Matter field Kähler metric in heterotic string theory from localisation

Journal of High Energy Physics, Apr 2018

Abstract 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 \( \mathcal{N} \) = 1 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 \( {\mathrm{\mathbb{P}}}^1\times {\mathrm{\mathbb{P}}}^3 \) and we obtain an explicit result for the matter field Kähler metric in this case.

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Matter field Kähler metric in heterotic string theory from localisation

HJE Matter eld Kahler metric in heterotic string theory from localisation Stefan Blesneag 0 1 2 5 6 Evgeny I. Buchbinder 0 1 2 3 6 Andrei Constantin 0 1 2 4 6 Andre Lukas 0 1 2 5 6 Eran Palti 0 1 2 6 0 SE-751 20 , Uppsala , Sweden 1 35 Stirling Highway , Crawley WA 6009 , Australia 2 1 Keble Road , Oxford, OX1 3NP , U.K 3 Department of Physics M013, The University of Western Australia 4 Department of Physics and Astronomy, Uppsala University 5 Rudolf Peierls Centre for Theoretical Physics, Oxford University 6 Fohringer Ring 6 , 80805 Munchen , Germany We propose an analytic method to calculate the matter eld Kahler metric in heterotic compacti cations on smooth Calabi-Yau three-folds with Abelian internal gauge elds. The matter eld Kahler metric determines the normalisations of the N = 1 chiral super elds, which enter the computation of the physical Yukawa couplings. We rst derive the general formula for this Kahler metric by a dimensional reduction of the relevant supergravity theory and nd that its T-moduli dependence can be determined in general. It turns out that, due to large internal gauge ux, the remaining integrals localise around certain points on the compacti cation manifold and can, hence, be calculated approximately without precise knowledge of the Ricci- at Calabi-Yau metric. In a nal 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 P1 Flux compacti cations; Superstrings and Heterotic Strings - 3 and we obtain an explicit result for the matter eld Kahler metric in this case. 1 Introduction 2 3 4 5 3.1 3.2 5.1 5.2 5.3 6 Conclusion 1 Introduction The matter eld Kahler metric in heterotic compacti cations Localisation of matter eld wave functions on projective spaces Wave functions on P1 Wave functions on products of projective spaces A local Calabi-Yau calculation Relating local and global quantities Kahler form and connection An example Wave functions and the matter eld Kahler metric has been made in this direction, both in the older literature [1{6] and more recently [7{ 27]. 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 ux [18, 19], the case we are focusing on in this note. One of the next important steps towards realistic particle physics from string theory is to nd 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 [28{31] or by methods rooted in di erential geometry [28, 32{34]. The second step is the calculation of the matter eld Kahler metric which determines the eld normalisation and the re-scaling required to convert the holomorphic into the physical Yukawa couplings. As a { 1 { non-holomorphic quantity, the matter eld Kahler metric is notoriously di cult to calculate since it requires knowledge of the Ricci- at Calabi-Yau metric for which analytical expressions are not available. This technical di culty 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 eld Kahler metric as a function of the moduli. The only class of heterotic Calabi-Yau models where an analytic expression for the matter eld Kahler metric is known is for models with standard embedding of the spin connection into the gauge connection. In this case, the matter eld Kahler metrics for the (1; 1) and (2; 1) matter elds are essentially given by the metrics on the corresponding moduli spaces [28, 35]. Recently, Candelas, de la Ossa and McOrist [36] (see also ref. [37]) have proposed an 0-correction of the heterotic moduli space metric, which includes bundle moduli. This information may be used to infer the Kahler metric of matter elds 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 elds which can account for the physical particles. There are two other avenues for calculating the matter eld Kahler metric suggested by results in the literature. The rst one relies on Donaldson's numerical algorithm to determine the Ricci- at Calabi-Yau metric [38{40] and subsequent work applying this algorithm to various explicit examples and to the numerical calculation of the Hermitian Yang-Mills connection on vector bundles [41{48]. 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 signi cant computational challenge, would be very worthwhile carrying out. A disadvantage of this method is that it will only provide the Yukawa couplings at speci c points in moduli space and that extracting information about their moduli dependence will be quite di cult. In this paper, we will focus on a di erent approach, based on localisation due to ux, which can lead to analytic results for the matter eld Kahler metric. This method is motivated by work in F-theory [49{53] where the localisation of matter elds 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- at Calabi-Yau metric. It is not immediately obvious whether and how this approach might transfer to the heterotic case, since heterotic compacti cations lack the intuitive local picture, related to intersecting D-brane models, which is available in Ftheory. In this paper, we will show, using methods from di erential geometry developed in refs. [32{34], 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. Su ciently large ux - in the heterotic case E8 E8 gauge ux - leads to a localisation of wave functions which allows calculating their normalisation locally, without recourse to the Ricciat Calabi-Yau metric. { 2 { To carry this out explicitly we will proceed in three steps. First, we derive the general formula