Small gauge transformations and universal geometry in heterotic theories
Jock McOrist1 and Roberto Sisca2
Department of Mathematics2
University of Surrey
Guildford, GU2 7XH, UK

11footnotetext<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

We explore the observation that small gauge transformations in heterotic
supergravity are related to a choice of connection on the moduli space. We
start by describing the action of small gauge transformations in heterotic
supergravity. We show a convenient gauge fixing is holomorphic gauge together
with residual gauge fixing involving the holomorphic top form. This is related
to a choice of holomorphic structure and Lee form on the universal bundle. The
field strength for this connection is related to second order deformation
theory and we point out it is generically the case that higher order
deformations do not commute. At the level of supergravity, this gauge fixing
allows us to determine the Hodge decomposition, and so a map between
parameters and field deformations. At large radius, we give a prescription for
how to compute the ${\alpha^{\backprime}\,}$–corrections to the Hodge
decomposition and apply this to the moduli space metric.

###### Contents

  1. 1 Introduction
  2. 2 Small gauge transformations and heterotic moduli
    1. 2.1 Complex structure $J$
    2. 2.2 Holomorphic $(3,0)$-form $\Omega$
    3. 2.3 The hermitian form $\omega$
    4. 2.4 The gauge field $F$
    5. 2.5 The three-form $H$
    6. 2.6 Summary of small transformations on fields
    7. 2.7 The equations of motion
  3. 3 Gauge fixing
    1. 3.1 A warm–up
    2. 3.2 Holomorphy and holomorphic gauge
    3. 3.3 Residual gauge symmetries
  4. 4 The Hodge decomposition and the moduli space metric
    1. 4.1 The equations of motion
    2. 4.2 The Hodge decomposition in holomorphic gauge
    3. 4.3 Comments
    4. 4.4 An application to the moduli space metric
  5. 5 First order universal geometry and gauge transformations
    1. 5.1 Why universal geometry?
    2. 5.2 Small gauge transformations on a fixed manifold $X$
    3. 5.3 Small gerbes
    4. 5.4 Small diffeomorphisms
    5. 5.5 Holomorphy
  6. 6 A taste of second order universal geometry
  7. 7 Conclusions
  8. A Complexified tangent spaces and gauge groups
  9. B Gauge fixing small diffeomorphisms on Calabi–Yau manifolds
    1. B.1 Real deformations
    2. B.2 Holomorphic deformations
    3. B.3 A third way
    4. B.4 Conformally balanced condition
  10. C Hodge duals and contractions
  11. D Connection symbols on $X$

##  1 Introduction

In a recent series of papers [1, 2, 3] we developed a theory of derivatives
for describing the moduli space $M$ of heterotic vacua realising $N=1$
supersymmetry in a $\mathbbl{R}^{1,3}$ spacetime, and used it to compute
various properties such as the natural Kähler metric on $M$. We work at large
radius, in which the supergravity approximation is valid and so the heterotic
flux $H$ is subleading in ${\alpha^{\backprime}\,}$. These theories are
defined by a complex $3$-fold $X$ with $c_{1}(X){\;=\;}0$ and a hermitian form
$\omega$ as well as a holomorphic vector bundle $\mathpzc E\to X$ with a
connection $A$ satisfying the Hermitian–Yang–Mills equation and a well-defined
three-form $H$. The anomaly relation yields a modified Bianchi identity for
$H$.

${\text{d}}H~{}=-\frac{{\alpha^{\backprime}\,}}{4}\Big{(}\text{Tr}\hskip
2.0pt(F^{2})-\text{Tr}\hskip 2.0pt(R^{2})\,\Big{)}~{},$ (1.1)

while the supersymmetry relation implies the manifold is non-Kähler 111In the
above equations $R$ is the curvature two-form evaluated with the Hull
connection $\Theta^{\text{H}}=\Theta^{\text{\tiny LC}}+\frac{1}{2}H$. We
denote by $x^{m}$ the real coordinates of $X$ and its complex coordinates by
$(x^{\mu},x^{\overline{\nu}})$. The coordinates along $M$ are denoted by
$y^{a}$, and complex coordinates by $(y^{\alpha},y^{\overline{\beta}})$.. In
the following we will generally omit the wedge product symbol ‘$\wedge$’
between forms, unless doing so would lead to ambiguity.

$H\;=\;{\text{d}}^{c}\omega~{},\quad{\text{d}}^{c}\omega\;=\;\frac{1}{3!}J^{m}J^{n}J^{p}({\text{d}}\omega)_{mnp}~{},$
(1.2)

where $J^{m}=J_{n}{}^{m}{\text{d}}x^{n}$ is a 1-form constructed out of the
complex structure. For a fixed topology, these equation have parameters which
describe families of heterotic vacua. The parameters are also coordinates for
the manifold $M$. As in [4] we take a constant dilaton, which as argued there,
is a consequence of demanding a well-defined supergravity approximation.

To warm-up, consider $X$ to be fixed so that $M$ is the moduli space of
connections $A$. This is closely related to the context of the
Kobayashi–Hitchin correspondence [5]. It is also the context of a heterotic
theory in the ${\alpha^{\backprime}\,}\to 0$ limit. The connection $A$
transforms under gauge symmetries

$A\to\;{}^{\Phi}A\;=\;\Phi\,A\,\Phi^{-1}-({\text{d}}\Phi)\,\Phi^{-1}~{}.$
(1.3)

We wish to study deformations $\delta A$. The background gauge principle
implies two types of transformation properties for $\delta A$:

$\delta A\to\;^{\Phi}\\!\delta A\;=\;\Phi\,\delta A\,\Phi^{-1}~{},\qquad\delta
A\to\;\\!\delta A+{\text{d}}_{A}\phi~{},\qquad\phi\ll 1~{}.$ (1.4)

The former is a classical symmetry of the background, while the latter is a
symmetry of the quantum theory, and we refer to as a small gauge
transformation.

To make progress in understanding the moduli space $M$, we need a relation
between variations of parameters and deformations of fields called the
Kodaira–Spencer map — see [6] for details — in such a way that the
transformation laws of $\delta A$ are respected. A solution implementing the
first symmetry property was described in [2]. One introduces a connection
$\Lambda{\;=\;}{\Lambda}_{a}\,{\text{d}}y^{a}$, so that it has legs along the
moduli space $M$ and also transforms in a manner parallel to $A$

$\Lambda\to\;^{\Phi}\Lambda_{a}\;=\;\Phi\,\Lambda_{a}\,\Phi^{-1}-({\partial}_{a}\Phi)\,\Phi^{-1}~{}.$
(1.5)

The deformation and its relation to parameters is given by $\delta A=\delta
y^{a}\,{\mathfrak{D}}_{a}A$ where ${\mathfrak{D}}_{a}A$ contains two
connections $A$ and $\Lambda$

${\mathfrak{D}}_{a}A\;=\;\
{\partial}_{a}A-{\text{d}}_{A}\Lambda_{a}~{},~{}~{}~{}\text{where}~{}~{}~{}{\text{d}}_{A}\Lambda_{a}\;=\;{\text{d}}\Lambda_{a}+[A,\,\Lambda_{a}]~{}.$
(1.6)

It is easy to check that under (1.3)–(1.5) the covariant derivative
${\mathfrak{D}}_{a}A$ transforms homogeneously as required. The small gauge
transformation property of (1.6) was not studied in [1, 2, 3] in any detail.
In this paper we explore this in more detail.

We observe that field fluctuations depending holomorphically on parameters,
such as ${\mathfrak{D}}_{\overline{\alpha}}\mathcal{A}=0$, is a gauge fixing
which we call holomorphic gauge. There is a residual gauge freedom which can
be gauge fixed by the condition $\delta\Omega^{(3,0)}=k\Omega$, where $\Omega$
is the holomorphic top–form; equivalently the $(2,1)$–forms describing
deformations of complex structure are ${\partial}$–closed in this gauge. We
then apply the equations of motion in this gauge. This allows us to determine
the Hodge decomposition, showing that holomorphic gauge is not the same as
harmonic gauge in general. The equations are highly coupled. We give a
prescription for solving these equations perturbatively in
${\alpha^{\backprime}\,}$ and apply this to the moduli space metric.

There is a close connection with gauge fixing and the theory of universal
geometry developed in [1]. We start from the observation that under a small
deformation $\Lambda_{a}\to\Lambda_{a}-\phi_{a}$, we have

${\mathfrak{D}}_{a}A\to{\mathfrak{D}}_{a}A+{\text{d}}_{A}\phi_{a}~{},$

which is exactly a small gauge transformation. So if we fix small gauge
transformations, we must have fixed $\Lambda$ and its field strength, at least
to some degree. In the space $\mathpzc A$ of all connections,222The manifold
$M$ is a subspace of the quotient $\mathpzc A/G$ where $G$ is the action of
the Lie group in (1.3).  background gauge transformations move a basepoint
$A\in\mathpzc A\to{}^{\Phi}A\in\mathpzc A$. The points ${}^{\Phi}A$ form the
gauge orbit. The tangent space at $A\in\mathpzc A$ has an orthogonal
decomposition into a vertical subspace tangential to gauge orbits and a
horizontal subspace spanned by $\\{{\mathfrak{D}}_{a}A\\}$, which are
precisely the deformations of $A$. Furthermore, the choice of horizontal is
determined by the connection $\Lambda$, and that choice is preserved under the
action of background gauge transformations. The choice of $\Lambda$ therefore
should be of no physical consequence in the spacetime theory. This is
illustrated in Figure 1.

Figure 1: The moduli space $M$ of connections and the gauge orbits. The change
of base point $A\to{}^{\Phi}A$ is a background gauge transformation; the
change in deformation $\delta A\to\delta A^{\prime}$ is a small gauge
transformation. The choice of connection on $M$ is associated with a choice of
deformation $\delta A$. The vertical $G$–orbits are generated by
${\text{d}}_{A}\phi_{a}$.

Consider $\mathbbl{X}=X\times M$ and on it a connection
$\mathbbl{A}=A+\Lambda$ for a universal bundle $\mathpzc U$, with a field
strength $\mathbbl{F}=\mathbbl{d}\mathbbl{A}+\mathbbl{A}^{2}$. Under
decomposition of legs, there is $F$ with legs purely along $X$; a mixed term
$\mathbbl{F}_{am}{\text{d}}x^{m}{\;=\;}{\mathfrak{D}}_{a}A$, which is exactly
the covariant derivative; and finally there is a third term
$\mathbbl{F}_{ab}$, whose legs lie purely along $M$. In light of the previous
paragraph, $\mathbbl{F}_{ab}$ influences the map between parameters $y^{a}$
and deformations $\delta A$. Indeed, in the toy example discussed above
holomorphic gauge at first order establishes
$\mathbbl{F}_{{\overline{\alpha}}{\overline{\mu}}}=0$. At second order in
deformations, we show there is an independent gauge fixing and that
holomorphic gauge here corresponds to
$\mathbbl{F}_{{\overline{\alpha}}{\overline{\beta}}}=0$. Hence, holomorphic
gauge to second order in deformation theory corresponds to the universal
bundle being holomorphic.

In this paper we extend this observation to the heterotic vacuum. The manifold
$X$ is not fixed and depends on parameters. We show this is forced upon us by
the equations of motion; the bundle never decouples. There are other gauge
symmetries to consider: diffeomorphisms of $X$ and transformations of the
$B$-field. The former imply $X$ is fibered over $M$, and in [1] we studied
this fibration labelled $\mathbbl{X}$. The latter imply the presence of a
2-gerbe $\mathbbl{B}$ on $\mathbbl{X}$. Consequently, we need to introduce
additional connections on $M$ and their role will be to define a relation
between parameters $y^{a}$ and deformations of fields. The connection
$c_{a}{}^{m}{\text{d}}y^{a}\otimes{\partial}_{m}$ is associated to
diffeomorphisms of $X$. Deformations of $c_{a}$ result in small
diffeomorphisms of tensor fields on $X$. Small gerbe transformations are
realised by deformations of an object
$\mathbbl{B}_{am}{\text{d}}y^{a}{\text{d}}x^{m}$.

##  2 Small gauge transformations and heterotic moduli

We derive the action of small diffeomorphisms, gauge transformations and small
gerbe transformations. Our derivation starts with small diffeomorphisms and
from the equations of motion we are able to deduce the action of small gauge
and small gerbes transformations on $\delta A$ and $\delta B$ respectively.

Suppose on $X$ we have a tensor field $T_{m_{1}\cdots m_{p}}$ which can
undergo a diffeomorphism

$T_{n_{1}\cdots
n_{k}}(\widetilde{x})\frac{\partial\widetilde{x}{}^{n_{1}}}{\partial
x^{m_{1}}}\cdots\frac{\partial\widetilde{x}{}^{n_{k}}}{\partial
x^{m_{k}}}\;=\;T_{m_{1}\cdots m_{k}}(x).$ (2.1)

For $\widetilde{x}=x+\varepsilon$ with $\varepsilon^{m}$ small

$\begin{split}T_{m_{1}\cdots m_{k}}(\widetilde{x})&~{}\simeq~{}T_{m_{1}\cdots
m_{k}}(x)+\varepsilon^{n}\,{\partial}_{n}T_{m_{1}\cdots
m_{k}}(x)+({\partial}_{m_{1}}\varepsilon^{n})T_{n\cdots
m_{k}}(x)+\cdots+({\partial}_{m_{k}}\varepsilon^{p})T_{m_{1}\cdots
p}(x)\\\\[3.0pt] &\;=\;T_{m_{1}\cdots
m_{k}}(x)+(\mathcal{L}_{\varepsilon}T)_{m_{1}\cdots m_{k}}(x)~{},\end{split}$
(2.2)

where $\mathcal{L}_{\varepsilon}$ is the Lie derivative taken with respect to
the vector $\varepsilon^{m}$. We have worked to first order in $\varepsilon$.
When studying deformations $\delta T$ of the tensor, the small diffeomorphisms
are regarded as unphysical and so in the physical theory we identify

$\delta T\sim\delta T+\mathcal{L}_{\varepsilon}\,T~{},$ (2.3)

Appendix §B carefully explains the gauge fixing of diffeomorphisms in the
study of the moduli space of Calabi-Yau manifolds in three different ways. We
now describe how this works for fields of heterotic to first order in
deformations.

###  2.1 Complex structure $J$

On $X$ there is an integrable complex structure $J$ and this facilitates
introducing holomorphic coordinates $x^{m}=(x^{\mu},x^{\overline{\nu}})$.
Deformations that modify complex structure can be expressed in terms of the
undeformed complex structure as

$\delta J\;=\;\delta
J_{\overline{\nu}}{}^{\mu}\,{\text{d}}x^{\overline{\nu}}\otimes{\partial}_{\mu}+\delta
J_{\nu}{}^{\overline{\mu}}\,{\text{d}}x^{\nu}\otimes{\partial}_{\overline{\mu}}~{}.$

To first order in deformation theory, demanding the Nijenhuis tensor be
preserved. When decomposed into type we get

${\overline{\partial}}(\delta J{}^{\mu})\;=\;0~{},\quad{\partial}(\delta
J{}^{\overline{\mu}})\;=\;0~{},$

where the notation here is as in the introduction $\delta J^{\mu}=\delta
J_{\overline{\nu}}{}^{\mu}{\text{d}}x^{\overline{\nu}}$. Small diffeomorphisms
induce an identification $\delta J\sim\delta J+\mathcal{L}_{\varepsilon}J$
where because $\delta J$ is a real tensor, the vector $\varepsilon$ must also
be real. Decomposing into complex type this equation becomes

$\delta J{}^{\mu}\sim\delta
J{}^{\mu}+2\text{i}\,{\overline{\partial}}\varepsilon^{\mu}\qquad\text{and}\qquad\delta
J{}^{\overline{\mu}}\sim\delta
J{}^{\overline{\mu}}-2\text{i}\,{\partial}\varepsilon^{\overline{\mu}}~{}.$

Hence, $\delta J_{{\overline{\nu}}}{}^{\mu}\,{\text{d}}x^{\overline{\nu}}\in
H_{\overline{\partial}}^{(0,1)}(X,\mathpzc T_{X}^{(1,0)})$. We expand in a
basis for the cohomology group and this defines the Kodaira–Spencer map
between tangent vectors and field variations

$\delta J_{{\overline{\nu}}}{}^{\mu}\;=\;\delta
y^{a}\big{(}2\text{i}\,\Delta_{a\,{\overline{\nu}}}{}^{\mu}\,\big{)}~{}.$

###  2.2 Holomorphic $(3,0)$-form $\Omega$

Consider the d-closed holomorphic $(3,0)$–form $\Omega$. A first order
deformation $\delta\Omega$ obeys three equations

${\partial}\,\delta\Omega^{(3,0)}\;=\;0~{},\qquad{\overline{\partial}}\,\delta\Omega^{(3,0)}+{\partial}\,\delta\Omega^{(2,1)}\;=\;0~{},\qquad{\overline{\partial}}\,\delta\Omega^{(2,1)}\;=\;0~{}.$

These equations are solved by

$\delta\Omega^{(3,0)}\;=\;\delta
y^{a}\left(k_{a}\Omega+{\partial}\xi^{(2,0)}_{a}\right)~{},\quad\delta\Omega^{(2,1)}\;=\;\delta
y^{a}\chi_{a}~{},\quad{\partial}\chi_{a}\;=\;\\!-{\overline{\partial}}{\partial}\xi_{a}^{(2,0)}~{},$
(2.4)

where $\xi^{(2,0)}_{a}$ are arbitrary $(2,0)$–forms, the $k_{a}$ are constant
over $X$ and $\chi_{a}=\Delta_{a}{}^{\mu}\,\Omega_{\mu}$ are
${\overline{\partial}}$–closed $(2,1)$–forms.

At this point it is convenient to describe a two–parameter family of
connections on $\mathpzc T_{X}$ given as follows

$\begin{split}\Theta^{(\epsilon,\rho)}{}_{\mu}{}^{\nu}{}_{\sigma}&=\Theta^{\text{\tiny
LC}}{}_{\mu}{}^{\nu}{}_{\sigma}+\frac{(\epsilon-\rho)}{2}\,H_{\mu}{}^{\nu}{}_{\sigma}~{},\\\\[2.84544pt]
\Theta^{(\epsilon,\rho)}{}_{\mu}{}^{\overline{\nu}}{}_{\overline{\sigma}}&=\Theta^{\text{\tiny
LC}}{}_{\mu}{}^{\overline{\nu}}{}_{\overline{\sigma}}+\frac{(\epsilon-\rho)}{2}\,H_{\mu}{}^{\overline{\nu}}{}_{\overline{\sigma}}~{},\\\\[2.84544pt]
\Theta^{(\epsilon,\rho)}{}_{\mu}{}^{\overline{\nu}}{}_{\sigma}&=0~{},\\\\[2.84544pt]
\Theta^{(\epsilon,\rho)}{}_{\mu}{}^{\nu}{}_{\overline{\sigma}}&=\Theta^{\text{\tiny
LC}}{}_{\mu}{}^{\nu}{}_{\overline{\sigma}}+\frac{(\epsilon+\rho)}{2}\,H_{\mu}{}^{\nu}{}_{\overline{\sigma}}\,~{},\end{split}$

where $\Theta^{\text{\tiny LC}}$ is the Levi–Civita connection. The Bismut
connection is given by $\Theta^{\text{\tiny B}}{=}\Theta^{(-1,0)}$, the Hull
connection by $\Theta^{\text{\tiny H}}{=}\Theta^{(1,0)}$ and the Chern
connection by $\Theta^{{\rm Ch}}{=}\Theta^{(0,-1)}$.

