EMPG-18-12
IPhT-t18/088
Finite deformations from a heterotic superpotential:
holomorphic Chern–Simons and an L∞ algebra
Anthony Ashmore,a Xenia de la Ossa,a Ruben Minasian,b Charles
Strickland-Constablec,d and Eirik Eik Svanese,f
aMathematical Institute, University of Oxford, Andrew Wiles Building,
Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
bInstitut de Physique The´orique, Universite´ Paris Saclay, CEA, CNRS, F-91191,
Gif-sur-Yvette, France
cSchool of Mathematics, University of Edinburgh, James Clerk Maxwell Building,
Peter Guthrie Tait Road, Edinburgh, EH9 3FD, UK
dSchool of Physics, Astronomy and Mathematics, University of Hertfordshire,
College Lane, Hatfield, AL10 9AB, UK
eDepartment of Physics, King’s College London, London, WC2R 2LS, UK
fThe Abdus Salam International Centre for Theoretical Physics, 34151 Trieste, Italy
Abstract: We consider finite deformations of the Hull–Strominger system. Starting from the
heterotic superpotential, we identify complex coordinates on the off-shell parameter space.
Expanding the superpotential around a supersymmetric vacuum leads to a third-order Maurer–
Cartan equation that controls the moduli. The resulting complex effective action generalises
that of both Kodaira–Spencer and holomorphic Chern–Simons theory. The supersymmetric
locus of this action is described by an L3 algebra.
Contents
1 Introduction 1
2 The Hull–Strominger system and a heterotic superpotential 4
2.1 N = 1 heterotic vacua and the Hull–Strominger system 4
2.2 The Atiyah algebroid and a holomorphic structure 6
3 The off-shell parameter space 7
3.1 F -term conditions from the superpotential 10
3.2 Constraints from the SU(3) structure and holomorphicity 10
4 Higher-order deformations 12
4.1 The superpotential 13
4.2 A Maurer–Cartan equation from the holomorphic structure 14
4.3 Including the bundle 15
4.4 Vanishing of the superpotential 17
5 Moduli and an L3 algebra 18
5.1 Quasi-isomorphism to a natural holomorphic L3 algebra 19
5.2 An L∞ field equation 20
6 A reduced L3 algebra and an effective action 21
6.1 Integrating out b 22
6.2 Effective field theory and Yukawa couplings 23
7 Conclusions 26
A Conventions 28
A.1 Heterotic supergravity 28
A.2 Holomorphic structure 29
A.3 L∞ structure 30
B Comments on heterotic flux quantisation 31
B.1 A toy example: The abelian bundle 32
B.2 The two-form gerbe example 33
B.3 Heterotic flux quantisation 35
B.4 Deforming the system and a well-defined global two-form 36
– i –
C The off-shell N = 1 parameter space and holomorphicity of Ω 38
C.1 SU(3)× SO(6) structures in the NS-NS sector 38
C.2 SU(3)× SO(6 + n) structures in heterotic supergravity 41
C.3 The off-shell hermitian structure on V 42
D Comments on D-terms 44
D.1 Massless deformations 44
D.2 Including bundle moduli 48
D.3 Polystable bundles 51
D.4 Full Maurer–Cartan equations 52
E Massless moduli 53
1 Introduction
A full understanding of the parameter space of string theory is an outstanding mathematical
challenge and would lead to powerful constraints on the landscape of string models. Of the
various limits of string theory, the heterotic string has been the focus of much phenomenology
thanks to the relative ease with which one can engineer four-dimensional theories with chiral
fermions and the Standard Model gauge group [1–8]. Much of this work has been on models
where the internal manifold is Calabi–Yau, mostly because such spaces can be constructed
using algebraic geometry and then used for compactifications without knowledge of their
explicit metrics.
Calabi–Yau compactifications are not the most general way to obtain an N = 1 theory
in four dimensions that admits a Minkowski vacuum. The general solution to O(α′) is given
by compactifying on a complex three-fold X with H flux and a gauge bundle that satisfy
an anomaly cancellation condition. The conditions on the geometry and fluxes for such a
solution are known as the Hull–Strominger system [9, 10]. Known solutions to this system
include Calabi–Yau spaces with bundles and a small number of honestly non-Ka¨hler geometries.
Generically, a given solution of the Hull–Strominger system will admit deformations of the
geometry, flux and bundle that remain N = 1 solutions – these deformations are known as
moduli. These moduli appear in the massless spectrum of the low-energy theory, so it is
important that we understand the moduli space of a given compactification.
The moduli spaces of Calabi–Yau compactifications at zeroth order in α′ are well un-
derstood using the language of special geometry. Until recently the general case had not
been tackled – this might come as a surprise. Certainly in type II theories the conditions
for an N = 1 Minkowski solution are sufficiently complicated (thanks to branes and other
ingredients) that their moduli spaces might not admit a general formulation. In the heterotic
case, the underlying geometry is relatively straightforward. One might have expected that
– 1 –
the gauge sector and anomaly conditions complicate matters somewhat, but that the moduli
space might still be understood. Starting with [5, 11–15], this gap is now being filled. (See
also [16–19] for worldsheet approaches.)
Infinitesimally, the moduli space is characterised by the existence of a holomorphic
structure D on a bundle Q over the three-fold X. The fact that the moduli space is finite
dimensional is intimately connected to this holomorphic structure and the Bianchi identity
for the flux. The infinitesimal moduli are captured by the cohomology H
(0,1)
D
(Q), where Q is
defined by a series of extensions. In this way, the complex structure, hermitian and bundle
moduli are combined in a single structure. Furthermore, one can define the analogue of special
geometry for these general heterotic compactifications and find the metric on the moduli
space [20, 21].
A natural question to ask is whether one can understand the moduli spaces to higher order.
If we think about deformations of a complex structure, we know the infinitesimal moduli are
given by H
(0,1)
∂
(T (1,0)X), while the higher-order deformations satisfy a Maurer–Cartan equation.
A similar thing happens for bundle deformations [22], and simultaneously deformations of the
bundle and complex structure [23, 24]. In this way, moduli can be obstructed at higher orders
and can give non-zero contributions to the superpotential of the four-dimensional effective
theory. The aim of this work is to derive the corresponding conditions on the moduli for the
Hull–Strominger system at higher orders. In other words, we want to derive the conditions on
the moduli when they describe a small but finite deformation of the original heterotic solution.
There are a number of ways one might go about this. One path would be to start with the
equations of the Hull–Strominger system and deform the various fields. The deformed fields
should still satisfy the Hull–Strominger system (as it describes the most general solution) so
one can rewrite the system of equations as conditions on the deformations themselves. This
is similar to the path taken in [20, 21]. Our approach will be complimentary. It has been
shown that supersymmetry of the heterotic system can be described using a four-dimensional
superpotential [21, 25, 26]. The vanishing of the superpotential and its first derivative imposes
the F -term conditions in the four-dimensional theory and leads to an N = 1 Minkowski
vacuum. Our plan is to deform the fields that appear in the superpotential and then read
off the conditions on the moduli for the superpotential and its derivative to vanish. These
two approaches will be shown to be equivalent in a future publication [27]. A particular
advantage of proceeding this way is that one can use the knowledge that the superpotential is
a holomorphic function of the moduli fields to streamline the problem.
In addition to the usual N = 1 lore that the superpotential is holomorphic, we give
an argument that the superpotential is holomorphic on the space of moduli fields without
requiring that they give a solution to the Hull–Strominger system. This is equivalent to
saying that the off-shell parameter space – the space of SU(3) structures, B fields and gauge
bundles – is a complex space and the superpotential is a holomorphic function of these
parameters. We outline how this follows from generalised geometry where N = 1 NS-NS
compactifications are described by a generalised SU(3)× SU(4) structure [28]. We identify the
invariant (holomorphic) object which characterises this structure and find that the complex
– 2 –
coordinates on the space of structures match with the usual complex structure, complexified
hermitian and bundle deformations.
We show that the conditions on the moduli fields from the vanishing of the superpotential
and its first derivative can be written as a pair of third-order Maurer–Cartan equations
using the holomorphic structure and a number of brackets. Moreover, we show that the
superpotential itself can be rewritten using these operators in a Chern–Simons-like form:
∆W =
∫
X
〈y,Dy − 13 [y, y]− ∂b〉 ∧ Ω, (1.1)
where y describes the complex, hermitian and bundle moduli, and b is a (0, 2)-form. ∆W is
written using a holomorphic structure D, a pairing 〈·, ·〉 on the moduli fields, and a bracket
[·, ·]. The brackets can be understood as coming from an underlying holomorphic Courant
algebroid that describes the combined deformations of the complex structure, metric and
fluxes [29, 30].
We then show that the supersymmetry conditions can be recast in terms of an L3 algebra.
We outline how the L3 algebra gives a C∞ resolution of the underlying holomorphic Courant
algebroid. The natural L3 field equation reproduces the supersymmetry conditions, and the
L3 structure gives the gauge symmetries of the moduli space in a compact form.
It is known that generic deformation problems have a description in terms of L∞ structures,
so it is not unexpected that our moduli fields are governed by one. What is unexpected is
that the structure truncates at finite order leaving us with an L3 algebra. Why does the
deformation truncate in our case? A generic deformation problem can be parametrised in
many equivalent ways – some may truncate at finite order while others do not. Essentially, the
structure of the heterotic system and its formulation using a superpotential guides us to pick a
“nice” parametrisation. Said another way, we know from supergravity that the superpotential
should be a holomorphic function of the parameters. Thus when we express the superpotential
in the obvious complex coordinates on the parameter space, we get the most natural way to
package the deformation problem.
We begin in section 2 with a review of the Hull–Strominger system and the description of
its infinitesimal moduli in terms of a holomorphic structure as in [14]. In section 3 we discuss
the off-shell parameter space of the theory and give the complex coordinates on the parameter
space. We show how the F -term conditions follow from a heterotic superpotential to set the
scene for the higher-order deformations. In section 4 we examine the higher-order deformation
problem and find the system of equations that govern the moduli of the Hull–Strominger
system. We show how this can be written in terms of the holomorphic structure D and
a bracket [·, ·] arising from a holomorphic Courant algebroid. In section 5 we rewrite the
equations that govern the moduli in terms of an L3 structure. We give the various multilinear
products `k that define the L3 structure and discuss how various properties, such as the
moduli equations and gauge symmetries, are naturally encoded in this L3 language. In section
6 we discuss how the system simplifies under various assumptions and comment on how the
– 3 –
effective field theory is encoded in our language. We finish with a discussion of some open
questions and avenues for future work.
In the appendices, we lay out our conventions, include a few comments on how flux
quantisation works in the heterotic theory, discuss the off-shell parameter space in terms of
generalised geometry, show that the D-term conditions do not affect the moduli problem
and review how the massless moduli are captured by the the cohomology of the holomorphic
structure.
2 The Hull–Strominger system and a heterotic superpotential
We begin with a review of the Hull–Strominger system [9, 10] and the description of its
infinitesimal moduli using a holomorphic structure [14, 15, 31].
2.1 N = 1 heterotic vacua and the Hull–Strominger system
The Hull–Strominger system is a set of equations whose solutions are supersymmetric
Minkowski vacua of heterotic string theory to order O(α′). The ten-dimensional solution is
a product of four-dimensional Minkowski space with a six-dimensional complex manifold X.
X admits a vector bundle V with connection A whose curvature F is valued in EndV . The
tangent bundle TX of X also admits a connection Θ whose curvature R is valued in EndTX.
X admits an SU(3) structure defined by a nowhere vanishing spinor η or, equivalently, a
non-degenerate two-form ω and a nowhere vanishing three-form Ψ that are compatible
ω ∧Ψ = 0, i‖Ψ‖2 Ψ ∧Ψ =
1
3!
ω ∧ ω ∧ ω. (2.1)
The invariant objects are defined by bilinears of the spinor as
ωmn = −i η†γmnη, Ψmnp = ηTγmnpη, (2.2)
where we are free to normalise the spinor so that ‖Ψ‖2 = 8. In what follows it will be useful
to define a three-form Ω which is related to Ψ by a dilaton factor as
Ω = e−2φΨ. (2.3)
Supersymmetry of the vacuum follows from the vanishing of the supersymmetry variations
of the fermionic fields, given in equations (A.7)–(A.9). To first order in α′, these conditions
are equivalent to the Hull–Strominger system:
dΩ = 0, (2.4)
i(∂ − ∂)ω = H := dB + α
′
4
(ωCS(A)− ωCS(Θ)), (2.5)
Ω ∧ F = 0, (2.6)
ωyF = 0, (2.7)
– 4 –
d(e−2φω ∧ ω) = 0, (2.8)
where ωCS is the Chern–Simons three-form for the connection,
ωCS(A) = tr(A ∧ dA+ 23A ∧A ∧A). (2.9)
The closure of Ω from (2.4) implies that the manifold is complex with a holomorphically trivial
canonical bundle, while condition (2.8) tells us that X is conformally balanced. Conditions
(2.6) and (2.7) mean V is a polystable holomorphic bundle1 so the curvature satisfies the
hermitian Yang–Mills equations. Finally, (2.5) defines the flux H in terms of the heterotic
B field and the anomaly cancellation condition, and links it with the intrinsic torsion of the
SU(3) structure. The corresponding Bianchi identity is2
dH =
α′
4
(trF ∧ F − trR ∧R). (2.10)
This set of conditions defines what one might call a heterotic SU(3) structure.
Upon considering the four-dimensional N = 1 theory that would follow from compactifying
on such a solution, the Hull–Strominger system naturally splits into F - and D-term conditions.
As discussed in [26], the F -term equations are
dΩ = 0,
i(∂ − ∂)ω = H,
Ω ∧ F = 0.
(2.11)
It is these equations that the heterotic superpotential reproduces. The remaining equations
of the Hull–Strominger system are the conformally balanced condition and the Yang–Mills
equations, referred to as the D-term equations.
Modulo certain mild assumptions on the geometry, the infinitesimal deformations are
parametrised by the cohomology H
(0,1)
D
(Q), where D and Q are to be defined below. This
cohomology is reviewed in appendix E. Under infinitesimal deformations, the D-term equations
fix a representative of a certain cohomology class [14], and so should be thought of as gauge
fixing conditions that do not affect the moduli problem. This is of course expected from the
four-dimensional N = 1 supergravity point of view [33–35]. In appendix D we show that
preserving the D-term conditions for finite deformations also amounts to fixing a gauge. One
might worry about Fayet–Iliopoulos terms appearing, but these are in fact accounted for by
modding out by D-exact terms, as shown in [14].
1More precisely, it is the complex vector bundle VC (defined in appendix C.3) that is a holomorphic bundle.
2The curvature R in the Bianchi identity is the curvature of a connection on TX, satisfying its own hermitian
Yang–Mills conditions in order for the equations of motion to be fulfilled [32]. To O(α′), this connection is ∇−,
given by taking the connection in (A.7) with the opposite sign for H.
– 5 –
2.2 The Atiyah algebroid and a holomorphic structure
The vector bundle V is hermitian in agreement with (0, 2) supersymmetry on the worldsheet [36].
The curvature F of the bundle is given by
F = dA+A ∧A, (2.12)
where A is a one-form connection valued in EndV . The exterior derivative on V twisted by A
is
dA := d + [A, ·], (2.13)
where the action of the bracket on a p-form β is
[A, β] = A ∧ β − (−1)pβ ∧A. (2.14)
A holomorphic structure on V is fixed by the (0, 1) component of dA, which we denote ∂A.
This operator squares to zero if the bundle is holomorphic, that is F(0,2) = 0. Moreover, the
Bianchi identity for the curvature is simply
∂AF = 0. (2.15)
A deformation of the Hull–Strominger system corresponds to simultaneous deformations
of the complex structure, hermitian structure and gauge bundle. Taking each of these in
isolation is not sufficient. In particular, deformations of the hermitian structure alone lead to
an infinite-dimensional moduli space. It is surprising that if one considers the full deformation
problem together with the anomaly cancellation condition, one finds a finite-dimensional
moduli space. Of course, this is what one would expect from string theory, but the precise
way in which this happens is rather remarkable.
As discussed in [14], the infinitesimal moduli of the Hull–Strominger system are captured
by deformations of a holomorphic structure. The holomorphic structure D acts on a bundle
Q. Locally Q is given by
Q ' T ∗(1,0)(X)⊕ EndV ⊕ EndTX ⊕ T (1,0)X. (2.16)
Globally, Q is defined by an extension3
0→ T ∗(1,0)(X)→ Q→ Q1 → 0, (2.17)
where the bundle Q1 is defined by
0→ EndV ⊕ EndTX → Q1 → T (1,0)(X)→ 0. (2.18)
3Full details can be found in [14].
