Matter Field Kähler Metric in Heterotic String Theory
We propose an analytic method to calculate the matter field Kähler metric in heterotic compactifications on smooth Calabi-Yau three-folds with Abelian internal gauge fields. The matter field Kähler metric determines the normalisations of the chiral superfields, which enter the computation of the physical Yukawa couplings. We first derive the general formula for this Kähler metric by a dimensional reduction of the relevant supergravity theory and find that its T-moduli dependence can be determined in general. It turns out that, due to large internal gauge flux, the remaining integrals localise around certain points on the compactification manifold and can, hence, be calculated approximately without precise knowledge of the Ricci-flat Calabi-Yau metric. In a final step, we show how this local result can be expressed in terms of the global moduli of the Calabi-Yau manifold. The method is illustrated for the family of Calabi-Yau hypersurfaces embedded in and we obtain an explicit result for the matter field Kähler metric in this case.
Rudolf Peierls Centre for Theoretical Physics, Oxford University,
1 Keble Road, Oxford, OX1 3NP, U.K.
[0.3cm] Department of Physics M013, The University of Western Australia,
35 Stirling Highway, Crawley WA 6009, Australia
[0.3cm] Department of Physics and Astronomy, Uppsala University,
SE-751 20, Uppsala, Sweden
[0.3cm] Max-Planck-Institut für Physik (Werner-Heisenberg-Institut)
Föhringer Ring 6, 80805 München, Germany
The computation of four-dimensional Yukawa couplings from string theory is notoriously difficult, mainly because methods to compute the matter field Kähler metric which enters the physical Yukawa couplings are lacking. In this note we report some progress in this direction. We outline a method to calculate the matter field Kähler metric in the context of Calabi-Yau compactifications of the heterotic string with Abelian internal gauge fluxes.
Models of particle physics derived from the heterotic string can be remarkably successful in accounting for the qualitative features of the Standard Model; much progress has been made in this direction, both in the older literature [Candelas:1985en, Greene:1986bm, Greene:1986jb, Distler:1987ee, Distler:1993mk, Kachru:1995em] and more recently [Braun:2005ux, Braun:2005bw, Braun:2005nv, Bouchard:2005ag, Blumenhagen:2006ux, Blumenhagen:2006wj, Anderson:2007nc, Anderson:2008uw, Anderson:2009mh, Braun:2009qy, Braun:2011ni, Anderson:2011ns, Anderson:2012yf, Anderson:2013xka, Buchbinder:2013dna, He:2013ofa, Buchbinder:2014qda, Buchbinder:2014sya, Buchbinder:2014qca, Constantin:2015bea, Braun:2017feb]. In fact, heterotic models with the correct spectrum of the (supersymmetric) standard model can now be obtained with relative ease and in large numbers, particularly in the context of Abelian internal gauge flux [Anderson:2011ns, Anderson:2012yf], the case we are focusing on in this note.
One of the next important steps towards realistic particle physics from string theory is to find models with the correct Yukawa couplings. The calculation of physical Yukawa coupings in string theory proceeds in three steps. First, the holomorphic Yukawa couplings, that is, the trilinear couplings in the superpotential have to be determined. As holomorphic quantities, their calculation can be accomplished either by algebraic methods [Candelas:1987se, Braun:2006me, Anderson:2009ge, Anderson:2010tc] or by methods rooted in differential geometry [Candelas:1987se, Blesneag:2015pvz, Blesneag:2016yag, Buchbinder:2016jqr]. The second step is the calculation of the matter field Kähler metric which determines the field normalisation and the re-scaling required to convert the holomorphic into the physical Yukawa couplings. As a non-holomorphic quantity, the matter field Kähler metric is notoriously difficult to calculate since it requires knowledge of the Ricci-flat Calabi-Yau metric for which analytical expressions are not available. This technical difficulty has held up progress in calculating Yukawa couplings from string theory for a long time and it will be the focus of the present paper.
The third step consists of stabilising the moduli and inserting their values into the moduli-dependent expressions for the physical Yukawa couplings to obtain actual numerical values. We will not address this step in the present paper, but rather focus on developing methods to calculate the matter field Kähler metric as a function of the moduli.
The only class of heterotic Calabi-Yau models where an analytic expression for the matter field Kähler metric is known is for models with standard embedding of the spin connection into the gauge connection. In this case, the matter field Kähler metrics for the and matter fields are essentially given by the metrics on the corresponding moduli spaces [Candelas:1987se, Candelas:1990pi]. Recently, Candelas, de la Ossa and McOrist [Candelas:2016usb] (see also Ref. [McOrist:2016cfl]) have proposed an -correction of the heterotic moduli space metric, which includes bundle moduli. This information may be used to infer the Kähler metric of matter fields that arise from bundle moduli. However, we will not pursue this method here, since our main interest is not in bundle moduli but in the gauge matter fields which can account for the physical particles.
