Published for SISSA by Springer
Received: November 22, 2012
Revised: December 6, 2012 Accepted: December 6, 2012
Published: January 2, 2013
Moduli stabilising in heterotic nearly Kahler compactications
Michael Klaput, Andre Lukas, Cyril Matti and Eirik E. SvanesRudolf Peierls Center for Theoretical Physics, Oxford University, 1 Keble Road, Oxford, OX1 3NP, U.K.
E-mail: mailto:[email protected]
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Abstract: We study heterotic string compactications on nearly Kahler homogeneous spaces, including the gauge eld e ects which arise at order [prime]. Using Abelian gauge elds, we are able to solve the Bianchi identity and supersymmetry conditions to this order. The four-dimensional external space-time consists of a domain wall solution with moduli elds varying along the transverse direction. We nd that the inclusion of [prime] corrections improves the moduli stabilization features of this solution. In this case, one of the dilaton and the volume modulus asymptotes to a constant value away from the domain wall. It is further shown that the inclusion of non-perturbative e ects can stabilize the remaining modulus and lift the domain wall to an AdS vacuum. The coset SU(3)/U(1)2 is used as an explicit example to demonstrate the validity of this AdS vacuum. Our results show that heterotic nearly Kahler compactications can lead to maximally symmetric four-dimensional space-times at the non-perturbative level.
Keywords: Flux compactications, Superstrings and Heterotic Strings, Superstring Vacua
ArXiv ePrint: 1210.5933
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JHEP01(2013)015
[circlecopyrt] SISSA 2013 doi:http://dx.doi.org/10.1007/JHEP01(2013)015
Web End =10.1007/JHEP01(2013)015
Contents
1 Introduction 1
2 Heterotic supergravity and Hitchin ow 32.1 Heterotic supergravity 32.2 Heterotic half-BPS domain wall solutions 52.3 G2 and SU(3) structure from the supersymmetry conditions 52.4 Half-at mirror geometry 8
3 Solutions on homogeneous spaces to lowest order in [prime] 9
3.1 SU(3) structure on coset spaces 93.2 Half-at mirror geometry of the cosets 103.3 Levi-Civita connection 123.4 Vector bundles on coset spaces 133.5 Line bundle sums 143.6 Solutions to lowest order in [prime] 15
3.6.1 Bianchi identity 163.6.2 Killing spinor equations 163.6.3 Hermitian Yang-Mills equations 173.6.4 Hitchin ow equations 173.7 Side issues: Kaluza-Klein gauge group and Wilson lines 18
4 Solutions on homogeneous spaces including [prime] corrections 194.1 Perturbative solution 194.2 Full solution Ansatz 214.3 Exact solution to the Bianchi identity 214.4 Hitchin ow revisited 234.5 Solving the ow equations 234.5.1 Case 1, B = 0 24
4.5.2 Case 2, B < 0 24
4.5.3 Case 3, B > 0 24
4.6 Discussion 25
5 The four-dimensional e ective theory 255.1 Four-dimensional supergravity and elds 255.2 Kahler potential and superpotential 275.3 D-terms 275.4 F-term conditions 285.5 Including a gaugino condensate 285.6 Supersymmetric AdS example 315.7 Search for non-supersymmetric vacua 31
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6 Discussion and outlook 32
A Conventions and SU(3)-structures 33A.1 Conventions 34A.2 SU(3)-structures 34
B The coset spaces 35B.1 SU(3)/U(1)2 35
B.2 Sp(2)/SU(2) [notdef] U(1) 37
B.3 G2/SU(3) 39
C Bianchi identity and related computations 41C.1 tr R^ R 41 C.1.1 SU(3)/U(1)2 42
C.1.2 Sp(2)/SU(2) [notdef] U(1) 42
C.1.3 G2/SU(3) 43C.2 tr F ^ F 43
C.2.1 SU(3)/U(1)2 43
C.2.2 Sp(2)/SU(2) [notdef] U(1) 44
C.2.3 G2/SU(3) 44C.3 Solving the Bianchi identity 44 C.3.1 SU(3)/U(1)2 44
C.3.2 Sp(2)/SU(2) [notdef] U(1) 45
D Index of the Dirac operator 46
1 Introduction
In the search for realistic models of particle physics, the E8 [notdef] E8 heterotic string [1],
compactied on Calabi-Yau manifolds, has long been an attractive approach to string model building [2], due to its appealing properties for gauge coupling unication [3] and the built-in exceptional gauge groups, among others. Indeed, large numbers of heterotic standard models have recently been constructed by compactifying on Calabi-Yau manifolds with Abelian gauge bundles [4, 5].
Moduli stabilization of heterotic Calabi-Yau compactications has been more di cult as compared to the type IIB string (although see [69] for some recent progress), mainly due to the absence of RR uxes. It is expected that the RR uxes are mapped into geometric uxes (torsion) on the heterotic side, motivating the study of heterotic compactications on non Calabi-Yau spaces. The rst such class of heterotic compactications, based on complex non-Kahler manifolds has been studied by Strominger [10] and was developed further in refs. [1117].
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A more general class of heterotic non Calabi-Yau compactications on half-at mirror manifolds manifolds which appear in the context of type II mirror symmetry with NS ux has been introduced some time ago in refs. [1820]. Although few half-at mirror manifolds are known explicitly some of their properties can be inferred from mirror symmetry, to the extent that compactication and computation of the associated e ective theories can be carried out reasonably explicitly, a feature which is non-trivial for non Calabi-Yau compactications. This considerable advantage comes at a price of introducing two major complications. First, half-at mirror manifolds do not solve the heterotic string in the presence of four-dimensional maximally symmetric space-time but rather have to be combined with a four-dimensional half BPS domain wall for a full 10-dimensional solution. Secondly, it is not clear in general how to construct the gauge bundles required for heterotic compactications on half-at mirror manifolds. Let us discuss these two issues in turn.
At rst sight a four-dimensional non maximally symmetric space-time such as a domain wall appears to be phenomenologically unviable. However, it was shown in refs. [18, 19] that heterotic compactications on half-at mirror manifolds can still be associated with a fully covariant four-dimensional N = 1 supergravity theory. Due to a superpotential and an associated runaway direction present in this theory it is not solved by Minkowski or AdS space but, in the simplest case, by a domain wall which forms the four-dimensional part of the aforementioned 10-dimensional solution. Obtaining a maximally symmetric four-dimensional space-time therefore becomes a matter of lifting a runaway direction in the scalar potential of the theory by additional contributions, for example of non-perturbative origin, a task frequently required in string compactications. In conclusion, heterotic half-at compactications are still potentially viable subject to such a lifting being carried out successfully.
The problem of constructing gauge bundles is technical rather than conceptual in nature. Progress in this direction has been made in refs. [2126] by focusing on nearly Kahler manifolds which are given as six-dimensional group or group coset manifolds. The most relevant example for our purpose is the coset SU(3)/U(1)2. The underlying group structure of these manifolds allows for an explicit construction of certain bundles, notably line bundles, and their associated connections. In particular, it has been shown that the coset SU(3)/U(1)2 with vector bundles constructed as sums of line bundles can lead to models with GUT-type symmetries and three chiral families.
This discussion suggests two important questions which have been left unanswered in the work on heterotic half-at compactications to date. Can the runaway direction indeed be lifted and a maximally symmetric four-dimensional space-time be achieved? Can we understand the back-reaction of the gauge elds, induced by the Bianchi identity at order [prime], onto the other elds?
These are the two main questions which we will address in the present paper, working within the context of nearly Kahler spaces and, in particular, the coset space SU(3)/U(1)2.
We will see that the answers are yes in both cases and that the two issues of moduli stabilisation and [prime] corrections are indeed related. For line bundle sums we are able to solve the Bianchi identity and compute the e ect of the resulting non-vanishing NS eld strength H at order [prime]. We nd that these [prime] e ects help with moduli stabilisation in that
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they lead to one of the relevant moduli (either the dilaton or the volume modulus) being asymptotically constant away from the domain wall. Adding non-perturbative e ects from gaugino condensation then leads to a complete stabilisation and a four-dimensional AdS vacuum. For appropriate choices of the parameters (in particular the gauge bundle uxes) we nd that the internal volume is su ciently large for the [prime] expansion to be justied.
The plan of the paper is as follows. In section 2 we start by reviewing half-at domain wall solutions of the heterotic string. Section 3 describes the specic form of these solutions for coset spaces at lowest order in [prime] and, in section 4, these results are extended to rst order in [prime]. In section 5 we introduce the associated four-dimensional theories and discuss moduli stabilization. We conclude in section 6. Conventions and details of the underlying calculations are provided in a number of technical appendices.
2 Heterotic supergravity and Hitchin ow
Before we describe the explicit solutions to order [prime] central to this paper, we briey discuss the general setting of N = 1 heterotic supergravity and domain wall solutions thereof. Half-at manifolds and, in particular, the nearly Kahler manifolds that we shall be concerned with later, form solutions to the heterotic equations at leading order in [prime] provided they are combined with a four-dimensional domain-wall solution [20, 26]. In this case, the variation of the half-at manifold along the direction transverse to the domain wall is described by Hitchin ow equations, as we will review.
2.1 Heterotic supergravity
The low-energy limit of heterotic E8 [notdef]E8 string theory is given by a 10-dimensional N = 1
supergravity theory coupled to 10-dimensional super Yang-Mills theory with E8 [notdef]E8 gauge group. Its bosonic eld content consists of the metric g, the dilaton , the two-form B and a E8 [notdef] E8 gauge eld A. The corresponding action can be obtained from sigma model
perturbation theory up to two loops1 [29] and its bosonic part in the string frame is given by
S =
[integraldisplay]
+ O( [prime]2) . (2.1)
Here 10 is the ten-dimensional Planck constant, F = dA + A ^ A is the gauge eld
strengths, R it the curvature scalar associated to the Levi-Civita connection ! and R is
the curvature two-form obtained from the connection
! KIJ = !
also known as Hull connection in the literature.
The three-form eld strength H is dened as
H = dB + [prime]
4 (wYM wGr) (2.3)
1See also ref. [27] which uses the supersymmetrisation of the Yang-Mills and Chern-Simons forms. A modern version of this derivation has been recently given in ref. [28].
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1
2 210
e2
R 1 4d ^ d +12H ^ H
+ [prime]
4 (tr F ^ F tr R ^ R)[bracketrightbigg]
K
IJ
1
2H
K
IJ , (2.2)
with the Yang-Mills and gravity Chern-Simons forms satisfying dwYM = tr F ^ F and dwGr = tr R ^ R, respectively. Taking the exterior derivative then leads to the
Bianchi identity
dH = [prime]
4 (tr F ^ F tr R ^ R) . (2.4)
The fermionic eld content of the supergravity is given by the gravitino M, the dilatino and the gaugino ~. The corresponding supersymmetry transformations are
M = rM +18HM[parenrightbigg]
" + O( [prime]2) (2.5)
=
/@ + 112H
[parenrightbigg]
" + O( [prime]2) (2.6)
~ = FMN MN " + O( [prime]2) . (2.7)
Here, M satisfy the Cli ord algebra in ten dimensions, HM = HMNP N P , H =
HMNP M N P , and " is a ten-dimensional Majorana-Weyl spinor.Hence, a supersymmetric solution of the theory, neglecting terms of order [prime]2 and
higher, satises
rM +18HM[parenrightbigg]
" = 0 (2.8)
/@ + 112H
[parenrightbigg]
" = 0 (2.9)
FMN MN " = 0 . (2.10)
Let us conclude this section with a few remarks on an integrability result and the di erent connections that appear in the action, Bianchi identity and supersymmetry conditions. Note rst that (2.8) can be written as
r+M " = 0 (2.11)
where r+ is the covariant derivative of the connection!+ KIJ = !
K
IJ + 12H
where ! is again the Levi-Civita connection. The connection !+ is commonly referred to as Bismut connection in the literature.
Hence, we encounter two di erent connections in action and Bianchi identity on the one hand and supersymmetry conditions on the other hand. This leads to an integrability result which was rst derived in ref. [30]. An alternative derivation using spinor methods can be found in ref. [17]. The integrability result states that the supersymmetry conditions imply the equations of motion if and only if the connection ! satises
RMNKL MN" = 0 . (2.13)
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K
IJ , (2.12)
It can be shown [17] in general that this condition is automatically satised up to corrections of rst order in [prime]. This means that a eld conguration which solves the supersymmetry conditions (2.5)(2.7) and the Bianchi identity (2.4) ignoring all O( [prime]) terms solves the
equations of motion derived from the action (2.1), again ignoring all terms O( [prime]). To see
this, denote by R[notdef](0) and H(0) solutions to the supersymmetry conditions and Bianchi identity ignoring O( [prime]) corrections, so that, in particular, dH(0) = 0. However, from the
denition of the connections ![notdef] we have (note the index structure)
R+(0)KLMN R(0)MNKL =
1
2(dH(0))KLMN (2.14)
and, therefore, equality of the two curvature forms, R+(0)KLMN = R(0)MNKL, follows. Combing this with
[r+M, r+N] " = R+MNKL KL " = 0 , (2.15)
a direct conclusion from the gravitino variation (2.11), the integrability condition (2.13) follows. This argument may, of course, break down at order [prime] since the ux need not be closed. For our purposes, it is su cient that the integrability condition is satised to lowest order. This guarantees that the equations of motion are satised to order [prime], the
order we are working to in this paper, provided, of course, Killing spinor equations and Bianchi identity are satised to the same order [17].