– 6 –
The holomorphic structure D on Q is a derivative4
D : Ω(0,p)(Q)→ Ω(0,p+1)(Q), (2.19)
where D
2
= 0 if and only if the Bianchi identities for H, F and R are satisfied. The Hull–
Strominger system is then equivalent to the data of the extension bundle Q, the nilpotent
holomorphic structure D, polystability of V and TX and the conformally balanced condition
on X.
The infinitesimal deformations of the holomorphic structure are simply elements of the
D-cohomology of Q-valued (0, 1)-forms – H(0,1)
D
(Q). As shown in [14], this is also the moduli
space of heterotic SU(3) structures. We give a short review of this in appendix E.
For the rest of the paper, we make a field redefinition to absorb the explicit α′ dependence
B → α
′
4
B, ω → α
′
4
ω. (2.20)
One can restore the proper factors of α′ by the inverse transformations. We will also suppress
the connection on TX – we can reintroduce it in what follows by treating TX as part of the
gauge bundle and defining the bundle metric on the TX subspace to be negative definite so
that the Bianchi identity comes with a negative sign for the tr R ∧R term.
The main aim of this work is to understand what happens for finite deformations. In
particular we will see the holomorphic structure is an important ingredient in describing
higher-order deformations. First let us discuss the off-shell parameter space and how the
heterotic SU(3) structure can be rephrased using a superpotential.
3 The off-shell parameter space
We now show that a subset of the system corresponding to F -term conditions can be derived
from a superpotential. We then discuss how deformations of the geometry, flux and bundle
can be parametrised using the observation that the superpotential is a holomorphic function.
The four-dimensional effective theory that one finds after compactifying the heterotic
string on an SU(3) structure manifold is controlled by a superpotential [21, 25, 26]. The
superpotential W is given in terms of the flux H and the SU(3) invariant forms by5
W =
∫
X
(H + i dω) ∧ Ω. (3.1)
As we will review, given the SU(3) structure relations (2.1) and the definition of H in (2.5),
W = δW = 0 reproduces the F -term conditions of the Hull–Strominger system [26].
Notice that δW = 0 requires us to vary the superpotential over some space of field
configurations. We need to understand what this space is in order to find how the superpotential
4A similar operator has appeared in the context of generalised Ka¨hler geometry [29].
5Here we have scaled away an overall factor of α′/4 that comes from the field redefinition in (2.20).
– 7 –
behaves when we perform a finite deformation of the background fields. We pause briefly
to distinguish between this parameter space and the moduli space of solutions to the Hull–
Strominger system.
The parameter space or space of field configurations Z is the space of SU(3) structures, B
fields and hermitian gauge bundles on the real manifold X. The SU(3) structure is equivalent
to the existence of a nowhere-vanishing spinor so that on this space of field configurations the
heterotic theory admits an “off-shell” N = 1 supersymmetry of the kind discussed in [37, 38].
These fields do not necessarily solve the Hull–Strominger system and so we often refer to them
as off-shell field configurations. This is the space over which the superpotential is varied.
The moduli space M of the Hull–Strominger system is a subspace of Z on which the
fields also solve the Hull–Strominger system. This set of fields is what one usually means by
moduli, and we will often refer to them as on-shell configurations. Another way of saying this
is that the superpotential and its derivatives vanish when evaluated on M.
As W is a superpotential for a four-dimensional N = 1 theory we expect it to be a
holomorphic function. The holomorphicity of W is a powerful tool for understanding de-
formations of the Hull–Strominger system and we will see later how its presence greatly
simplifies the problem. On the supersymmetric locus (on-shell), it is known that the superpo-
tential is a holomorphic function of the moduli fields [21] – this is simply the statement that
anti-holomorphic derivatives of the superpotential vanish on imposing the F -term conditions.
Physics goes further than this and insists that W is a holomorphic function of the off-shell
field configurations – the off-shell field space Z must admit complex coordinates and W must
be a holomorphic function of these coordinates. In other words, the three-form Ω and the
particular combinations of B + iω and A that appear in (3.1) must be parametrised by these
complex coordinates.
We outline a proof that such complex coordinates exist and that Ω is holomorphic on
the parameter space Z in appendix C using the formalism of generalised geometry. We also
discuss how the hermitian structure on V survives off-shell. For completeness, one should
really show that W itself can be expressed as a holomorphic function of the object χ˜ that we
define in appendix C – we leave this for a future work.
Note that on-shell, Ω is also holomorphic as a function of the complex coordinates of
X. When we talk of Ω being holomorphic we are instead referring to its dependence on the
coordinates of the off-shell parameter space.
Let t and t denote holomorphic and anti-holomorphic coordinates on the parameter space
Z. The corresponding holomorphic and anti-holomorphic variations are
∆ =
∞∑
n=1
1
n!
tnDnt , ∆ =
∞∑
n=1
1
n!
tnDnt , (3.2)
where D is a covariant derivative on the parameter space [20].
As we discuss in appendix C.3, off-shell the gauge bundle admits a real hermitian connection
valued in V . This decomposes into (1, 0)- and (0, 1)-forms, with a corresponding decomposition
– 8 –
of the Chern–Simons form. Not all components of the Chern–Simons form appear in W (as it
is wedged with Ω); only the (0, 1)-form components of A contribute. It is this component of
the connection that is the complex coordinate on the off-shell parameter space. A holomorphic
deformation of the connection is then given by a (0, 1)-form valued in EndV , which we denote
α:
A 7→ A+ ∆A = A+ α. (3.3)
We show in appendix C.3 that one can write the Chern–Simons form using connections valued
in V , VC, V C or a combination of these. From this it is clear that it is equivalent to work with
(0, 1)-forms valued in EndV or with (0, 1)-forms valued in EndVC.
A deformation of the complex structure is parametrised by a (1, 0)-vector valued (0, 1)-
form, µ ∈ Ω(0,1)(T (1,0)X), also known as a Beltrami differential. The complex coordinates on
X that define the complex structure deform to
dza → dza + ∆dza = dza + µa. (3.4)
Infinitesimally, the deformed complex structure is J + 2iµ. For a small but finite deformation,
the holomorphic three-form becomes [39, 40]
Ω→ Ω + ∆Ω = Ω + ıµΩ + 12 ıµıµΩ + 13! ıµıµıµΩ, (3.5)
where the variations of Ω in coordinates are
ıµΩ =
1
2Ωabcµ
a ∧ dzbc,
ıµıµΩ = Ωabcµ
a ∧ µb ∧ dzc,
ıµıµıµΩ = Ωabcµ
a ∧ µb ∧ µc.
(3.6)
The fact that a variation of Ω is completely captured by µ without needing µ is an indication
that Ω is a holomorphic function of the coordinates of the off-shell parameter space. Note that
the variation of Ω can in principle have a (3, 0) component. However, the (3, 0) part should be
interpreted as a Ka¨hler transformation and so is not part of the physical moduli. Another way
of saying this is that ∆ in (3.2) is built from covariant derivatives on the parameter space [20].
Restricted to variations of the complex structure, the (3, 0) component is attributed to a
connection in the usual way.
The holomorphic deformations of the hermitian and B field moduli are
(B + i ∆ω)(1,1) and B(0,2), (3.7)
where a subscript (p, q) denotes the type with respect to undeformed complex structure. Here
B is a combination of variations of the B field and the exact term in the variation of the
Chern–Simons term, given by
B = ∆B + tr ∆A ∧A, (3.8)
– 9 –
up to a d-closed two-form. The Green–Schwarz mechanism ensures B is gauge invariant. As
we show in appendix B, flux quantisation then implies that B is a globally defined two-form,
so it can indeed be a modulus. As we will see, the (0, 2) component of ∆ω is actually fixed in
terms of the other moduli by the SU(3) relations, but it will be convenient to package this
with B(0,2) into (B + i ∆ω)(0,2).
3.1 F -term conditions from the superpotential
Let us review how one derives the F -term conditions from the superpotential. We take δ
to be an infinitesimal deformation of the fields, leading to a corresponding variation of the
superpotential δW
δW =
∫
X
(
2 tr δA ∧ F + d(B + i δω)) ∧ Ω + ∫
X
(H + i dω) ∧ δΩ. (3.9)
The F -term conditions come from requiring that both W and δW vanish for generic values of
the moduli. For arbitrary B + i δω and δA, the vanishing of δW requires
∂Ω = dΩ = 0, F ∧ Ω = 0. (3.10)
With this in mind, the vanishing of δW for arbitrary δΩ of type (2, 1) implies H(1,2) = −i ∂ω.
Using the previous conditions, the vanishing of W itself reduces to H(0,3) = ∂γ for some
(0, 2)-form γ. The Bianchi identity for H then implies H(0,3) = 0, giving us the final F -term
condition:
H = i(∂ − ∂)ω. (3.11)
With this we see W = δW = 0 reproduces the F -term equations of the Hull–Strominger
system.
Our plan is to extend this discussion to understand finite deformations around a super-
symmetric solution. First we need to understand how the requirement of an SU(3) structure
and holomorphicity of the superpotential constrain the possible deformations.
3.2 Constraints from the SU(3) structure and holomorphicity
The existence of an SU(3) structure is part of the data that goes into the superpotential, so
the deformed geometry should also define an SU(3) structure. Another way of saying this is
that the off-shell parameter space on which the superpotential is varied is the space of SU(3)
structures (plus bundles, and so on). This means the SU(3) structure compatibility condition
must still hold:
(ω + ∆ω) ∧ (Ω + ∆Ω) = 0, (3.12)
where ∆ is a finite holomorphic variation. Expanding this out according to complex type, we
find
0 =
(
ω + (∆ω)(2,0) + (∆ω)(1,1) + (∆ω)(0,2)
) ∧ (Ω + ıµΩ + 12 ıµıµΩ + 13! ıµıµıµΩ). (3.13)
– 10 –
Upon contracting this with Ω, we see this equation fixes the (0, 2) component of ∆ω in terms
of the other deformations:
(∆ω)(0,2) = ıµω + ıµ(∆ω)(1,1) − 12 ıµıµ(∆ω)(2,0). (3.14)
Now consider an anti-holomorphic variation ∆ under which Ω does not vary as it is a
holomorphic on the parameter space. We then note that as an anti-holomorphic deformation
does not change the complex structure (as ∆Ω = 0) the SU(3) compatibility condition reduces
to
∆ω ∧ Ω = 0. (3.15)
From this we see (∆ω)(0,2) = 0 and so, taking a conjugate, (∆ω)(2,0) = 0. Combined with the
previous result of a holomorphic variation of the compatibility condition, we have
(∆ω)(2,0) = 0, (∆ω)(0,2) = ıµω + ıµ(∆ω)(1,1). (3.16)
From this we see the (0, 2) component of the variation of ω is fixed by the complex structure
and hermitian moduli.
We can also play the same trick with the superpotential itself. The superpotential is
a holomorphic function of the moduli so an arbitrary anti-holomorphic variation of it must
vanish without having to impose the supersymmetry conditions – it must vanish “off shell”.
Let us turn off the gauge sector for now, and consider an anti-holomorphic variation of W :
∆W =
∫
X
d(∆B + i ∆ω) ∧ Ω = −
∫
X
(∆B + i ∆ω) ∧ dΩ. (3.17)
As W is a holomorphic functional, this must vanish for all anti-holomorphic deformations
without needing to impose the F -term conditions. Without imposing integrability of the
complex structure, generically we have dΩ ∈ Ω(3,1)(X)⊕ Ω(2,2)(X). For an anti-holomorphic
variation of the superpotential to vanish, it is sufficient that
(∆B + i ∆ω)(1,1) = 0, (∆B)(0,2) = 0, (3.18)
where we have used the first condition from (3.16) to remove (∆ω)(0,2). This agrees with (3.7)
where we stated that the holomorphic combinations are the (1, 1) and (0, 2) components of
B + iω (see also discussion in Appendix C.1). Taking a conjugate of these conditions we have
(∆B − i ∆ω)(1,1) = 0, (∆B)(2,0) = 0. (3.19)
Taken together these give
(∆B)(1,1) = i (∆ω)(1,1), (∆B)(2,0) = (∆ω)(2,0) = 0. (3.20)
– 11 –
For what follows, it is useful to define
b˜ = (∆B + i ∆ω)(0,2), x = i (∆ω)(1,1) = (∆B)(1,1), (3.21)
which are our complex coordinates on the parameter space.
One can repeat this exercise with gauge sector turned on. W is a holomorphic function of
the moduli so it does not change for arbitrary anti-holomorphic variations ∆A. For ∆W = 0
to hold at a generic off-shell point in field space, we find it is sufficient that (∆A)(0,1) = 0.
This implies (∆A)(1,0) = 0, so the holomorphic deformations correspond to
∆A = α ∈ Ω(0,1)(EndV ). (3.22)
In other words, the holomorphic coordinate on the parameter space is α, in agreement with
(3.3). Furthermore, one sees that the holomorphic deformations of the complexified hermitian
moduli are the (1,1) and (0,2) components of B + i∆ω.
As an aside, we note that there is a schematic way to see that Ω is a holomorphic function
of the parameter space coordinates. Consider a generic anti-holomorphic deformation of the
superpotential around a point in moduli space where the holomorphic top-form is closed,
dΩ = 0:
∆W =
∫
X
(H + i dω) ∧∆Ω +
∫
X
d(∆B + i ∆ω) ∧∆Ω. (3.23)
For an infinitesimal deformation, the second term can be dropped, and a sufficient condition
for the first term to vanish for generic H and ω is that
∆Ω = 0, (3.24)
infinitesimally. Note that this also kills the second term in (3.23) at second order in perturbation
theory. A sufficient condition for ∆W to vanish at this order is hence that ∆Ω = 0 to this level
as well. This argument can be continued ad infinitum, and we are left with condition (3.24),
at least for finite deformations away from a closed Ω. We will assume that this condition
holds true at generic off-shell points in the parameter space. Stronger evidence for this is
provided by the generalised geometry formulation of the off-shell parameter space presented
in appendix C; we also find a matching between the complex coordinates and the natural
parametrisation from generalised geometry.
4 Higher-order deformations
The main aim of this paper is to derive the conditions on the moduli when we move from
infinitesimal to finite deformations of solutions to the Hull–Strominger system. In other words,
we consider higher-order deformations of the fields that parametrise the supersymmetric
Minkowski solution. As we have mentioned we only need to consider the F -term relations to
understand the moduli space. We show in appendix D that under some reasonable assumptions
– 12 –
the D-term conditions do not constrain the moduli problem and should be thought of as gauge
fixing conditions – we expect this to hold in general.
4.1 The superpotential
Let us consider the effect on the superpotential of a finite deformation of the background
fields away from a point on the supersymmetric locus. In other words, we start with a
supersymmetric vacuum solution described by a superpotential W which is a functional of the
SU(3) structure, H and the bundle. Let us denote the superpotential evaluated at this point
by W |0. The vacuum is supersymmetric if both the superpotential and its first derivative
vanish when evaluated on the solution. Now move a finite distance from this solution in
parameter space by deforming the background. The superpotential evaluated at this new
point is W |∆ = W |0 + ∆W . We have a supersymmetric solution if both W |∆ and its first
derivative vanish at that point in parameter space, which is equivalent to the vanishing of
∆W and δ∆W . Let us see how this works out. For clarity of presentation, let us ignore the
bundle moduli – we will reinstate these in section 4.3.
A finite holomorphic deformation of the parameters gives
∆W =
∫
X
[(
H + i dω + d(∆B + i ∆ω)
)∧(ıµΩ + 12 ıµıµΩ + 13! ıµıµıµΩ)
+ d(∆B + i ∆ω) ∧ Ω
]
.
(4.1)
As we are deforming about a supersymmetric point we have H = i(∂ − ∂)ω and dΩ = 0, so
the first term simplifies and the last term vanishes, giving
∆W =
∫
X
[
i ∂ω ∧ ıµıµΩ + d(∆B + i ∆ω) ∧ (ıµΩ + 12 ıµıµΩ)
]
=
∫
X
[
i ∂ω ∧ ıµıµΩ + 2 ∂x ∧ ıµΩ + ∂x ∧ ıµıµΩ + ∂b˜ ∧ ıµΩ
]
,
(4.2)
where the ıµıµıµΩ term vanishes due to the type of d(∆B + i ∆ω), and we have written the
second line in terms of b˜ and x, the (0, 2) and (1, 1) parts of the complexified hermitian
moduli (3.21). As ıµ satisfies a graded Leibniz identity, we can rewrite the above as
∆W = 2
∫
X
(−ıµ∂x+ 12 i ıµıµ∂ω + 12 ıµıµ∂x− 12 ıµ∂b˜) ∧ Ω
= 2
∫
X
(µd ∧ ∂xd + iµd ∧ µe ∧ ∂dωecec + µd ∧ µe ∧ ∂dxe − 12µd ∧ ∂db˜) ∧ Ω.