for the matter eld Kahler metric for heterotic Calabi-Yau compacti cations by a standard reduction of the 10-dimensional supergravity. This formula, which provides the matter eld Kahler metric in terms of an integral over harmonic bundle valued forms is not, in itself, new (see, for example, ref. [54]). Our re-derivation serves two purposes. First, we would like to x 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 eld Kahler metric is consistent with four-dimensional N = 1 supergravity. We observe that this consistency already determines the dependence of the matter eld Kahler 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) E8 E8 gauge ux can lead to a localisation of the matter eld wave functions around certain points of the Calabi-Yau manifold. We will rst demonstrate this for toy examples based on line bundles on P1 as well as on products of projective spaces and then show that the e ect generalises to Calabi-Yau manifolds. As a result, we obtain local matter eld wave functions on Calabi-Yau manifolds and explicit results for their normalisation integrals. The nal step is to express these results in terms of the global moduli of the CalabiYau 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 N = 1 supersymmetry. In this way, we can obtain explicit results for the matter eld Kahler metric as a function of the Calabi-Yau moduli and this is carried out for the CalabiYau hyper-surface in P 1 P3. We believe this is the rst time such a result for the matter eld Kahler metric as a function of the properly de ned moduli has been obtained in any geometrical string compacti cation, 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 eld Kahler metric and we discuss the implications from four-dimensional N = 1 supersymmetry. In section 3, we show how gauge ux leads to the localisation of matter eld wave functions, starting with toy examples on P 1 and then generalising to products of projective spaces. Section 4 contains the local calculation of the wave function normalisation on a patch of the CalabiYau manifold. In section 5, we express this result in terms of the properly de ned moduli by relating global and local quantities and we obtain an explicit result for the matter eld Kahler metric on Calabi-Yau hyper-surfaces in P 1 P3. We conclude in section 6. 2 The matter eld Kahler metric in heterotic compacti cations Our rst step is to derive a general formula for the matter eld Kahler 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. [54], and our re-derivation here serves two purposes. Firstly, we would like to x notations and conventions so that our result is accurate, as is required for a detailed calculation of { 3 { Yukawa couplings. Secondly, we would like to explore the constraints on the matter eld Kahler metric which arise from four-dimensional N = 1 supergravity. Starting point is the 10-dimensional N = 1 supergravity coupled to a 10-dimensional E8 super Yang-Mills theory. This theory contains two multiplets, namely the gravity multiplets which consists of the metric g, the NS two-form B, the dilaton as well as their fermionic partners, the gravitino and the dilatino, and an E8 E8 Yang-Mills multiplet with gauge eld A and associated eld strength F = dA+A^A as well as its superpartners, the gauginos. To rst order in 0 and at the two-derivative level, the bosonic part of the associated 10-dimensional action is given by S = 2 1 2 H2 0 4 TrF 2 ; H = dB 0 4 (!YM !L) ; (2.1) HJEP04(218)39 where is the ten-dimensional gravitational coupling constant and !YM and !L are the gauge and gravitational Chern-Simons forms, respectively. We consider the reduction of this action on a Calabi-Yau three-folds X, with Ricciat metric g(6) and a holomorphic bundle V ! X with a connection A(6) that satis es the Hermitian Yang-Mills equations, as usual. Let us introduce the Kahler form J on X, related to the Ricci- at metric g(6) on X by gm(6n) = of harmonic ( 1,1 )-forms. Then we can expand iJmn and a basis Ji, where i = 1; : : : ; h1;1(X), J = tiJi ; B = B( 4 ) + i Ji ; with the Kahler moduli ti, their axionic partners i and the four-dimensional two-form B( 4 ). In addition, we have the zero mode ( 4 ) of the 10-dimensional dilaton as well as complex structure moduli Za, where a = 1; : : : ; h2;1(X). It is well-known that, in the absence of matter elds, these bosonic elds t into four-dimensional N = 1 chiral multiplets as S = Ve 2 ( 4 ) + i ; T i = ti + i i ; with the volume V of X and the dual that the Calabi-Yau volume can be written as of the four-dimensional two-form B( 4 ). We note V = Z X d6x qg(6) = 1 6 K ; K = dijktitj tk ; dijk = Ji ^ Jj ^ Jk ; Z X where dijk are the triple intersection numbers of X. Further, the Kahler moduli space metric takes the form Gij = 2 Kij where Ki = dijktj tk and Kij = dijktk. The complex structure moduli Za each form the bosonic part of an N = 1 chiral multiplet which we denote by the same name. In addition, there are matter elds CI which arise from expanding the gauge eld as A = A(6) + I CI ; { 4 { (2.2) (2.3) (2.4) (2.5) (2.6) where I are harmonic one-forms which take values in the bundle V . It is important to stress that the correct matter eld metric has to be computed relative to harmonic forms I and this is, in fact, how the dependence on the Ricci- at metric and the Hermitian Yang-Mills connection comes about. The elds CI each form the bosonic part of an N = 1 chiral supermultiplet. It is known that the de nition of the T i super elds in eq. (2.3) has to be adjusted in the presence of these matter elds. In the universal case with only one T-modulus and one matter eld C, the required correction to eq. (2.3) has been found to be proportional to jCj2 (see, for example, ref. [55]). For our general case, we, therefore start by modifying the de nition of the T-moduli in eq. (2.3) by writing T i = ti + i i + iIJ CI