2.2 Heterotic half-BPS domain wall solutions
Our 10-dimensional solutions consist of a six-dimensional space with SU(3) structure (the half-at mirror or, more specically, nearly Kahler spaces) and a four-dimensional domain wall, as described in refs. [20, 26]. This amounts to choosing the 1 + 2 dimensions along the domain wall to be maximally symmetric and the remaining seven dimensions to form a non-compact G2-structure manifold. The associated metric takes the form
ds2 = dx dx + dy2 + guv(xm)dxudxv
[bracehtipupleft] [bracehtipdownright][bracehtipdownleft] [bracehtipupright]
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. (2.16)
Here , , . . . range from 0 to 2 and label the domain wall coordinates, y = x3 is the remaining four-dimensional direction, transverse to the domain wall, and u, v, . . . run from 4 to 9 and label coordinates of the internal compact manifold X. The indices m, n, . . . run from 3 to 9 and label all seven directions of the G2 structure manifold Y .
As evident from the above equation, the seven dimensional G2 structure manifold Y is a warped product of the y direction and the SU(3) structure manifold X. To describe this structure mathematically, it is most convenient to formulate the G2 and SU(3) structures in terms of di erential forms, which we will do in the next section.
2.3 G2 and SU(3) structure from the supersymmetry conditions
We now briey review how the conditions for unbroken supersymmetry, (2.5)(2.7), give rise to the G2 and SU(3) structures of the domain wall solution (2.16), mainly following ref. [20].
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X, SU(3) structure
[bracehtipupleft] [bracehtipdownright][bracehtipdownleft] [bracehtipupright]
Y, G2 structure
The general ten dimensional Majorana-Weyl spinor " which appears in the supersymmetry conditions (2.5)(2.7) is decomposed in accordance with our metric Ansatz (2.16) as
"(xm) = (xm) . (2.17)
Here is an eigenvector of the third Pauli matrix 3, (xm) is a seven dimensional spinor, and is a constant Majorana spinor in 2+1 dimensions and represents the two preserved supercharges of the solution. Hence, from the viewpoint of four-dimensional N = 1 super-gravity, the solution is 12-BPS.
The spinor (xm) can be used to dene a three-form
'mnp = i mnp (2.18)
and a four-form mnpq = mnpq (2.19)
where m...n := m . . . n is a product of seven dimensional Dirac matrices. The two forms
' and dene a G2-structure and are both Hodge dual to each other with respect to the metric g7 = dy2 + guv(xm)dxudxv, that is, ' = 7 . Therefore, this is the metric
compatible with the so dened G2-structure on [notdef]y[notdef] [multicloseleft] X [31].
Now, it can be shown that the rst two supersymmetry conditions2 (2.8) and (2.9) are satised if and only if [20, 3234]
d7' = 2d7 ^ ' 7H (2.20)
d7 7 ' = 2d7 ^ 7' (2.21)
' ^ H = 2 7 d7 , (2.22)
7' ^ H = 0 . (2.23) Here, 7 is the seven-dimensional Hodge-star with respect to the metric g7 and and d7 =
dxm@m is the seven-dimensional exterior derivative.
To focus on the compact space X, we will now decompose these equations by performing a 6 + 1 split. The forms ' and can be written in terms of six dimensional forms as
' = dy ^ J + (2.24)
7' = dy ^ + +
1
2J ^ J , (2.25)
where J is a two-form and = + + i a complex three-form which, together, dene an
SU(3)-structure on X. In terms of these forms, eqs. (2.20)(2.23), can be re-written as
d = 2d ^ (2.26)
dJ = 2@y @y 2d ^ J + H (2.27) J ^ dJ = J ^ J ^ d (2.28)
2Together with the requirement that the H-ux has only legs in the compact directions.
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d + = J ^ @yJ @y J ^ J + 2d ^ + (2.29) J ^ H = d (2.30)
^ H = (2@y) 1 (2.31)
+ ^ H = 0 (2.32)
where all symbols and forms are quantities on the six-dimensional compact internal space X. In particular, denotes the six-dimensional Hodge dual with respect to the metric
g6 = guv(xm)dxudxv.
An SU(3) structure can be characterised by the decomposition of the torsion tensor into irreducible SU(3) representations, as reviewed in appendix A.2. The structure decomposes into ve torsion classes, which are related to the exterior derivatives of J and via (A.5) and (A.6). Using these relations, it can be shown that the supersymmetry conditions (2.26)(2.32) restrict the torsion classes to
W 1 = 0 W 2 = 0 W4 = d W5 = 2d (2.33)
and the remaining classes arbitrary. For the special case H = 0, d = 0 this means that all but W +1 and W +2 vanish and such SU(3) structures are referred to as half-at. Such half-at SU(3) structures (J, ) can also be characterized by the relations d = 0 and
J ^dJ = 0. Without such a restriction, SU(3) structures satisfying (2.33) are often referred
to as generalised half-at.Recall that the Strominger system is characterized by the stronger conditions
W1 = 0 W2 = 0 W4 = d W5 = 2d . (2.34)
Therefore, the Strominger system which results from a metric Ansatz with a maximally symmetric four-dimensional space-time is seen to be a special case of the more general Ansatz (2.16), as one would have expected. Specializing (2.34) further and setting H = 0, d = 0 forces all torsion classes to vanish which corresponds to the case of Calabi-Yau manifolds times four-dimensional Minkowski space.
In addition to the above conditions which restrict the gravitational sector of the super-gravity, the instanton condition (2.10) for a gauge eld lying purely in the compact space X is equivalent to the conditions
[notdef]F = 0 (2.35)
J[notdef]F = 0 , (2.36) known as the Hermitean Yang-Mills equations (HYM). Solving these equations turns out to be a technical challenge in any heterotic compactication. For compactications on Calabi-Yau manifolds, these are usually solved using the Donaldson-Uhlenbeck-Yau theorem which, roughly, states that every holomorphic poly-stable bundle on a compact Kahler manifold admits a unique Hermitean-Yang Mills connection. The geometries (2.33) are in general not Kahler (and not even complex, since W1 [negationslash]= 0 and W2 [negationslash]= 0) and, therefore, this
aforementioned theorem does not apply. However, explicit solutions to the HYM equations
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for Abelian gauge elds on homogeneous half-at manifolds have been obtained in ref. [26]. Taking into account the order [prime] backreaction of these gauge elds via the Bianchi identity is one of the main purposes of this paper.
2.4 Half-at mirror geometry
Before we move to explicit domain wall solutions on homogeneous spaces, we would like to review a convenient language in which to formulate the fundamental equations discussed in the previous section. As we have seen, the supersymmetry conditions can be cast in terms of the SU(3) structure (J, ), see eqs. (2.26)(2.32). For half-at mirror manifolds this can be made more concrete by introducing a language analogous to Calabi-Yau manifolds. It turns out that this language also applies to the explicit examples of nearly Kahler coset spaces considered here [26].
Half-at mirror manifolds were introduced in refs. [18, 19, 35] in the context of type II mirror symmetry with NS uxes. These manifolds are equipped with a set, [notdef]!i[notdef], of two-forms, and a dual set, [notdef]~
!i[notdef], of four forms. They also have a symplectic set, [notdef] A, B[notdef],
of three-forms, as in the Calabi-Yau case. These forms satisfy the following relations
ZX !i ^ ~!j = ji, ZX A ^ B = 0, [integraldisplay]X A ^ B = 0, [integraldisplay]X A ^ B = BA, (2.37)
similar to the harmonic basis forms on a Calabi-Yau manifold. Furthermore, we dene intersection numbers dijk analogous to the Calabi-Yau case by writing (in cohomology)
!i ^ !j dijk ~
!k . (2.38)
In contrast to Calabi-Yau manifolds, however, these forms are not harmonic anymore in general. Instead, they satisfy the di erential relations
d!i = ei 0 , d 0 = ei~
!i , d~
!i = 0 , d 0 = 0 . (2.39)
The coe cients ei are constants on X and parametrize the intrinsic torsion of the manifolds. The SU(3) structure forms J and can be expanded in this basis
J = vi!i , = ZA A + i GA A , (2.40)
where the elds vi are analogous to the Kahler moduli, the ZA analogous to the complex structure moduli and GA analogous to the derivatives of the pre-potential. Taking the exterior derivative we getdJ = viei 0 , d = Z0ei~
!i . (2.41)
By comparing with eqs. (A.5) and (A.6), these results can be used to read o the torsion classes of half-at mirror manifolds. In particular, we see that the constants ei indeed measure the intrinsic torsion of the manifolds.
The explicit construction of the above forms for the case of nearly Kahler coset spaces will be reviewed in the following section and the technical details are provided in appendix B.
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3 Solutions on homogeneous spaces to lowest order in [prime]
In this section we will review heterotic string solutions on coset spaces to lowest order in [prime],
following ref. [26]. This will be preparing the ground for computing the order [prime] corrections to these backgrounds in the next section.
Of the known nearly Kahler homogeneous spaces SU(3)/U(1)2, Sp(2)/SU(2) [notdef] U(1),
G2/SU(3) and SU(2) [notdef] SU(2), only the rst two spaces allow for line bundles using the
construction method we employ. A study of the expected number of generations, using the index of the Dirac operator, shows that only SU(3)/U(1)2 admits bundles with three generations. Hence, in our analysis we will focus on the cases SU(3)/U(1)2 and Sp(2)/SU(2) [notdef] U(1), even though the results can be extended in a straightforward way to
include all four spaces.
We will start with a brief review of coset geometry, the construction of SU(3) structures on cosets and the relation to half-at mirror geometry. Then, we discuss the construction of vector bundles and, in particular, line bundles on coset spaces. By combining these ingredients with a four-dimensional domain wall, we construct, to lowest order in [prime], 10-
dimensional solutions with two supercharges to the heterotic string.
3.1 SU(3) structure on coset spaces
We begin with a review of coset space di erential geometry and, in particular, the construction of SU(3) structures. We refer to appendix B and refs. [26, 36] for further technical details.
A coset space G/H is obtained by identifying all elements of the Lie group manifold G which are related by the action of the subgroup H G. For the construction of bundles on
G/H later on, it will be useful to view G as a principal bre bundle over G/H with bre H, that is, G = G(G/H, H). The base space G/H admits a natural frame of vielbeins, which descend from the left-invariant Maurer-Cartan forms on G and will be denoted by ei [36]. These one-forms are, in general, no longer left-invariant under the action of G. However, in the cases of interest, there exist G-(left)-invariant two-, three- and four-forms.
The space of G-invariant two- and three-forms for SU(3)/U(1)2 is spanned by 3
[notdef] e12 , e34 , e56 [notdef] , [notdef] e136 e145 + e235 + e246 , e135 + e146 e236 + e245 [notdef] , (3.1)
for Sp(2)/SU(2) [notdef] U(1) by
[notdef] e12 + e56 , e34 , [notdef] , [notdef]e136 e145 + e235 + e246 , e135 + e146 e236 + e245 [notdef] , (3.2)
and for G2/SU(3) by
[notdef] e12 + e56 + e34[notdef] , [notdef] e136 + e145 e235 + e246 , e135 e146 + e236 + e245 [notdef] , (3.3)
where ei1...in := ei1 ^ [notdef] [notdef] [notdef] ^ ein.
3The G-invariant four-forms which can be obtained from the above G-invariant two-forms via Hodge duality can be found in appendix B.
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Requiring the SU(3) structure to be compatible with the given group structure of the coset implies that the structure forms J and can be expressed in terms of the above forms. Indeed, one nds that the most general G-invariant structure forms for SU(3)/U(1)2 are given by
J = R21e12 R22e34 + R23e56
= R1R2R3
[bracketleftBig][parenleftBigg][parenleftBigg]e136
(3.4)
with independent parameters R1, R2 and R3. By comparing the spaces (3.1) and (3.2) of G-invariant forms, we conclude that the most general G-invariant structure forms on Sp(2)/SU(2) [notdef] U(1) are still given by (3.4) provided we set R1 = R3. Similarly, the most
general G-invariant structure on G2/SU(3) corresponds to setting R R1 = R2 = R3 in
eq. (3.4), but, in addition, with the signs of e2 and e4 reversed.4 This leads to
J = R2e12 + R2e34 + R2e56
= R3
[bracketleftBig][parenleftBigg][parenleftBigg]e136
for the SU(3) structures on G2/SU(3).
From the above SU(3) structure forms we can construct a unique compatible metric [37], which coincides with the most general G-invariant metric on G/H. For all three cases it is given by
ds2 = R21 (e1 e1 + e2 e2) + R22 (e3 e3 + e4 e4) + R23 (e5 e5 + e6 e6) (3.6)
where for SU(3)/U(1)2 the parameters R1, R2 and R3 are independent, for Sp(2)/SU(2) [notdef]
U(1) they are restricted by R1 = R3 and for G2/SU(3) by R1 = R2 = R3. Hence, we recognise the parameters Ri as radii of the coset, determining the volume and shape of the space.
Having introduced G-invariant geometry and SU(3) structure on our cosets, all required tools to solve the geometric sector of the heterotic string, that is, the Killing spinor equations (2.26)(2.32), are available. This has been known for some time and was rst realised in ref. [38]. The additional technical di culty of heterotic string compactications is the construction of vector bundles which satisfy the Hermitean Yang-Mills equations (2.35), (2.36). In past works, this has usually been approached using an Ansatz similar to the standard embedding. We will adopt the bundle construction developed in ref. [26] which contains the standard embedding Ansatz as special case.
3.2 Half-at mirror geometry of the cosets
We would now like to review the half-at mirror geometry, in the sense of section 2.4, for the three cosets introduced in the previous subsection. Technical details can be found in appendix B. We recall that half-at mirror geometry, in analogy with Calabi-Yau manifolds, is dened by a set of two-forms, [notdef]!i[notdef], a set of dual four-forms, [notdef]~
!i[notdef], and a set [notdef] A, B[notdef]
4The sign reversal of e2 and e4 can be avoided by redening the structure constants appropriately.