(4.3)
Our first condition for the deformed background to be supersymmetric is ∆W = 0 when
evaluated on the solution. In other words, the terms in the brackets in (4.3) should be zero
up to a ∂-exact term.
We also need to impose the vanishing of the first derivative of ∆W . As ∆W is a functional,
– 13 –
this amounts to treating it as an action and finding the resulting equations of motion. Varying
∆W , one finds
∂xa − iµd ∧ (∂ω)dabeb + µd ∧ ∂axd − µd ∧ ∂dxa − 12∂ab˜ = 0, (4.4)
∂µd − 12 [µ, µ]d = 0, (4.5)
∂ıµΩ = 0. (4.6)
A few comments are in order. The condition in (4.5) is nothing but the Maurer–Cartan
equation for finite deformations of a complex structure. This is somewhat expected as we know
solutions to the Hull–Strominger system are manifolds with a complex structure. Notice that
we also have a second condition on µ in (4.6) which is not usually seen in discussions on the
moduli space of complex structures. This condition comes from requiring that the deformed
three-form Ω + ∆Ω is closed and thus holomorphic with respect to the new complex structure
– this is stronger than requiring a complex structure alone. Note that this same condition
that appears in [40] for Kodaira–Spencer gravity – there it is imposed as a constraint from
the outset but one should actually think of it as requiring that the deformed three-form Ω
remains d-closed. One could make a change of variables which solves this constraint explicitly
by taking ıµΩ = a+ ∂b where a is ∂-harmonic. This may be useful when investigating the
quantum theory defined by the superpotential but we do not use it in what follows.
4.2 A Maurer–Cartan equation from the holomorphic structure
The idea now is that these equations can be interpreted as a Maurer–Cartan equation for the
deformations. We know the infinitesimal moduli of the Hull–Strominger system are captured
by the D-cohomology of the holomorphic structure, so we expect D will be the differential that
appears in such a Maurer–Cartan equation. The other ingredient is a bracket. We introduce
the bundle Q′ as the sum of the holomorphic cotangent and tangent bundles:
Q′ ' T ∗(1,0)X ⊕ T (1,0)X. (4.7)
This is the bundle Q defined in (2.16) with the gauge sector suppressed for now. We write
(0, 1)-forms valued in this bundle as
y ∈ Ω(0,1)(Q′), y = (xaea, µaeˆa) ≡ x+ µ. (4.8)
The holomorphic structure for the Hull–Strominger system without the bundles is given by an
operator D that acts on sections of Q′. We can also introduce a bracket [·, ·] on forms valued
in Q′ and a pairing 〈·, ·〉 that traces over the Q′ indices:
[·, ·] : Ω(0,p)(Q′)× Ω(0,q)(Q′)→ Ω(0,p+q)(Q′), (4.9)
〈·, ·〉 : Ω(0,p)(Q′)× Ω(0,q)(Q′)→ Ω(0,p+q)(X). (4.10)
– 14 –
We give explicit expressions for D, the bracket [·, ·] and the pairing 〈·, ·〉 in equations (A.11),
(A.13) and (A.16). Using these we can rewrite ∆W , given in (4.3), as
∆W =
∫
X
〈y,Dy − 13 [y, y]− ∂b〉 ∧ Ω, . (4.11)
Note that we have redefined the (0, 2)-form field as b = b˜− µa ∧ xa. The equations of motion
that follow from varying ∆W can then be written compactly as
Dy − 12 [y, y]− 12∂b = 0, (4.12)
∂ıµΩ = 0. (4.13)
Let us make a few comments. Looking at ∆W in equation (4.11), we see it resembles
a Chern–Simons action. More specifically, the form of the action is that of a holomorphic
Chern–Simons theory for y with a Lagrange multiplier b that enforces a constraint for y. This
constraint is the same as the gauge choice that is imposed in Kodaira–Spencer theory [40, 41].
Note that the conventional Chern–Simons action has appeared as a superpotential in other
work [42]; we expect a similar analysis could be applied here.
Notice also that infinitesimal deformations are captured by
Dy = 12∂b. (4.14)
It follows from this and dH ∝ ∂∂ω = 0 that ∂b is ∂-closed.6 If the underlying manifold X
satisfies the ∂∂-lemma or H(0,2)(X) vanishes, ∂b is ∂-exact and can be absorbed in a redefinition
of the complexified Ka¨hler moduli x. We then see that infinitesimally the complexified Ka¨hler
moduli are counted by H(0,1)(T ∗(1,0)X).
4.3 Including the bundle
We now want to include the bundle degrees of freedom in the superpotential – we do this by
adding a Chern–Simons term ωCS(A) for the gauge connection A:
W =
∫
X
(
dB + ωCS(A) + i dω
) ∧ Ω. (4.15)
The B field transforms in the usual Green-Schwarz manner so that H is gauge invariant:
δB = tr(d ∧A). (4.16)
Consider a shift of the gauge connection by A→ A+ α. The corresponding change of the
Chern–Simons form is
∆ωCS = 2 tr(F ∧ α) + tr(α ∧ dAα) + 23 tr(α ∧ α ∧ α) + d tr(α ∧A). (4.17)
6Recall we have turned off the gauge sector in this subsection.
– 15 –
The exact term in this variation combines with the variation of the B field to give
B = ∆B + tr(α ∧A). (4.18)
As we show in appendix B.4, B is a globally defined two-form so it can be a modulus [14].
Expanding about a supersymmetric point, the extra terms in the variation of the super-
potential from the gauge sector are
∆Wα =
∫
X
∆ωCS ∧ (Ω + ıµΩ + ıµıµΩ)
=
∫
X
[
(tr(α ∧ ∂Aα) + 23 tr(α ∧ α ∧ α)) ∧ Ω
+ (2 tr(F ∧ α) + tr(α ∧ ∂Aα)) ∧ ıµΩ
+ d tr(α ∧A) ∧ (ıµΩ + 12 ıµıµΩ)
]
,
(4.19)
where the final terms combine with b in ∆W to give B = b+ tr(α ∧A). Upon replacing b by
B in (4.3), the extra terms in the variation of the superpotential are
∆Wα =
∫
X
(
tr(α ∧ ∂Aα) + 23 tr(α ∧ α ∧ α)− 2µd ∧ Fd ∧ α− α ∧ µd ∧ (∂Aα)d
) ∧ Ω. (4.20)
The full expression for ∆W is given by the sum of expressions (4.20) and (4.3) (with b→ B) .
The equations of motion that follow from varying the full superpotential are
∂xa − iµd ∧ (∂ω)dabeb − trFa ∧ α+ µd ∧ ∂axd − µd ∧ ∂dxa + 12 trα ∧ (∂Aα)a − 12∂ab = 0,
∂Aα+ α ∧ α+ Fdadza ∧ µd − µd ∧ (∂Aα)d = 0,
∂µd − 12 [µ, µ]d = 0,
∂ıµΩ = 0,
(4.21)
where we have redefined b to be
b = (B + i ∆ω)(0,2) − µa ∧ xa. (4.22)
Remarkably, the superpotential can still be written in a Chern–Simons fashion as
∆W =
∫
X
〈y,Dy − 13 [y, y]− ∂b〉 ∧ Ω, (4.23)
where y is now (0, 1)-form valued in Q ' T ∗(1,0)X⊕EndV ⊕T (1,0)X, and D, [·, ·] and 〈·, ·〉 are
now given by expressions (A.19), (A.20) and (A.21). The corresponding equations of motion
– 16 –
are
Dy − 12 [y, y]− 12∂b = 0, (4.24)
∂ıµΩ = 0. (4.25)
Together with the vanishing of ∆W , these are the conditions for a supersymmetric Minkowski
solution.
4.4 Vanishing of the superpotential
We now want to understand the condition ∆W = 0 in more detail. In what follows, we will
consider the moduli problem with the gauge sector turned off. Everything we say goes through
when we replace the D operator, bracket and pairing with those that include the gauge sector.
The deformed vacuum solution is supersymmetric if the equations of motion are satisfied
and the superpotential itself vanishes. As X is assumed to be compact, the superpotential
vanishes if the terms wedged with Ω are ∂-exact, that is
〈y,Dy − 13 [y, y]− ∂b〉 = ∂β, (4.26)
where β is an arbitrary (0, 2)-form. Upon substituting the first equation of motion (4.24) into
this expression, it simplifies to
1
3!〈y, [y, y]〉 − 12ya∂ab = ∂β. (4.27)
Now we use D
2
= 0 to constrain β. Taking D of the first equation of motion (4.24) gives
0 = D
2
y − [Dy, y]− 12D∂b
∝ ea(−[y, [y, y]]a − [y, ∂b]a + ∂∂ab)
= ea
(
1
3!∂a〈y, [y, y]〉 − 12∂a(yd∂db) + ∂∂ab
)
∝ (∂∂b− 12∂(yd∂db) + 13!∂〈y, [y, y]〉),
(4.28)
where we have used [y, [y, y]]a = − 13!∂a〈y, [y, y]〉 and [y, ∂ab] = 12∂a(yd∂db).7 We can integrate
this expression to give
kΩ = ∂b− 12yd∂db+ 13!〈y, [y, y]〉, (4.29)
where k is constant as it is anti-holomorphic and X is compact. Combining this with the
vanishing of the superpotential (4.27) gives
kΩ = ∂b+ ∂β. (4.30)
As Ω is not ∂ exact, we must have k = 0. We then identify β = −b up to a ∂-closed (0, 2)-form.
7These identities are easy to check using the explicit expressions for the bracket and pairing.
– 17 –
Putting this all together the full set of equations is
Dy − 12 [y, y]− 12∂b = 0, (4.31)
∂b− 12〈y, ∂b〉+ 13!〈y, [y, y]〉 = 0, (4.32)
∂ıµΩ = 0. (4.33)
These equations are equivalent to the vanishing of the superpotential and its derivative, and
so their solutions are a supersymmetric Minkowski vacuum. In other words, solutions (y, b) to
these equations are precisely the moduli of the Hull–Strominger system.
We pause to make a few comments. First note that these equations contain finitely many
powers of y and b; the equations do not give an infinite set of relations. This is somewhat
striking – generic deformations of geometric structures do not usually truncate at a given
order. In our case the fact that the equations depend on terms up to O(y3) is an indication
that there is more structure to the Hull–Strominger system than at first sight. This extra
structure is the existence of an underlying holomorphic Courant algebroid describing the
subsector of deformations given by simultaneous deformations of the complex structure and
the three-form flux. In deriving these equations and finding third-order equations for the
moduli, we might be encouraged to think there is some sort of algebroid underlying the full
heterotic system. Indeed, the ten-dimensional heterotic theory has a description in terms of
generalised geometry [31, 43, 44] and the Hull–Strominger system can be recast as in terms of
holomorphic Courant algebroid [30].
One might wonder if the form of these equations can survive α′ corrections. We derived
the equations for the moduli by starting from the superpotential for the four-dimensional
theory that one would get by compactifying on a solution to the Hull–Strominger system.
Part of the data of such solutions is a complex manifold. A complex manifold admits an SU(3)
structure whose torsion is constrained [45]. A special case of such manifolds are those with
vanishing torsion so they have SU(3) holonomy and are Calabi–Yau. If the solution to the
Hull–Strominger system admits an α′ → 0 limit, the α′ = 0 solution is simply Calabi–Yau. In
this case it is known that the superpotential receives no α′ corrections (to finite order) and so,
although the α′-corrected geometry is not longer Calabi–Yau, the tree-level superpotential is
exact [22, 46]. This means equations (4.31)–(4.33) will be correct even after α′ corrections. It
is not known what happens if there is no large-volume Calabi–Yau limit.
Up to this point there has been an asymmetry in the way we have treated the vanishing of
∆W and its derivative. In the next section we will see that we can combine these conditions
into a single Maurer–Cartan equation for an L3 algebra.
5 Moduli and an L3 algebra
So far we have derived the equations that determine the moduli for solutions to the Hull–
Strominger system for finite deformations. We will show in this section that these equations
can be reinterpreted as the Maurer–Cartan equation for an L3 algebra. At first sight there is
– 18 –
no obvious reason why the deformations of a system as complicated as the heterotic string
should be described by such a “nice” algebra structure. However it is not as surprising if
one remembers that the data of the Hull–Strominger system is equivalent to a holomorphic
Courant algebroid with a holomorphic vector bundle [30]. The L∞ structures that govern
deformations of Courant algebroids (or Dirac structures) have been found; in particular it is
known that the deformation complex of a Dirac structure is isomorphic to a cubic L∞ or L3
structure [47–50].
As we review in appendix A.3, an L∞ structure is specified by a choice of graded vector
spaces Yn and multilinear products `k [51, 52]. The idea is that the conditions from the
superpotential are most naturally written in terms of an L∞ structure that combines the
action of D on the moduli fields y and ∂ on the (0, 2) moduli b. We take the vector spaces Yn
to be
Yn = Ω(0,n)(Q)⊕ Ω(0,n+1)(X), (5.1)
so that an element of Y1 is Y = (y, b) where y is a (0, 1)-form valued in Q and b is a (0, 2)-form.
Using this notation, we write the multilinear products `k as
`1(Y ) := (Dy − 12∂b, ∂b),
`2(Y, Y ) := ([y, y], 〈y, ∂b〉),
`3(Y, Y, Y ) := (0,−〈y, [y, y]〉),
`k≥4 := 0.
(5.2)
We give expressions for the `k where the entries are arbitrary elements of Yn in (A.29). One
can check that these products have the correct symmetry properties and obey the L∞ relations,
which we write in (A.26). Note that this is highly non-trivial and is an indication of the
underlying holomorphic Courant algebroid.
5.1 Quasi-isomorphism to a natural holomorphic L3 algebra
We briefly remark that these structures have a nice mathematical interpretation: our L3
algebra (Y, `1, `2, `3) is L∞ equivalent to the underlying holomorphic algebra.
Neglecting the gauge bundle for a moment, we have the sheaf E of D-holomorphic sections
of Q′ ' T ∗(1,0)X ⊕ T (1,0)X and the sheaf of holomorphic functions OX . These form an L3
algebra, with underlying two-term complex
OX ∂−→ E , (5.3)
in precisely the same way that a Courant algebroid E together with the real C∞ functions
form an L3 algebra [53] with two-term complex
C∞(R) d−→ E ' T ∗X ⊕ TX. (5.4)
One can then consider the Dolbeault resolutions of the sheaves E and OX , and extend ∂ to a
– 19 –
morphism between them:
0 OX C∞(C) Ω(0,1) Ω(0,2)
0 E Q′ Ω(0,1)(Q′) Ω(0,2)(Q′)
ι ∂ ∂ ∂
ι D D D
∂ ∂ ∂ ∂
(5.5)
Our complex Y is then the total complex of the deleted resolution and the differential `1 is
the natural differential on this (see e.g. [54]). Our construction gives higher `n brackets on Y ,
providing an L3 algebra structure on the total complex.
As (5.5) is simply a resolution of (5.3), this construction essential provides us with a local
reformulation of the holomorphic L3 algebra (5.3) in terms of C∞ objects. Explicitly, one has
a map of complexes (of sheaves) as follows:
0 OX E 0 0
0 C∞(C) Γ(E)⊕ Ω(0,1) Ω(0,1)(E)⊕ Ω(0,2) Ω(0,2)(E)⊕ Ω(0,3)
∂
`1 `1 `1 `1
ι ι
(5.6)
As the cohomology of the total complex is the same as the cohomology of the complex it is
resolving, this is a quasi-isomorphism. (One can check this explicitly in our case.) However,
the morphism in (5.6) also respects the bracket structure of the L3 algebras on each complex,
thus it is a quasi-isomorphism of L3 algebras. We conclude that, in the L∞ sense, our L3
algebra (Y, `1, `2, `3) is locally equivalent to the holomorphic algebra (5.3).
Including the gauge bundle in this construction is straightforward; one simply replaces the
bundle Q′ above with the full holomorphic bundle (C.34) (this also recently appeared in [30]).
One finds an essentially identical two-term complex to (5.3) (see [55] for the analogue of (5.4)
including the gauge bundle), giving an L3 algebra on the local holomorphic sections. Via the
Dolbeault resolution, one sees that this is quasi-isomorphic to our L3 algebra (Y, `1, `2, `3)
(now including the gauge bundles) exactly as above.
5.2 An L∞ field equation
As explained in [52], there is a natural field equation that one can write down for a given L∞
structure. The constraint on the form of the field equation is that it is covariant under gauge
transformations of the fields Y. In terms of the L∞ products, the field equation is
F(Y ) = `1(Y )− 12`2(Y )− 13!`3(Y ) + . . . (5.7)
For us this expression truncates at third order as `k≥4 = 0.
