial term is
We have integrated by parts here. As we have gaugefixed , we obtain
We need the action of on a vector field. Now,
(29) We therefore obtain
(30)
Putting together the terms (28) and (30) considered so far, we have
(31) 
These represent the ‘extra’ terms that contribute to the vector action in addition to the naive term, which gives
(32) 
We can finally combine equations (31) and (32) to write down the equation of motion satisfied by the vector modes,
(33) 
Eigenmodes of the vector fields are obtained by finding solutions of eq. (33) for . Zero modes correspond to solutions with vanishing . For intrinsically massive modes (such as KK modes), we show in the appendix that the masses and profiles of vector modes can be derived from those of the scalar modes.
We note that, regarded as a adjointvalued vector in real coordinates, must satisfy and so the he modes determine the modes. Working with complex coordinates, we require . In this notation, zero modes correspond to complex scalars and zero modes correspond to scalars.
2.1.2 Scalars
There are two degrees of freedom transforming as scalars in the extra dimensions. One corresponds to the mode that transforms as a vector in Minkowksi space, and the other to the transverse scalar mode that is valued in the normal bundle. As for the vector, the scalar can be written as . In the presence of an Abelian magnetic flux background, and representations come from modes with in the upperright or lowerleft blocks.
On dimensional reduction the basic equation determinining 4dimensional scalar modes is the Laplace equation on the compact space,
(34) 
where is the gaugecovariant derivative, . For the mode, eq. (34) is sufficient. This mode has all degrees of freedom in the tangent bundle of the brane and so is untwisted. All eigenmodes can be found directly from solving (34).
In contrast, the transverse scalar mode is valued in the normal bundle and so must satisfy a twisted version of the Laplace equation. As the normal bundle has nontrivial curvature, the covariant derivatives must be modified to account for the curvature of the bundle. This modification is equivalent to assuming the existence of an additional flux background, proportional to the Kähler form, in the equations of motion. For cases where the cycle is rigid (i.e. the normal bundle has no holomorphic sections), in the absence of flux there are no solutions to , and all eigenmodes are massive.
In this paper we shall never encounter cases where transverse scalars have zero modes in a supersymmetric flux background (according to [12], this can never occur in the geometric regime). For nonsupersymmetric flux configurations, transverse scalars can have ‘zero modes’, although due to lack of supersymmetry these modes are not massless. The form of these ‘zero modes’ can be found by solving , where incorporates both the gauge connection and that due to twisting, to obtain the holomorphic section of the bundle.
2.1.3 Fermions
For the fermionic degrees of freedom the basic equation of motion is the twisted Dirac equation [26],
(35) 
incorporates both the spin connection and the gauge connection due to the fluxes. We can write the fermion field as
(36) 
Here and are as in eq. (12). Both modes are lefthanded in four dimensions but have opposite extradimensional chirality and opposite Rcharges. The twisting consists in a shift in the effective magnetic flux felt by the fermions: as and have different Rcharges the shift takes a different sign for the two modes.
The effect of the twisting is that the gauge connection is shifted by an amount equivalent to the normal bundle, in such a way that for a pure stack of wrapped branes a single constant zero mode exists in the absence of any magnetic flux. This zero mode, which comes from the sector, corresponds to the gaugino of 4d super YangMills.
In supersymmetric configurations, the modes are fermionic partners of the scalar modes that come from internal vector degrees of freedom. The modes are fermionic partners of either the transverse scalars or the 4dimensional vector bosons, . The equations of motion for the CPT partners and follow from those for and : the zero modes of these are determined by the modes as in eq. (11).
2.2 Yukawa Interactions
Four dimensional scalar fields can arise either from the or degrees of freedom, while fermions arise from either in the or structure. Yukawa couplings will combine two of the fermions with a scalar. From a higher dimensional perspective Yukawa couplings originate from the tendimensional fermion kinetic term term
To extract the gauge indices we write , , and , where are all species indices. Using , we get
(37) 
Once the species are specified, the Yukawa couplings can be directly evaluated. As , the basic integral is
(38) 
Here both and are 10dimensional MajoranaWeyl spinors and so will either be of or of type.
Neglecting the gauge indices in (38), we can now focus on the spinor structure. The basic integral we need to evaluate is
(39) 
This integral takes a different form depending on whether or , correspond to the 4dimensional scalar arising either from a vector mode valued in the D7 tangent bundle or a scalar valued in the normal bundle.
Transverse Scalars
We first consider , where the scalar mode corresponds to the transverse scalar. In this case
(40) 
which gives a chirality flip in both the and directions. To obtain a nonvanishing integral we therefore need and to be either both of the form or both of the form .
We first assume the form , when the total Yukawa interaction is
(41)  