# Dispersion and absorption effects in the linearized Born-Infeld
electrodynamics under an external magnetic field

G. R. Santos<EMAIL_ADDRESS>Departamento de Física, Universidade
Federal Rural do Rio de Janeiro, BR 465-07, 23890-971, Seropédica, RJ, Brazil
M. J. Neves<EMAIL_ADDRESS>Departamento de Física, Universidade Federal
Rural do Rio de Janeiro, BR 465-07, 23890-971, Seropédica, RJ, Brazil

###### Abstract

The effects of the Ohmic and magnetic density currents are investigated in the
linearized Born-Infeld electrodynamics. The linearization is introduced
through an external magnetic field, in which the vector potential of the Born-
Infeld electrodynamics is expanded around of a magnetic background field, that
we consider uniform and constant in this paper. From the Born-Infeld
linearized equations, we obtain the solutions for the refractive index
associated with the electromagnetic wave superposition, when the current
density is ruled by the Ohm law, and in the second case, when the current
density is set by a isotropic magnetic conductivity. These solutions are
functions of the magnetic background $({\bf B})$, of the wave propagation
direction $({\bf k})$, and also it depends on the material conductivity, and
on the wave frequency. As consequence, the dispersion and the absorption of
waves change when ${\bf B}$ is parallel to ${\bf k}$ in relation to the case
of ${\bf B}$ perpendicular to ${\bf k}$ in the material medium. These
characteristics of the refraction index related to directions of ${\bf B}$ and
${\bf k}$ open a discussion of the birefringence in the material medium.

Born-Infeld electrodynamics, Dispersion and absorption of waves,
Birefringence.

##  I Introduction

The Born-Infeld (BI) electrodynamics (ED) is one of most famous non-linear
extensions of the Maxwell ED in the literature [1]. Originally, it was
proposed to explain the classical electron self-energy, since that, in Maxwell
ED, the electric field for a rest like-point charge is not defined at the
origin. After some years, the Euler-Heisenberg ED was discovered through the
radiative corrections of the quantum electrodynamics, when it is submitted to
the external electromagnetic field [2]. Nowadays, several examples of non-
linear EDs have applications or solutions in many research areas, as anomalous
couplings in physics beyond the Standard Model, black holes, string theory,
Dirac materials and others, see the refs. [4, 3, 5, 6, 7, 8, 9]. The
investigation of phenomena under action of an external field also is a subject
of interest in non-linear EDs [10, 11]. The introduction of the external
electromagnetic field in non-linear EDs is an approach that allows the
investigation of propagation effects, as the dispersion relations, group
velocities, the refractive index, and also the characteristics of the material
medium under an external (uniform and constant) magnetic field [12, 13, 14].
The study of electromagnetic waves in materials, as conductors, is one of
known applications of the Ohm law at room temperature [15]. It allows to
obtain the dispersion and absorption of waves in the material medium. Other
current density discussed in the literature of material physics is known as
the magnetic current. The current density vector is proportional to the
magnetic field, in which the proportionality constant is called magnetic
conductivity [16, 17, 18]. This current density has origin from the systems
with asymmetry of left- and right-handed chiral fermions, that is known as the
Chiral Magnetic Effect. In Weyl semimetals, the CME is related to massless
fermions acquire velocity along the magnetic field [19, 20].

Meeting all these motivations, we investigate the dispersion and absorption of
the wave propagation in the linearized BI ED by an external magnetic field,
when the material medium is governed by the current density of the Ohm law,
and posteriorly, when the current density is the magnetic one. The refractive
index of the material medium depends on the conductivity, on the wave
frequency, and also on the magnetic background. We show as the birefringence
phenomenon emerges as consequence of the wave dispersion [21]. The paper is
organized as follows : In the section II, we show the linearization of BI ED
in the presence of an external (uniform and constant) magnetic field. In the
section III, we obtain the dispersion and absorption of waves for the case of
a Ohmic current. The section IV is dedicated to the wave dispersion for an
isotropic magnetic conductivity. In the section V, we show the birefringence
phenomenon from the dispersions relations. For end, the final considerations
are cast in the section VI.

Throughout this work, we adopt natural units $\hbar=c=1$ with
$4\pi\epsilon_{0}=1$, We use the conversion $1\,\mbox{m}=5\times
10^{12}\,\mbox{MeV}^{-1}$. The electric and magnetic fields have squared-
energy mass dimension, where the conversion of Volt/m and Tesla (T) to the
natural system is given by $1\,\mbox{Volt/m}=2.27\times
10^{-18}\,\mbox{MeV}^{2}$ and $1\,\mbox{T}=6.8\times
10^{-10}\,\mbox{MeV}^{2}$, respectively.

##  II The linearized Born-Infeld elecrodynamics

The non-linear BI ED in the presence of a source $J^{\mu}=(\rho,{\bf J})$ is
governed by the lagrangian

$\displaystyle{\cal L}_{BI}=\beta^{2}\left[\,1-\sqrt{\,1-2\,\frac{{\cal
F}_{0}}{\beta^{2}}-\frac{{\cal
G}_{0}^{2}}{\beta^{4}}\,}\,\right]-J_{\mu}A_{0}^{\;\mu}\;,$ (1)

where $J^{\mu}=(\rho,{\bf J})$ is a $4$-current density, ${\cal F}_{0}$ and
${\cal G}_{0}$ are the gauge and Lorentz invariant

$\displaystyle{\cal F}_{0}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\,F_{0\mu\nu}^{2}=\frac{1}{2}\,\left({\bf
E}_{0}^{2}-{\bf B}_{0}^{2}\right)\;,$ (2a) $\displaystyle{\cal G}_{0}$
$\displaystyle=$
$\displaystyle-\frac{1}{4}\,F_{0\mu\nu}\widetilde{F}_{0}^{\;\,\,\mu\nu}={\bf
E}_{0}\cdot{\bf B}_{0}\;,$ (2b)

and $\beta$ is the BI parameter with electromagnetic field dimension (energy
to the squared in natural units). The $\beta$-BI parameter is interpreted as a
critical field of the theory in which it is reduced to the usual Maxwell ED in
the limit $\beta\rightarrow\infty$. For the analysis of propagation effects,
we use $\beta=1.187\times 10^{20}\,\mbox{V}/\mbox{m}\simeq
256\;\mbox{MeV}^{2}$ associated with the finite self-energy for the electron
[1].

The prescription for the external and uniform magnetic field is introduced
through the gauge $4$-potential $A_{0\mu}=a_{\mu}+A_{B\mu}$, where $a_{\mu}$
is the $4$-potential propagates in the space-time, and $A_{B\mu}$ is the
potential associated with the magnetic background field ${\bf B}$. The field-
strength tensor is also decomposed as $F_{0\mu\nu}=f_{\mu\nu}+F_{B\mu\nu}$, in
which
$f^{\mu\nu}=\partial^{\mu}a^{\nu}-\partial^{\nu}a^{\mu}=\left(\,-e^{i}\,,\,-\epsilon^{ijk}\,b^{k}\,\right)$
denotes the EM field strength tensor of the propagating fields, whereas
$F_{B}^{\;\,\mu\nu}=\partial^{\mu}A_{B}^{\;\,\nu}-\partial^{\nu}A_{B}^{\;\,\mu}=\left(\,0\,,\,-\epsilon^{ijk}\,B^{\,\,k}\,\right)$
sets the field strength of the magnetic background. Using this approach, the
lagrangian density (1) up to second order in the propagation gauge field is
read below

$\displaystyle{\cal L}_{BI}^{(2)}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\,c_{1}\,f_{\mu\nu}^{\,2}-\frac{1}{4}\,c_{2}\,f_{\mu\nu}\,\widetilde{f}^{\mu\nu}$
(3)
$\displaystyle+\,\frac{1}{8}\,Q_{B\mu\nu\kappa\lambda}\,f^{\mu\nu}\,f^{\kappa\lambda}-J_{\mu}\,(a^{\mu}+A_{B}^{\mu})\;,$

where
$\widetilde{f}_{\mu\nu}=\varepsilon_{\mu\nu\alpha\beta}f^{\alpha\beta}/2$ is
the strength field dual tensor, and we have defined the tensor
$Q_{B\mu\nu\kappa\lambda}$ evaluated at the magnetic background as follows

$\displaystyle Q_{B\mu\nu\kappa\lambda}\\!$ $\displaystyle=$
$\displaystyle\\!d_{1}\,F_{B\mu\nu}F_{B\kappa\lambda}+d_{2}\,\widetilde{F}_{B\mu\nu}\widetilde{F}_{B\kappa\lambda}$
(4)
$\displaystyle+\,d_{3}\,F_{B\mu\nu}\widetilde{F}_{B\kappa\lambda}+d_{3}\,\widetilde{F}_{B\mu\nu}F_{B\kappa\lambda}\;.$

The coefficients of the expansion $c_{i}\,(i=1,2)$ and $d_{i}\,(i=1,2,3)$ are
defined by

$\displaystyle c_{1}=\left.\frac{\partial{\cal L}}{\partial{\cal
F}_{0}}\right|_{{\bf B}},\left.c_{2}=\frac{\partial{\cal L}}{\partial{\cal
G}_{0}}\right|_{{\bf B}},\left.d_{1}=\frac{\partial^{2}{\cal L}}{\partial{\cal
F}_{0}^{2}}\right|_{{\bf B}},$
$\displaystyle\left.d_{2}=\frac{\partial^{2}{\cal L}}{\partial{\cal
G}_{0}^{2}}\right|_{{\bf B}},\left.d_{3}=\frac{\partial^{2}{\cal
L}}{\partial{\cal F}_{0}\partial{\cal G}_{0}}\right|_{{\bf B}}\;.$ (5)

Substituting the lagrangian (1), the BI coefficients in a uniform magnetic
background are given by

$\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle\frac{\beta}{\sqrt{\beta^{2}+{\bf B}^{2}}}\hskip 8.5359pt,\hskip
8.5359ptc_{2}=0\;,$ (6a) $\displaystyle d_{1}$ $\displaystyle=$
$\displaystyle\frac{\beta}{(\beta^{2}+{\bf B}^{2})^{3/2}}\;,$ (6b)
$\displaystyle d_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\beta\sqrt{\beta^{2}+{\bf B}^{2}}}\hskip
8.5359pt,\hskip 8.5359ptd_{3}=0\;.$ (6c)

The non-null coefficients simplifies the lagrangian density (3) as

$\displaystyle{\cal L}_{BI}^{(2)}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\,c_{1}\,f_{\mu\nu}^{\,2}+\frac{d_{1}}{8}\,(F_{B\mu\nu}f^{\mu\nu})^{2}+\frac{d_{2}}{8}\,(\widetilde{F}_{B\mu\nu}f^{\mu\nu})^{2}$
(7) $\displaystyle-J_{\mu}\,(a^{\mu}+A_{B}^{\mu})\;.$

Notice that, in the limit $\beta\rightarrow\infty$, the coefficients are
reduced to $c_{1}=1$ and $d_{1}=d_{2}=0$, and the lagrangian (7) leads to the
Maxwell ED for the propagating field. Since that the magnetic background field
is constant and uniform, in this particular case the coefficients of the
expansion do not depend on the space-time coordinates.

The action principle applied to the lagrangian (7), in relation to $a^{\mu}$,
yields the linearized field equation

$\partial^{\mu}G_{\mu\nu}=J_{\nu}\;,$ (8)

where
$G_{\mu\nu}=(c_{1}\,\eta_{\kappa\mu}\eta_{\lambda\nu}-d_{1}\,F_{B\mu\nu}F_{B\kappa\lambda}/2-d_{2}\,\widetilde{F}_{B\mu\nu}\widetilde{F}_{B\kappa\lambda}/2)f^{\kappa\lambda}$,
and the dual tensor $\widetilde{f}^{\mu\nu}$ satisfies the Bianchi identity
$\partial_{\mu}\widetilde{f}^{\mu\nu}=0$. The quadri-current satisfies the
charge conservation equation $\partial_{\mu}J^{\mu}=0$. In vector notation,
the correspondent field equations are

$\displaystyle\nabla\cdot{\mathbf{e}}+f\;\mathbf{B}\cdot\nabla(\mathbf{B}\cdot\mathbf{e})\\!=\rho\,,$
(9a)
$\displaystyle\nabla\times{\mathbf{e}}+\frac{\partial{\mathbf{b}}}{\partial
t}={\mathbf{0}}\;\hskip 5.69046pt,\hskip
5.69046pt\nabla\cdot{\mathbf{b}}=0\,\hskip 5.69046pt,\hskip 5.69046pt$ (9b)
$\displaystyle\nabla\times{\mathbf{b}}+d\;{\mathbf{B}}\times\nabla(\mathbf{B}\cdot{\mathbf{b}})={\mathbf{J}}+\frac{\partial{\mathbf{e}}}{\partial
t}+$ $\displaystyle+f\,\mathbf{B}\,\frac{\partial}{\partial
t}({\mathbf{B}}\cdot{\mathbf{e}})\;,$ (9c)

in which we rewrite the coefficients as

$\displaystyle
d=\frac{d_{1}}{c_{1}}=\frac{1}{\beta^{2}+{\mathbf{B}}^{2}}\hskip
8.5359pt\mbox{and}\hskip 8.5359ptf=\frac{d_{2}}{c_{1}}=\frac{1}{\beta^{2}}\;.$
(10)

The limit $\beta\rightarrow\infty$ recovers the usual Maxwell equations in
(9a)-(9c) for ${\bf e}$ and ${\bf b}$. The ${\bf B}$-magnetic field is so
interpreted as a background vector in all the previous equations. The presence
of this background field modifies the dispersion relations associated with the
wave solutions. It will be explored in the next section for a Ohmic current
density.

##  III The dispersion and absorption in the presence of Ohmic current

Since it is known in classical electrodynamics for a class of materials, such
as the conductors, the current density is governed by Ohm law

$\displaystyle\mathbf{J}=\sigma\,\mathbf{e}\;,$ (11)

where $\sigma$ is the electric conductivity at room temperature, that is
characteristics of the material medium.

