# Teukolsky equations, twistor functions, and conformally self-dual spaces

Bernardo Araneda111Email<EMAIL_ADDRESS>
Max-Planck-Institut für Gravitationsphysik
(Albert-Einstein-Institut), Am Mühlenberg 1,
D-14476 Potsdam, Germany

###### Abstract

We prove a correspondence, for Riemannian manifolds with self-dual Weyl
tensor, between twistor functions and solutions to the Teukolsky equations for
any conformal and spin weights. In particular, we give a contour integral
formula for solutions to the Teukolsky equations, and we find a recursion
operator that generates an infinite family of solutions and leads to the
construction of Čech representatives and sheaf cohomology classes on twistor
space. Apart from the general conformally self-dual case, examples include
self-dual black holes, scalar-flat Kähler surfaces, and quaternionic-Kähler
metrics, where we map the Teukolsky equation to the conformal wave equation,
establish new relations to the linearised Przanowski equation, and find new
classes of quaternionic deformations.

##  1 Introduction

The Teukolsky equations [1] are scalar partial differential equations
describing perturbations of algebraically special space-times, notably
rotating black holes. They are instrumental to modern studies in mathematical
relativity and gravitational physics, including black hole stability,
classical and quantum fields on black holes, propagation of gravitational
waves, etc. In Newman-Penrose notation [2], the Teukolsky equation for the
extreme spin-weight component, $\chi$, of a spin-$|s|$ field (with
$s\in\frac{1}{2}\mathbb{Z}$), is

$\displaystyle{\rm T}_{s}\chi:={}$ $\displaystyle
2\left[\left(D-(2s-1)\varepsilon+\tilde{\varepsilon}-2s\rho-\tilde{\rho}\right)\left(D^{\prime}+2s\varepsilon^{\prime}-\rho^{\prime}\right)\right.$
(1.1)
$\displaystyle\left.-(\delta-(2s-1)\beta+\tilde{\beta}^{\prime}-2s\tau-\tilde{\tau})\left(\delta^{\prime}+2s\beta^{\prime}-\tau^{\prime}\right)-2(s-\tfrac{1}{2})(s-1)\Psi_{2}\right]\chi=0$

where $D,D^{\prime},\delta,\delta^{\prime}$ are derivatives along a principal
null tetrad, $\Psi_{2}$ is the middle component of the Weyl tensor with
respect to this tetrad, and the rest of the symbols in (1.1) are the spin
coefficients of the tetrad. For $s=0$, (1.1) reduces to the covariant scalar
wave equation.

More generally, the Teukolsky equation (1.1) is part of a larger family of
scalar equations associated to space-time perturbations. Other examples
include: the spin-weight zero component of an electromagnetic perturbation
satisfies the Fackerell-Ipser equation [3]; gravitational perturbations of
static black holes are described by the Regge-Wheeler and Zerilli equations
[4, 5]; and deformations of hyper-Kähler and quaternionic-Kähler 4-metrics are
encoded, respectively, in the linearised heavenly and Przanowski equations [6,
7, 8]. All of the above (including (1.1)) are particular cases of a
generalised “wave” equation,

$\displaystyle(\Box^{\mathscr{C}}+c\Psi_{2})\chi=0$ (1.2)

where $c$ is some constant222The only case in which $c$ is not a constant is
the Zerilli equation, where $c$ is an integro-differential operator. This is
not relevant for the present paper, because of the assumption (1.3). and
$\Box^{\mathscr{C}}=g^{ab}\mathscr{C}_{a}\mathscr{C}_{b}$ is the “wave”
operator associated to a conformally and GHP invariant connection
$\mathscr{C}_{a}$ [9, 10]. The field $\chi$ is a weighted scalar field that
carries both conformal weight $w\in\frac{\mathbb{Z}}{2}$ and spin weight
$s=\frac{p}{2}\in\frac{\mathbb{Z}}{2}$. For simplicity, we will refer to (1.2)
as the “Teukolsky equation for weights $(w,p)$” even though it includes not
only (1.1) but also all of the other cases mentioned above (simply by
adjusting $w$ and $p$).