CJ ; knowledge, no general expression for iIJ has been obtained in the literature so far. where iIJ is a set of (potentially moduli-dependent) coe cients to be determined.1 To our The kinetic terms of the above super elds derive from a Kahler potential of the general form K = log dijk(T i + T i)(T j + T j )(T k + T k) + GIJ CI CJ ; (2.8) where Kcs is the Kahler potential for the complex structure moduli Za whose explicit form is well-known but is not relevant to our present discussion and GIJ is the (modulidependent) matter eld Kahler metric we would like to determine. The general task is now to compute the kinetic terms which result from this Kahler potential, insert the de nitions of S in eq. (2.3) and of T i 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 GIJ and iIJ . A quick look at the Kahler potential (2.8) shows that achieving this match is by no means a trivial matter. The matter eld Kahler metric GIJ depends on the T-moduli and, hence, the kinetic terms from (2.8) can be expected to include cross terms of the reduction of the 10-dimensional action (2.1) and, hence, there must be non-trivial cancellations which involve the derivatives of GIJ and iIJ . We nd that this issue can be resolved and indeed a complete match between the reduced 10-dimensional action (2.1) and the four-dimensional Kahler potential (2.8) can be achieved provided the following three requirements are satis ed. The coe cients i IJ which appear in the de nition (2.7) of the T I super elds are given by (2.7) HJEP04(218)39 i IJ = (2.9) where Gij is the inverse of the Kahler moduli space metric Gij . 1The dilaton super eld S receives a similar correction in the presence of matter elds [55] but this arises at one-loop level and will not be of relevance here. { 5 { The matter eld Kahler metric is given by GIJ = 1 Z 2V X I ^ ?V ( J ) ; carried out as ? = can be re-written as where ?V refers to a Hodge dual combined with a complex conjugation and an action of the hermitian bundle metric on V . Since the Hodge dual on a Calabi-Yau manifold acting on a (1; 0) form can be the result (2.10) for the matter eld Kahler metric (2.10) (2.12) i and HJEP04(218)39 i J ^ J ^ 2 3ititj 2K Z X GIJ = ijIJ ; ijIJ = Ji ^ Jj ^ I ^ (H J ) ; (2.11) where H is the hermitian bundle metric on V . The nal 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 ijIJ do not explicitly depend on the Kahler moduli ti. The above result means that the Kahler moduli dependence of the matter eld metric is completely determined as indicated in the rst equation (2.11), while the remaining integrals ijIJ are ti-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 eld Kahler metric and the physical Yukawa couplings. Note that the ti dependence of GIJ in eq. (2.11) is homogeneous of degree 1, as expected on general grounds. It is worth noting that the Kahler potential (2.8) with the matter eld Kahler metric as given in eq. (2.11) can, alternatively, also be written in the form K = provided that terms of higher than quadratic order in the matter elds CI are neglected. This can be seen by expanding the logarithm in eq. (2.12) to leading order in GIJ is homogeneous of degree by using 3Ki iIJ = GIJ . (The latter identity follows from G K 4K 1 in ti and the result (2.9) for iIJ ). This form of the ij 3Kj = ti, the fact that Kahler potential, together with the de nition (2.7) of the elds T i, means that, in terms of the underlying geometrical Kahler moduli ti, the dependence on the matter elds CI cancels. Indeed, inserting the de nition (2.7) of the T i moduli into eq. (2.12) turns the last logarithm into ln(8K). That this part of the Kahler potential can be written as the negative logarithm of the Calabi-Yau volume is in fact expected and provides a check of our calculation. 3 Localisation of matter eld wave functions on projective spaces As a warm-up, we rst discuss wave function normalisation on Pn and products of projective spaces, beginning with the simplest case of P1. (For a related discussion, in the context of Ftheory, see ref. [53].) In doing so we have two basic motivations in mind. First, considering projective space and P1 in particular provides us with a toy model for the actual CalabiYau case which we will tackle later. From this point of view, the following discussion will provide some intuition as to when wave function localisation occurs and when it leads to a good approximation for the normalisation integrals. On the other hand, projective spaces and their products provide the ambient spaces for the Calabi-Yau manifolds of interest and, hence, this section will be setting up some of the requisite notation and results we will be using later. 3.1 Wave functions on P 1 Homogeneous coordinates on P1 are denoted by x0, x1, the a ne coordinates on the patch g by z = x1=x0 and we also de ne = 1 + jzj2. For simplicity, we will write all quantities in terms of the a ne coordinate z and we will ensure they are globally wellde ned by demanding the correct transformation property under the transition z ! 1=z. In terms of z, the standard Fubini-Study Kahler potential and Kahler form can be written as i 2 Here, J has the standard normalisation, that is, RP1 J = 1. The associated Fubini-Study metric is Kahler-Einstein and, hence, the closest analogue of a Ricci- at Calabi-Yau metric we can hope for on P1. We are interested in line bundles L = OP1 (k) on P1 with rst Chern class c1(L) = kJ on which we introduce a hermitian structure with the bundle metric and the associated (Chern) connection and eld strength given by H = k ; kz dz ; 2 ikJ : (3.2) The analogue of the harmonic forms I in eq. (2.6) associated to matter elds are harmonic L-valued forms , that is, forms satisfying the equations where the Hodge star is taken with respect to the Fubini-Study metric. We would like to compute their normalisation integrals Z P1 h ; i = ^ ?