For the analysis of the propagation fields, we substitute the Fourier
transforms in the linearized equations (9a)-(9c)

$\displaystyle{\mathbf{e}}(x)=\int\frac{d^{4}k}{(2\pi)^{4}}\;{\mathbf{e}}_{0}(k)\;e^{-ik\cdot
x}\;,$ (12a)
$\displaystyle{\mathbf{b}}(x)=\int\frac{d^{4}k}{(2\pi)^{4}}\;{\mathbf{b}}_{0}(k)\;e^{-ik\cdot
x}\;,$ (12b)
$\displaystyle\rho(x)=\int\frac{d^{4}k}{(2\pi)^{4}}\;\rho_{0}(k)\;e^{-ik\cdot
x}\;,$ (12c)

where the scalar product means $k\cdot x=\omega\,t-{\bf k}\cdot{\bf r}$, in
which ${\bf k}$ is the wave vector, $\omega$ is the wave frequency, ${\bf
e}_{0}(k)$ and ${\bf b}_{0}(k)$ are the electric and magnetic wave amplitudes,
respectively, and $\rho_{0}(k)$ is Fourier transform of the charge density.
Thereby, we obtain the equations fields in the momentum space

$\displaystyle\mathbf{k}\cdot{\mathbf{e_{0}}}+\frac{1}{\beta^{2}}\;(\mathbf{B}\cdot\mathbf{k})(\mathbf{B}\cdot\mathbf{e_{0}})\\!=\rho_{0}(k)\,,\hskip
14.22636pt$ (13a)
$\displaystyle\mathbf{k}\times{\mathbf{e_{0}}}=\omega\,\mathbf{b_{0}}\;\hskip
5.69046pt,\hskip 5.69046pt\mathbf{k}\cdot{\mathbf{b_{0}}}=0\;,\hskip
34.14322pt$ (13b)
$\displaystyle\mathbf{k}\times{\mathbf{b_{0}}}+\dfrac{(\mathbf{B}\times\mathbf{k})}{\mathbf{B}^{2}+\beta^{2}}\,(\mathbf{B}\cdot{\mathbf{b_{0}}})=-i\,\sigma\,{\mathbf{e_{0}}}$
$\displaystyle-\frac{\omega}{\beta^{2}}\,\mathbf{B}\,(\mathbf{B}\cdot\mathbf{e_{0}})-\omega\,\mathbf{e_{0}}\;.\hskip
28.45274pt$ (13c)

The equations (13a)-(13c) can be combined such that we obtain the wave
equation for the electric amplitude $\mathbf{e}_{0i}$

$\displaystyle M_{i\,j}\,\mathbf{e}_{0j}=0\;,$ (14)

where $M_{ij}$ is the wave matrix

$\displaystyle
M_{ij}=\delta_{ij}\left(1-\mathbf{n}^{2}+i\,\frac{\sigma}{\omega}\right)+n_{i}\,n_{j}$
$\displaystyle+\dfrac{(\mathbf{B}\times\mathbf{n})_{i}\,(\mathbf{B}\times\mathbf{n})_{j}}{\mathbf{B}^{2}+\beta^{2}}+\frac{1}{\beta^{2}}\,\mathbf{B}_{i}\,\mathbf{B}_{j}\;.$
(15)

We have written the symmetric matrix $M_{ij}$ in terms of the refractive index
components $n_{i}=k_{i}/\omega$, whose the refractive index is defined by
$n=\sqrt{n_{i}\,n_{i}}$. The non-trivial solutions of (III) impose that
$\det(M_{ij})=0$, which leads to the equations

$\displaystyle
1-n^{2}+\dfrac{(\mathbf{B}\times\mathbf{n})^{2}}{\mathbf{B}^{2}+\beta^{2}}-i\,\frac{\sigma}{\omega}$
$\displaystyle=$ $\displaystyle 0\,,$ (16a)
$\displaystyle\left(\mathbf{n}^{2}-1-i\,\frac{\sigma}{\omega}\right)\left(1+\dfrac{\mathbf{B^{2}}}{\beta^{2}}+i\,\frac{\sigma}{\omega}\right)$
$\displaystyle=$
$\displaystyle\left(\dfrac{\mathbf{B}}{\beta}\times\mathbf{n}\right)^{2}\,.$

The solution of (16a) is given by

$\displaystyle
n_{1}=\sqrt{\dfrac{1+i\,\frac{\sigma}{\omega}}{1-({\mathbf{B}}^{2}+\beta^{2})^{-1}\,(\mathbf{B}\times\mathbf{\hat{k}})^{2}}}\;,$
(17)

that can be written as

$n_{1}=\Re[n_{1}]+i\,\Im[n_{1}]\;,$ (18)

where the real and imaginary parts are given by

$\displaystyle\Re[n_{1}]$ $\displaystyle=$
$\displaystyle\sqrt{\dfrac{(\beta^{2}+\mathbf{B}^{2})\left(\sqrt{1+\frac{\sigma^{2}}{\omega^{2}}}+1\right)}{2\beta^{2}+2\,(\mathbf{B}\cdot\mathbf{\hat{k}})^{2}}}\;,$
(19a) $\displaystyle\Im[n_{1}]$ $\displaystyle=$
$\displaystyle\sqrt{\dfrac{(\beta^{2}+\mathbf{B}^{2})\left(\sqrt{1+\frac{\sigma^{2}}{\omega^{2}}}-1\right)}{2\beta^{2}+2\,(\mathbf{B}\cdot\mathbf{\hat{k}})^{2}}}\;.$
(19b)

Figure 1:  Left panel : The real (black line) and imaginary (red dashed line)
parts of the solution $n_{1}$ as functions of the $\omega$-frequency. Right
panel : The real (black line) and imaginary (red dashed line) parts of the
solution $n_{2}$ as functions of the $\omega$-frequency. In both plots, we use
$\sqrt{\beta}=16$ MeV, $\sqrt{|{\bf B}|}=8.25$ MeV, and
$\sigma=0.2\,\mu\mbox{eV}$ for silver.

The solution of (16) is

$n_{2}=\sqrt{\dfrac{1+\mathbf{B}^{2}/\beta^{2}-\sigma^{2}/\omega^{2}+i\,(2+\mathbf{B}^{2}/\beta^{2})\,(\sigma/\omega)}{1+(\mathbf{B}\cdot\mathbf{\hat{k}}/\beta)^{2}+i\,(\sigma/\omega)}}\;,$
(20)

that has the real and imaginary parts

$\displaystyle\Re[n_{2}]$ $\displaystyle=$
$\displaystyle\sqrt{\frac{|a|}{2}}\,\sqrt{\sqrt{1+\frac{b^{2}}{a^{2}}}+\mbox{sign}(a)}\;,$
(21a) $\displaystyle\Im[n_{2}]$ $\displaystyle=$
$\displaystyle\sqrt{\frac{|a|}{2}}\,\sqrt{\sqrt{1+\frac{b^{2}}{a^{2}}}-\mbox{sign}(a)}\;,$
(21b)

where $a$ e $b$ are read below

$\displaystyle a\\!$ $\displaystyle=$
$\displaystyle\\!\dfrac{(1+\mathbf{B}^{2}/\beta^{2}-\sigma^{2}/\omega^{2})(\mathbf{B}\cdot\mathbf{\hat{k}})^{2}/\beta^{2}}{1+\sigma^{2}/\omega^{2}+[\,2+(\mathbf{B}\cdot\mathbf{\hat{k}})^{2}/\beta^{2}\,](\mathbf{B}\cdot\mathbf{\hat{k}})^{2}/\beta^{2}}$
(22a)
$\displaystyle+\;\,\dfrac{(1+\mathbf{B}^{2}/\beta^{2})\,(1+\sigma^{2}/\omega^{2})}{1+\sigma^{2}/\omega^{2}+[\,2+(\mathbf{B}\cdot\mathbf{\hat{k}})^{2}/\beta^{2}\,](\mathbf{B}\cdot\mathbf{\hat{k}})^{2}/\beta^{2}}\;,\hskip
19.91684pt$ $\displaystyle b\\!$ $\displaystyle=$
$\displaystyle\\!\frac{\sigma}{\omega}\,\dfrac{1+\sigma^{2}/\omega^{2}+(\mathbf{B}\cdot\mathbf{\hat{k}})^{2}/\beta^{2}(2+\mathbf{B}^{2}/\beta^{2})}{1+\sigma^{2}/\omega^{2}+[2+(\mathbf{B}\cdot\mathbf{\hat{k}})^{2}/\beta^{2}](\mathbf{B}\cdot\mathbf{\hat{k}})^{2}/\beta^{2}}\;,\hskip
25.6073pt$ (22b)

and $\mbox{sign}(a)=a/|a|$ is the sign function of $a$. The real parts (19a)
and (21a) contains the solutions of the dispersion relation, the wavelength
and the group velocity of the wave. The imaginary parts (19b) and (21b) define
the wave penetration in a conductor medium as $d=(\Im[n_{i}])^{-1}\,(i=1,2)$.
The limit $\beta\rightarrow\infty$ in (19a)-(19b), and in (21a)-(21b),
recovers the known results of the Maxwell ED:

$\displaystyle\lim_{\beta\rightarrow\infty}\Re[n_{1}]=\Re[n_{2}]=\sqrt{\sqrt{\frac{1}{4}+\frac{\sigma^{2}}{4\omega^{2}}}+\frac{1}{2}}\;,$
(23a)
$\displaystyle\lim_{\beta\rightarrow\infty}\Im[n_{1}]=\Im[n_{2}]=\sqrt{\sqrt{\frac{1}{4}+\frac{\sigma^{2}}{4\omega^{2}}}-\frac{1}{2}}\;.$
(23b)

The results (19a)-(19b) and (21a)-(21b) show the dependence of the real and
imaginary parts with the angle $(\theta)$ that the magnetic background does
with the wave propagation direction, i.e., $\cos\theta=\hat{{\bf
k}}\cdot\hat{{\bf B}}$. In the case of ${\bf k}$ parallel to ${\bf B}$, the
contribution of the $\beta$-BI parameter disappears in (19a) and
$(\ref{Imn1})$. Under this particular condition, the solutions (21a) and (21b)
keep the non-linear contribution. Under an intense magnetic field, the
condition $|{\bf B}|\gg\beta$ reduces the results to expressions

$\displaystyle\Re[n_{1}]=\Re[n_{2}]=\frac{1}{|\cos\theta|}\,\sqrt{\sqrt{\frac{1}{4}+\frac{\sigma^{2}}{4\omega^{2}}}+\frac{1}{2}}\;,$
(24a)
$\displaystyle\Im[n_{1}]=\Im[n_{2}]=\frac{1}{|\cos\theta|}\,\sqrt{\sqrt{\frac{1}{4}+\frac{\sigma^{2}}{4\omega^{2}}}-\frac{1}{2}}\;.$
(24b)

These particular cases show that when the magnetic field is intense, the
results of the Maxwell ED in a conductor are modified by the $\theta$-angle,
and it does not depend on the magnetic field magnitude.

Thereby, we illustrate the real and imaginary parts of $n_{1}$ (left panel)
and $n_{2}$ (right panel) as functions of the $\omega$-frequency in the fig.
(1). The fig. (1) is set with the values of $\sqrt{\beta}=16\,\mbox{MeV}$ for
propagation effects in BI ED, $\sqrt{B}=8.25\,\mbox{MeV}$ for a magnetic field
of a neutron star, and $\sigma=0.2\,\mu\mbox{eV}$ for silver in natural units
111In natural units, the electric resistivity has the conversion $1\Omega\cdot
m=2.95\times 10^{23}\,\mbox{GeV}^{-1}$. Therefore, electrical conductivity has
energy dimension.. The left panel in (1) shows that, when the magnetic field
is perpendicular to the wave propagation direction, the $\Im[n_{1}]$ part
decays for high frequencies in relation to $\Re[n_{1}]$, i. e., the dispersion
and absorption of the wave go to zero in the high-energy limit. The right
panel in (1) illustrates the real and imaginary parts of the $n_{2}$-solution
as functions of the $\omega$-frequency. In this case, we choose the magnetic
field parallel to the direction of the wave propagation direction.

