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Published for SISSA by Springer
Received: August 22, 2018
Accepted: October 18, 2018
Published: October 29, 2018
Finite deformations from a heterotic superpotential:
holomorphic Chern-Simons and an L1 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,
Radclie Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, U.K.
bInstitut de Physique Theorique, 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, U.K.
dSchool of Physics, Astronomy and Mathematics, University of Hertfordshire,
College Lane, Hateld, AL10 9AB, U.K.
eDepartment of Physics, King's College London,
London, WC2R 2LS, U.K.
fThe Abdus Salam International Centre for Theoretical Physics,
34151 Trieste, Italy
E-mail: ashmore@maths.ox.ac.uk, delaossa@maths.ox.ac.uk,
ruben.minasian@ipht.fr, c.strickland-constable@herts.ac.uk,
eirik.svanes@kcl.ac.uk
Abstract: We consider nite deformations of the Hull-Strominger system. Starting from
the heterotic superpotential, we identify complex coordinates on the o-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 eective
action generalises that of both Kodaira-Spencer and holomorphic Chern-Simons theory.
The supersymmetric locus of this action is described by an L3 algebra.
Keywords: Superstring Vacua, Flux compactications, Superstrings and Heterotic Strings,
Dierential and Algebraic Geometry
ArXiv ePrint: 1806.08367
Open Access, c
 The Authors.
Article funded by SCOAP3.
https://doi.org/10.1007/JHEP10(2018)179
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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 5
3 The o-shell parameter space 7
3.1 F -term conditions from the superpotential 9
3.2 Constraints from the SU(3) structure and holomorphicity 10
4 Higher-order deformations 12
4.1 The superpotential 12
4.2 A Maurer-Cartan equation from the holomorphic structure 13
4.3 Including the bundle 14
4.4 Vanishing of the superpotential 16
5 Moduli and an L3 algebra 17
5.1 Quasi-isomorphism to a natural holomorphic L3 algebra 18
5.2 An L1 eld equation 19
6 A reduced L3 algebra and an eective action 20
6.1 Integrating out b 20
6.2 Eective eld theory and Yukawa couplings 22
7 Conclusions 24
A Conventions 26
A.1 Heterotic supergravity 27
A.2 Holomorphic structure 27
A.3 L1 structure 28
B Comments on heterotic 
ux quantisation 30
B.1 A toy example: the abelian bundle 30
B.2 The two-form gerbe example 31
B.3 Heterotic 
ux quantisation 32
B.4 Deforming the system and a well-dened global two-form 34
C The o-shell N = 1 parameter space and holomorphicity of 
 35
C.1 SU(3) SO(6) structures in the NS-NS sector 36
C.2 SU(3) SO(6 + n) structures in heterotic supergravity 39
C.3 The o-shell hermitian structure on V 40
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D Comments on D-terms 42
D.1 Massless deformations 42
D.2 Including bundle moduli 45
D.3 Polystable bundles 48
D.4 Full Maurer-Cartan equations 49
E Massless moduli 50
1 Introduction
A full understanding of the parameter space of string theory is an outstanding mathemat-
ical 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 compactications
without knowledge of their explicit metrics.
Calabi-Yau compactications 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(0) is
given by compactifying on a complex three-fold X with H 
ux and a gauge bundle that
satisfy an anomaly cancellation condition. The conditions on the geometry and 
uxes
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-Kahler geometries. Generically, a given solution of the Hull-Strominger system will
admit deformations of the geometry, 
ux 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
compactication.
The moduli spaces of Calabi-Yau compactications at zeroth order in 0 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 condi-
tions for an N = 1 Minkowski solution are suciently 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 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 lled. (See also [16{19] for worldsheet approaches.)
Innitesimally, 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 nite
dimensional is intimately connected to this holomorphic structure and the Bianchi identity
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for the 
ux. The innitesimal moduli are captured by the cohomology H
(0;1)
D
(Q), where Q is
dened by a series of extensions. In this way, the complex structure, hermitian and bundle
moduli are combined in a single structure. Furthermore, one can dene the analogue of
special geometry for these general heterotic compactications and nd 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 innitesimal
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 eective 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 nite
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 elds. The deformed
elds 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 compli-
mentary. 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 rst 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 elds that appear in
the superpotential and then read o 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 elds 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 elds without
requiring that they give a solution to the Hull-Strominger system. This is equivalent to
saying that the o-shell parameter space | the space of SU(3) structures, B elds 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 compactications are described by a generalised SU(3)  SU(4) structure [28]. We
identify the invariant (holomorphic) object which characterises this structure and nd that
the complex coordinates on the space of structures match with the usual complex structure,
complexied hermitian and bundle deformations.
We show that the conditions on the moduli elds from the vanishing of the superpoten-
tial and its rst 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
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superpotential itself can be rewritten using these operators in a Chern-Simons-like form:

W =
Z
X
hy;Dy  1
3
[y; y] @bi ^ 
; (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 h
; 
i on the moduli elds, 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

uxes [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 C1 resolution of the underlying holo-
morphic Courant algebroid. The natural L3 eld 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 L1
structures, so it is not unexpected that our moduli elds are governed by one. What is
unexpected is that the structure truncates at nite 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 nite order while oth-
ers 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 param-
eters. 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 innitesimal moduli in terms of a holomorphic structure as in [14]. In section 3
we discuss the o-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 nd 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 dene 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 simplies under
various assumptions and comment on how the eective eld theory is encoded in our
language. We nish 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 
ux
quantisation works in the heterotic theory, discuss the o-shell parameter space in terms of
generalised geometry, show that the D-term conditions do not aect the moduli problem
and review how the massless moduli are captured by the the cohomology of the holomor-
phic structure.
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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
innitesimal 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(0). 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  whose curvature R is valued
in EndTX. X admits an SU(3) structure dened by a nowhere vanishing spinor  or,
equivalently, a non-degenerate two-form ! and a nowhere vanishing three-form 	 that are
compatible
! ^	 = 0; ik	k2 	 ^	 =
1
3!
! ^ ! ^ !: (2.1)
The invariant objects are dened by bilinears of the spinor as
!mn = i y
mn; 	mnp = T
mnp; (2.2)
where we are free to normalise the spinor so that k	k2 = 8. In what follows it will be
useful to dene a three-form 
 which is related to 	 by a dilaton factor as

 = e2	: (2.3)
Supersymmetry of the vacuum follows from the vanishing of the supersymmetry vari-
ations of the fermionic elds, given in equations (A.7){(A.9). To rst order in 0, these
conditions are equivalent to the Hull-Strominger system:
d
 = 0; (2.4)
i(@  @)! = H := dB + 
0
4
(!CS(A) !CS()); (2.5)

 ^ F = 0; (2.6)
!yF = 0; (2.7)
d(e2! ^ !) = 0; (2.8)
where !CS is the Chern-Simons three-form for the connection,
!CS(A) = tr

A ^ dA+ 2
3
A ^A ^A

: (2.9)
The closure of 
 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.
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Conditions (2.6) and (2.7) mean V is a polystable holomorphic bundle1 so the curvature
satises the hermitian Yang-Mills equations. Finally, (2.5) denes the 
ux H in terms of
the heterotic B eld and the anomaly cancellation condition, and links it with the intrinsic
torsion of the SU(3) structure. The corresponding Bianchi identity is2
dH =
0
4
(trF ^ F  trR ^R): (2.10)
This set of conditions denes what one might call a heterotic SU(3) structure.
Upon considering the four-dimensional N = 1 theory that would follow from compact-
ifying 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
d
 = 0;
i(@  @)! = H;

 ^ 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 innitesimal deformations are
parametrised by the cohomology H
(0;1)
D
(Q), where D and Q are to be dened below. This
cohomology is reviewed in appendix E. Under innitesimal deformations, the D-term equa-
tions x a representative of a certain cohomology class [14], and so should be thought of as
gauge xing conditions that do not aect 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 nite deformations also amounts to xing
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].
2.2 The Atiyah algebroid and a holomorphic structure
The vector bundle V is hermitian in agreement with (0; 2) supersymmetry on the world-
sheet [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 End V . The exterior derivative on V twisted
by A is
dA := d + [A; 
]; (2.13)
1More precisely, it is the complex vector bundle VC (dened 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 fullled [32]. To O(0), this
connection is r, given by taking the connection in (A.7) with the opposite sign for H.
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where the action of the bracket on a p-form  is
[A; ] = A ^   (1)p ^A: (2.14)
A holomorphic structure on V is xed 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 deforma-
tions of the complex structure, hermitian structure and gauge bundle. Taking each of
these in isolation is not sucient. In particular, deformations of the hermitian structure
alone lead to an innite-dimensional moduli space. It is surprising that if one considers
the full deformation problem together with the anomaly cancellation condition, one nds
a nite-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 innitesimal moduli of the Hull-Strominger system are cap-
tured 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 dened by an extension3
0! T (1;0)(X)! Q! Q1 ! 0; (2.17)
where the bundle Q1 is dened by
0! EndV  EndTX ! Q1 ! T (1;0)(X)! 0: (2.18)
The holomorphic structure D on Q is a derivative4
D : 
(0;p)(Q)! 
(0;p+1)(Q); (2.19)
where D
2
= 0 if and only if the Bianchi identities for H, F and R are satised. 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 condi-
tion on X.
The innitesimal 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 eld redenition to absorb the explicit
0 dependence
B ! 
0
4
B; ! ! 
0
4
!: (2.20)
3Full details can be found in [14].
4A similar operator has appeared in the context of generalised Kahler geometry [29].
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One can restore the proper factors of 0 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 dening the bundle metric on the TX subspace to be negative
denite 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 nite deformations. In
particular we will see the holomorphic structure is an important ingredient in describing
higher-order deformations. First let us discuss the o-shell parameter space and how the
heterotic SU(3) structure can be rephrased using a superpotential.
3 The o-shell parameter space
We now show that a subset of the system corresponding to F -term conditions can be de-
rived from a superpotential. We then discuss how deformations of the geometry, 
ux and
bundle can be parametrised using the observation that the superpotential is a holomor-
phic function.
The four-dimensional eective theory that one nds 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 
ux H and the SU(3) invariant forms by5
W =
Z
X
(H + i d!) ^ 
: (3.1)
As we will review, given the SU(3) structure relations (2.1) and the denition of H in (2.5),
W = W = 0 reproduces the F -term conditions of the Hull-Strominger system [26].
Notice that W = 0 requires us to vary the superpotential over some space of eld
congurations. We need to understand what this space is in order to nd how the superpo-
tential behaves when we perform a nite deformation of the background elds. We pause
brie
y to distinguish between this parameter space and the moduli space of solutions to
the Hull-Strominger system.
The parameter space or space of eld congurations Z is the space of SU(3) structures,
B elds 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 eld
congurations the heterotic theory admits an \o-shell" N = 1 supersymmetry of the kind
discussed in [37, 38]. These elds do not necessarily solve the Hull-Strominger system and
so we often refer to them as o-shell eld congurations. 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
elds also solve the Hull-Strominger system. This set of elds is what one usually means by
moduli, and we will often refer to them as on-shell congurations. 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
deformations of the Hull-Strominger system and we will see later how its presence greatly
5Here we have scaled away an overall factor of 0=4 that comes from the eld redenition in (2.20).
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simplies the problem. On the supersymmetric locus (on-shell), it is known that the
superpotential is a holomorphic function of the moduli elds [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 o-shell eld congurations | the o-shell eld space Z must admit complex
coordinates and W must be a holomorphic function of these coordinates. In other words,
the three-form 
 and the particular combinations of B + i! 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 
 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 o-shell. For completeness, one
should really show that W itself can be expressed as a holomorphic function of the object
~ that we dene in appendix C | we leave this for a future work.
Note that on-shell, 
 is also holomorphic as a function of the complex coordinates of
X. When we talk of 
 being holomorphic we are instead referring to its dependence on
the coordinates of the o-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