10
e145 + e235 + e246) + i (e135 + e146 e236 + e245
+ e145 e235 + e246) + i (e135 e146 + e236 + e245
[parenrightbig] [bracketrightBig]
,
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(3.5)
[parenrightbig] [bracketrightBig]
of symplectic three-forms. Unlike in the Calabi-Yau case, these forms are, in general, no longer closed but instead satisfy a set of di erential relations (2.39) which involve the torsion parameters ei.
It turns out that for all three cosets under consideration, there is only a single pair,
{ 0, 0[notdef], of symplectic three-forms in addition to a certain number of two- and four-form
pairs, [notdef]!i, ~
!i[notdef]. A subset,[notdef]!r[notdef] of the two-forms which we label by indices r, s, . . . are, in
fact, closed. For SU(3)/U(1)2 these forms are explicitly given by
!1 =
1
2
e12 + 12e34 12e56[parenrightBig]~!1 = 4 3V0
2e1234 + e1256 e3456[parenrightBig]
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!2 =
1 4
e12 + e34
[parenrightBig]
~!2 = 4
V0
e1234 + e1256
[parenrightBig]
!3 = 1
3
e12 e34 + e56[parenrightBig]
~!3 =
0 e1234 e1256 + e3456[parenrightBig]
0 =
2V0
e136 e145 + e235 + e246[parenrightBig] 0 = 1 2
e135 + e146 e236 + e245[parenrightBig]
(3.7)
In particular, there are three pairs of two- and four-forms in this case. The exterior derivatives of !3 and 0 are given by d!3 = 0 and d 0 = ~
!3, while all other forms are closed. This means the closed two-forms are !r, where r = 1, 2. Comparing with the general di erential relations (2.39) for half-at mirror geometry this shows that the three torsion parameters are given by (e1, e2, e3) = (0, 0, 1).
The coset Sp(2)/SU(2)[notdef]U(1) has only two pairs of two- and four-forms and the explicit
expressions read
!1 = 1 2
e12 + 2e34 + e56
[parenrightBig]
~!1 = 3V0
e1234 + 2e1256 + e3456
[parenrightBig]
!2 = 1
6
e12 e34 + e56[parenrightBig]
~!2 = 2
0 e1234 e1256 + e3456[parenrightBig]
(3.8)
All but !2 and 0 are closed and the non-vanishing exterior derivatives d!2 = 0, d 0 = ~
!2
0 =
2V0
e136 e145 + e235 + e246[parenrightBig] 0 = 1 2
e135 + e146 e236 + e245[parenrightBig]
show that the two torsion parameters are given by (e1, e2) = (0, 1). Hence, there is only one closed two-form, !1.
Finally, for G2/SU(3), we have
!1 = 5
3
[parenleftBig]
e12 + e34 + e56
[parenrightBig]
!1 =
5V0
~
e1234 + e1256 e3456[parenrightBig]
0 = p3
40V0
(3.9)
In particular, there is only one pair of two- and four-forms. The non-closed forms are !1,
0 with exterior derivatives d!1 = 0, d 0 = ~
e136 + e145 e235 + e246[parenrightBig] 0 = 10p3 [parenleftBig]e135 e146 + e236 + e245[parenrightBig].
!1 so that the single torsion parameter is e1 = 1. Note that there is no closed two-form in this case.
In all the expressions above, V0 is the coordinate volume, a specic number whose
value for each of the cosets can be found in appendix B. It can be shown that the above
11
forms indeed satisfy all the relations for half-at mirror geometry given in section 2.4. In particular, the SU(3) structure forms on the coset spaces given in the previous subsection can be re-written in half-at mirror form as
J = vi!i , = Z 0 + i G 0 , (3.10)
where Z is the single complex structure modulus and G the derivative of the pre-potential. From appendix B we see that for the rst two cosets, these two quantities are related by5
Z = V0
2 G . (3.11)
It is also easy to verify from the above expressions for the forms that
!i ^ 0 = !i ^ 0 = 0 (3.12)
for all i, in analogy with the Calabi-Yau case. These relations are also expected from the absence of G-invariant 5-forms on our coset spaces. A further useful relation can be deduced from the SU(3) structure compatibility relation (A.3). Inserting the expansions (3.10) for J and into this relation leads to
dijkvivjvk =
32ZG =
3 V0
22 G2 . (3.13)
This shows that Z is determined by the Kahler moduli vi and, therefore, no independent complex structure moduli exist in our coset models.
3.3 Levi-Civita connection
The Levi-Civita connection is the unique torsion-free and metric compatible connection on the tangent bundle. On our spaces, with the most general G-invariant metric (3.6), the Levi-Civita connection one-form is
!(LC) ab = 12f
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aib "i . (3.14)
The "i are the Maurer-Cartan left-invariant one-forms on G along the directions of the generators Hi of the sub-group H. On G/H these can be written in terms of the forms ei, but, as we will see, the explicit expressions are not required.
The Levi-Civita connection enters the Bianchi identity (2.4) as part of the connection one-form ! dened in (2.2). As we will see below, our spaces do not allow for H-ux at lowest order in [prime] and, therefore, we can set ! = !(LC). For SU(3)/U(1)2 this means that the contribution to the Bianchi identity at lowest order is given as
tr R(LC) ^ R(LC) =
94 V0 ~
acb ec + f
!3 (3.15)
as can be seen from eq. (C.7) (with the ux parameter C set to zero in this equation). The
results for the other cosets can be found in appendix C. Eq. (3.15) will play a role when we solve the Bianchi identity iteratively, leading us to an Ansatz for an exact solution for non-vanishing H and R in the Bianchi identity.
5On G2/SU(3) the relation di ers by a factor of 400/3.
12
3.4 Vector bundles on coset spaces
We now turn to the problem of nding appropriate gauge bundles on the cosets, which can satisfy the Hermitean Yang-Mills equations (2.35), (2.36). Such bundles have been explicitly constructed in [26], based on the well-known relation between vector bundles and principal bre bundle. The principal bre bundle in our case is G = G(G/H, H) and any representation : H !
Cn uniquely denes a rank n vector bundle which is referred to as an associated vector bundle. Moreover, any connection dened on G uniquely denes a connection on every associated vector bundle. We shall require the structure of the bundle to be compatible with the group structure of G/H. This leads to a natural connection on G = G(G/H, H), related to the reductive decomposition of the Lie algebra, given by
A = "iHi . (3.16)
Recall that Hi are the generators of the Lie algebra of H and the "i are the Maurer-Cartan left-invariant one-forms on G along the directions of the generators Hi. As before, their explicit form in terms of the vielbein ei will not be required.
On an associated vector bundle dened by the representation , the connection associated to A is then
A = "i(Hi) (3.17)
with eld strength
F =
aib (3.19)
then the curvature (3.18) satises the HYM equations.6 This choice is also commonly referred to as standard embedding, even though the geometric connection and the gauge connection are not equal. Note that (3.19) does not solve the Bianchi identity for H = 0 anymore. However, since this connection only di ers from the Levi-Civita connection (3.14) by a torsion term, both choices yield the same cohomology class for tr F ^ F and tr R ^ R.
6This choice is known as the H-connection on homogeneous spaces and should not be confused with the Hull connection (2.2) which is, unfortunately, often referred to as the H-connection as well. To avoid confusion will we not use this terminology in the present paper.
13
JHEP01(2013)015
1
2fabi(Hi)ea ^ eb . (3.18)
Note that the one-forms "i have indeed dropped out. This construction holds in general for every representation of H.
We would like to add a few remarks on the standard embedding, a choice of gauge connection frequently made in the literature. For this choice, the bundle curvature F and the Riemann curvature R are set equal, which solves the Bianchi identity (2.4) for H = 0. However, in the present context, such a choice leads to a problem. Since our spaces are not Ricci-at, the so-chosen eld strength F does not satisfy the Hermitian Yang-Mills (HYM) equations, so that the solution is not supersymmetric. If we choose instead
(Hi) ab = f
This means that the topological constraint arising from the Bianchi identity is satised, while the exact identity is only satised to lowest order in [prime]. This has been the case for most heterotic bundle constructions in past works. In contrast, we will construct exact solutions to the the Bianchi identity and solutions to order [prime] of the supersymmetry constraints.
3.5 Line bundle sums
When constructing a solution to the E8 [notdef] E8 heterotic string, the structure group of a
vector bundle has to be embedded in E8 and the resulting low-energy gauge group will be given by the commutant of the structure group within E8. Recently, it has been noted that vector bundles which consist of sums of line bundles provide a fertile class of models which can be studied systematically [4]. Such line bundle sums have been used for the half-at compactications in ref. [26] and will also be the focus of the present paper.
Let us rst focus on a single line bundle, L, dened by a one-dimensional representation : H !
C. For SU(3)/U(1)2, such a representation is characterized by two integers, pr, where r = 1, 2, which correspond to the charges of the two U(1) symmetries. Writing
(H7) = i (p1 + p2/2) (H8) = i p2/(2p3) (3.20) and using eq. (3.18) the rst Chern class of such a line bundle becomes
c1(L) = i
2 [F ] = pr!r . (3.21) Hence, the integers p = (pr) label the rst Chern class of the line bundles and we can adopt the notation L = OX(p). A sum of line bundles
V =
n
Ma=1
Xa=1pra!r . (3.23)
The case Sp(2)/SU(2) [notdef] U(1) works analogously, with each line bundle characterized by a
single integer (so that r only takes the value 1 in all equations) which corresponds to the charge of the U(1) factor in H. For G2/SU(3) the sub-group H has no one-dimensional representations (except the trivial one) and no line bundles can be obtained by this construction. Given that there are no G-invariant exact two-forms on our spaces, it follows that the eld strength for the connection on V is given by
F = [F ] = 2i
Xapra!r . (3.24)
To ensure that the structure group of V can be embedded into E8, we impose the vanishing of the rst Chern class, c1(V ) = 0. This condition restricts the integers pra by
n
Xa=1pra = 0 8 r . (3.25)
14
JHEP01(2013)015
OX (pa) (3.22)
is, therefore, characterized by the set, [notdef]pra[notdef], of integers and its total rst Chern class is
given by
c1(V ) =
n
Then, the structure group of V is S(U(1)n) which is indeed a sub-group of E8 for 1 < n 8.
Further, for n = 3, 4, 5, the commutant of S(U(1)n) within E8 is given by S(U(1)3) [notdef] E6,
S(U(1)4) [notdef] SO(10) and S(U(1)5) [notdef] SU(5), respectively. These are the phenomenologically
interesting GUT gauge groups and for the visible E8 we will, therefore, focus on line bundle sums of rank 3, 4 or 5.
Subsequently, we will require the vector bundle contribution to the Bianchi identity (2.4). Focusing on the main case of interest, we evaluate this contribution for a sum of line bundles on SU(3)/U(1)2. Writing (pa, qa) = (p1a, p2a) for ease of notation, we nd
tr F ^ F = V0
8
[bracketleftbigg] [summationdisplay]
a
(6p2a + q2a + 6paqa)~
!1 +
Xapa(3pa + 2qa)~ !2
+ 4
3
(3.26)
Note that we will, of course, have two di erent bundles, one for each E8 factor, corresponding to the visible and hidden sectors of the theory. Hence, the Bianchi identity has two contributions of the form (3.26), each controlled by its own set of integers. As we will see, the hidden bundle contribution is important as it can be adjusted to cancel the other terms in the Bianchi identity.
Another basic phenomenological requirement on the visible vector bundle is the presence of three chiral generations. The number of generations is counted by the index of the bundle which can be computed using the Atiyah-Singer index theorem. For a sum of line bundles, V , this has been done in appendix D, leading to
ind(V ) =
1 6drst
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Xa(3p2a + q2a + 3paqa)~ !3
[bracketrightbigg]
n
Xa=1prapsapta , (3.27)
where dijk are the intersection numbers. Specializing to SU(3)/U(1)2 gives
ind(V ) =
n
Xa=1
p3a + 12paqa(qa + 3pa)[parenrightbigg]. (3.28)
3.6 Solutions to lowest order in [prime]
We have now collected all ingredients to solve the heterotic string on our coset spaces. In this section we will review the solution at lowest order in [prime] which has been found in ref. [26].
As discussed in section 2, nding a supersymmetric vacuum of the heterotic string is equivalent to nding elds which satisfy the Bianchi identity (2.4), the Killing spinor equations (2.26)(2.32), the HYM equations (2.35), (2.36) and the integrability condition (2.13).
The discussion below equation (2.13) shows that the integrability condition is satised to lowest order. This means solving the Killing spinor equations and the Bianchi identity implies that the equations of motion are satised to lowest order as well. For clarity, we will label the lowest order solution by (0), except for the bundle7 which we will still denote
7We will see later that the solution at rst order requires all elds to change apart from the gauge eld strength.
15
by F . The relevant objects are then H(0), (0), J(0), (0), g(0) and F and we will also
denote the Hodge star with respect to the metric g(0) as 0.
3.6.1 Bianchi identity
Let us consider the Bianchi identity rst. At lowest order in [prime] it is
dH(0) = 0 . (3.29)
Now, take a look at the rst two Killing spinor equations (2.26), (2.27) at this order
d(e2(0) (0)) = 0 (3.30)
d(e2(0)J(0)) = @y(e2(0) (0)) + 0H(0)e2(0). (3.31)
Since H3(X) = 0 for all the spaces we are considering, these equations show that
0H(0)e2(0) is the sum of two exact forms and, hence, an exact form itself. Using this,
we have
[notdef][notdef]H(0)e(0)[notdef][notdef]2 = [integraldisplay]X
after partial integration. It follows that
H(0) = 0 . (3.33)
In fact, our proof holds for all domain wall compactications on an internal manifold with H3(X) = 0 and, therefore, no nontrivial H-ux can be present in such geometries at lowest order.8 Note that this is very similar to ndings in [40], which studied no-go theorems for heterotic ux compactications with maximally symmetric four-dimensional spacetimes.
