On the effective field theory of heterotic vacua

The effective field theory of heterotic vacua that realise preserving \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {N}{=}1$$\end{document}N=1 supersymmetry is studied. The vacua in question admit large radius limits taking the form , with a smooth threefold with vanishing first Chern class and a stable holomorphic gauge bundle . In a previous paper we calculated the kinetic terms for moduli, deducing the moduli metric and Kähler potential. In this paper, we compute the remaining couplings in the effective field theory, correct to first order in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha ^{\backprime }\,}$$\end{document}α‵. In particular, we compute the contribution of the matter sector to the Kähler potential and derive the Yukawa couplings and other quadratic fermionic couplings. From this we write down a Kähler potential and superpotential .


Introduction
We are interested in heterotic vacua that realise N =1 supersymmetric field theories in R 3,1 . At large radius, these take form R 3,1 × X where X is a compact smooth complex threefold with vanishing first Chern class. We study the E 8 ×E 8 heterotic string, and so there is a holomorphic vector bundle E with a structure group H ⊂ E 8 × E 8 and a d = 4 spacetime gauge symmetry given by the commutant G = [E 8 ×E 8 , H]. The bundle E has a connection A, with field strength F satisfying the Hermitian Yang-Mills equation. The field strength F is related to a gauge-invariant three-form H and the curvature of X through anomaly cancellation. The triple (X, E, H ) forms a heterotic structure, and the moduli space of these structures is described by what we call heterotic geometry. In this paper, we compute the contribution of fields charged under the spacetime gauge group G to the heterotic geometry.
The challenge in studying heterotic vacua is the complicated relationship between H , the field strength F and the geometry of X. Supersymmetry relates the complex structure J and Hermitian form ω of X to the gauge-invariant three-form H : where x m are real coordinates on X. Green-Schwarz anomaly cancellation gives a modified Bianchi identity for H where in the second of these equations R is the curvature two-form computed with respect to a appropriate connection with torsion proportional to H . This means the tangent bundle T X has torsion if H is nonzero. Unless one is considering the standard embedding-in which E is identified with T X the tangent bundle to X-the right-hand side of (1.2) is nonzero even when X is a Calabi-Yau manifold at large radius. This means that H is generically non-vanishing, though subleading in α , and so even for large radius heterotic vacua X is non-Kähler. Torsion is inescapable. The effective field theory of the light fields for these vacua is described by a Lagrangian with N = 1 supersymmetry, whose bosonic sector is of the form (1.3) Here κ 4 is the four-dimensional Newton constant, R 4 is the four-dimensional Ricci scalar, F g is the spacetime gauge field strength, the A is range over the scalar fields of the field theory, and their kinetic term comes with a metric G AB . The fields A may be charged under g, the algebra of the gauge group G, with an appropriate covariant derivative D e . Finally V ( ,¯ ) is the bosonic potential for the scalars. When E ∼ = T X the moduli space of the heterotic theory reduces to that of a Calabi-Yau manifold and is described by special geometry. The unbroken gauge group in spacetime is E 6 , and the charged matter content consists of fields charged in the 27 and 27 representations. The Yukawa couplings were calculated in supergravity in, for example, [1,2]. The effective field theory of this compactification was described in a beautiful paper [3], in which relations between the Kähler potential and superpotential were computed using string scattering amplitudes, (2,2) supersymmetry and Ward identities. The Kähler and superpotential were shown to be related to each other and in fact were both determined in terms of a pair of holomorphic functions. These are known as the special geometry relations. For a review of special geometry in the language of this paper, see [4]. A key question is how these relations generalise to other choices of bundle E.
We work towards answering this question by computing the effective field theory couplings correct to first order in α . In a previous paper [5] we commenced a study of heterotic geometry using α -corrected supergravity. This is complementary to a series of papers [6][7][8][9][10] who identified the parameter space with certain cohomology groups. In the context of effective field theory (1.3), one of the results of [5] was to calculate the contribution of the bosonic moduli fields to the metric G AB . In this paper, we compute the contribution of the matter sector to the metric G AB , and the Yukawa couplings, correct to order α . We describe an ansatz for the superpotential and Kähler potential for effective field theory: (1.4) The superpotential is normalised by comparing with the Yukawa couplings computed in the dimensional reduction using the conventions of Wess-Bagger [31]. The moduli have a metric ds 2 = 2G αβ dy α ⊗ dy β , where Z α = B α + i∂ α ω is the α -corrected, gauge-invariant generalisation of the complexified Kahler form δ B + iδω, the χ α form a basis of closed (2, 1)-forms, and the last line is the Kobayashi metric, extended to the entire parameter space, including deformations of the spin connection on T X . The metric expressed this way is an inner product of tensors corresponding to complex structure α , Hermitian moduli Z α , and bundle moduli D α A. The role of the spin connection D α θ is presumably determined in terms of the other moduli as they do not correspond to independent physical fields. The tensors depend on parameters holomorphically through de la Ossa and Svanes [6] showed that there exists a choice of basis for the parameters in which each of the tensors in the metric are in an appropriate cohomology; 1 hence, the moduli space metric (5.2) is the natural inner product (Weil-Peterson) on cohomology classes. The matter fields are C ξ and D τ and appear in the Kähler potential trivially, as they do in special geometry. The matter metric is the Weil-Petersson inner product of corresponding cohomology elements where φ ξ , ψ ρ are (0, 1)-forms valued in a sum over representations of the structure group H. In some sense it was remarkable that one was able to find a compact closed expression for the Kähler potential for the moduli metric. This was not a priori obvious, especially given the nonlinear PDEs relating parameters in the anomalous Bianchi identity and supersymmetry relations (1.1)-(1.2). Indeed, it turned out that the Kähler potential for the moduli in (1.4) is of the same in form as that of special geometry, except where one has replaced the Kähler form by the Hermitian form ω. At first sight this is confusing as the only fields appearing in the Kähler potential are ω and . Nonetheless, the Kähler potential still depends on bundle moduli in precisely the right way through a non-trivial analysis of the supersymmetry and anomaly conditions. The Hermitian form ω contains, hidden within, information about both the bundle and Hermitian moduli. 2 The metric (5.2) is compatible with the result in [11], who studied the α and α 2 corrections to the moduli space metric in the particular case where the Hermitian part of the metric varies, while the remaining fields are fixed: (∂ a ω) 1,1 = 0, B a = a = D a A = 0. In general all fields vary with parameters and the metric is nonzero already The analysis in [5] focussed primarily on D-terms relevant to moduli. In this paper, we compute the remaining D-terms, including the metric terms for the bosonic matter fields charged under g. We also compute the F-terms up to cubic order in fields, exploiting the formalism constructed in [5]. The primary utility of this is to derive an expression for the Yukawa couplings in a manifestly covariant fashion. Together with the metrics discussed above, one is now finally able to compute properly normalised Yukawa couplings, relevant to any serious particle phenomenology. The F-terms are protected in α -perturbation theory, and so the only possible α -corrections are due to worldsheet instantons.
The fields neutral under g, the singlet fields, also do not have any mass or cubic Yukawa couplings. In fact, all singlet couplings necessarily vanish. They correspond to moduli which are necessarily free parameters and so the singlets need to have unconstrained vacuum expectation values. If there were a nonzero singlet coupling at some order in the field expansion, e.g. 1 n , or in a e 6 theory (27 ·27) 326 ·1 101 , then some parameter y α would have its value fixed, a contradiction on it being a free parameter. 3 The superpotential W in (1.4) is an ansatz designed to replicate these couplings. Its functional form can be partly argued by symmetry. There is a complex line bundle over the moduli space in which the holomorphic volume form on X, denoted , transforms with a gauge symmetry → μ where μ ∈ C * . The superpotential is also a section of this line bundle, and transforms in the same way W → μW. Hence, W has an integrand proportional to . To make the integrand a nice top-form we need to wedge it with a gauge-invariant three-form. The three-form needs to contain a dependence on the matter fields, and this can only occur through the ten-dimensional H field. The other natural gauge-invariant three-forms that are not defined in a given complex structure are dω and d c ω. W is also required not to give rise to any singlet couplings. So all derivatives of W with respect to parameters must vanish. The combination H − d c ω manifestly satisfies this request. Derivatives with respect to matter fields of W do not vanish. As these are charged in g, the only nonzero contributions come from H . This allows us to fix the normalisation of W by comparing with the dimensional reduction calculation of the Yukawa couplings. Finally, W must be a holomorphic function of chiral fields, which is straightforward to check. It is convenient that the single expression for the superpotential captures both the matter and moduli couplings, and fact seemingly not realised before.
A complementary perspective on W was studied by [8]. In that paper, one starts with an su(3)-structure manifold X, posits the existence of W, and uses it as a device to reproduce the conditions needed for the heterotic vacuum to be supersymmetry. This builds on earlier work in the literature, see, for example, [14][15][16]. The superpotential ansatzed in those papers is of a different form to that described here, and the cubic and higher-order singlet couplings nor Yukawa couplings were not consistently computed. We choose to work with the expression above as it manifestly replicates the vanishing of all singlet couplings.
The layout of this paper is the following. In Sect. 2 we review the necessary background to study heterotic vacua, reviewing the results of [5]. In Sect. 3, we dimensionally reduce the Yang-Mills sector to obtain a metric on the matter fields. In Sect. 4, the reduction is applied to the gaugino to get the quadratic fermionic couplings, including the Yukawa couplings. In Sect. 5, we summarise the results. In Sect. 6 we show how these couplings are represented in the language of a Kähler potential K and superpotential W. Table 1 A table of objects used   Quantity Definition Comment