(H ) ; the analogue of the matter eld Kahler metric (2.10). These harmonic forms are in one-toone correspondence with the bundle cohomologies Hp(P1; L) and, depending on the value of k, we should distinguish three case. k 0: in this case, the only non-vanishing cohomology of L is h0(P1; L) = k + 1, so that the relevant harmonic forms are L-valued zero forms. The relevant solutions to eqs. (3.3) are explicitly given by the degree k polynomials in z. k = 1: in this case, all cohomologies of L vanish so there are no harmonic forms. { 7 { (3.1) (3.3) (3.4) 2: in this case, the only non-vanishing cohomology of L is h1(P1; L) = and the corresponding L-valued (0; 1)-forms which solve eqs. (3.3) can be written as = kh(z)dz, where h is a polynomial of degree k 2 in z. In the following, it is useful to work with the monomial basis q = kzqdz ; q = 0; : : : ; k 2 for these forms. Given that the forms I which appear in the actual reduction (2.6) are (0; 1)-forms the most relevant case is the last one for k the normalisation integral (3.4) leads to 2. In this case, inserting the forms (3.5) into In physical terminology, the integer k quanti es the ux and the integer q labels the families of matter elds. It is clear that the above integrals receive their main contribution from a patch near the a ne origin z ' 0, provided that the ux jkj is su ciently large and the family number q is su ciently small. In this case, it seems that the above integrals can be approximately evaluated locally near z ' 0, by using the at metric instead of the Fubini-Study metric as well as the corresponding at counterparts of the bundle metric and the harmonic forms. Formally, these at space quantities can be obtained from the exact ones by setting to one in the expression (3.1) for the Kahler form and by the replacement k ! ekjzj2 in the other quantities. That is, we use the replacements J = 2 i i dz ^ dz ! 2 dz ^ dz ; k ! e kjzj2 ; q = kzqdz ! ekjzj2 zqdz : { 8 { (3.5) (3.6) (3.7) (3.8) (3.9) jkj the to work out the local version of the normalisation integrals which leads to i Z C h q; piloc = zqzpekjzj2 dz ^ dz = 2 q! ( k 1)q+1 qp : For the ratio of local to exact normalisation this implies h q; qiloc = h q; qi ( k 2) ( k ( k 1)q+1 2 q) = 1 O q 2 k 1 : Hence, as long as the ux jkj is su ciently large and the family number satis es q2 local versions of these integrals do indeed provide a good approximation. It is worth noting that a transformation z ! 1=z to the other standard coordinate patch of P1 transforms the monomial basis forms q into forms of the same type but with the family number changing as q ! ( k 1) q. This means that families with a large family number q close to k 1 in the patch fx0 6= 0g acquire a small family number when transformed to the patch fx1 6= 0g and, hence, localise at the a ne origin of this patch, that is near z = 1. From this point of view it is not surprising that families with large q in the patch fx0 6= 0g cannot be dealt with by a local calculation near z ' 0. Instead, for such modes, we can carry out a local calculation analogous to the above one but near the a ne origin of the patch fx1 6= 0g. In summary, the harmonic bundle valued (0; 1) forms for L = OP1 (k), where k 2, are given by q as in eq. (3.5). For su ciently large ux jkj the modes with small family number q localise near the a ne origin of the path fx0 6= 0g, that is at z ' 0 and their normalisation can be obtained from a local calculation near this point. The modes with large family number q localise near the a ne origin of the other path fx1 6= 0g, that is, near z = 1 and their normalisation can be obtained by a similar local calculation around this point. Wave functions on products of projective spaces The previous discussion for line bundles on P1 can be straightforwardly generalised to line bundles on arbitrary products of projective spaces. For the sake of keeping notation simple, we will now illustrate this for the case of A = P 1 P3 which is, in fact, the ambient space of the Calabi-Yau manifold on which we focus later. The situation for general products of projective spaces is easily inferred from this discussion. Homogeneous coordinates on A = P P3 are denoted by x0; x1 for the P1 factor and by y0; y1; y2; y3 for P2. The associated a ne coordinates on the patch fx0 6= 0; y0 6= 0 g are z1 = x1=x0 and z +1 = y =y0 for = 1; 2; 3 and we de ne 1 = 1 + jz1j2 and 2 = 1 + P4 =2 jz j2. The Fubini-Study Kahler forms for the two projective factors are2 z z ) dz ^ dz ; J^1 = J^2 = i i 2 2 and, more generally, we can introduce the Kahler forms 1 1 2 2 2 i i 2 dz1 ^ dz1 ; 4 X 2 ; =2 ( 2 J^ = t1J^1 + t2J^2 ; { 9 { (3.10) (3.11) (3.12) (3.13) (3.14) HJEP04(218)39 class c1(L^) = k1J^1 + k2J^2 can be equipped with the hermitian bundle metric with Kahler parameters t1 > 0, t2 > 0 on A. Line bundles L^ = OA(k1; k2) with rst Chern H^ = 1 2 i(k1J^1 + k2J^2) : Speci cally, we are interested in those line bundles L^ with a non-vanishing rst cohomology which are precisely those with k1 2 and k2 0. In these cases h1(A; OA(k1; k2)) = ( k1 1) (k2 + 3)(k2 + 2)(k2 + 1) 6 and a basis for the associated harmonic L^-valued (0; 1) forms is provided by ^q = 1k1 z1q^1 z2q^2 z3q^3 z4q^4 dz1 ; 2From now on we will denote quantities de ned on the \ambient space" A by a hat in order to distinguish them from their Calabi-Yau counterparts to be introduced later. where q^ = (q^1; q^2; q^3; q^4) is a positive integer vector which labels the families and whose entries are constrained by q^1 = 0; : : : ; k1 2 and q^2 + q^3 + q^4 k2. Given these quantities, the integrand of the normalisation integral is proportional to ^q^ ^ ?