Recall that we also assume a $g_{s}$–perturbative string background so that we
can trust the ${\alpha^{\backprime}\,}$–expansion. It would be interesting to
understanding deviations from this assumption but doing so would likely need
technology beyond supergravity. Supersymmetry implies that
$\nabla^{\text{\tiny B}}_{\mu}\Omega{\;=\;}\nabla^{\text{\tiny
B}}_{\overline{\mu}}\Omega{\;=\;}0$. From this it follows that

$H_{\mu\nu}{}^{\nu}\;=\;0~{},\qquad{\partial}_{\mu}\left\|\Omega\right\|^{2}\;=\;0~{}.$

Using $H={\text{d}}^{c}\omega$, this means geometrically the Lee form
$\frac{1}{2}\omega^{mn}({\text{d}}\omega)_{mn}$ vanishes. The Lee form is a
1–form measuring the non–primitive part of ${\text{d}}\omega$. Such manifolds
are called balanced manifolds.

A small diffeomorphism acts as on the pure part as333Due to
$H_{\mu\nu}{}^{\nu}=0$ divergences of this vector with respect to Levi–Civita
and Bismut coincide $\nabla^{\text{\tiny
LC}}_{\nu}\varepsilon^{\nu}=\nabla_{\nu}^{\text{\tiny B}}\varepsilon^{\nu}$.
Furthermore, with the choice of $\varepsilon^{\nu}$ we find
$\nabla^{\text{\tiny
LC}}_{\nu}\,\varepsilon_{a}{}^{\nu}~{}=\\!-\frac{1}{3!\left\|\Omega\right\|^{2}}{\overline{\Omega}}^{\nu\rho\sigma}({\partial}\xi_{a}^{(2,0)})_{\nu\rho\sigma}~{}.$

$\delta\Omega^{(3,0)}\to\delta\Omega^{(3,0)}+{\partial}(\varepsilon^{\nu}\Omega_{\nu})\;=\;\delta\Omega^{(3,0)}+(\nabla^{\text{\tiny
LC}}_{\nu}\,\varepsilon^{\nu})\Omega~{}.$

###  2.3 The hermitian form $\omega$

There is a compatible hermitian form $\omega$. A real deformation of $\omega$
can be written as

$\delta\omega^{(2,0)}\;=\;\delta
y^{a}\Delta_{a}{}^{\overline{\mu}}\omega_{\overline{\mu}}~{},\quad\delta\omega^{(1,1)}=\delta
y^{a}({\partial}_{a}\omega)^{(1,1)}~{},\quad\delta\omega^{(0,2)}\;=\;\delta
y^{a}\Delta_{a}{}^{\mu}\omega_{\mu}~{},\quad\omega_{m}\;=\;\omega_{mn}{\text{d}}x^{n}~{}.$

It is subject to small diffeomorphisms

$\delta\omega\sim\delta\omega+\mathcal{L}_{\varepsilon_{a}}\,\omega\;=\;\delta\omega+\varepsilon_{a}{}^{m}({\text{d}}\omega)_{m}+{\text{d}}(\varepsilon_{a}{}^{m}\,\omega_{m})~{}.$

If ${\text{d}}\omega=0$ the manifold is Kähler and small diffeomorphisms
generate d–exact shifts of $\delta\omega$. In heterotic theories this is not
the case.

###  2.4 The gauge field $F$

The gauge field $A$ and its field strength $F={\text{d}}A+A^{2}$ also
transform under small diffeomorphisms. However this case is complicated by
gauge symmetries in which ${}^{\Phi}F=\Phi\,F\,\Phi^{-1}$ and $A$ transforms
according to (1.3). So in relating $F(x+\varepsilon)$ and $F(x)$ we need a
covariant generalisation of a Lie derivative, (2.2), which is

$F(\widetilde{x})~{}\simeq~{}F(x)+\varepsilon^{m}({\text{d}}_{A}F)_{m}+{\text{d}}_{A}(\varepsilon^{m}F_{m})~{}.$
(2.5)

Using the Bianchi identity we find an identification

$\delta F(x)\sim\delta F(x)+{\text{d}}_{A}(\varepsilon^{m}F_{m})~{}.$ (2.6)

We write $\mathcal{A}=A^{(0,1)}$ so that
$A=\mathcal{A}-\mathcal{A}^{\dagger}$. Holomorphy of $\mathpzc E$ means
$F^{(0,2)}={\overline{\partial}}_{\mathcal{A}}^{\,2}=0$. Taking the $(0,2)$
part of (2.6) and solving for $\delta\mathcal{A}$ gives

$\delta\mathcal{A}\sim\delta\mathcal{A}+\varepsilon^{\mu}F_{\mu}+{\overline{\partial}}_{\mathcal{A}}\phi~{},$

where $\phi$ is a section of ${\rm End}\,\mathpzc E$. In principle the last
term could be closed but not exact. However, this would represent a change in
moduli space coordinates (see below). This term is interpreted as an
independent gauge symmetry, small gauge transformations, and we see we arrived
at it for free by studying small diffeomorphisms.

We can expand the deformation in terms of covariant derivatives with respect
to parameters

$\delta\mathcal{A}\;=\;\delta
y^{a}\,{\mathfrak{D}}_{a}\mathcal{A}~{},\quad\text{where}\quad{}^{\Phi}{\mathfrak{D}}_{a}\mathcal{A}\;=\;\Phi\,{\mathfrak{D}}_{a}\mathcal{A}\,\Phi^{-1}~{}.$

The covariant derivatives both ensure the transformation law (1.4) is
satisfied and are representation of the Kodaira–Spencer map relating tangent
vectors on the moduli space to deformations of fields [2].

The fluctuation ${\mathfrak{D}}_{a}\mathcal{A}$ satisfies the Atiyah equation

${\overline{\partial}}_{\mathcal{A}}\left({\mathfrak{D}}_{a}\mathcal{A}\right)\;=\;\Delta_{a}{}^{\mu}F_{\mu}~{}.$
(2.7)

In terms of covariant derivatives, the gauge symmetry action becomes444At this
point we can explain the last term being exact. Suppose we write
${\mathfrak{D}}_{a}\mathcal{A}\sim\widetilde{{\mathfrak{D}}}_{a}\mathcal{A}={\mathfrak{D}}_{a}\mathcal{A}+\varepsilon^{\mu}F_{\mu}+\gamma_{a}$
where ${\overline{\partial}}_{\mathcal{A}}\gamma_{a}=0$. Suppose it is
non–trivial in cohomology and can be expanded in a basis
$\gamma_{a}=\gamma_{a}{}^{b}\,{\mathfrak{D}}_{b}\mathcal{A}$, where
${\mathfrak{D}}_{b}\mathcal{A}$ are basis elements for the cohomology group
and $\gamma_{a}{}^{b}$ is a parameter dependent matrix. Then,
${\mathfrak{D}}_{a}\mathcal{A}\sim\widetilde{\mathfrak{D}}_{a}\mathcal{A}=(\delta_{a}{}^{b}+\gamma_{a}{}^{b})\,{\mathfrak{D}}_{b}\mathcal{A}+\varepsilon_{a}{}^{\mu}F_{\mu}+{\overline{\partial}}_{\mathcal{A}}\phi_{a}~{}.$
This is a transformation law for a change of parameters and can be absorbed by
a redefinition of the $\delta y^{a}$.

${\mathfrak{D}}_{a}\mathcal{A}\sim{\mathfrak{D}}_{a}\mathcal{A}+\varepsilon_{a}{}^{\mu}F_{\mu}+{\overline{\partial}}_{\mathcal{A}}\phi_{a}~{},$
(2.8)

and it is a symmetry of (2.7) provided
$\Delta_{a}{}^{\mu}\sim\Delta_{a}{}^{\mu}+{\overline{\partial}}\varepsilon_{a}{}^{\mu}$.

In the literature a bundle modulus is typically associated to a fluctuation
${\mathfrak{D}}_{a}\mathcal{A}\in H^{1}(X,{\rm End}\,\mathpzc E)$. From (2.7)
these correspond to $\Delta_{a}=0$. While it may obvious to some readers, we
note this is only true in a particular gauge. A more invariant statement is
that a bundle modulus satisfies
${\overline{\partial}}_{\mathcal{A}}({\mathfrak{D}}_{a}\mathcal{A})=({\overline{\partial}}\kappa_{a}{}^{\mu})F_{\mu}$
for any $\kappa_{a}{}^{\mu}$. We will see in the heterotic theory that in fact
fluctuations ${\mathfrak{D}}_{a}\mathcal{A}$ are coupled to deformations of
the complex structure and hermitian structure of $X$.

###  2.5 The three-form $H$

Consider the three-form $H$

$H\;=\;{\text{d}}B-\frac{{\alpha^{\backprime}\,}}{4}\Big{(}{\text{CS}}[A]-{\text{CS}}[\Theta]\Big{)}~{},\quad{\text{CS}}[A]\;=\;\text{Tr}\hskip
2.0pt\\!\left(A{\text{d}}A+\frac{2}{3}\,A^{3}\right)~{},$ (2.9)

defined so that it satisfies the Bianchi identity (1.1). Here $\Theta$ is the
gauge potential for frame transformations, which we suppress for now. Under
background gauge transformations

${}^{\Phi}B\;=\;B-\frac{{\alpha^{\backprime}\,}}{4}\left\\{\text{Tr}\hskip
2.0pt(AY)-U\right\\}~{},\quad\frac{1}{3}\,\text{Tr}\hskip
2.0pt(Y^{3})\;=\;{\text{d}}U~{},$ (2.10)

where $Y=\Phi^{-1}{\text{d}}\Phi$ and ${\text{d}}U=\frac{1}{3}\text{Tr}\hskip
2.0ptY^{3}$.

The three–form $H$ is well-defined on $X$. A variation of it is

${\partial}_{a}H\;=\;{\text{d}}\mathpzc
B_{a}-\frac{{\alpha^{\backprime}\,}}{2}\text{Tr}\hskip
2.0pt\big{(}{\mathfrak{D}}_{a}AF\big{)}~{},$ (2.11)

where $\mathpzc B_{a}$ is defined as

$\mathpzc
B_{a}\;=\;{\mathfrak{D}}_{a}B+\frac{{\alpha^{\backprime}\,}}{4}\text{Tr}\hskip
2.0pt{(A\,{\mathfrak{D}}_{a}A)}-{\text{d}}\mathbbl{B}_{a}~{},$ (2.12)

and the field $\mathbbl{B}_{a}$ is a $1$–form on $X$. The definition of
${\mathfrak{D}}_{a}B$ and the symmetry properties of $\mathbbl{B}_{a}$ become
clearer in universal geometry. The transformation property of
${\mathfrak{D}}_{a}B$ under background transformations mimics that of $B$ in
(2.10) and is discussed in [2]:

${}^{\Phi}{\mathfrak{D}}_{a}B\;=\;{\mathfrak{D}}_{a}B-\frac{{\alpha^{\backprime}\,}}{4}\Big{(}\text{Tr}\hskip
2.0pt{({\mathfrak{D}}_{a}A\,Y)}-\mathfrak{U}_{a}\Big{)}~{},\quad{\text{d}}\mathfrak{U}_{a}\;=\;0~{}.$
(2.13)

We infer the transformation law for $\mathpzc B_{a}$ by using (2.11) and the
transformation law for the gauge field (2.8) to find

${\partial}_{a}H\sim{\text{d}}\Big{(}\widetilde{\mathpzc}B_{a}-\frac{{\alpha^{\backprime}\,}}{2}\text{Tr}\hskip
2.0pt\big{(}\phi_{a}F\big{)}\Big{)}-\frac{{\alpha^{\backprime}\,}}{2}\text{Tr}\hskip
2.0pt\Big{(}({\mathfrak{D}}_{a}A+\varepsilon_{a}{}^{m}F_{m})F\Big{)}~{},$

where $\widetilde{\mathpzc}B_{a}$ is to be determined. On the other hand,
(2.3) implies

${\partial}_{a}H\sim{\partial}_{a}H-\frac{{\alpha^{\backprime}\,}}{2}\varepsilon_{a}{}^{m}\,\text{Tr}\hskip
2.0pt(F_{m}F)+{\text{d}}(\varepsilon_{a}{}^{m}H_{m})~{}.$ (2.14)

Comparing the last two equations gives

$\mathpzc B_{a}\sim\widetilde{\mathpzc}B_{a}\;=\;\mathpzc
B_{a}+\varepsilon_{a}{}^{m}H_{m}+\frac{{\alpha^{\backprime}\,}}{2}\text{Tr}\hskip
2.0pt\big{(}\phi_{a}F\big{)}+{\text{d}}\mathfrak{b}_{a}~{},$ (2.15)

where $\mathfrak{b}_{a}$ is a real 1–form. This equation holds up to a gauge
invariant d–closed term. We have taken this to be exact
${\text{d}}\mathfrak{b}_{a}$. As explained previously, any non-trivial element
of $H^{2}(X,\mathbbl{R})$ can be absorbed by a redefinition of the parameter
space coordinates. In the ${\alpha^{\backprime}\,}\to 0$ limit, fluctuations
of the $B$–field admit a symmetry $\delta B\sim\delta
B+{\text{d}}\mathfrak{b}$ for some $1$–form $\mathfrak{b}$. The shift by
${\text{d}}\mathfrak{b}_{a}$ is the generalisation to heterotic theories and
we label it a small gerbe transformation.

Consider again the covariant derivative ${\mathfrak{D}}_{a}B$ in (2.12)
constructed so that $\delta B=\delta y^{a}\,{\mathfrak{D}}_{a}B$. Just as we
could infer the small transformation law
${\mathfrak{D}}_{a}A\sim{\mathfrak{D}}_{a}A+{\text{d}}_{A}\phi_{a}$ by
considering the infinitesimal limit $\Phi=1-\phi$ in (1.3), we could apply the
same limit to (2.10) to read off

${\mathfrak{D}}_{a}B\sim{\mathfrak{D}}_{a}B+\frac{{\alpha^{\backprime}\,}}{4}\text{Tr}\hskip
2.0pt\Big{(}A\,{\text{d}}\phi_{a}\Big{)}~{}.$ (2.16)

For future reference, this means the following combination transforms as

${\mathfrak{D}}_{a}B+\frac{{\alpha^{\backprime}\,}}{4}\text{Tr}\hskip
2.0pt(A{\mathfrak{D}}_{a}A)\sim{\mathfrak{D}}_{a}B+\frac{{\alpha^{\backprime}\,}}{4}\text{Tr}\hskip
2.0pt(A{\mathfrak{D}}_{a}A)+\frac{{\alpha^{\backprime}\,}}{2}\text{Tr}\hskip
2.0pt(\phi_{a}F)-{\text{d}}\left(\frac{{\alpha^{\backprime}\,}}{2}\text{Tr}\hskip
2.0pt\\!(A\phi_{a})\right)~{}.$

We will return to this in the next section.

###  2.6 Summary of small transformations on fields

We now put all of this together. First, it is convenient to form the
complexified combinations

$\mathpzc Z_{a}\;=\;\mathpzc
B_{a}+\text{i}\,{\mathfrak{D}}_{a}\omega~{},\qquad{\overline{\mathpzc}Z}_{a}\;=\;\mathpzc
B_{a}-\text{i}\,{\mathfrak{D}}_{a}\omega~{},$

where we denote
${\mathfrak{D}}_{a}\omega^{(p,q)}=({\partial}_{a}\omega)^{(p,q)}$ while on a
real form ${\mathfrak{D}}_{a}\omega={\partial}_{a}\omega$.

We record for posterity, the action of a combined small diffeomorphism, gerbe
and gauge transformation on heterotic moduli fields:

$\begin{split}\Delta_{a}{}^{\mu}&\sim\Delta_{a}{}^{\mu}+{\overline{\partial}}\varepsilon_{a}{}^{\mu}\,,\quad{\mathfrak{D}}_{a}\mathcal{A}\sim{\mathfrak{D}}_{a}\mathcal{A}+\varepsilon_{a}{}^{\mu}F_{\mu}+{\overline{\partial}}_{\mathcal{A}}\phi_{a}\,,\\\\[7.0pt]
\mathpzc Z_{a}&\sim\mathpzc
Z_{a}+\varepsilon_{a}{}^{m}(H+\text{i}{\text{d}}\omega)_{m}+\frac{{\alpha^{\backprime}\,}}{2}\text{Tr}\hskip
2.0pt\\!\left(F\phi_{a}\right)+{\text{d}}\\!\left(\mathfrak{b}_{a}+\text{i}\varepsilon_{a}{}^{m}\omega_{m}\right)~{},\\\\[5.0pt]
{\overline{\mathpzc}Z}_{a}&\sim{\overline{\mathpzc}Z}_{a}+\varepsilon_{a}{}^{m}(H-\text{i}{\text{d}}\omega)_{m}+\frac{{\alpha^{\backprime}\,}}{2}\text{Tr}\hskip
2.0pt\\!\left(F\phi_{a}\right)+{\text{d}}\\!\left(\mathfrak{b}_{a}-\text{i}\varepsilon_{a}{}^{m}\omega_{m}\right)~{}.\\\
\end{split}$ (2.17)

###  2.7 The equations of motion

One equation of motion is that the complex structure is integrable. A second
is the Atiyah equation (2.7). We have already checked that the small
transformations (2.17) are a symmetry of these equations, see below (2.8). We
now check the remaining equations of motion.

We start with $H={\text{d}}^{c}\omega$. Using
${\text{d}}^{c}\omega=J^{m}{\partial}_{m}\omega-({\text{d}}J^{m})\omega_{m}$
we have

$\begin{split}({\partial}_{a}{\text{d}}^{c}\omega)^{(0,3)}&\;=\;\\!-\text{i}{\overline{\partial}}({\partial}_{a}\omega)^{(0,2)}~{},\\\\[3.0pt]
({\partial}_{a}{\text{d}}^{c}\omega)^{(1,2)}&\;=\;2\text{i}\Delta_{a}{}^{\mu}({\partial}\omega)_{\mu}-\text{i}{\partial}({\partial}_{a}\omega)^{(0,2)}-\text{i}{\overline{\partial}}({\partial}_{a}\omega)^{(1,1)}~{},\end{split}$
(2.18)

Projecting ${\partial}_{a}H$ in (2.11) onto type we find

$\begin{split}{\overline{\partial}}\mathpzc Z_{a}^{(0,2)}&\;=\;0~{},\\\
{\partial}\mathpzc Z_{a}^{(0,2)}+{\overline{\partial}}\mathpzc
Z_{a}^{(1,1)}&\;=\;2\text{i}\Delta_{a}{}^{\mu}({\partial}\omega)_{\mu}+\frac{{\alpha^{\backprime}\,}}{2}\text{Tr}\hskip
2.0pt\big{(}{\mathfrak{D}}_{a}\mathcal{A}F\big{)}~{}.\end{split}$ (2.19)

There are two other equations given by complex conjugation. It is now
straightforward to see that (2.17) is a symmetry of (2.19) provided the
Bianchi identity (1.1) holds.

