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 SE751 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 CalabiYau threefolds 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 Tmoduli 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 CalabiYau metric. In a nal step, we show how this local result can be expressed in terms of the global moduli of the CalabiYau manifold. The method is illustrated for the family of CalabiYau 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 CalabiYau 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
rescaling required to convert the holomorphic into the physical Yukawa couplings. As a
{ 1 {
nonholomorphic quantity, the matter eld Kahler metric is notoriously di cult to
calculate since it requires knowledge of the Ricci at CalabiYau 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
modulidependent 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 CalabiYau 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 0correction 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 CalabiYau metric [38{40] and subsequent work applying this
algorithm to various explicit examples and to the numerical calculation of the Hermitian
YangMills 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 Ftheory [49{53] where the localisation of matter
elds on the
intersection curves of D7branes and Yukawa couplings on intersections of such curves
facilitates local computations of the Yukawa couplings which do not require knowledge
of the Ricci at CalabiYau 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 Dbrane 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 Ftheory. 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 CalabiYau 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 CalabiYau compacti cations by a
standard reduction of the 10dimensional 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 rederivation 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 fourdimensional N = 1 supergravity.
We observe that this consistency already determines the dependence of the matter
eld
Kahler metric on the Tmoduli, 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 CalabiYau 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 CalabiYau manifolds. As
a result, we obtain local matter eld wave functions on CalabiYau 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 CalabiYau manifold and by using information from fourdimensional
N = 1 supersymmetry. In this way, we can obtain explicit results for the matter eld
Kahler metric as a function of the CalabiYau moduli and this is carried out for the
CalabiYau hypersurface 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 Ftheory.
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 fourdimensional 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 CalabiYau hypersurfaces 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 CalabiYau manifold and the gauge bundle. The
basic structure of this formula is wellknown for some time, see, for example ref. [54],
and our rederivation 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 fourdimensional N = 1 supergravity.
Starting point is the 10dimensional N = 1 supergravity coupled to a 10dimensional
E8 super YangMills theory. This theory contains two multiplets, namely the gravity
multiplets which consists of the metric g, the NS twoform B, the dilaton
as well as their
fermionic partners, the gravitino and the dilatino, and an E8
E8 YangMills 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 twoderivative level, the bosonic part of the
associated 10dimensional 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 tendimensional gravitational coupling constant and !YM and !L are the
gauge and gravitational ChernSimons forms, respectively.
We consider the reduction of this action on a CalabiYau threefolds X, with
Ricciat metric g(6) and a holomorphic bundle V ! X with a connection A(6) that satis es the
Hermitian YangMills 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 fourdimensional twoform B(
4
).
In addition, we have the zero mode (
4
) of the 10dimensional dilaton
as well as complex
structure moduli Za, where a = 1; : : : ; h2;1(X). It is wellknown that, in the absence of
matter elds, these bosonic elds t into fourdimensional 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 CalabiYau volume can be written as
of the fourdimensional twoform 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 oneforms 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
YangMills 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
Tmodulus 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 Tmoduli 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 modulidependent) 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 wellknown 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 10dimensional 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 Tmoduli
and, hence, the kinetic terms from (2.8) can be expected to include cross terms of the
reduction of the 10dimensional action (2.1) and, hence, there must be nontrivial
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 10dimensional action (2.1)
and the fourdimensional 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 oneloop 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 rewritten 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 CalabiYau 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 10dimensional and the fourdimensional
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 tiindependent 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 CalabiYau volume is in fact expected and provides a check of
our calculation.
3
Localisation of matter eld wave functions on projective spaces
As a warmup, 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 CalabiYau 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 FubiniStudy Kahler potential and Kahler form can be written as
i
2
Here, J has the standard normalisation, that is, RP1 J = 1. The associated FubiniStudy
metric is KahlerEinstein and, hence, the closest analogue of a Ricci at CalabiYau 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
Lvalued forms , that is, forms satisfying the equations
where the Hodge star is taken with respect to the FubiniStudy 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
onetoone 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 nonvanishing cohomology of L is h0(P1; L) = k + 1, so
that the relevant harmonic forms
are Lvalued 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 nonvanishing cohomology of L is h1(P1; L) =
and the corresponding Lvalued (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
FubiniStudy 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 CalabiYau 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 FubiniStudy 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 nonvanishing 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 CalabiYau 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 higherdimensional 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 CalabiYau manifolds embedded in
A. For example, localisation near a curve in A will typically lead to localisation near a
point on a CalabiYau hypersurface 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 CalabiYau calculation
So far, we have approached the problem of computing wave function normalisations on
CalabiYau 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 CalabiYau manifold. In the next section, we will show how to connect this local
CalabiYau calculation with the ambient space point of view in order to obtain results as
functions of globally de ned moduli.
We start with a CalabiYau threefold X and a line bundle L ! X with a nonvanishing
rst cohomology and associated Lvalued 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 CalabiYau 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 CalabiYau 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 nonnegative 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 locallycomputed normalisation
where the a are positive constants. (It is, of course, possible to set a equal to one by
further coordinate rede 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 CalabiYau moduli.
The Hermitian YangMills 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 CalabiYau 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 CalabiYau manifold, starting with the Kahler form and the
connection on the bundle and then proceeding to bundlevalued 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 CalabiYau threefold 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 tidependent  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 globallocal 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 YangMills equation is satis ed as long
as the slope (L) = kiKi of L vanishes.
To work out the above globallocal 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 pointwise 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 clearcut
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 fourdimensional supersymmetry demands, if the a are
tidependent. In the following, we will proceed on the assumption that the a are indeed
tiindependent.
5.2
To complete the above calculation we should consider a speci c CalabiYau 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 CalabiYau hypersurfaces 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 bidegree (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 FubiniStudy 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 nonvanishing 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 rescaling 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 CalabiYau 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 CalabiYau manifold X = fp = 0g intersects
the patch U^ at the a ne origin and near it X is approximately given by the hyperplane
equation P4
=1 p z = 0. By a further linear rede 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 CalabiYau 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 YangMill
equation is locally satis ed for F provided that the slope
k2(4t1 + t2)) vanishes. As usual, this is the case on a certain sublocus 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 globallocal 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 nonzero
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 CalabiYau manifold X with a nonvanishing 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 CalabiYau 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 modulidependent 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 nonnegative 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
CalabiYau manifold, where all remaining complex structure parameters can be absorbed
into coordinate rede 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 CalabiYau 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 CalabiYau 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 fourdimensional 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 CalabiYau 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 CalabiYau 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 globallocal
relationships and by expressing the local result in terms of global quantities. This has been
explicitly carried out for the example of CalabiYau hypersurfaces X in the ambient space
P
1
P3 but the method can be applied to other CalabiYau hypersurfaces (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 fourdimensional 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 globallocal 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 threefamily models with localisation of
all relevant matter elds can be constructed.
It is likely that some of our methods can be applied to Ftheory and be used to express
local Ftheory results in terms of global moduli of the underlying fourfold. 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 Ftheory 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.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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