3.6.2 Killing spinor equations
Having solved the integrability condition and the Bianchi identity, we now turn to solving the Killing spinor equations. To lowest order the two Killing spinor equations (2.30) and (2.31) read
0 = 0 d(0) (3.34) 0 = (2@y(0)) 0 1 , (3.35)
or equivalently d(0) = @y(0) = 0. This means that, in addition to vanishing H-ux, the
dilaton is constant. The Killing spinor equations (2.26)(2.32) then reduce to the Hitchin ow equations [37]
d (0) = 0 (3.36) dJ(0) = @y (0) (3.37)
8This result agrees with the ndings of [39], which performed an extensive search for ux compactications of the heterotic string on various known non-Calabi-Yau backgrounds including our cosets.
16
JHEP01(2013)015
H(0) ^ 0e2(0)H(0) = 0 (3.32)
J(0) ^ dJ(0) = 0 (3.38) d (0)+ = J(0) ^ @yJ(0) . (3.39)
As can be explicitly checked, the Hitchin ow equations (3.36)(3.39) are solved by the G-invariant SU(3) structures (3.4), (3.5), provided the parameters Ri assume a certain y-dependence to be examined shortly.
3.6.3 Hermitian Yang-Mills equations
The gauge bundle has to satisfy the equivalent of the HYM equations, that is, the instanton conditions J[notdef]F = 0 and [notdef]F = 0. The second of these condition is automatically
satised for the holomorphic three-form (3.4), (3.5) and eld strengths (3.18). The rst condition, however, leads to an additional constraint on the parameters appearing in the SU(3) structure [26]. To see this, note that J[notdef]F = 0 is equivalent to
F ^ J ^ J = 0 . (3.40) Inserting J = vi!i, with the G-invariant two-forms !i and the eld strength (3.24) into eq. (3.40) gives
drjk pravjvk = 0 for all a . (3.41)
Here drjk are the intersection numbers (see appendix B) and we recall that indices i, j, . . . run over all two-forms while indices r, s, . . . only run over the subset of closed two-forms . The solution to eqs. (3.41), for generic values of the integers pra, is to set all vr to zero. For
SU(3)/U(1)2 ( Sp(2)/SU(2) [notdef] U(1) ) this leaves us with one remaining non-zero modulus
v3 (v2) corresponding to the non-harmonic two-form d!3 [negationslash]= 0 (d!2 [negationslash]= 0). Therefore, from
the relations (B.8)(B.10) and (B.18)(B.19) between the Kahler moduli vi and the radii Ri we see that the HYM are solved if
R21 = R22 = R23 R2 for SU(3)/U(1)2 (3.42)
R21 = R22 R2 for Sp(2)/SU(2) [notdef] U(1) . (3.43) It then follows, using the relations (A.5), (A.6) between the torsion classes and the SU(3)
structure forms, that the only non-vanishing torsion class is the real part of the rst class W +1 = 1/R [23, 26]. This means that the SU(3) structure of X is nearly Kahler.
There is a subtlety in the case SU(3)/U(1)2. If qa = 2pa or qa = 0 for all a, the
parameters Ri do not have to be all equal. (The analogous subtlety pa = 0 in the case Sp(2)/SU(2) [notdef] U(1) corresponds to the trivial bundle). From now on we exclude these
special cases, unless otherwise stated and we will return to this possibility when we discuss
the four-dimensional e ective supergravity in section 5.
3.6.4 Hitchin ow equations
So far we have not determined the y-dependence of the SU(3) structure forms which is governed by the Hitchin ow equations (3.36)(3.39). To work this out, we insert the half-at mirror geometry expansion (which we introduced in sections 2.4 and 3.2) into these
17
JHEP01(2013)015
ow equations. The two equations (3.36) and (3.38) are automatically satised using the compatibility constraints (3.12). The other two equations become
viei 0 = @yG 0 (3.44) Z ei ~
!i = dijk vi(@yvj) ~
!k . (3.45)
Multiplying with ^(vl!l) on both sides of (3.45) and integrating givesZekvk = dijkvi(@yvj)vk . (3.46)
Now, using the compatibility relation (3.13) we can express this in terms of the complex structure modulus
ekvk = @yG , (3.47)
which shows that equations (3.44) and (3.45) are, in fact, equivalent. We have seen previously, that the presence of the gauge elds force all radii to be equal. The y-dependence should, therefore, reside in this overall modulus R = R(y) and we write the SU(3) structure forms as
J(0) = R2 ~vk!k , (0) = R3
[parenleftBig]
, (3.48)
with (~vk) = (0, 0, ~v) for SU(3)/U(1)2 and (~vk) = (0, ~v) for Sp(2)/SU(2) [notdef] U(1) and a
constant ~v. The values of and follow from this choice via eq. (3.13). From (3.47), the y-dependence of R is determined by
@yR =
= S3, the permutation group of three elements. Hence, a Kaluza-Klein gauge group in four dimensions does not arise.
The standard method to break GUT gauge groups in heterotic constructions is to include a Wilson line in the gauge bundle. This requires a non-trivial rst fundamental group of the underlying space. However, all coset spaces studied here are simply connected and, hence, do not admit any Wilson lines. Alternatively, if the space admits a freely-acting symmetry a closely related compactication can be dened on the quotient manifold which
18
0 + i 0
~v3 . (3.49)
Since the right-hand side of this equation is a non-zero constant the solutions for R are linear in y and diverging as y ! [notdef]1. We will see later that the [prime] corrections can remove
this divergent behaviour.
3.7 Side issues: Kaluza-Klein gauge group and Wilson lines
An obvious question is whether the symmetries of our coset spaces G/H lead to a Kaluza-Klein gauge group in four dimensions, in addition to the remnants of the E8 [notdef] E8 gauge group. It turns out [41] that Kaluza-Klein gauge elds from such spaces take values in the quotient N(H)/H where N(H) is the normaliser of H in G. For our cosets, this quotient is merely a discrete group. For example, for SU(3)/U(1)2, with H = U(1)2, one nds that N(H)/H
[parenrightBig]
JHEP01(2013)015
has a non-trivial rst fundamental group and, hence, allows for the inclusion of Wilson lines. However, for our cosets it has been shown [42] that only torsion-free discrete groups can have a free action on G/H, that is, groups which do not posses any cyclic elements. In particular, this excludes all nite groups. The mathematical literature provides an existence theorem for a freely acting innite but nitely generated discrete freely-acting group on every coset of compact groups G, H. However, we have not been able to nd such a group explicitly for one of our cosets. For this reason, Wilson line breaking of the GUT symmetry is not currently an option. Instead, ux in the standard hypercharge direction might be used. Such details of particle physics model building are not the primary concern of the present paper and will not be discussed further.
4 Solutions on homogeneous spaces including [prime] corrections
In the previous sections we have seen how to construct lowest order solutions to the heterotic string on homogeneous spaces, using the associated vector bundle construction on cosets. It turns out that the four-dimensional space time is a domain wall and that the radius, R, of the internal space varies linearly with y, the coordinate transverse to the domain wall.
How do we expect this to change if we include rst order [prime] corrections? In our discussion before, we saw that the Bianchi identity (2.4) at lowest order requires the three-form ux H to be closed, which forces H to vanish at lowest order. Now, at the next order the Bianchi identity is
dH = [prime]
4 tr F ^ F tr R ^ R [parenrightbig]
(4.1)
and we expect a non-zero H which is not closed. From a four-dimensional point of view, ux will contribute to the (super)-potential and we, therefore, expect some e ect on moduli. Of course, the non-zero H also feeds into the gravitino and dilatino Killing spinor equations and will change the gravitational background.
In order to work this out, we rst need to nd solutions to the Bianchi identity (4.1) and then solve the Killing spinor equations (2.26)(2.32), the Hermitean Yang-Mills equations (2.35), (2.36) and the integrability condition (2.13). Of those, only the Bianchi identity and the integrability condition are changed by [prime] e ects.
4.1 Perturbative solution
We begin by solving the Bianchi identity (4.1) iteratively, using the lowest order solution on the right-hand side, in order to get an intuition for what form the general solutions will take. For concreteness, let us perform the analysis on SU(3)/U(1)2. The results for the other cases are summarized in appendix B. Taking the quantities on the right-hand side of eq. (4.1) to be at zero in [prime] we can write
dH(1) = [prime]
4 tr F ^ F tr R(LC) ^ R(LC)[parenrightBig]. (4.2)
The explicit expressions for the terms in the bracket have already been computed in eqs. (3.26) and (3.15). However, the gauge eld contribution, tr F ^ F , includes both
19
JHEP01(2013)015
E8 sectors so we should add two terms of the form (3.26), one for the observable sector with bundle parameters pa, qa, where a = 1, . . . , n, and one for the hidden sector with bundle parameters ~pa, ~qa, where a = 1, . . . ,.
An integrability condition for the Bianchi identity (4.2) is that the right-hand side is trivial in cohomology. Noting that ~
!3 = d 0, we see from (3.15) that tr R(LC) ^ R(LC)
is already cohomologically trivial and, hence, the same should be required for tr F ^ F .
This leads to relations between the observable and hidden bundle parameters which can be written as
n
Xa=1(6p2a + q2a + 6paqa) +
n
Xa=1pa(3pa + 2qa) +
Xa=1(6~p2a + ~q2a + 6~pa~qa) = 0 (4.3)
Xa=1~pa(3~pa + 2~qa) = 0 . (4.4)
Clearly solutions to these equations exist and explicit examples will be considered later. Note that the presence of the hidden bundle is helpful in that is can be used to cancel the observable bundle contributions which may be somewhat constrained by model building considerations.
Assuming we have satised the above constraints, the Bianchi identity takes the form
dH(1) = V0
B(p, q, ~p, ~q) [prime] d 0 . (4.5)
with some function B of the bundle parameters whose specic form is not important for
now and will be stated later. Using that H3(X) = 0 for all our spaces, we can immediately integrate this equation and obtain9
H(1) = V0
B(p, q, ~p, ~q) [prime] 0 . (4.6)
Even though this was evaluated for SU(3)/U(1)2 the result is similar for the other cosets, although the precise form of B depends on the coset.
What back-reaction does this have on the geometry of the homogeneous spaces? This can be seen from the Killing spinor equations (2.26)(2.32), which we repeat for convenience.
d = 2d ^ (4.7)
dJ = 2@y @y 2d ^ J + H (4.8) J ^ dJ = J ^ J ^ d (4.9)
d + = J ^ @yJ @yJ ^ J + 2d ^ + (4.10) J ^ H = d (4.11)
^ H = (2@y) 1 (4.12)
+ ^ H = 0 . (4.13)
9To be more precise, H may still contain an exact (non-G-invariant) piece. However, from (4.9) it follows that d = 0, which together with (4.8) and (4.10) implies that this exact piece has to be zero.
20
JHEP01(2013)015
From the orthogonality of the forms 0 and 0, eq. (2.39), we see that the solution (4.6) automatically solves eq. (4.13). Using the orthogonality of !i and 0, eq. (3.12), eq. (4.11) immediately leads to d = 0 and, hence, conditions (4.7), (4.9) reduce to d = 0 and
J ^ dJ = 0. These are exactly the same relations as obtained at zeroth order in [prime] (see
eqs. (3.36) and (3.38)). This means that the internal geometry remains half-at even after switching on [prime] corrections. However, from (4.12) we see that now @y [negationslash]= 0, which will
impact on the Hitchin ow equations (4.8) and (4.10), leading to a di erent y dependence of R. If we were now to proceed to the second order in [prime], it seems likely that the right-hand side of the Bianchi identity at the next order only picks up G-invariant terms. Since 0 is the only non-closed G-invariant three form on all cosets, this forces H(2) / 0,
thereby keeping the geometry half-at at the second and only altering the functional form of (y) and R(y). It seems this process can be iterated, leading to an all order in [prime] solution to the Bianchi identity, which preserves the half-at geometry of the cosets but induces higher order contributions to (y), R(y). We will now verify that this expectation is indeed correct.
4.2 Full solution Ansatz
Motivated by the above discussion, we start with the following Ansatz
J = R(y)2 ~vi!i
= R(y)3 [parenleftBig]
H = C(R, [prime]) V0 0
= (y)
for [notdef](J, ), H, [notdef]. The bundle is dened to be the same as at lowest order since the only [prime]
e ects on J and are through the radius R(y) which does not a ect the HYM equations. The function C(R, [prime]) in the Ansatz for H also depends on the bundle parameters and,
along with R(y), it has to be determined for a full solution. The tilded parameters have been dened in section 3.6.4. In the following, we present explicit expressions for the space SU(3)/U(1)2. The solutions for Sp(2)/SU(2) [notdef] U(1) can be found in appendix C.
4.3 Exact solution to the Bianchi identity
Now, we will show that our Ansatz solves the full Bianchi identity
dH = [prime]
4 (tr F ^ F tr R ^ R) (4.15)
for a particular choice of C(R, [prime]). For this, we need to compute tr R ^ R where R is
the curvature two-form of the Hull connection
! cab = !
c ab
21
JHEP01(2013)015
0 + i 0
[parenrightBig]
(4.14)
1
2H
cab . (4.16)
On the coset SU(3)/U(1)2 we then obtain (see appendix C.1 for details and results for the other cosets)
tr R ^ R =
3
4
3 C2R4 + 2 C R2[parenrightbigg]
V0
d 0 . (4.17)
In the limit C ! 0 we recover the zeroth order result (3.15), as we should. For tr F ^ F we
get the same result (3.26) as before. Including observable and hidden sector and assuming that the integrability conditions (4.3) and (4.4) are satised it can be written as
tr F ^ F = A(p, q, ~p, ~q) V0 d 0 (4.18)
where
A(p, q, ~p, ~q) =
1 12
JHEP01(2013)015
[bracketleftBigg]
n
Xa=1q2a +
Xa=1~q2a
[bracketrightBigg]
. (4.19)
It may seem that this only depends on the bundle parameters qa, ~qa, but not on pa, ~pa. However, note that this result only hold for consistent bundles satisfying the integrability conditions (4.3) and (4.4), which relate pa, ~pa with qa, ~qa.