##  IV The wave dispersion in the presence of an magnetic current

In this section, we study the dispersion effects for the case of a magnetic
conductivity current. The nature of this current density is associated with
the chirality between left- and right-handed fermions when it is submitted to
an external magnetic field [19]. The magnetic current density is given by

$\displaystyle\mathbf{J}=\sigma_{b}\,\mathbf{B}_{0}\;,$ (25)

where $\sigma_{b}$ is the magnetic conductivity, that we consider isotropic
throughout the material. Using the prescription of the magnetic background,
with $\mathbf{B}_{0}=\mathbf{b}+\mathbf{B}$, the linearized equations in the
presence of the current density (25) are read below

$\displaystyle\mathbf{k}\cdot{\mathbf{e_{0}}}+\frac{1}{\beta^{2}}\;(\mathbf{B}\cdot\mathbf{k})(\mathbf{B}\cdot\mathbf{e_{0}})\\!$
$\displaystyle=$ $\displaystyle\rho_{0}(k)\,,$ (26a)
$\displaystyle\mathbf{k}\times{\mathbf{e_{0}}}$ $\displaystyle=$
$\displaystyle\omega\,\mathbf{b_{0}}\;,$ (26b)
$\displaystyle\mathbf{k}\cdot{\mathbf{b_{0}}}$ $\displaystyle=$ $\displaystyle
0\,\hskip 5.69046pt,\hskip 5.69046pt$ (26c)
$\displaystyle\mathbf{k}\times{\mathbf{b_{0}}}+\frac{(\mathbf{B}\times\mathbf{k})(\mathbf{B}\cdot{\mathbf{b_{0}}})}{\beta^{2}+{\bf
B}^{2}}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!-i\,\sigma_{b}\,\mathbf{b_{0}}$
$\displaystyle-i\,\sigma_{b}\,\mathbf{B}\,\delta^{3}(\mathbf{k})\,\delta(\omega)-\frac{\omega}{\beta^{2}}\,\mathbf{B}\,(\mathbf{B}\cdot\mathbf{e_{0}})-\omega\,\mathbf{e_{0}}\;,$
(26d)

where we have substituted the plane wave solutions via Fourier transform for
the propagating EM fields $\mathbf{e}$, $\mathbf{b}$ and the charge density
$\rho$. Combining these equations, we obtain the wave equation for the
electric amplitude $e_{0j}$ :

$\displaystyle O_{i\,j}\,e_{0j}=0\;,$ (27)

where the matrix elements $O_{ij}$ are read below

$\displaystyle O_{i\,j}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\left(1-\mathbf{n}^{2}\right)\,\delta_{ij}+n_{i}\,n_{j}-i\,\frac{\sigma_{b}}{\omega}\,\epsilon_{ijk}\,n_{k}$
(28)
$\displaystyle+\dfrac{(\mathbf{B}\times\mathbf{n})_{i}\,(\mathbf{B}\times\mathbf{n})_{j}}{\mathbf{B}^{2}+\beta^{2}}+\frac{1}{\beta^{2}}\,\mathbf{B}_{i}\,\mathbf{B}_{j}\;.$

The non-trivial solution of (27) requires the condition $\det(O_{ij})=0$. The
correspondent solutions are given by

Figure 2:  Left panel : The $n_{1}$-solution from (29a) as function of the
$\omega$-frequency. Right panel : The $n_{2}$-solution from (29b) as function
of the $\omega$-frequency. In both plots, the solid black line means the case
of ${\bf B}\cdot{\bf k}=0$ (perpendiculars), whereas the dashed red lines set
the case of ${\bf B}\times{\bf k}=0$ (parallels). We choose the parameters as
$\sqrt{\beta}=16$ MeV, $\sqrt{|{\bf B}|}=8.25$ MeV, and
$\sigma_{b}=0.1\,\mu\mbox{eV}$.

$\displaystyle n_{1}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\sqrt{\frac{(\beta^{2}+\mathbf{B}^{2})\,\left(1+\,\frac{\sigma_{b}^{2}}{2\omega^{2}}-\frac{\sigma_{b}}{2\omega}\sqrt{4+\frac{\sigma_{b}^{2}}{\omega^{2}}}\right)}{\beta^{2}+(\mathbf{B}\cdot\mathbf{\hat{k}})^{2}}}\;,\hskip
28.45274pt$ (29a) $\displaystyle n_{2}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\sqrt{\frac{(\beta^{2}+\mathbf{B}^{2})\,\left(1+\,\frac{\sigma_{b}^{2}}{2\omega^{2}}+\frac{\sigma_{b}}{2\omega}\sqrt{4+\frac{\sigma_{b}^{2}}{\omega^{2}}}\right)}{\beta^{2}+(\mathbf{B}\cdot\mathbf{\hat{k}})^{2}}}\;.\hskip
28.45274pt$ (29b)

Notice that, both solutions are reals refractive index, and there is no
absorption in this case of an isotropic magnetic conductivity. In $n_{1}$ and
$n_{2}$, also emerge the dependence on the $\theta$-angle that $\mathbf{B}$
does with $\mathbf{\hat{k}}$-direction. In the case of $\mathbf{B}$ parallel
to $\mathbf{\hat{k}}$, the non-linear contribution of the $\beta$-parameter is
canceled in (29a) and $(\ref{n2 b})$. For $\mathbf{B}$ perpendicular to
$\mathbf{\hat{k}}$, the $\beta$-parameter and the magnetic background
contribute to the solutions. The solution $n_{1}$, when
$\mathbf{B}\cdot\mathbf{k}=0$ (black line), and when
$\mathbf{B}\times\mathbf{k}={\bf 0}$ (red line) are both showed in the fig.
(2). In this plot, we choose $\sqrt{\beta}=16\,\mbox{MeV}$, $\sqrt{|{\bf
B}|}=8.25\,\mbox{MeV}$, and $\sigma_{b}=0.1\,\mu\mbox{eV}$. In high frequency
range, the curves goes to a maximum refractive index, that satisfies the
condition $n_{1\perp}>n_{1\parallel}$, where $n_{1\perp}\simeq 1.034$, and
$n_{1\parallel}\simeq 1.0$, when $\omega\rightarrow\infty$.

The $n_{2}$-solution is shown in the right panel from the fig. (2). The black
line corresponds to $\mathbf{B}$ perpendicular to $\mathbf{\hat{k}}$, and the
dashed red line is the case of $\mathbf{B}$ parallel to $\mathbf{\hat{k}}$. In
this figure, we also consider $\sqrt{\beta}=16\,\mbox{MeV}$, $\sqrt{|{\bf
B}|}=8.25\,\mbox{MeV}$ and $\sigma_{b}=0.1\,\mu\mbox{eV}$. In the right panel,
the curves have a horizontal asymptotes for high energies, where
$n_{2\perp}\simeq 1.034$ and $n_{2\parallel}\simeq 1.0$, for a high frequency
range.

##  V Birefringence phenomenon

Since the refractive index depend on the $\theta$-angle that we have defined
as $\cos\theta=\mathbf{B}\cdot\mathbf{\hat{k}}$, the birefringence phenomenon
emerges from the solutions in both electric and magnetic conductivity current
cases. In the case of the Ohmic current, we consider the real part of the
solutions for the analysis of the birefringence. We define it as the
difference between the parallel and perpendicular real part of the refractive
indices

$\displaystyle\delta n_{i}^{e}=\Re[n_{i\perp}^{e}]-\Re[n_{i\parallel}^{e}]\;,$
(30)

where $i=\\{\,1\,,\,2\,\\}$, $n_{i\perp}^{e}$ is the refractive index when
$\mathbf{B}$ is perpendicular to $\mathbf{\hat{k}}$, and $n_{i\parallel}^{e}$
denotes the refractive index when $\mathbf{B}$ is parallel to
$\mathbf{\hat{k}}$. Substituting the real parts of (19a) and (21a), we obtain

$\displaystyle\delta n_{1}^{e}\\!$ $\displaystyle=$
$\displaystyle\\!\left(\\!\sqrt{1+\frac{{\bf
B}^{2}}{\beta^{2}}}-1\\!\right)\\!\sqrt{\sqrt{\frac{1}{4}+\frac{\sigma^{2}}{4\omega^{2}}}+\frac{1}{2}}\;,$
(31a) $\displaystyle\delta n_{2}^{e}\\!$ $\displaystyle=$
$\displaystyle\\!\sqrt{\sqrt{\frac{1}{4}+\frac{{\bf
B}^{2}}{4\beta^{2}}+\frac{\sigma^{2}}{4\omega^{2}}}+\frac{1}{2}+\frac{{\bf
B}^{2}}{2\beta^{2}}}$ (31b)
$\displaystyle-\sqrt{\sqrt{\frac{1}{4}+\frac{\sigma^{2}}{4\omega^{2}}}+\frac{1}{2}}\;.$

It is worth to highlight that the birefringence disappears $\delta
n_{i}^{e}=0$, when $\beta\rightarrow\infty$.

In the polarization vacuum with laser (PVLAS) experiment [21], the magnetic
field is $|{\bf B}|=2.5\,\mbox{T}=1.7\times 10^{-9}\,\mbox{MeV}^{2}$ in which
$\beta\gg|{\bf B}|$ in this case. Thereby, we can apply the results (31a) and
(31b) in this approximation to obtain the result

$\displaystyle\frac{\delta n_{1}^{e}}{B^{2}}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{2\beta^{2}}\,\sqrt{\sqrt{\frac{1}{4}+\frac{\sigma^{2}}{4\omega^{2}}}+\frac{1}{2}}\;,$
(32a) $\displaystyle\frac{\delta n_{2}^{e}}{B^{2}}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{4\beta^{2}}\frac{\frac{2\sigma^{2}}{\omega^{2}}+2+\sqrt{1+\frac{\sigma^{2}}{\omega^{2}}}}{(1+\frac{\sigma^{2}}{\omega^{2}})\sqrt{2+2\sqrt{1+\frac{\sigma^{2}}{\omega^{2}}}}}\;.$
(32b)

Using that $\delta n_{i}^{e}/B^{2}=(19\pm 27)\times 10^{-24}\,\mbox{T}^{-2}$
from the PVLAS experiment [21], the expression (32a) yields the acceptable
numeric result for the electric conductivity over the frequency :

$\displaystyle\frac{\sigma}{\omega}\simeq 56.75\;.$ (33)

Similarly to (30), we define the birefringence in the case of magnetic
conductivity as

$\displaystyle\delta n_{i}^{b}=n_{i\perp}^{b}-n_{i\parallel}^{b}\;.$ (34)

Using the solutions (29a) and (29b), the birefringence of $\delta n_{i}$ is
given by solutions

$\displaystyle\delta n_{1}^{b}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\left(\sqrt{1+\frac{{\bf
B}^{2}}{\beta^{2}}}-1\right)\sqrt{1+\,\frac{\sigma_{b}^{2}}{2\omega^{2}}-\frac{\sigma_{b}}{2\omega}\,\sqrt{4+\frac{\sigma_{b}^{2}}{\omega^{2}}}}\;,$
$\displaystyle\delta n_{2}^{b}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\left(\sqrt{1+\frac{{\bf
B}^{2}}{\beta^{2}}}-1\right)\sqrt{1+\,\frac{\sigma_{b}^{2}}{2\omega^{2}}+\frac{\sigma_{b}}{2\omega}\,\sqrt{4+\frac{\sigma_{b}^{2}}{\omega^{2}}}}\;.\;\;\;$

Notice that, the birefringence disappears in the limit
$\beta\rightarrow\infty$, or when the magnetic background is turned off. The
birefringence curves as functions of the $\omega$-frequency are shown in
figure (3). In this plot, we choose $\sqrt{\beta}=16\,\mbox{MeV}$,
$\sqrt{|\mathbf{B}|}=8.25\,\mbox{MeV}$ and $\sigma_{b}=0.5\,\mu\mbox{eV}$.

Figure 3:  The birefringence curves for $\delta n_{1}^{b}$ (black line) and
$\delta n_{2}^{b}$ (dashed red line) as functions of the $\omega$-frequency.
We choose the values $\sqrt{\beta}=16\,\mbox{MeV}$,
$\sqrt{|\mathbf{B}|}=8.25\,\mbox{MeV}$ and $\sigma_{b}=0.5\,\mu\mbox{eV}$.

The $\delta n_{1}$-solution has a minimum of birefringence at $\delta
n_{1min}\simeq 0.034$, whereas $\delta n_{2}$ has a maximum birefringence at
$\delta n_{2max}\simeq 2.034$, for high frequency $\omega\rightarrow\infty$.

Applying these results for the PVLAS experiment, the expressions (36a) and
(36b) are simplified as

$\displaystyle\frac{\delta n_{1}^{b}}{B^{2}}\\!\\!$ $\displaystyle\simeq$
$\displaystyle\\!\\!\frac{1}{2\beta^{2}}\,\sqrt{1+\,\frac{\sigma_{b}^{2}}{2\omega^{2}}-\frac{\sigma_{b}}{2\omega}\,\sqrt{4+\frac{\sigma_{b}^{2}}{\omega^{2}}}}\;,$
(36a) $\displaystyle\frac{\delta n_{2}^{b}}{B^{2}}\\!\\!$
$\displaystyle\simeq$
$\displaystyle\\!\\!\frac{1}{2\beta^{2}}\,\sqrt{1+\,\frac{\sigma_{b}^{2}}{2\omega^{2}}+\frac{\sigma_{b}}{2\omega}\,\sqrt{4+\frac{\sigma_{b}^{2}}{\omega^{2}}}}\;,\;\;$
(36b)

in which we have used the condition $\beta\gg|{\bf B}|$. Using that $\delta
n_{i}^{b}/B^{2}=(19\pm 27)\times 10^{-24}\,\mbox{T}^{-2}$, for the case of the
magnetic conductivity, we obtain the numeric result

$\displaystyle\frac{\sigma_{b}}{\omega}\simeq 5.18\;.$ (37)

##  VI Concluding comments

In this paper, we study the dispersion and absorption of waves in the
linearized Born-Infeld (BI) electrodynamics governed by the Ohmic and magnet
current densities. The linearization of the BI ED is introduced through the
propagating electromagnetic field added to a uniform and constant magnetic
background field. The BI non-linear lagrangian in expanded up to second order
for small propagating effects, and around the magnetic background. Thus, we
substitute the wave plane superpositions for the linearized electromagnetic
field, and discuss the wave propagation properties in a material medium in the
presence of the electric and magnetic conductivities.

From the wave equation, we calculate the refractive index solutions in terms
of the magnetic background, of the BI parameter, and of the electric
conductivity for the material medium. In this first case, the refractive index
has a real and imaginary parts, that are interpreted as the dispersion and the
absorption of the wave, respectively. In the second case, the magnetic current
density for an isotropic magnetic conductivity is investigated in which there
is no wave absorption. One fact is important : in all these solutions, the
refractive index depends on the $\theta$-angle that the magnetic background
${\bf B}$ does with the wave propagation direction $\hat{{\bf k}}$, i. e.,
$\cos\theta=\hat{{\bf B}}\cdot\hat{{\bf k}}$. Thus, the solutions provide the
situations in which ${\bf B}$ is parallel to ${\bf k}$, and ${\bf B}$
perpendicular to ${\bf k}$. It opens the discussion of the birefringence
phenomenon, that depends on the BI parameter, and on the magnetic background.
When the BI parameter is removed, the birefringence is null, and as well, all
the results of the Maxwell ED are recovered.