Heterotic geometry
The purpose of this section is to establish conventions and notation through a review of heterotic moduli geometry, most of which is explained in [5]. In terms of notation, there are occasional refinements and new results towards the end of the section. We largely work in the notation of [5], with a few exceptions, most important of which is that real parameters are denoted by y a and complex parameters by y α , y β . The discussion both there and in this section refers to forms defined on the manifold X. This is generalised in later sections in order to account for the charged matter fields. A  table of notation is given in Tables 1 and 2. Basic results and a summary of conventions are found in Appendices. Hodge theory and forms are in "Appendix A"; spinors in "Appendix B"; and representation theory in "Appendix C". We consider a geometry R 3,1 × X with X smooth, compact, complex and vanishing first Chern class. While X is not Kähler in general, we take it to be cohomologically Kähler satisfying the ∂∂-lemma, meaning that its cohomology groups are that of a Calabi-Yau manifold.
The heterotic action is fixed by supersymmetry up to and including α 2 -corrections. In string frame with an appropriate choice of connection for T X , it takes the form [17,18]: (2.1) Our notation is such that μ, ν, . . . are holomorphic indices along X with coordinates x; m, n, . . . are real indices along X; while e, f, . . . are spacetime indices corresponding to spacetime coordinates X. The 10-dimensional Newton constant is denoted by κ 10 , g 10 = − det(g M N ), is the 10-dimensional dilaton, R is the Ricci scalar evaluated using the Levi-Civita connection and F is the Yang-Mills field strength with the trace taken in the adjoint of the gauge group.
We define an inner product on p-forms by and take the p-form norm as Thus, the curvature squared terms correspond to where the Riemann curvature is evaluated using a twisted connection We write the metric on X as The manifold X has a holomorphic (3, 0)-form where μνρ depends holomorphically on parameters and coordinates of X. is a section of a line bundle over the moduli space, meaning that there is a gauge symmetry in which → μ where μ(y α ) ∈ C * is a holomorphic function of parameters. There is a compatibility relation where || || is the norm of . For fixed X, this is often normalised so that || || 2 = 8. However, || || depends on moduli and is gauge dependant and so it is not consistent in moduli problems to do this.