With these results, the Bianchi identity reduces to a quadratic equation for C given by
C =
[prime] 4
A +3 4
C2R4 + 2 C
R2 + 3[parenrightbigg][parenrightbigg]. (4.20)
Its positive solution is10
C(R, [prime]) = 13 [prime]/8R4 1 +
3 8
[prime] R2 +
r1 34 [prime]
R2 +3A + 916 [prime]2 R4
[parenrightBigg]
. (4.21)
In the large radius limit, [prime]
R2
1, this function behaves as
C(R, [prime]) =
B +1 128
27 + 12A
[parenrightBig] [prime]
R2 34096[parenleftBig] 27 + 24A + 16A2[parenrightBig] [prime]2R4 + O
[prime]3R6[parenrightbigg][bracketrightbigg] [prime] .
(4.22)
In particular, we see that the ux is of order [prime] and that the proper expansion parameter is [prime]/R2, as expected. The leading term
B =
4A + 9
16 (4.23)
is determined by the bundle parameters A, eq. (4.19), and is, in fact, all we will need in the
following. While the above results were derived for the coset SU(3)/U(1)2, we will express all subsequent equations in terms of B. The case Sp(2)/SU(2) [notdef] U(1) can then be obtained
by setting B = 1/2, as can be seen from appendix C.3.
10Note that there also exists a solution of the Bianchi identity for vanishing ux (C = 0), which would
lead to an additional constraint on the bundle parameters. However, we are interested in solutions with ux and will not explore this solution further.
22
4.4 Hitchin ow revisited
Apart from a non-vanishing H and y-dependence of R, our Ansatz (4.14) remains unchanged from its lowest order form. This means that all equations (4.7)(4.13) which do not contain y derivatives or H are automatically satised.
The remaining three equations, (4.8), (4.10) and (4.12), lead to di erential equations for the y-dependence of R(y) and (y) and inserting the Ansatz (4.14) into these gives
R2~viei 0 =
2@y R3 3R2@yR
[parenrightBig] 0 + C 0 (4.24)
R3ei~
!i = dijk~vi~vj ~
!k 2R3@yR @yR4 [parenrightbig]
CR3 0 ^ 0 = 2 @y 1 . (4.26)
A direct evaluation yields J ^J ^J = 6 R6 1 and, therefore, eq. (3.13) yields the relation 1 = V0422 0 ^ 0. If we insert this last relation into the third ow equation (4.26) and
then use the result in eq. (4.25), we obtain
@y = C
@yR = =
~v + 3 B [prime]
R2[bracketrightbigg], (4.30)
where B = (4A + 9)/16 for SU(3)/U(1)2 (and B = 1/2 for Sp(2)/SU(2) [notdef] U(1)). The
structure of the solutions to these equations depends crucially on the sign of B and we
distinguish the three cases
Case 1: B = 0
Case 2: B < 0
Case 3: B > 0 .
23
(4.25)
JHEP01(2013)015
V0
(R, [prime])
R3 (4.27)
1 6
~v + 3 C(R, [prime])
R2[bracketrightbigg]. (4.28)
Here, we have set = 2, the value appropriate for SU(3)/U(1)2. These two equations already fully determine R(y) and (y) and eq. (4.24) yields no additional information. This can be seen after multiplying it with [notdef] ^ (~vk!k) and making use of the compatibility
relation (3.13), in complete analogy with the lowest order analysis in section 3.6.4.
4.5 Solving the ow equations
Solving the above di erential equations (4.27) and (4.28) for the y-dependence of the radius R and the dilaton , with the function C from eq. (4.21) inserted, leads to an exact solution
of the Bianchi identity. However, in the present paper, we are only interested in corrections up to order [prime]. For this reason and to avoid unnecessary complications, we will consider these di erential equations only to order [prime]. Inserting the leading term in C from eq. (4.22)
into eqs. (4.27) and (4.28) leads to
@y = B
R3 [prime] (4.29)
1 6
@yR =
(4.31)
Note from eq. (4.19), that B is a function of the bundle parameters and that, for
SU(3)/U(1)2, all three cases can indeed be realized for appropriate bundle choices. Let us now discuss the solution for each of these cases in turn.
4.5.1 Case 1, B = 0In this case, H = 0, and eqs. (4.27) and (4.28) revert to their zeroth order counterparts discussed in section 3.6.4. This means that, due to a special choice of bundle, the [prime] corrections vanish and we remain with a constant dilaton and a linearly diverging radius R.
4.5.2 Case 2, B < 0In this case, eq. (4.30) allows for a special y-independent solution where R assumes the constant value
R20 = 3 [notdef]B[notdef] [prime]
~v . (4.32)
For this static solution, the equation can then be easily integrated and we obtain a linear dilaton
(y) = [notdef]B[notdef]
R30
[prime] y . (4.33)
The behaviour of this solution is radically di erent from what we have seen at zeroth order. There, the radius R was linearly divergent and the dilaton constant. For the above solution, this situation is reversed with R constant and the dilaton linearly diverging.
We can integrate eq. (4.30) in general, to obtain the implicit solution
y y0 = 6
[bracketleftBigg]
R
~v +
JHEP01(2013)015
p3[notdef]B[notdef] [prime]~v3/2 arctanh
[radicalBigg]
~v 3[notdef]B[notdef] [prime]
R
[parenrightBigg] [bracketrightBigg]
. (4.34)
Here y0 is an arbitrary integration constant which corresponds to the position of the domain wall and will be set to zero for convenience. This solution has the generic form displayed in gure 1 (solid line) and exhibits a kink at y = y0 = 0, indicating the position of the domain wall. It approaches the above constant solution (4.32) for R as [notdef]y[notdef] ! 1, that is,
far away from the domain wall. In this limit, the dilaton asymptotes the linearly divergent behaviour (4.33).
4.5.3 Case 3, B > 0No constant solution for R exists in this case and integrating eq. (4.30) gives
y y0 = 6
"
R~v +p3B [prime]~v3/2 arctan[parenleftBigg][radicalbigg]~v3B [prime]
R[parenrightBigg][bracketrightBigg]. (4.35)
This solution is plotted in gure 1 (dashed line) for y0 = 0. For [notdef]y[notdef] ! 1, R diverges
linearly and in fact approaches the zeroth order solution (3.49), while the dilaton becomes constant. Hence, we see that, far away from the domain wall, we recover the zeroth order solution, with a constant dilaton and a linearly divergent radius R.
24
R2
2.5
2.0
1.5
1.0
0.5
[Minus]20 [Minus]10 10 20 y
Figure 1. Plot of the radial modulus R2 as a function of the distance, y, from the domain wall at y = 0 for B < 0 (solid line), and B > 0 (dashed line). For convenience, we have set R20 = 1.
4.6 Discussion
To summarize, we have seen that the qualitative behaviour of the moduli on y, the coordinate transverse to the domain wall, is controlled by the gauge bundle via the quantity
B = (4A + 9)/16 for the case of SU(3)/U(1)2, where A is dened in eq. (4.19). For
Sp(2)/SU(2) [notdef] U(1) there is no gauge bundle dependence and B = 1/2 always. For B = 0
the solution is, in fact, unchanged from the zeroth order one which has a constant dilaton and a linearly divergent radius R. For B > 0 the solution is modied due to [prime] e ects close
to the domain wall but approaches the zeroth order solution far away from the domain wall. The behaviour is quite di erent for B < 0 which, asymptotically, leads to a constant
radius R and a linearly diverging dilaton.
We see that [prime] e ect can have a signicant e ect on moduli and their stabilization. From a four-dimensional viewpoint this should be encoded in a (super-) potential which appears at order [prime]. We will now discuss this in detail by considering the four-dimenional
N = 1 supergravity associated to our solutions.
5 The four-dimensional e ective theory
Above, we have found O( [prime]) corrected solutions to the 10-dimensional heterotic string.
In this section, we will examine the corresponding four-dimensional e ective supergravity theories and their vacua. In particular, we would like to verify that our 10-dimensional results can be reproduced from the perspective.
5.1 Four-dimensional supergravity and elds
We will follow the conventions of four-dimensional supergravity laid out in ref. [50]. Mostly, we are interested in a set of chiral elds, ( X), with Kahler potential K = K( X,
X) and
25
JHEP01(2013)015
superpotential W = W ( X). The scalar potential is given by
V = 44e 2
KX Y FX FY 3 24[notdef]W [notdef]2[parenrightBig]+ 12DaDa , (5.1)
where the F-terms are dened as FX = @XW + KXW , with KX = @XK. Further, KX Y
@X@Y K is the Kahler metric, KX Y is its inverse and Da are the D-terms.For compactications on our coset spaces, the relevant moduli superelds are ( X) =
(S, T i) with the dilaton S and T-moduli T i. We recall that the number of T-moduli depends on the specic coset. For SU(3)/U(1)2 we have three T-moduli, so i = 1, 2, 3, while Sp(2)/SU(2) [notdef] U(1) has two moduli, so i = 1, 2. There are no moduli analogous to
Calabi-Yau complex structure moduli.
We should now explain the relation between four- and 10-dimensional quantities, following refs. [18, 19]. First, the four-dimensional Newton constant is given in terms of its 10-dimensional counterpart by 24 = 210/V0. A set of elds, vi, analogous to the Kahler
moduli of CY manifolds, appears in the expansion
J = vi!i . (5.2)
of the SU(3) structure form J with respect to the two-forms !i of the half-at mirror basis introduced in sections 2.4 and 3.2. We also introduce the standard quantity
V =
16dijkvivjvk , (5.3)
proportional to the volume of the coset space, with the intersection numbers dijk explicitly given in appendix B. This allows us to dene the four-dimensional dilaton s in terms of its 10-dimensional counterpart as
s = e2 V
V0 . (5.4)
For the expansion of the 10-dimensional three-form eld strength we have
H = biei 0 +
2
4K
JHEP01(2013)015
V0 0 dbi ^ !i + dB4 , (5.5)
where [notdef] 0, 0[notdef] is the basis of G-invariant three-forms introduced in section 2.4 and 3.2, bi
are real scalars and B4 is a two-form in four dimensions. The factor in front of the ux parameter is conventional in order to simplify later expressions. The rst term in this expansion is due to the non-vanishing torsion of the internal space and ei are the torsion parameters. We recall that they are given by (e1, e2, e3) = (0, 0, 1) for SU(3)/U(1)2 and (e1, e2) = (0, 1) for Sp(2)/SU(2) [notdef] U(1). The second term in eq. (5.5) is a result of the
non-vanishing H-ux induced via the Bianchi-identity. Its coe cient, , can be read o from eqs. (4.14), (4.22) and is explicitly given by
= [prime]B , (5.6)
26
where, for SU(3)/U(1)2, the quantity B = (4A+9)/16 depends on parameters of the gauge
bundle as in eq. (4.19). For Sp(2)/SU(2) [notdef] U(1) it is always given by B = 1/2. Given these
preparations, we can identify the (scalar parts of the) four-dimensional superelds as
S = a + is , T i = bi + i vi , (5.7)
where a is the four-dimensional Poincar-dual of the two-form B4.
5.2 Kahler potential and superpotential
The Kahler potential for the above set of elds is obtained from standard dimensional reduction [18, 19] as
K = ln i(
S S) [parenrightbig]
ln(V) . (5.8) The superpotential is obtained from the generalized Gukov-Vafa-Witten formula [18, 43]
W =
drijpravivj
V
, (5.11)
and similarly for the hidden sector. The D-at conditions, Da = 0, hence implement the slope conditions (3.41) (which follow from the HYM equations) from a four-dimensional viewpoint. Therefore, generically the D-at conditions imply that all but the last modulus, v = eivi, vanish as we have seen in section 3.6.3. The associated axions are absorbed by the gauge elds so we remain with a single T-modulus supereld T = eiT i = b + iv and,
of course, the dilaton S. In terms of these e ective elds the Kahler potential and superpotential read
K = ln(S +
S) 3 ln(T +
27
JHEP01(2013)015
1
Z
[integraldisplay]
~X ^ (H i dJ) . (5.9)
After inserting the various forms from eq. (3.10) and (5.5) and using eq. (3.13) as well as the properties of the half-at mirror basis given in section 2.4 this leads to
W = eiT i + i . (5.10)
The rst term arises from the non-vanishing torsion of the internal space and the second term is due to the non-vanishing H-ux induced by the gauge bundle.
5.3 D-terms
The S(U(1)n) and S(U(1)) structure groups of our observable and hidden line bundle sums also appear as gauge symmetries in the four-dimensional theory. Their associated D-terms have a Fayet-Illiopoulos (FI) terms and, in general, matter eld terms which involve gauge bundle moduli [44]. Switching on these moduli deforms the gauge bundle to a one with non-Abelian structure group, a possibility which we will not consider in this paper. Focusing on the FI terms, one nds that for the observable sector
Da
T) , W = T + , (5.12)
where we have switched to the phenomenological denition S = s + ia and T = v + ib of the superelds, obtained from the previous one by multiplying the superelds by i and
changing the signs of the axions.
It is worth noting that the above D-terms receive a dilaton-dependent correction at one loop [45, 46]. This correction is small in the relevant part of moduli space and will not change our conclusions, qualitatively. For simplicity, we will therefore neglect this correction.