In the birefringence analysis, we use the known result from the PVLAS
experiment for the vacuum birefringence $\Delta n/B^{2}=(19\pm 27)\times
10^{-24}\,\mbox{T}^{-2}$ to estimate the ratio of electric and magnetic
conductivities over the wave $\omega$-frequency. Under these conditions, we
consider a weak magnetic field of $2.5\,\mbox{T}$ and the BI parameter of
$\sqrt{\beta}=16$ MeV. The result for the birefringence associated with the
electric conductivity yields the numeric result $\sigma/\omega=56.75$, whereas
for the case of the magnetic conductivity, the birefringence yields the result
$\sigma_{b}/\omega=5.18$ for a large range of frequency. For end, this paper
opens the discussion of applications of these refractive index solutions in
others non-linear electrodynamics, as Euler-Heisenberg and ModMax EDs. Other
perspective is to investigate the effects of the linearization in optics
classical law where the medium is affected by the external magnetic field.
This is one of the discussions for a forthcoming project.

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${\mathfrak{D}}_{\overline{\beta}}{\mathfrak{D}}_{\alpha}\mathcal{A}\;=\;{\overline{\partial}}_{\mathcal{A}}\Phi_{{\overline{\beta}}\alpha}~{},\qquad\mathbbl{F}_{\alpha{\overline{\beta}}}\;=\;\\!-\Phi_{\alpha{\overline{\beta}}}+\Phi_{{\overline{\beta}}\alpha}~{}.$

That $\mathbbl{F}$ is antihermitian implies
$\Phi^{\dagger}_{\alpha{\overline{\beta}}}+\Phi_{\alpha{\overline{\beta}}}=\Phi^{\dagger}_{{\overline{\beta}}\alpha}+\Phi_{{\overline{\beta}}\alpha}$.

Finally, there is a purely holomorphic derivative. We write down a Hodge
decomposition

$\begin{split}{\mathfrak{D}}_{\alpha}{\mathfrak{D}}_{\beta}\mathcal{A}&\;=\;{\mathfrak{D}}_{\alpha}{\mathfrak{D}}_{\beta}\mathcal{A}^{harm}+{\overline{\partial}}_{\mathcal{A}}\Phi_{\alpha\beta}+{\overline{\partial}}_{\mathcal{A}}^{\dagger}\Psi_{\alpha\beta}~{},\end{split}$
(6.4)

where the harmonic term will be proportional to the first order variations
${\mathfrak{D}}_{\alpha}\mathcal{A}$ through some parameter-dependent
coefficients.

Recall, the action of small gauge transformations is described by
$\Lambda_{a}\to\Lambda_{a}-\phi_{a}$. The field $\phi_{a}$ evaluated at
$y_{0}\in M$ gauge fixes the first order derivative
${\mathfrak{D}}_{a}\mathcal{A}$, which as described previously, is holomorphic
gauge ${\mathfrak{D}}_{{\overline{\alpha}}}\mathcal{A}=0$. Given this gauge
fixing, the second order derivatives transform under
${\mathfrak{D}}_{a}\phi_{b}|_{y_{0}}{\;=\;}{\partial}_{a}\phi_{b}+[\Lambda_{a},\phi_{b}]|_{y_{0}}$
as follows

${\mathfrak{D}}_{b}{\mathfrak{D}}_{a}\mathcal{A}\sim{\mathfrak{D}}_{b}{\mathfrak{D}}_{a}\mathcal{A}+{\overline{\partial}}_{\mathcal{A}}\left({\mathfrak{D}}_{b}\phi_{a}\right)~{}.$
(6.5)

Holomorphic gauge at second order is given by setting
${\mathfrak{D}}_{{\overline{\alpha}}}\phi_{{\overline{\beta}}}=\\!-\Phi_{{\overline{\alpha}}{\overline{\beta}}}$
and
${\mathfrak{D}}_{\alpha}\phi_{{\overline{\beta}}}=\\!-\Phi_{\alpha{\overline{\beta}}}$
so that

${\mathfrak{D}}_{{\overline{\beta}}}{\mathfrak{D}}_{{\overline{\alpha}}}\mathcal{A}\;=\;0~{},\qquad{\mathfrak{D}}_{\beta}{\mathfrak{D}}_{{\overline{\alpha}}}\mathcal{A}\;=\;0~{}.$
(6.6)

Note that antihermiticity requires
${\mathfrak{D}}_{\overline{\beta}}\phi_{\alpha}=-({\mathfrak{D}}_{\beta}\phi_{{\overline{\alpha}}})^{\dagger}=\Phi_{{\overline{\beta}}\alpha}^{\dagger}$.
This, together with the fact that there are no
${\overline{\partial}}_{\mathcal{A}}$–closed scalars, completely fixes the
gauge freedom at this order.

From the commutator of the first equation in (6.6) we gather that
$\mathbbl{F}_{{\overline{\alpha}}{\overline{\beta}}}=0$ and the universal
bundle is holomorphic. In passing we observe that this is only true after
gauge-fixing second order deformations to holomorphic gauge. There is not
enough freedom, however, to get rid of
$\mathbbl{F}_{\alpha{\overline{\beta}}}$. A perhaps useful intuition for this
is a simple counting. Second derivatives are given by four indendent
quantities

${\mathfrak{D}}_{\alpha}{\mathfrak{D}}_{\beta}\mathcal{A}_{{\overline{\mu}}}~{},\quad{\mathfrak{D}}_{{\overline{\alpha}}}{\mathfrak{D}}_{{\overline{\beta}}}\mathcal{A}_{{\overline{\mu}}}~{},\quad{\mathfrak{D}}_{\alpha}{\mathfrak{D}}_{{\overline{\beta}}}\mathcal{A}_{{\overline{\mu}}}~{},\quad{\mathfrak{D}}_{{\overline{\alpha}}}{\mathfrak{D}}_{\beta}\mathcal{A}_{{\overline{\mu}}}~{},$
(6.7)

while small gauge transformations amount to two independent quantities

${\mathfrak{D}}_{\alpha}\phi_{\beta}\,\quad{\mathfrak{D}}_{\alpha}\phi_{{\overline{\beta}}}~{}.$
(6.8)

Hence, we can only gauge-fix only two among (6.7), which is what we did in
(6.6).

In this gauge we have

${\mathfrak{D}}_{{\overline{\beta}}}{\mathfrak{D}}_{\alpha}\mathcal{A}\;=\;{\overline{\partial}}_{\mathcal{A}}\mathbbl{F}_{\alpha{\overline{\beta}}}~{},\qquad\mathbbl{F}_{\alpha{\overline{\beta}}}=\Phi_{{\overline{\beta}}\alpha}+\Phi^{\dagger}_{{\overline{\beta}}\alpha}\;=\;\Phi_{\alpha{\overline{\beta}}}+\Phi^{\dagger}_{\alpha{\overline{\beta}}}~{}.$
(6.9)

We now turn to the equations of motion. A real variation of the Bianchi
identity (6.2) is

${\mathfrak{D}}_{a}{\mathfrak{D}}_{b}F\;=\;{\text{d}}_{A}{\mathfrak{D}}_{a}{\mathfrak{D}}_{b}A+\\{{\mathfrak{D}}_{a}A,{\mathfrak{D}}_{b}A\\}~{}.$
(6.10)

The manifold $X$ has fixed complex structure. This implies
${\mathfrak{D}}_{a}{\mathfrak{D}}_{b}F^{(0,2)}=0$ and projecting the previous
equation onto type we find

${\overline{\partial}}_{\mathcal{A}}\left({\mathfrak{D}}_{a}{\mathfrak{D}}_{b}\mathcal{A}\right)+\\{{\mathfrak{D}}_{a}\mathcal{A},{\mathfrak{D}}_{b}\mathcal{A}\\}\;=\;0~{}.$
(6.11)

When indices of mixed holomorphic type are considered, we do not learn
anything new from this. When $a,b$ are holomorphic, we recognise the second
order Maurer–Cartan equation

$\begin{split}{\overline{\partial}}_{\mathcal{A}}\left({\mathfrak{D}}_{\alpha}{\mathfrak{D}}_{\beta}\mathcal{A}\right)+\\{{\mathfrak{D}}_{\alpha}\mathcal{A},{\mathfrak{D}}_{\beta}\mathcal{A}\\}&\;=\;0~{}.\end{split}$
(6.12)

From this we can read the coefficients $\Psi_{\alpha\beta}$ in the Hodge
decomposition (6.4)

$\Box_{\mathcal{A}}\,\Psi_{\alpha\beta}~{}=-\\{{\mathfrak{D}}_{\alpha}\mathcal{A},{\mathfrak{D}}_{\beta}\mathcal{A}\\}~{}.$
(6.13)

Unobstructedness of deformations means that the bracket does not contain zero-
modes and this is the condition that allows to invert the Laplacian and solve
for $\Psi_{\alpha\beta}$.

There is another equation to consider, the HYM equation $\omega^{2}F=0$. A
second variation of this, while keeping $X$ fixed, gives

$\omega^{2}\Big{(}{\text{d}}_{A}{\mathfrak{D}}_{a}{\mathfrak{D}}_{b}A+\\{{\mathfrak{D}}_{a}A,{\mathfrak{D}}_{b}A\\}\Big{)}\;=\;0~{}.$
(6.14)

In holomorphic gauge (6.6) this gives two equations

${\overline{\partial}}_{\mathcal{A}}^{\dagger}\,{\mathfrak{D}}_{\alpha}{\mathfrak{D}}_{\beta}\mathcal{A}\;=\;0~{},\quad{\overline{\partial}}_{\mathcal{A}}^{\dagger}\,{\mathfrak{D}}_{{\overline{\beta}}}{\mathfrak{D}}_{\alpha}\mathcal{A}~{}=-\text{i}\,\omega^{\mu{\overline{\nu}}}\\{{\mathfrak{D}}_{\alpha}\mathcal{A},{\mathfrak{D}}_{{\overline{\beta}}}\mathcal{A}^{\dagger}\\}_{\mu{\overline{\nu}}}~{}.$
(6.15)

Substituting (6.4) and (6.9) inside these and using stability to invert the
Laplacian, we finally obtain

$\begin{split}&\Phi_{\alpha\beta}\;=\;0~{},\qquad\mathbbl{F}_{\alpha{\overline{\beta}}}\;=\;\\!\Box_{{\overline{\partial}}_{\mathcal{A}}}^{-1}\Big{(}-\text{i}\,\omega^{\mu{\overline{\nu}}}\\{{\mathfrak{D}}_{\alpha}\mathcal{A},{\mathfrak{D}}_{{\overline{\beta}}}\mathcal{A}^{\dagger}\\}_{\mu{\overline{\nu}}}\Big{)}~{}.\end{split}$
(6.16)

The non-primitive part of the bracket of mixed variations acts as a source for
$\mathbbl{F}_{\alpha{\overline{\beta}}}$. Therefore, deformations of stable
bundles over a fixed manifold do not necessarily commute.

It would be very interesting to apply this second order analysis with the
beautiful work in [27] who described ‘holomorphic’ data in terms of a
functional of $SU(3)$ structures. In that work the starting point at first
order in deformations implicitly assumes holomorphic gauge (together with the
residual gauge constraint $\delta\Omega^{(3,0)}$ is harmonic) and at second
order assumes holomorphic derivatives commute. The latter condition may or may
not be true. Determining this requires a complete analysis of gauge fixing
second order derivatives and determining if the curvature tensor
$S_{ab},\mathbbl{H}_{ab},\mathbbl{F}_{ab}$ are non–zero. Doing so is therefore
both interesting and important.

##  7 Conclusions

We started with the humble ambition of writing down the action of small gauge
transformations together with clarifying gauge fixing in heterotic theories.
We achieved this goal.

Along the way we clarified certain issues. We related the choice of gauge
fixing to a choice of connection on the moduli space. We found holomorphic
gauge corresponds to the universal bundle being holomorphic. We reiterated
that taking a holomorphic deformation of fields is itself a gauge fixing,
holomorphic gauge. There is a residual gauge freedom which we showed can be
used to gauge fix the deformation of the holomorphic top form to be harmonic
$\delta\Omega^{(3,0)}=k\Omega$ with $k$ a constant. This gauge fixing implies
${\partial}\chi_{\alpha}=0$.

We checked that the equations of motion are properly invariant under gauge
transformations and they do not fix the gauge. They can then be used to write
explicit expressions for terms in the Hodge decomposition of all the fields,
which as we have gauge fixed, are the physical degrees of freedom. The role of
quantum corrections is to generate exact and co–exact terms. The coupled
nature of the equations makes it very hard to disentangle the heterotic moduli
as ‘bundle’ or ‘complex structure’ or ‘hermitian moduli’. For example, in
heterotic theories one cannot deform the bundle without also deforming the
manifold. Nonetheless we give a prescription for how to compute through the
mess applying it to the moduli space metric.

We showed the choice of gauge is related to a choice of connection on the
moduli space $M$ corresponds to choice of small gauge. For example,
holomorphic gauge in spacetime corresponds to the universal bundle being
holomorphic $\mathbbl{F}^{(0,2)}=0$. At second order deformations do not
obviously commute and gauge fixing is related to the field strengths
$\mathbbl{F}_{ab}$, $S_{ab}$ and $\mathbbl{H}_{ab}$ on the moduli space. In a
toy example, we found gauge fixing allowed us to eliminate some, but not all
of these field strengths. Consequently, bundle deformations do not commute.

There are many interesting questions. We hope to publish a full analysis of
second order universal geometry in the near future. This has important
implications for any ‘heterotic special geometry’ such as relations between
the Yukawa couplings and the moduli space metrics. Is it possible to formulate
a description of the moduli space in a fully gauge invariant way? The
discussion in this paper and the literature makes use of a nice choice of
gauge. Is there a more gauge invariant formulation of these results in which
we do not need to gauge fix? Or can we show our results do not depend on our
choice of gauge?

## Acknowledgements

JM would like to thank the University of Sydney, Australia and ICMAT, Madrid,
Spain for very kind hospitality while this work was completed. We would like
to acknowledge conversations with A. Ashmore, M. Garcia–Fernandez, M. Graffeo,
C. Strickland–Constable, E. Svanes, G. Williamson, and M. Wolf.

