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Letters in Mathematical Physics

, Volume 108, Issue 4, pp 1031–1081 | Cite as

On the effective field theory of heterotic vacua

  • Jock McOrist
Open Access
Article

Abstract

The effective field theory of heterotic vacua that realise Open image in new window preserving \(\mathcal {N}{=}1\) supersymmetry is studied. The vacua in question admit large radius limits taking the form Open image in new window , with Open image in new window a smooth threefold with vanishing first Chern class and a stable holomorphic gauge bundle Open image in new window . 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 \({\alpha ^{\backprime }\,}\). 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 Open image in new window and superpotential Open image in new window .

Keywords

Superstring theories Supersymmetric field theories String and supergravity theories Kaluza-Klein and other higher-dimensional theories 

Mathematics Subject Classification

81T30 81T60 83E15 83E30 83E50 

1 Introduction

We are interested in heterotic vacua that realise \(\mathcal {N}{=}1\) supersymmetric field theories in Open image in new window . At large radius, these take form Open image in new window where Open image in new window is a compact smooth complex threefold with vanishing first Chern class. We study the \(E_8{\times } E_8\) heterotic string, and so there is a holomorphic vector bundle Open image in new window with a structure group Open image in new window and a \(d=4\) spacetime gauge symmetry given by the commutant Open image in new window . The bundle Open image in new window 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 Open image in new window through anomaly cancellation. The triple Open image in new window 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 \({\mathfrak G}\) to the heterotic geometry.

The challenge in studying heterotic vacua is the complicated relationship between H, the field strength F and the geometry of Open image in new window . Supersymmetry relates the complex structure J and Hermitian form \(\omega \) of Open image in new window to the gauge-invariant three-form H:
$$\begin{aligned} H = \text {d}^c \omega , \qquad \text {d}^c\omega = \frac{1}{2}J_{m_1}{}^{n_1}J_{m_2}{}^{n_2} J_{m_3}{}^{n_3} (\partial _{n_1}\omega _{n_2n_3})\, \text {d}x^{m_1} \text {d}x^{m_2} \text {d}x^{m_3}. \end{aligned}$$
(1.1)
where \(x^m\) are real coordinates on Open image in new window . Green–Schwarz anomaly cancellation gives a modified Bianchi identity for H
$$\begin{aligned} \text {d}H =- \frac{{\alpha ^{\backprime }\,}}{4} \left( \,\text {Tr}\,F^2 - \,\text {Tr}\,R^2\, \right) , \end{aligned}$$
(1.2)
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 Open image in new window has torsion if H is nonzero. Unless one is considering the standard embedding—in which Open image in new window is identified with Open image in new window the tangent bundle to Open image in new window —the right-hand side of (1.2) is nonzero even when Open image in new window is a Calabi–Yau manifold at large radius. This means that H is generically non-vanishing, though subleading in \({\alpha ^{\backprime }\,}\), and so even for large radius heterotic vacua Open image in new window is non-Kähler. Torsion is inescapable.
The effective field theory of the light fields for these vacua is described by a Lagrangian with \(\mathcal {N}=1\) supersymmetry, whose bosonic sector is of the form
$$\begin{aligned} \mathcal {L}= \frac{1}{2\kappa _4^2} \sqrt{-G_4} \left( \mathcal {R}_4 -\frac{1}{4}\,\text {Tr}\,|F_{\mathfrak {g}}|^2 - 2G_{A {\overline{B}}} \widehat{\mathcal {D}}_e \Phi ^A \widehat{\mathcal {D}}_e \Phi ^{\overline{B}} - V(\Phi ,\bar{\Phi }) + \cdots \right) .\nonumber \\ \end{aligned}$$
(1.3)
Here \(\kappa _4\) is the four-dimensional Newton constant, \(\mathcal {R}_4\) is the four-dimensional Ricci scalar, \(F_{\mathfrak {g}}\) is the spacetime gauge field strength, the \(\Phi ^A\) is range over the scalar fields of the field theory, and their kinetic term comes with a metric \(G_{A{\overline{B}}}\). The fields \(\Phi ^A\) may be charged under \({\mathfrak {g}}\), the algebra of the gauge group \({\mathfrak G}\), with an appropriate covariant derivative \(\widehat{\mathcal {D}}_e\). Finally \(V(\Phi ,\bar{\Phi })\) is the bosonic potential for the scalars.

When Open image in new window 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 \({{\varvec{27}}}\) and \({{\overline{\varvec{27}}}}\) 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 Open image in new window .

We work towards answering this question by computing the effective field theory couplings correct to first order in \({\alpha ^{\backprime }\,}\). In a previous paper [5] we commenced a study of heterotic geometry using \({\alpha ^{\backprime }\,}\)-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 \(G_{A{\overline{B}}}\). In this paper, we compute the contribution of the matter sector to the metric \(G_{A{\overline{B}}}\), and the Yukawa couplings, correct to order \({\alpha ^{\backprime }\,}\). We describe an ansatz for the superpotential and Kähler potential for effective field theory: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
$$\begin{aligned} \begin{aligned} \text {d}s^2&= 2G_{\alpha {\overline{\beta }}} \,\text {d}y^\alpha \otimes \text {d}y^{\overline{\beta }}, \\ G_{\alpha {\overline{\beta }}}&= \frac{1}{4V} \int \Delta _\alpha {}^\mu \star \Delta _{\overline{\beta }}{}^\nu \,\, g_{\mu {\bar{\nu }}} + \frac{1}{4V} \int \mathcal {Z}_\alpha \star \mathcal {Z}_{\overline{\beta }}\,+ \\&\qquad + \frac{ {\alpha ^{\backprime }\,}}{4V}\int \,\text {Tr}\,\Big ( D_\alpha A \star D_{\overline{\beta }}A \Big ) - \frac{ {\alpha ^{\backprime }\,}}{4V}\int \,\text {Tr}\,\Big (D_\alpha \Theta \,\star \, D_{\overline{\beta }}\Theta ^\dag \Big ), \end{aligned} \end{aligned}$$
(1.5)
where Open image in new window is the \({\alpha ^{\backprime }\,}\)-corrected, gauge-invariant generalisation of the complexified Kahler form \(\delta B + \text {i}\delta \omega \), the \(\chi _\alpha \) 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 Open image in new window . The metric expressed this way is an inner product of tensors corresponding to complex structure \(\Delta _\alpha \), Hermitian moduli Open image in new window , and bundle moduli \(D_\alpha A\). The role of the spin connection \(D_\alpha \theta \) is presumably determined in terms of the other moduli as they do not correspond to independent physical fields. The tensors depend on parameters holomorphically throughde 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 \(C^\xi \) and Open image in new window 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 elementswhere \(\phi _\xi , \psi _\rho \) are (0, 1)-forms valued in a sum over representations of the structure group \(\mathcal {H}\).

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 \(\omega \). At first sight this is confusing as the only fields appearing in the Kähler potential are \(\omega \) and \(\Omega \). 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 \(\omega \) 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 \({\alpha ^{\backprime }\,}\) and \({\alpha ^{\backprime }\,}^2\) corrections to the moduli space metric in the particular case where the Hermitian part of the metric varies, while the remaining fields are fixed: \((\partial _a \omega )^{1,1} \ne 0\), Open image in new window . In general all fields vary with parameters and the metric is nonzero already at \(\mathcal {O}({\alpha ^{\backprime }\,})\).

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 \({\mathfrak {g}}\). 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 \({\alpha ^{\backprime }\,}\)-perturbation theory, and so the only possible \({\alpha ^{\backprime }\,}\)-corrections are due to worldsheet instantons.

The fields neutral under \({\mathfrak {g}}\), 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. \({{\varvec{1}}}^n\), or in a \({\mathfrak {e}}_6\) theory \(({{\varvec{27}}}\cdot {{\overline{\varvec{27}}}})^{326}\cdot {{\varvec{1}}}^{101}\), then some parameter \(y^\alpha \) would have its value fixed, a contradiction on it being a free parameter.3

The superpotential Open image in new window 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 Open image in new window , denoted \(\Omega \), transforms with a gauge symmetry \(\Omega \rightarrow \mu \Omega \) where Open image in new window . The superpotential is also a section of this line bundle, and transforms in the same way Open image in new window . Hence, Open image in new window has an integrand proportional to \(\Omega \). 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 \(\text {d}\omega \) and \(\text {d}^c\omega \). Open image in new window is also required not to give rise to any singlet couplings. So all derivatives of Open image in new window with respect to parameters must vanish. The combination \(H - \text {d}^c \omega \) manifestly satisfies this request. Derivatives with respect to matter fields of Open image in new window do not vanish. As these are charged in \({\mathfrak {g}}\), the only nonzero contributions come from H. This allows us to fix the normalisation of Open image in new window by comparing with the dimensional reduction calculation of the Yukawa couplings. Finally, Open image in new window 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 Open image in new window was studied by [8]. In that paper, one starts with an \({\mathfrak {su}}(3)\)-structure manifold Open image in new window , posits the existence of Open image in new window , 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 Open image in new window and superpotential Open image in new window .

1.1 Tables of notation

Table 1

A table of objects used

Quantity

Definition

Comment

Open image in new window

\(d=4\) spacetime gauge algebra

Group is \({\mathfrak G}\)

\({\mathfrak {h}}\)

Structure algebra of Open image in new window

Group is Open image in new window

\({{\varvec{r}}}\)

Representation of \({\mathfrak {h}}\)

\(\dim {{\varvec{r}}}= r\)

\({{\varvec{R}}}\)

Representation of \({\mathfrak {g}}\)

\(\dim {{\varvec{R}}} = R\)

\(\Phi \)

\(d=10\) gauge field in \(({{\varvec{r}}},{{\overline{\varvec{R}}}})\) of \({\mathfrak {h}}\oplus {\mathfrak {g}}\)

\(\phi = \Phi ^{0,1}\),   \(\Phi = \phi - \psi ^\dag \)

\(\Psi \)

\(d=10\) gauge field in \(({{\overline{\varvec{r}}}},{{\varvec{R}}})\) of \({\mathfrak {h}}\oplus {\mathfrak {g}}\)

\(\psi = \Psi ^{0,1}\),    \(\Psi = \psi - \phi ^\dag \)

\(\phi _{\xi }\)

Basis for Open image in new window

Valued in \({{\varvec{r}}}\) of \({\mathfrak {h}}\)

\(\psi _{\rho }\)

Basis for Open image in new window

Valued in \({{\overline{\varvec{r}}}}\) of \({\mathfrak {h}}\)

\(C^{\xi }, D^{\rho }, Y^\alpha \)

\(d=4\) bosons in the \({{\overline{\varvec{R}}}}\), \({{\varvec{R}}}\), \({{\varvec{1}}}\) of \({\mathfrak {g}}\) (e.g. \({{\overline{\varvec{27}}}}\), \({{\varvec{27}}}\), \({{\varvec{1}}}\) of \({\mathfrak {e}}_6\))

\(\xi ,\tau ,\alpha \) label harmonic bases

\(\mathcal {C}^\xi , \mathcal {D}^\rho , \mathcal {Y}^\alpha \)