For later convenience, we note that a solution to the first equation of (2.19)
with $h^{(0,2)}=0$ is given by $\mathpzc
Z_{a}^{(0,2)}={\overline{\partial}}\beta_{a}^{(0,1)}$ for some complex
$(0,1)$–form $\beta_{a}^{(0,1)}$. The second equation then becomes

${\overline{\partial}}(\mathpzc
Z_{a}^{(1,1)}-{\partial}\beta_{a}^{(0,1)})\;=\;2\text{i}\,\Delta_{a}{}^{\mu}({\partial}\omega)_{\mu}+\frac{{\alpha^{\backprime}\,}}{2}\text{Tr}\hskip
2.0pt\big{(}{\mathfrak{D}}_{a}\mathcal{A}F\big{)}~{}.$ (2.20)

In $g_{s}$–perturbative heterotic vacua, the dilaton is a constant in the
supergravity limit. The conformally balanced condition becomes
${\text{d}}(\omega^{2})$. It has a variation

${\text{d}}({\partial}_{a}\omega\,\omega)\;=\;0~{}.$ (2.21)

The action of (2.17) on ${\partial}_{a}\omega$ is a Lie derivative
${\partial}_{a}\omega\sim{\partial}_{a}\omega+\mathcal{L}_{\varepsilon_{a}}\omega$.
This is a symmetry of (2.21). To see this, we use two properties of the
operator $\mathcal{L}$, namely Leibnitz rule and
$[\mathcal{L}_{\varepsilon},{\text{d}}]=0$. The calculation goes as follows

${\text{d}}\big{(}\mathcal{L}_{\varepsilon_{a}}\omega\,\omega\big{)}\;=\;(\mathcal{L}_{\varepsilon}\,{\text{d}}\omega)\,\omega+(\mathcal{L}_{\epsilon_{a}}\omega)\,{\text{d}}\omega\;=\;\mathcal{L}_{\varepsilon_{a}}(\omega\,{\text{d}}\omega)\;=\;\frac{1}{2}\,\mathcal{L}_{\varepsilon_{a}}\,{\text{d}}(\omega^{2})\;=\;0~{}.$
(2.22)

Finally, the HYM equation $\omega^{2}F=0$ has a variation

$\omega^{2}\,({\text{d}}_{A}{\mathfrak{D}}_{a}A)+2\,F\,\omega\,{\partial}_{a}\omega\;=\;0~{}.$
(2.23)

Note that

$({\text{d}}^{2}_{A}\phi_{a})\,\omega^{2}\;=\;[F,\phi_{a}]\,\omega^{2}\;=\;[F\,\omega^{2},\phi_{a}]\;=\;0~{}.$
(2.24)

Hence, under (2.17) this is invariant because

$\Big{(}{\text{d}}_{A}({\text{d}}_{A}\phi_{a}+\varepsilon_{a}{}^{m})\,F_{m}\Big{)}\,\omega^{2}+2\,F\,\omega\,\mathcal{L}_{\varepsilon_{a}}\omega\;=\;\mathcal{L}_{\varepsilon}(F\,\omega^{2})\;=\;0~{},$
(2.25)

where the Lie derivative acting on some gauge group $G$–charged object $\xi$
is defined

$\mathcal{L}_{\varepsilon}\xi\;=\;\varepsilon^{m}({\text{d}}_{A}\xi)_{m}+{\text{d}}_{A}(\varepsilon^{m}\,\xi_{m})~{}.$
(2.26)

This confirms is what, of course, obvious: the equations of motion are
invariant under gauge transformations. But it also serves as a useful
consistency check that (2.17) is the correct transformation law.

##  3 Gauge fixing

We now fix the gauge freedom. Our gauge fixing is holomorphic gauge together a
condition on $\delta\Omega$. That is, given some choice of complex structure
on the moduli space $M$ , we set
${\mathfrak{D}}_{\overline{\alpha}}\mathcal{A}=0$ and $\mathpzc
Z_{\overline{\alpha}}=0$. It is important to note that taking a holomorphic
variation of equations of motion secretly assumes this choice of gauge, a
point that can lead to confusion in the literature. There is a residual gauge
freedom which we use to fix $\delta\Omega^{(3,0)}$ to be harmonic; this also
implies ${\partial}\chi_{\alpha}=0$ and
$\nabla_{\mu}\Delta_{\alpha}{}^{\mu}=0$ for an appropriate choice of
$\nabla_{\mu}$.

###  3.1 A warm–up

Consider a deformation of the Hermitian Yang–Mills equation

$\omega^{2}F\;=\;0~{},$

where $F={\text{d}}A+A^{2}$ and $A=\mathcal{A}-\mathcal{A}^{\dagger}$. A
variation of this equation on a fixed manifold $X$ results in

${\overline{\partial}}_{\mathcal{A}}^{\dagger}(\delta\mathcal{A})\;=\;\\!-{\partial}_{\mathcal{A}}^{\dagger}(\delta\mathcal{A}^{\dagger})~{}.$

This provides no constraints on $\delta A$, but notice it does depend on both
$\delta\mathcal{A}$ and $\delta\mathcal{A}^{\dagger}$ and so is a real
equation. Suppose we can take a holomorphic variation of this equation in
which $\delta\mathcal{A}^{\dagger}=0$. Then we end up with a constraint

${\overline{\partial}}_{\mathcal{A}}^{\dagger}\left(\delta\mathcal{A}\right)\;=\;0~{}.$

It naively looks as though the HYM equation fixed us to harmonic gauge.
However, this conclusion is incorrect. The gauge fixing occurred earlier on,
in the assumption that we could take a holomorphic variation. Indeed, under a
small gauge transformation
$\delta\mathcal{A}^{\dagger}\sim\delta\mathcal{A}^{\dagger}+{\partial}_{\mathcal{A}^{\dagger}}\phi^{\dagger}$,
and so it is only true that $\delta\mathcal{A}^{\dagger}=0$ in a particular
gauge. We call this holomorphic gauge. So the correct statement is that in
holomorphic gauge the Hermitian Yang–Mills equation further constrains
$\delta\mathcal{A}$ to be in harmonic gauge. The aim of this section is to
systematically study the heterotic equations of motion and gauge fixing, and
then describe how these uniquely fix the physical degrees of freedom.

As a toy example to warm–up consider a $d=4$ supersymmetric $U(1)$ gauge
theory with $N+1$ chiral multiplets whose bosons are denoted $\phi^{i}$ can
carry charge $+1$ under the gauge symmetry. The scalar potential is
$V=\frac{1}{2}D^{2}$ where the D-term is
$D{\;=\;}\phi^{i}{\overline{\phi}}_{i}-r$ with $r$ the Fayet–Iliopoulos (FI)
parameter. We have set the coupling constant to unity. The fields $\phi^{i}$
can be interpreted as complex coordinates on $\mathbbl{C}^{N+1}$ with the flat
hermitian metric used in lowering the index ${\overline{\phi}}_{i}$. The space
of classical vacua, therefore, corresponds to the single D–term vanishing
modulo gauge transformations and is

$\mathbbl{P}^{N}\;=\;\mathbbl{C}^{N+1}/\\!\\!/U(1)~{},$

where the $/\\!\\!/$ denotes the symplectic quotient: the gauge quotient
$\phi^{i}\sim\phi^{i}e^{\text{i}\lambda}$, $\lambda\in\mathbbl{R}$ and the
moment map $\phi^{i}{\overline{\phi}}_{i}-r=0$ imposed simultaneously.555It is
well–known we can equally view this moduli space as a holomorphic quotient
$(\mathbbl{C}^{N+1}-{0})/\mathbbl{C}^{*}$. This amounts to complexifying the
gauge symmetry $U(1)\to\mathbbl{C}^{*}$ and forgetting about the D–terms. In a
$d=4$ supersymmetric field theory we can study either the real compact gauge
group or the complexified gauge group. Whether this holds in string field
theory is far from clear. That being so, we work with the real compact gauge
group in heterotic theories to guarantee self–consistency.

Given a point $\phi^{i}\in\mathbbl{P}^{N}$ we can study deformations
$\phi^{i}\to\phi^{i}+\delta\phi^{i}$. Deformations $\delta\phi^{i}$ inherit a
gauge symmetry
$\delta\phi^{i}\sim\delta\phi^{i}+\text{i}\phi^{i}\delta\lambda$ for some
small gauge parameter $\delta\lambda$ which is real. These are a simple
example of small gauge transformations which we discuss in this paper. The
D-terms impose an equation of motion on the fluctuations

$\delta\phi^{i}{\overline{\phi}}_{i}+\phi^{i}\delta{\overline{\phi}}_{i}{\;=\;}2\,\text{Re~{}}(\delta\phi^{i}{\overline{\phi}}_{i}){\;=\;}0~{}.$

The D–terms do not fix the $U(1)$ gauge symmetry. This is expected — after all
the equations of motion are gauge invariant. Instead a gauge fixing is an
additional condition such as
$\text{Im~{}}(\delta\phi^{i}{\overline{\phi}}_{i})=\xi$ for some
$\xi\in\mathbbl{R}$. The D–term together with the gauge fixing allow us to
determine the physical fluctuations. It is not hard to see they are exactly
tangent vectors to $\mathbbl{P}^{N}$ about a point. In the heterotic analysis
the equations of motion, including the HYM and balanced equation, together
with gauge fixing will allow us to determine physical degrees of freedom in
the same sort of way.

###  3.2 Holomorphy and holomorphic gauge

In §2, expressions were written in real coordinates with real parameters.
However, a supersymmetric theory always comes with a complex parameter space
$M$. Hence, its tangent space is complexified (see Appendix §A) and we gain
much computational power by introducing holomorphy. Loosely speaking,
holomorphy in this context means we need to decide a map between field
deformations and holomorphic tangent vectors $\delta y^{\alpha}$. Typically,
this relation is stated after (implicitly) gauge fixing. For this reason, we
first write down the relations without fixing a gauge. This will amount to
certain antiholomorphic combinations being exact. We will then show how to fix
the gauge freedom in §2 can be used to goto ‘holomorphic gauge’ in which we
recover the usual expressions given in the literature.

Let us illustrate this with integrable complex structure deformations. These
obey ${\overline{\partial}}\Delta_{a}{}^{\mu}=0$. Substituting $\delta
y^{a}\Delta_{a}=\delta y^{\alpha}\Delta_{\alpha}+\delta
y^{\overline{\beta}}\Delta_{\overline{\beta}}$ into this equation, we take
holomorphy to mean

$\Delta_{{\overline{\alpha}}}{}^{\mu}\;=\;{\overline{\partial}}\kappa_{\overline{\alpha}}{}^{\mu}~{}.$

In other words, the non–trivial elements of $H^{1}(X,\mathpzc T_{X})$ are
associated to holomorphic tangent vectors, but we have not set the
antiholomorphic deformations to zero. This is because under a small
diffeomorphism
$\Delta_{\overline{\alpha}}\sim\Delta_{\overline{\alpha}}+{\overline{\partial}}\varepsilon_{\overline{\alpha}}{}^{\mu}$
and so putting to ${\overline{\partial}}\kappa_{\overline{\alpha}}{}^{\mu}=0$
would amount to partially gauge fixing small diffeomorphism. This becomes more
involved once the remaining fields of heterotic are involved. Let us write out
how this works.

Consider the gauge field. Writing $\delta\mathcal{A}=\delta
y^{\alpha}\,{\mathfrak{D}}_{\alpha}\mathcal{A}+\delta
y^{\overline{\alpha}}\,{\mathfrak{D}}_{\overline{\alpha}}\mathcal{A}$ and
substituting into the Atiyah equation (2.7) the antiholomorphic deformation
satisfies
${\overline{\partial}}_{\mathcal{A}}{\mathfrak{D}}_{\overline{\alpha}}\mathcal{A}{\;=\;}{\overline{\partial}}_{\mathcal{A}}(\kappa_{\overline{\alpha}}{}^{\mu}F_{\mu})$
and holomorphy means we take the solution to be exact

${\mathfrak{D}}_{\overline{\alpha}}\mathcal{A}=\kappa_{\overline{\alpha}}{}^{\mu}F_{\mu}+{\overline{\partial}}_{\mathcal{A}}\Phi_{\overline{\alpha}}~{},$
(3.1)

for some section $\Phi_{\overline{\alpha}}$ of ${\rm End}\,\mathpzc E$. What
if the solution to this equation were not to be exact? The answer is that any
non–trivial elements of
$H^{(0,1)}_{{\overline{\partial}}_{\mathcal{A}}}(X,{\rm End}\,\mathpzc E)$ can
be absorbed by a change in complex structure of the moduli space [2]. This
will hold for other moduli.

The complexified hermitian form $\mathpzc Z_{a}$ is included through the
second equation of (2.19)

${\overline{\partial}}\Big{(}\mathpzc
Z_{\overline{\alpha}}^{(1,1)}-{\partial}\beta_{\overline{\alpha}}{}^{(0,1)}-2\text{i}\kappa_{\overline{\alpha}}{}^{\mu}({\partial}\omega)_{\mu}-\frac{{\alpha^{\backprime}\,}}{2}\text{Tr}\hskip
2.0pt\big{(}\Phi_{\overline{\alpha}}\,F\big{)}\Big{)}\;=\;0~{}.$

Holomorphy here amounts to the solution of this equation being exact. As
above, if we were to take this solution to be not exact, then any non–trivial
elements of $H^{(1,1)}(X,\mathbbl{C})$ on the right hand side can be absorbed
by a change in complex structure of the moduli space (see [2]). That being so,
we have

$\mathpzc
Z_{\overline{\alpha}}^{(1,1)}\;=\;2\text{i}\kappa_{\overline{\alpha}}{}^{\mu}({\partial}\omega)_{\mu}+\frac{{\alpha^{\backprime}\,}}{2}\text{Tr}\hskip
2.0pt\big{(}\Phi_{\overline{\alpha}}\,F\big{)}+{\partial}\beta_{\overline{\alpha}}{}^{(0,1)}+{\overline{\partial}}\xi_{\overline{\alpha}}^{(1,0)}~{}.$

Hence, without reference to a choice of gauge, holomorphy for the moduli
fields is defined as

$\begin{split}\Delta_{\overline{\alpha}}{}^{\mu}&\;=\;{\overline{\partial}}\kappa_{\overline{\alpha}}{}^{\mu}~{},\qquad{\mathfrak{D}}_{\overline{\alpha}}\mathcal{A}\;=\;\kappa_{\overline{\alpha}}{}^{\mu}F_{\mu}+{\overline{\partial}}_{\mathcal{A}}\Phi_{\overline{\alpha}}~{},\\\
\mathpzc
Z_{\overline{\alpha}}^{(1,1)}&\;=\;2\text{i}\kappa_{\overline{\alpha}}{}^{\mu}({\partial}\omega)_{\mu}+\frac{{\alpha^{\backprime}\,}}{2}\text{Tr}\hskip
2.0pt\big{(}\Phi_{\overline{\alpha}}F\big{)}+{\partial}\beta_{\overline{\alpha}}{}^{(0,1)}+{\overline{\partial}}\xi_{\overline{\alpha}}^{(1,0)}~{},\quad\mathpzc
Z_{\overline{\alpha}}^{(0,2)}\;=\;{\overline{\partial}}\beta_{\overline{\alpha}}^{(0,1)}~{}.\end{split}$
(3.2)

The fields in (3.2) transform under a small transformation as

$\begin{split}\Delta_{\overline{\alpha}}{}^{\mu}&~{}\sim~{}\Delta_{\overline{\alpha}}{}^{\mu}+{\overline{\partial}}\varepsilon_{\overline{\alpha}}{}^{\mu}~{},\\\\[8.0pt]
{\mathfrak{D}}_{\overline{\alpha}}\mathcal{A}&~{}\sim~{}{\mathfrak{D}}_{\overline{\alpha}}\mathcal{A}+\varepsilon_{\overline{\alpha}}{}^{\mu}F_{\mu}+{\overline{\partial}}_{\mathcal{A}}\phi_{\overline{\alpha}}~{},\\\\[5.0pt]
\mathpzc Z^{(1,1)}_{\overline{\alpha}}&~{}\sim~{}\mathpzc
Z^{(1,1)}_{\overline{\alpha}}+2\text{i}\varepsilon_{\overline{\alpha}}{}^{\mu}({\partial}\omega)_{\mu}+\frac{{\alpha^{\backprime}\,}}{2}\text{Tr}\hskip
2.0pt\\!\left(\phi_{\overline{\alpha}}F\right)+{\partial}\\!\left(\mathfrak{b}^{(0,1)}_{\overline{\alpha}}+\text{i}\varepsilon_{\overline{\alpha}}{}^{\mu}\omega_{\mu}\right)+{\overline{\partial}}\\!\left(\mathfrak{b}^{(1,0)}_{\overline{\alpha}}+\text{i}\varepsilon_{\overline{\alpha}}{}^{\overline{\mu}}\omega_{\overline{\mu}}\right)~{},\\\\[3.0pt]
\end{split}$

where we use (2.8) with $\delta\phi=\delta y^{\alpha}\phi_{\alpha}+\delta
y^{\overline{\beta}}\phi_{\overline{\beta}}$. We can equivalently express this
as

$\kappa_{\overline{\alpha}}{}^{\mu}\sim\kappa_{\overline{\alpha}}{}^{\mu}+\varepsilon_{\overline{\alpha}}{}^{\mu}~{},~{}~{}~{}\Phi_{\overline{\alpha}}\sim\Phi_{\overline{\alpha}}+\phi_{\overline{\alpha}}~{},~{}~{}~{}\beta^{(0,1)}_{\overline{\alpha}}\sim\beta^{(0,1)}_{\overline{\alpha}}+\mathfrak{b}^{(0,1)}_{\overline{\alpha}}+\text{i}\varepsilon_{\overline{\alpha}}{}^{\nu}\omega_{\nu}~{},~{}~{}~{}\xi_{\overline{\alpha}}^{(1,0)}\sim\xi_{\overline{\alpha}}^{(1,0)}+\mathfrak{b}^{(1,0)}_{\overline{\alpha}}+\text{i}\varepsilon_{\overline{\alpha}}{}^{\overline{\nu}}\omega_{\overline{\nu}}~{},$

where in fact the last two equations are correct up to
${\overline{\partial}}$– and ${\partial}$–closed term respectively.

We need to gauge fix. Holomorphic gauge amounts to
$\varepsilon_{\overline{\alpha}}{}^{\mu}=-\kappa_{\overline{\alpha}}{}^{\mu}$
and $\phi_{\overline{\alpha}}=-\Phi_{\overline{\alpha}}$ so that so that
$\Delta_{\overline{\alpha}}{}^{\mu}=0$ and
${\mathfrak{D}}_{\overline{\alpha}}\mathcal{A}=0$ respectively. Furthermore,
demand $\mathfrak{b},\varepsilon$ solve

$\mathfrak{b}^{(0,1)}_{\overline{\alpha}}-\text{i}\kappa_{\overline{\alpha}}{}^{\nu}\omega_{\nu}\;=\;\\!-\beta_{\overline{\alpha}}^{(0,1)}+{\overline{\partial}}\psi_{\overline{\alpha}}~{},\quad\mathfrak{b}^{(1,0)}_{\overline{\alpha}}+\text{i}\varepsilon_{\overline{\alpha}}{}^{\overline{\nu}}\omega_{\overline{\nu}}\;=\;\\!-\xi_{\overline{\alpha}}^{(1,0)}+{\partial}\psi_{\overline{\alpha}}~{},$
(3.3)

so that $\mathpzc Z_{\overline{\alpha}}^{(0,2)}=0$ and $\mathpzc
Z_{\overline{\alpha}}^{(1,1)}=0$ respectively. There is a residual gauge
freedom parameterised by $\psi_{\overline{\alpha}}$ a complex $(0,1)$–form on
the moduli space $M$. Finally, we also set

$\mathfrak{b}^{(0,1)}_{\alpha}+\text{i}\varepsilon_{\alpha}{}^{\nu}\omega_{\nu}\;=\;\\!-\beta_{\alpha}^{(0,1)}+{\overline{\partial}}\psi_{\alpha}~{},$
(3.4)

where $\psi_{\alpha}\,{\text{d}}y^{\alpha}$ is a complex $(1,0)$–form on $M$
and is independent from
$\psi_{\overline{\alpha}}\,{\text{d}}y^{\overline{\alpha}}$ in (3.3). This
amounts to $\mathpzc Z{}_{\alpha}^{(0,2)}=0$. The fact
$\Delta_{\overline{\alpha}}{}^{\mu}=0$ implies
$({\mathfrak{D}}_{\alpha}\omega)^{(2,0)}=0$ and so together with $\mathpzc
Z_{\overline{\alpha}}^{(0,2)}=0$ we see that $\mathpzc
B^{(2,0)}_{\alpha}={\mathfrak{D}}_{\alpha}\omega^{(2,0)}=0$. We can summarise
the gauge fixing as

$\Delta_{\overline{\alpha}}{}^{\mu}\;=\;0~{},\quad{\mathfrak{D}}_{\overline{\alpha}}\mathcal{A}\;=\;0~{},\quad{\overline{\mathpzc}Z}_{\overline{\alpha}}^{(2,0)}\;=\;0~{},\quad\mathpzc
Z_{\overline{\alpha}}^{(1,1)}\;=\;0~{},\quad\mathpzc
Z_{\overline{\alpha}}^{(0,2)}\;=\;{\overline{\mathpzc}Z}_{\overline{\alpha}}^{(0,2)}\;=\;0~{}.$
(3.5)

We have included ${\overline{\mathpzc}Z}_{\overline{\alpha}}^{(2,0)}$ as part
of holomorphy; this will be justified in the next section.