Despite a number of remarkable properties of the Teukolsky equation, the
construction of general solutions is a difficult problem. In this work, we are
interested in the study of this equation on a special class of geometries:
those with self-dual Weyl tensor,

$\displaystyle*C_{abcd}=C_{abcd},$ (1.3)

also called conformally self-dual. Even though the assumption (1.3) excludes
the standard Lorentzian black holes, the Teukolsky equation on a geometry
satisfying (1.3) is non-trivial in that it does not reduce to the ordinary
scalar wave equation. In terms of (1.1)-(1.2), the condition (1.3) implies
$\Psi_{2}=0$ (but no restrictions on the Ricci tensor).

Our interest in manifolds satisfying (1.3) is that they have a twistor space.
In a number of situations, twistor methods have led to remarkable
constructions of solutions to (linear and non-linear) differential equations
on space-time in terms of free holomorphic twistor functions, see e.g. [11]. A
prominent example at the linear level is the Penrose transform: an isomorphism
between the space of massless fields on Minkowski and certain twistor sheaf
cohomology groups [12]. In this spirit, our main result in connection to the
Teukolsky equation is the following:

######  Theorem 1.

Let $(\mathcal{M},g_{ab})$ be conformally self-dual (1.3), let $\mathcal{PT}$
be its twistor space, and let $w\in\frac{\mathbb{Z}}{2}$, $p\in\mathbb{Z}$.
Fix an integrable complex structure $J^{a}{}_{b}$ on $\mathcal{M}$, with
principal spinors $o_{A},\iota_{A}$.

  1. 1.

Let $\Psi$ be a representative of the Čech cohomology group
$\check{H}^{1}(\mathcal{PT},\mathcal{O}(2w))$, and let

$\displaystyle\chi(x^{a}):=\frac{1}{2\pi{\rm
i}}\oint_{\Gamma}\frac{\Psi(x^{a},\lambda)\lambda_{A}{\rm
d}\lambda^{A}}{(o_{B}\lambda^{B})^{w+1-\frac{p}{2}}(\iota_{B}\lambda^{B})^{w+1+\frac{p}{2}}},$
(1.4)

where $x^{a}\in\mathcal{M}$, $\Gamma\subset\mathbb{CP}^{1}$. Then $\chi$
solves the Teukolsky equation for weights $(w,p)$.

  2. 2.

Let $\chi$ be a solution to the Teukolsky equation for weights $(w,p)$. Then
$\chi$ generates a Čech representative $\Psi$ in
$\check{H}^{1}(\mathcal{PT},\mathcal{O}(2w))$, such that formula (1.4) holds.

  3. 3.

The space-time field $\chi$ of the previous item is invariant under changes of
Čech representative $\Psi$ if and only if $w\pm\frac{p}{2}\in-\mathbb{N}$.

In the statement of the above result, we are using the fact that Čech
representatives are local holomorphic sections of $\mathcal{O}(2w)$ over
twistor space. The correspondence with cohomology classes is more subtle as it
implies a particular relation between conformal and spin weights, as described
by item 3. We also see that the conformal weight characterising a Teukolsky
equation plays an important role in that it determines the Chern class of the
line bundle involved in the cohomology group
$\check{H}^{1}(\mathcal{PT},\mathcal{O}(2w))$. Finally, we note that even
though a space satisfying (1.3) admits many (local) integrable complex
structures, the construction in theorem 1 depends on a choice of a single one,
which we see as a desirable feature since non-self-dual black holes (which
motivate this study) have only one such structure (for a fixed orientation).

Theorem 1 will be established following a number of intermediate results.
After some preliminary material in section 2, in section 3 we will first see
that, for generic Hermitian manifolds, there is a close relationship between
the twistor distribution on the spin bundle and the connection
$\mathscr{C}_{a}$ on space-time (prop. 6). Then under the assumption (1.3),
the twistor distribution has a non-trivial kernel (twistor functions) and this
will lead to formula (1.4) via the relation to $\mathscr{C}_{a}$ (prop. 7). We
will then find a symmetry operator that generates an infinite family of
solutions to (1.2) for different weights (prop. 8), and this will in turn lead
to a recursive construction of twistor functions (prop. 9 and 10), following
pioneering methods in the twistor literature [13, 6, 7]. Having established
the above, a natural question is whether the correspondence between twistor
functions (thought of as Čech representatives) and solutions to (1.2) is one-
to-one, this will lead naturally to sheaf cohomology (prop. 11). In the
special case where the geometry is also conformally Kähler, the recursion
operator will allow us to transform the Teukolsky equation into the conformal
wave equation (prop. 12).