\(d=4\) fermions in \({{\overline{\varvec{R}}}}\), \({{\varvec{R}}}\) of \({\mathfrak {g}}\)

Calligraphic for anticommuting

\(B_e \,\text {d}X^e\)

\({\mathfrak {g}}\)-valued connection on Open image in new window

Occasionally embed in \(A_{{\mathfrak {e}}_8}\)

\(A_m\, \text {d}x^m\)

\({\mathfrak {h}}\)-valued connection for Open image in new window on Open image in new window

Occasionally embed in \(A_{{\mathfrak {e}}_8}\)

\(\delta \mathcal {A}\)

Fluctuation of connection for Open image in new window

Occasionally \(\delta A_{{\mathfrak {h}}}\)

\(\delta B\)

Fluctuation of connection for \({\mathfrak {g}}\)

Occasionally use \(\delta A_{{\mathfrak {g}}}\)

\(\varepsilon \)

Majorana–Weyl \({\mathfrak {so}}(9,1)\) spinor

 

\(\zeta \otimes \lambda \)

\({\mathfrak {so}}(3,1)\oplus {\mathfrak {so}}(6)\) spinors

\(\lambda , \lambda '\) positive/negative chirality

Table 2

A table of coordinates and indices

Coordinates

Holomorphic indices

Real indices

Calabi–Yau manifold

\(x^\mu \)

\(\mu ,\,\nu ,\ldots \)

\(m,n,\ldots \)

Open image in new window spacetime

\(X^e\)

\(e,\,f,\ldots \)

basis for rep \({{\varvec{r}}}\) of \({\mathfrak {h}}\)   e.g. \({\mathfrak {h}}= {\mathfrak {su}}(3)\)

\([T_{\mathfrak {h}}]^i{}_j \in {{\varvec{r}}}\)

\(i,j=1,\ldots , {r}\)

basis for rep \({{\varvec{R}}}\) of \({\mathfrak {g}}\)   e.g. \({\mathfrak {g}}= {\mathfrak {e}}_6\)

\([T_{\mathfrak {g}}]^{M}{}_N \in {{\varvec{R}}}\)

\(M,N=1,\ldots ,{R}\)

parameters of heterotic structure

\(y^\alpha \)

\(\alpha ,\,\beta , \gamma ,\ldots \)

\(a,\,b,c\ldots \)

indices for \(d=4\) spinors (occasional)

\(\zeta _a, {\overline{\zeta }}^{\dot{a}}\)

ab

2 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 \(y^\alpha , y^{\overline{\beta }}\). The discussion both there and in this section refers to forms defined on the manifold Open image in new window . 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 Open image in new window with Open image in new window smooth, compact, complex and vanishing first Chern class. While Open image in new window is not Kähler in general, we take it to be cohomologically Kähler satisfying the \(\partial \overline{\partial }\)-lemma, meaning that its cohomology groups are that of a Calabi–Yau manifold.

The heterotic action is fixed by supersymmetry up to and including \(\alpha '^2\)–corrections. In string frame with an appropriate choice of connection for Open image in new window , it takes the form [17, 18]:
$$\begin{aligned} S = \frac{1}{2\kappa _{10}^2} \int \! \text {d}^{10\,}\! X \sqrt{g_{10}}\, e^{-2\Phi } \Big \{ \mathcal {R}- \frac{1}{2}|H|^2 + 4(\partial \Phi )^2 - \frac{\alpha '}{4}\big ( \,\text {Tr}\,|F|^2 {-} \,\text {Tr}\,|R(\Theta ^+)|^2 \big ) \Big \}, \end{aligned}$$
(2.1)
Our notation is such that \(\mu ,\nu ,\ldots \) are holomorphic indices along Open image in new window with coordinates x; \(m,n,\ldots \) are real indices along Open image in new window ; while \(e,f,\ldots \) are spacetime indices corresponding to spacetime coordinates Open image in new window . The 10-dimensional Newton constant is denoted by \(\kappa _{10}\), \(g_{10}=-\det (g_{MN})\), \(\Phi \) is the 10-dimensional dilaton, \(\mathcal {R}\) 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
$$\begin{aligned} \langle S,\, T\rangle = \frac{1}{p!} \, g^{M_1 N_1} \ldots g^{M_p N_p}\, S_{M_1\ldots M_p} \,T_{N_1 \ldots N_p}. \end{aligned}$$
and take the p-form norm as
$$\begin{aligned} |T|^2 =\langle T,\, T\rangle . \end{aligned}$$
Thus, the curvature squared terms correspond to
$$\begin{aligned} \,\text {Tr}\,|F|^2 = \frac{1}{2}\,\text {Tr}\,F_{MN} F^{MN}~~~\text {and}~~~ \,\text {Tr}\,|R(\Theta ^+)|^2 = \frac{1}{2}\,\text {Tr}\,R_{MNPQ}(\Theta ^+) R^{MNPQ}(\Theta ^+), \end{aligned}$$
where the Riemann curvature is evaluated using a twisted connection
$$\begin{aligned} \Theta ^\pm _M = \Theta _M \pm \frac{1}{2}H_M, \end{aligned}$$
with \(\Theta _M\) is the Levi–Civita connection. The definition of the H field strength and its gauge transformations are given in Sect. 2.3.
We write the metric on Open image in new window as
$$\begin{aligned} \text {d}s^2 = 2g_{\mu {\bar{\nu }}} \text {d}x^\mu \text {d}x^{\bar{\nu }}. \end{aligned}$$
The manifold Open image in new window has a holomorphic (3, 0)-form
$$\begin{aligned} \Omega = \frac{1}{3!} \Omega _{\mu \nu \rho } \text {d}x^\mu \text {d}x^\nu \text {d}x^\rho , \end{aligned}$$
where \(\Omega _{\mu \nu \rho }\) depends holomorphically on parameters and coordinates of Open image in new window . \(\Omega \) is a section of a line bundle over the moduli space, meaning that there is a gauge symmetry in which \(\Omega \rightarrow \mu \Omega \) where Open image in new window is a holomorphic function of parameters.
There is a compatibility relation
$$\begin{aligned} \frac{\text {i}}{ ||\Omega ||^2} \Omega \, {\overline{ \Omega }}= \frac{1}{3!}\omega ^3, \qquad ||\Omega ||^2 = \frac{1}{3!} \Omega _{\mu \nu \rho } {\overline{ \Omega }}^{\mu \nu \rho }, \end{aligned}$$
(2.2)
where \({||\Omega ||}\) is the norm of \(\Omega \). For fixed Open image in new window , this is often normalised so that \(||\Omega ||^2 =8\). However, \({||\Omega ||}\) depends on moduli and is gauge dependant and so it is not consistent in moduli problems to do this.

2.1 Derivatives of \(\Omega \) and \(\Delta _\alpha \)

A variation of complex structure given by a parameter \(y^\alpha \) can be described in terms of the variation of the holomorphic three-form by noting that \(\partial _\alpha \Omega \in H^{(3,0)}\oplus H^{(2,1)}\) and writing
$$\begin{aligned} \frac{\partial \Omega }{\partial y^\alpha }=-\,k_\alpha \,\Omega + \chi _\alpha ~;~~~ \chi _\alpha =\frac{1}{2} \chi _{\alpha \,\kappa \lambda \bar{\nu }}\,\text {d}x^\kappa \text {d}x^\lambda \text {d}x^{\bar{\nu }}. \end{aligned}$$
(2.3)
Here \(\chi _\alpha \) are \(\overline{\partial }\)-closed (2, 1)-forms. Variations of complex structure Open image in new window can also be phrased in terms of (0, 1)-forms valued in Open image in new window
$$\begin{aligned} \partial _\alpha J = 2\text {i}\,\Delta _{\alpha \,{\bar{\nu }}}{}^\mu \text {d}x^{\bar{\nu }}\otimes \partial _\mu , \qquad \partial _\alpha \partial _\beta J = 2\text {i}\,\Delta _{\alpha \beta } = 2\text {i}\,\Delta _{\alpha \beta }{}_{\bar{\nu }}{}^\mu \text {d}x^{\bar{\nu }}\otimes \partial _\mu . \end{aligned}$$
We have denoted \(\partial _\beta \Delta _\alpha = \Delta _{\alpha \beta }\) which makes manifest the symmetry property \(\partial _\beta \Delta _\alpha = \partial _\alpha \Delta _\beta \). Occasionally we will denote parameter derivatives by Open image in new window .
The \(\Delta _\alpha \) and \(\chi _\alpha \) are relatedThe symmetric component of \(\Delta _\alpha {}^\mu \) appears in variations of the metric \(\delta g_{\bar{\mu }\bar{\nu }}=\Delta _{\alpha \,(\bar{\mu }\bar{\nu })}\, \delta y^\alpha \).
It is best to describe variations of complex structure through projectors P and Q onto holomorphic and antiholomorphic components, respectively,
$$\begin{aligned} P_m{}^n = \frac{1}{2}\left( \delta _m{}^n - \text {i}J_m{}^n\right) , \qquad Q_m{}^n = \frac{1}{2}\left( \delta _m{}^n + \text {i}J_m{}^n\right) . \end{aligned}$$
The projectors capture the implicit dependence on complex structure. For example, the operator \( \overline{\partial }=\text {d}x^m Q_m{}^n\, \partial _n \) undergoes a variation purely as a consequence of the implicit dependence on the complex structure:
$$\begin{aligned}{}[\partial _\alpha ,\overline{\partial }] =\left[ \partial _\alpha , (Q_m{}^n \text {d}x^m \otimes \partial _n)\right] = - \Delta _\alpha {}^\mu \partial _\mu \end{aligned}$$
(2.5)

2.2 The vector bundle Open image in new window

Let Open image in new window denote a vector bundle over Open image in new window , with structure group \(\mathcal {H}\), and A the connection on the associated principal bundle. That is, A is a gauge field valued in the adjoint representation \(\mathrm{ad }_{\mathfrak {h}}\) of the Lie algebra \({\mathfrak {h}}\) of \(\mathcal {H}\).