Derivatives of and α
A variation of complex structure given by a parameter y α can be described in terms of the variation of the holomorphic three-form by noting that ∂ α ∈ H (3,0) ⊕ H (2,1) and writing Here χ α are ∂-closed (2, 1)-forms. Variations of complex structure J : T X → T X can also be phrased in terms of (0, 1)-forms valued in T X We have denoted ∂ β α = αβ which makes manifest the symmetry property ∂ β α = ∂ α β . Occasionally we will denote parameter derivatives by ∂ α ω ∼ = ω ,α . The α and χ α are related The symmetric component of α μ appears in variations of the metric δgμν = α (μν) δy α .
It is best to describe variations of complex structure through projectors P and Q onto holomorphic and antiholomorphic components, respectively, The projectors capture the implicit dependence on complex structure. For example, the operator ∂ = dx m Q m n ∂ n undergoes a variation purely as a consequence of the implicit dependence on the complex structure:

The vector bundle E
Let E denote a vector bundle over X, with structure group H, and A the connection on the associated principal bundle. That is, A is a gauge field valued in the adjoint representation ad h of the Lie algebra h of H. Under a gauge transformation, A has the transformation rule where is a function on X that takes values in G. We take to be unitary and then d −1 and A are anti-Hermitian. The field strength is and this transforms in the adjoint of the gauge group: F → F −1 . Let A be the (0, 1) part of A then, since A is anti-Hermitian, On decomposing the field strength into type, we find F 0,2 = ∂A + A 2 . The bundle E is holomorphic if and only if there exists a connection such that F (0,2) = 0. The Hermitian Yang-Mills equation is

The B and H fields
There is a gauge-invariant three-form where CS denotes the Chern-Simons three-form and is the connection on T X for Lorentz symmetries.
With this transformation law, B is a 2-gerbe and H is invariant. An important constraint arising from supersymmetry is that H is related to the Hermitian form ω and complex structure J of X: which for an integrable complex structure reduces to We denote the real parameters of the compactification by y a and complex parameters by y α , y β . If the parameters are fixed to y = y 0 , the second term in (2.10) vanishes and the relation simplifies to However, when complex structure is varied ∂ α (dJ ) = 2i∂ α , the second term in (2.10) is nonzero and is important as it contributes to the equations satisfied by the moduli.

Derivatives of A
The heterotic structure (X, H, E) depends on parameters. This means the gauge connection A and its gauge transformations depend on parameters. As constructed in [5], the gauge covariant way of describing a deformation of A is given by introducing a covariant derivative where A = A a dy a is a connection on the moduli space with a transformation law With this transformation property D a A transforms homogeneously under (2.6): The moduli space M is complex, and we introduce a complex structure y a = (y α , y β ).
When parameters vary complex structure the holomorphic type of forms change, and the covariant derivatives D α A is no longer gauge covariant. This is remedied by defining a generalisation, termed the holotypical derivative D α A: where the vanishing of D β A follows from (2.5). It follows from the definition that under a gauge transformation the holotypical derivative transforms in the desired form Without the extra term − α μ A † μ in the holotypical derivative, this property does not hold as ∂ fails to commute with ∂ α .
The holotypical derivative can be extended to act on ( p, q)-forms. Define and understand W r,s m = 0 if r or s are negative or r +s > n−1. The holotypical derivatives are then given by The holotypical derivative has the nice feature that it preserves holomorphic type: We use D α to denote the covariant derivative to account for any gauge dependence of the real form W . For example, the covariant derivative of the field strength is related to that of A: However, the holotypical derivative of, for example, and this is known as the Atiyah constraint.

Derivatives of H
The second and third derivatives are defined to transform in a natural way inherited from that of D a B with db a an exact form. The exact form comes from the fact the physical quantity is dB, and so in writing B a there is a corresponding ambiguity. It is a simple exercise to note that ∂ a H is given by the expression In terms of holomorphic parameters, we introduce the holotypical derivative and find The second derivative is given by Despite appearances the right-hand side is symmetric in a, b after one uses that and so (2.24) The third derivative of H is given by where B cba = ∂ c ∂ b B a and in terms of the B-field: For similar reasons to the second derivative above, this is actually symmetric in a, b, c, but not made manifest in this expression for compactness.

Derivatives of d c ω
The derivative of d c ω in (2.10) with respect to parameters is We can evaluate (2.26) for a given complex structure, denoted | y 0 in the corresponding complex coordinates of X: For a given complex structure, we project onto holomorphic type then we need to use holotypical derivatives. Two cases we will need and then projecting onto (0, 3) and The second derivative of d c ω, given by differentiating (2.26) and then evaluated on a fixed complex structure, is We will have need for the (0, 3)-component: (2.28)
The heterotic structure are holomorphic functions of parameters. This can be compactly stated as

The matter field metric
In this section we dimensionally reduce the Yang-Mills term in (2.1) to obtain the metric for the matter fields. Our task divides into two steps. First, determine how A e 8 decomposes under e 8 ⊕ e 8 ⊃ g ⊕ h, using this to form a KK ansatz. For simplicity we will suppress writing the second e 8 sector. Second, use this to dimensionally reduce the d = 10 action thereby getting an effective field theory metric and Yukawa couplings for the matter fields and construct the Kähler potential.

Decomposing
The branching rule for g ⊕ h ⊂ e 8 is where ad denotes the relevant adjoint representation. The matter fields transform in representations of g ⊕ h. We denote these representations by r i for h and R i of g, respectively. Denote dimensions in the obvious way dim r i = r i and dim R i = R i . We have allowed for a sum over all the relevant representations r i and R i , including, for example, pseudo-real representations. For simplicity we will often suppress the sum and write a single matter field representations of g ⊕ h and its conjugate. The generalisation is obvious. The matrix presentation of the adjoint of e 8 is complicated. For the moment let us suppose we can write the generator in simplified form as We do this as a toy model to illustrate the key points of the calculation, and at the end of the day our results will not depend on this presentation. The background gauge field is where A h = A m (x)dx m is the h gauge field and we take it to have legs purely along the CYM; B g = B e (X )dX e is the R 3,1 spacetime gauge field valued in g. When there is no ambiguity we will drop the g, h subscripts. We indicate this combined Lorentz and gauge structure schematically in matrix notation Note that having off-diagonal terms turned on in the background would amount to Higgsing gauge group, which we do not want for the discussion in this paper. A fluctuation of A e 8 is of the form where again we have used matrix notation to indicate the structure. This gives us an intuition for understanding the transformation properties under a h ⊕ g rotation: We see that fields transform under g ⊕ h as: 1. δ A h transforms as (1, ad h ), and δ B g transforms as (ad g , 1); 2. is in the (R, r) and is a r × R matrix. Column vectors are in the fundamental; row vectors the antifundamental. For example, 1 , . . . , R are each column vectors transforming in the r of h.