Finally, one of the main applications of the Teukolsky equations is to
gravitational perturbations of vacuum spaces. Given our assumption (1.3), the
additional requirement of Ricci-flatness would give a hyper-Kähler metric. In
that case, a constant spin frame could be chosen and (1.1) would reduce to the
ordinary wave equation; to avoid this, one can introduce a cosmological
constant, so that there are no parallel spinors and (1.1) is non-trivial333One
can also study the Teukolsky equation on a hyper-Kähler metric by choosing a
non-constant spin frame, e.g. using a complex structure that is conformally
Kähler but not Kähler. Any Gibbons-Hawking space admits such a structure
(although other hyper-Kähler manifolds such as K3 or Atiyah-Hitchin do not)..
Such a space is called a quaternionic-Kähler 4-manifold, and in section 4 we
will apply our construction to study gravitational perturbations of these
spaces, via an operator identity (prop. 13). We will find new classes of
deformations that preserve the self-dual Einstein condition (prop. 14 and 15),
and we will also give contour integral formulae for them.

##  2 Preliminaries

###  2.1 Complex structures and connections

Let $(\mathcal{M},g_{ab})$ be a four-dimensional, orientable Riemannian
manifold, with Levi-Civita connection $\nabla_{a}$. Let $J^{a}{}_{b}$ be a
compatible almost-complex structure. The 2-form $g_{ac}J^{c}{}_{b}$ is
necessarily self-dual or anti-self-dual. Choosing the latter for concreteness,
$J^{a}{}_{b}$ can be written in terms of its principal spinors as

$\displaystyle J^{a}{}_{b}={\rm
i}(o^{A}\iota_{B}+\iota^{A}o_{B})\delta^{A^{\prime}}_{B^{\prime}},$ (2.1)

where we have chosen the normalisation $o_{A}\iota^{A}=1$. Notice that
$o_{A},\iota_{A}$ are only defined up to scale: we have the “GHP” freedom

$\displaystyle o_{A}\to zo_{A},\qquad\iota_{A}\to z^{-1}\iota_{A},\qquad
z\in\mathbb{C}^{*}.$ (2.2)

In Euclidean signature we have $\iota_{A}=o^{\dagger}_{A}$, and $z\in U(1)$.

The $+{\rm i}$ and $-{\rm i}$ eigenspaces of $J^{a}{}_{b}$ are respectively
$T^{+}={\rm span}\\{\iota^{A}\partial_{AA^{\prime}}\\}$ and $T^{-}={\rm
span}\\{o^{A}\partial_{AA^{\prime}}\\}$. The almost-complex structure is
integrable iff these eigenspaces are involutive, which can be shown to be
equivalent to the “shear-free” equations

$\displaystyle\iota^{A}\iota^{B}\nabla_{AA^{\prime}}\iota_{B}=0,\qquad
o^{A}o^{B}\nabla_{AA^{\prime}}o_{B}=0.$ (2.3)

In this case $T^{+}$ and $T^{-}$ are tangent to (complex) surfaces, which are
called twistor surfaces. Conversely, given two linearly independent spinor
fields satisfying (2.3) and $o_{A}\iota^{A}=1$, the tensor field (2.1) is an
integrable almost-complex structure.

The triple $(\mathcal{M},g_{ab},J^{a}{}_{b})$ is called an almost-Hermitian
manifold, and Hermitian if $J^{a}{}_{b}$ is integrable. The tensor field
$J^{a}{}_{b}$ is conformally invariant: if
$g_{ab}\to\hat{g}_{ab}=\Omega^{2}g_{ab}$ for some positive scalar field
$\Omega$, then $J^{a}{}_{b}$ is also a compatible almost-complex structure for
$\hat{g}_{ab}$. The principal spinors transform as

$\displaystyle\hat{o}_{A}=\Omega^{1/2}o_{A},\quad\hat{\iota}_{A}=\Omega^{1/2}\iota_{A},\qquad\hat{o}^{A}=\Omega^{-1/2}o^{A},\quad\hat{\iota}^{A}=\Omega^{-1/2}\iota^{A}.$
(2.4)

We note that, in a complexified picture, one can choose
$\hat{o}^{A}=\Omega^{w^{0}}o^{A}$ and
$\hat{\iota}^{A}=\Omega^{w^{1}}\iota^{A}$ where $w^{0},w^{1}$ are any real
numbers satisfying $w^{0}+w^{1}+1=0$ (see [2, Section 5.6]). The choice
$w^{0}=w^{1}=-1/2$ in (2.4) ensures that conformal transformations commute
with complex conjugation in Euclidean signature.