– 20 –
Remarkably, the L3 field equation coming from (5.2) reproduces the conditions from the
vanishing of the superpotential and its derivative:
F(Y ) = (Dy − 12∂b− 12 [y, y], ∂b− 12〈y, b〉+ 13!〈y, [y, y]〉). (5.8)
In other words, the conditions for a supersymmetric Minkowski vacuum are equivalent to
F(Y ) = 0, ∂ıµΩ = 0. (5.9)
A particularly nice property of this rewriting is that the L∞ structure gives us the gauge
transformations of the moduli for free and guarantees that the gauge algebra closes. The
gauge transformation of Y by a gauge parameter Λ = (λ, ξ) ∈ Y0 is
δΛY = `1(Λ) + `2(Λ, Y )− 12`3(Λ, Y, Y ), (5.10)
where the higher-order brackets vanish. In general the gauge transformations take field
equations to combinations of field equations – the field equations are covariant. If one could
construct an action that has the L3 field equation as its equations of motion, one would expect
that action to be invariant under the L3 gauge transformations. In contrast to [52], we have
not been able to find such a candidate action nor do we expect one to exist; this is due to
the fact that the supersymmetry conditions are the vanishing of the superpotential and its
first derivative. Note that the superpotential alone is not expected to be invariant under both
λ and ξ transformations – we will see in the next section that the superpotential is actually
invariant under the ξ transformations alone which correspond to shifts by ∂a-exact forms.
6 A reduced L3 algebra and an effective action
In this section we discuss some consequences of the L3 algebra. In particular we comment
on how the L3 algebra can be reduced by quotienting by ∂-exact forms and show that this is
equivalent to integrating out b. We also discuss the relation of the moduli (y, b) to the effective
theory one would find by compactifying on a solution to the Hull–Strominger system. As we
have mentioned, the form of the superpotential (4.23) closely resembles that of holomorphic
Chern–Simons theory, and is in fact a generalisation of this theory. Holomorphic Chern–Simons
theory has several interesting relations with mathematical disciplines such as open and closed
topological string theory, knot theory, Donaldson–Thomas invariants and so on, and it would
be interesting to look for heterotic generalisations of these relations. This will be the subject of
future work. For now, we will restrict ourselves to making some observations about the (semi-)
classical effective action (4.23), and its relation to the lower-dimensional effective physics.
– 21 –
6.1 Integrating out b
We want to integrate out the (0, 2)-form field b. Looking back at the form of the L3 gauge
transformations (5.10), taking Λ = (0, ξ) gives
δya =
1
2∂aξ, δb = ∂ξ − 12〈y, ∂ξ〉, (6.1)
for some (0, 1)-form ξ. One can check that the superpotential (4.23) is invariant under this
gauge transformation provided X is compact. From this we see that y is defined up to ∂a-exact
forms. Notice also that ∆W splits into two pieces:
∆W =
∫
X
〈y,Dy − 13 [y, y]〉 ∧ Ω−
∫
X
〈y, ∂b〉 ∧ Ω, (6.2)
where the second term can be written as∫
X
〈y, ∂b〉 ∧ Ω ∝
∫
X
b ∧ ∂ıµΩ. (6.3)
From this we see that b plays the role of a Lagrange multiplier. We shall see below that given
certain assumptions about the Hodge diamond of X, specifically h(2,0) = 0, then the field b
has no associated massless modes. We can then integrate out b, resulting in
∆W [y] =
∫
X
〈y,Dy − 13 [y, y]〉 ∧ Ω, (6.4)
where y now satisfies the constraint ∂ıµΩ = 0. We want to think of this functional as an
effective action. Note that for ∂ıµΩ = 0 there is a gauge symmetry of this action
δya = ∂aξ, (6.5)
where ξ ∈ Ω0,1(X). We are thus led to define Q˜ as a reduced sheaf of Q whose sections satisfy
the constraint and are defined up to ∂a-exact terms:
Γ(Q˜) = {Γ(Q) | ∂ıµΩ = 0, ya ∼ ya + ∂aξ}. (6.6)
One can check that the brackets on Q are well defined on Q˜, and that the L3 algebra descends
to a differential graded Lie algebra (DGLA). A very similar sheaf is also considered in the
discussion of βγ systems in [56]. Note that this DGLA is L∞ quasi-isomorphic to the algebra
on (5.1).
The second gauge symmetry of (6.4) is a generalisation of the Chern–Simons symmetry.
The superpotential is invariant under
δy = Dλ− [y, λ], (6.7)
where λ ∈ Ω0(Q˜) satisfies ∂ıλΩ = 0. Note that the gauge algebra generated by (5.10) is
– 22 –
reducible; a gauge transformation by Λ = (λ, ξ) is trivial if
λ = ∂w, ξ = −∂w + 12〈y, ∂w〉, (6.8)
for w ∈ Ω0(X).
From the effective superpotential (6.4) we derive the equation of motion
Dy − 12 [y, y] = 0. (6.9)
This should be interpreted as an equation on the sheaf Q˜. Note that under ya 7→ ya + ∂aξ,
this equation becomes
Dy − 12 [y, y] = ∂(∂ξ + yd∂dξ) ∼ 0, (6.10)
so it is well defined as an equation on the sheaf Q˜. Recall that we already know the
superpotential is invariant under δya = ∂aξ. One can show that equation (6.9), together with
the condition that the effective action vanishes, is equivalent to the Maurer–Cartan equations
(4.31)–(4.33). Indeed, note that (6.9) is equivalent to
Dy − 12 [y, y] = 12∂b, (6.11)
for some b ∈ Ω(0,2)(X). For solutions to this equation, the condition that the action vanishes
can then be written as 〈y, [y, y]〉 = ∂-exact, which can be rewritten as
1
3!〈y, [y, y]〉 − 12ya∂ab = ∂β, (6.12)
for some β ∈ Ω(0,2)(X). Here we used that the second term on the left-hand side is ∂-exact
– it integrates to zero against Ω due to the constraint ∂ıµΩ = 0. This then gives the same
starting point for our derivation of equations (4.31)–(4.33).
It is beyond the scope of the present paper to investigate general solutions to (6.9) and
∆W [y] = 0, i.e. integrable deformations of heterotic geometries. We will however make some
comments on the couplings derived from (6.4) in the four-dimensional effective field theory.
From this we make a conjecture about the obstructions that can appear in the Maurer–Cartan
equations.
6.2 Effective field theory and Yukawa couplings
Our starting point to derive the effective physics is the superpotential (4.23), where we have
re-introduced the field b. When dimensionally reducing the theory, it is common practice to
split our fields (y, b) into “massless” and “heavy” modes
y = y0 + yh, (6.13)
b = b0 + bh. (6.14)
– 23 –
We imagine performing a formal dimensional reduction of the theory to a four-dimensional
Minkowski background where we keep all the massive Kaluza–Klein modes for the time being.
The corresponding mass matrix of the reduced theory reads8
Vαβ = e
K∂α∂γW∂β∂κW Kγκ, (6.15)
where {α, β, . . .} denote holomorphic directions in the parameter space and K is the Ka¨hler
potential. Full knowledge of the Ka¨hler potential is not necessary at this point, but the curious
reader is referred to [20, 21] for more details. From the form of the mass matrix, it is easy to
see that a field direction α is massless if and only if
∂γ∂αW = 0 ∀ γ ⇒ (∂γ∂α∆W )|(y,b)=0 = 0 ∀ γ, (6.16)
where the field directions γ can in principle be massive.
In the end, we are interested in a reduced field theory of massless modes, where the massive
modes have been “integrated out”. It is easy to see that the field direction corresponding to b
is massive (although need not be an eigenmode of the mass matrix). Indeed, from (6.16) it
follows that b0 must satisfy
∂ab0 = 0, (6.17)
and so b is an anti-holomorphic section of Ω(0,2)(X). We restrict ourselves to geometries where
this bundle has no sections, in other words
H
(2,0)
∂
(X) = 0. (6.18)
This is true in particular for Calabi–Yau geometries. It follows that we can integrate out the b
field as far as the effective theory is concerned, leaving us with the effective superpotential
(6.4), where now the Beltrami differential component µ of y satisfies ∂ıµΩ = 0 as above. From
condition (6.16) it follows that the remaining massless fields y0 then satisfy
Dy0 = 0, (6.19)
where this should be viewed as an equation in the sheaf Q˜.
It is also natural to decompose the symmetry transformations (6.7) in terms of the massless
and massive modes. A suggestive decomposition, given the condition (6.19), is the following
δy0 = Dλ, (6.20)
δyh = −[y0, λ]− [yh, λ]. (6.21)
With this decomposition, we see that the massless modes are parametrised by cohomology
8In principle, there is also a potential coming from D-terms. However, as we show in appendix D, the
D-terms can be set to zero by a complexified gauge transformation and so they do not lift any moduli.
– 24 –
classes
[y0] ∈ H(0,1)D (Q˜). (6.22)
This cohomology is isomorphic to H
(0,1)
D
(Q) for manifolds satisfying either the ∂∂-lemma or
H(0,1)(X) = 0 [14, 15]. We give a brief review of this cohomology and its decomposition in
into more familiar cohomologies by means of long exact sequences in appendix E.
Decomposed in terms of massless and heavy modes, the effective action now reads
∆W =
∫
X
(〈yh, Dy0yh〉 − 13〈y0, [y0, y0]〉 − 〈yh, [y0, y0]〉 − 13〈yh, [yh, yh]〉) ∧ Ω, (6.23)
where we denote
Dy0yh = Dyh − [y0, yh]. (6.24)
We see that the heavy yh modes are the only ones that propagate internally. Note that
even though we take the expectation value of yh to vanish, by including internal quantum
corrections, we see that the coupling between y0 and yh can generate higher-order couplings
of the massless fields. These new couplings are however of quartic order and higher in y0, and
are hence non-renormalisable in the effective field theory. The only renormalisable coupling
we need to worry about from an effective field theory point of view is therefore the Yukawa
coupling
∆WYuk(y0) = −13
∫
X
〈y0, [y0, y0]〉 ∧ Ω. (6.25)
This argument is similar to and generalises Witten’s standard argument for the gauge sec-
tor [22].
Note that in addition to the standard Yukawa couplings between bundle moduli, the
Yukawa couplings (6.25) also contain couplings of gravity-gravity type (couplings of deforma-
tions of the geometry) and gravity-bundle type, often referred to as µ-terms in the literature.
It would be interesting to investigate the phenomenological implications of such couplings,
but it is beyond the scope of the present paper to do so.
Let αA ∈ H(0,1)(Q˜) denote a set of inequivalent cohomology classes spanning H(0,1)(Q˜),
and expand
y0 =
∑
A
CAα
A, (6.26)
where the CA now correspond to the four-dimensional fields, including in principle moduli
and matter fields. The Yukawa couplings then read
∆WYuk =
∑
A,B,C
CACBCC
∫
X
〈αA, [αB, αC ]〉 ∧ Ω. (6.27)
– 25 –
A massless field direction αA is then truly free if and only if
YABC =
∫
X
〈αA, [αB, αC ]〉 ∧ Ω = 0 ∀ αB, αC . (6.28)
In particular, this is true if
[αA, αB] = DβAB ∀ αB. (6.29)
Note that, starting from the Maurer–Cartan equation (6.9), this is simply the condition for
an infinitesimal deformation in the field direction αA to be unobstructed. The effective field
theory then prompts us to make the following conjecture: when H
(2,0)
∂
(X) = 0,9 the only
non-trivial obstructions coming from the Maurer–Cartan equations are given by the constraints
(6.29) on the infinitesimal moduli.
7 Conclusions
In this paper we have considered finite deformations of the Hull–Strominger system. Starting
with the four-dimensional N = 1 superpotential, we showed that integrable deformations
corresponding to holomorphic directions on the moduli space can be parametrised by solutions
of a Maurer–Cartan equation for an L3 algebra, which we described in detail.
There are many directions one could follow from this work. Firstly, one might wonder
which of the infinitesimal deformations parametrised by H
(0,1)
D
(Q˜) can be integrated to finite
deformations, corresponding to solutions of the L3 Maurer–Cartan equation. In particular,
are there some special cases where a generalisation of the Tian–Todorov lemma applies? It
would be interesting to apply our formalism to some explicit examples, and in this way work
out the spectrum of free fields in the low-energy four-dimensional theory.
The superpotential led to a generalisation of holomorphic Chern–Simons theory [57–59]
that couples hermitian and complex structure moduli. Following [57] and [40], it seems that
one should think of this theory by taking spacetime to be spanned by the anti-holomorphic
directions with holomorphic Ω-preserving generalised diffeomorphisms playing the role of a
gauge group.10 It would be interesting to study this further. In particular, by starting from
the superpotential as an effective action and investigating its quantisation one might hope it
defines a consistent quantum theory (cf. [61, 62]) and gives analogues of Donaldson–Thomas
or holomorphic Casson invariants for heterotic geometries. Note that the superpotential is
complex in general so the path integral will not be convergent. Such complex path integrals
have appeared before in the study of complex Chern–Simons theory [42, 63, 64] where they
are understood by analytic continuation. We foresee a similar treatment here.
9So we can integrate out b in the effective theory.
10Generalised diffeomorphisms are transformations generated by sections of Q via the Dorfman bracket, whose
antisymmetrisation is the Courant bracket. (In the real case, they are simply the diffeomorphisms together
with gauge transformations of the supergravity fields.) The Ω-preserving condition is just (4.6) which ensures
the deformed three-form is d-closed. Another Chern–Simons like theory featuring generalised diffeomorphisms
as a gauge algebra has recently appeared in [60].
– 26 –
As a step towards a complete understanding of the quantum heterotic moduli space, one
could construct a world-sheet AKSZ topological model [65, 66] or a topological string model
for the effective theory similar to Witten’s open string model for ordinary Chern–Simons
theory [67]. As a guide, one might start by comparing the heterotic moduli space with the
spectrum of holomorphic βγ systems and the chiral de Rham complex [56, 68–72]. Several
other approaches to the (0, 2) world-sheet have appeared over the years (see [46, 72–79] and
references therein). It would be interesting to investigate how these methods connect with the
approach outlined in the present paper. These are all interesting aspects which we hope to
explore in future publications.
One might also consider the moduli space of heterotic compactifications on more exotic
geometries, such as G2 or Spin(7) manifolds [80–82]. In the case of G2 compactifications, the
form of the moduli space is remarkably similar: for example, the infinitesimal deformations
are again captured by a cohomology. Despite this there are notable differences such as the
analogue of the bundle Q not appearing as an extension. It would be interesting to investigate
the finite deformation algebras in these cases, and in the process identify the corresponding
L∞ structure. This might give a G2 generalisation of Chern–Simons theory.
We have been concerned with finding the honest supersymmetric deformations of solutions
to the Hull–Strominger system. For this we only needed to consider the superpotential
in the four-dimensional theory. Of course, the four-dimensional theory also has a Ka¨hler
potential which is important for understanding the physical potential of the effective theory.
The Ka¨hler potential and the metric on the moduli space have been worked out in recent
publications [20, 21]. One might wonder how these objects appear in our formalism. It
seems that the cleanest description of these objects would follow from a proper analysis using
generalised geometry. As outlined in appendix C, the N = 1 heterotic structure is described
by an invariant object χ˜ so the Ka¨hler potential should be given by a functional of this object,
similar to the Hitchin functional for SU(3) and G2 structures [83]. Indeed, it seems that a
similar story applies to heterotic compactifications in other dimensions. We hope to make
progress on this in a future work.
Note that even though we have the invariant object χ˜ we do not have a natural integrability
condition for it – the generalised connection is not torsion-free in the heterotic string [43].
Curiously, it appears that when one looks at deformations of this structure there is a nice
integrability condition (given by the superpotential). It would be interesting to see if this
pattern persists for other generalised geometries.
Acknowledgements
We would like to thank Bobby Acharya, Dmitri Alekseevsky, Philip Candelas, Jose´ Figueroa-
O’Farrill, Marco Gualtieri, Brent Pym, Savdeep Sethi and Dan Waldram for helpful discussions.
AA is supported by a Junior Research Fellowship from Merton College, Oxford. XD is
supported in part by the EPSRC grant EP/J010790/1. CS-C has been supported by a Seggie
Brown Fellowship from the University of Edinburgh. ESS is supported by a grant from the
Simons Foundation (#488569, Bobby Acharya).
– 27 –
A Conventions
In this appendix we set out our conventions and notation. We will use (m,n, . . .) indices
to denote real coordinates and (a, b, . . . , a, b, . . .) to denote complex coordinates on the real
six-dimensional manifold X. Using this we can expand, for example, a vector as
v = vmeˆm = v
aeˆa + v
aeˆa. (A.1)
Our fields are form-valued so, to save space, we often omit the wedge symbol where it will
not lead to confusion. Our convention for the contraction of a vector-valued one-form with a
p-form is that the vector index is contracted as usual and the form components are wedged.