3.
is in the (R, r) and is a R × r matrix with 1 , . . . , R row vectors and so in the r of h. 4. To preserve the structure g ⊕ h, δ A h has legs only on the CYM, while δ B g has legs only in R 3,1 .

The reality condition is δ
Decomposing this condition according to the holomorphic type of X we have: while the (1, 0)-components are fixed by reality: 1,0 = −ψ † , 1,0 = − φ † . 4 The e 8 field strength is Its background value decomposes according to its orientation of legs: Consider δ F e 8 oriented along X. At this point we drop the h, g subscripts on A, B.
Then, the equations of motion require F e 8 be (1, 1) implying any (0, 2)-component must satisfy, The off-diagonal terms tell us the fields φ, ψ are holomorphic sections of E. We will occasionally introduce index to make manifest the fact that φ is in the Then, for example, We have introduced the ∂-cohomology group H 1 (X, E r ), with forms valued in the hsubbundle of E whose fibres are the representation r of h. The r index i,j is implicitly summed. 4 It may be useful to define¯ a μ given by¯ a μ = ( ā μ ) * so that a is a real index and a 1,0 = − ψ * a = −¯ a 1,0 .
We could also study the equations of motion for δ A e 8 d † For example, on the φ matter field this gives This has solution if φ is harmonic element of H 1 (X, E r ). This is slightly stronger than the cohomology relation (3.7).
Expand the fields φ and ψ in a harmonic basis for H 1 (X, E r ) and H 1 (X, E r ), respectively: while C ξ and D τ are valued in R and R, respectively. For example, consider the standard embedding. Then, . C ξ and D τ are in the R = 27 and R = 27.
We need to satisfy the reality condition † = − , which forces φ † = −ψ and so in terms of the (φ ξ , ψ τ ) basis: We denote conjugation through the barring of the indices. For example,

The matter field metric from reducing Yang-Mills, L F
The spirit of KK reduction is to promote the coefficients to spacetime fields: Y α (X ), C ξ (X ), D τ (X ), and integrate over the six-dimensional manifold to get an effective four-dimensional theory. With the conventions of [5], the d = 10 e 8 Yang-Mills field contribution to the d = 4 effective field theory is: We dimensionally reduce, doing a background field expansion. A small fluctuation of the field strength is given by (3.6), and so The first term involves just the bundle moduli, contributing to the moduli metric considered in [5]; the middle two terms involve the matter fields and the last term gives rise to the kinetic term for the d = 4 spacetime gauge field. The terms involving the matter fields are: where D e is the spacetime g-covariant derivative and d A the h-covariant derivative. Hence, using Tr |δ , and ignoring the moduli fields for the moment, we find the kinetic terms for the matter fields come from middle two terms in (3.12) and are Tr |δ F e 8 | 2 = −2 Tr D e μ D e †μ − 2 Tr D e μ D e †μ . (3.14) We have used the reality condition † = − . The matter fields have a KK ansatz, given by (3.10), which when substituted into each of the above terms gives where indices for the representation R and r are explicit. The trace projects onto invariants constructed by the Krönecker delta functions δ ij and δ M N . In the following we will suppress the indices and delta symbols where confusion will not arise. Substituting (3.14) and (3.15) into L F in (3.11), reintroducing the moduli contribution, calculated in [5], we find a kinetic term for both the matter fields and the moduli fields: from which we may identify the moduli space metric and matter field metric ds 2 = 2G αβ dy α dy β + 2G ξ η dC ξ dC η + 2G σ τ dD σ dD τ , (3.17) where we denote the coordinates of the moduli space M by y α , y β , and without wanting to clutter formulae, denote the coordinates of the matter fields by C ξ , D σany ambiguity with their corresponding fields will always be made explicit. The moduli fields have the metric computed in [5], given in (5.2). The matter fields also have a metric There is no trace as the integrands are written in the form r · r. Although we have indicated the result for a single representation r and r, the result generalises to a sum over representations of h. For any two-form F there is a relation ω F = 1 2 Fω 2 . Using this and ω μν = −ig μν we find We introduce the trace over r indices in order to be able to write φ x , ψ σ in any order. The matter field metrics are then expressible in a way closely resembling the moduli metric (3.20)

Fermions and Yukawa couplings
The fermionic couplings of interest to heterotic geometry derive from the kinetic term for the gaugino. We compute the quadratic and cubic fluctuation terms. The former are mass terms for the gauginos, which we show all vanish consistent with the vacuum being supersymmetric. The latter are the Yukawa couplings between two gauginos and a gauge boson.
In "Appendix B" all spinor conventions we used are explained. We also give a summary of results in spinors in d = 4, 6, 10 relevant to this section. We also derive some expressions for bilinears relevant to the dimensional reduction.

so(3, 1) ⊕ su(3) spinors
The gaugino is a Majorana-Weyl spinor ε which has zero modes on R 3,1 × X. The Lorentz algebra is so(3, 1) ⊕ so(6) ⊂ so(9, 1) under which ε is where λ and λ are in the 4 and 4 of so (6); ζ and ζ are in the 2 and 2 of so(3, 1). Where possible we use the 2-component Weyl notation for ζ, ζ and always leave the so (6) where λ c denotes taking the so(6) Majorana conjugate. The Majorana-Weyl spinor ε can now be written solely in terms of say ζ , λ : The presence of N = 1 spacetime supersymmetry means there is a globally welldefined spinor on X. This implies the existence of an su(3)-structure on X. Under so(6) → su(3) the spinors λ, λ decompose according to the branching rule 4 = 3⊕1, and 4 = 3 ⊕ 1, which we write as The spinors λ + , λ − are the nowhere vanishing su(3) invariant spinors. As established in appendix, we can express λ 3 , λ 3 in terms of λ ± and gamma matrices where {γ μ , γν} = g μν and μ , μ are components of 1-forms on X. Appendix details the construction of the su(3) bilinears: and where φ accounts for a relative phase difference between λ − and . Under the gauge symmetry → μ , the fermions λ ± transform as a phase: The bilinears above respect this gauge symmetry.