Under a gauge transformation, A has the transformation rule
$$\begin{aligned} ^{\Phi }A = \Phi (A - Y)\Phi ^{-1}, \quad Y=\Phi ^{-1}\text {d}\Phi , \end{aligned}$$
(2.6)
where \(\Phi \) is a function on Open image in new window that takes values in \({\mathfrak G}\). We take \(\Phi \) to be unitary and then \( \text {d}\Phi \,\Phi ^{-1}\) and A are anti-Hermitian. The field strength is
$$\begin{aligned} F=\text {d}A + A^2, \end{aligned}$$
and this transforms in the adjoint of the gauge group: \(F\rightarrow \Phi F\Phi ^{-1}\).
Let \(\mathcal {A}\) be the (0, 1) part of A then, since A is anti-Hermitian,
$$\begin{aligned} A = \mathcal {A}- \mathcal {A}^{\dag }. \end{aligned}$$
On decomposing the field strength into type, we find \(F^{0,2} = \overline{\partial }\mathcal {A}+ \mathcal {A}^2\). The bundle Open image in new window is holomorphic if and only if there exists a connection such that \(F^{(0,2)}=0\). The Hermitian Yang–Mills equation is
$$\begin{aligned} \omega ^2 F =0. \end{aligned}$$

2.3 The B and H fields

There is a gauge-invariant three-form
$$\begin{aligned} H=\text {d}B - \frac{{\alpha ^{\backprime }\,}}{4}\Big ({\text {CS}}[A] - {\text {CS}}[\Theta ]\Big ), \end{aligned}$$
(2.7)
where \({\text {CS}}\) denotes the Chern-Simons three-formand \(\Theta \) is the connection on Open image in new window for Lorentz symmetries. The three-form \(\text {d}B\) is defined so that H to be gauge invariant, and so \(\text {d}B\) itself has gauge transformations. Under a gauge transformation
$$\begin{aligned} {\text {CS}}[A]~\rightarrow ~{\text {CS}}[A] - \text {d}\,\text {Tr}\,\!(AY) + \frac{1}{3} \,\text {Tr}\,\left( Y^3\right) , \end{aligned}$$
together with the analogous rule for \({\text {CS}}[\Theta ]\). The integral of \(\,\text {Tr}\,(Y^3)\) over a three-cycle is a winding number, so it vanishes if the gauge transformation is continuously connected to the identity. The integral vanishes for every three-cycle and so \(\,\text {Tr}\,(Y^3)\) is exact \( \frac{1}{3}\,\,\text {Tr}\,(Y^3) =\text {d}U, \) for some globally defined two-form U. There are corresponding transformations for the connection \(\Theta \) in which Y is replaced by Z and U by W.
Anomaly cancellation condition means that the B field is assigned a transformation
$$\begin{aligned} B~\rightarrow ~B - \frac{{\alpha ^{\backprime }\,}}{4}\left\{ \,\text {Tr}\,(AY) - U - \,\text {Tr}\,(\Theta Z) + W\right\} . \end{aligned}$$
(2.8)
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 \(\omega \) and complex structure J of Open image in new window :
$$\begin{aligned} H = \text {d}^c\omega , \qquad \text {d}^c \omega = \frac{1}{2}J_{m_1}{}^{n_1}J_{m_2}{}^{n_2} J_{m_3}{}^{n_3} (\partial _{n_1}\omega _{n_2n_3})\, \text {d}x^{m_1} \text {d}x^{m_2} \text {d}x^{m_3}, \end{aligned}$$
(2.9)
which for an integrable complex structure reduces to
$$\begin{aligned} \text {d}^c \omega = J^m \partial _m \omega - (\text {d}J^m) \omega _m. \end{aligned}$$
(2.10)
We denote the real parameters of the compactification by \(y^a\) and complex parameters by \(y^\alpha ,y^{\overline{\beta }}\). If the parameters are fixed to \(y=y_0\), the second term in (2.10) vanishes and the relation simplifies to
$$\begin{aligned} \text {d}^c \omega |_{y=y_0} = \text {i}(\partial - \overline{\partial }) \omega . \end{aligned}$$
(2.11)
However, when complex structure is varied \(\partial _\alpha (\text {d}J) = 2\text {i}\partial \Delta _\alpha \), the second term in (2.10) is nonzero and is important as it contributes to the equations satisfied by the moduli.

2.4 Derivatives of A

The heterotic structure Open image in new window depends on parameters. This means the gauge connection A and its gauge transformations \(\Phi \) depend on parameters. As constructed in [5], the gauge covariant way of describing a deformation of A is given by introducing a covariant derivative
$$\begin{aligned} D_a A=\partial _a A - \text {d}_A A_a, \end{aligned}$$
(2.12)
where \(A^\sharp = A_a \text {d}y^a\) is a connection on the moduli space with a transformation law
$$\begin{aligned} {}^\Phi A_a =\Phi (A_a - Y_a)\Phi ^{-1},\qquad Y_a = \Phi ^{-1}\,\partial _a \Phi . \end{aligned}$$
(2.13)
With this transformation property \(D_a A\) transforms homogeneously under (2.6):
$$\begin{aligned} D_a A~\rightarrow ~ \Phi \,D_a A\,\Phi ^{-1}. \end{aligned}$$
The moduli space \(\mathcal {M}\) is complex, and we introduce a complex structure \(y^a = (y^\alpha , y^{\overline{\beta }})\). When parameters vary complex structure the holomorphic type of forms change, and the covariant derivatives \(D_\alpha \mathcal {A}\) is no longer gauge covariant. This is remedied by defining a generalisation, termed the holotypical derivative Open image in new window :where the vanishing of Open image in new window follows from (2.5). It follows from the definition that under a gauge transformation the holotypical derivative transforms in the desired formWithout the extra term \(- \Delta _\alpha {}^\mu \mathcal {A}^\dag _\mu \) in the holotypical derivative, this property does not hold as \(\overline{\partial }\) fails to commute with \(\partial _\alpha \).
The holotypical derivative can be extended to act on (pq)-forms. Define
$$\begin{aligned} W_m^{r,s} = \frac{1}{r!s!}\, W_{m \mu _1\cdots \mu _r {\bar{\nu }}_1\cdots {\bar{\nu }}_s} \text {d}x^{\mu _1\cdots \mu _r{\bar{\nu }}_1\cdots {\bar{\nu }}_s}, \end{aligned}$$
and understand \(W_m^{r,s} = 0\) if r or s are negative or \(r{+}s>n{-}1\). The holotypical derivatives are then given byThe holotypical derivative has the nice feature that it preserves holomorphic type:We use \(D_\alpha \) 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:
$$\begin{aligned} D_M F=\partial _M F + \left[ A^\sharp _M, F\right] = \text {d}_A (D_M A), \end{aligned}$$
However, the holotypical derivative of, for example, \(F^{(0,2)}\) givesand this is known as the Atiyah constraint.

2.5 Derivatives of H

It is of use to compute derivatives of H with respect to parameters. First define a gauge covariant derivative of B via
$$\begin{aligned} D_a B = \partial _a B -\frac{{\alpha ^{\backprime }\,}}{4} \,\text {Tr}\,( A_a\,\text {d}A), \end{aligned}$$
(2.16)
With this choice, we have a gauge transformation law for \(D_a B\) that is parallel to the gauge transformation (2.8) for B:
$$\begin{aligned} {}^\Phi D_a B= D_a B + \frac{{\alpha ^{\backprime }\,}}{4}\Big (\! \,\text {Tr}\,( Y D_a A ) + U_a \Big ). \end{aligned}$$
(2.17)
The second and third derivatives are defined to transform in a natural way inherited from that of \(D_a B\)
$$\begin{aligned} \begin{aligned} D_a D_b B&= \partial _a D_b B - \frac{{\alpha ^{\backprime }\,}}{4} \,\text {Tr}\,( D_b A \text {d}A_a),\\D_cD_a D_b B&= \partial _c D_a D_b B - \frac{{\alpha ^{\backprime }\,}}{4} \,\text {Tr}\,\left( D_b D_a A\, \text {d}A^\sharp _c\right) . \end{aligned} \end{aligned}$$
(2.18)
A gauge-invariant quantity Open image in new window is the formed from \(D_a B\) with \(\text {d}b_a\) an exact form. The exact form comes from the fact the physical quantity is \(\text {d}B\), and so in writing Open image in new window there is a corresponding ambiguity. It is a simple exercise to note that \(\partial _a H\) is given by the expressionIn terms of holomorphic parameters, we introduce the holotypical derivative and findThe second derivative is given bywhere Open image in new window and in terms of the B-fieldDespite appearances the right-hand side is symmetric in ab after one uses thatand soThe third derivative of H is given bywhere Open image in new window and in terms of the B-field:For similar reasons to the second derivative above, this is actually symmetric in abc, but not made manifest in this expression for compactness.

2.6 Derivatives of \(\text {d}^c \omega \)

The derivative of \(\text {d}^c \omega \) in (2.10) with respect to parameters isWe can evaluate (2.26) for a given complex structure, denoted \(|_{y_0}\) in the corresponding complex coordinates of Open image in new window :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 (1, 2) componentsThe second derivative of \(\text {d}^c\omega \), given by differentiating (2.26) and then evaluated on a fixed complex structure, is
$$\begin{aligned} \begin{aligned} \left( \partial _\alpha \partial _\beta \text {d}^c\omega \right) |_{y_0}&=~ 2\text {i}\left( \Delta _{\alpha \beta }{}^\mu \right) (\partial _\mu \omega ) + 2\text {i}\Delta _\alpha {}^\mu \partial _\mu (\partial _\beta \omega ) \\&\quad +\,2\text {i}\Delta _\beta {}^\mu \partial _\mu (\partial _\alpha \omega ) - 2\text {i}\,\left( \partial \Delta _{\alpha \beta }{}^\mu \right) \omega _\mu \\&\quad +\,\text {i}(\partial -\overline{\partial }) \omega _{,\alpha \beta } -2\text {i}\left( \partial \Delta _\alpha {}^\mu \right) \omega _{\mu ,\beta } - 2\text {i}\left( \partial \Delta _\beta {}^\mu \right) \omega _{\mu ,\alpha }. \end{aligned} \end{aligned}$$
We will have need for the (0, 3)-component:
$$\begin{aligned} \begin{aligned} \Big ( \partial _\alpha \partial _\beta \text {d}^c\omega \Big )|_{y_0}^{0,3}&= 2\text {i}( \Delta _\alpha {}^\mu \Delta _\beta {}^\nu + \Delta _\beta {}^\mu \Delta _\alpha {}^\nu )\partial _\mu \omega _\nu ^{0,1} -\text {i}\overline{\partial }( \omega _{,\alpha \beta })^{0,2}. \end{aligned} \end{aligned}$$
(2.28)

2.7 Supersymmetry relations

One can apply these results to compute how the supersymmetry condition \(\text {d}^c \omega = H\) relates the variations of fields. The parameter space coordinates are corrected at order \({\alpha ^{\backprime }\,}\). Differentiating with respect to these corrected coordinates, Eq. (2.9) gives rise to relations between first-order deformations of fields:where \(\gamma _\alpha ^{1,1}\) is \(\text {d}\)-closed (1, 1)-form, and \(k_\alpha ^{0,1}\) and \(l_\alpha ^{0,1}\) are some (0, 1)-forms. As discussed in [5] in \({\alpha ^{\backprime }\,}\)-perturbation theory, Open image in new window when appropriately gauge fixed.
The heterotic structure are holomorphic functions of parameters. This can be compactly stated as

3 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 \(A_{{\mathfrak {e}}_8}\) decomposes under \({\mathfrak {e}}_8\oplus {\mathfrak {e}}_8\supset {\mathfrak {g}}\oplus {\mathfrak {h}}\), using this to form a KK ansatz. For simplicity we will suppress writing the second \({\mathfrak {e}}_8\) 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.