Two of the equations $\mathpzc Z_{\overline{\alpha}}^{(1,1)}=\mathpzc
Z_{\overline{\alpha}}^{(0,2)}=0$ imply $\mathpzc
B_{\alpha}^{(0,2)}=\\!-\text{i}{\mathfrak{D}}_{\alpha}\omega^{(0,2)}$ and
$\mathpzc B_{\alpha}^{(1,1)}=\text{i}{\mathfrak{D}}_{\alpha}\omega^{(1,1)}$
respectively. Hence, the physical degrees of freedom are

${\mathfrak{D}}_{\alpha}\mathcal{A}~{},\quad\Delta_{\alpha}{}^{\mu}~{},\quad{\mathfrak{D}}_{\alpha}\omega^{(1,1)}~{},\quad{\mathfrak{D}}_{\alpha}\omega^{(0,2)}~{},$
(3.6)

though these are related by the equations of motion that we describe below. In
the following we will interchangably use $\mathpzc Z^{(1,1)}_{\alpha}$ and
${\overline{\mathpzc}Z}^{(0,2)}_{\alpha}$ for the last pair respectively; they
are equal up to a numerical factor.

In the physical theory $\delta J$, $\delta A$, ${\partial}_{a}\omega$ and
$\mathpzc B_{a}$ are real, and so we understand that we always pair
holomorphic and antiholmorphic deformations in such a way to give a real
deformation.

###  3.3 Residual gauge symmetries

Residual transformations must preserve holomorphic gauge. Firstly, there are
no solutions to
${\overline{\partial}}\varepsilon_{\overline{\alpha}}{}^{\mu}=0$ provided
$h^{(0,2)}=0$ as can be seen by contracting with $\Omega_{\mu}$. So any
residual diffeomorphisms must have form $\varepsilon_{\alpha}{}^{\mu}$.
Secondly, holomorphy completely fixes $\phi_{\alpha}$. To see this note that
we need ${\overline{\partial}}_{\mathcal{A}}\phi_{\overline{\alpha}}=0$ and
stability implies $H^{0}(X,{\rm End}\,\mathpzc E)$ is trivial and so
$\phi_{\overline{\alpha}}$ is a constant on $X$. We take the connection
$\Lambda_{a}$ to be antihermitian so that
$\phi_{\alpha}=-(\phi_{\overline{\alpha}})^{\dagger}$. Therefore small gauge
transformations for the bundle are fixed.666If we were considering a
complexified gauge group then $A$ is no longer antihermitian, and so
$\delta\mathcal{A}$ and $\delta\mathcal{A}^{\dagger}$ are independent degrees
of freedom. That being so, $\phi_{\alpha}$ is independent from
$\phi_{\overline{\alpha}}$ and so
${\mathfrak{D}}_{\overline{\alpha}}\mathcal{A}=0$ gauge fixes
$\phi_{\overline{\alpha}}$ but does nothing to $\phi_{\alpha}$.  With those
conditions in mind, we now explore the remaining gauge freedoms.

We still have a residual gauge freedom deriving from (3.3) and (3.4) of the
form

$\mathfrak{b}^{(0,1)}_{\overline{\alpha}}\;=\;{\overline{\partial}}\psi_{\overline{\alpha}}~{},\quad\mathfrak{b}^{(1,0)}_{\overline{\alpha}}+\text{i}\varepsilon_{\overline{\alpha}}{}^{\overline{\nu}}\omega_{\overline{\nu}}\;=\;{\partial}\psi_{\overline{\alpha}}~{},\quad\mathfrak{b}^{(1,0)}_{\overline{\alpha}}-\text{i}\varepsilon_{\overline{\alpha}}{}^{\overline{\nu}}\omega_{\overline{\nu}}\;=\;{\partial}{{\overline{\psi}}}_{\overline{\alpha}}~{},$
(3.7)

where ${{\overline{\psi}}}_{\overline{\alpha}}=(\psi_{\alpha})^{*}$. Define
$\varepsilon_{\overline{\alpha}}=\text{i}\varepsilon_{\overline{\alpha}}{}^{m}\omega_{m}$,
we can invert these equations

$\mathfrak{b}^{(1,0)}_{\overline{\alpha}}\;=\;\frac{1}{2}{\partial}(\psi_{\overline{\alpha}}+{{\overline{\psi}}}_{\overline{\alpha}})~{},\quad\varepsilon_{\overline{\alpha}}^{(1,0)}\;=\;\frac{1}{2}{\partial}(\psi_{\overline{\alpha}}-{{\overline{\psi}}}_{\overline{\alpha}})~{},\quad\mathfrak{b}^{(0,1)}_{\overline{\alpha}}\;=\;{\overline{\partial}}\psi_{\overline{\alpha}}~{},\quad\varepsilon_{\overline{\alpha}}^{(0,1)}\;=\;0~{}.$
(3.8)

Referring back to §2.2, in holomorphic gauge $\delta\Omega$ is a holomorphic
function of parameters and coordinates. Under a small diffeomorphism
$\delta\Omega^{(3,0)}\sim\delta\Omega^{(3,0)}+(\nabla^{\text{\tiny
LC}}_{\nu}\varepsilon^{\nu})\,\Omega$. Without loss of generality we can write
an arbitrary holomorphic vector as

$\varepsilon_{\alpha}{}^{\nu}\;=\;\\!-\frac{1}{2\left\|\Omega\right\|^{2}}\,{\overline{\Omega}}^{\nu\rho\sigma}(\xi_{\alpha}+{\partial}\zeta^{(1,0)}_{\alpha})_{\rho\sigma}~{},$
(3.9)

where ${\partial}\zeta_{\alpha}^{(1,0)}$ is a local representative of a closed
2-form. Then, note that
${\overline{\partial}}^{\dagger}\varepsilon^{(0,1)}{}=\\!-g^{\mu{\overline{\nu}}}{\partial}_{\mu}\varepsilon_{{\overline{\nu}}}=\\!-\nabla_{\nu}\,\varepsilon^{\nu}$.
Due to $H_{\mu\nu}{}^{\nu}=0$ we are free to use any of the Levi–Civita,
Bismut, Hull or Chern connections in that expression and that is why the label
is left blank. Hence,

$\nabla_{\nu}\,\varepsilon_{\alpha}{}^{\nu}\;=\;\\!-\frac{1}{3!\left\|\Omega\right\|^{2}}\,{\overline{\Omega}}^{\nu\rho\sigma}({\partial}\xi_{\alpha})_{\nu\rho\sigma}~{},$

and we kill the ${\partial}\xi^{(2,0)}$ in (2.4). Comparing with (3.8), the
existence of the gauge transformation amounts to solving a Poisson equation

$\Delta_{\overline{\partial}}({{\overline{\psi}}}_{\alpha}-\psi_{\alpha})\;=\;\\!-\frac{\text{i}}{3!\left\|\Omega\right\|^{2}}\,{\overline{\Omega}}^{\nu\rho\sigma}({\partial}\xi_{\alpha})_{\nu\rho\sigma}~{}.$

As discussed in Appendix §B.1 the obstruction to solving this Poisson equation
is that the source be orthogonal to zero modes. On $X$ these are constants and
so we see orthogonality is always satisfied:

$\frac{\text{i}}{3!\left\|\Omega\right\|^{2}}\int_{X}\,\mbox{vol}\,{\overline{\Omega}}^{\nu\rho\sigma}({\partial}\xi_{\alpha})_{\nu\rho\sigma}\sim\text{i}\int_{X}{\partial}\xi^{(2,0)}{\overline{\Omega}}\;=\;0~{}.$

Consequently, we can find a ${{\overline{\psi}}}_{\alpha}-\psi_{\alpha}$ so
that $\delta\Omega^{(3,0)}=(\delta k)\Omega$ is harmonic for some parameter
dependent constant $\delta k$. The closed form
${\partial}\zeta_{\alpha}^{(1,0)}$ corresponds to $\nabla^{\text{\tiny
LC}}_{\nu}\,\varepsilon_{\alpha}{}^{\nu}=0$. However, from (3.8) we see that
such gauge transformations correspond to zero modes on $X$ viz.
$\Box_{\overline{\partial}}(\psi_{\alpha}-{{\overline{\psi}}}_{\alpha})=0$.
The only zero modes on $X$ are constants, which for reasons explained in
Appendix §B.1, must be zero and so
$\psi_{\alpha}={{\overline{\psi}}}_{\alpha}$ and ${\partial}\zeta^{(0,1)}$
must vanish. The only residual freedom, therefore, is contained with
$\mathfrak{b}_{\alpha}={\text{d}}\psi_{\alpha}$. Such gauge transformations do
not affect any of the fields because the pertiant transformation law is
$\mathpzc B_{\alpha}\sim\mathpzc B_{\alpha}+{\text{d}}\mathfrak{b}_{\alpha}$
which clearly leaves $\mathpzc B_{\alpha}$ invariant. Hence, we have
completely fixed the gauge, the aim of this section.

A final comment is that in this gauge

${\partial}\chi_{\alpha}\;=\;0~{}.$

Using $\chi_{\alpha}{\;=\;}\Delta_{\alpha}{}^{\mu}\,\Omega_{\mu}$, the
identity
$({\partial}\Delta_{\alpha}{}^{\mu})\Omega_{\mu}={\partial}_{\mu}\Delta_{\alpha}{}^{\mu}\,\Omega$
and the fact
$\Omega=\frac{1}{3!}f\,\epsilon_{\mu\nu\rho}\,{\text{d}}x^{\mu\nu\rho}$ with
$|f|^{2}=\left\|\Omega\right\|^{2}\sqrt{g}$ we have

$\begin{split}{\partial}(\Delta_{\alpha}{}^{\mu}\,\Omega_{\mu})&\;=\;\\!-\left({\partial}_{\mu}\Delta_{\alpha}{}^{\mu}+\Delta_{\alpha}{}^{\mu}\,{\partial}_{\mu}\log{\sqrt{g}}\right)\Omega\;=\;\\!-\big{(}\nabla_{\mu}^{\rm
H/Ch}\Delta_{\alpha}{}^{\mu}\big{)}\Omega\;=\;0~{},\end{split}$

where in the last equality $\nabla_{\mu}^{\rm H/Ch}$ are a family of
connections with $\epsilon-\rho{\;=\;}1$. This, in particular, includes the
Hull and Chern connections. Hence, in this gauge we find

$\nabla_{\mu}^{\rm H/Ch}\Delta_{\alpha{\overline{\nu}}}{}^{\mu}\;=\;0~{}.$
(3.10)

Under a small diffeomorphism this condition is preserved. Seeing this is an
interesting exercise as it requires the condition
${\partial}_{\mu}{\partial}_{\overline{\nu}}\log\sqrt{g}=0$, a consequence of
$|f|^{2}=\sqrt{g}\left\|\Omega\right\|^{2}$, the norm
$\left\|\Omega\right\|^{2}$ being constant and $f$ being holomorphic. As the
metric is not Kähler this condition is not the same as Ricci flatness.

We can rephrase this gauge fixing in terms of
${\mathfrak{D}}_{a}\omega^{(0,2)}$ as follows. This will be useful in the next
subsection. First, the codifferential as given in Appendix C can be used to
write

${\overline{\partial}}^{\dagger}\Delta_{\alpha}{}^{\mu}\;=\;-\nabla^{{\rm
Ch}\,\,{\overline{\nu}}}\,\Delta_{\alpha{\overline{\nu}}}{}^{\mu}~{}=-\nabla^{{\rm
Ch}}_{\nu}\,\Delta_{\alpha}{}^{(\mu\nu)}+\nabla^{{\rm
Ch}}_{\nu}\,\Delta_{\alpha}{}^{[\mu\nu]}~{}.$

Second, combine this equation with (3.10) and the relation
${\mathfrak{D}}_{\alpha}\omega_{{\overline{\mu}}{\overline{\nu}}}=2\text{i}\,\Delta_{\alpha[{\overline{\mu}}{\overline{\nu}}]}$
to give

${\overline{\partial}}^{\dagger}\Delta_{\alpha}{}^{\mu}\;=\;\text{i}g^{\mu{\overline{\rho}}}\,\nabla^{{\rm
Ch}\,\,{\overline{\nu}}}\big{(}{\mathfrak{D}}_{\alpha}\omega_{{\overline{\nu}}{\overline{\rho}}}\big{)}~{}.$
(3.11)

This is in addition to the algebraic relation
${\mathfrak{D}}_{\alpha}\omega^{(0,2)}=\Delta_{\alpha}{}^{\mu}\,\omega_{\mu}$.

##  4 The Hodge decomposition and the moduli space metric

We now show how gauge fixing together with all of the equations of motion,
including the hermitian Yang–Mills equation and balanced equation, allow us to
solve for the terms in the Hodge decomposition of the heterotic moduli in
(3.6).

Mathematically we are determining the Kodaira–Spencer map which associates
parameters with field deformations. We do not a priori assume any structure
about the moduli space, for example, we do not label parameters as complex
structure or bundle moduli. We find that even at first order in
${\alpha^{\backprime}\,}$, the moduli fields are incredibly coupled and there
is not invariant distinction between moduli.

When the background is Calabi-Yau at large radius, so that supergravity is
guaranteed to be a good approximation, we show how one may go about solving
these coupled differential equations for the field deformations in (3.6). As
an application of this work, we give a prescription for how to compute the
moduli space metric derived in [1, 2].

###  4.1 The equations of motion

The physical degrees of freedom in (3.6) are
${\mathfrak{D}}_{\alpha}\omega^{(0,2)}$,
${\mathfrak{D}}_{\alpha}\omega^{(1,1)}$, ${\mathfrak{D}}_{\alpha}\mathcal{A}$
and $\Delta_{\alpha}$. The equations of motion from the Nijenhuis tensor,
Atiyah equation and (2.19) are

$\begin{split}{\overline{\partial}}\Delta_{\alpha}{}^{\mu}&\;=\;0~{},\\\\[4.26773pt]
{\overline{\partial}}_{\mathcal{A}}{\mathfrak{D}}_{\alpha}\mathcal{A}&\;=\;\Delta_{\alpha}{}^{\mu}F_{\mu}~{},\\\
{\overline{\partial}}\mathpzc
Z_{\alpha}^{(1,1)}&\;=\;2\text{i}\,\Delta_{\alpha}{}^{\mu}({\partial}\omega)_{\mu}+\frac{{\alpha^{\backprime}\,}}{2}\text{Tr}\hskip
2.0pt\big{(}{\mathfrak{D}}_{\alpha}\mathcal{A}\,F\big{)}-{\alpha^{\backprime}\,}\,\big{(}\nabla_{\nu}\Delta_{\alpha}{}^{\mu}+\text{i}\,\nabla^{\mu}\,{\mathfrak{D}}_{\alpha}\omega_{\nu}^{(0,1)}\big{)}R^{\nu}{}_{\mu}~{},\end{split}$
(4.1)

where
$R^{\mu}{}_{\nu}=R_{\rho{\overline{\sigma}}}{}^{\mu}{}_{\nu}\,{\text{d}}x^{\rho{\overline{\sigma}}}$
is the Riemann tensor. We have now included the contribution of the spin
connection ${\mathfrak{D}}_{\alpha}\Theta$. This depends, to zeroth order in
${\alpha^{\backprime}\,}$, on the metric moduli with the relation derived in
[1]:

${\mathfrak{D}}_{\alpha}\Theta{}_{\overline{\mu}}{}^{\nu}{}_{\sigma}\;=\;\nabla_{\sigma}\,\Delta_{\alpha{\overline{\mu}}}{}^{\nu}+\text{i}\,\nabla^{\nu}\,{\mathfrak{D}}_{\alpha}\omega_{\sigma{\overline{\mu}}}~{},$
(4.2)

which together with the symmetry
${\mathfrak{D}}_{\alpha}\Theta{}_{\overline{\mu}}{}^{\overline{\nu}}{}_{{\overline{\sigma}}}\;=\;\\!-g^{{\overline{\nu}}\lambda}\,{\mathfrak{D}}_{\alpha}\Theta_{{\overline{\mu}}}{}^{\rho}{}_{\lambda}\,g_{\rho{\overline{\sigma}}}$
allows us to write

$\text{Tr}\hskip
2.0pt\big{(}{\mathfrak{D}}_{\alpha}\Theta^{(0,1)}R\big{)}\;=\;2\,{\mathfrak{D}}_{\alpha}\Theta^{\mu}{}_{\nu}\,R^{\nu}{}_{\mu}\;=\;2\big{(}\nabla_{\nu}\Delta_{\alpha}{}^{\mu}+\text{i}\,\nabla^{\mu}\,{\mathfrak{D}}_{\alpha}\omega_{\nu}^{(0,1)}\big{)}R^{\nu}{}_{\mu}~{},$
(4.3)

where we have used $R_{{\overline{\mu}}\nu}=-R_{\nu{\overline{\mu}}}$. Note
that in [1] we checked that the spin connection is holomorphic
${\mathfrak{D}}_{\alpha}\Theta_{\mu}=0$ and that this holds provided we are in
the gauge fixing discussed here, at least to this order in
${\alpha^{\backprime}\,}$.

A motivation for this paper is that the equations (4.1), in particular the
last one, are exactly captured by the universal bundle in [1]. We explore this
further in §5, and understand this result as being the statement that the
universal bundle being holomorphic corresponds to field deformations of
heterotic string being in holomorphic gauge as constructed in the previous
subsection. It is, mathematically speaking, natural for the universal bundle
to be holomorphic and justifies a posteri our choice of holomorphic gauge in
§3.2 in studying heterotic supergravity.

The balanced condition means ${\text{d}}(\omega^{2})=0$. Taking a real
deformation and decomposing into type there is one non-trivial equation

${\partial}({\mathfrak{D}}_{a}\omega^{(0,2)}\omega)+{\overline{\partial}}({\mathfrak{D}}_{a}\omega^{(1,1)}\omega)\;=\;0~{},$

with the remaining equation given by complex conjugation. In the gauge in
which $\delta\Omega^{(3,0)}$ is harmonic, it follows that
$\delta\log\sqrt{g}{\;=\;}\omega^{\mu{\overline{\nu}}}\delta\omega_{\mu{\overline{\nu}}}$
is a function of parameters only. This contraction appears in the Hodge dual
of $\delta\omega^{(1,1)}$, see (LABEL:eq:Hodge2form). As a reminder, in
holomorphic gauge ${\mathfrak{D}}_{\alpha}\omega^{(2,0)}=0$ and
${\overline{\mathpzc}Z}_{\alpha}^{(1,1)}=0$ meaning $\mathpzc
Z{}_{\alpha}^{(1,1)}=2\text{i}\,{\mathfrak{D}}_{\alpha}\omega^{(1,1)}$.