In coordinates, for a vector-valued one-form w and a p-form ρ, we have
ıwρ = e
m ∧ ıwmρ = wn ∧ ρn, (A.2)
where ρm is defined as
ρm =
1
(p− 1)!ρmn1...np−1e
n1...np−1 . (A.3)
It follows that ıw satisfies a Leibniz rule:
ıw(ρ ∧ σ) = ıwρ ∧ σ + ρ ∧ ıwσ. (A.4)
The interior product of a vector with a one-form is extended to p-vectors and p-forms using
the y operation, defined as
uyρ = 1p!u
m1...mpρm1...mp , (A.5)
where u is a p-vector and ρ is a p-form. We indicate the p-vector obtained by raising the
indices of a p-form with the metric g by a superscript ] – for example
(ρ])m1...mp = gm1n1 . . . gmpnpρn1...np . (A.6)
A.1 Heterotic supergravity
The Hull–Strominger system follows from setting the supersymmetry variations of the ten-
dimensional gravitino ψ, dilatino λ and gaugino χ to zero. In our conventions these are
δψM = ∇+Mε = ∇LCM ε+ 18HMNPΓNP ε+O(α′2), (A.7)
δλ = (ΓM∂Mφ+
1
12HMNPΓ
MNP )ε+O(α′2), (A.8)
δχ = −12FMNΓMNε+O(α′2), (A.9)
where ε is a ten-dimensional Majorana–Weyl spinor and ∇LC denotes the Levi-Civita connec-
tion.
– 28 –
A.2 Holomorphic structure
Ignoring the gauge sector for the moment, the relevant fields are (0, 1)-forms taking values in
Q′ = T ∗(1,0)X ⊕ T (1,0)X. We write (0, 1)-forms valued in this bundle as
y = (xae
a, µaeˆa) = x+ µ, (A.10)
where xa and µ
a are (0, 1)-forms. More generally we will write yp = (xp, µp) ∈ Ω(0,p)(Q′) – we
will often drop the subscript denoting the form degree.
The holomorphic structure is defined by a D operator that is nilpotent. The action of D
on y ∈ Ω(0,p)(Q′) is
(Dy)a = ∂xa + i(∂ω)eace
c ∧ µe,
(Dy)a = ∂µa.
(A.11)
One can check that D
2
= 0 follows from dH ∝ ∂∂ω = 0. Note also that this convention
implies
(D∂ρ)a = ∂(∂aρ) = ∂a∂ρ, (A.12)
for a form ρ (not valued in Q′).
The bracket [·, ·] : Ω(0,p)(Q′)× Ω(0,q)(Q′)→ Ω(0,p+q)(Q′) is
[y, y′]a = µd ∧ ∂dx′a − ∂dxa ∧ µ′d − 12µd ∧ ∂ax′d + 12∂axd ∧ µ′d + 12∂aµd ∧ x′d − 12xd ∧ ∂aµ′d,
[y, y′]a = µb ∧ ∂bµ′a − ∂bµa ∧ µ′b.
(A.13)
The bracket is graded commutative and satisfies [yp, yq] = (−)1+pq[yq, yp]. Furthermore, D
satisfies a graded Leibniz identity with the bracket
D[yp, y
′
q] = [Dyp, y
′
q] + (−1)p[yp, Dy′q]. (A.14)
The bracket does not satisfy a graded Jacobi identity. As is the case for a Courant algebroid,
the Jacobi identity holds up to a ∂-exact term. Evaluated for y ∈ Ω(0,1)(Q′), one finds
[y, [y, y]] = − 1
3!
∂〈y, [y, y]〉, (A.15)
so that only the (1, 0)-form valued component is non-zero. Here we have defined a pairing
between two sections as
〈y, y′〉 = µd ∧ x′d + xd ∧ µ′d. (A.16)
– 29 –
More generally one finds
[ym, [y
′
n, y
′′
p ]] + (−1)m(n+p)[y′n, [y′′p , ym]] + (−1)p(m+n)[y′′p , [ym, y′n]]
= − 13!
[
∂a〈y, [y′, y′′]〉+ (−1)m(n+p)∂a〈y′, [y′′, y]〉+ (−1)p(m+n)∂a〈y′′, [y, y′]〉
]
.
(A.17)
When we include the gauge sector we write sections as
y = (xae
a, α, µaeˆa) = x+ α+ µ ∈ Ω(0,1)(Q), (A.18)
where xa and µ
a are (0, 1)-forms. The D operator, bracket and pairing are extended to include
the gauge field component. The D operator is
(Dy)a = ∂xa + i(∂ω)eace
c ∧ µe − tr(Fa ∧ α),
(Dy)a = ∂µa,
(Dy)α = ∂Aα+ Fb ∧ µb,
(A.19)
where the final component is the gauge field piece and Fa = Fabdz
b. The bracket is
[y, y′]a = . . .− 12 trα ∧ (∂Aα′)a + 12 tr(∂Aα)a ∧ α′,
[y, y′]a = . . . ,
[y, y′]α = −α ∧ α′ − (−)1+αα′α′ ∧ α+ µb ∧ (∂Aα′)b − (∂Aα)b ∧ µ′b
= −[α, α′] + µb ∧ (∂Aα′)b − (∂Aα)b ∧ µ′b,
(A.20)
where we have written only the extra terms that appear in the bracket. Again, the bracket
obeys a Jacobi identity up to a ∂-exact term. The pairing between sections is given by
〈y, y′〉 = µd ∧ x′d + xd ∧ µ′d + tr(α ∧ α′). (A.21)
A.3 L∞ structure
We follow [52] for the conventions of an L∞ algebra in the “`-picture”.11 We start with a
graded vector space Y
Y =
⊕
n
Yn, n ∈ Z, (A.22)
where the Yn are of degree n. The L∞ algebra admits multilinear products `1, `2, . . ., where
`k has degree 2− k. This means `1 is degree 1, `2 is degree 0, `3 is degree −1, and so on. The
products are graded commutative so that for example
`2(Y, Y
′) = (−1)1+Y Y ′`2(Y ′, Y ), (A.23)
11We have swapped n→ −n so that the degree of Yn matches the form degree of y
– 30 –
where a superscript denotes the degree of Y ∈ Yn. More generally we have
`k(Y
σ(1), . . . , Y σ(k)) = (−1)σ(σ;Y )`k(Y 1, . . . , Y k), (A.24)
where Y 1 = Y , Y 2 = Y ′, etc. The sign has two contributions: (−1)σ gives a plus if the
permutation is even and a minus if the permutation is odd; (σ;Y ) is determined by
Y 1 ∧ . . . ∧ Y k = (σ;Y )Y σ(1) ∧ . . . ∧ Y σ(k), (A.25)
where Y ∧ Y ′ = (−1)Y Y ′Y ′ ∧ Y .
In these conventions, the first few L∞ identities are
`1(`1(Y )) = 0,
`1(`2(Y, Y
′)) = `2(`1(Y ), Y ′) + (−1)Y `2(Y, `1(Y ′)),
`1(`3(Y, Y
′, Y ′′)) = −`3(`1(Y ), Y ′, Y ′′)− (−1)Y `3(Y, `1(Y ′), Y ′′)
− (−1)Y+Y ′`3(Y, Y ′, `1(Y ′′))− `2(`2(Y, Y ′), Y ′′)
− (−1)(Y+Y ′)Y ′′`2(`2(Y ′′, Y ), Y ′)− (−1)(Y ′+Y ′′)Y `2(`2(Y ′, Y ′′), Y )
(A.26)
The field equations and gauge transformations are
F(Y ) =
∞∑
n=1
(−1)n(n−1)/2
(n)!
`n(Y
n) = `1(Y )− 12`2(Y, Y )− 13!`3(Y, Y, Y ) + . . . , (A.27)
δΛY = `1(Λ) + `2(Λ, Y )− 12`3(Λ, Y, Y )− 13!`4(Λ, Y, Y, Y ) + . . . , (A.28)
where Y ∈ Y1 and Λ ∈ Y0.
The multilinear products `k for the moduli of the heterotic system are
`1(Y ) := (Dy +
1
2(−1)Y ∂b, ∂b),
`2(Y, Y
′) := ([y, y′], 12(〈y, ∂b′〉+ (−1)1+Y Y
′〈y′, ∂b〉)),
`3(Y, Y
′, Y ′′) := 13(−1)Y+Y
′+Y ′′(0, 〈y, [y′, y′′]〉+ (−1)Y (Y ′+Y ′′)〈y′, [y′′, y]〉
+ (−1)Y ′′(Y+Y ′) 〈y′′, [y, y′]〉),
`k≥4 := 0.
(A.29)
B Comments on heterotic flux quantisation
In this appendix we comment on flux quantisation in the heterotic string, and its relation
to the global nature of the deformation of certain quantities that appear in this paper. Our
discussion applies to general ten-dimensional heterotic supergravity and can thus be applied
to solutions other than four-dimensional Minkowski compactifications. Note that we are not
– 31 –
saying anything new here; understanding flux quantisation in the heterotic string is still an
open problem for the simple reason that H is not d-closed in general.
We proceed in steps, beginning with a toy example of the quantisation of the flux of
an abelian gauge bundle. We then present a similar treatment of the two-form gerbe as a
warm-up for the case of the heterotic gauge fields. We follow [84] for much of the early part of
the discussion.
B.1 A toy example: The abelian bundle
Consider a vector bundle over a manifold X and let A denote an abelian connection with
curvature F = dA. Let {U i} denote an open cover of X. We denote the overlaps by
U ij = U i ∩ U j and so on for higher intersections. We assume that the cover is “good” so
that the U i and their intersections are contractible. We employ the standard notion for
the Cˇech co-boundary operator where appropriate: for some sheaf F , if fi ∈ F(U i) then
(∂f)ij = fi − fj ∈ F(U ij), and so on.
For the curvature to be well defined, we require that on U ij we have
d(Ai −Aj) = 0, (B.1)
where Ai and Aj denotes the connections on U i and U j . As U ij is contractible, by the Poincare´
lemma, we must therefore have
(∂A)ij = Ai −Aj = dγij , (B.2)
on U ij for some zero-forms γij . On triple overlaps U ijk we have
d(∂γ)ijk = d(γij + γjk + γki) = (∂
2A)ijk = 0, (B.3)
which is often referred to as taking the co-cycle of dγij . It follows that cijk = (∂γ)ijk are
constants:
cijk = (∂γ)ijk = γij + γjk + γki ∈ R(U ijk) ⊂ C∞(U ijk;R). (B.4)
Clearly from (B.4) we have ∂c = 0, so the cijk define a class of the sheaf cohomology
[cijk] ∈ Hˇ2(X;R) ∼= H2(X;R), (B.5)
which represents the two-form flux F = dA. From (B.4) this might look like a trivial co-cycle,
but this is deceptive since the class is trivial only if the individual γij can be chosen to be
constant. In this case, we see from (B.2) that the connection A can be made global so that
the flux is trivial. In this language flux quantisation is the statement that the cijk are in fact
integers, cijk ∈ Z(U ijk), so that they define an integral class [cijk] ∈ Hˇ2(X;Z) ∼= H2(X;Z)
Gauge transformations of the connection do not change the class [cijk] ∈ Hˇ2(X;R). To
– 32 –
see this consider a gauge transformation which preserves the curvature F :
A′i = Ai + di, i ∈ C∞(U i;R). (B.6)
This transformation induces a deformation of the γij as
γ′ij = γij + i − j + κij = γij + (∂)ij + κij , (B.7)
where κij ∈ R(U ij) are constants. Thus c′ = ∂γ′ = ∂γ + ∂κ = c + ∂κ is shifted by a Cˇech
co-boundary, and so [c′ijk] = [cijk] ∈ Hˇ2(X;R) is unchanged. Note that in the case of a
quantised flux, the integrality of cijk holds only in a preferred set of gauges for γij under shifts
by real constants κij .
Now consider a general deformation of the above system. We denote the variations by
∆(. . .): for example A′ = A+ ∆A. We fix the (integral) cohomology class of the quantised flux
so that ∆cijk = 0. This means that we must have (∂∆γ)ijk = 0 and thus as C∞(R) is acyclic
we can find i ∈ C∞(U i;R) with ∆γij = (∂)ij = i−j . We see the γij can be deformed only by
a gauge transformation, as in (B.7). Performing a global gauge transformation A′′i = A
′
i − di,
we find that in the new gauge we have
∆˜Ai = A
′′
i −Ai = ∆Ai − di, (B.8)
so that on U ij we have
∆˜Ai − ∆˜Aj = d(∆γij − i + j) = 0. (B.9)
We have shown there exists a gauge in which the variation of the connection is a global
one-form ∆˜Ai = ∆˜Aj .
Note that we could have performed the deformation requiring only that ∆cijk = (∂κ)ijk
for κij ∈ R(U ij) – this fixes the real cohomology class [cijk] ∈ Hˇ2(X;R). We would then have
deduced that ∆γij = (∂)ij + κij , leading to the same gauge transformation of A as above.
This would correspond to deforming away from the gauge (choice of γij) in which cijk are
explicitly integral, while above we restricted ourselves to the gauge with cijk ∈ Z(U ijk).
The story becomes more intricate for non-abelian bundles. Recall that a non-abelian
connection A on overlaps U ij transforms as
Aj = gijAig
−1
ij + gijdg
−1
ij . (B.10)
For the purpose of the present paper we will assume without proof that we can take ∆gij = 0,
as in the abelian case.
B.2 The two-form gerbe example
The case of a two-form gerbe is a direct generalisation of the abelian bundle. The gerbe is
specified by a set of two-forms Bi ∈ Ω2(U i) covering the manifold. The field strength H = dB
– 33 –
is globally defined. As before, this means that on overlaps U ij we have
Bi −Bj = dλij , (B.11)
for some one-forms λij ∈ Ω1(U ij). Again we have (∂B)ij = dλij on U ij so that (∂λ)ijk is
d-closed
d(∂λ)ijk = (∂
2B)ijk = 0. (B.12)
As U ij is contractible (∂λ)ijk is actually exact
(∂λ)ijk = dζijk, (B.13)
for some ζijk ∈ C∞(U ijk;R). Finally we take the co-cycle of dζijkl on quadruple overlaps U ijkl:
d(∂ζ)ijkl = (∂
2λ)ijkl = 0. (B.14)
It follows that the functions cijkl = (∂ζ)ijkl are constants. As before, if the flux is quantised
the cijkl can be made integral by appropriate choice of ζijk. Indeed, the cijkl represent a
co-cycle in the sheaf cohomology
[cijkl] ∈ Hˇ3(X;Z) ∼= H3(X;Z), (B.15)
where the class in H3(X;Z) is given by the flux H = dB.
Consider deformations of the above system. From the quantisation of the constants cijkl
and the acyclicity of C∞(R) we have
∆cijkl = (∂∆ζ)ijk = 0 ⇒ ∆ζijk = (∂κ)ijk, (B.16)
for some κij ∈ C∞(U ij ;R). This leads immediately to12
(∂∆λ)ijk = d∆ζijk = (∂dκ)ijk ⇒ ∆λij = dκij + (∂)ij , (B.17)
for some i ∈ Ω1(U i). Putting this all together we have
∆Bi −∆Bj = d∆λij = di − dj , (B.18)
so that
∆Bi − di = ∆Bj − dj . (B.19)
Looking at these formulae, we see that the variations are forced to take precisely the form
needed to constitute a gauge transformation of B and its descendants λ and ζ. Thus, in an
appropriately chosen gauge, we have that ∆B is a global two-form.
12Note that again this relation would be unchanged if we simply fixed the real cohomology class [c] ∈ Hˇ3(X;R)
so that ∆cijkl = (∂r)ijkl for some rijk ∈ R(U ijk). We would then have ∆ζijk = rijk + (∂κ)ijk.
– 34 –
B.3 Heterotic flux quantisation
We now come to the example relevant for the present paper: the B field of the heterotic string.
Recall that the anomaly cancellation condition reads
H = dB + ωCS(A), (B.20)
where ωCS(A) denotes the Chern–Simons three-form and we set
α′
4 = 1. A gauge transformation
of A of the form
A′ = gAg−1 + gdg−1 (B.21)
induces the following transformation of the Chern–Simons form [43]
ωCS(A
′) = ωCS(A) + d tr(g−1dg ∧A)− dµ(g), (B.22)
where
dµ(g) = tr(g dg−1g dg−1g dg−1). (B.23)
Hence we have
ωCS(A
′) = ωCS(A) + dω2(g,A), (B.24)
where
ω2(g,A) = tr(g−1dg ∧A)− µ(g). (B.25)
Let us now consider the patching on U ij . From equation (B.20) we get
d(Bi −Bj) + dω2ij(g,A) = 0, (B.26)
where
ω2ij(g,A) = tr
(
g−1ji dgji ∧Aj
)− µ(gji). (B.27)
It follows that
Bi −Bj + ω2ij(g,A) = dλij , (B.28)
for some one-forms λij . Taking a co-cycle of this on triple intersections U ijk gives the relation
(∂ω2)ijk = d(∂λ)ijk. (B.29)
It can further be shown that
(∂ω2)ijk(g,A) = dω
1
ijk(g), (B.30)
for some one-forms ω1ijk(g) ∈ Ω1(U ijk).