Moreover, recall that for specic choices of the bundle parameters it is possible to satisfy (5.11) and leave more than just one of the moduli non-zero, as we pointed out at the end of section 3.6.3. However, the corresponding F-terms
FTs /
1 W @Ts V / dsijvivj . (5.13)
for these moduli drive the model back to the nearly-Kahler locus where only the last vi is
non-zero. Therefore, starting from this locus covers already the most general case.
5.4 F-term conditions
The superpotential (5.12) is S-independent and it is, therefore, expected that the dilaton cannot be stabilized. Below we will add a gaugino condensation term to W in order to improve on this. However, it is still instructive at this stage to consider the F-term equations which follow from (5.12). For the T modulus we have
FT =
1
2
JHEP01(2013)015
3
2v
3ib
2v . (5.14)
Hence, FT = 0 implies a vanishing T-axion, b = 0, and
v = 3 . (5.15)
Since v > 0 this solution is only physical provided that B < 0 and we have seen that this
can be achieved for appropriate bundle choices. Indeed, this is precisely the case discussed in section 4.5.2 which led to a domain solution with an asymptotically constant volume given by eq. (4.32). This asymptotic value is, in fact, identical to our four-dimensional result (5.15), as one would expect. Of course, FS W [negationslash]= 0 for this value of v so that we
do not have a full solution to the F-term conditions but, rather, a runaway in the dilaton direction. The simplest solution for this type of potential is a domain wall which is precisely what we have found previously from a 10-dimensional viewpoint.
5.5 Including a gaugino condensate
We will now attempt to lift the dilaton runaway by adding a gaugino condensate term to the superpotential, so that W in eq. (5.12) is replaced by
W = T + + kecS . (5.16)
28
Here, is dened in eq. (5.6), k is a constant of order one and c is a constant depending on the condensing gauge group, with typical values cSU(5) = 2/5, cE6 = 2/12, cE7 = 2/18 and cE8 = 2/30. In the following, it will be useful to introduce the re-scaled components
x = cs , y = ca (5.17)
of the dilaton supereld. With those variables, the dilaton F-term equations, FS = 0,
then read
v + + (1 + 2x)kexcos(y) = 0 (5.18) b (1 + 2x)kexsin(y) = 0 , (5.19)
while FT = 0 leads to
v + 3 + 3kexcos(y) = 0 (5.20) b kexsin(y) = 0 . (5.21)
The vanishing of the superpotential, W = 0, is equivalent to the conditions
v + + kexcos(y) = 0 (5.22) b kexsin(y) = 0 . (5.23)
The simplest type of vacuum is a supersymmetric Minkowski vacuum, that is a solution of FS = FT = W = 0. It is easy to see that this can only be achieved for s = 0 which corresponds to the limit of innite gauge coupling at the string scale and is, therefore, discarded.
Next, we should consider supersymmetric AdS vacua, which are stable by the Breitenlohner-Freedman criterion. These are solutions of FS = FT = 0. It follows immediately that the axions are xed by cos(y) = sign(k) and b = 0 while x and v are
determined by
f(x) (1 x)ex =
k , v =
JHEP01(2013)015
3x1 x
. (5.24)
Normally, we require a solution with x > 1 in order to be at su ciently weak coupling and we will focus on this case. Then, for a positive v we need the ux parameter to be negative and, hence, the constant k to be positive. A negative value for is indeed possible for SU(3)/U(1)2 but not for Sp(2)/SU(2) [notdef] U(1). Provided this choice of signs,
the equations (5.24) have two solutions, one with a value of x satisfying 1 < x < 2 which is an AdS saddle and another one with x > 2 which is an AdS minimum. The cosmological constant at those vacua is given by
=
3c2
4v3x
1 + x 1 x
2. (5.25)
We note that v is stabilized perturbatively while stabilization of the dilaton involves the gaugino condensation term. It has of course been observed some time ago [51] that the dilaton in heterotic CY compactications can be stabilized by a combination of a constant, arising from H-ux, and gaugino condensation in the superpotential. The situation here is di erent from these early considerations in two ways.
29
Figure 2. Plot of the consistent values for [notdef][notdef]. The shaded part is dened by the conditions (5.26)
and (5.27). The other three lines represent the condition (5.29) for values kmax = 10 (bottom line),
kmax = 20 (middle line) and kmax = 100 (top line). Consistent values for the ux [notdef][notdef] are, hence,
dened by the shaded part located below the line for the value of kmax under consideration.
There is an additional T-dependent term in the superpotential which arises from the
non-vanishing torsion of the internal space.
The ux term in the superpotential does not arise from harmonic H-ux but from
bundle ux.
It is important to check that the above vacuum can be in a acceptable region of eld space where all consistency conditions are satised. To discuss this we set [prime] to one from hereon.
We need that s > 1 to be at weak coupling, v 1 so that the [prime] expansion is sensible,
k exp(x) < 1 so that the condensate is small and [notdef] [notdef] < 1 for a small vacuum energy.
eqs. (5.24) immediately point to a tension in satisfying the rst two of these constraints. While v is proportional to the bundle ux and, hence, prefers a large value of , a large value of the dilaton requires to be small.
Let us consider this in more detail. For concreteness we use a minimum value of v = 9, a su ciently large value for the [prime] expansion to be sensible. This implies the constraint
|[notdef]
3(x 1)
x (5.26)
on the ux . We also require the non-perturbative e ects to be weak, that is k exp(x) <
1, which leads to the condition
|[notdef] x 1. (5.27)
Combining both conditions, it follows that x 3 and then < 1. Hence, the two condi
tions (5.26) and (5.27) are necessary and su cient to guarantee a consistent vacuum.
There is a further condition, concerning the constant k in the gaugino condensation term, whose value for a given vacuum is given by
k = [notdef][notdef]ex
x 1
. (5.28)
JHEP01(2013)015
30
The general expectation is for k not to be too large, so requiring it to be less than some maximum value kmax implies
|[notdef] kmax(x 1)ex . (5.29)
Figure 2 shows the restriction on [notdef][notdef] for di erent values of kmax. We see that simultaneous solutions to (5.26), (5.27) and (5.29) only exist if kmax 20. For kmax = O(100) the consistent ux values are in the range 2 [notdef][notdef] 4.
5.6 Supersymmetric AdS example
We would now like to show that the required values for the ux can indeed be obtained for appropriate choices of the gauge bundle. On the coset SU(3)/U(1)2 we choose observable and hidden line bundle sums dened by the parameters
(pi) = (2, 0, 0, 0, 2) (qi) = (1, 2, 1, 2, 2)
(~pi) = (2, 2, 0, 2, 2) (~qi) = (3, 4, 1, 4, 4) .
For this choice, the anomaly constraints (4.3) and (4.4) are satised and the chiral asymmetry in the observable sector is three. Since both line bundle sums have rank ve the gauge group in both sectors is S(U(1)5) [notdef] SU(5). Computing the ux = B from eq. (4.23) for
this bundle choice leads to
=
15
16 2.95. (5.30)
This value is negative, as required, and indeed within the consistent range for [notdef][notdef]. Both the
AdS saddle and the AdS minimum can be realized for this value of , as can also be seen from gure 3. Many more consistent examples can be found on the coset SU(3)/U(1)2.
However, the situation is di erent for Sp(2)/SU(2) [notdef] U(1). In this case, a line bundle is
specied by a single integer and anomaly cancellation already xes = /2. Since this value is positive it leads to x < 1 so that weak coupling is di cult to achieve.
5.7 Search for non-supersymmetric vacua
We conclude the section by adding some remarks regarding non-supersymmetric vacua. A general search for non-supersymmetric vacua for SU(3)/U(1)2 becomes di cult due to the presence of four complex moduli. However, one can perform an exhaustive search at the nearly Kahler locus the locus of vanishing D-terms where only two moduli, S and T , remain as at directions. On this locus, the scalar potential from the Kahler potential (5.12) and the superpotential (5.16), after minimizing and integrating out the axion directions, is given by
V /
1 sv3
JHEP01(2013)015
2 2v 53v2 + (2x + 1)2k2e2x [notdef] 2(2vx v + + 2x)kex[parenrightbigg]. (5.31)
The sign of the last term equals the value of cos(y) = [notdef]1. A contour plot of the potential for
specic values of < 0 , k > 0 and cos(y) = 1 (ensuring the existence of a supersymmetric
AdS vacuum) is given in gure 3 and the two supersymmetric vacua, one AdS minimum and
31
JHEP01(2013)015
Figure 3. Contour plot of the potential (5.31) with cos(y) = 1 for k = 53.4, = 15/16. This
potential has a supersymmetric AdS minimum at (x, v) [similarequal] (4, 11.8), and also a supersymmetric AdS
saddle at (x, v) [similarequal] (1.18, 58).
on AdS saddle, are clearly visible. For the choice k > 0 and cos(y) = +1 no supersymmetric vacua exist but we nd two classes of non-supersymmetric extrema, which can be both either dS and AdS, depending on the values of k, . Checking the Breitenlohner-Freedman criterion, we nd that all these non-supersymmetric extrema are unstable. This means that at the locus of vanishing D-terms (the nearly Kahler locus) only supersymmetric stable AdS vacua exist. It is still conceivable that stable non-supersymmetric vacua exist away from the nearly Kahler locus, but our attempts to nd such vacua have remained unsuccessful. This seems to agree with recent ndings in ref. [47] where compactications on SU(3)/U(1)2 have been studied from a slightly di erent point of view.
6 Discussion and outlook
In this paper, we have studied heterotic domain wall compactications on half-at manifolds, with particular emphasis on the inclusion of [prime] corrections and moduli stabilization.
In particular, we have tried to address the question as to whether the domain wall can be lifted to a maximally symmetric vacuum via stabilization of all moduli. For the examples studied the answer is a cautious yes. A combination of [prime] and non-perturbative e ects can indeed lift the runaway directions of the original, lowest-order perturbative potential and lead to a supersymmetric AdS vacuum. For appropriate bundle choices this stabilization does arise in a consistent part of moduli space, that is, at weak coupling and for
32
moderately large internal volume. However, there is a tension in that it is not possible, for the specic examples analysed, to make the volume very large (so that there is no doubt about the validity of the [prime] expansion) and keep the theory at weak coupling.
An explicit study of [prime] corrections and the required construction of gauge elds requires an explicit and accessible set of half-at manifolds. For this reason, we have focused on the coset spaces which admit half-at structures and, specically, on SU(3)/U(1)2 which provides the greatest exibility among those cosets for building gauge elds via the associated bundle construction. Following ref. [26], we have constructed explicit gauge bundles consisting of sums of line bundles. The conditions for these gauge elds to be supersym-metric the D-term conditions from a four-dimensional point of view x two of the three T-moduli, thereby restricting the half-at structure to be nearly Kahler. We have shown that the anomaly condition can be satised for appropriate bundle choices and we have solved the Bianchi identity explicitly for such choices. This results in a non-harmonic H-ux, induced by the bundle ux, which leads to a correction to the metric and the dilaton prole at order [prime]. These corrections preserve the nearly Kahler structure on the coset space.
From a four-dimensional point of view, the bundle-induced H-ux leads to an additional, constant term in the superpotential. This term can stabilize the remaining T-modulus but the dilaton is still left a runaway direction. Upon inclusion of gaugino condensation all moduli can indeed be stabilized in a supersymmetric AdS vacuum.
These results provide the rst concrete indication that maximal symmetry at lowest order in a string solution might not be a necessary condition for a physically acceptable vacuum. This, in turn, would mean that much larger classes of internal manifolds, such as half-at manifolds and their generalizations, are relevant in string phenomenology. A central question in this context is, of course, how the domain wall tension, essentially set by the torsion of the manifold, can be made su ciently small so that other e ects can compete and lift the vacuum. In our examples, this can be arranged at a marginal level by a choice of gauge bundles, although it is not possible to stabilize the theory at parametrically large volume. However, it has to be kept in mind that the coset spaces under consideration have a rather limited pattern of torsion and ux parameters available. It remains to be seen whether other half-at manifolds o er more exibility in this regard.
Acknowledgments
We would like to thank Andrei Constantin, Sao Grozdanov and James Sparks for useful discussions. M. K. is supported by a Lamb & Flag scholarship of St Johns College Oxford and by an STFC scholarship. A. L. is supported by the EC 6th Framework Programme MRTN-CT-2004-503369 and by the EPSRC network grant EP/l02784X/1. E. E. S. is supported by a Oxford University Clarendon scholarship.
A Conventions and SU(3)-structures
In this appendix we summarize our conventions and provide a brief review of the SU(3) structure formalism and the various classes of SU(3)-structure manifolds relevant to us.
33
JHEP01(2013)015
A.1 Conventions
We decompose ten-dimensional space-time as M2,1 [notdef] R [notdef] X, where M2,1 is the three-
dimensional Minkowski space, R denotes the y-direction transverse to the domain wall, and X is the compact coset space. The index conventions are then
10d : M, N, . . . = 0, 1, . . . , 97d : m, n, . . . = 3, 4, . . . , 96d : u, v, . . . = 0, 1, . . . , 9 (A.1) 4d : , , . . . = 0, 1, . . . , 33d : , , . . . = 0, 1, 2.
Group indices are denoted by
G : A, B, . . . = 1, 2, . . . , dim(G),G/H : a, b, . . . = 1, 2, . . . , 6 (A.2)
H : i, j, . . . = 7, 8, . . . , dim(G), .
Note that the indices a, b, . . . correspond to the vielbein frame of the six-dimensional internal geometry labeled by the above u, v, . . . indices.
A.2 SU(3)-structures
A six-dimensional manifold X has an SU(3)-structure if there exists a real two-form J and a complex three-form satisfying the relations
J ^ J ^ J =
3
4i ^
JHEP01(2013)015
, ^ J = 0, (A.3)
where both sides of the rst equation are non-zero everywhere.