Kaluza Klein ansatz
ε is in the adjoint of e 8 . Consequently, it decomposes under g ⊕ h ⊂ e 8 and the expectation from supersymmetry is that we find a natural pairing between fluctuations of the gauge field and the fermions. As the background is bosonic, all fermionic fields are fluctuations; we aim to study the effective field theory of those fluctuations that are massless. The massless fluctuations are zero modes of an appropriate Dirac operator. The gaugino and gauge field have decomposition under so(3, 1)⊕su(3) ⊂ so(9, 1) and for the gauge algebra e 8 ⊃ g ⊕ h: We organise our study of the zero modes according to their representations under so(3, 1) ⊕ su(3) and the gauge algebra g ⊕ h. We continue the mnemonic of indicating the gauge structure through block matrices 1) .
In the second line we have indicated the representations of the individual components of the gaugino by a subscript, and these are regarded as independent field fluctuations.
We will now drop the sum over representations ⊕ i in order to simplify notation and as the generalisation to include a sum over representations is obvious. We classify the zero modes by the symmetries under su(3)-structure and g ⊕ h. The first type of zero modes is su(3)-structure singlets and transforms in the (ad g , 1) with KK ansatz where we use the first line of (B.36). The second type of zero modes transforms as 3 under su(3)-structure and as (1, ad h ) ⊕ (R, r) ⊕ (R, r) under g ⊕ h. There is a natural pairing between δ A 0,1 e 8 and ζ ⊗ λ 3 . Using the KK ansatz (3.3), (3.10) for bosons there is a corresponding KK ansatz for the spinors: where the matrices in the last line are related to the μ in (4.3). We have denoted the anticommuting R 3,1 spinors by calligraphic letters Y α , C ξ and D τ ; superpartners to Y α , C ξ , D τ . C ξ and D τ are in the R and R of g, while Y α are neutral. The spinor in the 3 of su(3)-structure is determined by Majorana conjugation ζȧ ⊗ Altogether, the Majorana-Weyl spinor is given by substituting the above two expressions into the first line of (B.37) and combining with (4.9) giving (4.12) The ⊕ reflects the embedding of the 2-component Weyl spinors into a 4-component notation used in (B.36), (B.37).

Dimensional reduction of Tr ε M D M ε
The ten-dimensional kinetic term for the gaugino is now dimensionally reduced, with action: The quadratic fluctuations give the kinetic terms as well as any mass terms; the cubic fluctuations give Yukawa interactions. We split the bilinear into two terms where ε = ε † 0 is the Dirac conjugate. M are the d = 10 gamma matrices, given as e = γ e ⊗ γ (6) , μ = 1 ⊗ γ μ , as described in appendix. Here μ is a holomorphic index along X.

Quadratic couplings
As the background is bosonic, we take A to be the background gauge field We start with the derivative operator i Tr ε M ∂ M ε = i Tr ε e ∂ e ε + i Tr ε m ∂ m ε .
The first term Tr (ε e ∂ e ε) is computed using (4.2) and the third lines of (B.36), (B.37),  The first terms follows the calculation of (4.17) after using (4.15) and Tr B e = 0 (4.20) We now put the terms together. The / D 4 term comes from adding (4.17) and (4.19): whereD e is a spacetime covariant derivative appropriate to whatever representation if it acts on The matter and moduli fields have kinetic terms with non-trivial metrics: The fermions have identical metrics to their bosonic superpartners. The bundle moduli appear with a metric that coincides with that derived by Kobayashi and Itoh [19,20]. The mass terms come from adding together (4.17) and (4.20). We normalise the mass term to be compatible with the convention in [31]: where K is the Kähler potential, which on our background evaluates to be e K/2 = (2 √ 2V || ||) −1/2 . The mass terms are (4.24) The last term is normalised with a factor of 2 as the two indices are distinguished. As before, we do not write the trace, understanding the indices contracted in the natural way.
Recall that the equations of motion are Substituting this in we find The vanishing of m αβ is guaranteed when ∂(D α A) = 0, that is for bundle or Hermitian moduli. For complex structure parameters, if one can find a basis for parameters in which ∂(D α A) = 0 for complex structure moduli, so that α μ F μ = 0, then this is also satisfied. If this is not possible, we exploit the last line of the supersymmetry relation (2.29) We have used some results familiar from special geometry, see [21], which apply in this general heterotic context. They are: Here D α χ β is covariant with respect to gauge transformations χ α → μ χ α . Also, and In special geometry a αβγ is a Yukawa coupling for 27 3 fields, playing the role of an intersection quantity relating derivatives of χ α to χ γ . G 0 the metric on complex structures, used to raise and lower indices. The same relation applies in heterotic geometry with the understanding that the complex structures may be reduced by the Atiyah constraint. Consequently, we see that the Atiyah condition gives m αβ = 0.

Cubic fluctuations and Yukawa couplings
We now compute the cubic order fluctuations to get the Yukawa couplings. The calculation proceeds in a similar fashion to the above. The cubic interaction only comes from: (4.29) In the last two lines, we use the appropriate symmetric invariants to construct R 3 and R 3 .
Putting it together, normalising to agree with [31], we find (4.30) where c.c denotes the complex conjugate and the Yukawa couplings are given by The result is straightforwardly extended to vacua with more complicated branching rules involving multiple representations ⊕ p R p . More Yukawa couplings appear-one for each invariant computed via the trace-but the integrand is of the same form as above. The 1 3 coupling vanishes classically. To see this, write δA to second order in deformations δA = δy α D α A + δy α δy β D α D β A + · · · .
Note that D α D β A = D β D α A and so the second term is appropriately symmetric in indices α, β. A standard deformation theory argument related to the Kuranishi map implies the second-order deformation is unobstructed provided Substituting into (5.4) one finds Y αβγ = 0, when α μ F μ = 0. When α = 0, that is complex structure is varying, the coupling still vanishes with exactly the same argument as for the singlet mass term in (4.27).