By definition of the adjoint operators and a result in (LABEL:eq:Hodge2form),
we find two conditions derive from the balanced equation

${\overline{\partial}}^{\dagger}{\mathfrak{D}}_{\alpha}\omega^{(1,1)}\;=\;0~{},\qquad{\overline{\partial}}^{\dagger}{\mathfrak{D}}_{\alpha}\omega^{(0,2)}\;=\;\\!{\partial}^{\dagger}{\mathfrak{D}}_{\alpha}\omega^{(1,1)}~{}.$
(4.4)

In virtue of the first equation, the Hodge decomposition for
${\mathfrak{D}}_{\alpha}\omega^{(1,1)}$ with respect to the
${\overline{\partial}}$–operator is

${\mathfrak{D}}_{\alpha}\omega^{(1,1)}\;=\;{\mathfrak{D}}_{\alpha}\omega^{(1,1)}{}^{harm}+{\overline{\partial}}^{\dagger}\xi_{\alpha}^{(1,2)}~{},$
(4.5)

that is, there is no ${\overline{\partial}}$–exact term. The first term is
${\overline{\partial}}$–harmonic and is completely determined via Hodge theory
of $X$ up to a parameter dependent matrix. This matrix can be accounted for by
a holomorphic change of coordinates on the moduli space. The
${\overline{\partial}}$-coexact bit is determined by the moduli equation
(4.1). To see this note that $\mathpzc
Z{}_{\alpha}^{(1,1)}=2\text{i}\,{\mathfrak{D}}_{\alpha}\omega^{(1,1)}$ and so

${\overline{\partial}}{\overline{\partial}}^{\dagger}\xi_{\alpha}^{(1,2)}\;=\;\Delta_{\alpha}{}^{\mu}({\partial}\omega)_{\mu}-\frac{\text{i}{\alpha^{\backprime}\,}}{4}\text{Tr}\hskip
2.0pt({\mathfrak{D}}_{\alpha}\mathcal{A}F)+\frac{\text{i}{\alpha^{\backprime}\,}}{2}\,\big{(}\nabla_{\nu}\Delta_{\alpha}{}^{\mu}+\text{i}\,\nabla^{\mu}\,{\mathfrak{D}}_{\alpha}\omega_{\nu}^{(0,1)}\big{)}R^{\nu}{}_{\mu}~{}.$

The Hodge decomposition of the $(1,2)$–form is

$\xi_{\alpha}^{(1,2)}\;=\;\xi_{\alpha}^{(1,2)}{}^{harm}+{\overline{\partial}}\rho_{\alpha}^{(1,1)}+{\overline{\partial}}^{\dagger}\lambda_{\alpha}^{(1,3)}~{}.$

The terms $\xi_{\alpha}^{(1,2)}{}^{harm}$ and
${\overline{\partial}}^{\dagger}\lambda_{\alpha}^{(1,3)}$ do not contribute to
${\mathfrak{D}}_{\alpha}\omega^{(1,1)}$ and so are not physical degree of
freedom. The remaining term is determined by substituting into the previous
equation, and inverting the Laplacian

$\displaystyle\xi_{\alpha}^{(1,2)}$
$\displaystyle\;=\;\Box_{\overline{\partial}}^{-1}\Big{(}\Delta_{\alpha}{}^{\mu}({\partial}\omega)_{\mu}-\frac{\text{i}{\alpha^{\backprime}\,}}{4}\text{Tr}\hskip
2.0pt({\mathfrak{D}}_{\alpha}\mathcal{A}F)$ $\displaystyle\hskip
71.13188pt+\frac{\text{i}{\alpha^{\backprime}\,}}{2}\,\big{(}\nabla_{\nu}\Delta_{\alpha}{}^{\mu}+\text{i}\,\nabla^{\mu}\,{\mathfrak{D}}_{\alpha}\omega_{\nu}^{(0,1)}\big{)}R^{\nu}{}_{\mu}\Big{)}+{\overline{\partial}}^{\dagger}\text{--\,closed}~{}.$
(4.6)

The term in the bracket is not a zero mode of Laplacian by assumption and we
have not written a co–closed term as it does not contribute to (4.5). We will
come back to this equation in a moment.

Returning to (4.4), the second equation can be combined with (3.11) to gain
further information about $\Delta_{\alpha}$. Start with the left hand side

${\overline{\partial}}^{\dagger}{\mathfrak{D}}_{\alpha}\omega^{(0,2)}~{}=-\star{\partial}({\mathfrak{D}}_{\alpha}\omega^{(0,2)}\,\omega)~{}=-\star\left({\partial}{\mathfrak{D}}_{\alpha}\omega^{(0,2)}\right)\omega-\star\left({\mathfrak{D}}_{\alpha}\omega^{(0,2)}\,{\partial}\omega\right)~{},$

and make use of (C.12) to express this as

${\overline{\partial}}^{\dagger}{\mathfrak{D}}_{\alpha}\omega^{(0,2)}~{}=-\nabla^{{\rm
Ch}\,\,{\overline{\nu}}}({\mathfrak{D}}_{\alpha}\omega_{{\overline{\nu}}}^{(0,1)})-\frac{\text{i}}{2}\,{\mathfrak{D}}_{\alpha}\omega_{{\overline{\mu}}{\overline{\nu}}}\,({\partial}\omega)^{{\overline{\mu}}{\overline{\nu}}}~{}.$

The second equation in (4.4) can now be written as

$\nabla^{{\rm
Ch}\,\,{\overline{\nu}}}({\mathfrak{D}}_{\alpha}\omega_{{\overline{\nu}}}^{(0,1)})~{}=-{\partial}^{\dagger}{\mathfrak{D}}_{\alpha}\omega^{(1,1)}-\frac{\text{i}}{2}\,{\mathfrak{D}}_{\alpha}\omega_{{\overline{\mu}}{\overline{\nu}}}\,({\partial}\omega)^{{\overline{\mu}}{\overline{\nu}}}~{}.$
(4.7)

The gauge fixing in (3.11) becomes

$\begin{split}{\overline{\partial}}^{\dagger}\Delta_{\alpha}{}^{\mu}&\;=\;\\!-\text{i}\,\big{(}{\partial}^{\dagger}{\mathfrak{D}}_{\alpha}\omega^{(1,1)}\big{)}_{{\overline{\nu}}}~{}g^{{\overline{\nu}}\mu}+\frac{1}{2}\,{\mathfrak{D}}_{\alpha}\omega_{{\overline{\nu}}{\overline{\rho}}}\,({\partial}\omega)^{{\overline{\nu}}{\overline{\rho}}\mu}~{}.\end{split}$
(4.8)

The Hodge decomposition of $\Delta_{\alpha}$ with respect to the
${\overline{\partial}}$ operator is

$\Delta_{\alpha}\;=\;\Delta_{\alpha}{}^{harm}+{\overline{\partial}}\kappa_{\alpha}~{},\quad\kappa_{\alpha}\;=\;\kappa_{\alpha}{}^{\mu}\,{\partial}_{\mu}~{}.$
(4.9)

The first term is determined by Hodge theory as was the case for the hermitian
form. We have used that $\Delta_{\alpha}$ is integrable
${\overline{\partial}}\Delta_{\alpha}=0$ and so there is no
${\overline{\partial}}$-coexact term. Substituting into (4.8) and using that
the tangent bundle is stable $H^{0}_{{\overline{\partial}}}(X,\mathpzc
T_{X})=0$ to invert the Laplacian, we can solve for the vector

$\kappa_{\alpha}{}^{\mu}\;=\;\\!\Box_{{\overline{\partial}}}^{-1}\Big{(}-\text{i}\,\big{(}{\partial}^{\dagger}{\mathfrak{D}}_{\alpha}\omega^{(1,1)}\big{)}_{{\overline{\nu}}}~{}g^{{\overline{\nu}}\mu}+\frac{1}{2}\,{\mathfrak{D}}_{\alpha}\omega_{{\overline{\nu}}{\overline{\rho}}}\,({\partial}\omega)^{{\overline{\nu}}{\overline{\rho}}\mu}\Big{)}~{}.$
(4.10)

We will return to this equation shortly.

Finally, consider the Hermitian–Yang–Mills (HYM) equation. The
Donaldson–Uhlenbeck–Yau (DUY) theorem asserts that given the bundle is stable,
we find a unique connection $A$ such that

$\omega^{2}F\;=\;0~{}.$

Suppose this is the case. Under a deformation of the gauge field, with $X$
fixed, we end up

${\overline{\partial}}_{\mathcal{A}}^{\dagger}{\mathfrak{D}}_{a}\mathcal{A}\;=\;\\!-{\partial}_{\mathcal{A}^{\dagger}}^{\dagger}{\mathfrak{D}}_{a}\mathcal{A}^{\dagger}~{},$

which is an equation of motion of the fluctuations.

Suppose parameters also vary the manifold $X$. In that case a holomorphic
variation of the HYM is

$\text{i}({\mathfrak{D}}_{\alpha}\omega_{\mu{\overline{\nu}}})F^{\mu{\overline{\nu}}}+{\overline{\partial}}_{\mathcal{A}}^{\dagger}{\mathfrak{D}}_{\alpha}\mathcal{A}\;=\;0~{},$
(4.11)

where we use (LABEL:eq:Hodge2form). Write a Hodge decomposition

${\mathfrak{D}}_{\alpha}\mathcal{A}\;=\;{\mathfrak{D}}_{\alpha}\mathcal{A}^{harm}+{\overline{\partial}}_{\mathcal{A}}\Phi_{\alpha}+{\overline{\partial}}_{\mathcal{A}}^{\dagger}\Psi_{\alpha}^{(0,2)}~{},$
(4.12)

for some ${\rm End}\,\mathpzc E$–valued $(0,2)$–form $\Psi_{\alpha}$ and
scalar $\Phi_{\alpha}$. The harmonic representative is determined purely by
Hodge theory. The remaining pieces are determined by the Atiyah equation and
the HYM equation as we now show.

Firstly, the Atiyah equation gives

${\overline{\partial}}_{\mathcal{A}}{\overline{\partial}}_{\mathcal{A}}^{\dagger}\Psi_{\alpha}^{(0,2)}\;=\;\Delta_{\alpha}{}^{\mu}F_{\mu}~{}.$
(4.13)

As we have done previously, the Hodge decomposition of
$\Psi_{\alpha}^{(0,2)}={\overline{\partial}}_{\mathcal{A}}\Xi_{\alpha}^{(0,1)}+{\overline{\partial}}^{\dagger}(\cdots)$,
where the second term does not contribute to
${\mathfrak{D}}_{\alpha}\mathcal{A}$ and so is not written explicitly. The
Atiyah equation becomes a Laplacian which we can invert for unobstructed
deformations
${\overline{\partial}}_{\mathcal{A}}\Xi_{\alpha}^{(0,1)}=\Box^{-1}_{{\overline{\partial}}_{\mathcal{A}}}(\Delta_{\alpha}{}^{\mu}F_{\mu})$.
Hence, we determine the last term in (4.12)

$\Psi_{\alpha}^{(0,2)}\;=\;\Box^{-1}_{{\overline{\partial}}_{\mathcal{A}}}(\Delta_{\alpha}{}^{\mu}F_{\mu})+{\overline{\partial}}_{\mathcal{A}}^{\dagger}\text{--\,closed}~{},$
(4.14)

where we have not written the
${\overline{\partial}}_{\mathcal{A}}^{\dagger}$–closed term as it does not
contribute to ${\mathfrak{D}}_{a}\mathcal{A}$.

Secondly, because $({\overline{\partial}}_{\mathcal{A}}^{\dagger})^{2}=0$ we
find

$\Box_{{\overline{\partial}}_{\mathcal{A}}}\Phi_{\alpha}\;=\;\\!-\text{i}({\mathfrak{D}}_{\alpha}\omega_{\mu{\overline{\nu}}})F^{\mu{\overline{\nu}}}~{},$
(4.15)

The bundle satisfies the HYM and so is stable, thence
$\Box_{{\overline{\partial}}_{\mathcal{A}}}$ has trivial kernel. This means we
can invert the Laplacian for $\Phi_{\alpha}$.

###  4.2 The Hodge decomposition in holomorphic gauge

We collate the results (4.5), (4.9), and (4.12) together in one place

$\begin{split}{\mathfrak{D}}_{\alpha}\omega^{(1,1)}&\;=\;{\mathfrak{D}}_{\alpha}\omega^{(1,1)\,harm}+{\overline{\partial}}^{\dagger}\xi_{\alpha}^{(1,2)}~{},\\\
\Delta_{\alpha}&\;=\;\Delta_{\alpha}{}^{harm}+{\overline{\partial}}\kappa_{\alpha}~{},\\\
{\mathfrak{D}}_{\alpha}\mathcal{A}&\;=\;{\mathfrak{D}}_{\alpha}\mathcal{A}^{harm}+{\overline{\partial}}_{\mathcal{A}}\Phi_{\alpha}+{\overline{\partial}}_{\mathcal{A}}^{\dagger}\Psi_{\alpha}^{(0,2)}~{}.\\\
\end{split}$ (4.16)

The field
${\mathfrak{D}}_{\alpha}\omega^{(0,2)}=\Delta_{\alpha}{}^{\mu}\,\omega_{\mu}$
obeys the following equation
${\overline{\partial}}^{\dagger}\Delta_{\alpha}{}^{\mu}=\text{i}g^{\mu{\overline{\rho}}}\,\nabla^{{\rm
Ch}\,\,{\overline{\nu}}}\big{(}{\mathfrak{D}}_{\alpha}\omega_{{\overline{\nu}}{\overline{\rho}}}\big{)}$.

The gauge fixing and equations of motion allow us to determine the exact and
co-exact terms. Collecting (4.1), (4.10), (4.14), (4.15) together in one
place:

$\displaystyle\xi_{\alpha}^{(1,2)}$
$\displaystyle\;=\;\Box_{\overline{\partial}}^{-1}\Big{(}\Delta_{\alpha}{}^{\mu}({\partial}\omega)_{\mu}-\frac{\text{i}{\alpha^{\backprime}\,}}{4}\text{Tr}\hskip
2.0pt({\mathfrak{D}}_{\alpha}\mathcal{A}F)$ $\displaystyle\hskip
71.13188pt+\frac{\text{i}{\alpha^{\backprime}\,}}{2}\,\big{(}\nabla_{\nu}\Delta_{\alpha}{}^{\mu}+\text{i}\,\nabla^{\mu}\,{\mathfrak{D}}_{\alpha}\omega_{\nu}^{(0,1)}\big{)}R^{\nu}{}_{\mu}\Big{)}+{\overline{\partial}}^{\dagger}\text{--\,closed}~{},$
(4.17a) $\displaystyle\kappa_{\alpha}{}^{\mu}$
$\displaystyle\;=\;\Box_{{\overline{\partial}}}^{-1}\Big{(}-\text{i}\,\big{(}{\partial}^{\dagger}{\mathfrak{D}}_{\alpha}\omega^{(1,1)}\big{)}^{\mu}+\frac{1}{2}({\mathfrak{D}}_{\alpha}\omega_{{\overline{\nu}}{\overline{\rho}}})({\partial}\omega)^{{\overline{\nu}}{\overline{\rho}}\mu}\Big{)}~{},$
(4.17b) $\displaystyle\Psi_{\alpha}^{(0,2)}$
$\displaystyle\;=\;\Box^{-1}_{{\overline{\partial}}_{\mathcal{A}}}(\Delta_{\alpha}{}^{\mu}F_{\mu})+{\overline{\partial}}_{\mathcal{A}}^{\dagger}\text{--\,closed}~{},\qquad\Phi_{\alpha}\;=\;\Box_{{\overline{\partial}}_{\mathcal{A}}}^{-1}\Big{(}-\text{i}({\mathfrak{D}}_{\alpha}\omega^{\mu{\overline{\nu}}})F_{\mu{\overline{\nu}}}\Big{)}~{}.$
(4.17c)

We have assumed the Laplacian is invertible: that is the sources have no zero
modes of the Laplacian. The conditions for this to be the case are written in
the previous subsection, we repeat here. Consider the first equation of
(4.17c). We have $\Delta_{\alpha}{}^{\mu}F_{\mu}$ is
${\overline{\partial}}_{\mathcal{A}}$–closed. The Laplacian is invertible if
and only if $\Delta_{\alpha}{}^{\mu}F_{\mu}$ is
${\overline{\partial}}_{\mathcal{A}}$–exact, which is of course, exactly the
Atiyah condition (2.7). Similarly, the source in (4.17a) needs to be
${\overline{\partial}}$–exact which is exactly the last line of (4.1). On the
other Laplacians for $\kappa_{\alpha}$ and $\Phi_{\alpha}$ are always
invertible provided the bundles are stable. So we really view (4.17a)–(4.17c)
as being supplemented by (4.1).

We have gauge fixed. Hence, the ${\overline{\partial}}_{\mathcal{A}}$ and
${\overline{\partial}}$–exact pieces in (4.16) are physical. We could try to
discard them by a gauge transformation, but as we have fixed the gauge
completely, this would inevitably lead to a condition such as
${\mathfrak{D}}_{\alpha}\mathcal{A}^{\dagger}\neq 0$ or
${\partial}\chi_{\alpha}\neq 0$. Such terms were set to zero in deriving the
equations of above. This is just as in electromagnetism: in say Coulomb gauge,
the potential $A$ is physical as the gauge has been fixed, and we no longer
have the freedom to perform gauge transformations.

Turning to (4.17a)–(4.17c), supplemented by (4.1), these are highly coupled
equations, and finding a direct solution is very hard.777We would like to
thank Mario Garcia–Fernandez for discussion on this point. These equations are
another, rather explicit, echo of the intuition that there is no clear
decoupling of parameters between $X$ and $\mathpzc E$. One approach is to
proceed perturbatively in ${\alpha^{\backprime}\,}$. Expand fields in
${\alpha^{\backprime}\,}$ with a square bracket denoting the order

${\mathfrak{D}}_{\alpha}\omega\;=\;[{\mathfrak{D}}_{\alpha}\omega]_{0}+{\alpha^{\backprime}\,}[{\mathfrak{D}}_{\alpha}\omega]_{1}+\ldots~{},~{}~{}~{}\xi_{\alpha}\;=\;[\xi_{\alpha}]_{0}+{\alpha^{\backprime}\,}[\xi_{\alpha}]_{1}+\cdots$

At zeroth order in ${\alpha^{\backprime}\,}$, as we are assuming a large
radius limit so that SUGRA embeds into string theory, we have
$[{\text{d}}\omega]_{0}=0$. The right hand side of the first equation in
(4.17a) is therefore $\mathcal{O}({\alpha^{\backprime}\,})$ and so
$[{\overline{\partial}}^{\dagger}\xi_{\alpha}^{(1,2)}]_{0}=0$ to this order.
Hence, $[{\mathfrak{D}}_{\alpha}\omega^{(1,1)}]_{0}$ is harmonic with respect
to the zeroth order Calabi-Yau metric $[g_{\mu{\overline{\nu}}}]_{0}$. Also,
$[{\mathfrak{D}}_{\alpha}\omega^{(0,2)}]_{0}=0$ as follows by the calculation
in appendix §B.3. Using equivalence of Kähler Laplacians, the right hand side
of (4.17b) also vanishes and so $[\Delta_{\alpha}{}]_{0}$ is harmonic with
respect to $[g_{\mu{\overline{\nu}}}]_{0}$. Finally, the sources in (4.17c)
completely determines $[{\mathfrak{D}}_{\alpha}\mathcal{A}]_{0}$. In summary,
$[{\mathfrak{D}}_{\alpha}\omega]_{0}$, $[\Delta_{\alpha}{}]_{0}$ are both
harmonic with respect to the Calabi-Yau metric while
$[{\mathfrak{D}}_{\alpha}\mathcal{A}]_{0}$ is harmonic at zeroth order only
when $[\Delta_{\alpha}]_{0}=[{\mathfrak{D}}_{\alpha}\omega^{(1,1)}]_{0}=0$.