The expression for ω1ijk(g) is not relevant in the present context, but it can be taken to
be independent of the gauge connection A, depending on the transition functions gij . To see
this, simply vary ω2ij(g,A) with respect to A. We find that
∆ω2ij(g,A) = tr(αi ∧Ai)− tr(αj ∧Aj), (B.31)
– 35 –
where we have defined ∆A = α, which transforms appropriately in the adjoint representation
when the transition functions gij are kept constant. From this it is clear that
∆(∂ω2)ijk(g,A) = 0, (B.32)
which shows that we may take ω1ijk(g) to be independent of A without loss of generality.
Putting together (B.29) and (B.30), we then get the relation
(∂λ)ijk − ω1ijk(g) = dζijk, (B.33)
for some functions ζijk ∈ C∞(U ijk;R). We again co-cycle this relation on quadruple intersec-
tions U ijkl to get
d
(
ω0ijkl(g) + (∂ζ)ijkl
)
= 0, (B.34)
where we have used the fact that
(∂ω1)ijkl(g) = dω
0
ijkl(g), (B.35)
for some functions ω0ijkl(g). We can co-cycle this relation again on quintuple overlaps U ijklm
to get
d(∂ω0)ijklm(g) = 0, (B.36)
and thus the numbers kijklm = (∂ω
0)ijklm(g) define a class
[kijklm] ∈ Hˇ4(X;R) ∼= H4(X;R). (B.37)
This represents the Pontryagin class trF ∧ F in the cohomology of the sheaf R. Note however
from (B.34) we have for our particular bundle
ω0ijkl(g) + (∂ζ)ijkl = cijkl, (B.38)
for some constants cijkl. Computing the co-cycle (∂ω
0)ijklm(g) we thus find
kijklm = (∂c)ijklm, (B.39)
and so [kijklm] = 0 ∈ Hˇ4(X;R). This is the sheaf cohomology version of the statement that
the bundle in question has a trivial first Pontryagin class as trF ∧ F = dH is exact.
B.4 Deforming the system and a well-defined global two-form
We now consider variations of the above heterotic story. The goal is to show that the heterotic
quantisation condition leads us naturally to a global two-form
B = ∆B + tr(α ∧A). (B.40)
– 36 –
This is an essential part of the moduli of the main text, where α = ∆A is the global one-form
variation of the gauge connection. The story is very similar to that of the two gerbe, with
some subtleties. Note that as for the abelian bundle, we will assume that we can choose to
keep the transition functions gij constant under deformations, that is
∆gij = 0, (B.41)
even in the case where the bundle is non-abelian. With this assumption, any deformation of
the bundle connection α = ∆A can be assumed to be a section of Ω1(EndV ).
We begin by noting that, imposing (B.41), a deformation of (B.38) gives
(∂∆ζ)ijkl = ∆cijkl. (B.42)
Hence the constants ∆cijkl define a sheaf cohomology class in Hˇ
3(X;R), even though the
original cijkl did not. We will require that this class vanishes [∆cijkl] = 0 ∈ Hˇ3(X;R), so that
the deformation does not produce any new third cohomology. One could think of this as
fixing the topological data associated to the original numbers cijkl, assuming that they have
some notion of “integrality” which cannot be continuously deformed, even though they do not
explicitly define such a class themselves. Indeed, this condition also seems to agree with the
general world-sheet arguments on the heterotic B field and flux quantisation made in [85],
where one thinks of the B field as a torsor. The lack of the notion of zero in the space of B
fields is reflected in our setup by the lack of an explicit cohomology class associated to the
cijkl. However, the variations of B about a given starting point do define a cohomology class
naturally.
From this requirement we have
(∂∆ζ)ijkl = (∂r)ijkl ⇒ ∆ζijk = rijk + (∂κ)ijk, (B.43)
for some rijk ∈ R(U ijk) and κij ∈ C∞(U ij). Next we take the variation of (B.33), again
imposing (B.41), and use the acyclicity of Ω1(U ijk) to find
(∂∆λ)ijk = d∆ζijk = (∂dκ)ijk ⇒ ∆λij = dκij + (∂)ij , (B.44)
exactly as for the simple two gerbe case. Finally, we take the variation of (B.28) to obtain
∆Bi − di + tr(αi ∧Ai) = ∆Bj − dj + tr(αj ∧Aj). (B.45)
We see we can absorb the d terms via a global gauge transformation of B as before, so that
on the overlaps U ij we have
Bi = Bj . (B.46)
Thus B defines a global two-form. Note in particular that B is gauge invariant with respect to
gauge transformations of the bundle [20]. This follows from the Green–Schwarz mechanism
– 37 –
wherein the B field transforms as
B → B − ω2(g,A). (B.47)
C The off-shell N = 1 parameter space and holomorphicity of Ω
In this appendix, we present a description of the space of off-shell scalar field configurations
that appears when we rewrite the ten-dimensional theory with manifest four-dimensional
N = 1 supersymmetry. This rewriting is done in the same spirit as the rewriting of eleven-
dimensional supergravity as a four-dimensional N = 8 theory in [86] and the rewriting of type
II supergravities as four-dimensional N = 2 theories in [37, 38, 87–89]. Here we focus only on
the scalars, which are described as an (infinite-dimensional) space of generalised geometric
structures of a type we will specify. We then show how to recover the three-form Ω from the
generalised geometric structure and outline how one can see that it is holomorphic on the
off-shell field space, as we claim in section 3.
C.1 SU(3)× SO(6) structures in the NS-NS sector
As a warm up, we consider the common NS-NS sector of ten-dimensional string theories,
which is well known to admit a description in the language of SO(10, 10)× R+ generalised
geometry [90] (see also [91, 92]). When rewriting the theory as a four-dimensional theory, we
consider the spacetime to admit a product structure, breaking the ten-dimensional Lorentz
symmetry to SO(3, 1)×SO(6). The fields which are naturally SO(3, 1) scalars can be packaged
as the generalised metric of an SO(6, 6)× R+ generalised geometry on the six internal spatial
dimensions (see for example [38, 87]).
As discussed in [28], the conditions for an N = 1 supersymmetric vacuum can be phrased
as the existence of an integrable SU(3) × SO(6) structure on the internal six-dimensional
generalised tangent space E ' TX⊕T ∗X. Such a structure encodes the generalised metric and
thus the physical fields. If we require only the presence of an off-shell N = 1 supersymmetry,
then one merely restricts the fields to admit a globally defined spinor, corresponding to a
(possibly non-integrable) SU(3)× SO(6) structure. The configuration space of off-shell scalar
fields that we require in our four-dimensional N = 1 description of the ten-dimensional theory
is thus the (infinite-dimensional) space of SU(3)× SO(6) structures on the generalised tangent
space E. This is the N = 1 NS-NS sector analogue of the discussion of the N = 2 vector- and
hyper-multiplet structures given for the full type II theories in [37, 88].
An SU(3) × SO(6) ⊂ SO(6, 6) × R+ structure on E is specified by a particular type of
complex section χ˜ of
W = L⊗ ∧3EC, (C.1)
where L ' ∧6T ∗X is the auxiliary R+ bundle transforming in the 1+1 representation of
SO(6, 6)× R+ as introduced in [90]. It is convenient to write χ˜ as
χ˜ = Φχ, χ ∈ Γ(∧3EC), (C.2)
– 38 –
where Φ ∈ Γ(L) is the generalised density with Φ = √ge−2φ in the coordinate frame (see [90]).
In order for χ˜ to have the correct stabiliser, χ must lie in a particular orbit under the action
of SO(6, 6) on ∧3EC at each point of the manifold X.
One can alternatively describe such a generalised structure in terms of a generalised
metric, which defines an SO(6)× SO(6) structure on E, which then splits as E ' C+ ⊕ C−,
so that
∧3E −→ ∧3C+ ⊕ (∧2C+ ⊗ C−)⊕ (C+ ⊗ ∧2C−)⊕ ∧3C− (C.3)
To this, one must add a non-vanishing section of the spin bundle for C+ which we denote
η ∈ Γ(S(C+)). Using local bases Eˆ+m for C+ and Eˆ−m˜ for C− defined by13
Eˆ+m = eˆ
+
m + e
+
m − ieˆ+mB,
Eˆ+m˜ = eˆ
−
m˜ − e−m˜ − ieˆ−m˜B,
(C.5)
one can write an explicit formula for the object χ
χ = 13!(η
Tγmnpη)Eˆ+m ∧ Eˆ+n ∧ Eˆ+p = 13! ΨmnpEˆ+m ∧ Eˆ+n ∧ Eˆ+p , (C.6)
where Ψ is the three-form spinor bilinear of η with itself. Writing the object χ in this way, it
is guaranteed that it will lie in the correct orbit and its stabiliser is SU(3)× SO(6).
Given an N = 2 structure parametrised by two complex pure spinors Φ± ∈ S(E)⊗ L1/2
(see [38, 93]), one can also build an N = 1 structure via the expression
χ˜MNP = Φ
+
ΓMNPΦ−, (C.7)
thus providing a third description of the structure.
Having defined the structure, we will now show how to extract the ordinary complex
three-form on the manifold from it. Recall that the generalised tangent space is an extension
0 −→ T ∗X −→ E pi−→ TX −→ 0, (C.8)
with the classes of such extensions labelled by the cohomology class of the three-form flux
[H] ∈ H3(M ;R). The map pi is referred to as the anchor map of the Courant algebroid E.
This anchor map induces further maps on tensor products of E. In particular we obtain an
13These are the split frames of [90], but constructed from an arbitrary local frame for the tangent bundle
rather than a vielbein for the ordinary metric. However, the ordinary metric g is still used to lower the indices
on the one-form frames e±m, such that the O(6, 6) metric has components
η =
(
g 0
0 −g
)
(C.4)
in these frames.
– 39 –
induced map, which we also label pi,
pi : ∧3E −→ ∧3TX. (C.9)
Acting on the bundle L⊗ ∧3E and using L ' detT ∗X we have
pi : L⊗ ∧3E → detT ∗X ⊗ ∧3TX ∼= ∧3T ∗X. (C.10)
Thus, applying the anchor map to the generalised structure χ˜ we obtain an ordinary three-form
on the manifold.
We now calculate this three-form explicitly. We first note that
pi(Eˆ±m) = eˆ
±
m, (C.11)
which immediately gives us
pi(χ) = 13! Ψ
mnp(eˆm ∧ eˆn ∧ eˆp) ∈ ∧3TXC (C.12)
However, we are interested in the object χ˜ = Φχ and in our frame we have Φ =
√
ge−2φ so
that
pi(χ˜) = 13!
√
ge−2φΨmnp(eˆm ∧ eˆn ∧ eˆp) ∈ ∧6T ∗X ⊗ ∧3TXC (C.13)
Now we use the natural isomorphism
∧6T ∗X ⊗ ∧3TX −→ ∧3T ∗X
1
3!X
mnp(eˆm ∧ eˆn ∧ eˆp) 7−→ 1(3!)2 mnpm′n′p′Xm
′n′p′(em ∧ en ∧ ep) (C.14)
where m1...m6(= ±1) is the Levi-Civita symbol. The standard metric volume form is then
εm1...m6 =
√
gm1...m6 and we find that under the identification (C.14) we have
pi(χ˜) = e−2φ 1
(3!)2
εmnpm′n′p′Ψ
m′n′p′(em ∧ en ∧ ep) = e−2φ(?Ψ) = −i e−2φΨ. (C.15)
To conclude our discussion of the SU(3)× SO(6) structure in SO(6, 6)× R+ generalised
geometry, we note that standard group theoretical arguments give us that the homogeneous
space
SO(6, 6)× R+
SU(3)× SO(6) (C.16)
is diffeomorphic to the orbit of χ˜ (at a point in M) under the action of SO(6, 6)× R+ on the
complex 220C representation. The homogeneous space (C.16) has a complex structure, with
respect to which the element χ˜ of the 220C is holomorphic, that is the embedding of (C.16)
into C220 is a holomorphic map. If we imagine that this complex structure naturally extends
to the infinite-dimensional space of SU(3)× SO(6) structures on E, then we expect that our
generalised three-form χ˜ will be holomorphic on that space. As the anchor map pi is linear
– 40 –
and fixed by the topology of E, we have that pi(χ˜) = −i e−2φΨ = −i Ω is also holomorphic on
the space of such structures.
Note that the decomposition of the Lie algebras appearing here is
so(6, 6)→ su(3)⊕ u(1)⊕ so(6)⊕
[
(3,1)+2 ⊕ (3,1)−2
]
R
⊕
[
(3,6)+1 ⊕ (3,6)−1
]
R
, (C.17)
so that we can locally parametrise the space (C.16) by exponentiating the action of the complex
Lie algebra elements which do not annihilate χ˜. Taking out the overall scale we have
χ˜′ = eceα+β · χ˜, c ∈ C∗ α ∈ (3,1)+2 β ∈ (3,6)+1. (C.18)
The parameters c, α and β then become local complex coordinates on the space (C.16).
Note that as SU(3) objects, α and β carry the same degrees of freedom as the parameters
(µ, (∆ω + i ∆B)(1,1), (∆ω + i ∆B)(0,2)) used to parametrise the space of N = 1 field configura-
tions in sections 3 and 3.2 of the main text, while the parameter c is associated to Ka¨hler
transformations.
C.2 SU(3)× SO(6 + n) structures in heterotic supergravity
We can perform a similar analysis to the above in SO(6, 6 + n) × R+ generalised geometry
for heterotic supergravity [43, 44], where n is the dimension of the gauge group G. In that
geometry the generalised tangent space E is an extension of the real Atiyah algebroid A
0 −→ C∞(g) −→ A −→ TX −→ 0, (C.19)
by the cotangent bundle
0 −→ T ∗X −→ E −→ A −→ 0, (C.20)
with the composition of the maps above still giving an anchor map pi : E → TX.
The structure group of the bundle E is then the geometric subgroup (GL(6,R)×G)n
((T ∗X⊗g)n∧2T ∗X) of SO(6, 6+n)×R+, though as usual in generalised geometry we think of
E as an SO(6, 6 +n)×R+ vector bundle. The generalised metric defines an SO(6)×SO(6 +n)
structure on E, with a local frame (Eˆ+m, Eˆ
−
m˜, Eˆ
−
α ) which can be built from the physical fields
(g,B, φ,A) (see [43] for details). The presence of a single globally defined spinor on M breaks
the SO(6) factor to SU(3), so that an N = 1 structure is an SU(3)× SO(6 + n) structure on
E. Using the split frame, one can again write an explicit formula for an object χ˜ ∈ L⊗ ∧3E
defining such an N = 1 structure:
χ˜ = 13!Φ(η
Tγmnpη)Eˆ+m ∧ Eˆ+n ∧ Eˆ+p = 13!
√
ge−2φΨmnpEˆ+m ∧ Eˆ+n ∧ Eˆ+p . (C.21)
As we still have pi(Eˆ+m) = eˆ
+
m, the argument of appendix C.1 still holds to give us that
pi(χ˜) = −i e−2φΨ, and similar group theoretical reasoning leads us to conclude that both χ˜
– 41 –
and e−2φΨ = Ω are holomorphic on the coset space
SO(6, 6 + n)× R+
SU(3)× SO(6 + n) (C.22)
at each point of X. In this case, the corresponding Lie algebra decomposition reads
so(6, 6 + n)→ su(3)⊕ u(1)⊕ so(6 + n)⊕
[
(3,1)+2 ⊕ (3,1)−2
]
R
⊕
[
(3,6 + n)+1 ⊕ (3,6 + n)−1
]
R
(C.23)
so that we can locally parametrise the orbit of χ˜ via
χ˜′ = eceα+β · χ˜ c ∈ C∗ α ∈ (3,1)+2 β ∈ (3,6 + n)+1. (C.24)
As SU(3) × G objects, we have the same degrees of freedom as in appendix C.1, but now
augmented by βa
α, that is the (0, 1)-form part of the deformation of the gauge field. These
then match the parametrisation of the off-shell N = 1 field space of section 3 as employed in
section 4.3.
C.3 The off-shell hermitian structure on V
The off-shell parameter space of the Hull–Strominger system is the space of SU(3) structures,
B fields and gauge fields for G. On-shell, we know that the gauge bundle V must be a
polystable holomorphic bundle. We outline which parts of this structure on the gauge bundle
survive off-shell.
The gauge group G is a compact unitary real Lie group, which has complex representations.
