On the effective field theory of heterotic vacua
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 KaluzaKlein and other higherdimensional theoriesMathematics Subject Classification
81T30 81T60 83E15 83E30 83E501 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 gaugeinvariant threeform 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.
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 .
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 nontrivial 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 Dterms relevant to moduli. In this paper, we compute the remaining Dterms, including the metric terms for the bosonic matter fields charged under \({\mathfrak {g}}\). We also compute the Fterms 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 Fterms 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 topform we need to wedge it with a gaugeinvariant threeform. The threeform needs to contain a dependence on the matter fields, and this can only occur through the tendimensional H field. The other natural gaugeinvariant threeforms 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 higherorder 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
A table of objects used
Quantity  Definition  Comment 

\(d=4\) 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 
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)  \(\zeta _a, {\overline{\zeta }}^{\dot{a}}\)  −  a, b, 
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.
2.1 Derivatives of \(\Omega \) and \(\Delta _\alpha \)
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 \(\mathcal {H}\).
2.3 The B and H fields
2.4 Derivatives of A
2.5 Derivatives of H
2.6 Derivatives of \(\text {d}^c \omega \)
2.7 Supersymmetry relations
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 \( {\mathfrak {g}}\oplus {\mathfrak {h}}\subset {\mathfrak {e}}_8\)
 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.
\(\Phi \) 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.
\(\Psi \) 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.
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 .
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 \({{\varvec{R}}} = {{\varvec{27}}}\).
3.2 The matter field metric from reducing YangMills, \(\mathcal {L}_F\)
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
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.
4.2 Dimensional reduction of Open image in new window
4.2.1 Quadratic couplings
4.2.2 Cubic fluctuations and Yukawa couplings
5 The final result: moduli, matter metrics and Yukawa couplings

\({\mathfrak {g}}\)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 \({\mathfrak {g}}\);
6 The superpotential and Kähler potential
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 Fterm 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 \(\Omega \).
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 nonperturbatively in \({\alpha ^{\backprime }\,}\). For Open image in new window obtained by deforming Open image in new window , some of these nonperturbative 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 nonvanishing would imply the vacuum does not exist, and thereby shrink the moduli space of heterotic vacua quantum mechanically.
Footnotes
 1.
I would like to thank Xenia de la Ossa for explaining this choice of basis to me.
 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.
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.
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.
We do not consider the universal multiplet, the \(d=4\) dilaton and Bfield, which decouples.
 6.
The form of this integrand is due to Xenia de la Ossa who suggested to me in private conversation.
 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.
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.
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.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.
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