##  Appendix A Complexified tangent spaces and gauge groups

In this short appendix we write down some well-known mathematical facts,
largely drawn from [28], about complexification of vector spaces and groups,
and relate these facts to gauge transformations.

Let $X$ be a differentiable manifold of dimension $m$. Let $g,h\in\mathpzc
F(X)$ be two real–valued smooth functions. Then,
$f=g+\text{i}h:X\to\mathbbl{C}$ is a complex valued function and is an element
of the complexified set of smooth functions of $X$ denoted as $\mathpzc
F(X)^{\,\mathbbl{C}}$. Its complex conjugate is ${\overline{f}}=g-\text{i}h$
and it is real if and only if ${\overline{f}}=f$.

Let $\mathpzc V$ be a real vector space with real dimension $m$. . Its
complexification $\mathpzc V^{\,\mathbbl{C}}$ is a vector space of complex
dimension $m$ whose elements consist of $V+\text{i}W$ with $V,W\in\mathpzc V$,
and multiplication by a scalar is enhanced from real scalars to complex scalar
quantities. $\mathpzc V\subset\mathpzc V^{\,\mathbbl{C}}$ with the
identification of $V+\text{i}0\in\mathpzc V^{\,\mathbbl{C}}$ and $V\in\mathpzc
V$. Such vectors are said to be real. The complex conjugate of $Z=V+\text{i}W$
is denoted ${\overline{Z}}=V-\text{i}W$ and $Z$ is real if and only if
$Z={\overline{Z}}$. A linear operator $e$ acts on $\mathpzc V^{\,\mathbbl{C}}$
is the obvious way: $e(V+\text{i}W)=e(V)+\text{i}e(W)$. If $e\to\mathbbl{R}$
is a linear function, so that $e\in\mathpzc V^{*}$, then its complexification
acts on $\mathpzc V^{\,\mathbbl{C}}$ as $e:\mathpzc
V^{\,\mathbbl{C}}\to\mathbbl{C}$. The quantity $e$ is said to be real if
${\overline{e(V+\text{i}W)}}=e(V-\text{i}W)$. Tensors are complexified in the
obvious way: $T=T_{1}+\text{i}T_{2}$ where $T_{1},T_{2}$ are tensors of the
same type. A tensor is real if ${\overline{T}}=T$.

Let $e_{m}$ be a basis for $\mathpzc V$. If we regard these as complex vectors
the same basis $\\{e_{m}\\}$ becomes a basis for $\mathpzc V^{\,\mathbbl{C}}$.
That is, if $V=V^{m}e_{m}$ and $W=W^{m}e_{m}$ are both real vectors then
$V+\text{i}W=(V^{m}+\text{i}W^{m})e_{m}\in\mathpzc V^{\,\mathbbl{C}}$, where
${\overline{e_{m}}}=e_{m}$. Hence, $\dim{R}\mathpzc V=\dim{C}\mathpzc V{C}$.

The tangent space $\mathpzc
T_{p}X=\\{V^{m}{\partial}_{m}\,|\,V^{m}\in\mathbbl{R}\\}$ is an example of a
real vector space. Its complexification is therefore $\mathpzc
T_{p}X^{\,\mathbbl{C}}=\\{(V^{m}+\text{i}W^{m}){\partial}_{m}\,|\,V^{m},W^{m}\in\mathbbl{R}\\}$.
As above, the basis elements for $\mathpzc T_{p}X$ are regarded as basis
elements for the complexified tangent space. The co-tangent space is
complexified in the obvious way $\mathpzc
T^{*}_{p}X=\\{\omega+\text{i}\eta\,|\,\omega,\eta\in\mathpzc T^{*}_{p}X$.
Tensors are similarly complexified, and the extension to the tangent bundle
$\mathpzc T_{X}{C}$ and co-tangent bundle $\mathpzc T_{X}^{*}{C}$ follows.

If $X$ has $\dim{R}X=2k$ and admits an integrable complex structure then we
can find a split of the coordinates so that
$\frac{{\partial}}{{\partial}x^{1}},\cdots,\frac{{\partial}}{{\partial}x^{k}},\frac{{\partial}}{{\partial}y^{1}},\cdots\frac{{\partial}}{{\partial}y^{k}}$
are a basis for $\mathpzc T_{p}X$ so that
$J(\frac{{\partial}}{{\partial}x^{m}})=\frac{{\partial}}{{\partial}y^{m}}$ and
$J(\frac{{\partial}}{{\partial}y^{m}})=-\frac{{\partial}}{{\partial}x^{m}}$.
These vectors are also a basis for the complexified tangent space $\mathpzc
T_{p}X{C}$, so that $\dim{C}\mathpzc T_{p}X{C}=2k$. Furthermore, we can
redefine the basis so that

$\frac{{\partial}}{{\partial}z^{\mu}}\;=\;\frac{1}{2}\Big{(}\frac{{\partial}}{{\partial}x^{\mu}}-\text{i}\frac{{\partial}}{{\partial}y^{\mu}}\Big{)}~{},\qquad\frac{{\partial}}{{\partial}z^{\overline{\mu}}}\;=\;\frac{1}{2}\Big{(}\frac{{\partial}}{{\partial}x^{\overline{\mu}}}+\text{i}\frac{{\partial}}{{\partial}y^{\overline{\mu}}}\Big{)}~{},\qquad\mu,{\overline{\mu}}\;=\;1,\cdots
k~{},$

is also basis for $\mathpzc T_{p}X{C}$ but one in which $J$ is constant and
diagonal. Note that
${\overline{{\partial}/{\partial}z^{\mu}}}={\partial}/{\partial}z^{\overline{\mu}}$.
With this choice of basis we have a canonical split of the vector fields

$\mathpzc T_{X}{C}\;=\;\mathpzc T_{X}^{(1,0)}+\mathpzc T_{X}^{(0,1)}~{}.$

and so a vector $V\in\mathpzc T_{X}{C}$ is

$V^{\mu}{\partial}_{\mu}+W^{\overline{\mu}}{\partial}_{\overline{\mu}}~{},\qquad
V^{\mu},W^{\overline{\mu}}\in\mathbbl{C}~{},\qquad\mu,{\overline{\mu}}=1,\cdots,k~{}.$

This is real if and only ${\overline{V^{\mu}}}=W^{\overline{\mu}}$. We often
write this condition as ${\overline{V^{\mu}}}=V^{\overline{\mu}}$ or as
${\overline{V^{(1,0)}}}=V^{(0,1)}$. Note that $\dim{C}\mathpzc
T_{X}^{(1,0)}=\dim{C}\mathpzc T_{X}^{(0,1)}=k$.

##  Appendix B Gauge fixing small diffeomorphisms on Calabi–Yau manifolds

Suppose the manifold $X$ is Kähler so that ${\text{d}}\omega=0$ and
$h^{(0,2)}=h^{(0,1)}=0$. On a Kähler manifold, there is a relation between the
Laplacians

$\Box_{{\partial}}\;=\;\Box_{{\overline{\partial}}}\;=\;\frac{1}{2}\Box_{{\text{d}}}~{}.$
(B.1)

###  B.1 Real deformations

Consider a real deformation. By this we mean that $\delta=\delta
y^{\alpha}{\partial}_{\alpha}+\delta
y^{\overline{\beta}}{\partial}_{\overline{\beta}}$ so that when applied to the
Kähler form, the deformation is manifestly real
$(\delta\omega)^{*}=\delta\omega$.

Using (B.1), $h^{0,2}=h^{2,0}=0$ and the Hodge decomposition being unique, we
can Hodge decompose the equation of motion ${\text{d}}\delta\omega=0$ with
respect to the d–operator

$\delta\omega\;=\;\gamma+{\text{d}}\beta~{}.$

where $\gamma$ is d–harmonic $(1,1)$–form.

Small diffeomorphisms act as

$\delta\omega\sim\delta\omega+{\text{d}}(\varepsilon^{m}\omega_{m})~{}.$

We can partially fix this freedom by setting
$\varepsilon^{m}\omega_{m}=-\beta+{\text{d}}\psi$, where $\psi$ is an
arbitrary function on $X$. In this gauge $\delta\omega=\gamma^{(1,1)}$ is
harmonic. There is a residual gauge freedom parameterised by $\psi$.

There is also the holomorphic $(3,0)$ form $\Omega$. Its deformation
decomposed into type is

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

The Hodge decomposition of $\delta\Omega^{(3,0)}$ with respect to the
${\partial}$–operator is

$\delta\Omega^{(3,0)}\;=\;k\,\Omega+{\partial}\zeta^{(2,0)}\;=\;(k+k_{\zeta}(x))\Omega~{},$
(B.3)

where in the first equality we use $h^{(3,0)}=1$ and so $\Omega$ is the unique
harmonic $(3,0)$–form and $k$ is a parameter dependent constant. In the second
equality we have shown the ${\partial}$–exact term is proportional to
$\Omega$, except where the coefficient
$k_{\zeta}(x)=\frac{1}{3!\left\|\Omega\right\|^{2}}({\partial}\xi)_{\mu\nu\rho}{\overline{\Omega}}^{\mu\nu\rho}\Omega$
is not a constant. The equation of motion ${\text{d}}\delta\Omega=0$ implies

${\partial}\chi\;=\;{\overline{\partial}}{\partial}\zeta^{(2,0)}~{},\qquad{\overline{\partial}}\chi\;=\;0~{},$

where $\chi=\delta\Omega^{(2,1)}$.

We can also use
$\Omega=\frac{1}{3!}f(x,y)\epsilon_{\mu\nu\rho}{\text{d}}x^{\mu\nu\rho}$ where
$f=e^{\text{i}\phi}|f|$ and $|f|^{2}=\left\|\Omega\right\|^{2}\sqrt{g}$. The
pure part of the deformation is

$\delta\Omega^{(3,0)}\;=\;\Big{(}\text{i}\delta\phi+\frac{1}{2}(\delta\log\left\|\Omega\right\|^{2}+\omega^{\mu{\overline{\nu}}}\delta\omega_{\mu{\overline{\nu}}}\big{)}\Big{)}\Omega~{},$

where $\delta\phi$ is a globally defined function on $X$. Supersymmetry
implies $\left\|\Omega\right\|$ is a constant. Using the fact the Levi–Civita
connection on a Kähler manifold is metric and hermitian so that
$\nabla_{\rho}\omega{\;=\;}0$ and torsionless

$\begin{split}{\partial}_{\rho}\left(\omega^{\mu{\overline{\nu}}}\delta\omega_{\mu{\overline{\nu}}}\right)&\;=\;\left(\nabla_{\rho}\omega^{\mu{\overline{\nu}}}\right)\delta\omega_{\mu{\overline{\nu}}}+\omega^{\mu{\overline{\nu}}}\left(\nabla_{\rho}\delta\omega_{\mu{\overline{\nu}}}\right)\\\
&\;=\;-\text{i}\nabla^{\overline{\nu}}\delta\omega_{\rho{\overline{\nu}}}-\text{i}g^{\mu{\overline{\nu}}}({\partial}\delta\omega^{(1,1)})_{\rho\mu{\overline{\nu}}}~{}.\end{split}$
(B.4)

With $\delta\omega$ harmonic, it is both ${\partial}$-closed and
${\overline{\partial}}$–coclosed and so both terms vanish. Hence
$\omega^{\mu{\overline{\nu}}}\delta\omega_{\mu{\overline{\nu}}}$ is a
constant.

We still have the residual gauge freedom parameterised by
$\varepsilon^{m}\omega_{m}={\text{d}}\psi$. While $\delta\omega$ is invariant,
$\delta\Omega^{(3,0)}$ transforms

$\delta\Omega^{(3,0)}\sim\delta\Omega^{(3,0)}-\text{i}(\nabla_{\mu}\nabla^{\mu}\psi)\Omega~{},$
(B.5)

where we use
${\partial}(\varepsilon^{m}\Omega_{m}){\;=\;}(\nabla_{\mu}\varepsilon^{\mu})\Omega$
and $\varepsilon^{\mu}=\\!-\text{i}\nabla^{\mu}\psi$. This implies an action
on $\delta\phi$ given by

$\delta\phi\sim\delta\phi-\nabla^{\mu}\nabla_{\mu}\psi~{}.$ (B.6)

At this point, we pause to recount some properties of the Laplacian acting on
a scalar $\Psi$ with a source $\Phi$ on the compact Kähler manifold $X$:

$\Box_{\partial}\Psi\;=\;\nabla^{\mu}\nabla_{\mu}\Psi\;=\;\Phi~{}.$ (B.7)

We divide $\Psi$ into a formal sum of zero modes
$\Box_{\partial}\Psi_{0}{\;=\;}0$ and non-zero modes
$\Box_{\partial}\Psi_{i}=\lambda_{i}\Psi_{i}$, where $\lambda_{i}\neq 0$:

$\Psi=\Psi_{0}+\sum_{i}\Psi_{i}~{}.$

The obstruction to solving the equation (B.7) is that the source $\Phi$ be
orthogonal to the zero modes $\int_{X}\Psi_{0}\Phi=0$. On $X$ the zero modes
are constants. Hence, pick a $\Psi_{0}\neq 0$ and evaluate the following
integral in two different ways

$\int\Psi_{0}\nabla^{\mu}\nabla_{\mu}\Psi_{i}\;=\;\lambda_{i}\Psi_{0}\int\Psi_{i}\;=\;\\!-\int\nabla^{\mu}\Psi_{0}\nabla_{\mu}\Psi_{i}\;=\;0~{},$

we see that the non-zero modes have vanishing integral
$\int_{X}\Psi_{i}{\;=\;}0$. If $\Phi=k$ for some constant $k$ then does (B.7)
have a solution? The integral of a total divergence on a compact manifold
vanishes

$\int_{X}\nabla^{\mu}\nabla_{\mu}\Psi\;=\;k\,V\;=\;0~{},$

where $V$ is the volume of $X$ and so we see that $k=0$.