 =
1X
n=1
1
n!
tnDnt ; 
 =
1X
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, o-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 of the Chern-Simons form. Not all components of the Chern-Simons form
appear in W (as it is wedged with 
); only the (0; 1)-form components of A contribute. It is
this component of the connection that is the complex coordinate on the o-shell parameter
space. A holomorphic deformation of the connection is then given by a (0; 1)-form valued
in EndV , which we denote :
A 7! A+ 
A = A+ : (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 End V or with (0; 1)-forms valued in End VC.
A deformation of the complex structure is parametrised by a (1; 0)-vector valued (0; 1)-
form,  2 
(0;1)(T (1;0)X), also known as a Beltrami dierential. The complex coordinates
on X that dene the complex structure deform to
dza ! dza + 
dza = dza + a: (3.4)
Innitesimally, the deformed complex structure is J + 2i. For a small but nite deforma-
tion, the holomorphic three-form becomes [39, 40]

! 
 + 

 = 
 + {
 + 1
2
{{
 +
1
3!
{{{
; (3.5)
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where the variations of 
 in coordinates are
{
 =
1
2

abc
a ^ dzbc;
{{
 = 
abc
a ^ b ^ dzc;
{{{
 = 
abc
a ^ b ^ c:
(3.6)
The fact that a variation of 
 is completely captured by  without needing  is an indication
that 
 is a holomorphic function of the coordinates of the o-shell parameter space. Note
that the variation of 
 can in principle have a (3; 0) component. However, the (3; 0)
part should be interpreted as a Kahler 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 eld moduli are
(B + i 
!)(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 eld and the exact term in the variation of
the Chern-Simons term, given by
B = 
B + tr 
A ^A; (3.8)
up to a d-closed two-form. The Green-Schwarz mechanism ensures B is gauge invariant.
As we show in appendix B, 
ux quantisation then implies that B is a globally dened
two-form, so it can indeed be a modulus. As we will see, the (0; 2) component of 
! is
actually xed 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 
!)(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 
to be an innitesimal deformation of the elds, leading to a corresponding variation of the
superpotential W
W =
Z
X

2 tr A ^ F + d(B + i !)
 ^ 
 + Z
X
(H + i d!) ^ 
: (3.9)
The F -term conditions come from requiring that both W and W vanish for generic values
of the moduli. For arbitrary B + i ! and A, the vanishing of W requires
@
 = d
 = 0; F ^ 
 = 0: (3.10)
With this in mind, the vanishing of W for arbitrary 
 of type (2; 1) impliesH(1;2) = i @!.
Using the previous conditions, the vanishing of W itself reduces to H(0;3) = @
 for some
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(0; 2)-form 
. The Bianchi identity for H then implies H(0;3) = 0, giving us the nal F -term
condition:
H = i(@  @)!: (3.11)
With this we see W = W = 0 reproduces the F -term equations of the Hull-
Strominger system.
Our plan is to extend this discussion to understand nite deformations around a su-
persymmetric 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 dene an SU(3) structure. Another way of saying this
is that the o-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:
(! + 
!) ^ (
 + 

) = 0; (3.12)
where 
 is a nite holomorphic variation. Expanding this out according to complex type,
we nd
0 =

! + (
!)(2;0) + (
!)(1;1) + (
!)(0;2)

 ^ 
 + {
 + 1
2
{{
 +
1
3!
{{{


: (3.13)
Upon contracting this with 
, we see this equation xes the (0; 2) component of 
! in
terms of the other deformations:
(
!)(0;2) = {! + {(
!)(1;1) 
1
2
{{(
!)(2;0): (3.14)
Now consider an anti-holomorphic variation 
 under which 
 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 

 = 0) the SU(3) compatibility
condition reduces to

! ^ 
 = 0: (3.15)
From this we see (
!)(0;2) = 0 and so, taking a conjugate, (
!)(2;0) = 0. Combined with
the previous result of a holomorphic variation of the compatibility condition, we have
(
!)(2;0) = 0; (
!)(0;2) = {! + {(
!)(1;1): (3.16)
From this we see the (0; 2) component of the variation of ! is xed 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
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\o shell". Let us turn o the gauge sector for now, and consider an anti-holomorphic
variation of W :

W =
Z
X
d(
B + i 
!) ^ 
 = 
Z
X
(
B + i 
!) ^ d
: (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 d
 2 
(3;1)(X)
(2;2)(X). For an anti-holomorphic
variation of the superpotential to vanish, it is sucient that
(
B + i 
!)(1;1) = 0; (
B)(0;2) = 0; (3.18)
where we have used the rst condition from (3.16) to remove (
!)(0;2). This agrees
with (3.7) where we stated that the holomorphic combinations are the (1; 1) and (0; 2)
components of B + i! (see also discussion in appendix C.1). Taking a conjugate of these
conditions we have
(
B  i 
!)(1;1) = 0; (
B)(2;0) = 0: (3.19)
Taken together these give
(
B)(1;1) = i (
!)(1;1); (
B)(2;0) = (
!)(2;0) = 0: (3.20)
For what follows, it is useful to dene
~b = (
B + i 
!)(0;2); x = i (
!)(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 o-shell point in eld space, we nd it is sucient that
(
A)(0;1) = 0. This implies (
A)(1;0) = 0, so the holomorphic deformations correspond to

A =  2 
(0;1)(EndV ): (3.22)
In other words, the holomorphic coordinate on the parameter space is , in agreement
with (3.3). Furthermore, one sees that the holomorphic deformations of the complexied
hermitian moduli are the (1,1) and (0,2) components of B + i
!.
As an aside, we note that there is a schematic way to see that 
 is a holomorphic func-
tion 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, d
 = 0:

W =
Z
X
(H + i d!) ^

 +
Z
X
d(
B + i 
!) ^

: (3.23)
For an innitesimal deformation, the second term can be dropped, and a sucient condition
for the rst term to vanish for generic H and ! is that


 = 0; (3.24)
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innitesimally. Note that this also kills the second term in (3.23) at second order in
perturbation theory. A sucient condition for 
W to vanish at this order is hence that


 = 0 to this level as well. This argument can be continued ad innitum, and we
are left with condition (3.24), at least for nite deformations away from a closed 
. We
will assume that this condition holds true at generic o-shell points in the parameter
space. Stronger evidence for this is provided by the generalised geometry formulation of
the o-shell parameter space presented in appendix C; we also nd 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
innitesimal to nite deformations of solutions to the Hull-Strominger system. In other
words, we consider higher-order deformations of the elds that parametrise the supersym-
metric 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 reason-
able assumptions the D-term conditions do not constrain the moduli problem and should
be thought of as gauge xing conditions | we expect this to hold in general.
4.1 The superpotential
Let us consider the eect on the superpotential of a nite deformation of the background
elds 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 j0. The vacuum is supersymmetric if both the superpotential and its rst
derivative vanish when evaluated on the solution. Now move a nite distance from this
solution in parameter space by deforming the background. The superpotential evaluated
at this new point is W j
 = W j0 + 
W . We have a supersymmetric solution if both W j

and its rst derivative vanish at that point in parameter space, which is equivalent to the
vanishing of 
W and 
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 nite holomorphic deformation of the parameters gives

W =
Z
X

H + i d! + d(
B + i 
!)