3.1 Decomposing A under \( {\mathfrak {g}}\oplus {\mathfrak {h}}\subset {\mathfrak {e}}_8\)

The branching rule for \( {\mathfrak {g}}\oplus {\mathfrak {h}}\subset {\mathfrak {e}}_8\) is
$$\begin{aligned} {\mathrm{ad}}_{{\mathfrak {e}}_8} = ({{\varvec{1}}}, {\mathrm{ad}}_{{\mathfrak {h}}}) \oplus ({\mathrm{ad}}_{{\mathfrak {g}}}, {{\varvec{1}}}) \oplus _i ({{\overline{\varvec{R_i}}}}, {{\varvec{r_i}}}) \oplus ({{\varvec{R_i}}}, {{\overline{\varvec{r_i}}}}). \end{aligned}$$
(3.1)
where \(\mathrm{ad }\) denotes the relevant adjoint representation. The matter fields transform in representations of \({\mathfrak {g}}\oplus {\mathfrak {h}}\). We denote these representations by \({{\varvec{r_i}}}\) for \({\mathfrak {h}}\) and \({{\varvec{R_i}}}\) of \({\mathfrak {g}}\), respectively. Denote dimensions in the obvious way \(\dim {{\varvec{r_i}}} = r_i\) and \(\dim {{\varvec{R_i}}} = R_i\). We have allowed for a sum over all the relevant representations \({{\varvec{r_i}}}\) and \({{\varvec{R_i}}}\), including, for example, pseudo-real representations. For simplicity we will often suppress the sum and write a single matter field representations of \({\mathfrak {g}}\oplus {\mathfrak {h}}\) and its conjugate. The generalisation is obvious.
The matrix presentation of the adjoint of \({\mathfrak {e}}_8\) is complicated. For the moment let us suppose we can write the generator in simplified form as
$$\begin{aligned} T_{{\mathfrak {e}}_8} = \begin{pmatrix} T_{{\mathfrak {h}}} &{} \quad 0\\ 0 &{} \quad T_{{\mathfrak {g}}} \end{pmatrix}. \end{aligned}$$
(3.2)
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
$$\begin{aligned} A_{{\mathfrak {e}}_8} = A_{\mathfrak {h}}\oplus B_{\mathfrak {g}}, \end{aligned}$$
where \(A_{\mathfrak {h}}= A_m(x) \text {d}x^m\) is the \({\mathfrak {h}}\) gauge field and we take it to have legs purely along the CYM; \(B_{\mathfrak {g}}= B_e(X) \text {d}X^e\) is the Open image in new window spacetime gauge field valued in \({\mathfrak {g}}\). When there is no ambiguity we will drop the \({\mathfrak {g}}, {\mathfrak {h}}\) subscripts. We indicate this combined Lorentz and gauge structure schematically in matrix notation
$$\begin{aligned} A_{{\mathfrak {e}}_8} = \begin{pmatrix} A_{{\mathfrak {h}}} &{} \quad 0 \\ 0 &{} \quad B_{{\mathfrak {g}}} \\ \end{pmatrix} = A_m \text {d}x^m + B_e \text {d}X^e. \end{aligned}$$
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 \(A_{{\mathfrak {e}}_8}\) is of the form
$$\begin{aligned} \begin{aligned} \delta A_{{\mathfrak {e}}_8}&= \delta A_{(\mathrm{ad }_{\mathfrak {h}},{{\varvec{1}}})} \oplus _i \delta A_{({{\varvec{r_i}}},{{\overline{\varvec{R_j}}}})}\oplus _j \delta A_{({{\overline{\varvec{r_j}}}},{{\varvec{R_j}}})} \oplus \delta A_{({{\varvec{1}}}, \mathrm{ad }_{\mathfrak {g}})} \\&=~ \begin{pmatrix} \delta A_{{\mathfrak {h}}} &{} \quad \Phi \\ \Psi &{} \quad \delta B_{{\mathfrak {g}}} \\ \end{pmatrix}, \qquad \Phi _{r\times R} = \begin{pmatrix} \Phi ^1&\cdots&\Phi ^R \end{pmatrix}, \qquad \Psi _{R\times r} = \begin{pmatrix} \Psi ^1 \\ \vdots \\ \Psi ^R \end{pmatrix}, \end{aligned}\qquad \end{aligned}$$
(3.3)
where again we have used matrix notation to indicate the structure. This gives us an intuition for understanding the transformation properties under a \({\mathfrak {h}}\oplus {\mathfrak {g}}\) rotation:
$$\begin{aligned} \begin{aligned} \delta A_{{\mathfrak {e}}_8} ~&\rightarrow ~ \delta A_{{\mathfrak {e}}_8} + [T_{{\mathfrak {e}}_8} , \,\delta A_{{\mathfrak {e}}_8}] \\&=~ \delta A_{{\mathfrak {e}}_8} + \begin{pmatrix} [ T_{{\mathfrak {h}}},\, \delta A_{\mathfrak {h}}] &{} \quad T_{{\mathfrak {h}}} \,\Phi - \Phi \, T_{{\mathfrak {g}}} \\ T_{{\mathfrak {g}}}\, \Psi - \Psi \, T_{{\mathfrak {h}}} &{} \quad [ T_{{\mathfrak {g}}}, \,\delta B_{\mathfrak {g}}]. \end{pmatrix} \end{aligned} \end{aligned}$$
(3.4)
We see that fields transform under \({\mathfrak {g}}\oplus {\mathfrak {h}}\) as:
  1. 1.

    \(\delta A_{\mathfrak {h}}\) transforms as \(({{\varvec{1}}},\mathrm{ad }_{{\mathfrak {h}}})\), and \(\delta B_{\mathfrak {g}}\) transforms as \((\mathrm{ad }_{\mathfrak {g}}, {{\varvec{1}}})\);

     
  2. 2.

    \(\Phi \) is in the \(({{\overline{\varvec{R}}}},{{\varvec{r}}})\) and is a \(r\times R\) matrix. Column vectors are in the fundamental; row vectors the antifundamental. For example, \(\Phi ^{1},\ldots ,\Phi ^R\) are each column vectors transforming in the \({{\varvec{r}}}\) of \({\mathfrak {h}}\).

     
  3. 3.

    \(\Psi \) is in the \(({{\varvec{R}}},{{\overline{\varvec{r}}}})\) and is a \(R\times r\) matrix with \(\Psi ^1,\ldots ,\Psi ^R\) row vectors and so in the \({{\overline{\varvec{r}}}}\) of \({\mathfrak {h}}\).

     
  4. 4.

    To preserve the structure \({\mathfrak {g}}\oplus {\mathfrak {h}}\), \(\delta A_{\mathfrak {h}}\) has legs only on the CYM, while \(\delta B_{\mathfrak {g}}\) has legs only in Open image in new window .

     
The reality condition is \(\delta A_{{\mathfrak {e}}_8}^\dag = - \,\delta A_{{\mathfrak {e}}_8}\) which implies
$$\begin{aligned} \delta A_{\mathfrak {h}}^\dag = - \,\delta A_{\mathfrak {h}}, \qquad \Phi ^\dag = - \,\Psi , \qquad \delta B_{\mathfrak {g}}^\dag = - \,\delta B_{\mathfrak {g}}. \end{aligned}$$
Decomposing this condition according to the holomorphic type of Open image in new window we have:
$$\begin{aligned} \delta A_{\mathfrak {h}}= \delta \mathcal {A}- \delta \mathcal {A}^\dag , \qquad \Phi = \phi - \psi ^\dag , \quad \Psi =\psi - \phi ^\dag , \end{aligned}$$
where
$$\begin{aligned} \phi = \Phi _{0,1}, \quad \psi = \Psi _{0,1}, \end{aligned}$$
while the (1, 0)-components are fixed by reality: \(\Phi _{1,0} = -\psi ^\dag \), \(\Psi _{1,0} = -\,\phi ^\dag \).4
The \({\mathfrak {e}}_8\) field strength is
$$\begin{aligned} F_{{\mathfrak {e}}_8} = \text {d}A_{{\mathfrak {e}}_8} + A_{{\mathfrak {e}}_8}^2. \end{aligned}$$
Its background value decomposes according to its orientation of legs:
$$\begin{aligned} \begin{aligned} F_{{\mathfrak {e}}_8 ef} \,\text {d}X^e \,\text {d}y^f&=\text {d}B_{\mathfrak {g}}+ B_{\mathfrak {g}}^2,\\ F_{{\mathfrak {e}}_8 mn} \,\text {d}x^m \,\text {d}x^n&= \text {d}A_{\mathfrak {h}}+ A_{\mathfrak {h}}^2,\\ F_{{\mathfrak {e}}_8 me} \,\text {d}x^m \,\text {d}X^e&= \left( \partial _m B_{{\mathfrak {g}}\,e} - \partial _e A_{{\mathfrak {h}}\,m} + A_{{\mathfrak {h}}\,m} B_{{\mathfrak {g}}\,e} - B_{{\mathfrak {g}}\,e} A_{{\mathfrak {h}}\,m}\right) \,\text {d}x^m \,\text {d}X^e = 0.\\ \end{aligned} \end{aligned}$$
(3.5)
Under \(A_{{\mathfrak {e}}_8} \rightarrow A_{{\mathfrak {e}}_8} + \delta A_{{\mathfrak {e}}_8}\),
$$\begin{aligned} \begin{aligned} \delta F_{{\mathfrak {e}}_8}&= \text {d}_{A_{{\mathfrak {e}}_8}} \delta A_{{\mathfrak {e}}_8} \\&=~ \begin{pmatrix} \text {d}_A( \delta A_{\mathfrak {h}}) &{} \quad \text {d}\Phi + A_{\mathfrak {h}}\Phi + \Phi B_{\mathfrak {g}}\\ \text {d}\Psi + \Psi A_{\mathfrak {h}}+ B_{\mathfrak {g}}\Psi &{} \quad \text {d}_{B_{\mathfrak {g}}} (\delta B_{\mathfrak {g}}) \end{pmatrix}. \\ \end{aligned} \end{aligned}$$
(3.6)
Consider \(\delta F_{{\mathfrak {e}}_8}\) oriented along Open image in new window . At this point we drop the \({\mathfrak {h}}, {\mathfrak {g}}\) subscripts on AB. Then, the equations of motion require \(F_{{\mathfrak {e}}_8}\) be (1, 1) implying any (0, 2)-component must satisfy,The off-diagonal terms tell us the fields \(\phi ,\psi \) are holomorphic sections of Open image in new window . We will occasionally introduce index to make manifest the fact that \(\phi \) is in the \(({{\overline{\varvec{R}}}}, {{\varvec{r}}})\) of \({\mathfrak {g}}\oplus {\mathfrak {h}}\) by writing \(\phi ^{i\,{\overline{M}}}\) where \(i,{\bar{\jmath }}=1, \ldots , {r}\) for representations \({{\varvec{r}}}\) of \({\mathfrak {h}}\); \(M,{\overline{N}}=1,\ldots , R\) for representations \({{\varvec{R}}}\) of \({\mathfrak {g}}\). Then, for example,We have introduced the \(\overline{\partial }\)-cohomology group Open image in new window , with forms valued in the \({\mathfrak {h}}\)-subbundle of Open image in new window whose fibres are the representation \({{\varvec{r}}}\) of \({\mathfrak {h}}\). The \({{\varvec{r}}}\) index \(i,{\bar{\jmath }}\) is implicitly summed.
We could also study the equations of motion for \(\delta A_{{\mathfrak {e}}_8}\) \( \text {d}^\dag _A (\text {d}_A \delta A_{{\mathfrak {e}}_8}) = 0. \) Choosing the gauge \(\text {d}_A^\dag \delta A_{{\mathfrak {e}}_8} = 0\) we see \( \Box _A \delta A_{{\mathfrak {e}}_8} = 0. \) For example, on the \(\phi \) matter field this gives
$$\begin{aligned} \left( \overline{\partial }_\mathcal {A}\overline{\partial }_\mathcal {A}^\dag + \overline{\partial }_\mathcal {A}^\dag \overline{\partial }_\mathcal {A}\right) \phi = 0. \end{aligned}$$
This has solution if \(\phi \) is harmonic element of Open image in new window . This is slightly stronger than the cohomology relation (3.7).
Expand the fields \(\phi \) and \(\psi \) in a harmonic basis for Open image in new window and Open image in new window , respectively:where Open image in new window and Open image in new window are harmonic formswhile \(C^\xi \) and Open image in new window are valued in \({{\overline{\varvec{R}}}}\) and \({{\varvec{R}}}\), respectively.