We now apply these lessons to the gauge fixing (B.6). We divide up the source
$\delta\phi$ into zero modes $\delta\phi_{0}$ and non-zero modes
$\delta\phi_{nz}$. The facts above indicate that $\int_{X}\delta\phi_{nz}=0$
and so we can find a $\psi$ so that
$\nabla^{\mu}\nabla_{\mu}\psi{\;=\;}\delta\phi_{nz}$. However, we cannot find
a $\psi$ in which $\nabla^{\mu}\nabla_{\mu}\psi{\;=\;}\delta\phi_{0}$ for
$\delta\phi_{0}$ a non-zero constant. Hence, by a choice of gauge, we can kill
the non-zero modes of $\delta\phi$ but not the zero modes, which are
constants. 999We have completely fixed the gauge freedom as
$\nabla^{\mu}\nabla_{\mu}\psi=k$ has solution only for $k=0$, in which case
$\psi$ are constants and acts on the fields only through its derivatives.

With the choice of gauge, we have $\delta\omega$ being harmonic and
$\delta\Omega^{(3,0)}=k\Omega$ for some parameter-dependent constant $k$. The
equation ${\text{d}}\,\delta\Omega{\;=\;}0$ gives

${\partial}\chi\;=\;{\partial}{\overline{\partial}}\zeta^{(2,0)}\;=\;0~{},\quad\text{and}\quad{\overline{\partial}}\,\chi\;=\;0~{},$

and we have $\nabla_{\mu}\Delta_{\alpha}^{\mu}=0$.

###  B.2 Holomorphic deformations

Let us now repeat this analysis in which $\delta_{h}=\delta
y^{\alpha}{\partial}_{\alpha}$ is a holomorphic deformation. This means
$\delta_{h}\omega$ is no longer real. The equation
${\text{d}}\delta_{h}\omega=0$ gives three parts

${\overline{\partial}}\delta_{h}\omega^{(0,2)}\;=\;0~{},\quad{\partial}\delta_{h}\omega^{(0,2)}+{\overline{\partial}}\delta_{h}\omega^{(1,1)}\;=\;0~{},\quad{\partial}\delta_{h}\omega^{(1,1)}\;=\;0~{}.$

The first equation is solved by

$\delta_{h}\omega^{(0,2)}\;=\;{\overline{\partial}}\xi^{(0,1)}~{}.$

The second equation becomes
${\overline{\partial}}(\delta_{h}\omega^{(1,1)}-{\partial}\xi^{(0,1)}){\;=\;}0$
which has a solution

$\delta_{h}\omega^{(1,1)}\;=\;\gamma^{(1,1)}+{\partial}\xi^{(0,1)}+{\overline{\partial}}\beta^{(1,0)}~{},$

where $\gamma^{(1,1)}$ is ${\overline{\partial}}$–harmonic. The final equation
gives ${\overline{\partial}}{\partial}\beta^{(1,0)}=0$ and after using
$h^{1,0}=h^{2,0}=0$ we have $\beta^{(1,0)}={\partial}\Phi$. Hence,

$\delta_{h}\omega^{(1,1)}\;=\;\gamma^{(1,1)}+{\partial}(\xi^{(0,1)}-{\overline{\partial}}\Phi)~{}.$

The gauge symmetries are small holomorphic diffemorphisms, in which
$\varepsilon^{\overline{\nu}}=0$ and $\varepsilon^{\mu}$ is free. These
preserve $\delta_{h}\omega^{(2,0)}=0$. The action on the remaining parts is

$\delta_{h}\omega^{(0,2)}\sim\delta_{h}\omega^{(0,2)}+\text{i}{\overline{\partial}}\varepsilon^{(0,1)}~{},\qquad\delta_{h}\omega^{(1,1)}\sim\delta_{h}\omega^{(1,1)}+\text{i}{\partial}\varepsilon^{(0,1)}~{}.$
(B.8)

We can choose a gauge in which $\delta_{h}\omega^{(0,2)}=0$ by setting

$\text{i}\varepsilon^{(0,1)}\;=\;-\xi^{(0,1)}+{\overline{\partial}}\psi~{}.$

The remaining term $\psi$ can be fixed by demanding $\delta_{h}\omega^{(1,1)}$
be harmonic

$\psi\;=\;\Phi~{}.$

While $\delta_{h}\omega$ is harmonic it is not necessarily real.

The holomorphic top form $\delta_{h}\Omega$ has a decomposition as before in
(B.2)–(B.3). The utility of $\delta$ being a holomorphic deformation is that
$\delta_{h}{\overline{\Omega}}=0$ and so

$\delta_{h}\Omega\;=\;(\delta_{h}\log
f)\Omega\;=\;(\delta_{h}\log|f|^{2})\Omega\;=\;\left(\delta_{h}\log\left\|\Omega\right\|^{2}+\omega^{\mu{\overline{\nu}}}\delta_{h}\omega_{\mu{\overline{\nu}}}\right)\Omega~{}.$

As before, the Levi–Civita connection satisfies $\nabla_{\rho}\omega=0$ and
${\partial}$–closure of $\delta_{h}\omega^{(1,1)}$ implies
$\nabla_{\rho}\delta_{h}\omega_{\mu{\overline{\nu}}}=\nabla_{\mu}\delta_{h}\omega_{\rho{\overline{\nu}}}$.
Hence,

${\partial}_{\rho}(\omega^{\mu{\overline{\nu}}}\delta_{h}\omega_{\mu{\overline{\nu}}})\;=\;\omega^{\mu{\overline{\nu}}}\nabla_{\mu}\delta_{h}\omega_{\rho{\overline{\nu}}}\;=\;-\text{i}\nabla^{\overline{\nu}}\delta_{h}\omega_{\rho{\overline{\nu}}}~{}.$

In the gauge in which $\delta_{h}\omega^{(1,1)}$ is harmonic it follows that
$\omega^{\mu{\overline{\nu}}}\delta_{h}\omega_{\mu{\overline{\nu}}}$ is a
constant. Hence, $\delta_{h}\Omega^{(3,0)}=k\Omega$ for some
parameter–dependent constant $k$ and ${\partial}\zeta^{(2,0)}=0$. This is the
same gauge as the previous subsection.

###  B.3 A third way

To satisfy ourselves the previous calculation is correct, we repeat the
calculation this time starting from a holomorphic variation of $\Omega$. The
decomposition of $\delta\Omega^{(3,0)}$ with respect to the
${\partial}$–operator gives

$\delta\Omega^{(3,0)}\;=\;k\Omega+{\partial}\xi^{(2,0)}~{},$

where ${\partial}\xi^{(2,0)}=k_{\xi}\Omega$. A gauge transformation acts as

$\delta\Omega\to\delta\Omega+{\text{d}}(\varepsilon^{\nu}\Omega_{\nu})\;=\;\delta\Omega+(\nabla_{\nu}\varepsilon^{\nu})\Omega~{}.$

We choose

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

where $\zeta^{(1,0)}$ is an arbitrary one form. With this choice
$\nabla_{\nu}\varepsilon^{\nu}=\\!-\frac{1}{3!\left\|\Omega\right\|^{2}}{\overline{\Omega}}^{\nu\rho\sigma}({\partial}\xi^{(2,0)})_{\nu\rho\sigma}$
and $\delta\Omega^{(3,0)}=k\Omega$ for some constant $k$. We see that
$\zeta^{(1,0)}$ is a residual gauge freedom that does not affect
$\delta\Omega$.

The variation of the Kähler form is holomorphic, and so the equation of motion
${\text{d}}\delta\omega=0$ implies

${\overline{\partial}}\delta\omega^{(0,2)}\;=\;0~{},\quad{\partial}\delta\omega^{(0,2)}+{\overline{\partial}}\delta\omega^{(1,1)}\;=\;0~{},\quad{\partial}\delta\omega^{(1,1)}\;=\;0~{}.$

The solution to these equations is

$\delta\omega^{(0,2)}\;=\;{\overline{\partial}}\alpha^{(0,1)}~{},\quad\delta\omega^{(1,1)}=\gamma^{(1,1)}+{\partial}\alpha^{(0,1)}~{},\quad\delta\omega^{(2,0)}\;=\;0~{},$

where $\gamma^{(1,1)}$ is a harmonic $(1,1)$–form. There is a condition on
$\alpha^{(0,1)}$ that derives from the gauge fixing of $\delta\Omega$. To see
this, we first turn to a calculation used several times above

${\partial}(\omega^{\mu{\overline{\nu}}}\delta\omega_{\mu{\overline{\nu}}})\;=\;\text{i}{\overline{\partial}}^{\dagger}\delta\omega^{(1,1)}\;=\;0~{},\quad{\overline{\partial}}(\omega^{\mu{\overline{\nu}}}\delta\omega_{\mu{\overline{\nu}}})\;=\;\\!-\text{i}g^{\mu{\overline{\nu}}}({\overline{\partial}}\delta\omega^{(1,1)})_{\mu{\overline{\nu}}}+\text{i}{\partial}^{\dagger}\delta\omega^{(1,1)}\;=\;0~{},$

The vanishing of these equation is the gauge fixing
$\delta\Omega^{(3,0)}=k\Omega$. Consider the first equation:

${\overline{\partial}}^{\dagger}\delta\omega^{(1,1)}\;=\;(\nabla^{\overline{\nu}}\nabla_{\rho}\alpha_{\overline{\nu}}){\text{d}}x^{\rho}\;=\;{\partial}(\nabla^{\overline{\nu}}\alpha_{\overline{\nu}})\;=\;0~{}.$

There are no holomorphic functions on $X$ and so this implies
$\nabla^{\overline{\nu}}\alpha_{\overline{\nu}}$ is a constant. In fact, the
constant must vanish, as can be seen by integrating it over the manifold and
using that a total divergance on a compact manifold has vanishing integral.
Hence,

${\overline{\partial}}^{\dagger}\alpha^{(0,1)}\;=\;0~{}.$

On the other hand, we write this in terms of the Hodge dual

${\overline{\partial}}^{\dagger}\alpha^{(0,1)}\;=\;\\!-\star\,{\partial}\star\alpha^{(0,1)}\;=\;\\!-\star\,{\partial}(\alpha_{\overline{\nu}}\Omega^{\overline{\nu}})\frac{\text{i}{\overline{\Omega}}}{\left\|\Omega\right\|^{2}}\;=\;0~{},$

The fact this vanishes implies
$\alpha_{\overline{\nu}}\Omega^{\overline{\nu}}={\partial}\beta^{(1,0)}$ for
some $(1,0)$–form. This equation is invertible

$\alpha^{(0,1)}\;=\;\frac{1}{2\left\|\Omega\right\|^{2}}{\overline{\Omega}}^{\rho\sigma}({\partial}\beta^{(1,0)})_{\rho\sigma}~{}.$

We finally, see that $\alpha^{(0,1)}$ takes the same form as the residual
gauge freedom described above, and that by setting
$\zeta^{(1,0)}=-\beta^{(1,0)}$ we fix $\delta\omega^{(1,1)}=\gamma^{(1,1)}$ to
be harmonic.

We have now illustrated how to gauge fix to harmonic gauge in three seperate
ways, and this serves as a useful guide for gauge fixing the heterotic system.

###  B.4 Conformally balanced condition

A Calabi–Yau manifold is in particular a balanced manifold and satisfies

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

A holomorphic variation gives a pair of independent equations

${\partial}(\delta_{h}\omega^{(0,2)}\omega)+{\overline{\partial}}(\delta_{h}\omega^{(1,1)}\omega)\;=\;0~{},\qquad{\partial}(\delta_{h}\omega^{(1,1)}\omega)\;=\;0~{}.$

Under a small diffeomorphism (B.8) these equations are preserved provided we
use ${\text{d}}\\!\omega=0$. Hence, a holomorphic variation of the balanced
condition on a Kähler manifold is not a gauge fixing.

Using the following Hodge duals

$\star\,\delta_{h}\omega^{(1,1)}\;=\;(\omega^{\mu{\overline{\nu}}}\delta_{h}\omega_{\mu{\overline{\nu}}})\omega^{2}-\delta_{h}\omega^{(1,1)}\omega~{},\qquad\star\,\delta_{h}\omega^{(0,2)}\;=\;\delta_{h}\omega^{(0,2)}\omega~{},$

we get

${\partial}^{\dagger}\delta_{h}\omega^{(1,1)}\;=\;{\overline{\partial}}^{\dagger}\delta_{h}\omega^{(0,2)}+\text{i}\,{\overline{\partial}}(\omega^{\mu{\overline{\nu}}}\delta_{h}\omega_{\mu{\overline{\nu}}})~{},\qquad{\overline{\partial}}^{\dagger}\delta_{h}\omega^{(1,1)}\;=\;\text{i}\,{\partial}(\omega^{\mu{\overline{\nu}}}\delta_{h}\omega_{\mu{\overline{\nu}}})~{}.$

##  Appendix C Hodge duals and contractions

We establish a set of notations and conventions, as well enumerate several
useful relations.

### Real Riemannian manifolds

We start with a $D$-dimensional compact Riemannian manifold $X$ with metric
$g$.

### Inner products for forms

The pointwise inner product on (exterior powers of) the cotangent space uses
the inverse metric $g^{-1}=g^{mn}\,{\partial}_{m}\otimes{\partial}_{n}$. Given
$k$-forms $\eta$, $\xi$ it is defined as

$g^{-1}(\eta,\xi)\;=\;\frac{1}{k!}\,\eta_{m_{1}...m_{k}}\,g^{m_{1}n_{1}}...g^{m_{k}n_{k}}\,\xi_{n_{1}...n_{k}}\;=\;\frac{1}{k!}\,\eta^{m_{1}...m_{k}}\,\xi_{m_{1}...m_{k}}~{},$
(C.1)

which is a real function over $X$, whose integral defines an inner product for
forms

$(~{}\cdot~{},~{}\cdot~{})~{}~{}:~{}~{}\Omega^{k}(X)\times\Omega^{k}(X)\to\mathbb{R}~{},$

with

$(\eta,\xi)\;=\;\frac{1}{V}\int_{X}\mbox{vol}\,g^{-1}(\eta,\xi)\;=\;\frac{1}{V\,k!}\int_{X}\mbox{vol}\,\eta^{m_{1}...m_{k}}\,\xi_{m_{1}...m_{k}}~{}.$

This is positive definite and with the definition above the zero-form $1$ and
the $D$-form vol have unit norm. Recall, the Riemannian volume form is

$\mbox{vol}\;=\;\frac{1}{D!}\,\sqrt{g}\,\epsilon_{m_{1}...m_{D}}\,{\text{d}}x^{m_{1}...m_{D}}~{},$

where $\epsilon_{m_{1}...m_{D}}$ is the constant antisymmetric symbol.