^{
 + 1
2
{{
 +
1
3!
{{{


+ d(
B + i 
!) ^ 


:
(4.1)
As we are deforming about a supersymmetric point we have H = i(@  @)! and d
 = 0,
so the rst term simplies and the last term vanishes, giving

W =
Z
X

i @! ^ {{
 + d(
B + i 
!) ^

{
 +
1
2
{{


=
Z
X
h
i @! ^ {{
 + 2 @x ^ {
 + @x ^ {{
 + @~b ^ {

i
;
(4.2)
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where the {{{
 term vanishes due to the type of d(
B+i 
!), and we have written the
second line in terms of ~b and x, the (0; 2) and (1; 1) parts of the complexied hermitian
moduli (3.21). As { satises a graded Leibniz identity, we can rewrite the above as

W = 2
Z
X

 {@x+ 1
2
i {{@! +
1
2
{{@x 1
2
{@~b

^ 

= 2
Z
X

d ^ @xd + id ^ e ^ @d!ecec + d ^ e ^ @dxe  1
2
d ^ @d~b

^ 
:
(4.3)
Our rst 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 rst derivative of 
W . As 
W is a
functional, this amounts to treating it as an action and nding the resulting equations of
motion. Varying 
W , one nds
@xa  id ^ (@!)dabeb + d ^ @axd  d ^ @dxa 
1
2
@a~b = 0; (4.4)
@d  1
2
[; ]d = 0; (4.5)
@{
 = 0: (4.6)
A few comments are in order. The condition in (4.5) is nothing but the Maurer-Cartan
equation for nite 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  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 
 + 

 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 
 remains d-closed. One could make a change of variables
which solves this constraint explicitly by taking {
 = a+ @b where a is @-harmonic. This
may be useful when investigating the quantum theory dened 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 innitesimal moduli of the Hull-Strominger system
are captured by the D-cohomology of the holomorphic structure, so we expect D will be
the dierential that appears in such a Maurer-Cartan equation. The other ingredient is
a bracket. We introduce the bundle Q0 as the sum of the holomorphic cotangent and
tangent bundles:
Q0 ' T (1;0)X  T (1;0)X: (4.7)
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This is the bundle Q dened in (2.16) with the gauge sector suppressed for now. We write
(0; 1)-forms valued in this bundle as
y 2 
(0;1)(Q0); y = (xaea; ae^a)  x+ : (4.8)
The holomorphic structure for the Hull-Strominger system without the bundles is given by
an operator D that acts on sections of Q0. We can also introduce a bracket [
; 
] on forms
valued in Q0 and a pairing h
; 
i that traces over the Q0 indices:
[
; 
] : 
(0;p)(Q0) 
(0;q)(Q0)! 
(0;p+q)(Q0); (4.9)
h
; 
i : 
(0;p)(Q0) 
(0;q)(Q0)! 
(0;p+q)(X): (4.10)
We give explicit expressions for D, the bracket [
; 
] and the pairing h
; 
i in equa-
tions (A.11), (A.13) and (A.16). Using these we can rewrite 
W , given in (4.3), as

W =
Z
X
hy;Dy  1
3
[y; y] @bi ^ 
; : (4.11)
Note that we have redened the (0; 2)-form eld as b = ~b  a ^ xa. The equations of
motion that follow from varying 
W can then be written compactly as
Dy  1
2
[y; y] 1
2
@b = 0; (4.12)
@{
 = 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 specically, the form of the action is that of a holomor-
phic 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 super-
potential in other work [42]; we expect a similar analysis could be applied here.
Notice also that innitesimal deformations are captured by
Dy =
1
2
@b: (4.14)
It follows from this and dH / @@! = 0 that @b is @-closed.6 If the underlying manifold
X satises the @@-lemma or H(0;2)(X) vanishes, @b is @-exact and can be absorbed in a
redenition of the complexied Kahler moduli x. We then see that innitesimally the
complexied Kahler 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 !CS(A) for the gauge connection A:
W =
Z
X

dB + !CS(A) + i d!

 ^ 
: (4.15)
6Recall we have turned o the gauge sector in this subsection.
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The B eld transforms in the usual Green-Schwarz manner so that H is gauge invariant:
B = tr(d ^A): (4.16)
Consider a shift of the gauge connection by A! A+ : The corresponding change of
the Chern-Simons form is

!CS = 2 tr(F ^ ) + tr( ^ dA) + 2
3
tr( ^  ^ ) + d tr( ^A): (4.17)
The exact term in this variation combines with the variation of the B eld to give
B = 
B + tr( ^A): (4.18)
As we show in appendix B.4, B is a globally dened two-form so it can be a modulus [14].
Expanding about a supersymmetric point, the extra terms in the variation of the
superpotential from the gauge sector are

W =
Z
X

!CS ^ (
 + {
 + {{
)
=
Z
X
h
(tr( ^ @A) + 2
3
tr( ^  ^ )) ^ 

+ (2 tr(F ^ ) + tr( ^ @A)) ^ {

+ d tr( ^A) ^ ({
 + 1
2
{{
)
i
;
(4.19)
where the nal terms combine with b in 
W to give B = b+ tr( ^ A). Upon replacing b
by B in (4.3), the extra terms in the variation of the superpotential are

W =
Z
X

tr(^ @A) + 2
3
tr(^ ^ ) 2d ^Fd ^  ^ d ^ (@A)d

^
: (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  id ^ (@!)dabeb  trFa ^ + d ^ @axd
d ^ @dxa + 1
2
tr ^ (@A)a  1
2
@ab = 0;
@A+  ^ + Fdadza ^ d  d ^ (@A)d = 0;
@d  1
2
[; ]d = 0;
@{
 = 0;
(4.21)
where we have redened b to be
b = (B + i 
!)(0;2)  a ^ xa: (4.22)
Remarkably, the superpotential can still be written in a Chern-Simons fashion as

W =
Z
X
hy;Dy  1
3
[y; y] @bi ^ 
; (4.23)
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where y is now (0; 1)-form valued in Q ' T (1;0)XEndV T (1;0)X, and D, [
; 
] and h
; 
i
are now given by expressions (A.19), (A.20) and (A.21). The corresponding equations of
motion are
Dy  1
2
[y; y] 1
2
@b = 0; (4.24)
@{
 = 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 o. 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 satis-
ed and the superpotential itself vanishes. As X is assumed to be compact, the superpo-
tential vanishes if the terms wedged with 
 are @-exact, that is
hy;Dy  1
3
[y; y] @bi = @; (4.26)
where  is an arbitrary (0; 2)-form. Upon substituting the rst equation of motion (4.24)
into this expression, it simplies to
1
3!
hy; [y; y]i  1
2
ya@ab = @: (4.27)
Now we use D
2
= 0 to constrain . Taking D of the rst equation of motion (4.24) gives
0 = D
2
y  [Dy; y] 1
2
D@b
/ ea([y; [y; y]]a  [y; @b]a + @@ab)
= ea

1
3!
@ahy; [y; y]i  1
2
@a(y
d@db) + @@ab

/

@@b 1
2
@(yd@db) +
1
3!
@hy; [y; y]i

;
(4.28)
where we have used [y; [y; y]]a =  13!@ahy; [y; y]i and [y; @ab] = 12@a(yd@db).7 We can inte-
grate this expression to give
k
 = @b 1
2
yd@db+
1
3!
hy; [y; y]i; (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
k
 = @b+ @: (4.30)
7These identities are easy to check using the explicit expressions for the bracket and pairing.
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As 
 is not @ exact, we must have k = 0. We then identify  = b up to a @-closed
(0; 2)-form. Putting this all together the full set of equations is
Dy  1
2
[y; y] 1
2
@b = 0; (4.31)
@b 1
2
hy; @bi+ 1
3!
hy; [y; y]i = 0; (4.32)
@{
 = 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 nitely
many powers of y and b; the equations do not give an innite 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 rst
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 
ux. In deriving these equations and nding 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 0 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 0 ! 0 limit, the 0 = 0 solution is simply Calabi-
Yau. In this case it is known that the superpotential receives no 0 corrections (to nite
order) and so, although the 0-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 0 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 nite deformations. We will show in this section that these equations
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can be reinterpreted as the Maurer-Cartan equation for an L3 algebra. At rst sight there is
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 L1 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 L1
or L3 structure [47{50].
As we review in appendix A.3, an L1 structure is specied 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 L1 structure that combines the
action of D on the moduli elds y and @ on the (0; 2) moduli b. We take the vector spaces
Yn to be
Yn = 
(0;n)(Q) 
(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  1
2
@b; @b

;
`2(Y; Y ) := ([y; y]; hy; @bi);
`3(Y; Y; Y ) := (0;hy; [y; y]i);
`k4 := 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 L1
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 brie
y remark that these structures have a nice mathematical interpretation: our L3
algebra (Y; `1; `2; `3) is L1 equivalent to the underlying holomorphic algebra.
Neglecting the gauge bundle for a moment, we have the sheaf E of D-holomorphic
sections of Q0 ' 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 C1 functions
form an L3 algebra [53] with two-term complex
C1(R) d! E ' T X  TX: (5.4)
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One can then consider the Dolbeault resolutions of the sheaves E and OX , and extend @
to a morphism between them:
0 OX C1(C) 
(0;1) 
(0;2)
0 E Q0 
(0;1)(Q0) 
(0;2)(Q0)
 @ @ @
 D D D
@ @ @ @
(5.5)
Our complex Y is then the total complex of the deleted resolution and the dierential `1
is the natural dierential 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 C1 objects. Explicitly,
one has a map of complexes (of sheaves) as follows:
0 OX E 0 0
0 C1(C) (E) 
(0;1) 
(0;1)(E) 
(0;2) 
(0;2)(E) 
(0;3)
@
`1 `1 `1 `1
 
(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 L1
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 Q0 above with the full holomorphic bundle (C.34) (this also recently appeared
in [30]). One nds 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 L1 eld equation
As explained in [52], there is a natural eld equation that one can write down for a given
L1 structure. The constraint on the form of the eld equation is that it is covariant under
gauge transformations of the elds Y. In terms of the L1 products, the eld equation is
F(Y ) = `1(Y ) 1
2
`2(Y ) 1
3!
`3(Y ) + : : : (5.7)
For us this expression truncates at third order as `k4 = 0.
Remarkably, the L3 eld equation coming from (5.2) reproduces the conditions from
the vanishing of the superpotential and its derivative:
F(Y ) =