For dJ = d = 0, the above SU(3)-structure reduces to an SU(3)-holonomy for X. In general, J and are not closed and the deviation from SU(3)-holonomy is measured by the intrinsic torsion which transforms in the SU(3) representation
2 (1 + 1) (8 + 8) (6 +
6) (3 +
3) (3 +
3) . (A.4)
The ve irreducible parts of this representation correspond to the ve torsion classes Wi,
i = 1, . . . , 5. They can also be explicitly read o from dJ and d via the relations
dJ =
3 2Im(W1
) + W4 ^ J + W3, (A.5)
d = W1J ^ J + W2 ^ J +
W5 ^ , (A.6)
where
W3 ^ J = W3 ^ = W2 ^ J ^ J = 0, (A.7)
34
in order for the SU(3)-relations (A.3) to be satised. For a given SU(3)-structure (J, ) there is a unique SU(3)-invariant metric g and an associated almost complex structure
J vu = gvwJuw. This almost complex structure is integrable i W1 = W2 = 0.
Some specic classes of SU(3)-structures, relevant for the present paper, are characterized as follows.
nearly Kahler 2 W1, almost Kahler 2 W2,
Kahler 2 W5,half-at 2 W +1 W +2 W3 (A.8)
where the subscript, +, denotes the real part of the torsion classes. Since W1 and W2 are non-zero the above classes of manifolds are, in general, not complex.
B The coset spaces
This appendix provides a short summary of all relevant data for the coset spaces considered in this paper, namely SU(3)/U(1)2, Sp(2)/SU(2) [notdef] U(1) and G2/SU(3). More details and
derivations can be found in ref. [26] and references therein. Although the space G2/SU(3) does not seem to allow for phenomenologically interesting models in our context, it is included for completeness. The data given here includes the generators of the Lie-group, relevant topological data and the half-at mirror structure dened by the two-forms [notdef]!i[notdef], their four-form duals [notdef]~
!i[notdef] and the symplectic set [notdef] 0, 0[notdef]. In accordance with our index
convention (A.1), the reductive decomposition of the Lie algebra of G is given by [notdef]TA[notdef] =
{Ka, Hi[notdef], where the Ka, a = 1, . . . , 6 denote the coset generators and Hi the generators of
the sub-group H.
B.1 SU(3)/U(1)2
This coset is isomorphic to F3, the space of ags of C3. It is also the twistor space of CP 2 and has been studied extensively in the mathematical literature. Of particular interest is the fact that it admits two almost complex structures, one of which is integrable and the other nearly Kahler. This is true in general for every six-dimensional manifold that is the twistor space of a four-dimensional manifold [48]. The latter is induced by the coset structure of SU(3)/U(1)2 and given below.
A possible choice of SU(3) generators is provided by the Gell-Mann matrices
1 =
i 2
JHEP01(2013)015
0
B
@
0 1 0 1 0 0 0 0 0
1[parenleftBigg], 2 =12 0
B
@
0 1 0
1 0 0 0 0 0
1
C
A
, 3 =
i 2
0
B
@
1 0 0 0 1 0
0 0 0
1
C
A
,
4 =
i 2
0
B
@
0 0 1 0 0 0 1 0 0
1[parenleftBigg], 5 =12 0
B
@
0 0 1
0 0 0
1 0 0
1
C
A
, 6 =
i 2
0
B
@
0 0 0 0 0 1 0 1 0
1
C
A
,
7 =12 0
B
@
0 0 0 0 0 1
0 1 0
1
C
A
, 8 =
i 2p3 0
B
@
1 0 0 0 1 0 0 0 2
1
C
A
.
35
The two U(1) sub-groups are generated by 3 and 8. Hence, we choose as generators the re-labelled Gell-Mann matrices
K1 = 1 K2 = 2 K3 = 4 K4 = 5
K5 = 6 K6 = 7 H7 = 3 H8 = 8 .
(B.1)
The geometry of the homogeneous space SU(3)/U(1)2 is determined by the structure constants which, relative to the basis [notdef]Ka, Hi[notdef], are given by
f 7
12 = 1 f 6
13 = f
514 = f
523 = f
624 = f
473 = f
6
75 = 1/2 (B.2)
f 8
34 = f
JHEP01(2013)015
856 = p3/2 .
A basis of G-invariant two-, three- and four-forms is given by
!1 =
1
2
e12 + 12e34 12e56[parenrightBig]~!1 = 4 3V0
2e1234 + e1256 e3456[parenrightBig]
!2 =
1 4
e12 + e34
[parenrightBig]
~!2 = 4
V0
e1234 + e1256
[parenrightBig]
!3 = 1
3
e12 e34 + e56[parenrightBig]
~!3 =
0 e1234 e1256 + e3456[parenrightBig]
0 =
2V0
e136 e145 + e235 + e246[parenrightBig] 0 = 1 2
e135 + e146 e236 + e245[parenrightBig]
(B.3)
where ei1...in := ei1 ^ [notdef] [notdef] [notdef] ^ ein and the dimensionless volume V0 is given by
V0 =
ZX e123456 = 4(2)3 . (B.4)
This G-invariant basis forms full the half-at mirror relations in section 2.4 with torsion parameters (e1, e2, e3) = (0, 0, 1) and intersection numbers
d111 = 6 d112 = 3 d113 = 4 d122 = 1 d123 = 2 d133 = 0
d222 = 0 d223 = 43 d233 = 0 d333 =
64
9 .
(B.5)
The only non-zero Betti numbers are b0 = 1, b2 = 2, b4 = 2 and b6 = 1 so that the Euler number is ~ = 6. The most general G-invariant SU(3) structure forms are given by
J = R21e12 R22e34 + R23e56 = vi!i, = R1R2R3
(e136 e145 + e235 + e246) + i (e135 + e146 e236 + e245)[parenrightBig]
= Z 0 + i G 0
(B.6)
with associated G-invariant metrics
ds20 = R21 (e1 e1 + e2 e2) + R22 (e3 e3 + e4 e4) + R23 (e5 e5 + e6 e6) . (B.7)
36
In these relations, the Ri are three arbitrary radii of the coset space which are related to the moduli vi by
v1 =
4
3 (R21 + R22 2R23) (B.8)
v2 = 4(R22 R23) (B.9)
v3 = (R21 + R22 + R23) (B.10)
and to (Z, G) by
Z = 2V0
R1R2R3 , G = 2R1R2R3 . (B.11)
B.2 Sp(2)/SU(2) U(1)
As a topological space this coset is isomorphic to CP 3. Another coset realisation of CP 3 is
SU(4)/S(U(3) [notdef] U(1)) which may be more familiar to the reader.
CP 3 is the twistor space of S4 and, therefore, admits two almost complex structures: one integrable and the other nearly Kahler. The rst corresponds to the invariant structure on the coset SU(4)/S(U(3)[notdef]
U(1)) while the latter corresponds to the invariant structure on Sp(2)/SU(2) [notdef] U(1) and
is given below.A possible choice for the generators of the Lie-group Sp(2) is
K1 = 1 p2
0
B
B
B
@
JHEP01(2013)015
0 0 1 0 0 0 0 1
1 0 0 0
0 1 0 0
1
C
C
C
A
, K2 = i p2
0
B
B
B
@
0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0
1
C
C
C
A
,
K3 =
0
B
B
B
@
i 0 0 0 0 i 0 0
0 0 0 0 0 0 0 0
1
C
C
C
A
, K4 =
0
B
B
B
@
0 1 0 0
1 0 0 0
0 0 0 0 0 0 0 0
1
C
C
C
A
, K5 = 1 p2
0
B
B
B
@
0 0 0 1 0 0 1 0
0 1 0 0
1 0 0 0
1
C
C
C
A
K6 = i p2
0
B
B
B
@
0 0 1 0
0 0 0 1
1 0 0 0
0 1 0 0
1
C
C
C
A
, H7 =
0
B
B
B
@
0 0 0 0 0 0 0 0 0 0 i 0 0 0 0 i
1
C
C
C
A
,
H8 =
0
B
B
B
@
0 0 0 0 0 0 0 0 0 0 0 1
0 0 1 0
1
C
C
C
A
, H9 =
0
B
B
B
@
0 0 0 0 0 0 0 0 0 0 0 i
0 0 i 0
1
C
C
C
A
, H10 =
0
B
B
B
@
0 i 0 0 i 0 0 0 0 0 0 0 0 0 0 0
1
C
C
C
A
.
There are two possible reductive decompositions of Sp(2) leading to two di erent cosets. The decomposition into [notdef]Ka, Hi[notdef] given above corresponds to the non-maximal embedding
of SU(2)[notdef]U(1). The other choice, the maximal embedding, leads to a di erent coset which
does not admit a half-at Sp(2)-invariant SU(3)-structure.
37
The structure constants in the given basis are
f 6
13 = f
514 = f
523 = f
624 = 1
f 6
71 = f
572 = f
581 = f
682 = f
291 = f
695 = f
210 1 = f
610 5 = 1 (B.12)
f 9
78 = f
410 3 = 2 .
A basis of G-invariant two-, three- and four-forms is given by
!1 = 1 2
e12 + 2e34 + e56
[parenrightBig]
~!1 = 3V0
e1234 + 2e1256 + e3456
[parenrightBig]
JHEP01(2013)015
!2 = 1
6
e12 e34 + e56[parenrightBig]
~!2 = 2
0 e1234 e1256 + e3456[parenrightBig]
0 =
2V0
e136 e145 + e235 + e246[parenrightBig] 0 = 1 2
e135 + e146 e236 + e245[parenrightBig]
(B.13)
with
ZX e123456 =(2)312 . (B.14)
As before, these G-invariant forms satisfy the half-at mirror geometry relations in section 2.4 for torsion parameters (e1, e2) = (0, 1) and intersection numbers
d111 = 1 d112 = 16 d122 = 0 d222 =
227 . (B.15)
The only non-zero Betti numbers are b0 = b2 = b4 = b6 = 1 and, hence, the Euler number is ~ = 4. The most general G-invariant SU(3)-structure forms are
J = R21 e12 R22 e34 + R21 e56 = vi!i = R21R2
(e136 e145 + e235 + e246) + i (e135 + e146 e236 + e245)[parenrightBig]
= Z 0 + i G 0
(B.16)
V0 =
with associated G-invariant metrics
ds20 = R21 (e1 e1 + e2 e2) + R22 (e3 e3 + e4 e4) + R21 (e5 e5 + e6 e6) . (B.17)
The two coset radii Ri are related to half-at mirror moduli by
v1 =
2
3 (R21 R22) (B.18)
v2 = 2(2R21 + R22) (B.19)
and
Z = 2V0
R21R2 G = 2 R21R2 . (B.20)
38
B.3 G2/SU(3)
This coset is topologically a sphere
G2/SU(3)
= S6 . (B.21)
Like the the other spaces the sphere admits di erent realisations as coset, for example SO(7)/SO(6)
= S6. However, in contrast to the other cases there is no known integrable almost complex structure on S6. The conjecture that no such almost complex structure exists is known as Cherns last theorem. There is a well known nearly Kahler structure on S6 which arises from the octonions (the sphere S6 can be regarded as a subset of the octonions) and is invariant under the action of G2. This structure will be presented below.
Our choice of G2 generators and their reductive decomposition is
K1 = 1 p3
0
B
B
B
B
B
B
B
B
B
B
@
0 2 0 0 0 0 0
2 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0
0 0 0 1 0 0 0
1
C
C
C
C
C
C
C
C
C
C
A
, K2 = 1 p3
0
B
B
B
B
B
B
B
B
B
B
@
0 0 2 0 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0 0 1
0 0 0 1 0 0 0
0 0 0 0 1 0 0
1
C
C
C
C
C
C
C
C
C
C
A
,
JHEP01(2013)015
K3 = 1 p3
0
B
B
B
B
B
B
B
B
B
B
@
0 0 0 0 2 0 0
0 0 0 0 0 1 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 1 0 0 0 0 0
0 0 1 0 0 0 0
1
C
C
C
C
C
C
C
C
C
C
A
, K4 = 1 p3
0
B
B
B
B
B
B
B
B
B
B
@
0 0 0 2 0 0 0
0 0 0 0 0 0 1 0 0 0 0 0 1 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
0 1 0 0 0 0 0
1
C
C
C
C
C
C
C
C
C
C
A
,
K5 = 1 p3
0
B
B
B
B
B
B
B
B
B
B
@
0 0 0 0 0 0 2 0 0 0 1 0 0 0 0 0 0 0 1 0 0
0 1 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0
1
C
C
C
C
C
C
C
C
C
C
A
, K6 = 1 p3
0
B
B
B
B
B
B
B
B
B
B
@
0 0 0 0 0 2 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 2 0 0 0 0 0 0
0 0 0 0 0 0 0
1
C
C
C
C
C
C
C
C
C
C
A
,
H7 =
0
B
B
B
B
B
B
B
B
B
B
@
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0 0 0 0 1 0 0
0 0 0 1 0 0 0
1
C
C
C
C
C
C
C
C
C
C
A
, H8 =
0
B
B
B
B
B
B
B
B
B
B
@
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 1
0 0 0 1 0 0 0 0 0 0 0 1 0 0
1
C
C
C
C
C
C
C
C
C
C
A
,
39
H9 =
0
B
B
B
B
B
B
B
B
B
B
@
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0
1
C
C
C
C
C
C
C
C
C
C
A
, H10 =
0
B
B
B
B
B
B
B
B
B
B
@
0 0 0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0
1
C
C
C
C
C
C
C
C
C
C
A
,
H11 =
0
B
B
B
B
B
B
B
B
B
B
@
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0
1
C
C
C
C
C
C
C
C
C
C
A
, H12 =
0
B
B
B
B
B
B
B
B
B
B
@
0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0
0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0
1
C
C
C
C
C
C
C
C
C
C
A
,
H13 =
0
B
B
B
B
B
B
B
B
B
B
@
0 0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 1 0 0 0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1
C
C
C
C
C
C
C
C
C
C
A
, H14 = 1 p3
0
B
B
B
B
B
B
B
B
B
B
@
0 0 0 0 0 0 0 0 0 2 0 0 0 0
0 2 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0 0 0 0 0 1 0
1
C
C
C
C
C
C
C
C
C
C
A
.