(H^ ^q^ ) su ciently small, we expect localisation on a patch U^ around the a ne origin z Hence, provided the uxes jk1j and k2 are su ciently large and the family numbers q this case, we can again work with the at limit where the above quantities turn into large, localisation on U^ near the a ne origin z A few general conclusions can be drawn from this. First, localisation near a point in A does require all uxes jkij to be large. If one of the uxes is not large then localisation will happen near a higher-dimensional variety in A. For example, if jk1j is not large then the wave function will localise near P1 times a point in P3. We note that such a partial localisation may actually be su cient when we come to discuss Calabi-Yau manifolds embedded in A. For example, localisation near a curve in A will typically lead to localisation near a point on a Calabi-Yau hyper-surface embedded in A. Secondly, provided all jkij are indeed ' 0, for = 1; 2; 3; 4, requires all q^ to be su ciently small. If a certain q^ is large localisation may still arise near another point in A. For example, if q^1 is large while the other q^ are small, then localisation occurs near z1 = 1, z2 = z3 = z4 = 0. 4 A local Calabi-Yau calculation So far, we have approached the problem of computing wave function normalisations on Calabi-Yau manifolds from the viewpoint of the prospective ambient embedding spaces. In this section, we will take the complementary point of view and carry out a local calculation on a Calabi-Yau manifold. In the next section, we will show how to connect this local Calabi-Yau calculation with the ambient space point of view in order to obtain results as functions of globally de ned moduli. We start with a Calabi-Yau three-fold X and a line bundle L ! X with a non-vanishing rst cohomology and associated L-valued harmonic (0; 1) forms. Our goal is to determine the normalisation of these harmonic forms by a local calculation, assuming, at this stage, that localisation indeed occurs. To do this, we focus on a patch U X with local complex coordinates Za, where a = 1; 2; 3, chosen such that the Kahler form J , associated to the Ricci- at Calabi-Yau metric, is locally on U well approximated by3 3We will denote local quantities, de ned on the patch U , by script symbols. J = i 2 3 X a=1 adZa ^ dZa ; (3.15) ' 0. In (3.16) (4.1) J ^ J ^ F = 0 , 1 2K3 + 1 3K2 + 2 3K1 = 0 : The resulting equation for the Ka will translate into a constraint on the Calabi-Yau moduli in a way that will become more explicit later. For now we should note that it implies not all Ka can have the same sign (given that the a need to be positive). Consider harmonic (0; 1)-forms v 2 H1(X; L). On U they are approximated by (0; 1)-forms satisfy the local version of the harmonic equations which must In analogy with the projective case, speci cally eq. (3.16), we assume that K1 < 0 and K2; K3 > 0. Wether these sign choices are actually realised cannot be checked locally but requires making contact with the global picture - we will come back to this later. If they are, potentially localising solutions to these equations are of the form = eK1jZ1j2 P (Z1; Z2; Z3)dZ1, where P is an arbitrary function of the variables indicated. Localisation of these solution still depends on the precise form of the function P which cannot be determined from a local calculation. We will return to this issue in the next section when we discuss the relation to the global picture. For now, we take a practical approach and work with a monomial basis of solutions given by q = eK1jZ1j2 Z1q1 Z2q2 Z3q3 dZ1 ; (4.2) (4.3) (4.4) (4.5) (4.6) (4.7) where q = (q1; q2; q3) is a vector with non-negative integers. The normalisation of these monomial solutions can be explicitly computed and is given by Mq;p := h q; piloc = q ^ ?(H p) = J ^ J ^ q ^ (H q) i 2 q;p Z U i ' 4 2 2 3 q;p dZa ^ dZajZaj2qa e jKajjZaj2 After performing the integration we nd for the locally-computed normalisation where the a are positive constants. (It is, of course, possible to set a equal to one by further coordinate re-de nitions but, for later purposes, we nd it useful to keep these explicitly.) On U , we can approximate the hermitian bundle metric H and the associated eld strength F of L by H = e P3a=1 KajZaj2 ) 3 X KadZa ^ dZa ; where Ka are constants which will ultimately become functions of the Calabi-Yau moduli. The Hermitian Yang-Mills equation, J ^J ^F = 0, should be satis ed locally which leads to Y qa! jKaj qa 1 : The appearance of the exponential in each of the integrals in the second line indicates that there is indeed a chance for localisation to occur. However, the validity and practical usefulness of this result depends on a number of factors which are impossible to determine in the local picture. First of all, we should indeed have K1 < 0 and K2; K3 > 0 for localisation to happen, but these conditions can only be veri ed by relating to the global picture. Secondly, families are de ned as cohomology classes in H1(X; L) and at this stage it is not clear precisely how these relate to the monomial basis forms (4.5). The above calculation shows that the smaller the integers in q = (q1; q2; q3) the better the localisation and this ties in with the result on projective spaces in the previous section. Finding the relation between the elements of H1(X; L) and the local basis forms q is, therefore, crucial in deciding the validity and accuracy of the approximation for the physical families. Finally, we would like to express the local result (4.7) in term of the properly de ned global CalabiYau moduli. We will now address these issues by relating the above local calculation to the full Calabi-Yau manifold. 5 Relating local and global quantities We will start by relating the local quantities which have entered the previous calculation to global quantities on the Calabi-Yau manifold, starting with the Kahler form and the connection on the bundle and then proceeding to bundle-valued forms. This will allows us to express the result (4.7) for the wave function normalisation in terms of properly de ned moduli. 