There are two other avenues for calculating the matter field Kähler metric suggested by results in the literature. The first one relies on Donaldson’s numerical algorithm to determine the Ricci-flat Calabi-Yau metric [Donaldson:2001, Donaldson:2005, DonaldsonNumerical] and subsequent work applying this algorithm to various explicit examples and to the numerical calculation of the Hermitian Yang-Mills connection on vector bundles [Wang:2005, Headrick:2005ch, Douglas:2006hz, Doran:2007zn, Headrick:2009jz, Douglas:2008es, Anderson:2010ke, Anderson:2011ed]. At present, this approach has not been pushed as far as numerically calculating physical Yukawa couplings. However, it appears that this is possible in principle and, while constituting a very significant computational challenge, would be very worthwhile carrying out. A disadvantage of this method is that it will only provide the Yukawa couplings at specific points in moduli space and that extracting information about their moduli dependence will be quite difficult.
In this paper, we will focus on a different approach, based on localisation due to flux, which can lead to analytic results for the matter field Kähler metric. This method is motivated by work in F-theory [Heckman:2008qa, Font:2009gq, Conlon:2009qq, Aparicio:2011jx, Palti:2012aa] where the localisation of matter fields on the intersection curves of D7-branes and Yukawa couplings on intersections of such curves facilitates local computations of the Yukawa couplings which do not require knowledge of the Ricci-flat Calabi-Yau metric. It is not immediately obvious whether and how this approach might transfer to the heterotic case, since heterotic compactifications lack the intuitive local picture, related to intersecting D-brane models, which is available in F-theory. In this paper, we will show, using methods from differential geometry developed in Refs. [Blesneag:2015pvz, Blesneag:2016yag, Buchbinder:2016jqr], that localisation of wave functions can nevertheless arise in heterotic models. The underlying mechanism is, in fact, similar to the one employed in F-theory. Sufficiently large flux - in the heterotic case gauge flux - leads to a localisation of wave functions which allows calculating their normalisation locally, without recourse to the Ricci-flat Calabi-Yau metric.
To carry this out explicitly we will proceed in three steps. First, we derive the general formula for the matter field Kähler metric for heterotic Calabi-Yau compactifications by a standard reduction of the 10-dimensional supergravity. This formula, which provides the matter field Kähler metric in terms of an integral over harmonic bundle valued forms is not, in itself, new (see, for example, Ref. [Lukas:1999yn]). Our re-derivation serves two purposes. First, we would like to fix conventions and factors as this will be required for an accurate calculation of the physical Yukawa couplings and, secondly, we will show explicitly how this formula for the matter field Kähler metric is consistent with four-dimensional supergravity. We observe that this consistency already determines the dependence of the matter field Kähler metric on the T-moduli, a result which, to our knowledge, has not been pointed out in the literature so far.
The second step is to show how (Abelian) gauge flux can lead to a localisation of the matter field wave functions around certain points of the Calabi-Yau manifold. We will first demonstrate this for toy examples based on line bundles on as well as on products of projective spaces and then show that the effect generalises to Calabi-Yau manifolds. As a result, we obtain local matter field wave functions on Calabi-Yau manifolds and explicit results for their normalisation integrals.
The final step is to express these results in terms of the global moduli of the Calabi-Yau manifold. We show that this can indeed be accomplished by relating global to local quantities on the Calabi-Yau manifold and by using information from four-dimensional supersymmetry. In this way, we can obtain explicit results for the matter field Kähler metric as a function of the Calabi-Yau moduli and this is carried out for the Calabi-Yau hyper-surface in . We believe this is the first time such a result for the matter field Kähler metric as a function of the properly defined moduli has been obtained in any geometrical string compactification, including F-theory.
The plan of the paper is as follows. In the next section, we sketch the supergravity calculation which leads to the general formula for the matter field Kähler metric and we discuss the implications from four-dimensional supersymmetry. In Section LABEL:sec:proj, we show how gauge flux leads to the localisation of matter field wave functions, starting with toy examples on and then generalising to products of projective spaces. Section LABEL:sec:loc contains the local calculation of the wave function normalisation on a patch of the Calabi-Yau manifold. In Section LABEL:sec:global, we express this result in terms of the properly defined moduli by relating global and local quantities and we obtain an explicit result for the matter field Kähler metric on Calabi-Yau hyper-surfaces in . We conclude in Section LABEL:sec:con.
2 The matter field Kähler metric in heterotic compactifications
Our first step is to derive a general formula for the matter field Kähler metric, in terms of the underlying geometrical data of the Calabi-Yau manifold and the gauge bundle. The basic structure of this formula is well-known for some time, see, for example Ref. [Lukas:1999yn], and our re-derivation here serves two purposes. Firstly, we would like to fix notations and conventions so that our result is accurate, as is required for a detailed calculation of Yukawa couplings. Secondly, we would like to explore the constraints on the matter field Kähler metric which arise from four-dimensional supergravity.