The vector bundle VC is a complex vector bundle with structure group GC, the complexification
of G. On-shell (on a solution of the Hull-Strominger system), this complex vector bundle has
two structures on it: a holomorphic structure and a hermitian structure.
The hermitian structure on VC is a reduction of the structure group GC to the compact
real form G. This is given by a set of local GC frames for VC on patches of X, which, on the
overlaps of patches, are related by transition functions in G ⊂ GC only. The holomorphic
structure on VC is given by a set of GC frames with respect to which the transition functions
are holomorphic functions into the complexified group GC. Clearly these two sets of frames
are different for the simple reason that there are no holomorphic maps into the real group G.
From the complex vector bundle VC, we can define a real vector bundle
V = [VC ⊕ V C]R, (C.25)
which also admits an action of the complex group GC. The hermitian structure is a positive
definite metric on V which pairs VC and V C, such that in the special frames alluded to above
– 42 –
its component matrix takes the canonical form
h =
1
2
(
0 δαβ
δαβ 0
)
(C.26)
where α, β = 1, . . . ,dimC VC are indices for VC and α, β = 1, . . . ,dimC VC are indices for V C.
Off-shell, the manifold X admits an almost complex structure as part of the SU(3)
structure. As X is not honestly complex, we lose the holomorphic structure on VC – we cannot
define holomorphic maps without an integrable complex structure. Nevertheless, physics
tells us we have a real connection on VR and that the physical gauge group G is compact
and unitary. This means we have the hermitian structure, even off-shell.14 This hermitian
structure simply says that VC defines a real vector bundle V with a compact unitary structure
group.
The physical connection A is a local section of
Ω1(X; EndV ) ∼ Ω1(X; g). (C.27)
The almost complex structure defines a split of this into (1, 0)- and (0, 1)-form parts. As the
(1, 0)- and (0, 1)-form summands are intrinsically complex, it is natural to see them as living
in the complexified Lie algebra, i.e. Ω(1,0)(X; gC) and Ω
(0,1)(X; gC), as these are defined via
tensor products over the field C. Naively, the connection has four parts with indices as
(Aa)
α
β (Aa)
α
β
(Aa)
α
β (Aa)
α
β
(C.28)
However, as the connection is real hermitian, any one of them defines the rest via complex
conjugation and multiplication by the hermitian metric on VC. Explicitly, the real condition
fixes [
(Aa)
α
β
]∗
= (Aa)
α
β,
[
(Aa)
α
β
]∗
= (Aa)
α
β, (C.29)
while the hermitian condition fixes[
(Aa)
α
β
]∗
= −hββ(Aa)βαhαα,
[
(Aa)
α
β
]∗
= −hββ(Aa)βαhαα. (C.30)
Together, these allow us to determine all parts of the connection given only (Aa)
α
β; for
example we have
(Aa)
α
β = −hββ(Aa)βαhαα. (C.31)
On-shell, the holomorphic structure means that, in each patch, one can choose a GC
gauge where (Aa)
α
β = 0. Off-shell we cannot do this in general, but we can use the previous
identities to write any formula purely in terms of (Aa)
α
β. For example, the terms appearing
14As X is only almost complex, one might be tempted to call this an almost hermitian structure to emphasise
that the holomorphic structure is not present.
– 43 –
in the superpotential involve traces, which simplify using identities like
(Aa)
β
α(Ab)
α
β = (Ab)
β
α(Aa)
α
β. (C.32)
These enable us to write all of the needed expressions using only the objects with indices for
VC (eliminating the appearance of objects with indices for V C); for example
ωCS(A) = tr(A ∧ dA+ 23A ∧A ∧A) = 2
[
Aαβ ∧ dAαβ + 23Aαβ ∧Aβγ ∧Aγα
]
. (C.33)
so that ωCS(A) ∧Ω features only the components (Aa)αβ . Equivalently we could have written
this expression purely in terms of V C indices or some combination of the two – this freedom
reflects the fact that it is V that appears in the heterotic system, so a split into VC and V C is
somewhat arbitrary. The important point is that it is the (0,1) part of the gauge field that
appears. This mirrors the generalised geometry argument from the previous section.
Using the freedom to write the Chern–Simons form using only VC indices, one can take
the bundle appearing in the deformation complex to be the holomorphic bundle
T ∗(1,0)X ⊕ EndVC ⊕ T (1,0)X. (C.34)
which also appears in [30].
D Comments on D-terms
In addition to satisfying the F -term conditions derived from the superpotential, a solution
should also satisfy the D-term conditions:
d
(
e−2φω ∧ ω) = 0, (D.1)
ω ∧ ω ∧ F = 0. (D.2)
The first condition is referred to as the conformally balanced condition, while the second
condition is the Yang–Mills condition. In this appendix we want to show that these conditions
impose no extra constraints on the heterotic moduli, given some mild assumptions on the
geometry and bundle. This is of course common knowledge from the supergravity point of
view [33].15
D.1 Massless deformations
We begin by considering compactifications without bundles. The D-term condition of relevance
is the conformally balanced condition (D.1). Consider first a massless deformation of this
15As we mention in the main text, for an N = 1 supersymmetric theory in four dimensions, supersymmetry
breaking is controlled completely by the F -terms when there are no FI parameters. Given a solution to the
F -term conditions, one can always make a complex gauge transformation to find a solution to the D-term
conditions on the same orbit. In the heterotic case, this is equivalent to assuming the bundles V and TX are
stable.
– 44 –
condition so that y0 = (x0, µ0) satisfies Dy0 = 0. Our plan is to show that the D-term
conditions can be solved order-by-order using the gauge symmetries of x0 so that they do not
further constrain the moduli.
First we note that deformations of the hermitian form ω and the dilaton φ are linked via
the SU(3) normalisation condition
i
8Ω ∧ Ω = 16e−4φω ∧ ω ∧ ω. (D.3)
Remembering ∆ω(0,2) = 0, ∆ω(1,1) = −ix and ∆ω(0,2) = ıµω − i ıµx, a massless holomorphic
variation of this condition gives
e4∆0φ
1
3!
ω ∧ ω ∧ ω = 1
3!
ω ∧ ω ∧ ω − i
2
ω ∧ ω ∧ x0 − 1
2
ω ∧ x0 ∧ x0 + i
3!
x0 ∧ x0 ∧ x0. (D.4)
Now we expand the massless deformation in terms of a small parameter :
y0 =  y(1) + 
2 y(2) + . . . (D.5)
∆0φ =  φ(1) + 
2 φ(2) + . . . (D.6)
with a corresponding expansion for x and µ. At first and second order the SU(3) normalisation
condition fixes
φ(1) = −
3
4
iωyx(1), (D.7)
φ(2) = −
3
4
iωyx(2) +
1
8
x(1)yx(1). (D.8)
This continues to higher order – the deformation of the dilaton at nth order is fixed by the
non-primitive part of x(n) and the lower-order fields x(i) for i < n.
Now consider a massless holomorphic deformation of the conformally balanced condition.
We expand in  and separate into complex type. At O(1) the deformation of the conformally
balanced condition reduces to
∂
(
i e−2φω ∧ x(1) + φ(1)e−2φω ∧ ω
)
= 0, (D.9)
∂
(
i e−2φω ∧ x(1) + φ(1)e−2φω ∧ ω
)
= ∂
(
e−2φω ∧ ıµ(1)ω
)
. (D.10)
Note now that the Gauduchon metric defined by ω˜ = e−φω is balanced as ω˜ ∧ ω˜ is d-closed.
We do this as on a hermitian manifold the various Laplace operators for a balanced metric
agree on functions [94]:
∆˜df = 2∆˜∂f = 2∆˜∂f. (D.11)
– 45 –
We also use that the Hodge stars16 on a p-form are related by
? = e(3−p)φ?˜. (D.12)
Using this we can write the previous equations as
∂
†˜
X(1) = 0, (D.13)
∂ †˜X(1) = i ∂
†˜
ıµ(1)ω˜, (D.14)
where we have used the relation between the trace of x(1) and φ(1) given in (D.7), and we have
defined
X(1) = e
−φ(x(1) − 12 ω˜yx(1) ω˜). (D.15)
The Hodge decomposition for Aeppli cohomology implies that (D.14) and (D.13) determine
the (∂ + ∂)-exact part of X(1). Indeed an equivalent set of equations is
∂ †˜∂ †˜X1 = 0, (D.16)
∂∂
†˜
X1 = 0, (D.17)
∂∂ †˜X1 = i ∂∂
†˜
ıµ1ω˜. (D.18)
We now want to argue that these conditions are simply gauge conditions and so do not impose
extra conditions on the moduli. Recalling the form of Dy0 from (A.11) we see shifts of x0
by ∂-exact terms drop out explicitly and that shifts by ∂-exact terms fall out as we are
working modulo ∂-exact forms.17 A gauge choice for x0 then amounts to a choice of element in
∂Ω(0,1)(X) + ∂Ω(1,0)(X). We make a simplification: let us assume the following cohomologies
vanish
H
(1,0)
∂
(X) = H
(2,0)
∂
(X) = H
(0,1)
∂
(X) = H
(0,2)
∂
(X) = 0, (D.19)
so that we get a Hodge decomposition of the space of (∂ + ∂)-exact forms as
∂Ω(0,1) + ∂Ω(1,0) = ∂∂Ω(0,0) ⊕ ∂∂ †˜Ω(2,0) ⊕ ∂∂ †˜Ω(0,2). (D.20)
Note that it is important that we include the dilaton degrees of freedom: shifts of x(1) are
generically not primitive and so they will change the SU(3) normalisation condition, but we
can compensate for this by shifting φ(1) (which does not appear explicitly in the first-order
conformally balanced condition).
We start by shifting x(1) by −∂∂κ(1), where κ(1) is a function. A short calculation shows
16We are using the convention for the Hodge star where α∧?β = αyβ vol so that ?ω = 1
2
ω∧ω and ?Ω = −i Ω.
The dual of a primitive (1, 1)-form αp satisfying ωyαp is ?αp = −ω ∧ αp. We also have ?α20 = α20 ∧ ω where
α20 is a (2, 0)-form. This choice satisfies ?
2 = (−1)p on a p-form. The adjoint Dolbeault operators are defined
by ∂† = − ? ∂?, and we denote the corresponding operators for the Gauduchon metric with a tilde.
17One can also do this calculation with y and b so that the shift by ∂-exact forms is explicit too.
– 46 –
that equation (D.16) becomes
∂ †˜∂ †˜
(
e−φ(x(1) − 12 ω˜yx(1) ω˜)
)
= ∂ †˜∂ †˜
(
e−φ∂∂κ(1)
)
+ 12∆˜∂
(
e−φ∆˜∂κ(1)
)
. (D.21)
One can check that the operator acting on κ(1) is a positive semi-definite self-adjoint elliptic
operator whose image is given by non-constant functions. This means that (D.21) can always
be solved by an appropriate choice of κ(1). This fixes the ∂∂-gauge symmetry of x(1).
Next consider a shift x(1) by −∂∂ †˜α(1),18 where α(1) is a (2, 0)-form. A short calculation
shows that (D.17) becomes
∂∂
†˜(
e−φ(x(1) − 12 ω˜yx(1) ω˜)
)
= ∂∂
†˜
(e−φ∂∂ †˜α(1)). (D.22)
Again, one can check that the operator acting on α(1) is positive semi-definite, self-adjoint and
elliptic so that (D.1) can always be solved for by a choice of α(1). This fixes the ∂∂
†˜-gauge
symmetry of x(1).
Finally consider a shift of x(1) by −∂∂ †˜β(1), where β(1) is a (0, 2)-form. A short calculation
shows that (D.18) becomes
∂∂ †˜
(
e−φ(x(1) − 12 ω˜yx(1) ω˜)
)
− i ∂∂ †˜(ıµ(1)ω˜) = ∂∂ †˜
(
e−φ∂∂ †˜β(1)
)
. (D.23)
As ıµ(1)ω is a (0, 2)-form and we assume H
(0,2)(X) vanishes, ∂∂
†˜
ıµ(1)ω is actually ∂∂
†˜-exact.
Thus when we shift x(1), this equation can be solved providing the operator acting on the
gauge parameter is elliptic and positive semi-definite as before – it is simple to check that this
is the case. This fixes the ∂∂
†˜
-gauge symmetry of x(1).
What happens at higher orders in ? At second order we have
∂
†˜
(X(2) +A(2)) = 0, (D.24)
∂ †˜(X(2) +A(2)) = i ∂
†˜
ıµ(2)ω˜ + ∂
†˜
B(2), (D.25)
where
X(2)(x(2)) = e
−φ(x(2) − 12 ω˜yx(2) ω˜). (D.26)
A(2)(x(1)) = i e
−φ
(
−38(ωyx(1))2ω + 14x(1)yx(1) ω + 12ωyx(1) x(1) + ?(x(1) ∧ x(1))
)
,
(D.27)
B(2)(x(1), µ(1)) = e
−φ
(
−12ωyx(1) ıµ(1)ω + ıµ(1)x(1) + ?(x(1) ∧ ıµ(1)ω)
)
. (D.28)
We see that X(2) depends only on the second-order correction to x0 while A(2) and B(2) are
fixed by the first-order terms (which should be thought of as gauge transformed to solve the
18One should think of this x(1) as already gauge transformed to solve the previous condition.
– 47 –
first-order conditions). Again (D.24) and (D.25) are equivalent to
∂ †˜∂ †˜(X(2) +A(2)) = 0, (D.29)
∂∂
†˜
(X(2) +A(2)) = 0, (D.30)
∂∂ †˜(X(2) +A(2)) = i ∂∂
†˜
ıµ(2)ω˜ + ∂∂
†˜
B(2). (D.31)
As X(2) is a function of x(2) alone, we can perform gauge transformations of x(2) without
affecting A(2) and B(2). Generically these gauge transformations will break the SU(3) normal-
isation condition, but we can always shift φ(2) to compensate for this (which is what we have
implicitly done by eliminating φ(2) from the equations). An analogous argument to the one we
gave previously then shows that we can always solve these conditions using the gauge freedom
of x(2).
From this it is simple to see that this process can be continued to all orders. The
conformally balanced condition at order n is a set of equations for x(n) with x(i<n) fixed.
Again, one can always find a gauge transformation of x(n) that solves these conditions. From
this we conclude that the D-term conditions are gauge fixing conditions for the moduli and
do not further constrain the moduli problem.
D.2 Including bundle moduli
We now want to show that when we include the bundle moduli we can solve the D-terms and
Yang–Mills equations simultaneously. As stated at the start of section 3, we have the freedom
to work with either EndV or EndVC. We choose to work with EndVC in this appendix
and appendix E, so that the Donaldson–Uhlenbeck–Yau and Li–Yau theorems directly apply
[95–97].
The Yang–Mills condition is
ω ∧ ω ∧ F ∝ ρ˜ ∧ F = 0. (D.32)
where we have defined ρ˜ = 12 ω˜ ∧ ω˜. A massless holomorphic deformation of this condition
reads
∆0ρ˜ ∧ F + (ρ˜+ ∆0ρ˜) ∧ ∂Aα0 = 0. (D.33)
Note that ∆0ρ˜ is completely determined by x0.
For now we assume H0(EndVC) = 0. This is true for irreducible bundles with hermitian
connections satisfying the Yang–Mills equation – such bundles are stable. If H0(EndVC) 6= 0
the Yang–Mills condition can give rise to Fayet–Iliopoulos D-terms in the lower-dimensional
supergravity. Such terms have been studied extensively in heterotic compactifications (see
[8, 11] and references therein). Their appearance can be understood in the context of the
infinitesimal moduli problem as modding out by D-exact terms [14, 26], see also appendix E.
– 48 –
The equations of motion Dy0 = 0 admit gauge transformations of x0 and α0:
δx0 = ∂a0 + ∂b0 + trFγ0,
δα0 = ∂Aγ0.