We now do the first order in ${\alpha^{\backprime}\,}$ analysis. Due to
${\partial}\omega=\mathcal{O}({\alpha^{\backprime}\,})$ the right hand side of
the first equation in (4.17a) is determined by $[\Delta_{\alpha}]_{0}$ and
$[{\mathfrak{D}}_{\alpha}\mathcal{A}]_{0}$. Hence,
$[{\mathfrak{D}}_{\alpha}\omega]_{1}$ can be computed by inverting the
Laplacian and using the ${\alpha^{\backprime}\,}$–corrected harmonic
representative ${\mathfrak{D}}_{\alpha}\omega^{(1,1)\,harm}$. We have
determined $[{\mathfrak{D}}_{\alpha}\omega^{(1,1)}]_{1}$. This can then be
used in (4.17b). Neglecting the term
$({\mathfrak{D}}_{\alpha}\omega_{{\overline{\mu}}{\overline{\nu}}})({\partial}\omega)^{{\overline{\mu}}{\overline{\nu}}\rho}=\mathcal{O}({\alpha^{\backprime}\,}^{2})$,
we can solve for $[\kappa_{\alpha}{}^{\mu}]_{1}$ and determine
$[\Delta_{\alpha}{}^{\mu}]_{1}$. Finally, substitute
$[{\mathfrak{D}}_{\alpha}\omega^{(1,1)}]_{1}$ and
$[\Delta_{\alpha}{}^{\mu}]_{1}$ into (4.17c) and solve for
$[{\mathfrak{D}}_{\alpha}\mathcal{A}]_{1}$. We are finished to
$\mathcal{O}({\alpha^{\backprime}\,})$.

###  4.3 Comments

We pause to make some comments on results so far. First, we now see how
quantum corrections modify the field deformations: they do so by the exact and
co-exact pieces. These pieces are determined by contractions of zeroth order
harmonic representatives with the background fields. The harmonic
representatives are a linear combination of elements of $H^{1}(X,\mathpzc
T_{X}^{(0,1)})\oplus H^{1}(X,\mathpzc T_{X}^{(1,0)})\oplus
H^{1}(X,\mbox{End}\,V)$. The role of Atiyah equation and anomaly cancellation
is to determine which linear combinations to take. For example, we need
conditions like

$\begin{split}&[\Delta_{\alpha}{}^{\mu}]_{0}\,[F_{\mu}]_{0}\quad\text{and}\quad[\Delta_{\alpha}{}^{\mu}]_{1}\,[F_{\mu}]_{0}+[\Delta_{\alpha}{}^{\mu}]_{0}\,[F_{\mu}]_{1}\;=\;{\overline{\partial}}_{\mathcal{A}}\text{--\,exact}~{},\\\
&[\Delta_{\alpha}{}^{\mu}]_{0}\,[({\partial}\omega)_{\mu}]_{1}-\frac{\text{i}}{4}\Big{[}\text{Tr}\hskip
2.0pt({\mathfrak{D}}_{\alpha}\mathcal{A}\,F)-2\,\big{(}\nabla_{\nu}\Delta_{\alpha}{}^{\mu}+\text{i}\,\nabla^{\mu}\,{\mathfrak{D}}_{\alpha}\omega_{\nu}^{(0,1)}\big{)}R^{\nu}{}_{\mu}\Big{]}_{0}\;=\;{\overline{\partial}}\text{--\,exact}~{}.\end{split}$
(4.18)

The elements that satisfy these equations form a vector space which is related
to a first cohomology of an appropriate bundle, see [7, 8, 9, 10, 11] building
on the work of [12] who describe the zeroth order in ${\alpha^{\backprime}\,}$
deformations at the level of a non–linear sigma models, taking into account
gauge symmetries. However, in constructing these bundles it is convenient to
treat deformations of the spin connection ${\mathfrak{D}}_{\alpha}\Theta$ as
independent degrees of freedom. Doing so gives rise to additional but
unphysical directions in the fibres of the bundle and one needs to presumably
perform an additional quotient to get the physical degrees of freedom. In
(4.18) we have not treated ${\mathfrak{D}}_{\alpha}\Theta$ as an independent
degree of freedom. Instead we have used (4.2) relating
${\mathfrak{D}}_{\alpha}\Theta$ to the metric moduli. This should amount to a
quotient of the cohomology but seeing this remains future work.

Second, the analysis performed here is complementary to finding how the
background fields are corrected perturbatively in ${\alpha^{\backprime}\,}$,
see for example [13, 14, 15, 4]. In these works the gauge-fixing for the
corrections to the background is discussed. Recall that we assume a
perturbative string background in which the 10–dimensional dilaton is a
constant, see [4] for more details.

Third, the field $\Delta_{\alpha}$ describes a holomorphic deformation of
complex structure. Note that it is perturbatively corrected in
${\alpha^{\backprime}\,}$. This is unlike the non–renormalisation that occurs
in some type II theories.

Fourth, it is unlikely that one can keep $X$ fixed and vary the bundle at
first order in ${\alpha^{\backprime}\,}$. To see this, suppose we try to hold
$X$ fixed. Then $[\Delta_{\alpha}]_{0}=[{\mathfrak{D}}_{\alpha}\omega]_{0}=0$
and $[{\mathfrak{D}}_{\alpha}\mathcal{A}]_{0}$ are harmonic. At first order in
${\alpha^{\backprime}\,}$, (4.17a) implies that
$[{\mathfrak{D}}_{\alpha}\omega^{(1,1)}]_{1}$ is turned on and sources
(4.17b)–(4.17c), unless the following cancellation happens

$\Big{[}\text{Tr}\hskip
2.0pt{({\mathfrak{D}}_{\alpha}\mathcal{A}^{harm}\,F)}\Big{]}_{0}\;=\;0~{},$

which is a much stronger condition than (4.18). This phenomenon arose because
the conformally balanced and HYM equation, together with gauge fixing
holomorphic gauge, coupled these degrees of freedom together.

Fifth, the fixing of small gauge transformations is a necessary part of the
map between parameters and physical field deformations. Our choice of
holomorphic gauge is natural for a supersymmetric theory in which the moduli
space is complex. Whether there exists choice of gauge in which (4.17c)
further simplify is far from clear.

###  4.4 An application to the moduli space metric

We can apply these results to the Kähler moduli space metric derived in [1, 2]

$\begin{split}g^{\sharp}_{\alpha{\overline{\beta}}}&\;=\;\frac{1}{V}\int_{X}\Big{(}\Delta_{\alpha}{}^{\mu}\star\Delta_{{\overline{\beta}}}{}^{{\overline{\nu}}}\,g_{\mu{\overline{\nu}}}+{\mathfrak{D}}_{\alpha}\omega^{(1,1)}\star{\mathfrak{D}}_{{\overline{\beta}}}\omega^{(1,1)}+\frac{{\alpha^{\backprime}\,}}{4}\text{Tr}\hskip
2.0pt{({\mathfrak{D}}_{\alpha}A\star{\mathfrak{D}}_{{\overline{\beta}}}A)}\Big{)}\\\\[2.84544pt]
&\hskip
85.35826pt~{}+~{}\frac{{\alpha^{\backprime}\,}}{2V}\int_{X}\mbox{vol}~{}\Big{(}\Delta_{\alpha{\overline{\mu}}{\overline{\nu}}}\,\Delta_{{\overline{\beta}}\rho\sigma}+{\mathfrak{D}}_{\alpha}\omega_{\rho{\overline{\mu}}}\,{\mathfrak{D}}_{{\overline{\beta}}}\omega_{\sigma{\overline{\nu}}}\Big{)}\,R^{{\overline{\mu}}\rho{\overline{\nu}}\sigma}+\mathcal{O}({\alpha^{\backprime}\,}^{2})~{}.\end{split}$
(4.19)

This metric is a series of inner products together with an
${\alpha^{\backprime}\,}$–correction involving a contraction with the Riemann
tensor. The term involving the Riemann tensor appears because of (4.2) as
mentioned above.

A nice thing that happens is that the exact and co–exact terms in (4.16)
either cancel or contribute in a way that is higher order in
${\alpha^{\backprime}\,}$. This is due to harmonic, exact and co–exact forms
being mutually orthogonal. The ${\alpha^{\backprime}\,}$–corrections to the
harmonic forms are also orthogonal. For example, writing

$[\Delta_{\alpha}{}^{harm}]\;=\;[\Delta_{\alpha}{}^{harm}]_{0}+{\alpha^{\backprime}\,}[\Delta_{\alpha}{}^{harm}]_{1}~{}.$

We may as well take $[\Delta_{\alpha}{}^{harm}]_{1}$ to be orthogonal to
$[\Delta_{\alpha}{}^{harm}]_{0}$ as is done in say [4]. Any contribution which
is “parallel” is just a change of basis and represents a vacuous change.
Hence,

$\int_{X}\Delta_{\alpha}{}^{\mu}\star\Delta_{\overline{\beta}}{}^{\overline{\nu}}\,g_{\mu{\overline{\nu}}}\;=\;\int_{X}[\Delta_{\alpha}{}^{\mu}]_{0}\star[\Delta_{\overline{\beta}}{}^{\overline{\nu}}]_{0}\,g_{\mu{\overline{\nu}}}~{}+~{}\mathcal{O}({\alpha^{\backprime}\,}^{2})~{},$

A similar result applies for the other inner products. Perhaps what was not
obvious before the analysis in this section was whether there could be
corrections to ${\mathfrak{D}}_{\alpha}\omega$ or $\Delta_{\alpha}$ which
would contribute to the metric at $\mathcal{O}({\alpha^{\backprime}\,})$. We
now see the answer is no; the ${\alpha^{\backprime}\,}$–corrections come
either from $\star$ or are written explicitly as above. It would be
interesting to see how if this structure persists at next order in
${\alpha^{\backprime}\,}$.

We can give a concrete prescription for computing the metric

$\begin{split}g^{\sharp}_{\alpha{\overline{\beta}}}&=\frac{1}{V}\int_{X}\Big{(}[\Delta_{\alpha}{}^{\mu}]_{0}\star[\Delta_{{\overline{\beta}}}{}^{{\overline{\nu}}}]_{0}~{}g_{\mu{\overline{\nu}}}+[{\mathfrak{D}}_{\alpha}\omega^{(1,1)}]_{0}\star[{\mathfrak{D}}_{\overline{\beta}}\omega^{(1,1)}]_{0}+\frac{{\alpha^{\backprime}\,}}{4}\,\text{Tr}\hskip
2.0pt{\big{(}[{\mathfrak{D}}_{\alpha}A]_{0}\star[{\mathfrak{D}}_{\overline{\beta}}A]_{0}\big{)}}\Big{)}\\\\[2.84544pt]
&\hskip
28.45274pt~{}+~{}\frac{{\alpha^{\backprime}\,}}{2V}\int_{X}\Big{[}\mbox{vol}\,\Big{(}\Delta_{\alpha\,{\overline{\mu}}{\overline{\nu}}}\,\Delta_{{\overline{\beta}}\rho\sigma}+{\mathfrak{D}}_{\alpha}\omega_{\rho{\overline{\mu}}}\,{\mathfrak{D}}_{\overline{\beta}}\omega_{\sigma{\overline{\nu}}}\Big{)}R^{{\overline{\mu}}\rho{\overline{\nu}}\sigma}\Big{]}_{0}+\mathcal{O}({\alpha^{\backprime}\,}^{2})~{}.\end{split}$
(4.20)

The terms $[\Delta_{\alpha}{}^{\mu}]_{0}$ and
$[{\mathfrak{D}}_{\alpha}\omega^{(1,1)}]_{0}$ are the zeroth order harmonic
forms. The only ${\alpha^{\backprime}\,}$–correction in the first two terms
comes from background metric $g_{\mu{\overline{\nu}}}$, and their computation
are discussed in [14, 15, 4].

We are to view this as the metric on the quantum corrected cohomology ring
corresponding to heterotic moduli. For models connected to the standard
embedding this ring is computed using gauged linear sigma models in [16, 17,
18], while a mathematics discussion is in [19, 20]; see [21, 22, 23] for some
reviews. Note that this is physical string theory metric, in which the spin
connection is determined in terms of the remaining moduli. The lowest order
contribution was determined in [1] and from a different point of view in [4].

##  5 First order universal geometry and gauge transformations

In the previous section we gauge-fixed and used the equations of motion to
isolate the physical degrees of freedom. In this section, we put this into the
context of the universal geometry constructed in [1, 2]. We will find that a
small transformations correspond to a first order deformation of a connection
on the moduli space. This means that the the choice of connection on $M$
corresponds to a choice of gauge. This makes sense: neither the connection on
$M$ nor the choice of gauge are physical quantities and so should not enter
the physical theory.

So far we have not been too detailed about parameter space derivatives. They
are needed for general covariance of the theory as symmetries can depend on
coordinates of $X$ but also on parameters.

###  5.1 Why universal geometry?

In the classic works on complex structure moduli space for CY manifolds, such
as [24, 25], the authors benefited from the existence of special coordinates
called periods of $\Omega$. In those coordinates $\Omega$ takes a simple form

$\Omega(x,z)\;=\;z^{a}\,\alpha_{a}(x)-\mathcal{G}_{a}(z)\,\beta^{a}(x)~{},$
(5.1)

where $\alpha_{a},\beta^{a}$ are a symplectic basis for $H^{3}(X)$. Within
this choice of special coordinates it is sensible to compute deformations
through a partial derivative ${\partial}_{a}$. This is because the tensorial
part of $\Omega$ does not depend on parameters: the forms $\alpha_{a}$ and
$\beta^{a}$ are “constant” across $M$. This feature remains true as long as
diffeomorphisms and symplectic rotations do not depend on parameters.

The same story goes for the field $B+\text{i}\,\omega$, the so-called Kähler
cone. There exist special coordinates on $M$ in which this depends linearly on
parameters

$(B+\text{i}\,\omega)(x,t)=t^{a}\,\omega_{a}(x)~{},$ (5.2)

with $\omega_{a}$ a choice of “constant” harmonic $(1,1)$-forms.

The existence of coordinates on $M$ of this type is a remarkable feature. One
might wonder, given the intricate analysis of the previous section, if similar
coordinates exist for heterotic moduli. If one wants to investigate without
assuming that special coordinates like above exist, a more covariant formalism
is needed. Universal geometry allows to do that.

The caveat – or the gain, according to one’s view – of universal geometry is
that one needs to study connections on the moduli space and their associated
curvatures, as well as higher order tensors. In particular, we cannot really
assume that deformations commute $[{\mathfrak{D}}_{a},{\mathfrak{D}}_{b}]=0$
as a starting point. We will see in the next section how important is the role
of gauge fixing: curvatures have their own small gauge transformations and
these affect commutativity.

###  5.2 Small gauge transformations on a fixed manifold $X$

We begin with one of the simplest cases: the moduli space of bundle
deformations on a fixed manifold $X$. In that case, we consider the action of
only small gauge transformations. In the notation of [1, 2], we denote
$\mathbbl{X}=X\times M$ and $\mathbbl{A}=A+\Lambda$, where
$\Lambda=\Lambda_{a}\,{\text{d}}y^{a}$, is the connection for the universal
bundle $\mathpzc U\to X\times M$. This discussion at the end of the last
section makes it clear that this is only really valid in the
${\alpha^{\backprime}\,}\to 0$ limit of a heterotic theory.

The covariant parameter derivative of $A$ on a fixed manifold $X$ is

${\mathfrak{D}}_{a}A\;=\;{\partial}_{a}A-{\text{d}}_{A}\Lambda_{a}~{},$ (5.3)

and as observed in [1, 2] this coincides with a component of the field
strength $\mathbbl{F}_{am}{\text{d}}x^{m}={\mathfrak{D}}_{a}A$. For the moment
we consider real coordinates and forms; holomorphy will follow shortly. This
derivative was constructed in [1, 2] to be manifestly covariant under
background gauge transformations. However, as we discovered in the previous
section, there is an action of small gauge transformations

${\mathfrak{D}}_{a}A~{}\to~{}{\mathfrak{D}}_{a}A+{\text{d}}_{A}\phi_{a}~{},$
(5.4)

for some End$\,\mathpzc E$-valued $\phi_{a}$. The key observation is that by
comparing (5.3)-(5.4) we see that a small gauge transformation precisely
corresponds to a deformation of the connection

$\Lambda_{a}~{}\to~{}\Lambda_{a}-\phi_{a}~{}.$ (5.5)

This deformation, in general, modifies the field strength
$\mathbbl{F}_{ab}={\partial}_{a}\Lambda_{b}-{\partial}_{b}\Lambda_{a}+[\Lambda_{a},\Lambda_{b}]$
as

$\mathbbl{F}_{ab}~{}\to~{}\mathbbl{F}_{ab}-{\mathfrak{D}}_{a}\phi_{b}+{\mathfrak{D}}_{b}\phi_{a}~{},\qquad{\mathfrak{D}}_{a}\phi_{b}\;=\;{\partial}_{a}\phi_{b}+[\Lambda_{a},\phi_{b}]~{}.$
(5.6)

The geometric picture is the following. Consider the space of all connections
$\mathpzc A$ and pick a base point $A$. The space $\mathpzc A$ can be thought
of as a fibration over $M$ [5], whose fibres are the gauge orbits generated by
(1.3). Deformations of the basepoint $A$ along the gauge orbit are regarded as
trivial, that is $\delta A\sim\delta A+{\text{d}}_{A}\phi$. Non–trivial
deformations are defined by introducing a connection, which is a 1-form on the
base $M$ of the fibration. The Kodaira–Spencer map identifies gauge
inequivalent deformations of connections $\delta A$ with the tangent space
$\mathpzc T_{M}$ [26]. In fact, the mixed component $\mathbbl{F}_{am}$ is an
explicit realisation of the Kodaira–Spencer map $T_{y}M\to\Omega^{1}(X,{\rm
End}\,\mathpzc E)$. This is the same in spirit as the Donaldson map used in
Donaldson theory. This mixed component is precisely the covariant derivative
in (5.3). We now see that the ambiguity in the choice of deformation $\delta
A$ in $\mathpzc A$ corresponds to a choice of connection on $M$. Viewed in the
language of $\mathpzc A$ therefore, it is not surprising that we have
discovered that small gauge transformations correspond to deformations of the
connection $\Lambda_{a}{\text{d}}y^{a}$ on the moduli space $M$.

We now introduce holomorphy. In [1, 2] we took this to mean that the bundle
$\mathpzc U\to\mathbbl{X}$ is itself a holomorphic bundle, and so
$\mathbbl{F}^{(0,2)}=0$. The mixed component of this equation is
${\mathfrak{D}}_{\overline{\alpha}}\mathcal{A}=0$. We now know this is
actually part of holomorphic gauge. This is not surprising: asserting
$\mathpzc U$ is a holomorphic bundle involves a choice of connection
$\Lambda_{\overline{\alpha}}$ and the space of such connections is related to
gauge fixing. Indeed, if we deform
$\Lambda_{\overline{\alpha}}\to\Lambda_{\overline{\alpha}}-\phi_{\overline{\alpha}}$,
then we see that holomorphy is preserved if

${\overline{\partial}}_{\mathcal{A}}\phi_{\overline{\alpha}}\;=\;0~{},\qquad{\mathfrak{D}}_{\overline{\alpha}}\phi_{\overline{\beta}}\;=\;{\mathfrak{D}}_{\overline{\beta}}\phi_{\overline{\alpha}}~{}.$

The first equation is discussed in the first paragraph of (3.3), and the only
possible solutions are parameter dependent constants. In particular, it means
that $\phi_{\overline{\alpha}}$ must commute with the algebra ${\mathfrak{g}}$
as alluded to above. The second equation we will interpret in the next
section. It is a further constraint on the parameter dependence of gauge
transformations $\phi_{\overline{\alpha}}$, which are associated with gauge
fixing second order deformations. We note that
$\phi_{\overline{\alpha}}\sim\phi_{\overline{\alpha}}+{\mathfrak{D}}_{\overline{\alpha}}\psi$
is a symmetry of this equation.