We define a weighted spinor field, with weights $(w,p)$ where $w$ is conformal
weight and $p$ is GHP weight, to be a spinor field $\varphi^{\mathcal{I}}$
($\mathcal{I}$ represents an arbitrary collection of abstract tensor/spinor
indices) which transforms under (2.2) and (2.4) as

$\displaystyle\varphi^{\mathcal{I}}\to\Omega^{w}z^{p}\varphi^{\mathcal{I}}.$
(2.5)

For example: $o_{A}$ and $\iota_{A}$ have weights $(\frac{1}{2},1)$ and
$(\tfrac{1}{2},-1)$ respectively, and $o^{A}$ and $\iota^{A}$ have weights
$(-\frac{1}{2},1)$ and $(-\tfrac{1}{2},-1)$. A weighted field is more properly
thought of as a section of a complex vector bundle $E_{(w,p)}$ over
$\mathcal{M}$. A connection $\mathscr{C}_{a}$ in this bundle was constructed
in [9, 10]. If $\psi,\varphi_{A},v_{b}$ are respectively a scalar, spinor, and
vector field, all of them with weights $(w,p)$, then $\mathscr{C}_{a}$ is
defined by

$\displaystyle\mathscr{C}_{a}\psi$
$\displaystyle=\nabla_{a}\psi+(wf_{a}+pP_{a})\psi,$ (2.6a)
$\displaystyle\mathscr{C}_{a}\varphi_{B}$
$\displaystyle=\nabla_{a}\varphi_{B}+(wf_{a}+pP_{a})\varphi_{B}-f_{A^{\prime}B}\varphi_{A},$
(2.6b) $\displaystyle\mathscr{C}_{a}v_{b}$
$\displaystyle=\nabla_{a}v_{b}+(wf_{a}+pP_{a})v_{b}-(f_{a}\delta^{c}_{b}+f_{b}\delta^{c}_{a}-g_{ab}f^{c})v_{c},$
(2.6c)

where $f_{a}$ is the Lee form and $P_{a}$ is a conformally invariant version
of the GHP connection 1-form $\omega_{a}$:

$\displaystyle f_{a}:=-\tfrac{1}{2}J^{b}{}_{c}\nabla_{b}J^{c}{}_{a},\qquad
P_{a}:=\omega_{a}-\tfrac{1}{2}{\rm
i}J^{b}{}_{a}f_{b},\qquad\omega_{a}:=\iota_{B}\nabla_{a}o^{B}.$ (2.7)

The action of $\mathscr{C}_{a}$ on fields with a different index structure is
deduced from (2.6) by standard spinor/tensor rules.

######  Remark 2.

We collect some identities that will be useful later, cf. [10] for more
details. Let $J^{a}{}_{b}$ be an almost-complex structure, with principal
spinors $o_{A},\iota_{A}$.

  1. 1.

The following identities hold (see e.g. [9, Eq. (2.53)]):

$\displaystyle\mathscr{C}_{AA^{\prime}}o_{B}$
$\displaystyle=\sigma^{0}_{A^{\prime}}\iota_{A}\iota_{B},\qquad\sigma^{0}_{A^{\prime}}:=o^{A}o^{B}\nabla_{AA^{\prime}}o_{B},$
(2.8a) $\displaystyle\mathscr{C}_{AA^{\prime}}\iota_{B}$
$\displaystyle=\sigma^{1}_{A^{\prime}}o_{A}o_{B},\qquad\sigma^{1}_{A^{\prime}}:=\iota^{A}\iota^{B}\nabla_{AA^{\prime}}\iota_{B}.$
(2.8b)

In particular, from (2.3) we see that $J^{a}{}_{b}$ is integrable iff
$\sigma^{0}_{A^{\prime}}=\sigma^{1}_{A^{\prime}}=0$.

  2. 2.