For example, consider the standard embedding. Then, Open image in new window and Open image in new window ; Open image in new window with the Open image in new window . \(C^\xi \) and Open image in new window are in the \({{\overline{\varvec{R}}}} = {{\overline{\varvec{27}}}}\) and \({{\varvec{R}}} = {{\varvec{27}}}\).

We need to satisfy the reality condition \(\Phi ^\dag = -\Psi \), which forces \(\phi ^\dag =- \psi \) and so in terms of the Open image in new window basis:We denote conjugation through the barring of the indices. For example, \(\phi _{\overline{\xi }}= (\phi _\xi )^\dag \) is a (1, 0)-form valued in \({{\overline{\varvec{r}}}}\) of \({\mathfrak {h}}\) and \(C^{\overline{\xi }}= (C^\xi )^\dag \) is in the \({{\varvec{R}}}\) of \({\mathfrak {g}}\).

3.2 The matter field metric from reducing Yang-Mills, \(\mathcal {L}_F\)

The spirit of KK reduction is to promote the coefficients to spacetime fields: \(Y^\alpha (X)\), Open image in new window , and integrate over the six-dimensional manifold to get an effective four-dimensional theory. With the conventions of [5], the \(d=10\) \({\mathfrak {e}}_8\) 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
$$\begin{aligned} \begin{aligned} \,\text {Tr}\,|\delta F_{{\mathfrak {e}}_8}|^2&= \,\text {Tr}\,\left( \text {d}_{A}( \delta A )\,\star \,\text {d}_{A}( \delta A) \right) + \,\text {Tr}\,\left( \text {d}_{A+B} \Phi \, \star \,\text {d}_{A+B} \Psi \right) \\&\quad +\,\,\text {Tr}\,\left( \text {d}_{A+B} \Psi \,\star \, \text {d}_{A+B} \Phi \right) + \,\text {Tr}\,\left( \text {d}_{B} \delta B \, \star \,\text {d}_{B} \delta B \right) , \end{aligned} \end{aligned}$$
(3.12)
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:
$$\begin{aligned} \begin{aligned} \text {d}_{A+B} \Phi&= (\partial _e \Phi + \Phi B_e) \,\text {d}X^e + (\partial _M \Phi _N + A_M \Phi _N)\, \text {d}x^M \text {d}x^N \\&= \widehat{\mathcal {D}}_e \Phi \,\text {d}X^e+ \text {d}_A \Phi ,\\ \text {d}_{A+B} \Psi&= \widehat{\mathcal {D}}_e \Psi \, \text {d}X^e + \text {d}_A \Psi ,\\ \end{aligned} \end{aligned}$$
(3.13)
where \(\widehat{\mathcal {D}}_e\) is the spacetime \({\mathfrak {g}}\)-covariant derivative and \(\text {d}_A\) the \({\mathfrak {h}}\)-covariant derivative. Hence, using \( \,\text {Tr}\,|\delta F|^2 = \frac{1}{2}\,\text {Tr}\,\delta F_{MN} \delta F^{MN} = 2 \,\text {Tr}\,\delta F_{e\mu } \delta F^{e\mu }, \) where \(\,\text {Tr}\,(\delta F_{e\mu } \delta F^{e\mu } ) = \,\text {Tr}\,(\delta F_{e{\bar{\nu }}} \delta F^{e{\bar{\nu }}})\), 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
$$\begin{aligned} \begin{aligned} \,\text {Tr}\,|\delta F_{{\mathfrak {e}}_8}|^2 = - 2 \,\text {Tr}\,\left( \widehat{\mathcal {D}}_e \Phi _{\bar{\mu }}\, \widehat{\mathcal {D}}^e \Phi ^{\dagger \,{\bar{\mu }}}\right) - 2\,\text {Tr}\,\left( \widehat{\mathcal {D}}_e \Psi _{\bar{\mu }}\, \widehat{\mathcal {D}}^e \Psi ^{\dagger \,{\bar{\mu }}}\right) . \end{aligned} \end{aligned}$$
(3.14)
We have used the reality condition \(\Phi ^\dag = -\Psi \). The matter fields have a KK ansatz, given by (3.10), which when substituted into each of the above terms giveswhere indices for the representation \({{\varvec{R}}}\) and \({{\varvec{r}}}\) are explicit. The trace projects onto invariants constructed by the Krönecker delta functions \(\delta _{i{\bar{\jmath }}}\) and \(\delta _{M{\overline{N}}}\). In the following we will suppress the indices and delta symbols where confusion will not arise.
Substituting (3.14) and (3.15) into \(\mathcal {L}_F\) in (3.11), reintroducing the moduli contribution, calculated in [5], we find a kinetic term for both the matter fields and the moduli fields:
$$\begin{aligned} \begin{aligned} \mathcal {L}_{F}&= -2G_{\alpha {\overline{\beta }}} \partial _e Y^\alpha \,\partial ^e Y^{\overline{\beta }}- 2G_{\xi {\overline{\eta }}} \, \widehat{\mathcal {D}}_e C^{\xi }\, \widehat{\mathcal {D}}^e C^{{\overline{\eta }}} - 2G_{\sigma {\overline{\tau }}} \, \widehat{\mathcal {D}}_e D^{{\overline{\tau }}}\, \widehat{\mathcal {D}}^e D^{\sigma }, \end{aligned} \end{aligned}$$
(3.16)
from which we may identify the moduli space metric and matter field metric
$$\begin{aligned} \text {d}s^2 = 2G_{\alpha {\overline{\beta }}} \text {d}y^\alpha \,\text {d}y^{\overline{\beta }}+ 2G_{\xi {\overline{\eta }}} \, \text {d}C^{\xi }\text {d}C^{{\overline{\eta }}} + 2G_{\sigma {\overline{\tau }}} \, \text {d}D^{\sigma }\, \text {d}D^{{\overline{\tau }}}, \end{aligned}$$
(3.17)
where we denote the coordinates of the moduli space \(\mathcal {M}\) by \(y^\alpha , y^{\overline{\beta }}\), and without wanting to clutter formulae, denote the coordinates of the matter fields by \(C^\xi , D^\sigma \)—any 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 metricThere is no trace as the integrands are written in the form \({{\overline{\varvec{r}}}}\cdot {{\varvec{r}}}\). Although we have indicated the result for a single representation \({{\varvec{r}}}\) and \({{\overline{\varvec{r}}}}\), the result generalises to a sum over representations of \({\mathfrak {h}}\).
For any two-form \(\mathcal {F}\) there is a relation \( \omega \star \mathcal {F}= \frac{1}{2}\mathcal {F}\omega ^2. \) Using this and \(\omega ^{\mu {\bar{\nu }}} = -\text {i}g^{\mu {\bar{\nu }}}\) we find
$$\begin{aligned} \begin{aligned} \,\phi _{\overline{\eta }}\star \phi _\xi&=-\text {i}\omega \star \phi _\xi ^i \, \phi ^{\bar{\jmath }}_{\overline{\eta }}\, \delta _{i{\bar{\jmath }}} =- \frac{\text {i}}{2} \omega ^2 \,\,\text {Tr}\,(\phi _\xi \,\phi _{\overline{\eta }}),\\ \,\psi _\sigma \star \psi _{\overline{\tau }}&=-\text {i}\omega \star \psi _\sigma ^{\bar{\jmath }}\, \psi ^i_{\overline{\tau }}\, \delta _{i{\bar{\jmath }}}= -\frac{\text {i}}{2} \omega ^2 \,\,\text {Tr}\,(\psi _\sigma \,\psi _{\overline{\tau }}). \end{aligned} \end{aligned}$$
(3.19)
We introduce the trace over \({{\varvec{r}}}\) indices in order to be able to write \(\phi _x, \psi _\sigma \) in any order. The matter field metrics are then expressible in a way closely resembling the moduli metric

4 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.