The final result: moduli, matter metrics and Yukawa couplings
The effective field theory has N = 1 supersymmetry, with a gravity multiplet and a gauge symmetry g. The N = 1 chiral multiplets consist of 5 • g-neutral scalar fields Y α and fermions Y α corresponding to moduli; • g-charged bosons C ξ and fermions C ξ in the R of g; • g-charged bosons D ρ and fermions D ρ in the R of g; The final result is expressed as a Lagrangian with normalisation conventions matching [31] The kinetic terms for fields contain metrics. The metric for fermions and bosons are identical, consistent with supersymmetry. The moduli metric, derived in [5], is: The metric terms for the fermionic superpartners to moduli Y α are fixed by supersymmetry from the bosonic result. The matter field metrics are given in (3.20), The mass terms written in (4.24) vanish m αβ = m ξτ = 0. 5 We do not consider the universal multiplet, the d = 4 dilaton and B-field, which decouples.

The superpotential and Kähler potential
The effective field theory has N = 1 supersymmetry in R 3,1 , and so the couplings ought to be derivable from a superpotential and Kähler potential. The Kähler potential for the moduli metric couplings was proposed in [5], and checked against a dimensional reduction of the α -corrected supergravity action. It is in which ω is the Hermitian form of X. The α -corrections preserved the form of the special geometry Kähler potential, and the second term remains classical. The Kähler potential for the matter field metric is trivial and given by where a, b = 1, . . . , R label the R representation and the trace is taken with respect to the delta function. The F-term couplings for the d = 4 chiral multiplets are described by a superpotential. In the language of d = 4 effective field theory, this superpotential takes the general form where the Tr projects onto the appropriate R-invariant and we are to view these as chiral multiplets in N = 1 d = 4 superspace in the usual way. The omitted terms are the quartic and higher-order couplings and non-perturbative corrections. It is important that W gives no singlet couplings, and this means all parameter derivatives of W vanish.
We would like to study a superpotential in a similar vein to the Kähler potential proposal (6.1). As ten-dimensional fields A e 8 and H depend on both parameters and matter fields. The fields d c ω and are valued on X and depend only on moduli fields. The spirit of the dimensional reduction is to promote the parameters to d = 4 fields.
In this vein define a superpotential 6 (6.4) in which the fields are regarded as functionals of the d = 4 chiral multiplets. The couplings in the effective field theory are specified by differentiating W and evaluating the integral after fixing the parameters y = y 0 . The rules for differentiating fields in the expressions for K and W with respect to parameters have been described in [5], which is complicated by virtue of h gauge transformations being parameter-and coordinate-dependent. These transformations are, however, independent of matter fields, and so the rule for matter field differentiation is simple It is important that we have written the ten-dimensional e 8 gauge field A e 8 , and not A h , as this is the functional of the matter fields-C ξ , D τ -as illustrated in, for example, (3.3) and (3.10). The integrand in W is a functional of the ten-dimensional H so that it depends on matter fields. The rule is to differentiate as noted above and then evaluate the integral on the fields' vacuum expectation values (VEV). Note that it is the VEV of H that satisfies d c ω = H , and the matter fields VEVs vanish C ξ = D τ = 0. For example, the tadpole matter and moduli couplings for a vacuum at the point y = y 0 are where we use ∂ α H in (2.20) and ∂ α d c ω in (2.26), and we evaluate them on some fixed y = y 0 .
As an ansatz W must satisfy a number of tests: it must be a section of a line bundle over the moduli space; any derivative with respect to parameters must vanish viz. ∂ α ∂ β ∂ γ · · · W = 0; be a holomorphic function of chiral fields; tadpole and mass terms for the matter fields must vanish; capture the F-term couplings derived through dimensional reduction in this paper. The expression (6.4) passes these tests. 7 W is a section of the line bundle transforming under the gauge symmetry → μ(y) as W → μW where μ(y) is a holomorphic function of parameters. This is necessary in order to consistently couple to gravity [22]. This fixes the integrand to be proportional to .
The supersymmetry relation H = d c ω holds for all y 0 ∈ M. Hence, derivatives of it vanish: 8 where y α 1 , . . . , y α n are any collection of parameters and we evaluate on a supersymmetric vacuum, denoted by y = y 0 . It then follows that any derivative of the superpotential with respect to parameters vanishes. This is what is used in (6.5) to show that all tadpole terms vanish. The argument clearly extends to higher order. Consider the kth derivative This vanishes on any supersymmetric background: W is independent of moduli fields, and so W does not give rise to any singlet couplings in agreement with the dimensional reduction. An analogous argument, together with being holomorphic, shows that despite neither H nor d c ω being holomorphic, W is a holomorphic function of fields. For example, the first-order derivative is Using (6.6) all higher-order antiholomorphic derivatives of (H − d c ω) vanish. It is also the case that (∂ ξ ) n W = 0 for all n ≥ 1. So, W is a holomorphic function of chiral fields.
The expression for the masses can be written as derivatives of W where for the second term we use that d c ω, do not depend on C ξ , D τ , while ∂ ξ ∂ τ H is given by (2.22) with D a A → ∂ ξ A = φ ξ . As A depends linearly on the matter fields, all second derivatives vanish.
The Yukawa couplings Y are also all derived from W. Using (2.25), we find agreement with the functional forms in (4.31), of which the non-vanishing terms are Even though the singlet couplings vanish, one can check that their functional form is correctly derivable from W. The fact of 1/2 is in order to agree with the convention given in [31]. It is satisfying that the superpotential consistently captures the couplings derived in the dimensional reduction, both involving moduli and matter fields. Furthermore, it manifestly does not give rise to any singlet couplings.