Define the following Dolbeault-like operators

$\displaystyle\tilde{\mathscr{C}}_{A^{\prime}}:=o^{A}\mathscr{C}_{AA^{\prime}},\qquad\mathscr{C}_{A^{\prime}}:=\iota^{A}\mathscr{C}_{AA^{\prime}}.$
(2.9)

If $J^{a}{}_{b}$ is integrable and $\Psi_{ABCD}$ is algebraically special
along $o^{A},\iota^{A}$, then

$\displaystyle\tilde{\mathscr{C}}^{A^{\prime}}\tilde{\mathscr{C}}_{A^{\prime}}=0,\qquad\mathscr{C}^{A^{\prime}}\mathscr{C}_{A^{\prime}}=0$
(2.10)

when acting on any weighted spinor field. See [10, Proposition 2.2].

  3. 3.

Suppose that $J^{a}{}_{b}$ is integrable and $\Psi_{ABCD}$ is algebraically
special along $o^{A},\iota^{A}$. Let $A_{A^{\prime}},B_{A^{\prime}}$ be
weighted spinor fields such that

$\displaystyle\tilde{\mathscr{C}}^{A^{\prime}}A_{A^{\prime}}=0,\qquad\mathscr{C}^{A^{\prime}}B_{A^{\prime}}=0.$
(2.11)

Then there exist, locally, scalar fields $a,b$ such that (cf. [10, Lemma 2.4])

$\displaystyle A_{A^{\prime}}=\tilde{\mathscr{C}}_{A^{\prime}}a,\qquad
B_{A^{\prime}}=\mathscr{C}_{A^{\prime}}b.$ (2.12)

###  2.2 Teukolsky operators

The connection $\mathscr{C}_{a}$ defines a “wave” operator on generic weighted
spinor fields:

$\displaystyle\Box^{\mathscr{C}}:=g^{ab}\mathscr{C}_{a}\mathscr{C}_{b}.$
(2.13)

######  Proposition 3.

Let $(\mathcal{M},g_{ab},J^{a}{}_{b})$ be an almost-Hermitian manifold.

  1. 1.

The operator (2.13) acting on arbitrary weighted scalar fields with weights
$(w,p)$ has the following Newman-Penrose expression:

$\displaystyle\tfrac{1}{2}\Box^{\mathscr{C}}={}$
$\displaystyle[D-(p-1)\varepsilon+\tilde{\varepsilon}+(w-\tfrac{p}{2}+1)\rho-\tilde{\rho}][D^{\prime}+p\varepsilon+(w+\tfrac{p}{2})\rho^{\prime}]$
(2.14)
$\displaystyle-[\delta-(p-1)\beta+\tilde{\beta}^{\prime}+(w-\tfrac{p}{2}+1)\tau-\tilde{\tau}^{\prime}][\delta^{\prime}+p\beta^{\prime}+(w+\tfrac{p}{2})\tau^{\prime}]$
$\displaystyle+\tfrac{3}{2}p\Psi_{2}-p(\sigma\sigma^{\prime}-\kappa\kappa^{\prime}).$

  2. 2.

Suppose that $(\mathcal{M},g_{ab},J^{a}{}_{b})$ is Hermitian. Let ${\rm
T}_{s}$ be the spin-weight $s$ Teukolsky operator (1.1). Then for any scalar
field $\chi$ with weights $w=-s-1$, $p=2s$, we have

$\displaystyle{\rm
T}_{s}\chi=\left[\Box^{\mathscr{C}}-2(2s^{2}+1)\Psi_{2}\right]\chi.$ (2.15)

  3. 3.

Suppose that $(\mathcal{M},g_{ab},J^{a}{}_{b})$ is conformally Kähler, where
the Kähler metric is $\hat{g}_{ab}=\phi^{2}g_{ab}$. Then for any scalar field
$\eta$ with weights $(w,0)$, we have

$\displaystyle\Box^{\mathscr{C}}\eta=\phi^{-(w+1)}(\Box+2\Psi_{2}+\tfrac{R}{6})[\phi^{(w+1)}\eta].$
(2.16)

  4. 4.