### Contraction of forms

Given two forms $\eta_{k}$ and $\xi_{l}$, where the subscript denotes their
degree and $k\leq l$, we can form their contraction, or interior product. The
symbol that denotes this operation is

$\lrcorner:\Omega^{k}(X)\times\Omega^{l}(X)\to\Omega^{l-k}(X)~{},$

and acts on forms as follows

$\eta_{k}\,\lrcorner\,\xi_{l}\;=\;\frac{1}{k!(l-k)!}\,\eta^{m_{1}...m_{k}}\,\xi_{m_{1}...m_{k}\,n_{1}...n_{l-k}}\,{\text{d}}x^{n_{1}...n_{l-k}}\;=\;\frac{1}{k!}\,\eta^{m_{1}...m_{k}}\,\xi_{m_{1}...m_{k}}~{}.$

The inner product (C.1) becomes then just a special case of contraction

$g^{-1}(\eta,\xi)\;=\;\eta\,\lrcorner\,\xi~{}.$

An interesting feature of this operator is that it is the adjoint of the wedge
product

$(\sigma_{k-l}\,\lrcorner\,\eta_{k},\xi_{l})\;=\;(\eta_{k},\sigma_{k-l}{\,\wedge\,}\xi_{l})~{}.$
(C.2)

### Hodge star operator

The Hodge dual operator

$\star:\Omega^{k}(X)\to\Omega^{D-k}(X)~{},$

defined as

$\begin{split}\star\,\eta&\;=\;\frac{\sqrt{g}}{k!(D-k)!}\,\eta^{m_{1}...m_{k}}\,\epsilon_{m_{1}..m_{k}n_{1}...n_{D-k}}\,{\text{d}}x^{n_{1}...n_{D-k}}~{}.\end{split}$

Hence, $\eta\,\star\xi=(\eta\,\lrcorner\,\xi)\,\mbox{vol}$.

The $\star$ operator satisfies the identities

$\begin{split}\star^{2}\,\eta_{k}&\;=\;(-)^{k(D-k)}\,\eta_{k}~{},\\\\[5.69046pt]
g^{-1}(\eta,\xi)&\;=\;g^{-1}(\star\eta\,,\star\xi)~{}.\end{split}$

The first line means that $\star$ is invertible with possible eigenvalues $\pm
1$ or $\pm i$ according to the degree of the form and the number of
dimensions. The second property means $\star$ is an isometry.

### Codifferential

The codifferential is denoted

${\text{d}}^{\dagger}:\Omega^{k}(X)\to\Omega^{k-1}(X)~{},$

and is defined as the adjoint of the de Rham operator. That is,

$({\text{d}}\eta_{k-1},\xi_{k})\;=\;(\eta_{k-1},{\text{d}}^{\dagger}\xi_{k})~{}.$

In order to find its explicit expression we need to perform an integration by
parts and use the first line in (C). Boundary terms will be neglected because
we assume $X$ has no boundary. The calculation goes as follows

$\begin{split}({\text{d}}\eta_{k-1},\xi_{k})&\;=\;\frac{1}{V}\int_{X}{\text{d}}\eta_{k-1}\star\xi_{k}\;=\;\\!(-)^{k}\,\frac{1}{V}\int_{X}\eta_{k-1}~{}{\text{d}}\star\xi_{k}\\\\[5.69046pt]
&\;=\;(-)^{k}\,\frac{1}{V}\int_{X}\eta_{k-1}\,(-)^{(D-k+1)(k-1)}\star^{2}{\text{d}}\star\xi_{k}\;=\;\frac{1}{V}\int_{X}\eta_{k-1}\,\star\big{(}-(-)^{D(k+1)}\star{\text{d}}\star\xi_{k}\big{)}~{},\end{split}$

and we end up with

${\text{d}}^{\dagger}\xi_{k}\;=\;\\!-(-)^{D(k-1)}\star{\text{d}}\star\xi_{k}~{}.$

Another expression, very useful in calculations, is written in terms of the LC
connection. It can be obtained very quickly by reminding that d has a
representation

${\text{d}}\;=\;{\text{d}}x^{m}\,\nabla^{\text{\tiny LC}}_{m}~{},$

using property (C.2) and an integration by parts, as follows

$({\text{d}}\eta,\xi)\;=\;({\text{d}}x^{m}\,\nabla^{\text{\tiny
LC}}_{m}\eta,\xi)\;=\;(\nabla^{\text{\tiny
LC}}_{m}\eta,\xi^{m})\;=\;(\eta,-\nabla^{\text{\tiny LC}}_{m}\xi^{m})~{}.$

We gather that

${\text{d}}^{\dagger}\xi_{k}\;=\;\\!-\nabla^{\text{\tiny
LC}}_{m}\xi^{m}\;=\;\\!-\frac{1}{(k-1)!}\,\nabla^{\text{\tiny
LC}}_{n}\xi^{n}{}_{m_{1}...m_{k-1}}\,{\text{d}}x^{m_{1}...m_{k-1}}~{}.$

Proving this identity without using (C.2) involves rather cumbersome
calculations.

### Complex manifolds

We now take $X$ to be complex with complex dimension $N$. We denote
holomorphic coordinates $x^{\mu},x^{{\overline{\nu}}}$. The manifold is
equipped with a hermitian metric

$g\;=\;g_{\mu{\overline{\nu}}}\,({\text{d}}x^{\mu}\otimes{\text{d}}x^{{\overline{\nu}}}+{\text{d}}x^{{\overline{\nu}}}\otimes{\text{d}}x^{\mu})~{},\quad{\text{d}}s^{2}\;=\;2\,g_{\mu{\overline{\nu}}}~{}{\text{d}}x^{\mu}{\text{d}}x^{{\overline{\nu}}}~{},$
(C.3)

and compatible hermitian form $\omega$

$\omega\;=\;\omega_{\mu{\overline{\nu}}}\,{\text{d}}x^{\mu{\overline{\nu}}}\;=\;\text{i}g_{\mu{\overline{\nu}}}\,{\text{d}}x^{\mu{\overline{\nu}}}~{}.$

The pointwise inner product for forms respects hermitianity: given forms
$\eta,\xi$ of holomorphic type $(p,q)$ we define

$g^{-1}(\eta,\overline{\xi})\;=\;\eta\,\lrcorner\,\overline{\xi}\;=\;\frac{1}{p!q!}\,\eta^{{\overline{\mu}}_{1}...{\overline{\mu}}_{p}\nu_{1}...\nu_{q}}\,\overline{\xi}_{{\overline{\mu}}_{1}...{\overline{\mu}}_{p}\nu_{1}...\nu_{q}}~{}.$

When integrated over the manifold this gives

$(\eta,\overline{\xi})\;=\;\frac{1}{V}\int_{X}\eta\star\overline{\xi}~{}.$

Observe how the Hodge $\star$ operator acts on type

$\star:\Omega^{(p,q)}(X)\to\Omega^{(N-q,N-p)}(X)~{}.$

The de Rham differential splits into the sum of Dolbeault operators
${\text{d}}={\partial}+{\overline{\partial}}$. Analogously, the codifferential
also splits
${\text{d}}^{\dagger}={\partial}^{\dagger}+{\overline{\partial}}^{\dagger}$
where

$\begin{split}&{\partial}^{\dagger}:\Omega^{(p,q)}(X)\to\Omega^{(p-1,q)}(X)\qquad,\qquad{\partial}^{\dagger}\;=\;\\!-\star{\overline{\partial}}\,\star~{},\\\\[2.84544pt]
&{\overline{\partial}}^{\dagger}:\Omega^{(p,q)}(X)\to\Omega^{(p,q-1)}(X)\qquad,\qquad{\overline{\partial}}^{\dagger}\;=\;\\!-\star{\partial}\,\star~{}.\end{split}$
(C.4)

Let us discuss the volume form on $X$. The holomorphic coordinates have a
writing $x^{\mu}=u^{\mu}+\text{i}\,v^{\mu}$ with $u,v$ real coordinates. In
terms of the latter, the volume form is defined as

$\mbox{vol}\;=\;\sqrt{g}~{}{\text{d}}u^{1}\,{\text{d}}v^{1}...{\text{d}}u^{N}\,{\text{d}}v^{N}~{}.$

Expressed in terms of the complex coordinates:

$\mbox{vol}\;=\;\frac{\text{i}^{N^{2}}}{N!^{2}}\,\sqrt{|g|}\,\epsilon_{\mu_{1}...\mu_{N}}\,\epsilon_{{\overline{\nu}}_{1}...{\overline{\nu}}_{N}}\,{\text{d}}x^{\mu_{1}...\mu_{N}}\,{\text{d}}x^{{\overline{\nu}}_{1}...{\overline{\nu}}_{N}}~{}.$
(C.5)

We can also write this in terms of the holomorphic volume form

$\mbox{vol}\;=\;\frac{\text{i}\,\Omega{\overline{\Omega}}}{\left\|\Omega\right\|^{2}}~{}.$

It coincides with the $N$-th power of the hermitian form

$\begin{split}\frac{1}{N!}\omega^{N}&\;=\;\frac{(-)^{\frac{N(N-1)}{2}}\,\text{i}^{N}}{N!}\,g_{\mu_{1}{\overline{\nu}}_{1}}...\,g_{\mu_{N}{\overline{\nu}}_{N}}\,\epsilon^{\mu_{1}...\mu_{N}}\,\epsilon^{{\overline{\nu}}_{1}...{\overline{\nu}}_{N}}\,{\text{d}}x^{12...N}\,{\text{d}}x^{\bar{1}\bar{2}...\bar{N}}\\\\[5.69046pt]
&\;=\;\text{i}^{N^{2}}\det{g_{\mu{\overline{\nu}}}}\,{\text{d}}x^{12...N}\,{\text{d}}x^{\bar{1}\bar{2}...\bar{N}}\\\\[2.0pt]
&\;=\;\frac{\text{i}^{N^{2}}}{N!^{2}}\,\sqrt{|g|}\,\epsilon_{\mu_{1}...\mu_{N}}\,\epsilon_{{\overline{\nu}}_{1}...{\overline{\nu}}_{N}}\,{\text{d}}x^{\mu_{1}...\mu_{N}}\,{\text{d}}x^{{\overline{\nu}}_{1}...{\overline{\nu}}_{N}}~{}.\end{split}$

### Codifferential for $\Delta_{\alpha}$

Consider the space $\Omega^{(0,q)}(X,\mathpzc T_{X}^{(1,0)})$. We are mostly
interested in $q=1$ but it is not much harder to work in generality. Elements
of this space are

$\eta^{\mu}\;=\;\frac{1}{q!}\,\eta_{{\overline{\nu}}_{1}...{\overline{\nu}}_{q}}{}^{\mu}\,{\text{d}}x^{{\overline{\nu}}_{1}...{\overline{\nu}}_{q}}~{}.$

There is a hermitian metric

$(\eta^{\mu},\xi^{{\overline{\nu}}})\;=\;\frac{1}{V}\int_{X}\eta^{\mu}\star\xi^{{\overline{\nu}}}\,g_{\mu{\overline{\nu}}}~{},\quad\xi^{{\overline{\nu}}}\
\;=\;\frac{1}{q!}\,\xi_{\rho_{1}..\rho_{q}}{}^{{\overline{\nu}}}\,{\text{d}}x^{\rho_{1}...\rho_{q}}~{},$
(C.6)

where we understand that $\star$ treats $\eta^{\mu},\xi^{{\overline{\nu}}}$ as
forms. For example, when $q=0,1$

$\star\,v^{\mu}\;=\;v^{\mu}~{}\mbox{vol}~{},\quad\star\,\Delta^{\mu}\;=\;\text{i}\,\Delta^{\mu}~{}\frac{\omega^{2}}{2}~{}.$

There is also a differential operator

${\overline{\partial}}:\Omega^{(0,q)}(X,\mathpzc
T_{X}^{(1,0)})\to\Omega^{(0,q+1)}(X,\mathpzc T_{X}^{(1,0)})~{},$

that raises the degree of one. We use holomorphic coordinates, and this acts
covariantly.

With the help of the metric (C.6) we can define its adjoint

${\overline{\partial}}^{\dagger}:\Omega^{(0,q)}(X,\mathpzc
T_{X}^{(1,0)})\to\Omega^{(0,q-1)}(X,\mathpzc T_{X}^{(1,0)})~{},$

which satisfies the property

$(\eta^{\mu},{\partial}\xi^{{\overline{\nu}}})\;=\;({\overline{\partial}}^{\dagger}\eta^{\mu},\xi^{{\overline{\nu}}})~{}.$

The calculation uses an integration by parts – in which we neglect boundary
terms – and standard properties of $\star$. We end up with

$\begin{split}(\eta^{\mu},{\partial}\xi^{{\overline{\nu}}})\;=\;\frac{1}{V}\int_{X}\Big{(}-\star{\partial}(\star\eta^{\rho}\,g_{\rho{\overline{\sigma}}})\,g^{{\overline{\sigma}}\mu}\Big{)}\star\xi^{{\overline{\nu}}}\,g_{\mu{\overline{\nu}}}~{}.\end{split}$

From this we read the expression for the codifferential

${\overline{\partial}}^{\dagger}\eta^{\mu}~{}=-\star{\partial}(\star\eta^{\rho}\,g_{\rho{\overline{\sigma}}})\,g^{{\overline{\sigma}}\mu}~{}.$
(C.7)

For $q=0$ this vanishes trivially. When $q=1$, the case we are most interested
in, we find using also the balanced condition

${\overline{\partial}}^{\dagger}\Delta^{\mu}~{}=-\star\frac{\text{i}\omega^{2}}{2}\,\big{(}{\partial}\Delta^{\mu}+g^{\mu{\overline{\sigma}}}{\partial}g_{{\overline{\sigma}}\rho}~{}\Delta^{\rho}\big{)}~{}=-\nabla_{{\rm
Ch}}^{{\overline{\nu}}}\Delta_{{\overline{\nu}}}{}^{\mu}~{}.$ (C.8)

Observe how this is different from the gauge-fixing condition (3.10), due to a
different ordering of the indices. The two expressions coincide when
$\Delta_{[{\overline{\mu}}{\overline{\nu}}]}=0$.