Outlook
We have calculated the effective field theory of heterotic vacua of the form R 3,1 × X at large radius, correct to order α . The field theory is specified by a Kähler potential and superpotential. Supersymmetry forbids W from being corrected perturbatively in α , but is in general corrected non-perturbatively in α . For E obtained by deforming T X , some of these non-perturbative corrections have been computed as functions of moduli using linear sigma models, see, for example, [23][24][25][26][27]. One can now use the results obtained here and those in [5] to determine the normalised quantum corrected Yukawa couplings, in examples that may be of phenomenological interest, see, for example, [28]. Although the Kähler potential is corrected perturbatively in α , it was conjectured in [5] that the form of the Kähler potential does not change to all orders in perturbation theory, and that the α -corrections are contained within the Hermitian form ω. This conjecture is consistent with the work in [6,7], and it would be very interesting to prove this conjecture, at least to second order in α . Although we have derived this result using a single pair of matter fields, the result clearly generalises to a sum over representations ⊕ p R p ⊕ p R p . The main burden of the generalisation is to evaluate the trace using the appropriate branching rules.
Many questions arise. For example, are there any special geometry type relations between K and W? Finding a prepotential analogous to special geometry looks difficult, partly because it involved analysis related on the geometry of the standard embedding and Calabi-Yau manifold's. Nonetheless, it is likely K and W are related.
It would be interesting to compute the field theory couplings in specific examples. For E attained by deforming T X one might be able to compare with the linear sigma model parameter space studied in say [24,29,30] and study the quantum corrections to the 27 3

and 27
3 couplings using the correctly normalised fields. We showed using deformation theory arguments that the 1 3 coupling vanishes classically. A pressing question is to what extent these couplings vanish exactly. Any non-vanishing would imply the vacuum does not exist, and thereby shrink the moduli space of heterotic vacua quantum mechanically.

A Hodge theory on real and complex manifolds
We establish some notation and results for forms on real and complex manifolds to be used in the text. Coordinates for R 3,1 are denoted X e , while real coordinates on X are denoted by x m . Complex coordinates are denoted x μ , xν. We need to write coordinate expressions for forms more than metrics, and so our convention is to omit the wedge symbol ∧ except where confusion may arise. We write metrics as ds 2 = g mn dx m ⊗ dx n , only occasionally omitting the ⊗ only where confusion will not arise.

A.1 Real manifolds
The volume form on a n-dimensional Riemannian manifold is where 12···n = 12···n = 1 is the permutation symbol. The determinant of the metric is g = 1 n! p 1 ··· p n q 1 ···q n g p 1 q 1 · · · g p n q n .
If ω is a top-form, then The Hodge dual of a p-form A p is The inner product of two p-forms is then

A.2 Complex manifolds
On a complex manifold the metric is Hermitian with det(g mn )=g. In addition to the Hodge dual , which contracts a ( p, q) with a (q, p) form, on a complex manifold we can define a which contracts a pair of ( p, q)-forms, and so forming an inner product. If α, β are two ( p, q)-forms, then it is defined as It has a complex volume form, which is nowhere vanishing and globally well defined: where || || is a coordinate scalar, and so a constant for a fixed manifold, but depends on parameters, denoted y. We can write where e μνρ is the permutation symbol and f (x, y) is a holomorphic function of coordinates and parameters. μνρ is not a tensor and consequently f transforms like g 1/4 under holomorphisms (holomorphic diffeomorphisms) The complex volume form transform like sections of a complex line bundle on M. Under a gauge transformation → μ with μ ∈ C * , we have || || 2 → |μ| 2 || || 2 while g is invariant. It is sometimes convenient to isolate the phases of f and μ: If A, B, C are (0, 1)-forms, then where we have the compatibility relation It is also useful to note In coordinates

B Spinors
We establish some conventions and results for spinors in d = 4, d = 6 and d = 10.
We define the Pauli matrices as and we denote 1 n the n × n identity matrix. We also define We denote σ e f = 1 2 (σ e σ f − σ f σ e ). A similar definition applies for the gamma matrices.
Let ε be Weyl spinor that is of positive chirality ε = ± ε. As ( M ) * and −( M ) * both satisfy the same Lorentz algebra as M , there are two similarity transformations preserving the Lorentz algebra under which the spinor transforms to ε → B i ε. Hence, ε * and B i ε transform in the same way under Lorentz transformations, and we can define Majorana conjugation to be and the Majorana condition is ε = ε c . Applying Majorana conjugation twice gives a consistency condition B * i B i = 1, no sum on the i, which must be satisfied. For so(9, 1) it is possible to find both B (1) and B (2) satisfying (B.2) that also satisfy the consistency condition; this is not true for so (3,1) and so (6). In the text we utilise B (2) , which, with our choice of basis, gives a manifestly consistent Majorana condition for so(3, 1) ⊕ so (6).
We utilise the convention of complex conjugation of a pair of spinors interchanging their order without introducing a sign.
The 4 of so(3, 1) admits a pair of Weyl representations of positive and negative chirality, respectively, and in the second equality, expressed as spinors in the basis (B.3). This is sometimes denoted = ζ ⊕ ζ . Where possible, we adopt the 2-component spinor notation, see, for example, [31,32]. The indices on Weyl spinors are denoted byȧ and a. The rule for raising and lowering is through the permutation symbol where 12 = 21 = 1 and 21 = 12 = −1: Complex conjugation exchanges dots on indices (ζ a ) * = ζȧ, ( ab ) * = ȧḃ , etc. These spinors are assigned the Grassmann odd property and so anticommute. However, when complex conjugating a pair of spinors, the order is interchanged without a sign: The indices on σ e and σ e are related σ e aȧ = ab σ fḃb ȧḃ , and the index structure on is When indices are not written there is an implicit contraction through the symbol. For example, Analogous conventions exist for dotted indices, given by complex conjugating the above equation. Some useful spinor relations to be used inside actions are: In this basis, a Majorana spinor satisfies ζ a = ζ a . We will not impose this and choose to work in the Weyl basis.