If the geometry is conformally self-dual, then the operators (2.9), (2.13)
acting on any weighted scalar field satisfy

$\displaystyle\mathscr{C}^{A^{\prime}}\tilde{\mathscr{C}}_{A^{\prime}}=\tfrac{1}{2}\Box^{\mathscr{C}},\qquad[\mathscr{C}_{A^{\prime}},\tilde{\mathscr{C}}^{A^{\prime}}]=0.$
(2.17)

###### Proof.

For the first item, note that
$\Box^{\mathscr{C}}=\mathscr{C}_{a}(g^{ab}\mathscr{C}_{b})$ and write the
metric as $g^{ab}=2(\ell^{a}n^{b}-m^{a}\tilde{m}^{b})-{\rm i}J^{ab}$, where
$\ell^{a},n^{a},m^{a},\tilde{m}^{a}$ is the null tetrad determined by the
principal spinors $o^{A},\iota^{A}$ of $J^{a}{}_{b}$ and an arbitrarily chosen
primed spin frame (the operator (2.13) is independent of this choice). From
(2.8), we always have $\mathscr{C}_{a}J^{ab}=0$. Expressions for
$\mathscr{C}_{a}\ell^{a}$ and $\mathscr{C}_{a}m^{a}$ are easily obtained from
(2.6c), and for the contractions of $\mathscr{C}_{a}$ with the null tetrad one
uses (2.6a).

The second item follows from (2.14) by comparing this expression to (1.1) and
noticing that on a Hermitian manifold,
$\kappa=\sigma=0=\kappa^{\prime}=\sigma^{\prime}$ (cf. item 2.8 in remark 2).

For the third item, the proof is very similar to the proof of Lemma 3.2 in
[9], by using the fact that in a conformally Kähler manifold, the Lee form is
$f_{a}=\partial_{a}\log\phi$.

Finally, the fourth item follows from equations (B.12), (B.13) and (B.26) in
[10], after noticing that a conformally self-dual geometry has $\Psi_{2}=0$. ∎

The above identities for the operator (2.13) motivate our assertion in the
introduction concerning the relation of eq. (1.2) to other equations in the
literature. Since in this work we are interested in geometries with
$\Psi_{2}=0$, we make the following definition:

######  Definition 4.

Let $\chi$ be a weighted scalar field, with conformal weight $w$ and GHP
weight $p$. For a conformally self-dual manifold $(\mathcal{M},g_{ab})$, the
second order equation

$\displaystyle\Box^{\mathscr{C}}\chi=0$ (2.18)

will be called the Teukolsky equation for weights $(w,p)$.

###  2.3 The twistor distribution

For any Riemannian 4-manifold $(\mathcal{M},g_{ab})$, we define the
correspondence space $\mathcal{F}$ to be the projective (unprimed) spin
bundle, so that locally $\mathcal{F}$ is identified with
$\mathcal{M}\times\mathbb{CP}^{1}$. A local section of $\mathcal{F}$ is a
spinor field $\lambda^{A}$ up to scale. A choice of compatible almost-complex
structure $J^{a}{}_{b}$ gives two preferred local trivialisations of
$\mathcal{F}$, in the form of the spinor fields $o^{A},\iota^{A}$ subject to
(2.2). Any local section $\lambda^{A}$ can be written as

$\displaystyle\lambda^{A}=\lambda_{1}o^{A}-\lambda_{0}\iota^{A},$ (2.19)

where $\lambda_{0}=\lambda_{A}o^{A}$, $\lambda_{1}=\lambda_{A}\iota^{A}$. We
can use $(\lambda_{0},\lambda_{1})$ as homogeneous coordinates in the fibres
of $\mathcal{F}$. Note, however, that due to the freedom (2.2)-(2.4), the
coordinates $(\lambda_{0},\lambda_{1})$ have weights: if $\lambda_{A}$ is
conformally invariant, then

$\displaystyle w(\lambda_{0})=-\frac{1}{2},\quad p(\lambda_{0})=1,\qquad
w(\lambda_{1})=-\frac{1}{2},\quad p(\lambda_{1})=-1.$ (2.20)

Any point $(x,\lambda_{0},\lambda_{1})$ in $\mathcal{F}$ defines a two-
dimensional subspace $\\{\lambda^{A}\partial_{AA^{\prime}}\\}\subset
T_{x}\mathcal{M}$, called a twistor plane. Using the spin connection, these
vectors can be lifted horizontally to $T_{(x,\lambda)}\mathcal{F}$; the
resulting two-dimensional distribution in $\mathcal{F}$ is called the twistor
distribution (cf. [13, Section 13.3] and [14, Section 10.1]): it can be
written abstractly as