Consider the role of gauge symmetries and the structure group of $\mathpzc U$.
The connection $\mathbbl{A}$ transforms under a gauge transformation

$\mathbbl{A}\to\;^{\Phi U}\mathbbl{A}\;=\;{\Phi
U}(\mathbbl{A}-\mathbbl{Y}){\Phi U}^{-1}~{},\qquad\mathbbl{Y}\;=\;{\Phi
U}^{-1}\mathbbl{d}{\Phi U}~{}.$ (5.7)

where ${\Phi U}$ is an element of the structure group of the universal bundle
$\mathpzc U$. If we fix $y\in M$ then the universal bundle $\mathpzc U$ is
such that it reduces to $\mathpzc E\to X$ with structure algebra
${\mathfrak{g}}$. We take the structure algebra of $\mathpzc U$ to be
${\mathfrak{g}U}={\mathfrak{g}}\oplus{\mathfrak{g}}_{b}$, where we are leaving
${\mathfrak{g}}_{b}$ arbitrary apart from it being a real, semi-simple and
compact Lie algebra and $[{\mathfrak{g}U},{\mathfrak{g}}_{b}]={\mathfrak{g}}$.
The connection $A$ is taken to be valued in ${\mathfrak{g}}$, while $\Lambda$
is valued in ${\mathfrak{g}U}$. We also demand that
${\mathfrak{D}}_{a}A\in\Omega^{1}(X,{\rm End}\mathpzc E)$ and so is in
particular valued in ${\rm ad}_{\mathfrak{g}}$.

We see that (5.7) decomposes into two separate transformation laws

$A\to\;^{\Phi}\\!A\;=\;\Phi(A-Y)\Phi^{-1}~{},\quad\Lambda\to\;^{\Phi
U}\Lambda_{a}\;=\;{\Phi U}(\Lambda_{a}-Y_{a}){\Phi U}^{-1}~{},\quad
Y_{a}\;=\;{\Phi U}^{-1}{\partial}_{a}{\Phi U}~{},$

where $\Phi$ is exponentiation of the Lie algebras ${\mathfrak{g}}$. If we
take the limit $\Phi=1-\phi$ and ${\Phi U}=1-{\phi U}$, then we can gain some
intuition for small gauge transformations. The transformation law reduces to

$A\to
A+{\text{d}}_{A}\phi~{},\qquad\Lambda_{a}\to\Lambda_{a}+{\mathfrak{D}}_{a}{\phi
U}~{}.$

We have already studied the deformations in the first equation. So let us for
now restrict ourselves to those of the form
${\text{d}}_{A}\phi={\text{d}}_{A}{\phi U}\;=\;0$. We take this to mean that
$\phi,{\phi U}$ depend only on parameters and commute with ${\mathfrak{g}}$.
This means they are ${\mathfrak{g}}_{b}$–valued. If we denote
$\phi_{a}={\mathfrak{D}}_{a}{\phi U}$, then we see that $\mathbbl{F}$
transforms as

$F\to
F~{},\quad{\mathfrak{D}}_{a}A\to{\mathfrak{D}}_{a}A~{},\quad\mathbbl{F}_{ab}\to\mathbbl{F}_{ab}+[{\mathfrak{D}}_{a},{\mathfrak{D}}_{b}]{\phi
U}\;=\;\mathbbl{F}_{ab}+[\mathbbl{F}_{ab},{\phi U}]~{}.$

where we use ${\text{d}}_{A}({\mathfrak{D}}_{a}{\phi
U})={\mathfrak{D}}_{a}({\text{d}}_{A}{\phi U})-[{\mathfrak{D}}_{a}A,{\phi
U}]=0$. Hence, there are deformations of $\Lambda_{a}$ that do not modify $A$
nor ${\mathfrak{D}}_{a}A$; these are like gauge-for-gauge transformations.

An curious aside: recall the space of deformations of holomorphic bundles
$\mathpzc E\to X$ is a moduli space $M$. We see here that the space of
deformations of $\mathpzc U$ with $\delta A=0$ reduces to the space of
deformations of holomorphic $\mathfrak{g}_{b}$–bundles on $M$. We leave this
for further work.

###  5.3 Small gerbes

Just as $\mathbbl{F}_{a}={\mathfrak{D}}_{a}A$ came from the mixed term of a
field strength, the form $\mathpzc B_{a}$ in (2.12) is most elegantly defined
as most easily follows by considering the mixed component of $\mathbbl{H}$:

$\mathpzc
B_{a}\;=\;\mathbbl{H}_{a}\;=\;{\mathfrak{D}}_{a}B+\frac{{\alpha^{\backprime}\,}}{4}\text{Tr}\hskip
2.0pt{(A\,{\mathfrak{D}}_{a}A)}-{\text{d}}\mathbbl{B}_{a}~{},\quad{\mathfrak{D}}_{a}B\;=\;{\partial}_{a}B-\frac{{\alpha^{\backprime}\,}}{4}\text{Tr}\hskip
2.0pt(\Lambda_{a}\,{\text{d}}A)~{}.$ (5.8)

This gives an interpretation to the 1–form $\mathbbl{B}_{a}$ as the mixed
component $\mathbbl{B}_{am}$ of the universal $\mathbbl{B}$–field. Under
background gauge transformations
$\mathbbl{B}\sim\mathbbl{B}+\frac{{\alpha^{\backprime}\,}}{4}\text{Tr}\hskip
2.0pt\mathbbl{Y}\mathbbl{A}+\frac{{\alpha^{\backprime}\,}}{4}\mathbbl{U}$, and
applying this to $\mathbbl{B}_{a}$ and $\mathpzc B_{a}$ we find exactly the
transformation law (2.13).

We immediately see that small gerbe transformations $\mathpzc B_{a}\to\mathpzc
B_{a}+{\text{d}}\mathfrak{b}_{a}$ are realised by a deformation

$\mathbbl{B}_{a}\to\mathbbl{B}_{a}-\mathfrak{b}_{a}~{},$ (5.9)

and so we view $\mathbbl{B}_{a}$ in an analogous fashion to $\Lambda_{a}$ for
the gauge transformations: it is a ‘gerbe–connection’ on the moduli space $M$.
The tangibility $[2,1]$ of $\mathbbl{H}$ is also deformed

$\mathbbl{H}_{ab}~{}\to~{}\mathbbl{H}_{ab}-{\mathfrak{D}}_{a}\mathfrak{b}_{b}+{\mathfrak{D}}_{b}\mathfrak{b}_{a}~{}.$
(5.10)

If we compare this with the gauge transformation law for the the field
strength $\mathbbl{F}_{ab}$ in (5.6), then we see that $\mathbbl{B}_{a}$ is
naturally viewed as a connection for gerbe transformations and
$\mathbbl{H}_{ab}$ is its related field strength.

In the previous section we deduced that small gauge transformations,
parameterised by $\phi_{a}$, result in a transformation law for $\mathpzc
B_{a}$ in (2.15) and that for ${\mathfrak{D}}_{a}B$ in (2.16). The realisation
of this in universal geometry is more subtle. While a deformation
$\Lambda_{a}\to\Lambda_{a}-\phi_{a}$ gives the correct law for
${\mathfrak{D}}_{a}A$, we have

${\mathfrak{D}}_{a}B\to{\mathfrak{D}}_{a}B+\frac{{\alpha^{\backprime}\,}}{4}\text{Tr}\hskip
2.0pt(\phi_{a}{\text{d}}A)~{},$

which does not agree with (2.15). We are missing the fact that
$\mathbbl{B}_{a}$ is to be simultaneously deformed. This is a manifestation of
the fact the $B$–field is charged under the gauge symmetry. The correct
deformation turns out to be

$\Lambda_{a}\to\Lambda_{a}-\phi_{a}~{},\quad{\rm
and}\quad\mathbbl{B}_{a}\to\mathbbl{B}_{a}-\frac{{\alpha^{\backprime}\,}}{4}\text{Tr}\hskip
2.0pt\left(A\,\phi_{a}\right)~{}.$ (5.11)

It is an exercise to see that $\mathbbl{H}_{a}$ transforms in the same manner
as (2.15).

To reconcile with (2.16) one could redefine ${\mathfrak{D}}_{a}B$ as follows

$\mathpzc
B_{a}\;=\;{\mathfrak{D}}_{a}B+\frac{{\alpha^{\backprime}\,}}{4}\text{Tr}\hskip
2.0pt{(A\,{\mathfrak{D}}_{a}A)}-{\text{d}}\left(\mathbbl{B}_{a}+\frac{{\alpha^{\backprime}\,}}{4}(\text{Tr}\hskip
2.0pt\Lambda_{a}A)\right)~{},\quad{\mathfrak{D}}_{a}B\;=\;{\partial}_{a}B-\frac{{\alpha^{\backprime}\,}}{4}\text{Tr}\hskip
2.0pt(A\,{\text{d}}\Lambda_{a})~{}.$

If we do so, we see that under (5.11) the derivative ${\mathfrak{D}}_{a}B$ has
the correct transformation law. We note that this definition of the covariant
derivative of the B-field differs from [2] by a d–exact term. The symmetry
property (2.16) gave us no choice in this matter. Physically, there is no
difference as ${\mathfrak{D}}_{a}B$ is only ever defined up to d–closed forms,
a manifestation of its parent ${\text{d}}B$.

Higher tangibilities of the three-form $\mathbbl{H}$ also transform. For
example,

$\begin{split}\mathbbl{H}_{ab}&~{}\to~{}\mathbbl{H}_{ab}+\frac{{\alpha^{\backprime}\,}}{2}\text{Tr}\hskip
2.0pt{(\phi_{a}\,{\mathfrak{D}}_{b}A-\phi_{b}\,{\mathfrak{D}}_{a}A)}~{},\end{split}$

where in analogy with (5.11) we defined the rule

$\mathbbl{B}_{ab}~{}\to~{}\mathbbl{B}_{ab}-\frac{{\alpha^{\backprime}\,}}{4}\text{Tr}\hskip
2.0pt{(\phi_{a}\,\Lambda_{b}-\phi_{b}\,\Lambda_{a})}~{}.$

###  5.4 Small diffeomorphisms

We now suppose that $X$ and the bundle $\mathpzc E$ can simultaneously deform.
As in [1] we interpret diffeomorphisms as a gauge symmetry. This necessitates
introducing an invariant basis of forms $e^{m},{\text{d}}y^{a}$ and vectors
${\partial}_{m},e_{a}$ where

$e^{m}={\text{d}}x^{m}+c_{a}{}^{m}{\text{d}}y^{a},\qquad
e_{a}={\partial}_{a}-c_{a}{}^{m}{\partial}_{m}~{},$

which factorise the hermitian form on $\mathbbl{X}$ into a block diagonal form

$\mathbbl{\hskip
1.0pt\bbomega}\;=\;\frac{1}{2}\omega_{mn}e^{m}e^{n}+\frac{1}{2}\omega^{\sharp}_{ab}{\text{d}}y^{a}{\text{d}}y^{b}~{}.$
(5.12)

The symbol $c_{a}{}^{m}$ transforms like a connection under diffeomorphisms
and its role will be analogous to $\Lambda_{a}$. As described in [1] we need
to express quantities in the ‘e’–basis, and if there is any ambiguity will
denote the result by a superscript ${\sharp}$ symbol. A key example is the
gauge field $\mathbbl{A}$. When expressed in the ‘e’–basis it has a
transformed component

$\mathbbl{A}=A_{m}e^{m}+A^{\sharp}_{a}{\text{d}}y^{a}~{},\qquad
A^{\sharp}_{a}\;=\;\Lambda_{a}-A_{m}c_{a}{}^{m}~{}.$

Consider a form $\eta$ without any additional gauge symmetries and whose legs
lie purely along the manifold $X$. Then a result derived in [1] is that its
deformation $\delta\eta$ is a Lie derivative with respect to the vectors
$e_{a}$:

$\delta\eta\;=\;\delta y^{a}\mathcal{L}_{e_{a}}\eta~{}.$ (5.13)

Due to the fibration structure of $\mathbbl{X}$, the Lie derivative functions
as a directional direction, ${\mathfrak{D}}_{a}=\mathcal{L}_{e_{a}}\eta$ and
it is helpful to write out the expression explicitly

${\mathfrak{D}}_{a}\eta\;=\;\frac{1}{q!}\Big{(}e_{a}(\eta_{m_{1}\cdots
m_{q}})-c_{a}{}^{n}{}_{,\,m_{1}}\,\eta_{nm_{2}\cdots
m_{q}}-c_{a}{}^{n}{}_{,\,m_{2}}\,\eta_{m_{1}n\cdots
m_{q}}-\cdots\Big{)}\,e^{m_{1}\cdots m_{q}}~{}.$ (5.14)

If we have a tensor of the form

$\xi\;=\;\xi{}_{n_{1}\cdots n_{s}}{}^{m_{1}\cdots m_{r}}\
e^{n_{1}}\otimes\cdots\otimes
e^{n_{s}}\otimes{\partial}_{m_{1}}\\!\otimes\cdots\otimes{\partial}_{m_{r}}~{},$

then its derivative is defined as

$\begin{split}{\mathfrak{D}}_{a}\,\xi{}_{n_{1}\cdots n_{s}}{}^{m_{1}\cdots
m_{r}}&\;=\;e_{a}(\xi{}_{n_{1}\cdots n_{s}}{}^{m_{1}\cdots
m_{r}})+c_{a}{}^{m_{1}}{}_{,\,k}\,\xi{}_{n_{1}\cdots n_{s}}{}^{km_{2}\cdots
m_{r}}+\cdots+c_{a}{}^{m_{r}}{}_{,\,k}\,\xi_{n_{1}\cdots n_{s}}{}^{m_{1}\cdots
m_{r-1}k}\\\\[2.84544pt] &\hskip 30.0pt-
c_{a}{}^{k}{}_{,\,n_{1}}\,\xi{}_{kn_{2}\cdots n_{s}}{}^{m_{1}\cdots
m_{r}}-\cdots-c_{a}{}^{k}{}_{,\,n_{s}}\,\xi{}_{n_{1}\cdots
n_{s-1}k}{}^{m_{1}\cdots m_{r}}~{}.\end{split}$ (5.15)

As we introduce additional gauge symmetries and associated connections we will
generalise this covariant derivative. For ease of notation, we will use the
same symbol except where a possible ambiguity may arise.

A small diffeomorphism,
$\delta\eta\to\delta\eta+\mathcal{L}_{\varepsilon}\eta$ is realised by a
deformation of the connection $c_{a}$:

$c_{a}~{}\to~{}c_{a}-\varepsilon_{a}~{},\qquad\varepsilon_{a}\;=\;\varepsilon_{a}{}^{m}{\partial}_{m}~{}.$
(5.16)

It is elementary to see that

${\mathfrak{D}}_{a}\eta~{}\to~{}{\mathfrak{D}}_{a}\eta+\mathcal{L}_{\varepsilon_{a}}\eta~{}.$
(5.17)

The connection $c_{a}$ has a curvature tensor defined as the Lie bracket of
two vertical vectors $S_{ab}{}^{m}=[e_{a},e_{b}]^{m}$. Under the deformation
(5.16) we find

$S_{ab}{}^{m}~{}\to~{}S_{ab}{}^{m}+{\mathfrak{D}}_{a}\varepsilon_{b}{}^{m}-{\mathfrak{D}}_{b}\varepsilon_{a}{}^{m}~{}.$

As derived in [1], this curvature tensor appears in forms on $\mathbbl{X}$
with two or more legs along the moduli space. For example, in complex
coordinates

$(\mathbbl{d}^{c}\mathbbl{\hskip
1.0pt\bbomega})_{\alpha\beta}\;=\;-\text{i}\,S{}_{\alpha\beta}{}^{\mu}\,\omega_{\mu}~{},\quad(\mathbbl{d}^{c}\mathbbl{\hskip
1.0pt\bbomega})_{{\overline{\alpha}}{\overline{\beta}}}\;=\;\text{i}\,S{}_{{\overline{\alpha}}{\overline{\beta}}}{}^{\overline{\mu}}\,\omega_{\overline{\mu}}~{},\quad(\mathbbl{d}^{c}\mathbbl{\hskip
1.0pt\bbomega})_{\alpha\overline{\beta}}\;=\;-\text{i}\,S{}_{\alpha{\overline{\beta}}}{}^{{\overline{\mu}}}\,\omega_{{\overline{\mu}}}+\text{i}\,S{}_{\alpha\overline{\beta}}{}^{\mu}\,\omega_{\mu}\
.$

If we start with $S=0$ and demand it is preserved then

${\mathfrak{D}}_{a}\varepsilon_{b}{}^{m}-{\mathfrak{D}}_{b}\varepsilon_{a}{}^{m}\;=\;0~{}.$
(5.18)

Locally, this amounts to $\varepsilon_{a}{}^{m}={\mathfrak{D}}_{a}v{}^{m}$ for
some vertical vector field $v^{m}\,{\partial}_{m}$. See [1] for the definition
of ${\mathfrak{D}}_{a}$ acting on mixed component tensors.

The connection $c_{a}$ determines tangibility, and so is also modified by
deformations. In the following discussion it is convenient to compare the
deformed frame, denoted by a tilde with the undeformed frame. The basis of
1-forms deforms as

$e^{m}~{}\to~{}{{\widetilde{e}}\,}^{m}\;=\;e^{m}-{\text{d}}y^{a}\,\varepsilon_{a}{}^{m}~{}.$
(5.19)

This has interesting consequences. Consider the field strength $\mathbbl{F}$.
Firstly

$\begin{split}\mathbbl{F}&\;=\;\frac{1}{2}F_{mn}\,e^{mn}+\mathbbl{F}_{an}{\text{d}}y^{a}e^{n}+\frac{1}{2}\mathbbl{F}_{ab}{\text{d}}y^{ab}\\\\[4.0pt]
&\;=\;\frac{1}{2}F_{mn}\,{{\widetilde{e}}\,}^{mn}+(\mathbbl{F}_{an}+\varepsilon_{a}{}^{m}F_{mn}){\text{d}}y^{a}{{\widetilde{e}}\,}^{n}+\frac{1}{2}(\mathbbl{F}_{ab}+\mathbbl{F}_{an}\varepsilon_{b}{}^{n}-\mathbbl{F}_{bn}\varepsilon_{a}{}^{n}){\text{d}}y^{ab}\\\\[3.0pt]
\end{split}$ (5.20)

Comparing with $\mathbbl{F}$ expressed in terms of its tilded components:

$\mathbbl{F}\;=\;\frac{1}{2}F_{mn}\,{{\widetilde{e}}\,}^{mn}+\widetilde{\mathbbl}{F}_{an}{\text{d}}y^{a}{{\widetilde{e}}\,}^{n}+\frac{1}{2}\widetilde{\mathbbl}{F}_{ab}{\text{d}}y^{ab}~{},$

and identifying $\mathbbl{F}_{an}={\mathfrak{D}}_{a}A_{n}$ and
$\widetilde{\mathbbl{F}}_{an}=\widetilde{{\mathfrak{D}}}_{a}A_{n}$ we find a
transformation law

$\widetilde{{\mathfrak{D}}}_{a}A\;=\;{\mathfrak{D}}_{a}A+\varepsilon_{a}{}^{m}F_{m}~{},\qquad\widetilde{\mathbbl{F}}_{ab}\;=\;\mathbbl{F}_{ab}-\varepsilon_{a}{}^{n}{\mathfrak{D}}_{b}A_{n}+\varepsilon_{b}{}^{n}{\mathfrak{D}}_{a}A_{n}~{}.$

The second equation implies the curvature on $M$ is also deformed in a
non–trivial way. The first equation is exactly the transformation law under a
small diffeomorphisms as derived in supergravity. When applied to
${\mathfrak{D}}_{\alpha}\mathcal{A}$ and combined with $A^{\sharp}_{\alpha}\to
A^{\sharp}_{\alpha}-\phi_{\alpha}$ — note that we use $A^{\sharp}$ and not
$\Lambda$ as we are in the ‘e’–basis — we recover exactly (2.8)

$\widetilde{{\mathfrak{D}}}_{\alpha}\mathcal{A}\;=\;{\mathfrak{D}}_{\alpha}\mathcal{A}+\varepsilon_{\alpha}{}^{\mu}F_{\mu}+{\overline{\partial}}_{\mathcal{A}}\phi_{\alpha}~{}.$

In the same way, we determine that field strength $\mathbbl{H}$ transforms and
from this we deduce the transformation law for $\mathbbl{H}_{a}=\mathpzc
B_{a}$ under small diffeomorphisms

$\widetilde{\mathpzc}B_{a}=\mathpzc B_{a}+\varepsilon_{a}{}^{m}H_{m}~{}.$

Again this is in exact agreement with supergravity.