4.1 Fermion zero modes on Open image in new window

4.1.1 \({\mathfrak {so}}(3,1)\oplus {\mathfrak {su}}(3)\) spinors

The gaugino is a Majorana–Weyl spinor \(\varepsilon \) which has zero modes on Open image in new window . The Lorentz algebra is \({\mathfrak {so}}(3,1)\oplus {\mathfrak {so}}(6) \subset {\mathfrak {so}}(9,1)\) under which \(\varepsilon \) is
$$\begin{aligned} \varepsilon = \zeta ' \otimes \lambda '\oplus \zeta \otimes \lambda ~\cong ~ \begin{pmatrix} \zeta '\\ 0 \end{pmatrix}\otimes \lambda ' + \begin{pmatrix} 0\\ {\overline{\zeta }}\end{pmatrix}\otimes \lambda , \end{aligned}$$
where \(\lambda \) and \(\lambda '\) are in the \({{\varvec{4}}}\) and \({{\varvec{4}}}'\) of \({\mathfrak {so}}(6)\); \(\zeta \) and \(\zeta '\) are in the \({{\varvec{2}}}\) and \({{\varvec{2}}}'\) of \({\mathfrak {so}}(3,1)\). Where possible we use the 2-component Weyl notation for \(\zeta ,\zeta '\) and always leave the \({\mathfrak {so}}(6)\) spinor indices implicit. We write \(\oplus \) in this context to reflect the embedding of 2-component spinors into a 4-component notation as shown by the second equality. The barring of 2-component spinors, and dotting the spinor index, comes with complex conjugation \((\zeta ^a)^* = {\overline{\zeta }}^{\dot{a}}\) as described in appendix. The fermions \(\zeta ,\zeta '\) are Grassmann odd and under complex conjugation are interchanged without paying the price of a sign.
The Majorana condition implies \(\zeta \otimes \lambda \) are determined in terms of \(\zeta '\otimes \lambda '\). With our conventions this means
$$\begin{aligned} {\overline{\zeta }}^{\dot{a}}\otimes \lambda = {\overline{\zeta }}'{}^{\dot{a}}\otimes \lambda '^c, \end{aligned}$$
(4.1)
where \(\lambda '^c\) denotes taking the \({\mathfrak {so}}(6)\) Majorana conjugate. The Majorana–Weyl spinor \(\varepsilon \) can now be written solely in terms of say \(\zeta ',\lambda '\):
$$\begin{aligned} \varepsilon = \begin{pmatrix} \zeta _a'\\ 0 \end{pmatrix}\otimes \lambda '~ + \begin{pmatrix} 0\\ {\overline{\zeta }}'^{\dot{a}}\end{pmatrix}\otimes \lambda '{}^c . \end{aligned}$$
(4.2)
The presence of \(\mathcal {N}=1\) spacetime supersymmetry means there is a globally well-defined spinor on Open image in new window . This implies the existence of an \({\mathfrak {su}}(3)\)-structure on Open image in new window . Under \({\mathfrak {so}}(6) \rightarrow {\mathfrak {su}}(3)\) the spinors \(\lambda ,\lambda '\) decompose according to the branching rule \({{\varvec{4}}} = {{\varvec{3}}} \oplus {{\varvec{1}}}\), and \( {{\varvec{4}}}' = {{\overline{\varvec{3}}}} \oplus {{\varvec{1}}}\), which we write as
$$\begin{aligned} \lambda = \lambda _{{{\varvec{3}}}} \oplus \lambda _+,\qquad \qquad \lambda ' = \lambda _{{{\overline{\varvec{3}}}}} \oplus \lambda _-. \end{aligned}$$
The spinors \(\lambda _+, \lambda _-\) are the nowhere vanishing \({\mathfrak {su}}(3)\) invariant spinors. As established in appendix, we can express \(\lambda _{{{\varvec{3}}}}, \lambda _{{{\overline{\varvec{3}}}}}\) in terms of \(\lambda _\pm \) and gamma matrices
$$\begin{aligned} \lambda _{{{\varvec{3}}}} = \Lambda _\mu \gamma ^\mu \lambda _-, \qquad \lambda _{{{\overline{\varvec{3}}}}} = \Lambda '_{\bar{\mu }}\gamma ^{\bar{\mu }}\lambda _+, \end{aligned}$$
(4.3)
where \(\{\gamma ^\mu , \gamma ^{\bar{\nu }}\} = g^{\mu {\bar{\nu }}}\) and \(\Lambda _\mu , \Lambda '_{\bar{\mu }}\) are components of 1-forms on Open image in new window . Appendix details the construction of the \({\mathfrak {su}}(3)\) bilinears:
$$\begin{aligned} \lambda _+^\dag \gamma ^\mu \gamma ^{\bar{\nu }}\lambda _+ = g^{\mu {\bar{\nu }}}, \qquad \lambda _-^\dag \gamma ^{\bar{\nu }}\gamma ^\mu \lambda _- = g^{\mu {\bar{\nu }}}, \end{aligned}$$
(4.4)
and
$$\begin{aligned} \Omega _{\mu \nu \rho }= - e^{-\text {i}\phi }\,||\Omega ||\, \lambda _-^\dag \gamma _{\mu \nu \rho } \lambda _+,\qquad {\overline{ \Omega }}_{\overline{\mu \nu \rho }}= e^{\text {i}\phi }\,||\Omega ||\, \lambda _+^\dag \gamma _{\overline{\mu \nu \rho }} \lambda _-, \end{aligned}$$
(4.5)
where \(\phi \) accounts for a relative phase difference between \(\lambda _-\) and \(\Omega \). Under the gauge symmetry \(\Omega \rightarrow \mu \Omega \), the fermions \(\lambda _\pm \) transform as a phase:
$$\begin{aligned} \lambda _\pm \rightarrow e^{\pm \text {i}\xi /2} \lambda _\pm , \quad \mu = |\mu | e^{\text {i}\xi }. \end{aligned}$$
The bilinears above respect this gauge symmetry.
Given (4.3), the action of Majorana conjugation is
$$\begin{aligned} \begin{aligned} \lambda _\pm ^c&= -\text {i}\lambda _\mp , \qquad \lambda _{{{\overline{\varvec{3}}}}}^c = \text {i}\Lambda '{}_\mu ^\dag \gamma ^\mu \lambda _-, \qquad \lambda _{{{\varvec{3}}}}^c = \text {i}\Lambda _{\bar{\mu }}^\dag \gamma ^{\bar{\mu }}\lambda _+. \end{aligned} \end{aligned}$$
(4.6)

4.1.2 Kaluza Klein ansatz

\(\varepsilon \) is in the adjoint of \({\mathfrak {e}}_8\). Consequently, it decomposes under \({\mathfrak {g}}\oplus {\mathfrak {h}}\subset {\mathfrak {e}}_8\) 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 \({\mathfrak {so}}(3,1) \oplus {\mathfrak {su}}(3) \subset {\mathfrak {so}}(9,1)\)
$$\begin{aligned} \begin{aligned} \varepsilon :\quad {{\varvec{16}}}&= \left( {{\varvec{2}}} \otimes {{\varvec{1}}} \oplus {{\varvec{2}}}'\otimes {{\varvec{1}}}\right) \oplus \left( {{\varvec{2}}}\otimes {{\varvec{3}}} \right) \oplus \left( {{\varvec{2}}}'\otimes {{\overline{\varvec{3}}}}\right) ,\\A:\quad {{\varvec{10}}}&= {{\varvec{4}}} \oplus {{\varvec{3}}} \oplus {{\overline{\varvec{3}}}}, \end{aligned} \end{aligned}$$
(4.7)
and for the gauge algebra \({\mathfrak {e}}_8 \supset {\mathfrak {g}}\oplus {\mathfrak {h}}\):
$$\begin{aligned} {\mathrm{ad}}_{{\mathfrak {e}}_8} = ({{\varvec{1}}}, {\mathrm{ad}}_{{\mathfrak {h}}}) \oplus ({\mathrm{ad}}_{{\mathfrak {g}}}, {{\varvec{1}}}) \oplus _i ({{\overline{\varvec{R_i}}}}, {{\varvec{r_i}}}) \oplus _i({{\varvec{R_i}}}, {{\overline{\varvec{r_i}}}}). \end{aligned}$$
We organise our study of the zero modes according to their representations under \({\mathfrak {so}}(3,1)\oplus {\mathfrak {su}}(3)\) and the gauge algebra \({\mathfrak {g}}\oplus {\mathfrak {h}}\). We continue the mnemonic of indicating the gauge structure through block matrices
$$\begin{aligned} \begin{aligned} \delta A&= \delta A_{(\mathrm{ad }_{\mathfrak {g}},{{\varvec{1}}})} \oplus _i \delta A_{({{\varvec{R_i}}},{{\overline{\varvec{r_i}}}})}\oplus _j \delta A_{({{\overline{\varvec{R_j}}}},{{\varvec{r_j}}})} \oplus \delta A_{({{\varvec{1}}}, \mathrm{ad }_{\mathfrak {h}})} \\&=~ \begin{pmatrix} \delta A_{\mathrm{ad }_{\mathfrak {h}}} &{} \quad \oplus _j \delta A_{({{\overline{\varvec{R_j}}}},{{\varvec{r_j}}})} \\ \oplus _i\delta A_{({{\varvec{R_i}}},{{\overline{\varvec{r_i}}}})} &{} \quad \delta A_{\mathrm{ad }_{{\mathfrak {g}}}} \end{pmatrix} = \begin{pmatrix} \delta \mathcal {A}- \delta \mathcal {A}^\dag &{} \quad \Phi \\ \Psi &{} \quad \delta B \end{pmatrix}, \\ \varepsilon&=~ \varepsilon _{(\mathrm{ad }_{\mathfrak {g}},{{\varvec{1}}})} \oplus _i \varepsilon _{({{\varvec{R_i}}},{{\overline{\varvec{r_j}}}})}\oplus _j \varepsilon _{({{\overline{\varvec{R_j}}}},{{\varvec{r_j}}})} \oplus \varepsilon _{({{\varvec{1}}}, \mathrm{ad }_{\mathfrak {h}})}\\&=~ \begin{pmatrix} ( \zeta '\otimes \lambda ' \oplus \zeta '{}^c\otimes \lambda '{}^c)_{({{\varvec{1}}},\mathrm{ad }_{\mathfrak {h}})} &{}\quad \oplus _j( \zeta '\otimes \lambda ' \oplus \zeta '{}^c\otimes \lambda '{}^c)_{({{\overline{\varvec{R_j}}}},{{\varvec{r_j}}})} \\ \oplus _i ( \zeta '\otimes \lambda ' \oplus \zeta '{}^c\otimes \lambda '{}^c)_{({{\varvec{R_i}}},{{\overline{\varvec{r_i}}}})} &{} \quad (\zeta '\otimes \lambda ' \oplus \zeta '{}^c\otimes \lambda '{}^c)_{(\mathrm{ad }_{\mathfrak {g}},{{\varvec{1}}})} \end{pmatrix}. \end{aligned} \end{aligned}$$
(4.8)
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 \(\oplus _i\) 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 \({\mathfrak {su}}(3)\)-structure and \({\mathfrak {g}}\oplus {\mathfrak {h}}\). The first type of zero modes is \({\mathfrak {su}}(3)\)-structure singlets and transforms in the \((\mathrm{ad }_{\mathfrak {g}},{{\varvec{1}}})\) with KK ansatz
$$\begin{aligned} \begin{aligned} \delta A_{{\mathfrak {e}}_8\,e}\, \text {d}X^e&= \begin{pmatrix} 0 &{} \quad 0 \\ 0 &{}\quad \delta B_e \text {d}X^e \end{pmatrix},\\ \varepsilon _{ (\mathrm{ad }_{\mathfrak {g}},{{\varvec{1}}})}&= \begin{pmatrix} 0 &{} \quad 0\\ 0 &{} \quad \zeta _{\mathrm{ad }_{\mathfrak {g}}} \end{pmatrix} \otimes \lambda _- \oplus \begin{pmatrix} 0 &{} \quad 0\\ 0 &{} \quad \text {i}{\overline{\zeta }}_{\mathrm{ad }_{\mathfrak {g}}} \end{pmatrix} \otimes \lambda _+, \end{aligned}~ \end{aligned}$$
(4.9)
where we use the first line of (B.36).
The second type of zero modes transforms as \({{\overline{\varvec{3}}}}\) under \({\mathfrak {su}}(3)\)-structure and as \( ({{\varvec{1}}},\mathrm{ad }_{{\mathfrak {h}}}) \oplus ({{\varvec{R}}},{{\overline{\varvec{r}}}}) \oplus ({{\overline{\varvec{R}}}},{{\varvec{r}}})\) under \({\mathfrak {g}}\oplus {\mathfrak {h}}\). There is a natural pairing between \(\delta A^{0,1}_{{\mathfrak {e}}_8}\) and \(\zeta '\otimes \lambda _{{{\overline{\varvec{3}}}}}\). 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 \(\Lambda _{\bar{\mu }}'\) in (4.3). We have denoted the anticommuting Open image in new window spinors by calligraphic letters \(\mathcal {Y}^\alpha \), \(\mathcal {C}^{\xi }\) and Open image in new window ; superpartners to Open image in new window . \(\mathcal {C}^{\xi }\) and Open image in new window are in the \({{\overline{\varvec{R}}}}\) and \({{\varvec{R}}}\) of \({\mathfrak {g}}\), while \(\mathcal {Y}^\alpha \) are neutral.
The spinor in the \({{\varvec{3}}}\) of \({\mathfrak {su}}(3)\)-structure is determined by Majorana conjugation \({\overline{\zeta }}^{\dot{a}}\otimes \lambda _{{{\varvec{3}}}} = {\overline{\zeta }}'^\dag {}^{\dot{a}}\otimes \lambda _{{{\overline{\varvec{3}}}}}^c\), expressed through (4.1) and (4.6):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) givingThe \(\oplus \) reflects the embedding of the 2-component Weyl spinors into a 4-component notation used in (B.36), (B.37).