Dy  1
2
@b 1
2
[y; y]; @b 1
2
hy; bi+ 1
3!
hy; [y; y]i

: (5.8)
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In other words, the conditions for a supersymmetric Minkowski vacuum are equivalent to
F(Y ) = 0; @{
 = 0: (5.9)
A particularly nice property of this rewriting is that the L1 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  = (; ) 2 Y0 is
Y = `1() + `2(; Y ) 1
2
`3(; Y; Y ); (5.10)
where the higher-order brackets vanish. In general the gauge transformations take eld
equations to combinations of eld equations | the eld equations are covariant. If one
could construct an action that has the L3 eld 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 nd 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 rst derivative. Note that the superpotential alone is not expected
to be invariant under both  and  transformations | we will see in the next section that
the superpotential is actually invariant under the  transformations alone which correspond
to shifts by @a-exact forms.
6 A reduced L3 algebra and an eective 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 eective theory one would nd 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 dis-
ciplines 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 eective action (4.23), and its
relation to the lower-dimensional eective physics.
6.1 Integrating out b
We want to integrate out the (0; 2)-form eld b. Looking back at the form of the L3 gauge
transformations (5.10), taking  = (0; ) gives
ya =
1
2
@a; b = @  1
2
hy; @i; (6.1)
for some (0; 1)-form . 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 dened up to
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@a-exact forms. Notice also that 
W splits into two pieces:

W =
Z
X
hy;Dy  1
3
[y; y]i ^ 

Z
X
hy; @bi ^ 
; (6.2)
where the second term can be written asZ
X
hy; @bi ^ 
 /
Z
X
b ^ @{
: (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, specically h(2;0) = 0, then the
eld b has no associated massless modes. We can then integrate out b, resulting in

W [y] =
Z
X
hy;Dy  1
3
[y; y]i ^ 
; (6.4)
where y now satises the constraint @{
 = 0. We want to think of this functional as an
eective action. Note that for @{
 = 0 there is a gauge symmetry of this action
ya = @a; (6.5)
where  2 
0;1(X). We are thus led to dene ~Q as a reduced sheaf of Q whose sections
satisfy the constraint and are dened up to @a-exact terms:
( ~Q) = f(Q) j @{
 = 0; ya  ya + @ag: (6.6)
One can check that the brackets on Q are well dened on ~Q, and that the L3 algebra
descends to a dierential graded Lie algebra (DGLA). A very similar sheaf is also considered
in the discussion of 
 systems in [56]. Note that this DGLA is L1 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
y = D [y; ]; (6.7)
where  2 
0( ~Q) satises @{
 = 0. Note that the gauge algebra generated by (5.10) is
reducible; a gauge transformation by  = (; ) is trivial if
 = @w;  = @w + 1
2
hy; @wi; (6.8)
for w 2 
0(X).
From the eective superpotential (6.4) we derive the equation of motion
Dy  1
2
[y; y] = 0: (6.9)
This should be interpreted as an equation on the sheaf ~Q. Note that under ya 7! ya + @a,
this equation becomes
Dy  1
2
[y; y] = @(@ + yd@d)  0; (6.10)
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so it is well dened as an equation on the sheaf ~Q. Recall that we already know the
superpotential is invariant under ya = @a. One can show that equation (6.9), together
with the condition that the eective action vanishes, is equivalent to the Maurer-Cartan
equations (4.31){(4.33). Indeed, note that (6.9) is equivalent to
Dy  1
2
[y; y] =
1
2
@b; (6.11)
for some b 2 
(0;2)(X). For solutions to this equation, the condition that the action
vanishes can then be written as hy; [y; y]i = @-exact, which can be rewritten as
1
3!
hy; [y; y]i  1
2
ya@ab = @; (6.12)
for some  2 
(0;2)(X). Here we used that the second term on the left-hand side is @-exact
| it integrates to zero against 
 due to the constraint @{
 = 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 eective eld
theory. From this we make a conjecture about the obstructions that can appear in the
Maurer-Cartan equations.
6.2 Eective eld theory and Yukawa couplings
Our starting point to derive the eective physics is the superpotential (4.23), where we have
re-introduced the eld b. When dimensionally reducing the theory, it is common practice
to split our elds (y; b) into \massless" and \heavy" modes
y = y0 + yh; (6.13)
b = b0 + bh: (6.14)
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
V = e
K@@
W@@W K
; (6.15)
where f; ; : : :g denote holomorphic directions in the parameter space and K is the Kahler
potential. Full knowledge of the Kahler 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 eld direction  is massless if and only if
@
@W = 0 8 
 ) (@
@
W )j(y;b)=0 = 0 8 
; (6.16)
where the eld directions 
 can in principle be massive.
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 complexied gauge transformation and so they do not lift any moduli.
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In the end, we are interested in a reduced eld theory of massless modes, where the
massive modes have been \integrated out". It is easy to see that the eld direction corre-
sponding 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 
(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 eld as far as the eective theory is concerned, leaving us with the eective superpo-
tential (6.4), where now the Beltrami dierential component  of y satises @{
 = 0 as
above. From condition (6.16) it follows that the remaining massless elds 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
y0 = D; (6.20)
yh = [y0; ] [yh; ]: (6.21)
With this decomposition, we see that the massless modes are parametrised by cohomol-
ogy classes
[y0] 2 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 eective action now reads

W =
Z
X

hyh; Dy0yhi 
1
3
hy0; [y0; y0]i  hyh; [y0; y0]i  1
3
hyh; [yh; yh]i

^ 
; (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 elds. These new couplings are however of quartic order and higher in y0,
and are hence non-renormalisable in the eective eld theory. The only renormalisable
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coupling we need to worry about from an eective eld theory point of view is therefore
the Yukawa coupling

WYuk(y0) = 1
3
Z
X
hy0; [y0; y0]i ^ 
: (6.25)
This argument is similar to and generalises Witten's standard argument for the gauge
sector [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 de-
formations of the geometry) and gravity-bundle type, often referred to as -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 A 2 H(0;1)( ~Q) denote a set of inequivalent cohomology classes spanning H(0;1)( ~Q),
and expand
y0 =
X
A
CA
A; (6.26)
where the CA now correspond to the four-dimensional elds, including in principle moduli
and matter elds. The Yukawa couplings then read

WYuk =
X
A;B;C
CACBCC
Z
X
hA; [B; C ]i ^ 
: (6.27)
A massless eld direction A is then truly free if and only if
YABC =
Z
X
hA; [B; C ]i ^ 
 = 0 8 B; C : (6.28)
In particular, this is true if
[A; B] = DAB 8 B: (6.29)
Note that, starting from the Maurer-Cartan equation (6.9), this is simply the condition
for an innitesimal deformation in the eld direction A to be unobstructed. The eective
eld 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 innitesimal moduli.
7 Conclusions
In this paper we have considered nite deformations of the Hull-Strominger system. Start-
ing with the four-dimensional N = 1 superpotential, we showed that integrable deforma-
tions 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 innitesimal deformations parametrised by H
(0;1)
D
( ~Q) can be integrated to nite
deformations, corresponding to solutions of the L3 Maurer-Cartan equation. In particular,
9So we can integrate out b in the eective theory.
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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 elds 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 
-preserving generalised dieomorphisms 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 eective action and investigating its quantisation one might hope it
denes 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.
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 eective 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 
 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 compactications on more exotic
geometries, such as G2 or Spin(7) manifolds [80{82]. In the case of G2 compactications, the
form of the moduli space is remarkably similar: for example, the innitesimal deformations
are again captured by a cohomology. Despite this there are notable dierences such as
the analogue of the bundle Q not appearing as an extension. It would be interesting to
investigate the nite deformation algebras in these cases, and in the process identify the
corresponding L1 structure. This might give a G2 generalisation of Chern-Simons theory.
We have been concerned with nding the honest supersymmetric deformations of solu-
tions 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 Kahler
potential which is important for understanding the physical potential of the eective the-
ory. The Kahler 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
10Generalised dieomorphisms 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 dieomorphisms
together with gauge transformations of the supergravity elds.) The 
-preserving condition is just (4.6)
which ensures the deformed three-form is d-closed. Another Chern-Simons like theory featuring generalised
dieomorphisms as a gauge algebra has recently appeared in [60].
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described by an invariant object ~ so the Kahler 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 compactications in other dimensions. We
hope to make progress on this in a future work.
Note that even though we have the invariant object ~ we do not have a natural inte-
grability 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.
Acknowledgments
We would like to thank Bobby Acharya, Dmitri Alekseevsky, Philip Candelas, Jose
Figueroa-O'Farrill, Mario Garcia-Fernandez, 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).
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 = vme^m = v
ae^a + v
ae^a: (A.1)
Our elds 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 , we have
{w = e
m ^ {wm = wn ^ n; (A.2)
where m is dened as
m =
1
(p 1)!mn1:::np1e
n1:::np1 : (A.3)
It follows that {w satises a Leibniz rule:
{w( ^ ) = {w ^  +  ^ {w: (A.4)
The interior product of a vector with a one-form is extended to p-vectors and p-forms using
the y operation, dened as
uy = 1
p!
um1:::mpm1:::mp ; (A.5)
where u is a p-vector and  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
(])m1:::mp = gm1n1 : : : gmpnpn1:::np : (A.6)
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A.1 Heterotic supergravity
The Hull-Strominger system follows from setting the supersymmetry variations of the ten-
dimensional gravitino  , dilatino  and gaugino  to zero. In our conventions these are
 M = r+M" = rLCM "+
1
8
HMNP
NP "+O(02); (A.7)
 =