There is a complex structure on $\mathbbl{X}$

$\mathbbl{J}\;=\;J_{m}{}^{n}\,e^{m}\otimes{\partial}_{n}+J^{\sharp}{}_{a}{}^{b}\,{\text{d}}y^{a}\otimes
e_{b}~{}.$ (5.21)

The manifold $\mathbbl{X}$ being complex means the indices must be pure [1]

$c_{\alpha}{}^{{\overline{\mu}}}\;=\;0~{}.$ (5.22)

It is interesting to study how $\mathbbl{J}$ transforms under the change of
‘e’–basis in (5.19):

$\mathbbl{J}=J_{m}{}^{n}\,{{\widetilde{e}}\,}^{m}\otimes{\partial}_{n}+J^{\sharp}_{a}{\,}^{b}\,{\text{d}}y^{a}\otimes\widetilde{e}_{b}+(\varepsilon_{a}{}^{m}\,J_{m}{}^{n}-J^{\sharp}{}_{a}{}^{b}\,\varepsilon_{b}{}^{n})\,{\text{d}}y^{a}\,{\partial}_{n}~{}.$
(5.23)

At first we see the structure is no longer block diagonal, which is not
surprising considering with have deformed $c_{a}$. However, as the complex
structure is integrable, the Frobenius theorem tells us we can find a
compatible set of holomorphic coordinates and in these coordinates we see the
off–diagonal terms vanish when $\varepsilon_{\alpha}{}^{\overline{\mu}}=0$. So
deformations that preserve integrability of $\mathbbl{J}$ are identified with
ones that preserve its block diagonal nature. This has an echo in
supergravity. In the last section, we saw that part of holomorphic gauge is
$\Delta_{{\overline{\alpha}}{\overline{\nu}}}{}^{\mu}=0$. Small diffemorphisms
that preserve holomorphic gauge are of the form
$\varepsilon_{\overline{\alpha}}{}^{\mu}=0$. So we learn that holomorphic
gauge amounts to $\mathbbl{J}$ being integrable and block diagonal.

Furthermore, in [1] it was shown that
$\Delta_{\overline{\nu}}{}^{\alpha}=-{\partial}_{\overline{\nu}}c_{\alpha}{}^{\mu}$.
Hence, under a deformation $c_{\alpha}{}^{\mu}\to
c_{\alpha}{}^{\mu}-\varepsilon_{\alpha}{}^{\mu}$ we have exactly the correct
transformation law

$\Delta_{\alpha{\overline{\nu}}}{}^{\mu}~{}\to~{}\widetilde{\Delta}_{\alpha{\overline{\nu}}}{}^{\mu}\;=\;\Delta_{\alpha{\overline{\nu}}}{}^{\mu}+{\partial}_{\overline{\nu}}\varepsilon_{\alpha}{}^{\mu}~{}.$
(5.24)

Before we proceed we pause to introduce some technical details, which while
maybe mildy painful, but nonetheless are necessary to account for the
diffeomorphism symmetries correctly. Recall a form with $p$ legs along $M$ and
$q$ legs along $X$ is said to have tangibility $[p,q]$. We denote the set of
such forms by $\Omega^{[p,q]}(\mathbbl{X})$. The diffeomorphic invariant way
to distinguish legs in this way is through the ‘e’–basis. This means however,
we need to introduce a generalisation of the de Rahm operator that acts like
the de Rahm operator on $X$, but annihilates the bases $e^{m}$. Neither of the
operators $e^{m}{\partial}_{m}$ or ${\text{d}}x^{m}{\partial}_{m}$ achieve
this. So with a slight abuse of notation888In [1] we used the Iceland ‘deth’
symbol Ð for derivatives of corpus while here we denote this by the usual
d–operator; here we use ${\mathfrak{D}}$ which was denoted $\text{\DH}\hskip
1.0pt^{\sharp}$ in [1]. We hope by simplifying the number of derivative
symbols used, the paper will be easier to digest., we redefine d and
${\mathfrak{D}}$ to act as

$\displaystyle{\text{d}}\,f$
$\displaystyle\;=\;({\partial}_{m}f)\,e^{m}~{},\qquad$
$\displaystyle{\text{d}}\,e^{m}$ $\displaystyle\;=\;0~{},\quad$
$\displaystyle{\text{d}}\,({\text{d}}y^{a})$ $\displaystyle\;=\;0~{},$ (5.25)
$\displaystyle{\mathfrak{D}}f$
$\displaystyle\;=\;e_{a}(f)\,{\text{d}}y^{a}~{},\qquad$
$\displaystyle{\mathfrak{D}}e^{m}$
$\displaystyle\;=\;\\!-c_{a}{}^{m}{}_{,\,n}\,{\text{d}}y^{a}e^{n}~{},\hskip
10.0pt$ $\displaystyle{\mathfrak{D}}({\text{d}}y^{a})$
$\displaystyle\;=\;0~{}.$

It is straightforward to check that

${\text{d}}^{2}\;=\;0~{}~{}~{}\text{and}~{}~{}~{}\\{{\text{d}},\,{\mathfrak{D}}\\}\;=\;0~{}.$
(5.26)

Note that d now maps a $[p,q]$–form to a $[p,q+1]$–form and ${\mathfrak{D}}$
maps a $[p,q]$–form to a $[p+1,q]$ form. There are strong similarities with
complex manifolds. A form of definite holomorphic type, say $(r,s)$, is acted
upon by the ${\partial}$– and ${\overline{\partial}}$–operators that map this
form to $(r+1,s)$ and $(r,s+1)$. In the limit where $c_{a}\to 0$, which
includes the case of a fixed manifold $X$, the operators above reduce to what
has been used in previous sections.

The de Rham operator $\mathbbl{d}$ acting on a vertical form $\eta$ decomposes

$\mathbbl{d}\eta\;=\;{\text{d}}\eta+{\mathfrak{D}}\eta-\frac{1}{2(q-1)!}\,\eta_{m_{1}\cdots
m_{q-1}\,n}S{}_{bc}{}^{n}{\text{d}}y^{b}{\text{d}}y^{c}\,\,e^{m_{1}\cdots
m_{q-1}}~{}.$ (5.27)

The term involving $S_{ab}=[e_{a},e_{b}]$ measures the lack of integrability
of tangibility. In general a tangibility $[p,q]$ form with $q\geq 1$ is mapped
to a form with three different tangibilities divided up into branches as shown

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The hermitian form is given in (5.12) and its deformation, derived from
(5.13), is given by ${\mathfrak{D}}_{a}\omega=\mathcal{L}_{e_{a}}\omega$. Just
as $\mathbbl{F}$ includes ${\mathfrak{D}}_{a}A$ as its mixed component, the
deformation of $\omega$ is a mixed component of $\mathbbl{d}\mathbbl{\hskip
1.0pt\bbomega}$:

$\mathbbl{d}\mathbbl{\hskip
1.0pt\bbomega}\;=\;\frac{1}{3!}({\text{d}}\omega)_{mnp}\,e^{mnp}+\frac{1}{2}{\mathfrak{D}}_{a}\omega_{mn}\,{\text{d}}y^{a}e^{mn}-\frac{1}{2}S_{bc}{}^{n}\omega_{nm}{\text{d}}y^{bc}e^{m}~{},$
(5.28)

where

$({\text{d}}\omega)_{mnp}\;=\;{\partial}_{m}\omega_{np}+{\partial}_{n}\omega_{pm}+{\partial}_{p}\omega_{mn}~{},$

and ${\mathfrak{D}}_{a}\omega$ is of the form (5.14) with $q=2$. The
transformation law under $c_{a}\to\widetilde{c}_{a}=c_{a}-\varepsilon_{a}$
follows directly from (5.14):

${\mathfrak{D}}_{a}\omega~{}\to~{}\widetilde{\mathfrak{D}}_{a}\omega\;=\;{\mathfrak{D}}_{a}\omega+\varepsilon_{a}{}^{m}\,({\text{d}}\omega)_{m}+{\text{d}}(\varepsilon_{a}{}^{m}\,\omega_{m})~{}.$
(5.29)

where the notation here means

$({\text{d}}\omega)_{m}\;=\;\frac{1}{2}({\partial}_{m}\omega_{np}+{\partial}_{n}\omega_{pm}+{\partial}_{p}\omega_{mn})e^{np}~{},\qquad{\text{d}}(\varepsilon_{a}{}^{m}\omega_{m})\;=\;{\partial}_{p}(\varepsilon_{a}{}^{m}\omega_{mn})e^{pn}~{}.$

This is exactly the transformation law of a small diffeomorphism. It is also
possible to deduce this transformation law by studying how
$\mathbbl{d}\mathbbl{\hskip 1.0pt\bbomega}$ transforms under the change of
frame.

###  5.5 Holomorphy

This section opened with a discussion of the role of holomorphy for the gauge
field. In universal geometry this is the fact $\mathbbl{F}^{(0,2)}=0$ and
results in gauge fixing $A$ to holomorphic gauge. We now extend this to other
universal fields.

We begin with complex structure $J$. Holomorphic gauge in supergravity is the
condition $\Delta_{{\overline{\alpha}}{\overline{\nu}}}{}^{\mu}=0$. As
described above this amounts to the connection $c_{a}$ being pure in indices,
or equivalently, the fibration $\mathbbl{X}$ being complex.

As for the moduli space of Calabi-Yau manifolds the deformations of $B$ and
$\omega$ are combined and holomorphy is gauge fixed to be

${\overline{\mathpzc}Z}_{{\overline{\alpha}}}^{(2,0)}\;=\;0~{},\qquad\mathpzc
Z_{\overline{\alpha}}^{(1,1)}\;=\;0~{},\qquad\mathpzc
Z_{\overline{\alpha}}^{(0,2)}\;=\;{\overline{\mathpzc}Z}_{\overline{\alpha}}^{(0,2)}\;=\;0~{}.$

For the hermitian form $\mathbbl{\hskip 1.0pt\bbomega}$ and
$\mathbbl{B}$–field, holomorphy is captured by

$\mathbbl{H}-\mathbbl{d}^{c}\mathbbl{\hskip 1.0pt\bbomega}\;=\;0~{}.$

To see this we need only consider the tangibility $[1,2]$ component, which has
one leg along the moduli space $M$ and the remainder along $X$. Then, note
that $\frac{1}{2}\mathbbl{H}_{{\overline{\alpha}}mn}e^{mn}=\mathpzc
B_{{\overline{\alpha}}}$ and $\frac{1}{2}(\mathbbl{d}^{c}\mathbbl{\hskip
1.0pt\bbomega})_{{\overline{\alpha}}\,mn}e^{mn}=\\!-\text{i}{\mathfrak{D}}_{\overline{\alpha}}\omega^{(1,1)}+\text{i}{\mathfrak{D}}_{\overline{\alpha}}\omega^{(2,0)}$.
Hence, we find

$\Delta_{\overline{\alpha}}{}^{\mu}\;=\;0~{},\quad{\mathfrak{D}}_{\overline{\alpha}}\mathcal{A}\;=\;0~{},\quad{\overline{\mathpzc}Z}_{\overline{\alpha}}^{(2,0)}\;=\;0~{},\quad\mathpzc
Z_{\overline{\alpha}}^{(1,1)}\;=\;0~{},\quad\mathpzc
Z_{\overline{\alpha}}^{(0,2)}\;=\;{\overline{\mathpzc}Z}_{\overline{\alpha}}^{(0,2)}\;=\;0~{}.$

As promised this is precisely the hermitian moduli part of holomorphic gauge
summarised in (3.5).

In §3.3 we observed there is a residual gauge freedom which can be used to set
$\delta\Omega^{(3,0)}$ to be harmonic. Equivalently,
$\omega^{\mu{\overline{\nu}}}{\mathfrak{D}}_{a}\omega_{\mu{\overline{\nu}}}$
is a constant on $X$. There is a geometric echo of this gauge fixing in the
universal geometry. We can consider the Lee form for the universal geometry

$\frac{1}{2}\mathbbl{\hskip 1.0pt\bbomega}^{mn}(\mathbbl{d}\mathbbl{\hskip
1.0pt\bbomega})_{mn}+\frac{1}{2}\mathbbl{\hskip
1.0pt\bbomega}^{ab}(\mathbbl{d}\mathbbl{\hskip
1.0pt\bbomega})_{ab}\;=\;\frac{1}{2}{\text{d}}y^{a}\omega^{mn}{\mathfrak{D}}_{a}\omega_{mn}+\frac{1}{2}\omega^{mn}({\text{d}}\omega)_{mn}-\frac{1}{2}\omega^{{\sharp}ab}(S_{ab}{}^{m}\omega_{m})~{},$

The gauge fixing implies the first term, which has a leg along the moduli
space, is a constant over manifold $X$; it depends only on parameters. The
second and third terms have legs along the manifold. The first term is the
usual Lee form and the second term is an extrinsic contribution coming from
the embedding of $X$ in $\mathbbl{X}$. For heterotic theories with constant
dilaton the second term vanishes, and if we do as in [1] and set $S_{ab}=0$
then the final term also vanishes. So the residual gauge fixing of §3.3
amounts to the Lee form on the universal bundle being closed. It would be
interesting to explore this geometric condition in the case where the dilaton
is not constant.

##  6 A taste of second order universal geometry

The study of higher order deformations is necessary if one wants to fully
understand the moduli space. They can capture quantities such as the metric
$g^{\sharp}_{ab}$ or the Yukawa couplings $\kappa_{abc}$, and will be crucial
to any heterotic generalisation of special geometry. Furthermore, the work of
[27] suggests they govern the underlying algebra of heterotic moduli spaces.

Our aim here is to give a taste of what is to be learnt at second order with
an example which is the study of holomorphic bundles on a fixed manifold, a
situation which occurs at the standard embedding or at zeroth order in
heterotic. We will find that studying this moduli space in the context of
universal geometry will actually allow us to learn something useful about the
role of curvature $\mathbbl{F}_{ab}$ on the moduli space. We will not achieve
the ultimate aim of describing the theory of higher order covariant
derivatives for heterotic theories. This we hope to publish on soon. Indeed,
even at first order when one fleshes out the details of gauge fixing, this is
already extremely complicated as demonstrated in §4.2.

We study the moduli of holomorphic stable vector bundles $\mathpzc E\to X$
over a fixed complex manifold with a balanced metric
${\text{d}}(\omega^{2})=0$. The structure group of $\mathpzc E$ is a compact
real group $G$, with Lie algebra ${\mathfrak{g}}$. This example is discussed
in §5.2. It is relevant to the mathematics of the Kobayashi–Hitchin
correspondence, see e.g. [5]. It is also relevant to heterotic theories to
lowest order in ${\alpha^{\backprime}\,}$. To next order in
${\alpha^{\backprime}\,}$, the example becomes more complicated. As we have
shown in section §4.2 to first order in ${\alpha^{\backprime}\,}$ the
heterotic equations of motion demand the complex structure and hermitian
structure of $X$ also vary with a change in the bundle.

The manifold $X$ is fixed so we do not worry about $c_{a}$ or $S_{ab}$.
Therefore, we consider a universal bundle $\mathcal{U}\to M\times X$. We are
ignoring ${\alpha^{\backprime}\,}$–corrections, which as discussed in §4.2
will force $X$ to vary, and so introduce a connection $c_{a}$. In this limit,
the bundle is equipped with a connection:

$\mathbbl{A}\;=\;\Lambda_{a}\,{\text{d}}y^{a}+A~{},\quad\mathbbl{F}\;=\;\frac{1}{2}\,\mathbbl{F}_{ab}\,{\text{d}}y^{ab}+{\text{d}}y^{a}\,\mathbbl{F}_{a}+F~{},\quad\mathbbl{F}_{a}\;=\;{\mathfrak{D}}_{a}A~{},$
(6.1)

with Bianchi identity

$\begin{split}{\mathfrak{D}}_{a}F\;=\;{\text{d}}_{A}\left({\mathfrak{D}}_{a}A\right)~{},\quad[{\mathfrak{D}}_{a},{\mathfrak{D}}_{b}]A~{}=-{\text{d}}_{A}\mathbbl{F}_{ab}~{}.\end{split}$
(6.2)

A deformation of the gauge field can be written in real coordinates as

$\delta\mathcal{A}\;=\;\delta y^{a}{\mathfrak{D}}_{a}\mathcal{A}+(\delta
y^{a}\otimes\delta
y^{b})\,{\mathfrak{D}}_{a}{\mathfrak{D}}_{b}\mathcal{A}+\cdots.$

Recall the first order analysis in §3.2–§4.2. Holomorphy on a fixed manifold
means that
${\mathfrak{D}}_{\overline{\alpha}}\mathcal{A}{\;=\;}{\overline{\partial}}_{\mathcal{A}}\Phi_{\overline{\alpha}}$
and fixing to holomorphic gauge means
${\mathfrak{D}}_{\overline{\alpha}}\mathcal{A}=0$. This completely fixes the
gauge at first order. The deformation above becomes

$\delta\mathcal{A}\;=\;\delta
y^{\alpha}{\mathfrak{D}}_{\alpha}\mathcal{A}+(\delta y^{\alpha}\otimes\delta
y^{\beta})\,{\mathfrak{D}}_{\alpha}{\mathfrak{D}}_{\beta}\mathcal{A}+(\delta
y^{\overline{\alpha}}\otimes\delta
y^{\beta})\,{\mathfrak{D}}_{\overline{\alpha}}{\mathfrak{D}}_{\beta}\mathcal{A}+\cdots.$

The commutators imply
$[{\mathfrak{D}}_{\beta},{\mathfrak{D}}_{\alpha}]\mathcal{A}={\overline{\partial}}_{\mathcal{A}}\mathbbl{F}_{\alpha\beta}$
and
$[{\mathfrak{D}}_{\overline{\beta}},{\mathfrak{D}}_{\alpha}]\mathcal{A}={\overline{\partial}}_{\mathcal{A}}\mathbbl{F}_{\alpha{\overline{\beta}}}$
and we immediately see that deformations do not necessarily commute. However,
a key point is that the choice of connection $\Lambda_{a}$ is closely related
to gauge fixing. What we show here that under the assumption $X$ is fixed we
can find a gauge in which $\mathbbl{F}_{\alpha\beta}=0$. But the term
$\mathbbl{F}_{\alpha{\overline{\beta}}}$ is not necessarily zero. So
deformations of the gauge field do not necessarily commute.

To see this we start by defining holomorphy at second order in same way as
first order:

${\mathfrak{D}}_{{\overline{\alpha}}}{\mathfrak{D}}_{{\overline{\beta}}}\mathcal{A}\;=\;{\overline{\partial}}_{\mathcal{A}}\Phi_{{\overline{\alpha}}{\overline{\beta}}}~{},\qquad{\mathfrak{D}}_{\alpha}{\mathfrak{D}}_{{\overline{\beta}}}\mathcal{A}\;=\;{\overline{\partial}}_{\mathcal{A}}\Phi_{\alpha{\overline{\beta}}}~{}.$
(6.3)

As ${\mathfrak{D}}_{\overline{\beta}}{\mathfrak{D}}_{\alpha}\mathcal{A}$ is
${\overline{\partial}}_{\mathcal{A}}$–exact and we find two additional
equations