### Hodge dual relations on a three-fold

We can finally enumerate some Hodge dual relations for various types of forms
on $X$.

One-forms, type $(1,0)$:

$\star\,\eta^{(1,0)}\;=\;\\!-\text{i}\,\eta^{(1,0)}~{}\frac{\omega^{2}}{2}~{}.$
(C.9)

Two-forms, types $(2,0)$ and $(1,1)$:

$\begin{split}&\star\,\eta^{(2,0)}\;=\;\eta^{(2,0)}\,\omega~{},\\\\[2.84544pt]
&\star\,\eta^{(1,1)}\;=\;\\!-\text{i}\,\eta_{\mu}{}^{\mu}~{}\frac{\omega^{2}}{2}-\eta^{(1,1)}\,\omega\;=\;(\omega\,\lrcorner\,\eta^{(1,1)})~{}\frac{\omega^{2}}{2}-\eta^{(1,1)}\,\omega~{},\end{split}$
(C.10)

Three-forms, types $(3,0)$ and $(2,1)$:

$\begin{split}&\star\,\eta^{(3,0)}\;=\;\\!-\text{i}\,\eta^{(3,0)}~{},\\\\[2.84544pt]
&\star\,\eta^{(2,1)}\;=\;\text{i}\,\eta^{(2,1)}-\eta_{\mu}{}^{\mu}\,{}^{(1,0)}\,\omega\;=\;\text{i}\,\eta^{(2,1)}-\text{i}\,(\omega\,\lrcorner\,\eta^{(2,1)})\,\omega~{},\end{split}$
(C.11)

Type $(2,3)$:

$\star\eta^{(2,3)}\;=\;\frac{\text{i}}{2}~{}\eta_{\mu\nu}{}^{\mu\nu}\,{}^{(0,1)}\;=\;\frac{\text{i}}{2}~{}\omega\,\lrcorner\,(\omega\,\lrcorner\,\eta^{(2,3)})~{}.$
(C.12)

A useful special case of (LABEL:eq:Hodge2form) is
$\frac{1}{2}\omega^{2}=\star\omega$. Further relations such as that for
$\eta^{(0,1)}$ can be easily determined using complex conjugation.

##  Appendix D Connection symbols on $X$

We enumerate some commonly used connection symbols in heterotic theories. We
list the components in complex coordinates. We also give expressions for
various divergences which are useful for calculations in the paper.

Levi-Civita

$\begin{split}&\Gamma^{\text{\tiny
LC}}{}_{\mu}{}^{\nu}{}_{\rho}\;=\;\frac{1}{2}\,g^{\nu{\overline{\sigma}}}({\partial}_{\mu}g_{\rho{\overline{\sigma}}}+{\partial}_{\rho}g_{\mu{\overline{\sigma}}})\;=\;g^{\nu{\overline{\sigma}}}\,{\partial}_{\mu}g_{\rho{\overline{\sigma}}}-\frac{1}{2}H_{\mu}{}^{\nu}{}_{\rho}\;=\;g^{\nu{\overline{\sigma}}}\,{\partial}_{\rho}g_{\mu{\overline{\sigma}}}+\frac{1}{2}H_{\mu}{}^{\nu}{}_{\rho}~{},\\\\[2.84544pt]
&\Gamma^{\text{\tiny
LC}}{}_{\mu}{}^{{\overline{\nu}}}{}_{\rho}\;=\;0~{},\\\\[2.84544pt]
&\Gamma^{\text{\tiny
LC}}{}_{\mu}{}^{\nu}{}_{{\overline{\rho}}}\;=\;\frac{1}{2}\,g^{\nu{\overline{\sigma}}}({\partial}_{{\overline{\rho}}}g_{\mu{\overline{\sigma}}}-{\partial}_{{\overline{\sigma}}}g_{\mu{\overline{\rho}}})\;=\;\frac{1}{2}H_{\mu}{}^{\nu}{}_{{\overline{\rho}}}~{},\\\\[2.84544pt]
&\Gamma^{\text{\tiny
LC}}{}_{\mu}{}^{{\overline{\nu}}}{}_{{\overline{\rho}}}\;=\;\frac{1}{2}\,g^{{\overline{\nu}}\sigma}({\partial}_{\mu}g_{{\overline{\rho}}\sigma}-{\partial}_{\sigma}g_{\mu{\overline{\rho}}})\;=\;\\!-\frac{1}{2}H_{\mu}{}^{{\overline{\nu}}}{}_{{\overline{\rho}}}~{}.\end{split}$
(D.1)

Bismut

$\begin{split}&\Gamma^{{\text{\tiny
B}}}{}_{\mu}{}^{\nu}{}_{\rho}\;=\;g^{\nu{\overline{\sigma}}}\,{\partial}_{\rho}g_{\mu{\overline{\sigma}}}\;=\;g^{\nu{\overline{\sigma}}}\,{\partial}_{\mu}g_{\rho{\overline{\sigma}}}-H_{\mu}{}^{\nu}{}_{\rho}~{},\\\\[2.84544pt]
&\Gamma^{{\text{\tiny
B}}}{}_{\mu}{}^{{\overline{\nu}}}{}_{\rho}\;=\;0~{},\\\\[2.84544pt]
&\Gamma^{{\text{\tiny
B}}}{}_{\mu}{}^{\nu}{}_{{\overline{\rho}}}\;=\;0~{},\\\\[2.84544pt]
&\Gamma^{{\text{\tiny
B}}}{}_{\mu}{}^{{\overline{\nu}}}{}_{{\overline{\rho}}}\;=\;g^{{\overline{\nu}}\sigma}({\partial}_{\mu}g_{\sigma{\overline{\rho}}}-{\partial}_{\sigma}g_{\mu{\overline{\rho}}})\;=\;\\!-H_{\mu}{}^{{\overline{\nu}}}{}_{{\overline{\rho}}}~{}.\end{split}$
(D.2)

Hull

$\begin{split}&\Gamma^{{\text{\tiny
H}}}{}_{\mu}{}^{\nu}{}_{\rho}\;=\;g^{\nu{\overline{\sigma}}}\,{\partial}_{\mu}g_{\rho{\overline{\sigma}}}\;=\;g^{\nu{\overline{\sigma}}}\,{\partial}_{\rho}g_{\mu{\overline{\sigma}}}+H_{\mu}{}^{\nu}{}_{\rho}~{},\\\\[2.84544pt]
&\Gamma^{{\text{\tiny
H}}}{}_{\mu}{}^{{\overline{\nu}}}{}_{\rho}\;=\;0~{},\\\\[2.84544pt]
&\Gamma^{{\text{\tiny
H}}}{}_{\mu}{}^{\nu}{}_{{\overline{\rho}}}\;=\;g^{\nu{\overline{\sigma}}}({\partial}_{{\overline{\rho}}}g_{\mu{\overline{\sigma}}}-{\partial}_{{\overline{\sigma}}}g_{\mu{\overline{\rho}}})\;=\;H_{\mu}{}^{\nu}{}_{{\overline{\rho}}}~{},\\\\[2.84544pt]
&\Gamma^{{\text{\tiny
H}}}{}_{\mu}{}^{{\overline{\nu}}}{}_{{\overline{\rho}}}\;=\;0~{}.\end{split}$
(D.3)

Chern

$\begin{split}&\Gamma^{{\rm
Ch}}{}_{\mu}{}^{\nu}{}_{\rho}\;=\;g^{\nu{\overline{\sigma}}}\,{\partial}_{\mu}g_{\nu{\overline{\sigma}}}\;=\;g^{\nu{\overline{\sigma}}}\,{\partial}_{\rho}g_{\mu{\overline{\sigma}}}+H_{\mu}{}^{\nu}{}_{\rho}~{},\\\\[2.84544pt]
&\Gamma^{{\rm
Ch}}{}_{\mu}{}^{{\overline{\nu}}}{}_{\rho}\;=\;0~{},\\\\[2.84544pt]
&\Gamma^{{\rm
Ch}}{}_{\mu}{}^{\nu}{}_{{\overline{\rho}}}\;=\;0~{},\\\\[2.84544pt]
&\Gamma^{{\rm
Ch}}{}_{\mu}{}^{{\overline{\nu}}}{}_{{\overline{\rho}}}\;=\;0~{}.\end{split}$
(D.4)

### Divergences

The divergence of a vector $\varepsilon^{\mu}$ taken with respect to a generic
connection

$\nabla_{\mu}\varepsilon^{\mu}\;=\;{\partial}_{\mu}\varepsilon^{\mu}+\varepsilon^{\mu}\,\Gamma_{\nu}{}^{\nu}{}_{\mu}~{}.$
(D.5)

To compute this we need the following contraction

$\begin{split}\Gamma^{\text{\tiny
LC}}{}_{\nu}{}^{\nu}{}_{\mu}&\;=\;{\partial}_{\mu}\log{\sqrt{g}}+\frac{1}{2}H_{\mu\nu}{}^{\nu}~{},\\\\[2.84544pt]
\Gamma^{{\text{\tiny
B}}}{}_{\nu}{}^{\nu}{}_{\mu}&\;=\;{\partial}_{\mu}\log{\sqrt{g}}~{},\\\\[5.69046pt]
\Gamma^{{\text{\tiny H}}}{}_{\nu}{}^{\nu}{}_{\mu}\;=\;\Gamma^{{\rm
Ch}}{}_{\nu}{}^{\nu}{}_{\mu}&\;=\;{\partial}_{\mu}\log{\sqrt{g}}+H_{\mu\nu}{}^{\nu}~{}.\end{split}$
(D.6)

The four choices above give

$\begin{split}\nabla_{\mu}^{\text{\tiny
LC}}\varepsilon^{\mu}&\;=\;{\partial}_{\mu}\varepsilon^{\mu}+\varepsilon^{\mu}\,{\partial}_{\mu}\log{\sqrt{g}}+\frac{1}{2}\varepsilon^{\mu}\,H_{\mu\nu}{}^{\nu}~{},\\\\[2.84544pt]
\nabla_{\mu}^{{\text{\tiny
B}}}\varepsilon^{\mu}&\;=\;{\partial}_{\mu}\varepsilon^{\mu}+\varepsilon^{\mu}\,{\partial}_{\mu}\log{\sqrt{g}}~{},\\\\[5.69046pt]
\nabla_{\mu}^{{\text{\tiny H}}}\varepsilon^{\mu}\;=\;\nabla_{\mu}^{{\rm
Ch}}\varepsilon^{\mu}&\;=\;{\partial}_{\mu}\varepsilon^{\mu}+\varepsilon^{\mu}\,{\partial}_{\mu}\log{\sqrt{g}}+\varepsilon^{\mu}\,H_{\mu\nu}{}^{\nu}\end{split}$
(D.7)

As for the vector-valued form $\Delta_{{\overline{\nu}}}{}^{\mu}$ we have, for
a generic connection

$\nabla_{\mu}\Delta_{{\overline{\nu}}}{}^{\mu}\;=\;{\partial}_{\mu}\Delta_{{\overline{\nu}}}{}^{\mu}+\Delta_{{\overline{\nu}}}{}^{\mu}\,\Gamma_{\rho}{}^{\rho}{}_{\mu}-\Gamma_{\mu}{}^{{\overline{\rho}}}{}_{{\overline{\nu}}}\,\Delta_{{\overline{\rho}}}{}^{\mu}~{},$
(D.8)

and the four choices above give

$\begin{split}\nabla_{\mu}^{\text{\tiny
LC}}\Delta_{{\overline{\nu}}}{}^{\mu}&\;=\;{\partial}_{\mu}\Delta_{{\overline{\nu}}}{}^{\mu}+\Delta_{{\overline{\nu}}}{}^{\mu}\,{\partial}_{\mu}\log{\sqrt{g}}+\frac{1}{2}\Delta_{{\overline{\nu}}}{}^{\mu}\,H_{\mu\rho}{}^{\rho}-\frac{1}{2}\Delta^{\mu\rho}\,H_{\mu\rho{\overline{\nu}}}~{},\\\\[2.84544pt]
\nabla_{\mu}^{{\text{\tiny
B}}}\Delta_{{\overline{\nu}}}{}^{\mu}&\;=\;{\partial}_{\mu}\Delta_{{\overline{\nu}}}{}^{\mu}+\Delta_{{\overline{\nu}}}{}^{\mu}\,{\partial}_{\mu}\log{\sqrt{g}}-\Delta^{\mu\rho}\,H_{\mu\rho{\overline{\nu}}}~{},\\\\[5.69046pt]
\nabla_{\mu}^{{\text{\tiny
H}}}\Delta_{{\overline{\nu}}}{}^{\mu}\;=\;\nabla_{\mu}^{{\rm
Ch}}\Delta_{{\overline{\nu}}}{}^{\mu}&\;=\;{\partial}_{\mu}\Delta_{{\overline{\nu}}}{}^{\mu}+\Delta_{{\overline{\nu}}}{}^{\mu}\,{\partial}_{\mu}\log{\sqrt{g}}+\Delta_{{\overline{\nu}}}{}^{\mu}\,H_{\mu\rho}{}^{\rho}~{}.\end{split}$
(D.9)

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