4.2 Dimensional reduction of Open image in new window

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 termswhere Open image in new window is the Dirac conjugate. Open image in new window are the \(d=10\) gamma matrices, given asas described in appendix. Here \(\mu \) is a holomorphic index along Open image in new window .

4.2.1 Quadratic couplings

As the background is bosonic, we take A to be the background gauge field
$$\begin{aligned} A = \begin{pmatrix} \mathcal {A}- \mathcal {A}^\dag &{} \quad 0 \\ 0 &{} \quad B_e \text {d}X^e \end{pmatrix}. \end{aligned}$$
(4.15)
We start with the derivative operatorThe first term Open image in new window is computed using (4.2) and the third lines of (B.36), (B.37),Spinor and representation indices are contracted in the natural way.
The second term Open image in new window follows from the fourth lines of (B.36), (B.37) together with the relation in (A.5):Next we compute the reduction ofThe first terms follows the calculation of (4.17) after using (4.15) and \(\,\text {Tr}\,B_e = 0\) The second term Open image in new window mirrors the calculation of (4.20), using the background (4.15)We now put the terms together. The Open image in new window term comes from adding (4.17) and (4.19):where \({\hat{D}}_e\) is a spacetime covariant derivative appropriate to whatever representation if it acts onThe 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 Open image in new window is the Kähler potential, which on our background evaluates to be Open image in new window . The mass terms areThe 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 areSubstituting this in we findThe vanishing of \(m_{\alpha \beta }\) is guaranteed when Open image in new window , that is for bundle or Hermitian moduli. For complex structure parameters, if one can find a basis for parameters in which Open image in new window for complex structure moduli, so that \(\Delta _\alpha {}^\mu F_\mu = 0\), 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 \(\mathcal {D}_\alpha \chi _\beta \) is covariant with respect to gauge transformations \(\chi _\alpha \rightarrow \mu \, \chi _\alpha \). Also,
$$\begin{aligned} \mathfrak {a}_{\alpha \beta \gamma } = - \int \Omega \Delta _\alpha {}^\mu \Delta _\beta {}^\nu \Delta _\gamma {}^\rho \ \Omega _{\mu \nu \rho }, \qquad \mathfrak {a}_{\alpha \beta }{}^{\overline{\gamma }}= \mathfrak {a}_{\alpha \beta \gamma } G_{0}{}^{\gamma {\overline{\gamma }}}, \quad G_{0\,\alpha {\overline{\beta }}} ~=-\frac{\int \chi _\alpha \chi _{\overline{\beta }}}{\int \Omega {\overline{ \Omega }}}. \end{aligned}$$
andIn special geometry \(\mathfrak {a}_{\alpha \beta \gamma }\) is a Yukawa coupling for \({{\varvec{27}}}^3\) fields, playing the role of an intersection quantity relating derivatives of \(\chi _\alpha \) to \(\overline{\chi }_{\overline{\gamma }}\). \(G_0\) 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 \( m_{\alpha \beta } = 0. \)

4.2.2 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:The fluctuations only occur on the internal space Open image in new window . The gauge structure of \(\delta A\) is specified in (4.8). The calculation is a simple generalisation of the result (4.20) using the fourth line of (B.37).In the last two lines, we use the appropriate symmetric invariants to construct \({{\varvec{R}}}^3\) and \({{\overline{\varvec{R}}}}^3\).
Putting it together, normalising to agree with [31], we findwhere \(\mathrm{c.c}\) denotes the complex conjugate and the Yukawa couplings are given byThe result is straightforwardly extended to vacua with more complicated branching rules involving multiple representations \(\oplus _p {{\varvec{R}}}_p\). More Yukawa couplings appear—one for each invariant computed via the trace—but the integrand is of the same form as above.
The \({{\varvec{1}}}^3\) coupling vanishes classically. To see this, write \(\delta \mathcal {A}\) to second order in deformationsNote that Open image in new window and so the second term is appropriately symmetric in indices \(\alpha ,\beta \). A standard deformation theory argument related to the Kuranishi map implies the second-order deformation is unobstructed providedSubstituting into (5.4) one finds Open image in new window when \(\Delta _\alpha {}^\mu F_\mu = 0\). When \(\Delta _\alpha \ne 0\), 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 coupling Open image in new window also vanishes by demanding that \(\phi _\xi \), equivalently, Open image in new window , remain solutions of the equation of motion under a bundle deformation Open image in new window : Open image in new window . Hence, the singlet couplings vanish.

5 The final result: moduli, matter metrics and Yukawa couplings

The effective field theory has \(\mathcal {N}=1\) supersymmetry, with a gravity multiplet and a gauge symmetry \({\mathfrak {g}}\). The \(\mathcal {N}=1\) chiral multiplets consist of 5
  • \({\mathfrak {g}}\)-neutral scalar fields \(Y^\alpha \) and fermions \(\mathcal {Y}^\alpha \) corresponding to moduli;

  • \({\mathfrak {g}}\)-charged bosons \(C^\xi \) and fermions \(\mathcal {C}^\xi \) in the \({{\overline{\varvec{R}}}}\) of \({\mathfrak {g}}\);

  • \({\mathfrak {g}}\)-charged bosons \(D^\rho \) and fermions \(\mathcal {D}^\rho \) in the \({{\varvec{R}}}\) of \({\mathfrak {g}}\);

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:
$$\begin{aligned} \begin{aligned} \text {d}s^2&= 2G_{\alpha {\overline{\beta }}} \,\text {d}y^\alpha \otimes \text {d}y^{\overline{\beta }}, \\ G_{\alpha {\overline{\beta }}}&= \frac{1}{4V} \int \Delta _\alpha {}^\mu \star \Delta _{\overline{\beta }}{}^\nu \,\, g_{\mu {\bar{\nu }}} + \frac{1}{4V} \int \mathcal {Z}_\alpha \star \mathcal {Z}_{\overline{\beta }}\,+ \\&\quad + \,\frac{ {\alpha ^{\backprime }\,}}{4V}\int \,\text {Tr}\,\Big ( D_\alpha A \star D_{\overline{\beta }}A \Big ) - \frac{ {\alpha ^{\backprime }\,}}{4V}\int \,\text {Tr}\,\Big (D_\alpha \Theta \,\star \, D_{\overline{\beta }}\Theta ^\dag \Big ). \end{aligned} \end{aligned}$$
(5.2)
The metric terms for the fermionic superpartners to moduli \(\mathcal {Y}^\alpha \) 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 Open image in new window
The Yukawa nonzero couplings in (4.31) are

6 The superpotential and Kähler potential

The effective field theory has \(\mathcal {N}=1\) supersymmetry in Open image in new window , 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 \({\alpha ^{\backprime }\,}\)-corrected supergravity action. It isin which \(\omega \) is the Hermitian form of Open image in new window . The \({\alpha ^{\backprime }\,}\)-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 bywhere \(a,b=1,\ldots , R\) label the \({{\varvec{R}}}\) 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 formwhere the \(\,\text {Tr}\,\) projects onto the appropriate \({{\varvec{R}}}\)-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 Open image in new window gives no singlet couplings, and this means all parameter derivatives of Open image in new window vanish.
We would like to study a superpotential in a similar vein to the Kähler potential proposal (6.1). As ten-dimensional fields \(A_{{\mathfrak {e}}_8}\) and H depend on both parameters and matter fields. The fields \(\text {d}^{\text{ c }}\omega \) and \(\Omega \) are valued on Open image in new window 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 6in which the fields are regarded as functionals of the \(d=4\) chiral multiplets. The couplings in the effective field theory are specified by differentiating Open image in new window and evaluating the integral after fixing the parameters \(y=y_0\).
The rules for differentiating fields in the expressions for Open image in new window and Open image in new window with respect to parameters have been described in [5], which is complicated by virtue of \({\mathfrak {h}}\) 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
$$\begin{aligned} \partial _\xi A_{{\mathfrak {e}}_8} = \frac{\partial A_{{\mathfrak {e}}_8}}{\partial C^\xi } = \phi _\xi . \end{aligned}$$
It is important that we have written the ten-dimensional \({\mathfrak {e}}_8\) gauge field \(A_{{\mathfrak {e}}_8}\), and not \(A_{\mathfrak {h}}\), as this is the functional of the matter fields— Open image in new window —as illustrated in, for example, (3.3) and (3.10). The integrand in Open image in new window 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 \(\text {d}^{\text{ c }}\omega = H\), and the matter fields VEVs vanish Open image in new window .
For example, the tadpole matter and moduli couplings for a vacuum at the point \(y=y_0\) arewhere we use \(\partial _\alpha H\) in (2.20) and \(\partial _\alpha \text {d}^{\text{ c }}\omega \) in (2.26), and we evaluate them on some fixed \(y=y_0\).

As an ansatz Open image in new window 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. Open image in new window ; 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

Open image in new window is a section of the line bundle transforming under the gauge symmetry \(\Omega \rightarrow \mu (y) \Omega \) as Open image in new window where \(\mu (y)\) 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 \(\Omega \).

The supersymmetry relation \(H= \text {d}^{\text{ c }}\omega \) holds for all \(y_0\in \mathcal {M}\). Hence, derivatives of it vanish:8
$$\begin{aligned} \partial _{\alpha _1} \cdots \partial _{\alpha _n} (H-\text {d}^{\text{ c }}\omega )|_{y=y_0}= 0, \qquad \partial _{{\overline{\beta }}_1} \cdots \partial _{{\overline{\beta }}_n} (H-\text {d}^{\text{ c }}\omega )|_{y=y_0} = 0, \end{aligned}$$
(6.6)
where \(y^{\alpha _1}, \ldots , y^{\alpha _n}\) 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 derivativeThis vanishes on any supersymmetric background: Open image in new window is independent of moduli fields, and so Open image in new window does not give rise to any singlet couplings in agreement with the dimensional reduction.
An analogous argument, together with \(\Omega \) being holomorphic, shows that despite neither H nor \(\text {d}^c\omega \) being holomorphic, Open image in new window is a holomorphic function of fields. For example, the first-order derivative isUsing (6.6) all higher-order antiholomorphic derivatives of \(\Omega (H-\text {d}^{\text{ c }}\omega )\) vanish. It is also the case that Open image in new window for all \(n\ge 1\). So, Open image in new window 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 \( \text {d}^c \omega , \Omega \) do not depend on Open image in new window , while Open image in new window is given by (2.22) with \(D_a A \rightarrow \partial _\xi A = \phi _\xi \). As A depends linearly on the matter fields, all second derivatives vanish.
The Yukawa couplings Open image in new window are also all derived from Open image in new window . Using (2.25), we find agreement with the functional forms in (4.31), of which the non-vanishing terms areEven though the singlet couplings vanish, one can check that their functional form is correctly derivable from Open image in new window . 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.