B.1.3 so(6)
We describe so(6) spinors first in flat space, before coupling them to a curved manifold with su(3)-structure in the next section.
A Dirac spinor decomposes into a pair of Weyl representations A basis compatible with this is The chirality operator is We define raising and lowering operators, introducing holomorphic and antiholomorphic indices These are real (γ μ ) * = γ μ , related by (γ μ ) † = γμ and satisfy {γ μ , γν} = δ μν . In this basis only so(2) ⊕ so(2) ⊕ so(2) ⊂ is manifestly preserved. The conjugation matrix B 6 satisfies and in this basis is the product of all imaginary matrices so that which satisfies B −1 6 =B * 6 = − B 6 and so an so(6) spinor λ satisfies (λ c ) c = λ. In the second equality, we have written B 6 in terms of raising and lowering operators. The conjugation matrix changes chirality B 6 γ μ ± =γ μ ∓ B 6 .
The complex conjugation matrix where B 4 is in (B.4) and B 6 is in (B.14). B is imaginary, unitary and anti-Hermitian B † = B −1 = −B and satisfies the property that The Majorana conjugate of a spinor ε in the 32 is ε c = B −1 ε * . Expand ε in terms of so(3, 1) ⊕ so (6) spinors where λ and λ are in the 4 and 4 , respectively, while ζ and ζ are in the 2 and 2 , respectively. In the second line we have written this in terms of the four-dimensional so(3, 1) spinors, with their spinor indices explicit; we will always leave the so (6) spinor indices implicit. The Majorana condition ε c = ε is simplified by using (B.11), It is sometimes more convenient to write this simply as The Majorana-Weyl spinor ε can now be written solely in terms of, for example, ζ , λ :

B.3 Spinors on a complex manifold X with su(3)-structure
The manifold X is endowed with an su(3)-structure meaning that there is a globally well-defined non-vanishing spinor implying a reduction of the structure group so(6) → su(3) under which and the spinors decompose, respectively, as The spinors λ + , λ − are the su(3) invariant spinors that are nowhere vanishing on the manifold X that define the su(3)-structure.
With respect to the basis (B.12), the raising and lowering matrices γ μ + and γν − are real and related by Hermitian conjugation (γ μ + ) † = γμ − . This reality property is consequence of our choice of basis, and any physical result will not depend on this choice. Care must be taken when interpreting the holomorphy of indices, and where any ambiguity may arise, we will keep the ± subscript. Nonetheless, at the end of a calculation we will be able to interpret the indices in terms of holomorphic or antiholomorphic indices of T 1,0 X ⊕ T 0,1 X . The matrices satisfy an algebra where μ,ν are coordinate indices and the right-hand side is the inverse metric. 9 Majorana conjugation is defined using a covariant version of B in (B.14). On an su(3) manifold, B is a coordinate scalar, gauge invariant, and satisfies the property that This fixes This is the main example where confusion can arise in holomorphy of indices, and so we use the ± subscript for clarity. We build spinor representations by lowering and raising operators. Denote the lowest weight state λ − , satisfying γμλ − = 0. We define the remaining spinors as follows: (B.20) where γ μν = 1 2 (γ μ γ ν − γ ν γ μ ) and γ μνρ = 1 3! (γ μ γ ν γ ρ − γ μ γ ρ γ ν + · · · ). Note that γ μν λ − = γ μ γ ν λ − . Here μνρ is the permutation symbol with 123 = 1 and + is a tensor density to be fixed.
To identify + we study its transformation properties under symmetries of the moduli space and under holomorphisms. First note that + transforms like g 1/4 under holomorphisms. Hence, + ∝ g 1/4 up to a parameter-dependent coordinate scalar. Second, recall the gauge symmetry → μ where μ = |μ|e iξ ∈ C * . Under this symmetry the fermions λ ± are charged transforming under the U (1) ⊂ C * as 10 λ ± → λ ± e iξ/2 , (B.21) and so + transforms as + → + e iξ . Hence, + ∝ ( f √ g)/| f |, now fixed up to a gauge-neutral coordinate scalar. If we demand that λ † + λ + = λ † − λ − this fixes the constant to be a phase and we can write the final result as for some phase e iφ . There are three phases of interest: ψ ± = arg λ ± and ζ the phase of f = | f |e iζ . The gauge symmetry eliminates one of these degrees of freedom, and we can form two gauge-invariant combinations φ = ψ + − ψ − − ζ and = ψ + + ψ − . Using one of these global symmetries we could choose φ = 0, which in the gauge where ζ = 0 amounts to fixing the relative phases of λ ± equal.

B.4 Spinors charged under gauge symmetries
Sometimes the spinors carry additional structure, for example being charged in a representation of g ⊕ h. In that case complex conjugation is promoted to Hermitian conjugation. Consider first ζ, ζ ; these spinors may be charged in a representation of g. In computing a Majorana conjugate, the complex conjugate in (B.11) is promoted to Hermitian conjugate on the gauge structure. It is normally easy to do this with the spinor indices explicit; Majorana conjugation does not transpose the spinor structure.
As a way of illustration, there are three relevant cases to the text. The first are when ζ, ζ are singlets under ad g , in which Majorana conjugation is unchanged from the previous subsection. The second case, ζ is in a representation R, denoted ζ R , and Hermitian conjugation acts as (ζ a R ) † ∼ = ζȧ R . The third case is when ζ is in the adjoint of g, in which it is anti-Hermitian (ζ a ad g ) † = −ζȧ ad g . The Majorana conjugates of the last two cases are explicitly: The Majorana condition (B.17) implies that if ζ is in the R, then ζ is in the R. Of course, if the representation is real, then R = R in the above. Similar comments apply when λ, λ carry representations of h. Only λ 3 and λ 3 turn out to carry non-trivial representations of h, and this is through the object μ and μ in (B.20) and (B.25). The singlets λ ± are always gauge singlets. The generalisation of (B.29) is where μ is charged in a representation and †μ is the appropriate Hermitian conjugate. Putting this into (B.17) determines ζ ⊗ λ in terms of ζ ⊗ λ : ζȧ ⊗ λ + = −iζ ȧ † ⊗ λ + , ζȧ ⊗ μ γ μ λ − = iζ ȧ † ⊗ † μ γ μ λ − . (B.34)

B.5 Some useful spinor bilinears
We express the Majorana-Weyl spinor ε in terms of ζ , λ and list some bilinears relevant to the main text.

(B.35)
We have left the four-dimensional spinor indices for clarity, but will now drop them in spinor contractions, using convention (B.7).
We can now evaluate these relations for some specific examples relevant to the text when the spinors are charged in representations of g ⊕ h:

C Some representation theory
Consider some Lie algebra e and for this subsection only denote a = 1, . . . , dim e. We choose A to be anti-Hermitian A † = −A. In terms of adjoint generators T a : We identify Let χ be in the fundamental and in the antifundamental. The covariant derivatives are Decompose into type: For A a to be a real form, we require