M@M+
1
12
HMNP
MNP

"+O(02); (A.8)
 = 1
2
FMN
MN"+O(02); (A.9)
where " is a ten-dimensional Majorana-Weyl spinor and rLC denotes the Levi-Civita
connection.
A.2 Holomorphic structure
Ignoring the gauge sector for the moment, the relevant elds are (0; 1)-forms taking values
in Q0 = T (1;0)X  T (1;0)X. We write (0; 1)-forms valued in this bundle as
y = (xae
a; ae^a) = x+ ; (A.10)
where xa and 
a are (0; 1)-forms. More generally we will write yp = (xp; p) 2 
(0;p)(Q0)
| we will often drop the subscript denoting the form degree.
The holomorphic structure is dened by a D operator that is nilpotent. The action of
D on y 2 
(0;p)(Q0) is
(Dy)a = @xa + i(@!)eace
c ^ e;
(Dy)a = @a:
(A.11)
One can check that D
2
= 0 follows from dH / @@! = 0. Note also that this convention
implies
(D@)a = @(@a) = @a@; (A.12)
for a form  (not valued in Q0).
The bracket [
; 
] : 
(0;p)(Q0) 
(0;q)(Q0)! 
(0;p+q)(Q0) is
[y; y0]a = d ^ @dx0a  @dxa ^ 0d 
1
2
d ^ @ax0d
+
1
2
@axd ^ 0d + 1
2
@a
d ^ x0d 
1
2
xd ^ @a0d;
[y; y0]a = b ^ @b0a  @ba ^ 0b:
(A.13)
The bracket is graded commutative and satises [yp; yq] = ()1+pq[yq; yp]. Furthermore, D
satises a graded Leibniz identity with the bracket
D[yp; y
0
q] = [Dyp; y
0
q] + (1)p[yp; Dy0q]: (A.14)
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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 2 
(0;1)(Q0), one nds
[y; [y; y]] =  1
3!
@hy; [y; y]i; (A.15)
so that only the (1; 0)-form valued component is non-zero. Here we have dened a pairing
between two sections as
hy; y0i = d ^ x0d + xd ^ 0d: (A.16)
More generally one nds
[ym; [y
0
n; y
00
p ]] + (1)m(n+p)[y0n; [y00p ; ym]] + (1)p(m+n)[y00p ; [ym; y0n]]
=  1
3!
h
@ahy; [y0; y00]i+ (1)m(n+p)@ahy0; [y00; y]i+ (1)p(m+n)@ahy00; [y; y0]i
i
:
(A.17)
When we include the gauge sector we write sections as
y = (xae
a; ; ae^a) = x+ +  2 
(0;1)(Q); (A.18)
where xa and 
a are (0; 1)-forms. The D operator, bracket and pairing are extended to
include the gauge eld component. The D operator is
(Dy)a = @xa + i(@!)eace
c ^ e  tr(Fa ^ );
(Dy)a = @a;
(Dy) = @A+ Fb ^ b;
(A.19)
where the nal component is the gauge eld piece and Fa = Fabdz
b. The bracket is
[y; y0]a = : : : 1
2
tr ^ (@A0)a + 1
2
tr(@A)a ^ 0;
[y; y0]a = : : : ;
[y; y0] =  ^ 0  ()1+00 ^ + b ^ (@A0)b  (@A)b ^ 0b
= [; 0] + b ^ (@A0)b  (@A)b ^ 0b;
(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
hy; y0i = d ^ x0d + xd ^ 0d + tr( ^ 0): (A.21)
A.3 L1 structure
We follow [52] for the conventions of an L1 algebra in the \`-picture".11 We start with a
graded vector space Y
Y =
M
n
Yn; n 2 Z; (A.22)
11We have swapped n! n so that the degree of Yn matches the form degree of y
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where the Yn are of degree n. The L1 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
0) = (1)1+Y Y 0`2(Y 0; Y ); (A.23)
where a superscript denotes the degree of Y 2 Yn. More generally we have
`k(Y
(1); : : : ; Y (k)) = (1)(;Y )`k(Y 1; : : : ; Y k); (A.24)
where Y 1 = Y , Y 2 = Y 0, etc. The sign has two contributions: (1) gives a plus if the
permutation is even and a minus if the permutation is odd; (;Y ) is determined by
Y 1 ^ : : : ^ Y k = (;Y )Y (1) ^ : : : ^ Y (k); (A.25)
where Y ^ Y 0 = (1)Y Y 0Y 0 ^ Y .
In these conventions, the rst few L1 identities are
`1(`1(Y )) = 0;
`1(`2(Y; Y
0)) = `2(`1(Y ); Y 0) + (1)Y `2(Y; `1(Y 0));
`1(`3(Y; Y
0; Y 00)) =  `3(`1(Y ); Y 0; Y 00) (1)Y `3(Y; `1(Y 0); Y 00)
 (1)Y+Y 0`3(Y; Y 0; `1(Y 00)) `2(`2(Y; Y 0); Y 00)
 (1)(Y+Y 0)Y 00`2(`2(Y 00; Y ); Y 0) (1)(Y 0+Y 00)Y `2(`2(Y 0; Y 00); Y )
(A.26)
The eld equations and gauge transformations are
F(Y ) =
1X
n=1
(1)n(n1)=2
(n)!
`n(Y
n) = `1(Y ) 1
2
`2(Y; Y ) 1
3!
`3(Y; Y; Y ) + : : : ; (A.27)
Y = `1() + `2(; Y ) 1
2
`3(; Y; Y ) 1
3!
`4(; Y; Y; Y ) + : : : ; (A.28)
where Y 2 Y1 and  2 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
0) :=

[y; y0];
1
2
(hy; @b0i+ (1)1+Y Y 0hy0; @bi)

;
`3(Y; Y
0; Y 00) :=
1
3
(1)Y+Y 0+Y 00(0; hy; [y0; y00]i+ (1)Y (Y 0+Y 00)hy0; [y00; y]i
+ (1)Y 00(Y+Y 0) hy00; [y; y0]i);
`k4 := 0:
(A.29)
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B Comments on heterotic 
ux quantisation
In this appendix we comment on 
ux 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 compactications. Note that
we are not saying anything new here; understanding 
ux 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 
ux 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 elds. 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 fU ig 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 Cech co-boundary operator where appropriate: for some sheaf F , if fi 2 F(U i) then
(@f)ij = fi  fj 2 F(U ij), and so on.
For the curvature to be well dened, 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 = d
ij ; (B.2)
on U ij for some zero-forms 
ij . On triple overlaps U ijk we have
d(@
)ijk = d(
ij + 
jk + 
ki) = (@
2A)ijk = 0; (B.3)
which is often referred to as taking the co-cycle of d
ij . It follows that cijk = (@
)ijk are
constants:
cijk = (@
)ijk = 
ij + 
jk + 
ki 2 R(U ijk)  C1(U ijk;R): (B.4)
Clearly from (B.4) we have @c = 0, so the cijk dene a class of the sheaf cohomology
[cijk] 2 H2(X;R) = H2(X;R); (B.5)
which represents the two-form 
ux 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 
ij 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 
ux is trivial. In this language 
ux quantisation is the statement
that the cijk are in fact integers, cijk 2 Z(U ijk), so that they dene an integral class
[cijk] 2 H2(X;Z) = H2(X;Z)
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Gauge transformations of the connection do not change the class [cijk] 2 H2(X;R). To
see this consider a gauge transformation which preserves the curvature F :
A0i = Ai + di; i 2 C1(U i;R): (B.6)
This transformation induces a deformation of the 
ij as

0ij = 
ij + i  j + ij = 
ij + (@)ij + ij ; (B.7)
where ij 2 R(U ij) are constants. Thus c0 = @
0 = @
 + @ = c+ @ is shifted by a Cech
co-boundary, and so [c0ijk] = [cijk] 2 H2(X;R) is unchanged. Note that in the case of a
quantised 
ux, the integrality of cijk holds only in a preferred set of gauges for 
ij under
shifts by real constants ij .
Now consider a general deformation of the above system. We denote the variations
by 
(: : :): for example A0 = A + 
A. We x the (integral) cohomology class of the
quantised 
ux so that 
cijk = 0. This means that we must have (@

)ijk = 0 and thus as
C1(R) is acyclic we can nd i 2 C1(U i;R) with 

ij = (@)ij = i  j . We see the 
ij
can be deformed only by a gauge transformation, as in (B.7). Performing a global gauge
transformation A00i = A
0
i  di, we nd that in the new gauge we have
~
Ai = A
00
i Ai = 
Ai  di; (B.8)
so that on U ij we have
~
Ai  ~
Aj = d(

ij  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 = (@)ijk
for ij 2 R(U ij) | this xes the real cohomology class [cijk] 2 H2(X;R). We would then
have deduced that 

ij = (@)ij + ij , leading to the same gauge transformation of A
as above. This would correspond to deforming away from the gauge (choice of 
ij) in
which cijk are explicitly integral, while above we restricted ourselves to the gauge with
cijk 2 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 specied by a set of two-forms Bi 2 
2(U i) covering the manifold. The eld strength
H = dB is globally dened. As before, this means that on overlaps U ij we have
Bi Bj = dij ; (B.11)
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for some one-forms ij 2 
1(U ij). Again we have (@B)ij = dij on U ij so that (@)ijk is
d-closed
d(@)ijk = (@
2B)ijk = 0: (B.12)
As U ij is contractible (@)ijk is actually exact
(@)ijk = dijk; (B.13)
for some ijk 2 C1(U ijk;R). Finally we take the co-cycle of dijkl on quadruple over-
laps U ijkl:
d(@)ijkl = (@
2)ijkl = 0: (B.14)
It follows that the functions cijkl = (@)ijkl are constants. As before, if the 
ux is quantised
the cijkl can be made integral by appropriate choice of ijk. Indeed, the cijkl represent a
co-cycle in the sheaf cohomology
[cijkl] 2 H3(X;Z) = H3(X;Z); (B.15)
where the class in H3(X;Z) is given by the 
ux H = dB.
Consider deformations of the above system. From the quantisation of the constants
cijkl and the acyclicity of C1(R) we have

cijkl = (@
)ijk = 0 ) 
ijk = (@)ijk; (B.16)
for some ij 2 C1(U ij ;R). This leads immediately to12
(@
)ijk = d
ijk = (@d)ijk ) 
ij = dij + (@)ij ; (B.17)
for some i 2 
1(U i). Putting this all together we have