5.1 Kahler form and connection We begin, somewhat generally, with a Calabi-Yau three-fold X, a basis Ji, where i = 1; : : : ; h1;1(X) of its second cohomology and Kahler forms with the Kahler moduli t = (ti) restricted to the Kahler cone. Further, we assume that all the forms Ji, and, hence, J are chosen to be harmonic relative to the Ricci- at metric on X speci ed by the Kahler class [J ]. Note that, despite what eq. (5.1) might seem to suggest, the harmonic forms Ji are typically ti-dependent | all we know is that their cohomology classes [Ji] do not change with the Kahler class so they are allowed to vary by exact forms. X, we would like to introduce the forms Ji, where i = which are local (1; 1)-forms with constant coe cients which approximate their global counterparts Ji and J on U . How are these global and local forms related? We rst note that the top forms J ^ J ^ J and Ji ^ J ^ J are harmonic and must, therefore be proportional J = X tiJi J = X tiJi i i Ji ^ J ^ J = ci(t)J ^ J ^ J ; which must involve the same constants ci(t). Inserting at forms into eq. (5.5) then allows us to determine the ci(t) in terms of the parameters in these forms and equating these expressions to the global result (5.4) leads to constraints on the local forms Ji. This global-local correspondence has an immediate implication for bundles on X and their local counterparts on U . Consider a line bundle L ! X with rst Chern class c1(L) = kiJi and eld strength F = 2 i P kiJi. Then, for the local version F = i 2 i P kiJi of i the eld strength we nd, using eqs. (5.5) and (5.4), that (5.4) (5.5) (5.6) (5.8) F ^ J ^ J = J ^ J ^ J kiKi K and, hence, that the local version of the Hermitian Yang-Mills equation is satis ed as long as the slope (L) = kiKi of L vanishes. To work out the above global-local correspondence more explicitly, we consider a case with two Kahler moduli, so h1;1(X) = 2. In this case, we can choose complex coordinates za, where a = 1; 2; 3, on the patch U X such that J1 = i 2 3 X a=1 adza^dza ; where ci(t) are functions of the Kahler moduli but independent of the coordinates of X. By inserting eq. (5.1) and integrating over X we can easily compute these constants as where the quantities and i were de ned in and around eq. (2.4). On the other hand, the relation (5.3) holds point-wise and, hence, has a local counterpart ci(t) = i ; where the a are constants. (More speci cally, starting with two arbitrary (1; 1) forms J1 and J2 with constant coe cients, by standard linear algebra, we can always diagonalise J2 into \unit matrix form" and then further diagonalise J1 without a ecting J2.) Inserting the above forms into eq. (5.5) gives and equating these results to the global ones in eq. (5.4) imposes constraints on the unknown local coe cients a. However, it is not obvious that the a are Kahler moduli independent, particularly since the forms Ji do, in general, depend on Kahler moduli. In the following, we will assume that this is indeed the case, although we do not, at present, have a clear-cut proof. There are two pieces of evidence which support this assumption. First, it is not obvious that equating (5.8) with (5.4) allows for a solution with constant a (valid for all t) but we nd that, in all cases which we have checked, that it does. Secondly, it is hard to see how a local calculation of the integrals in eq. (2.11) can lead to Kahler moduli independent results for ijIJ , as four-dimensional supersymmetry demands, if the a are ti-dependent. In the following, we will proceed on the assumption that the a are indeed ti-independent. 5.2 To complete the above calculation we should consider a speci c Calabi-Yau manifold. As before, we focus on the ambient space A = P 1 P3, discussed in section 3.2, and use the same notation for coordinates, Kahler forms and Kahler potentials as introduced there. The Calabi-Yau hyper-surfaces X A we would like to consider are then de ned as the manifold has Hodge numbers h1;1(X) = 2, h2;1(X) = 86 and Euler number (X) = zero loci of bi-degree (2; 4) polynomials p, that is sections of the bundle N^ = OA(2; 4). This Its second cohomology is spanned by the restrictions J^ijX , where i = 1; 2, of the two ambient space Kahler forms and, relative to this basis, the second Chern class of the tangent bundle is c2(T X) = (24; 44). The Kahler class on X can be parametrised by the restricted ambient space Kahler forms J jX = t1J^1jX + t2J^2jX ; ^ where t1; t2 > 0 are the two Kahler parameters. Of course neither of these forms is harmonic relative to the Ricci- at metric on X associated to the class [J^jX ] (as they are obtained by restricting the ambient space Fubini-Study Kahler forms) but there exist forms Ji and J in the same cohomology classes which are. In other words, J and Ji are the harmonic forms introduced in eq. (5.1) and we demand that their cohomology classes satisfy [J ] = [J^jX ], [Ji] = [J^ijX ]. The non-vanishing triple intersection numbers of this manifold are given by Inserting these results into eq. (5.4) we nd ; 1 = 6 ; 2 = 3 = 0 ; and equating these expressions to the local results (5.8) leads to the solution which is unique, up to permutations of the coordinates za. This means, from eqs. (5.7), the local forms Ji and J can (after another coordinate re-scaling z1 ! z1=p6) be written as We note that J is of the form (4.1) used in our local calculation and we can match expressions by setting za = Za and 1 = t1 + 6 t2 ; 1 2 = 3 = t2 : (5.9) (5.10) (5.11) (5.12) (5.13) (5.14) (5.15) (5.16) Another interesting observation is that these forms satisfy Ji ^ Jj ^ Jk = 1 16 3 dijk 3 a=1 ^ dza ^ dza ; where dijk are the intersection numbers (5.10) of the manifold in question, that is, our local forms \intersect" on the global intersection numbers. They also relate in an interesting way to the ambient space Kahler