(D.34)
Upon expanding x0 and α0 in powers of , the first order contributions to (D.33) are
0 = −i e−φx(1) ∧ ω˜ ∧ F + ρ˜ ∧ ∂Aα(1). (D.35)
We now want to show that we can solve this and the conformally balanced condition by an
appropriate gauge choice. We denote these conditions schematically as
Dy(1) = 0, Cy(1) = 0, Y y(1) = 0, (D.36)
where D is the usual operator, and C and Y are the operators that act on y(1) to give the
relevant equations. Explicitly (to first order) we have
Cy(1) =
[
∂
†˜(
e−φ(x(1) − 12 ω˜yx(1) ω˜)
)]
+
[
∂ †˜
(
e−φ(x(1) − 12 ω˜yx(1) ω˜)
)− i ∂ †˜ıµ(1)ω˜], (D.37)
Y y(1) = −i e−φx(1) ∧ ω˜ ∧ F + ρ˜ ∧ ∂Aα(1). (D.38)
Let us start with a solution to the equations of motion
Dy(1) = 0. (D.39)
The equations of motion have a gauge symmetry under shifts by ∂a+ ∂b+ trFγ:
Dy1 = 0 = D(y(1) + ∂a(1) + ∂b(1) + trFγ(1)). (D.40)
Generically a solution y(1) to Dy(1) = 0 will not solve either Cy(1) = 0 or Y y(1) = 0 on the
nose. Let us make a gauge transformation of y(1):
y′(1) = y(1) + ∂α(1) + ∂β(1). (D.41)
This still solves the equations of motion – Dy′(1) = 0 – but affects the remaining two conditions:
Cy′(1) = Cy(1) + C(∂α(1) + ∂β(1)). (D.42)
We showed in the previous section that there is always a choice of α and β that solves this
condition, that is
Cy(1) = −C(∂α+ ∂β) ⇒ Cy′(1) = 0. (D.43)
At this point we have a solution y′(1) that solves Dy
′
(1) = Cy
′
(1) = 0 but generically has
Y y′(1) 6= 0. Note also that the forms α and β are fixed by y(1). We now want to show that we
– 49 –
can always find a solution to Y y = 0 using gauge transformations. Let us define
y′′(1) = y
′
(1) + ∂a(1) + ∂b(1) + trFγ(1) + ∂Aγ(1). (D.44)
This solves Dy′′(1) = 0 – what happens to the other two conditions?
Cy′′(1) = C(∂a(1) + ∂b(1)) + C(trFγ(1)), (D.45)
Y y′′(1) = Y y
′
(1) + Y (∂a(1) + ∂b(1)) + Y (trFγ(1)) + Y (∂Aγ(1)). (D.46)
Again, we can solve Cy′′(1) = 0 for any γ(1) by choosing a and b appropriately. We now want
to see if we can solve Y y′′(1) = 0 using the gauge freedom in γ(1).
Note that C picks out the (∂ + ∂)-exact terms of the input (and has no kernel acting on
this) so that Cy′′(1) = 0 is equivalent to
∂a(1) + ∂b(1) +
[
trFγ(1)
]
∂+∂
= 0. (D.47)
Using this we have
Y y′′(1) = Y y
′
(1) + Y (P[trFγ(1)]) + Y (∂Aγ(1)), (D.48)
where P projects onto Ω(1,1)(X)\{im ∂ + im ∂}. Explicitly we have19
Y y′′(1) = i ?˜
(
e−φF ]˜yx′(1) + ∂
†˜
Aα
′
(1) + e
−φF ]˜yP[trFγ(1)] + ∆˜Aγ(1)
)
. (D.49)
One can check that the operator acting on γ(1) is a positive semi-definite elliptic operator with
trivial kernel, so one can always find a γ(1) that solves Y y
′′
(1) = 0. Again, one can see that this
argument will hold to all orders in .
Recall that we included TX as part of the gauge bundle V . Strictly speaking one should
show you can solve the analogue of (D.49) for TX as well. Naively an extra minus sign
will appear in the trace term, leading to a question of whether the operator acting on γ(1)
has trivial kernel. Physically we expect this equation will not cause any problems – there is
evidence that the degrees of freedom in the connection for TX are not real moduli as they
can be eliminated using field redefinitions (at least to O(α′)) [98]. Another way of saying
this is that the hermitian Yang–Mills condition for TX is automatically satisfied provided
the other supersymmetry conditions are solved and one chooses the appropriate connection,
which at O(α′) is the connection ∇−. Though we do not have a proof, we expect this will
hold to higher order in α′ when one chooses the connection on TX to be an instanton.
19Here we have used ?˜1 = 1
6
ω˜3, ?˜(a11 ∧ ω˜ ∧ b11) = −b]˜11ya11 if b11 is a primitive (1, 1)-form (like F ), and
?˜(ρ˜ ∧ a11) = ω˜ya11. We use ]˜ to denote raising with the Gauduchon metric.
– 50 –
D.3 Polystable bundles
Let us now consider the situation where the bundle is polystable, that is we have a reducible
bundle where each factor is stable of slope zero. If a bundle factor has non-vanishing first
Chern class, which is possible for a sum of irreducible components, then it admits a single
section proportional to the identity isomorphism on the bundle factor. This can give rise to a
non-trivial Fayet–Iliopoulos term in the lower-dimensional supergravity. We now show that
equation (D.49) can still be solved when such sections are present.
A polystable bundle is a sum of irreducible components
VC =
⊕
i
V
(i)
C , (D.50)
where each factor V
(i)
C is a stable slope-zero bundle with non-zero trace. The trace-less part of
(D.49) can be solved by the methods outlined in the previous subsection. The non-trivial part
comes from the trace, which we isolate by taking the trace to get
− i ?˜ tr(Y y′′(1))(i) = e−φ trF (i) ]˜yx′(1) +∂
†˜
A trα
′
(1) +e
−φ trF (i) ]˜yP[trF (i)γ(i)(1)]+∆˜ tr γ
(i)
(1). (D.51)
It is simplest to show that this can be solved by reintroducing the α′ parameter, which we
have neglected until this point. To simplify the notation, we imagine the bundle VC is abelian
for the remainder of the section. Reintroducing α′, equation (D.51) becomes
− i ?˜ Y y′′(1) = e−φF ]˜yx′(1) + ∂
†˜
Aα
′
(1) +
1
4α
′ e−φ F ]˜yP[Fγ(1)] + ∆˜γ(1). (D.52)
We can now expand this equation in α′. We expand the deformations x′(1) and α
′
(1), together
with the gauge parameter γ(1), while keeping the background geometry fixed. The potential
non-zero contributions to γ(1) are then
γ(1) =
1
α′
γ
(−1)
(1) + γ
(0)
(1) + α
′ γ(1)(1) + . . . (D.53)
It follows from (D.52) that γ
(−1)
(1) must be constant. The expression we get at zeroth order in
α′ is
− i ?˜ Y y′′(0)(1) = e−φF ]˜yx′
(0)
(1) + ∂
†˜
Aα
′(0)
(1) +
1
4 e
−φ F ]˜yF γ(−1)(1) + ∆˜Aγ
(0)
(1) = 0, (D.54)
where we note that
P[Fγ(−1)(1) ] = Fγ
(−1)
(1) , (D.55)
for constant γ
(−1)
(1) . The constant part of (D.54) can now be obtained by integrating over X.
– 51 –
Doing so, we get
− i
∫
X
?˜ Y y′′(0)(1) =
∫
X
e−φF ]˜yx′(0)(1) +
1
4γ
(−1)
(1)
∫
X
e−φ F ]˜yF, (D.56)
This determines γ
(−1)
(1) as a function of x
′(0)
(1). The remaining non-constant part can then be
solved for modulo a constant term, which determines γ
(0)
(1) .
It is easy to see that this can be continued to higher orders in α′. At the next order, we
use the so-far undetermined constant piece of γ
(0)
(1) to fix the constant piece of the first-order
equation. This determines γ
(0)
(1) , which is now independent of γ
(1)
(1) . We then solve for the
non-constant piece, determining γ
(1)
(1) modulo a new constant. It is also easy to check that a
similar argument will hold at higher orders in the expansion parameter .
The argument we have given here for a single abelian factor is easily generalised to the
general polystable case. This shows that the D-term equations can be solved even when there
are Fayet–Iliopoulos terms present. The reader may have expected that the D-term equations
would obstruct some hermitian moduli when FI terms are present. This is not quite right;
instead the expected obstructed moduli will form part of the gauge transformations and so
should not be included in the first place. We give more detail on this in appendix E.
D.4 Full Maurer–Cartan equations
The arguments we have given for the massless deformations can be generalised to solutions to
the full set of Maurer–Cartan equations:
Dy − 12 [y, y]− 12∂b = 0, (D.57)
∂b− 12ya∂ab+ 13!〈y, [y, y]〉 = 0, (D.58)
∂ıµΩ = 0. (D.59)
What is the gauge symmetry of these equations? The L3 algebra structure gives us the gauge
transformation of Y by a gauge parameter Λ = (λ, ξ) ∈ Y0
δΛY = `1(Λ) + `2(Λ, Y )− 12`3(Λ, Y, Y )
=
(
Dλ+ 12∂ξ + [λ, y], 0
)
+
(
0, ∂ξ + 12(〈λ, ∂b〉 − 〈y, ∂ξ〉)− 13(12〈λ, [y, y]〉+ 〈y, [y, λ]〉)
)
.
(D.60)
Now expand Y and Λ in powers of . At first order the equations of motion and gauge
transformations are
Dy(1) − 12∂b(1) = 0, δy(1) = Dλ(1) + 12∂ξ(1), (D.61)
∂b(1) = 0, δb(1) = ∂ξ(1), (D.62)
∂ıµ(1)Ω = 0. (D.63)
– 52 –
Exactly as before, x(1) and α(1) have a gauge symmetries that allow us to solve the D-term
conditions to first order.
At second order we have
Dy(2) − 12 [y(1), y(1)]− 12∂b(2) = 0, δy(2) = Dλ(2) + 12∂ξ(2) + [λ1, y1], (D.64)
∂b(2) − 12ya(1)∂ab(1) = 0, δb(2) = ∂ξ(2) + 12
(〈λ(1), ∂b(1)〉 − 〈y(1), ∂ξ(1)〉), (D.65)
∂ıµ(2)Ω = 0. (D.66)
Here (y(1), b(1)) are the gauge-transformed fields that solve the first-order D-term conditions.
Furthermore λ1 is fixed by these first-order conditions. We note that the second order field
y(2) is shifted by the first order gauge transformation by the term [λ1, y1]. This means it is
y(2) − [λ(1), y(1)] that appears in the second-order D-term condition, which then admits shifts
by terms of the form Dλ(2) +
1
2∂ξ(2) which we may use to solve the D-term conditions just as
in the first order case, fixing λ(2) and ξ(2) in the process. We can see that this structure will
continue to hold to all orders. Assuming we have solved the D-terms at ith order where i < n,
y(n) is shifted by gauge transformations involving the y(i). We can then use the shifts of y(n)
by (∂ + ∂)-exact forms to solve the D-terms at nth order. Solving the D-term conditions thus
fixes the gauge symmetries (though there are residual gauge-of-gauge transformations).
E Massless moduli
In this appendix we clarify the meaning of the massless moduli
[y0] ∈ H(0,1)D (Q˜), y0 = (µ0, α0, x0). (E.1)
The massless moduli have been analysed in great detail in the [14, 15, 31] – what follows is a
short review. The first thing to notice is that
∂µ0 = 0. (E.2)
The fields µ0 parametrise the complex structure moduli which, modulo gauge symmetries,
take values in the cohomology
[µ0] ∈ H(0,1)∂ (T
(1,0)X) ∼= H(2,1)
∂
(X). (E.3)
Recall that we also impose the extra condition
∂ıµ0Ω = 0. (E.4)
This condition can be viewed as an equation of motion for the field b appearing in the
superpotential (4.11). It is also a necessary condition for preserving the d-closure of the
holomorphic three-form, which in general is a stronger condition then preserving an integrable
– 53 –
complex structure alone. However, with the extra assumption that H
(0,1)
∂
(X) = 0, it can be
shown that each class in H
(0,1)
∂
(T (1,0)X) has a representative satisfying this condition [14].
Next we must solve the equation
∂Aα0 + Fb ∧ µb0 = 0. (E.5)
This is then the familiar condition that the complex structure deformations µa should be in
the kernel of the Atiyah map
Fµ : H(0,1)∂ (T
(1,0)X)→ H(0,2)
∂A
(EndVC), (E.6)
where Fµ is given by contraction with F as in (E.5). The fields α0 now decompose as
α0 = α
c
0 + α
µ0
0 , (E.7)
where αc0 is closed and α
µ0
0 is uniquely determined by the complex structure moduli through
equation (E.5). Modulo gauge symmetries, we have
[αc0] ∈ H(0,1)∂A (EndVC), (E.8)
which we refer to as gauge moduli.
The final equation we must solve is
∂x0 a + i(∂ω)eac dz
c ∧ µe0 − tr(Fa ∧ α0) = 0. (E.9)
Note that we need only solve this equation modulo ∂a-exact two-forms. In particular, the
equation we should solve is
∂x0 a + i(∂ω)eac dz
c ∧ µe0 − tr(Fa ∧ α0) = ∂ab, (E.10)
for some b ∈ Ω(0,2)(X) which depends on the particular sheaf representative. Note however
that if H
(0,2)
∂ (X) 6= 0, then we can always add an element of H(0,2)∂ (X) to b without changing
this equation. Hence H
(0,2)
∂ (X), i.e. anti-holomorphic sections of Ω
(0,2)(X), should in principle
be considered as part of the massless moduli. We have throughout this paper assumed that
this cohomology vanishes.
Let us define the maps
H : kerF → H(0,2)(T ∗(1,0)X /∂-exact), (E.11)
Fα : H(0,1)∂A (EndVC)→ H
(0,2)(T ∗(1,0)X /∂-exact). (E.12)
The maps are given by
H(µ0)a = i(∂ω)eac dzc ∧ µe0 − tr(Fa ∧ αµ00 ), (E.13)
– 54 –
Fα(αc0)a = − tr(Fa ∧ αc0). (E.14)
It is easy to check that the right-hand side of both of these equations is ∂-closed. The condition
imposed by (E.9) is that the sum should be exact. The number of remaining massless moduli
is then given by
| kerH|+ | kerFα|+ | imH ∩ imFα|. (E.15)
We have the freedom to view the extra moduli in the last term as either complex structure
moduli, bundle moduli, or something in between. Viewing it as complex structure moduli
means that the moduli µ0 which remain massless are those which satisfy
H(µ0) ⊆ imFα. (E.16)
The remaining massless bundle moduli αc0 then need be in the kernel of Fα.
Equation (E.9) now determines the non-closed part of x0 uniquely in terms of the massless
complex structure and bundle moduli. The remaining hermitian moduli xc0 are ∂-closed
modulo ∂a-exact terms, and hence take values in the cohomology
[xc0] ∈ H(0,1)∂ (T
∗(1,0)X /∂-exact). (E.17)
These are the fields conventionally referred to as Ka¨hler moduli or hermitian moduli. As it
turns out, this cohomology forms a subset of the Aeppli cohomology [99]. Indeed, in order to
define an element of H
(0,1)
∂
(T ∗(1,0)X /∂-exact), we need that
∂x0 = ∂κ, (E.18)
for some κ ∈ Ω(0,2)(X). In particular, ∂∂x0 = 0. Modulo the gauge symmetries of x0 – shifts
by ∂- and ∂-exact terms – we see that x0 is an element in the Aeppli cohomology.
For the geometries we are considering in this appendix, it is true that H
(0,1)
∂
(T ∗(1,0)X)
is isomorphic to H
(1,1)
∂
(X). Let us see why. Note that (E.18) implies that ∂κ = 0, which by
H(0,2)(X) = 0 implies that κ = ∂β for some β ∈ Ω(0,1)(X). Let us now use the freedom to
shift x0 by a ∂-exact term ∂α to get
∂x0 = ∂(∂κ+ ∂α) . (E.19)
We can choose α such that the right-hand side vanishes. This is equivalent to ∂κ+ ∂α = 0,
which by the Hodge decomposition (D.20) fixes the ∂∂
†˜
-exact part of ∂α. The remaining
∂-exact part of ∂α is also ∂-exact. We conclude that for the geometries under consideration,
we have
H
(0,1)
∂
(T ∗(1,0)X /∂-exact) ∼= H(0,1)
∂
(T ∗(1,0)X) = H(1,1)
∂
(X) , (E.20)
and the hermitian moduli are counted by h(1,1) as in a Calabi–Yau compactification.
We now compare this with the cohomology H
(0,1)
D
(Q˜). Using long exact sequences in
– 55 –
cohomology, it is a straightforward exercise to show that
H
(0,1)
D
(Q˜) ∼=
H
(0,1)
∂
(T ∗(1,0)X /∂-exact)
imFα ⊕ ker(H+ Fα). (E.21)
This is almost as expected, except that we are modding out the hermitian moduli by elements
in imFα, where we now view Fα as
Fα : H0(EndVC)→ H(0,1)∂ (T
∗(1,0)X /∂-exact). (E.22)
We can understand this from considering gauge transformations by a D-exact term. For
polystable bundles, part of these transformations are formed by sections of H0(EndVC). Again
we have VC =
⊕
i V
(i)
C for stable slope-zero factors V
(i)
C . Thus we have
imFα = span{c1F1 + . . .+ cnFn}, (E.23)
for constants ci such that
∑
i nici = 0, where ni is the dimension of the i
th bundle, so that the
total section remains traceless. As we saw in the previous section, these are precisely the gauge
transformations needed to solve the D-term conditions in the presence of Fayet–Iliopoulos
terms, and so hermitian elements of this form should not be thought of as moduli.
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