JHEP01(2013)015
The structure constants in this basis read
f 13
7 10 = f
127 11 = f
6
73 = f
574 = 1
f 12
8 10 = f
138 11 = f
583 = f
684 = f
119 10 = f
139 12 = f
493 = f
695 = 1
f 6
10 1 = f
510 2 = f
511 1 = f
611 2 = f
412 1 = f
312 2 = f
313 1 = f
413 2 = 1
f 14
10 11 = f
1412 13 = p3, f
978 = 2 (B.22)
f 2
14 1 = f
613 = f
514 = f
523 = f
624 = 2/p3
614 5 = 1/p3 .
A basis of G-invariant two-, three- and four-forms is given by
!1 = 5
3
[parenleftBig]
e12 + e34 + e56
f 4
14 3 = f
[parenrightBig]
!1 =
5V0
~
e1234 + e1256 e3456[parenrightBig]
0 = p3
40V0
e136 + e145 e235 + e246[parenrightBig] 0 = 10p3 [parenleftBig]e135 e146 + e236 + e245[parenrightBig].
(B.23)
where
ZX e123456 =9(2)320 . (B.24)
These G-invariant forms satisfy the half-at mirror relations in section 2.4 with torsion parameter e1 = 1 and intersection number d111 = 100. The non-vanishing Betti numbers
40
V0 =
are b0 = b6 = 1 which yields the Euler number ~ = 2, as we would have expected from G2/SU(3)
= S6.
The most general G-invariant SU(3)-structures are
J = R2e12 + R2e34 + R2e56 = v !1 (B.25) = R3
(e136 e145 + e235 + e246) + i (e135 + e146 e236 + e245)[parenrightBig]
= Z 0 + i G 0 . (B.26)
with associated metrics
ds20 = R2 (e1 e1 + e2 e2) + R2 (e3 e3 + e4 e4) + R2 (e5 e5 + e6 e6) . (B.27)
The single coset radius R is related to the half-at mirror moduli by
v = 3
5 R2 (B.28)
and
Z = 40V0
p3 R3 G =
C Bianchi identity and related computations
This section gives a summary of the calculation involved in solving the Bianchi identity (2.4) for the three homogeneous spaces considered. We will rst focus on the connection on the tangent bundle and the computation of tr R ^ R. Then we will present the results for
tr F ^F . Finally, we insert everything into the Bianchi identity and determine the constant C in the Ansatz (4.14) for H.
C.1 tr R^ R
For all three spaces we have the metric of the form
ds20 = R21 (e1 e1 + e2 e2) + R22 (e3 e3 + e4 e4) + R23 (e5 e5 + e6 e6) . (C.1)
Since we are interested in performing the calculation at the nearly Kahler locus we set R R1 = R2 = R3 so that the metric becomes the same for all three spaces. R is
calculated from the Hull connection
! ab = ! ab
acb = Hcbdgda. Furthermore, ! is the Levi-Civita connection
acb ec + f
41
JHEP01(2013)015
p3
10 R3 . (B.29)
1
2H
acb ec . (C.2)
where H = 1
given by
3! Habc eabc and H
! ab = 12f
aib "i . (C.3)
Here, the "i are the coset descendants of the left-invariant Maurer-Cartan forms on G in the direction of H. On G/H they can be expressed in terms of the basis forms ea. However, we will not need these relations explicitly since the "i will drop out of the expression for tr R ^ R.
The curvature two-form R is given by
(R) ab = (d!) ab (!) cb ^ (!) ac , (C.4) where the uncommon minus sign stems from our index convention for the connection one-form.
The H-ux in our solution is proportional to 0 for each example. Using the structure constants given in appendix B and the denition for 0, this means that we can write for each coset
Habc = C f
dab dc . (C.5)
with a constant C.
Let us now state the result for each case.
C.1.1 SU(3)/U(1)2
Here, the Ansatz (C.5) for the H-ux takes the form
H = C(R, [prime]) V0 0 (C.6)
Evaluating tr R^ R at the nearly Kahler locus gives
tr R^ R =
3
4
JHEP01(2013)015
[parenleftbigg]
C2
R4 2 C
R2 3
[parenrightbigg]
0 !3 . (C.7)
Recall that ~
!3 = d 0 and, hence, this lies in the trivial cohomology class of H4(X). Consequently, the rst Pontryagin class of SU(3)/U(1)2 is p1(T X) = 0.
C.1.2 Sp(2)/SU(2) U(1)
Here, the Ansatz (C.5) for the H-ux reads
H = C(R, [prime])
2 V0
0 (C.8)
and we nd
tr R^ R = [parenleftbigg]
48 ~
!1 [parenleftbigg]
6 C2
R4 12 C
R2 10
[parenrightbigg]
~
!2
[parenrightbigg]
V0
. (C.9)
Unlike for SU(3)/U(1)2, this this represents a non-trivial cohomology class of H4(X) given by
p1(T X) =
1 8 2
tr R^ R
[bracketrightbig]= 4 ~!1 . (C.10)
42
C.1.3 G2/SU(3)
The Ansatz (C.5) for the H-ux is
H = C(R, [prime])
80V0
3 0 . (C.11)
so that
tr R^ R =
64
3
2C
R2 C2R4[parenrightbigg]5V0 ~!1 . (C.12)
Since d 0 = ~
!1 the rst Pontryagin class is again trivial, p1(T X) = 0.
C.2 tr F ^ FA line bundle L over the coset G/H is dened by the dim H2(G/H) integer numbers p = (pr) such that its rst Chern class is given by c1(L) = pr!r. Such a line line bundle is also denoted by L = OX(p). The curvature of a connection on L is given by
F = (2i) pr!r . (C.13)
The vector bundles we construct are direct sums of line bundles
V =
n
Ma=1
JHEP01(2013)015
OX (pa) , (C.14)
and are, hence, characterized by an integer matrix (pra). We impose that c1(V )
Pa pa = 0 so that the structure group of V is S(U(1)n). Using the mirror half-at geometric structure, we can express tr F ^ F in terms of the intersection numbers by
tr F ^ F = 42 drst
n
Xa=1psapta ~!r . (C.15)
C.2.1 SU(3)/U(1)2
Here, there are two integers dening every line bundle and, for ease of notation we write (pa , qa) = (p1a , p2a). The intersection numbers are given in eq. (B.5) and from direct evaluation of (C.15) we nd
tr F ^ F = V0
8
[bracketleftbigg] [summationdisplay]
a
(6p2a + q2a + 6paqa)~
!1 +
Xapa(3pa + 2qa)~ !2
+ 43
Xa(3p2a + q2a + 3paqa)~ !3
[bracketrightbigg]
(C.16)
This means that the second chern class of the bundle is
ch2(V ) =
182 tr F ^ F =
1
2
[bracketleftBigg][summationdisplay]
a
(C.17)
Note, in general, this represents a di erent cohomology class than tr R ^ R so that solving the Bianchi identity imposes restrictions on the bundle integers (pa, qa).
43
(6p2a + q2a + 6paqa)~
!1 +
Xapa(3pa + 2qa)~ !2
[bracketrightBigg]
C.2.2 Sp(2)/SU(2) U(1)
Here, a line bundle is dened by a single integer and we write pa = p1a. Inserting the intersection numbers from (B.15) into eq. (C.15) we obtain
tr F ^ F = V0Xap2a(6 ~!1 + ~!2) (C.18)
Hence, the second Chern class of the bundle is given by
ch2(V ) =
182 tr F ^ F =
1
2
Xap2a ~!1 . (C.19)
C.2.3 G2/SU(3)
In this case, the second Betti number is zero and, hence, there are no non-trivial line bundles on this coset space. However, it is still possible to solve the Bianchi identity with non-Abelian gauge bundles. An obvious choice is the (quasi) standard embedding as described in ref. [26]. This choice has already been studied in the early work [38] where it was realised that the Dirac index of such bundles is
ind(Vstandard) = 12~ = 1 , (C.20)
implying one chiral family only. Another possible choice is the natural G-invariant connection [26] which yields a rank three bundle and solves the Hermitean Yang-Mills equations. However, this vector bundle has a Dirac index of zero and no chiral families are possible.
C.3 Solving the Bianchi identity
We now combine the previous results to solve the Bianchi identity
dH = [prime]
4 (tr F ^ F tr R ^ R) . (C.21)
We will omit the case G2/SU(3) which is of no phenomenological interest in the context of our bundle construction as we have pointed out in the previous section.
We solve the Bianchi identity in three steps. Firstly, the Hermitean Yang-Mills equations are solved for the nearly Kahler locus R Ri, 8 i and we will focus on this case.
Secondly, since dH is exact tr R^R and tr F ^F have to lie in the same cohomology class.
This yield restrictions on the line bundle integers which involve the observable line bundle sum, V =
Ln a=1
OX (pa) and the hidden line bundle sum, ~V =
JHEP01(2013)015
OX (~pa). Thirdly, using these restrictions, we compute both sides of the Bianchi identity and determine the unknown constant C in the Ansatz (C.5) for H.
C.3.1 SU(3)/U(1)2
First, we note that the cohomology class of tr R^ R in eq. (C.7) is trivial. This means
that the class of tr F ^F in eq. (C.17) needs to be trivial as well which leads to the conditions
n
Xa=1(6p2a + q2a + 6paqa) +
L a=1
Xa=1(6~p2a + ~q2a + 6~pa~qa) = 0 (C.22)
n
Xa=1pa(3pa + 2qa) +
Xa=1~pa(3~pa + 2~qa) = 0 . (C.23)
44
Together with the Ansatz (C.5) for the ux H, the Bianchi identity reduces to a quadratic equation
C =
[prime] 4
A +3 4
C2R4 + 2 C
R2 + 3[parenrightbigg][parenrightbigg]. (C.24)
for C where
A(p, q, ~p, ~q) =
1 6
[bracketleftBigg]
n
Xa=1(3p2a + q2a + 3paqa) +
Xa=1(3~p2a + ~q2a + 3~pa~qa)
[bracketrightBigg]
. (C.25)
JHEP01(2013)015
Its positive solution is
C(R, [prime]) = 8R43 [prime] 1 +
3 8
[prime] R2 +
r1 34 [prime]
R2 +3A + 916 [prime]2 R4
[parenrightBigg]
. (C.26)
In the large radius limit, [prime]/R2 1, this solution can be expanded as
C(R, [prime]) =
B +(27 + 12A)128 [prime]
R2 (27 + 24A + 16A2)4096 [prime]2R4 + O
[prime]3R6[parenrightbigg][bracketrightbigg] [prime] , (C.27)
where B = (4A + 9)/16. From this we obtain the H-ux relevant for the four-dimensional
theory is
H = 0 where = V0 [prime]
B . (C.28)
C.3.2 Sp(2)/SU(2) U(1)
Comparing the cohomology classes of tr R^R from eq. (C.9) and tr F ^F from eq. (C.19),
we see that equality implies
n
Xa=1p2a +
Xa=1~p2a = 8 , (C.29)
Together with the Ansatz (C.5) for H the Bianchi identity reduces to the quadratic equation
C =
[prime] 8
A + 10 + 12 C
R2 6 C2 R4[parenrightbigg]
(C.30)
for C where
Xa=1~p2a = 8 . (C.31)
Even though A is only a constant, we will still include it in the expressions below in order
to have a notation similar to the rest of the paper.
Its positive solution is
C(R, [prime]) = 2R43 [prime] 1 +
3 2
A(p, ~p) =
n
Xa=1p2a
[prime] R2 +
r1 3 [prime]
R2 +3 [prime]2 R4
[parenrightBigg]
. (C.32)
45
This can once more be expanded in the large radius limit, [prime]/R2 1, which gives
C(R, [prime]) =
[prime]2
R4[parenrightbigg][bracketrightBigg] [prime] , (C.33)
where A = 8 and B = (A + 10)/4 = 1/2. From this we obtain the H-ux as
H = 0 where = V0 [prime]
B . (C.34)
D Index of the Dirac operator
The chiral asymmetry of the e ective four-dimensional theory is given by the index of the Dirac operator and it is, therefore, expected to count the net number of families. Hence, its knowledge is an important phenomenological constraint. In this section we will derive an expression for the index for a sum of line bundles.
On a six dimensional manifold the index is given by [49]
ind(V ) =
ZX
1 6drstprpspt +
1 6drst
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SISSA, Trieste, Italy 2013
Abstract
We study heterotic string compactifications on nearly Kähler homogeneous spaces, including the gauge field effects which arise at order [alpha]'. Using Abelian gauge fields, we are able to solve the Bianchi identity and supersymmetry conditions to this order. The four-dimensional external space-time consists of a domain wall solution with moduli fields varying along the transverse direction. We find that the inclusion of [alpha]' corrections improves the moduli stabilization features of this solution. In this case, one of the dilaton and the volume modulus asymptotes to a constant value away from the domain wall. It is further shown that the inclusion of non-perturbative effects can stabilize the remaining modulus and "lift" the domain wall to an AdS vacuum. The coset SU(3)/U(1)^sup 2^ is used as an explicit example to demonstrate the validity of this AdS vacuum. Our results show that heterotic nearly Kähler compactifications can lead to maximally symmetric four-dimensional space-times at the non-perturbative level.
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