forms J^i. So far, we have considered an arbitrary patch U on X but from now on let us focus on a speci c choice, starting with the ambient space patch U^ A near the a ne origin z ' 0. This patch is of obvious interest since we know from the ambient space discussion in section 3.2 that some wave functions localise on it. If it is su ciently small, the de ning equation of the Calabi-Yau manifold on U^ can be approximated by 4 =1 p = p0 + X p z + O(z2) ; where p0 and p are some of the parameters in p. It is possible, by linear transformations of the homogeneous coordinates on P1 and P3, to eliminate the p0 term and, in the following, we assume that this has been done. Then, the Calabi-Yau manifold X = fp = 0g intersects the patch U^ at the a ne origin and near it X is approximately given by the hyper-plane equation P4 =1 p z = 0. By a further linear re-de nition of coordinates on the P3 factor of the ambient space this equation can be brought into the simpler form (5.17) (5.18) (5.19) (5.20) (5.21) (5.22) where a is a constant. If we restrict the at versions of the ambient space Kahler forms, as given in eq. (3.16), to U using eq. (5.19) we nd that the de ning equation (5.19) with a = 1=p6 we nd provided we set a = 1=p6. This means on the patch U we understand the relation between ambient space Kahler froms J^i, local Kahler forms Ji and their global counterparts Ji on X. We can now extend this correspondence to (line) bundles and their connections. As in section 3.2 we consider line bundles L^ = OA(k1; k2) and we restrict these to line bundles L = OX (k1; k2) := L^jX on the Calabi-Yau manifold X. (Of course, the line bundle L should be thought o as merely part of the full vector bundle of the compacti cation in question.) The hermitian bundle metric H^ for L^ was given in eq. (3.12) and its local approximation on U^ in eq. (3.16). If we restrict this local bundle metric on U^ to U , using ) H = H^ jU = exp We note that this expression of H is of the general form (4.2) used in the local calculation, provided we set za = Za and identify z4 = az1 ; J^ijU = Ji ; K1 = k1 + K2 = K3 = k2 : 1 6 k2 ; From the discussion around eq. (5.6) we also conclude that the Hermitian Yang-Mill equation is locally satis ed for F provided that the slope k2(4t1 + t2)) vanishes. As usual, this is the case on a certain sub-locus of Kahler moduli (L) = dijkkitj tk = 2t2(2k1t2 + space, provided that k1 and k2 have opposite signs. Wave functions and the matter eld Kahler metric As the last step, we should work out the global-local correspondence for wave functions. As in section 3.2 we consider line bundles L^ = OA(k1; k2) with k1 2 and k2 > 0 with a non-zero rst cohomology H1(A; L^) whose dimension is given in eq. (3.13) and with harmonic basis forms ^q^ introduced in eq. (3.14). These line bundles restrict to line bundle L = OX (k1; k2) := L^jX on the Calabi-Yau manifold X with a non-vanishing rst cohomology (see, for example, ref. [34]) HJEP04(218)39 H1(X; L) = p(H1(A; N^ H1(A; L^) L^)) : Explicit representatives for this cohomology can be obtained by restrictions ^q^ jX although these forms are not necessarily harmonic with respect to any particular metric. (Also, they have to be suitably identi ed due to the quotient in eq. (5.23). As long as k2 < 4 the cohomology in the denominator of eq. (5.23) vanishes so that the quotient is trivial and the restrictions ^q^ jX form a basis of H1(X; L) as stands.) Finally, we have the monomial basis q of locally harmonic forms de ned in eq. (4.5). In summary, we are dealing with three sets of basis forms and their linear combinations, namely ^q^ = ek1jz1j2 z1q^1 z2q^2 z3q^3 z4q^4 dz1 ~q~ = ek1jz1j2 z1q~1 z2q~2 z3q~3 z1q~4 dz1 q = eK1jzj2 z1q1 z2q2 z3q3 dz1 ^(a^) = X a^q^ ^q^ ^ q ~(a~) = X a~q~ ~q~ ~ q (a) = X aq q : q To be clear, hatted wave functions ^q^ are de ned on the ambient space A, wave functions ~q~ refer to their restrictions to the Calabi-Yau patch U and the q are the harmonic wave functions on the patch U . Recall that we need K1 < 0 as a necessary condition for the harmonic solutions q to have a nite norm and, by virtue of the identi cation (5.22), this translates into K1 < 0 , k1 > k2 6 : Hence, for this particular example, the condition K1 < 0 is not moduli-dependent and can be satis ed by a suitable choice of line bundle. We would like to determine the relation between the above three types of forms, or, equivalently, the relation between the coe cients a^, a~ and a, given that ~(a~) = ^(a^)jU are related by restriction and that ~(a~) and (a) are in the same cohomology class so must The rst of these correspondences between a^ and a~ is easy to establish. Given the relation is by restriction, there is a matrix S such that a~ = Sa^ and using the approximate (5.23) (5.24) (5.25) de ning equation (5.19) we nd that To establish the correspondence between a and a~ we rst de ne the matrix T by where M is the local normalisation matrix computed in eq. (4.7). Since (a) and ~(a~) di er by an exact form we know that h (a); (b)i = ayM b and h (a); ~(b~)i = ayM T b ~ must be equal to each other and, since this holds for all a, it follows that The explicit form of the matrix T , from its de nition (5.27), is Tq;p~ = q1;p~1 p~4 q2;p~2 q3;p~3 q1! jK1j q1 1 : As discussed earlier, the families correspond to cohomology classes in H1(X; L) and in view of eq. (5.23) and subject to possible identi cations it, therefore, makes sense to label families by the hatted basis ^q^ on the ambient space. For simplicity of notation, we write the hated indices as I = q^ form now on. We also recall from section 3.2 that these indices are non-negative and further constrained by I1 = 0; : : : ; k1 2 and I2 + I3 + I4 k2. With this notation, the matter eld Kahler metric is given by the general expression Sq~;p^ = q~;p^ 6q^4=2 : h q; ~p~ i = (M T )q;p~ b = T b~ : (5.26) (5.27) (5.28) (5.29) (5.30) (5.31) (5.32) (5.33) (I1 I4) I1 I4;J1 J4 I2;J2 I3;J3 : 1 2V GI;J := (SyT yM T S)I;J : Inserting the above results for S and T as well as the local normalisation matrix (4.7) we nd explicitly, GI;J = NI;J where the constants NI;J are given by NI;J = J1! I1! I2! I3! jk1 + k2=6jI1 I4+1 6I4=2+J4=2+1 2(I1 I4)! jk1jJ1+1kI2+I3+2 2 For the lowest mode, I = 0, this number specialises to N0;0 = 3 jk1 + k2=6j : k2 2 A few remarks about this result are in order. First, we note that the Kahler moduli dependence in eq. (5.31) is in line with the result (2.11) from dimensional reduction. In general, the matter eld Kahler metric is also a function of complex structure moduli. For our example, this dependence has dropped out completely, that is, the quantities NI;J are constants. This feature results from our linearised local approximation (5.19) of the Calabi-Yau manifold, where all remaining complex structure parameters can be absorbed into coordinate re-de nitions. We do expect complex structure dependence to appear at the next order, that is, if we approximate the de ning equation locally by a quadric in a ne coordinates. Also, our result (5.31) has an implicit complex structure dependence in that its validity depends on the choice of complex structure. Whether neglecting the quadratic and higher terms in z in eq. (5.18) does indeed provide a good approximation depends, among other things, on the choice of coe cient in the de ning equation p, that is, on the choice of complex structure. Another feature of our result (5.31) is that it is diagonal in family space and, formally, this happens because the matrices M , S and T are all diagonal. We have seen in section 4 that this is a general feature of the matrix M . However, S and T do not need to be diagonal in general. In our example, this happens due to the simple form (5.19) of the local Calabi-Yau de ning equation and the resulting diagonal form of the local Kahler form J in eq. (5.15). Finally, we remind the reader that the result (5.31) can only be trusted if the line bundle L = OX (k1; k2) satis es the condition (5.25), if the ux parameters jkij are su ciently large and if the family numbers I are su ciently small, in line with our discussion in section 3. 6 In this note, we have reported progress on computing the matter eld Kahler metric in heterotic Calabi-Yau compacti cations. Three main results have been obtained. First, by dimensional reduction we have derived a general formula (2.11) for the matter eld Kahler metric and we have argued that constraints from four-dimensional supersymmetry already fully determine the Kahler moduli dependence of this metric. Secondly, provided large ux leads to localisation of the matter eld wave function, we have shown how the matter eld Kahler metric can be obtained from a local computation on the Calabi-Yau manifold, leading to the general result (4.7). This result, while quite general, is unfortunately of limited use, mainly since it is not expressed in terms of the global moduli of the Calabi-Yau manifold. This makes it di cult to identify the conditions for its validity and it falls short of the ultimate goal of obtaining the matter eld Kahler metric as a function of the properly de ned moduli super elds. We have attempted to address these problems by working out a global-local relationships and by expressing the local result in terms of global quantities. This has been explicitly carried out for the example of Calabi-Yau hyper-surfaces X in the ambient space P 1 P3 but the method can be applied to other Calabi-Yau hyper-surfaces (and, possibly complete intersections) as well. Our main result in this context is the Kahler metric for matter elds from line bundles L = OX (k1; k2) on X given in eqs. (5.31), (5.32), which is expressed as a function of the proper four-dimensional moduli elds. We have also stated the conditions for this result to be trustworthy, namely the constraint (5.25) on the line bundle L as well as large uxes jkij and small family numbers. More details and examples will be given in a forthcoming paper. The global-local relationship established in this way points to two problems of localised calculations both of which are intuitively plausible. First, the large ux values demanded by localisation typically also lead to large numbers of families. For this reason, there is HJEP04(218)39 a tension between localisation and the phenomenological requirement of three families. Secondly, large ux typically leads to a \large" second Chern class c2(V ) of the vector bundle which might violate the anomaly constraint c2(V ) c2(T X). Hence, there is also a tension between localisation and consistency of the models. It remains to be seen and is a matter of ongoing research whether consistent three-family models with localisation of all relevant matter elds can be constructed. It is likely that some of our methods can be applied to F-theory and be used to express local F-theory results in terms of global moduli of the underlying four-fold. It would be interesting to carry this out explicitly and check if the tension between localisation on the one hand and the phenomenological requirement of three families and cancelation of anomalies on the other hand persists in the F-theory context. Acknowledgments A.L. is partially supported by the EPSRC network grant EP/N007158/1 and by the STFC grant ST/L000474/1. E. I. B. and A. C. would like to thank the Department of Physics of the University of Oxford where some of this work has been carried out for warm hospitality. Open Access. 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Ştefan Blesneag, Evgeny I. Buchbinder, Andrei Constantin, Andre Lukas, Eran Palti. Matter field Kähler metric in heterotic string theory from localisation, Journal of High Energy Physics, 2018, 139, DOI: 10.1007/JHEP04(2018)139