Bi 
Bj = d
ij = 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  and . Thus, in
an appropriately chosen gauge, we have that 
B is a global two-form.
B.3 Heterotic 
ux quantisation
We now come to the example relevant for the present paper: the B eld of the heterotic
string. Recall that the anomaly cancellation condition reads
H = dB + !CS(A); (B.20)
12Note that again this relation would be unchanged if we simply xed the real cohomology class [c] 2
H3(X;R) so that 
cijkl = (@r)ijkl for some rijk 2 R(U ijk). We would then have 
ijk = rijk + (@)ijk.
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where !CS(A) denotes the Chern-Simons three-form and we set
0
4 = 1. A gauge transfor-
mation of A of the form
A0 = gAg1 + gdg1 (B.21)
induces the following transformation of the Chern-Simons form [43]
!CS(A
0) = !CS(A) + d tr(g1dg ^A) d(g); (B.22)
where
d(g) = tr(g dg1g dg1g dg1): (B.23)
Hence we have
!CS(A
0) = !CS(A) + d!2(g;A); (B.24)
where
!2(g;A) = tr(g1dg ^A) (g): (B.25)
Let us now consider the patching on U ij . From equation (B.20) we get
d(Bi Bj) + d!2ij(g;A) = 0; (B.26)
where
!2ij(g;A) = tr

g1ji dgji ^Aj

 (gji): (B.27)
It follows that
Bi Bj + !2ij(g;A) = dij ; (B.28)
for some one-forms ij . Taking a co-cycle of this on triple intersections U ijk gives the
relation
(@!2)ijk = d(@)ijk: (B.29)
It can further be shown that
(@!2)ijk(g;A) = d!
1
ijk(g); (B.30)
for some one-forms !1ijk(g) 2 
1(U ijk).
The expression for !1ijk(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 !2ij(g;A) with respect to A. We nd that

!2ij(g;A) = tr(i ^Ai) tr(j ^Aj); (B.31)
where we have dened 
A = , which transforms appropriately in the adjoint representa-
tion when the transition functions gij are kept constant. From this it is clear that

(@!2)ijk(g;A) = 0; (B.32)
which shows that we may take !1ijk(g) to be independent of A without loss of generality.
Putting together (B.29) and (B.30), we then get the relation
(@)ijk  !1ijk(g) = dijk; (B.33)
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for some functions ijk 2 C1(U ijk;R). We again co-cycle this relation on quadruple inter-
sections U ijkl to get
d

!0ijkl(g) + (@)ijkl


= 0; (B.34)
where we have used the fact that
(@!1)ijkl(g) = d!
0
ijkl(g); (B.35)
for some functions !0ijkl(g). We can co-cycle this relation again on quintuple overlaps U ijklm
to get
d(@!0)ijklm(g) = 0; (B.36)
and thus the numbers kijklm = (@!
0)ijklm(g) dene a class
[kijklm] 2 H4(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
!0ijkl(g) + (@)ijkl = cijkl; (B.38)
for some constants cijkl. Computing the co-cycle (@!
0)ijklm(g) we thus nd
kijklm = (@c)ijklm; (B.39)
and so [kijklm] = 0 2 H4(X;R). This is the sheaf cohomology version of the statement that
the bundle in question has a trivial rst Pontryagin class as tr F ^ F = dH is exact.
B.4 Deforming the system and a well-dened 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( ^A): (B.40)
This is an essential part of the moduli of the main text, where  = 
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  = 
A can be assumed to be a section of 
1(EndV ).
We begin by noting that, imposing (B.41), a deformation of (B.38) gives
(@
)ijkl = 
cijkl: (B.42)
Hence the constants 
cijkl dene a sheaf cohomology class in H
3(X;R), even though the
original cijkl did not. We will require that this class vanishes [
cijkl] = 0 2 H3(X;R), so
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that the deformation does not produce any new third cohomology. One could think of this
as xing 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 dene such a class themselves. Indeed, this condition also seems to
agree with the general world-sheet arguments on the heterotic B eld and 
ux quantisation
made in [85], where one thinks of the B eld as a torsor. The lack of the notion of zero in
the space of B elds is re
ected 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 dene
a cohomology class naturally.
From this requirement we have
(@
)ijkl = (@r)ijkl ) 
ijk = rijk + (@)ijk; (B.43)
for some rijk 2 R(U ijk) and ij 2 C1(U ij). Next we take the variation of (B.33), again
imposing (B.41), and use the acyclicity of 
1(U ijk) to nd
(@
)ijk = d
ijk = (@d)ijk ) 
ij = dij + (@)ij ; (B.44)
exactly as for the simple two gerbe case. Finally, we take the variation of (B.28) to obtain

Bi  di + tr(i ^Ai) = 
Bj  dj + tr(j ^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 denes 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 wherein the B eld transforms as
B ! B  !2(g;A): (B.47)
C The o-shell N = 1 parameter space and holomorphicity of 

In this appendix, we present a description of the space of o-shell scalar eld congurations
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 (innite-dimensional) space of generalised
geometric structures of a type we will specify. We then show how to recover the three-
form 
 from the generalised geometric structure and outline how one can see that it is
holomorphic on the o-shell eld space, as we claim in section 3.
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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 elds 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 elds. If we require only the presence of an
o-shell N = 1 supersymmetry, then one merely restricts the elds to admit a globally
dened spinor, corresponding to a (possibly non-integrable) SU(3) SO(6) structure. The
conguration space of o-shell scalar elds that we require in our four-dimensional N = 1
description of the ten-dimensional theory is thus the (innite-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 specied by a particular type of
complex section ~ 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 ~ as
~ = ;  2 (^3EC); (C.2)
where  2 (L) is the generalised density with  = pge2 in the coordinate frame
(see [90]). In order for ~ to have the correct stabiliser,  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 denes 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
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 2 (S(C+)). Using local bases E^+m for C+ and E^~m for C dened by13
E^+m = e^
+
m + e
+
m  ie^+mB;
E^+~m = e^

~m  e~m  ie^~mB;
(C.5)
one can write an explicit formula for the object 
 =
1
3!
(T
mnp)E^+m ^ E^+n ^ E^+p =
1
3!
	mnpE^+m ^ E^+n ^ E^+p ; (C.6)
where 	 is the three-form spinor bilinear of  with itself. Writing the object  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  2 S(E)
L1=2
(see [38, 93]), one can also build an N = 1 structure via the expression
~MNP = 
+
MNP; (C.7)
thus providing a third description of the structure.
Having dened 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 ! TX ! 0; (C.8)
with the classes of such extensions labelled by the cohomology class of the three-form 
ux
[H] 2 H3(M ;R). The map  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 induced map, which we also label ,
 : ^3E ! ^3TX: (C.9)
Acting on the bundle L
 ^3E and using L ' detT X we have
 : L
 ^3E ! detT X 
 ^3TX = ^3T X: (C.10)
Thus, applying the anchor map to the generalised structure ~ we obtain an ordinary three-
form on the manifold.
We now calculate this three-form explicitly. We rst note that
(E^m) = e^

m; (C.11)
which immediately gives us
() =
1
3!
	mnp(e^m ^ e^n ^ e^p) 2 ^3TXC (C.12)
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 em, such that the O(6; 6) metric has components
 =
 
g 0
0 g
!
(C.4)
in these frames.
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However, we are interested in the object ~ =  and in our frame we have  =
p
ge2 so
that
(~) =
1
3!
p
ge2	mnp(e^m ^ e^n ^ e^p) 2 ^6T X 
 ^3TXC (C.13)
Now we use the natural isomorphism
^6T X 
 ^3TX ! ^3T X
1
3!
Xmnp(e^m ^ e^n ^ e^p) 7! 1
(3!)2
mnpm0n0p0X
m0n0p0(em ^ en ^ ep) (C.14)
where m1:::m6(= 1) is the Levi-Civita symbol. The standard metric volume form is then
"m1:::m6 =
p
gm1:::m6 and we nd that under the identication (C.14) we have
(~) = e2
1
(3!)2
"mnpm0n0p0	
m0n0p0(em ^ en ^ ep) = e2(?	) = i e2	: (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 homoge-
neous space
SO(6; 6) R+
SU(3) SO(6) (C.16)
is dieomorphic to the orbit of ~ (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 ~ 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 innite-dimensional space of SU(3)SO(6) structures on E, then
we expect that our generalised three-form ~ will be holomorphic on that space. As the
anchor map  is linear and xed by the topology of E, we have that (~) = i e2	 = i 

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)
h
(3;1)+2  (3;1)2
i
R

h
(3;6)+1  (3;6)1
i
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 ~. Taking out the overall scale
we have
~0 = ece+ 
 ~; c 2 C  2 (3;1)+2  2 (3;6)+1: (C.18)
The parameters c;  and  then become local complex coordinates on the space (C.16).
Note that as SU(3) objects,  and  carry the same degrees of freedom as the parameters
(; (
! + i 
B)(1;1); (
! + i 
B)(0;2)) used to parametrise the space of N = 1 eld con-
gurations in sections 3 and 3.2 of the main text, while the parameter c is associated to
Kahler transformations.
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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 ! C1(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  : 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 denes an
SO(6)  SO(6 + n) structure on E, with a local frame (E^+m; E^~m; E^ ) which can be built
from the physical elds (g;B; ;A) (see [43] for details). The presence of a single globally
dened 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 ~ 2 L
 ^3E dening such an N = 1 structure:
~ =
1
3!
(T
mnp)E^+m ^ E^+n ^ E^+p =
1
3!
p
ge2	mnpE^+m ^ E^+n ^ E^+p : (C.21)
As we still have (E^+m) = e^
+
m, the argument of appendix C.1 still holds to give us that
(~) = i e2	, and similar group theoretical reasoning leads us to conclude that both ~
and e2	 = 
 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)
h
(3;1)+2  (3;1)2
i
R

h
(3;6 + n)+1  (3;6 + n)1
i
R
(C.23)
so that we can locally parametrise the orbit of ~ via
~0 = ece+ 
 ~ c 2 C  2 (3;1)+2  2 (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 a
, that is the (0; 1)-form part of the deformation of the gauge eld. These
then match the parametrisation of the o-shell N = 1 eld space of section 3 as employed
in section 4.3.
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C.3 The o-shell hermitian structure on V
The o-shell parameter space of the Hull-Strominger system is the space of SU(3) struc-
tures, B elds and gauge elds 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 o-shell.
The gauge group G is a compact unitary real Lie group, which has complex represen-
tations. The vector bundle VC is a complex vector bundle with structure group GC, the
complexication 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 complexied group GC. Clearly these two sets of frames
are dierent for the simple reason that there are no holomorphic maps into the real group G.
From the complex vector bundle VC, we can dene 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
denite metric on V which pairs VC and V C, such that in the special frames alluded to
above its component matrix takes the canonical form
h =
1
2
 