7 Outlook

We have calculated the effective field theory of heterotic vacua of the form Open image in new window at large radius, correct to order \({\alpha ^{\backprime }\,}\). The field theory is specified by a Kähler potential and superpotential. Supersymmetry forbids Open image in new window from being corrected perturbatively in \({\alpha ^{\backprime }\,}\), but is in general corrected non-perturbatively in \({\alpha ^{\backprime }\,}\). For Open image in new window obtained by deforming Open image in new window , 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 \({\alpha ^{\backprime }\,}\), it was conjectured in [5] that the form of the Kähler potential does not change to all orders in perturbation theory, and that the \({\alpha ^{\backprime }\,}\)-corrections are contained within the Hermitian form \(\omega \). 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 \({\alpha ^{\backprime }\,}\).

Although we have derived this result using a single pair of matter fields, the result clearly generalises to a sum over representations \(\oplus _p {{\varvec{R}}}_p \oplus _p {{\overline{\varvec{R}}}}_p\). 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 Open image in new window and Open image in new window ? 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 Open image in new window and Open image in new window are related.

It would be interesting to compute the field theory couplings in specific examples. For Open image in new window attained by deforming Open image in new window 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 \({{\varvec{27}}}^3\) and \({{\overline{\varvec{27}}}}^3\) couplings using the correctly normalised fields. We showed using deformation theory arguments that the \({{\varvec{1}}}^3\) 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.

Footnotes

  1. 1.

    I would like to thank Xenia de la Ossa for explaining this choice of basis to me.

  2. 2.

    It is important to note that the derivation here and in [5], no assumption is made about expanding around the standard embedding. \(\mathcal {E}\) is not related to the tangent bundle.

  3. 3.

    An important open question is, when are singlet couplings are generated by worldsheet instantons? At least for vacua derived from linear sigma models, there are arguments that suggest that after summing over all worldsheet instantons all the singlet couplings vanish [12, 13]. Here we assume the vacua is well defined with a large radius limit, and so all singlet couplings vanish.

  4. 4.

    It may be useful to define \(\bar{\Psi }_\mu ^a\) given by \( \bar{\Psi }_\mu ^a =( \Psi _{\bar{\mu }}^a)^* \) so that a is a real index and \(\Phi _{1,0}^a = -\, \psi ^{*\,a} = -\, \bar{\Psi }^a_{1,0} \).

  5. 5.

    We do not consider the universal multiplet, the \(d=4\) dilaton and B-field, which decouples.

  6. 6.

    The form of this integrand is due to Xenia de la Ossa who suggested to me in private conversation.

  7. 7.

    In the literature a different ansatz is proposed for the superpotential: Open image in new window After careful calculation one can check Open image in new window , and so there are no \({{\varvec{1}}}, {{\varvec{1}}}^2, {{\varvec{1}}}^3\) couplings. To what extent this reproduces singlet couplings to higher order is an interesting question.

  8. 8.

    Many examples of relations involving complex structure do not hold for all \(y_0\in \mathcal {M}\). A simple example is \(\text {d}J\). Although for any fixed complex structure \(\text {d}J|_{y=y_0} = 0\), differentiating we get something nonzero \(\partial _\alpha \text {d}J = \partial \Delta _\alpha |_{y=y_0} \ne 0\).

  9. 9.

    We can phrase this in terms of tangent space indices, and then use the vielbein to go to coordinate indices, but for succinctness have skipped this step.

  10. 10.
    This charge assignment is determined by studying the Kähler transformations of the Kähler potential.under \(\Omega \rightarrow \mu \Omega \). As described in [31], in order to couple \(d=4\) chiral fields to gravity preserving \(\mathcal {N}=1\) supersymmetry the Open image in new window fermions must transform, which in order for the \({\mathfrak {so}}(9,1)\) fermions to remain neutral, implies the transformation law (B.21).

Notes

Acknowledgements

It is a pleasure to thank Philip Candelas, Emily Carter and Xenia de la Ossa for many interesting and helpful conversations related to this work. I would like to acknowledge the hospitality of Mathematical Institute, University of Oxford, where part of this work was completed. I am supported by STFC Grant ST/L000490/1.

References

  1. 1.
    Strominger, A., Witten, E.: New manifolds for superstring compactification. Commun. Math. Phys. 101, 341 (1985)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Strominger, A.: Yukawa couplings in superstring compactification. Phys. Rev. Lett. 55, 2547 (1985)ADSCrossRefGoogle Scholar
  3. 3.
    Dixon, L.J., Kaplunovsky, V., Louis, J.: On effective field theories describing (2,2) vacua of the heterotic string. Nucl. Phys. B329, 27–82 (1990)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Candelas, P., de la Ossa, X.C.: Moduli space of Calabi–Yau manifolds. Prepared for XIII International School of Theoretical Physics: The Standard Model and Beyond, Szczyrk, Poland, pp. 19–26 (1989)Google Scholar
  5. 5.
    Candelas, P., de la Ossa, X., McOrist, J.: A metric for heterotic moduli. arXiv:1605.05256 [hep-th]
  6. 6.
    de la Ossa, X., Svanes, E.E.: Holomorphic bundles and the moduli space of N = 1 supersymmetric heterotic compactifications. JHEP 10, 123 (2014). arXiv:1402.1725 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    de la Ossa, X., Svanes, E.E.: Connections, field redefinitions and heterotic supergravity. JHEP 12, 008 (2014). arXiv:1409.3347 [hep-th]CrossRefGoogle Scholar
  8. 8.
    de la Ossa, X., Hardy, E., Svanes, E.E.: The heterotic superpotential and moduli. JHEP 01, 049 (2016). arXiv:1509.08724 [hep-th]MathSciNetCrossRefGoogle Scholar
  9. 9.
    Anderson, L.B., Gray, J., Sharpe, E.: Algebroids, heterotic moduli spaces and the Strominger system. JHEP 07, 037 (2014). arXiv:1402.1532 [hep-th]ADSCrossRefGoogle Scholar
  10. 10.
    Garcia-Fernandez, M., Rubio, R., Tipler, C.: Infinitesimal moduli for the Strominger system and killing spinors in generalized geometry. arXiv:1503.07562 [math.DG]
  11. 11.
    Anguelova, L., Quigley, C., Sethi, S.: The leading quantum corrections to stringy Kahler potentials. JHEP 1010, 065 (2010). arXiv:1007.4793 [hep-th]”ADSCrossRefzbMATHGoogle Scholar
  12. 12.
    Beasley, C., Witten, E.: Residues and world sheet instantons. JHEP 10, 065 (2003). arXiv:hep-th/0304115 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Silverstein, E., Witten, E.: Criteria for conformal invariance of (0,2) models. Nucl. Phys. B444, 161–190 (1995). arXiv:hep-th/9503212 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Becker, K., Becker, M., Dasgupta, K., Green, P .S.: Compactifications of heterotic theory on nonKahler complex manifolds. 1. JHEP 04, 007 (2003). arXiv:hep-th/0301161 [hep-th]ADSCrossRefGoogle Scholar
  15. 15.
    Gurrieri, S., Lukas, A., Micu, A.: Heterotic on half-flat. Phys. Rev. D70, 126009 (2004). arXiv:hep-th/0408121 [hep-th]ADSGoogle Scholar
  16. 16.
    Lopes Cardoso, G., Curio, G., Dall’Agata, G., Lust, D.: BPS action and superpotential for heterotic string compactifications with fluxes. JHEP 10, 004 (2003). arXiv:hep-th/0306088 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Bergshoeff, E., de Roo, M.: The Quartic effective action of the heterotic string and supersymmetry. Nucl. Phys. B328, 439 (1989)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Bergshoeff, E., de Roo, M.: Supersymmetric Chern-Simons terms in ten-dimensions. Phys. Lett. B218, 210 (1989)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Kobayashi, S.: Differential Geometry of Complex Vector Bundles. Iwanami Shoten, Tokyo (1987)CrossRefzbMATHGoogle Scholar
  20. 20.
    Itoh, M.: Geometry of anti-self-dual connections and Kuranishi map. J. Math. Soc. Jpn. 40(1), 9–33 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Candelas, P., de la Ossa, X.: Moduli space of Calabi–Yau manifolds. Nucl. Phys. B355, 455–481 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Witten, E., Bagger, J.: Quantization of Newton’s constant in certain supergravity theories. Phys. Lett. B115, 202–206 (1982)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    McOrist, J., Melnikov, I.V.: Half-twisted correlators from the Coulomb branch. JHEP 04, 071 (2008). arXiv:0712.3272 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    McOrist, J., Melnikov, I.V.: Summing the instantons in half-twisted linear sigma models. JHEP 02, 026 (2009). arXiv:0810.0012 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    McOrist, J., Melnikov, I.V.: Old issues and linear sigma models. arXiv:1103.1322 [hep-th]
  26. 26.
    McOrist, J.: The revival of (0,2) linear sigma models. Int. J. Mod. Phys. A26, 1–41 (2011). arXiv:1010.4667 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Melnikov, I., Sethi, S., Sharpe, E.: Recent developments in (0,2) mirror symmetry. SIGMA 8, 068 (2012). arXiv:1209.1134 [hep-th]”MathSciNetzbMATHGoogle Scholar
  28. 28.
    Anderson, L.B., Constantin, A., Gray, J., Lukas, A., Palti, E.: A comprehensive scan for heterotic SU(5) GUT models. JHEP 1401, 047 (2014). arXiv:1307.4787 [hep-th]. ”ADSCrossRefGoogle Scholar
  29. 29.
    Melnikov, I.V., Plesser, M.: A (0,2) mirror map. JHEP 1102, 001 (2011). arXiv:1003.1303 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Kreuzer, M., McOrist, J., Melnikov, I.V., Plesser, M.R.: (0,2) deformations of linear sigma models. JHEP 07, 044 (2011). arXiv:1001.2104 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Wess, J., Bagger, J.: Supersymmetry and Supergravity. Princeton University Press, Princeton (1992)zbMATHGoogle Scholar
  32. 32.
    Argyres, P.C.: An introduction to global supersymmetry. http://homepages.uc.edu/~argyrepc/cu661-gr-SUSY/susy1996.pdf. Accessed 9 Nov 2017

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of SurreyGuildfordUK

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