Starting point is the 10-dimensional supergravity coupled to a 10-dimensional super Yang-Mills theory. This theory contains two multiplets, namely the gravity multiplets which consists of the metric , the NS two-form , the dilaton as well as their fermionic partners, the gravitino and the dilatino, and an Yang-Mills multiplet with gauge field and associated field strength as well as its superpartners, the gauginos. To first order in and at the two-derivative level, the bosonic part of the associated 10-dimensional action is given by
where is the ten-dimensional gravitational coupling constant and and are the gauge and gravitational Chern-Simons forms, respectively.
We consider the reduction of this action on a Calabi-Yau three-folds , with Ricci-flat metric and a holomorphic bundle with a connection that satisfies the Hermitian Yang-Mills equations, as usual. Let us introduce the Kähler form on , related to the Ricci-flat metric on by and a basis , where , of harmonic (1,1)-forms. Then we can expand
with the Kähler moduli , their axionic partners and the four-dimensional two-form . In addition, we have the zero mode of the 10-dimensional dilaton as well as complex structure moduli , where . It is well-known that, in the absence of matter fields, these bosonic fields fit into four-dimensional chiral multiplets as
with the volume of and the dual of the four-dimensional two-form . We note that the Calabi-Yau volume can be written as
where are the triple intersection numbers of . Further, the Kähler moduli space metric takes the form
where and . The complex structure moduli each form the bosonic part of an chiral multiplet which we denote by the same name.
In addition, there are matter fields which arise from expanding the gauge field as
where are harmonic one-forms which take values in the bundle . It is important to stress that the correct matter field metric has to be computed relative to harmonic forms and this is, in fact, how the dependence on the Ricci-flat metric and the Hermitian Yang-Mills connection comes about. The fields each form the bosonic part of an chiral supermultiplet. It is known that the definition of the superfields in Eq. (2.3) has to be adjusted in the presence of these matter fields. In the universal case with only one T-modulus and one matter field , the required correction to Eq. (2.3) has been found to be proportional to (see, for example, Ref. [Lukas:1997fg]). For our general case, we, therefore start by modifying the definition of the T-moduli in Eq. (2.3) by writing
where is a set of (potentially moduli-dependent) coefficients to be determined 111The dilaton superfield receives a similar correction in the presence of matter fields [Lukas:1997fg] but this arises at one-loop level and will not be of relevance here. . To our knowledge, no general expression for has been obtained in the literature so far.
The kinetic terms of the above superfields derive from a Kähler potential of the general form
where is the Kähler potential for the complex structure moduli whose explicit form is well-known but is not relevant to our present discussion and is the (moduli-dependent) matter field Kähler metric we would like to determine. The general task is now to compute the kinetic terms which result from this Kähler potential, insert the definitions of in Eq. (2.3) and of in Eq. (2.7) and compare the result with what has been obtained from the reduction of the 10-dimensional action (2.1). This comparison should lead to explicit expressions for and .
A quick look at the Kähler potential (2.8) shows that achieving this match is by no means a trivial matter. The matter field Kähler metric depends on the T-moduli and, hence, the kinetic terms from (2.8) can be expected to include cross terms of the form . However, such cross terms can clearly not arise from the dimensional reduction of the 10-dimensional action (2.1) and, hence, there must be non-trivial cancellations which involve the derivatives of and . We find that this issue can be resolved and indeed a complete match between the reduced 10-dimensional action (2.1) and the four-dimensional Kähler potential (2.8) can be achieved provided the following three requirements are satisfied.
The coefficients which appear in the definition (2.7) of the superfields are given by
where is the inverse of the Kähler moduli space metric .
The matter field Kähler metric is given by
where refers to a Hodge dual combined with a complex conjugation and an action of the hermitian bundle metric on .
Since the Hodge dual on a Calabi-Yau manifold acting on a form can be carried out as the result (2.10) for the matter field Kähler metric can be re-written as
where is the hermitian bundle metric on . The final requirement for a match between the dimensionally reduced 10-dimensional and the four-dimensional theory (2.8) can then be stated by saying that the above integrals do not explicitly depend on the Kähler moduli .
The above result means that the Kähler moduli dependence of the matter field metric is completely determined as indicated in the first equation (2.11), while the remaining integrals are -independent but can still be functions of the complex structure moduli. To our knowledge this is a new result which is of considerable relevance for the structure of the matter field Kähler metric and the physical Yukawa couplings. Note that the dependence of in Eq. (2.11) is homogeneous of degree , as expected on general grounds.
It is worth noting that the Kähler potential (2.8) with the matter field Kähler metric as given in Eq. (2.11) can, alternatively, also be written in the form