0 
 0
!
(C.26)
where ;  = 1; : : : ; dimC VC are indices for VC and ;  = 1; : : : ; dimC VC are indices for V C.
O-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 dene 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 o-shell.14 This
hermitian structure simply says that VC denes a real vector bundle V with a compact
unitary structure group.
The physical connection A is a local section of

1(X; EndV )  
1(X; g): (C.27)
The almost complex structure denes 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 complexied Lie algebra, i.e. 
(1;0)(X; gC) and 

(0;1)(X; gC), as these are
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.
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dened via tensor products over the eld C. Naively, the connection has four parts with
indices as
(Aa)

 (Aa)


(Aa)

 (Aa)


(C.28)
However, as the connection is real hermitian, any one of them denes the rest via com-
plex conjugation and multiplication by the hermitian metric on VC. Explicitly, the real
condition xes 
(Aa)



= (Aa)

 ;

(Aa)



= (Aa)

 ; (C.29)
while the hermitian condition xes

(Aa)



= h(Aa)h;

(Aa)



= h(Aa)h: (C.30)
Together, these allow us to determine all parts of the connection given only (Aa)

 ; for
example we have
(Aa)

 = h(Aa)h: (C.31)
On-shell, the holomorphic structure means that, in each patch, one can choose a GC
gauge where (Aa)

 = 0. O-shell we cannot do this in general, but we can use the
previous identities to write any formula purely in terms of (Aa)

 . For example, the terms
appearing in the superpotential involve traces, which simplify using identities like
(Aa)

(Ab)

 = (Ab)

(Aa)

 : (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
!CS(A) = tr

A ^ dA+ 2
3
A ^A ^A

= 2

A ^ dA + 2
3
A ^A
 ^A


: (C.33)
so that !CS(A) ^ 
 features only the components (Aa) . Equivalently we could have
written this expression purely in terms of V C indices or some combination of the two |
this freedom re
ects 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 eld 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].
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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

e2! ^ !
 = 0; (D.1)
! ^ ! ^ F = 0: (D.2)
The rst 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 con-
ditions 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 compactications without bundles. The D-term condition of rel-
evance is the conformally balanced condition (D.1). Consider rst a massless deformation
of this condition so that y0 = (x0; 0) satises 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 ! and the dilaton  are linked
via the SU(3) normalisation condition
i
8

 ^ 
 = 1
6
e4! ^ ! ^ !: (D.3)
Remembering 
!(0;2) = 0, 
!(1;1) = ix and 
!(0;2) = {! i {x, a massless holomorphic
variation of this condition gives
e4
0
1
3!
! ^ ! ^ ! = 1
3!
! ^ ! ^ !  i
2
! ^ ! ^ x0  1
2
! ^ 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)

0 =  (1) + 
2 (2) + : : : (D.6)
with a corresponding expansion for x and . At rst and second order the SU(3) normali-
sation condition xes
(1) = 
3
4
i!yx(1); (D.7)
(2) = 
3
4
i!yx(2) +
1
8
x(1)yx(1): (D.8)
15As we mention in the main text, for an N = 1 supersymmetric theory in four dimensions, supersym-
metry 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 nd 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.
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This continues to higher order | the deformation of the dilaton at nth order is xed by
the non-primitive part of x(n) and the lower-order elds x(i) for i < n.
Now consider a massless holomorphic deformation of the conformally balanced condi-
tion. We expand in  and separate into complex type. At O(1) the deformation of the
conformally balanced condition reduces to
@

i e2! ^ x(1) + (1)e2! ^ !

= 0; (D.9)
@

i e2! ^ x(1) + (1)e2! ^ !

= @

e2! ^ {(1)!

: (D.10)
Note now that the Gauduchon metric dened by ~! = e! is balanced as ~!^ ~! 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)
We also use that the Hodge stars16 on a p-form are related by
? = e(3p)~?: (D.12)
Using this we can write the previous equations as
@
~y
X(1) = 0; (D.13)
@
~yX(1) = i @
~y
{(1) ~!; (D.14)
where we have used the relation between the trace of x(1) and (1) given in (D.7), and we
have dened
X(1) = e


x(1) 
1
2
~!yx(1) ~!

: (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
@
~y@
~y
X1 = 0; (D.16)
@@
~y
X1 = 0; (D.17)
@@
~yX1 = i @@
~y
{1 ~!: (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
16We are using the convention for the Hodge star where  ^ ? = y vol so that ?! = 1
2
! ^ ! and
?
 = i 
. The dual of a primitive (1; 1)-form p satisfying !yp is ?p = ! ^ p. We also have
?20 = 20^! where 20 is a (2; 0)-form. This choice satises ?2 = (1)p on a p-form. The adjoint Dolbeault
operators are dened by @y = ?@?, 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.
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element in @
(0;1)(X) + @
(1;0)(X). We make a simplication: 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
@
(0;1) + @
(1;0) = @@
(0;0)  @@~y
(2;0)  @@~y
(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 (1) (which does not appear explicitly in the
rst-order conformally balanced condition).
We start by shifting x(1) by @@(1), where (1) is a function. A short calculation
shows that equation (D.16) becomes
@
~y@
~y

e

x(1) 
1
2
~!yx(1) ~!

= @
~y@
~y
e@@(1)

+
1
2
~
@

e ~
@(1)

: (D.21)
One can check that the operator acting on (1) is a positive semi-denite 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 (1). This xes the @@-gauge symmetry
of x(1).
Next consider a shift x(1) by @@~y(1),18 where (1) is a (2; 0)-form. A short calculation
shows that (D.17) becomes
@@
~y

e

x(1) 
1
2
~!yx(1) ~!

= @@
~y
(e@@~y(1)): (D.22)
Again, one can check that the operator acting on (1) is positive semi-denite, self-adjoint
and elliptic so that (D.1) can always be solved for by a choice of (1). This xes the
@@
~y-gauge symmetry of x(1).
Finally consider a shift of x(1) by @@
~y
(1), where (1) is a (0; 2)-form. A short calcu-
lation shows that (D.18) becomes
@@
~y

e

x(1) 
1
2
~!yx(1) ~!

 i @@~y({(1) ~!) = @@
~y

e@@
~y
(1)

: (D.23)
As {(1)! is a (0; 2)-form and we assume H
(0;2)(X) vanishes, @@
~y
{(1)! is actually @@
~y-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-denite as before | it is simple to check that
this is the case. This xes the @@
~y
-gauge symmetry of x(1).
What happens at higher orders in ? At second order we have
@
~y
(X(2) +A(2)) = 0; (D.24)
@
~y(X(2) +A(2)) = i @
~y
{(2) ~! + @
~y
B(2); (D.25)
18One should think of this x(1) as already gauge transformed to solve the previous condition.
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where
X(2)(x(2)) = e
(x(2) 
1
2
~!yx(2) ~!): (D.26)
A(2)(x(1)) = i e


3
8
(!yx(1))2! +
1
4
x(1)yx(1) ! +
1
2
!yx(1) x(1) + ?(x(1) ^ x(1))

;
(D.27)
B(2)(x(1); (1)) = e


1
2
!yx(1) {(1)! + {(1)x(1) + ?(x(1) ^ {(1)!)

: (D.28)
We see that X(2) depends only on the second-order correction to x0 while A(2) and B(2) are
xed by the rst-order terms (which should be thought of as gauge transformed to solve
the rst-order conditions). Again (D.24) and (D.25) are equivalent to
@
~y@
~y
(X(2) +A(2)) = 0; (D.29)
@@
~y
(X(2) +A(2)) = 0; (D.30)
@@
~y(X(2) +A(2)) = i @@
~y
{(2) ~! + @@
~y
B(2): (D.31)
As X(2) is a function of x(2) alone, we can perform gauge transformations of x(2) without
aecting A(2) and B(2). Generically these gauge transformations will break the SU(3)
normalisation condition, but we can always shift (2) to compensate for this (which is what
we have implicitly done by eliminating (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) xed.
Again, one can always nd a gauge transformation of x(n) that solves these conditions.
From this we conclude that the D-term conditions are gauge xing 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 End V or EndVC. We choose to work with End VC 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
! ^ ! ^ F / ~ ^ F = 0: (D.32)
where we have dened ~ = 12 ~! ^ ~!. A massless holomorphic deformation of this condi-
tion reads

0 ~ ^ F + (~+ 
0 ~) ^ @A0 = 0: (D.33)
Note that 
0 ~ is completely determined by x0.
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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 com-
pactications (see [8, 11] and references therein). Their appearance can be understood in
the context of the innitesimal moduli problem as modding out by D-exact terms [14, 26],
see also appendix E.
The equations of motion Dy0 = 0 admit gauge transformations of x0 and 0:
x0 = @a0 + @b0 + trF
0;
0 = @A
0:
(D.34)
Upon expanding x0 and 0 in powers of , the rst order contributions to (D.33) are
0 = i ex(1) ^ ~! ^ F + ~ ^ @A(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 rst order) we have
Cy(1) =

@
~y

e(x(1) 
1
2
~!yx(1) ~!)

+

@
~y

e

x(1) 
1
2
~!yx(1) ~!

 i @~y{(1) ~!

;
(D.37)
Y y(1) = i ex(1) ^ ~! ^ F + ~ ^ @A(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+ trF
:
Dy1 = 0 = D(y(1) + @a(1) + @b(1) + trF
(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):
y0(1) = y(1) + @(1) + @(1): (D.41)
This still solves the equations of motion | Dy0(1) = 0 | but aects the remaining two
conditions:
Cy0(1) = Cy(1) + C(@(1) + @(1)): (D.42)
We showed in the previous section that there is always a choice of  and  that solves this
condition, that is
Cy(1) = C(@+ @) ) Cy0(1) = 0: (D.43)
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At this point we have a solution y0(1) that solves Dy
0
(1) = Cy
0
(1) = 0 but generically has
Y y0(1) 6= 0. Note also that the forms  and  are xed by y(1). We now want to show that
we can always nd a solution to Y y = 0 using gauge transformations. Let us dene
y00(1) = y
0
(1) + @a(1) + @b(1) + trF
(1) + @A
(1): (D.44)
This solves Dy00(1) = 0 | what happens to the other two conditions?
Cy00(1) = C(@a(1) + @b(1)) + C(trF
(1)); (D.45)
Y y00(1) = Y y
0
(1) + Y (@a(1) + @b(1)) + Y (trF
(1)) + Y (@A
(1)): (D.46)
Again, we can solve Cy00(1) = 0 for any 
(1) by choosing a and b appropriately. We now
want to see if we can solve Y y00(1) = 0 using the gauge freedom in 
(1).
Note that C picks out the (@ + @)-exact terms of the input (and has no kernel acting
on this) so that Cy00(1) = 0 is equivalent to
@a(1) + @b(1) +

trF
(1)

@+@
= 0: (D.47)
Using this we have
Y y00(1) = Y y
0
(1) + Y (P[trF
(1)]) + Y (@A
(1)); (D.48)
where P projects onto 
(1;1)(X)nfim @ + im @g. Explicitly we have19
Y y00(1) = i ~?

eF~]yx0(1) + @
~y
A
0
(1) + e
F~]yP[trF
(1)] + ~
A
(1)


: (D.49)
One can check that the operator acting on 
(1) is a positive semi-denite elliptic operator
with trivial kernel, so one can always nd a 
(1) that solves Y y
00
(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

(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 eld redenitions (at least to O(0)) [98]. Another way
of saying this is that the hermitian Yang-Mills condition for TX is automatically satised
provided the other supersymmetry conditions are solved and one chooses the appropriate
connection, which at O(0) is the connection r. Though we do not have a proof, we
expect this will hold to higher order in 0 when one chooses the connection on TX to be
an instanton.
19Here we have used ~?1 = 1
6
~!3, ~?(a11 ^ ~! ^ b11) = b~]11ya11 if b11 is a primitive (1; 1)-form (like F ), and
~?(~ ^ a11) = ~!ya11. We use ~] to denote raising with the Gauduchon metric.
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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 rst
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 =
M
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 y00(1))(i) = e trF (i)
~]yx0(1) + @
~y
A tr
0
(1) + e
 trF (i)~]yP[trF (i)
(i)(1)] + ~
 tr 
(i)
(1):
(D.51)
It is simplest to show that this can be solved by reintroducing the 0 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 0, equation (D.51) becomes
 i ~? Y y00(1) = eF
~]yx0(1) + @
~y
A
0
(1) +
1
4
0 e F~]yP[F
(1)] + ~

(1): (D.52)
We can now expand this equation in 0. We expand the deformations x0(1) and 
0
(1), together
with the gauge parameter 
(1), while keeping the background geometry xed. The potential
non-zero contributions to 
(1) are then

(1) =
1
0

(1)
(1) + 
(0)
(1) + 
0 
(1)(1) + : : : (D.53)
It follows from (D.52) that 
(1)
(1) must be constant. The expression we get at zeroth order
in 0 is
 i ~? Y y00(0)(1) = eF
~]yx0(0)(1) + @
~y
A
0(0)
(1) +
1
4
e F~]yF 
(1)(1) + ~
A
(0)
(1) = 0; (D.54)
where we note that
P[F
(1)(1) ] = F
(1)
(1) ; (D.55)
for constant 
(1)
(1) . The constant part of (D.54) can now be obtained by integrating over
X. Doing so, we get
 i
Z
X
~? Y y00(0)(1) =
Z
X
eF~]yx0(0)(1) +
1
4

(1)
(1)
Z
X
e F~]yF; (D.56)
This determines 
(1)
(1) as a function of x
0(0)
(1). The remaining non-constant part can then be
solved for modulo a constant term, which determines 
(0)
(1) .
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It is easy to see that this can be continued to higher orders in 0. At the next order, we
use the so-far undetermined constant piece of 
(0)
(1) to x the constant piece of the rst-order
equation. This determines 
(0)
(1) , which is now independent of 
(1)
(1) . We then solve for the
non-constant piece, determining 
(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 rst 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  1
2
[y; y] 1
2
@b = 0; (D.57)
@b 1
2
ya@ab+
1
3!
hy; [y; y]i = 0; (D.58)
@{
 = 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  = (; ) 2 Y0
Y = `1() + `2(; Y ) 1
2
`3(; Y; Y )
=

D+
1
2
@ + [; y]; 0


+

0; @ +
1
2
(h; @bi  hy; @i) 1
3
(
1
2
h; [y; y]i+ hy; [y; ]i)
:
(D.60)
Now expand Y and  in powers of . At rst order the equations of motion and gauge
transformations are
Dy(1) 
1
2
@b(1) = 0; y(1) = D(1) +
1
2
@(1); (D.61)
@b(1) = 0; b(1) = @(1); (D.62)
@{(1)
 = 0: (D.63)
Exactly as before, x(1) and (1) have a gauge symmetries that allow us to solve the D-term
conditions to rst order.
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At second order we have
Dy(2) 
1
2
[y(1); y(1)]
1
2
@b(2) = 0; y(2) = D(2) +
1
2
@(2) + [1; y1]; (D.64)
@b(2) 
1
2
ya(1)@ab(1) = 0; b(2) = @(2) +
1
2
h(1); @b(1)i  hy(1); @(1)i
; (D.65)
@{(2)
 = 0: (D.66)
Here (y(1); b(1)) are the gauge-transformed elds that solve the rst-order D-term condi-
tions. Furthermore 1 is xed by these rst-order conditions. We note that the second
order eld y(2) is shifted by the rst order gauge transformation by the term [1; y1]. This
means it is y(2) [(1); y(1)] that appears in the second-order D-term condition, which then
admits shifts by terms of the form D(2) +
1
2@(2) which we may use to solve the D-term
conditions just as in the rst order case, xing (2) and (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 n
th order.
Solving the D-term conditions thus xes 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] 2 H(0;1)D ( ~Q); y0 = (0; 0; x0): (E.1)
The massless moduli have been analysed in great detail in the [14, 15, 31] | what follows
is a short review. The rst thing to notice is that
@0 = 0: (E.2)
The elds 0 parametrise the complex structure moduli which, modulo gauge symmetries,
take values in the cohomology
[0] 2 H(0;1)@ (T
(1;0)X) = H(2;1)
@
(X): (E.3)
Recall that we also impose the extra condition
@{0
 = 0: (E.4)
This condition can be viewed as an equation of motion for the eld 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 inte-
grable 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 con-
dition [14].
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Next we must solve the equation
@A0 + Fb ^ b0 = 0: (E.5)
This is then the familiar condition that the complex structure deformations a should be
in the kernel of the Atiyah map
F : H(0;1)@ (T
(1;0)X)! H(0;2)
@A
(EndVC); (E.6)
where F is given by contraction with F as in (E.5). The elds 0 now decompose as
0 = 
c
0 + 
0
0 ; (E.7)
where c0 is closed and 
0
0 is uniquely determined by the complex structure moduli through
equation (E.5). Modulo gauge symmetries, we have
[c0] 2 H(0;1)@A (EndVC); (E.8)
which we refer to as gauge moduli.
The nal equation we must solve is
@x0 a + i(@!)eac dz
c ^ e0  tr(Fa ^ 0) = 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(@!)eac dz
c ^ e0  tr(Fa ^ 0) = @ab; (E.10)
for some b 2 
(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 chang-
ing this equation. Hence H
(0;2)
@ (X), i.e. anti-holomorphic sections of 

(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 dene the maps
H : kerF ! H(0;2)(T (1;0)X =@-exact); (E.11)
F : H(0;1)@A (EndVC)! H
(0;2)(T (1;0)X =@-exact): (E.12)
The maps are given by
H(0)a = i(@!)eac dzc ^ e0  tr(Fa ^ 00 ); (E.13)
F(c0)a =  tr(Fa ^ c0): (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
j kerHj+ j kerFj+ j imH \ imFj: (E.15)
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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 0 which remain massless are those which satisfy
H(0)  imF: (E.16)
The remaining massless bundle moduli c0 then need be in the kernel of F.
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] 2 H(0;1)@ (T
(1;0)X =@-exact): (E.17)
These are the elds conventionally referred to as Kahler moduli or hermitian moduli. As it
turns out, this cohomology forms a subset of the Aeppli cohomology [99]. Indeed, in order
to dene an element of H
(0;1)
@
(T (1;0)X =@-exact), we need that
@x0 = @; (E.18)
for some  2 
(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 @ = 0, which
by H(0;2)(X) = 0 implies that  = @ for some  2 
(0;1)(X). Let us now use the freedom
to shift x0 by a @-exact term @ to get
@x0 = @(@+ @) : (E.19)
We can choose  such that the right-hand side vanishes. This is equivalent to @+@ = 0,
which by the Hodge decomposition (D.20) xes the @@
~y
-exact part of @. The remaining @-
exact part of @ 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 compactication.
We now compare this with the cohomology H
(0;1)
D
( ~Q). Using long exact sequences in
cohomology, it is a straightforward exercise to show that
H
(0;1)
D
( ~Q) =
H
(0;1)
@
(T (1;0)X =@-exact)
imF  ker(H+ F): (E.21)
This is almost as expected, except that we are modding out the hermitian moduli by
elements in imF, where we now view F as
F : H0(EndVC)! H(0;1)@ (T
(1;0)X =@-exact): (E.22)
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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 =
L
i V
(i)
C for stable slope-zero factors V
(i)
C . Thus we have
imF = spanfc1F1 + : : :+ cnFng; (E.23)
for constants ci such that
P
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.
Open Access. This article is distributed under the terms of the Creative Commons
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