$\displaystyle\lambda^{A}L_{AA^{\prime}}=\lambda^{A}\partial_{AA^{\prime}}-\Gamma_{AA^{\prime}B}{}^{C}\lambda^{A}\lambda^{B}\tfrac{\partial}{\partial\lambda^{C}},$
(2.21)

where $\Gamma_{AA^{\prime}B}{}^{C}$ is the unprimed spin connection
1-form444Recall that given a spin frame $\varepsilon^{A}_{i}$, with dual frame
$\theta^{i}_{A}$, the spin connection 1-form is the matrix-valued 1-form
$\Gamma_{AA^{\prime}i}{}^{j}$ defined by
$\nabla_{AA^{\prime}}\varepsilon^{C}_{i}=\Gamma_{AA^{\prime}i}{}^{j}\varepsilon^{C}_{j}$.
The object (2.22) is
$\Gamma_{AA^{\prime}B}{}^{C}:=\theta^{i}_{B}\nabla_{AA^{\prime}}\varepsilon^{C}_{i}$
(sum over $i$).,

$\displaystyle\Gamma_{AA^{\prime}B}{}^{C}=o_{B}\nabla_{AA^{\prime}}\iota^{C}-\iota_{B}\nabla_{AA^{\prime}}o^{C}.$
(2.22)

The twistor distribution is integrable iff $(\mathcal{M},g_{ab})$ is
conformally self-dual (eq. (1.3), or in spinor terms $\Psi_{ABCD}\equiv 0$),
see e.g. [13, Proposition 13.4.1]. In this case, (2.21) defines a foliation of
the correspondence space $\mathcal{F}$ by integral surfaces. The space of
leaves of the foliation is a three-dimensional complex manifold called twistor
space:

$\displaystyle\mathcal{PT}\equiv\mathcal{F}\diagup\\{\lambda^{A}L_{AA^{\prime}}\\}.$
(2.23)

The projection into $\mathcal{M}$ of each integral surface in $\mathcal{F}$ is
a twistor surface, so $\mathcal{PT}$ can also be viewed as the moduli space of
twistor surfaces in $\mathcal{M}$. Recalling the discussion below (2.3), the
existence of twistor surfaces defined by independent spinor fields implies
that, locally, one can find integrable almost-complex structures on
$\mathcal{M}$.

##  3 Twistor Theory

###  3.1 The correspondence space

Let $(\mathcal{M},g_{ab},J^{a}{}_{b})$ be almost-Hermitian, with the almost-
complex structure given by (2.1). Let $\mathcal{F}$ be the correspondence
space as in section 2.3. The local trivialisations defined by
$o^{A},\iota^{A}$ give a covering of $\mathcal{F}$ by two open sets,
$U_{0}=\\{\lambda_{0}\neq 0\\}$ and $U_{1}=\\{\lambda_{1}\neq 0\\}$. The
variables

$\displaystyle\zeta=\lambda_{1}/\lambda_{0},\qquad\tilde{\zeta}=\lambda_{0}/\lambda_{1},$
(3.1)

are inhomogeneous coordinates on the fibres over $U_{0}$ and over $U_{1}$
respectively. On the intersection $U_{0}\cap U_{1}$ we can use either of them;
the relation is $\tilde{\zeta}=\zeta^{-1}$.

The tautological bundle $\mathcal{O}(-1)\to\mathcal{F}$ is the line bundle
whose fibre over a point is the set of multiples of the corresponding
projective spinor. It is determined by the transition function $\zeta$ from
$U_{1}$ to $U_{0}$. Similarly, the bundles $\mathcal{O}(k)$, $k\in\mathbb{Z}$,
are determined by the transition function $\zeta^{-k}$. A holomorphic section
of $\mathcal{O}(k)$ is a function that is holomorphic and homogeneous of
degree $k$ in $(\lambda_{0},\lambda_{1})$, that is
$\Psi(x,c\lambda_{0},c\lambda_{1})=c^{k}\Psi(x,\lambda_{0},\lambda_{1})$ for
any non-zero complex number $c$. We are interested in holomorphic sections of
$\mathcal{O}(k)$ over the intersection $U_{0}\cap U_{1}$; the set of such
sections is denoted $\Gamma(U_{0}\cap U_{1},\mathcal{O}(k))$.

In the trivialisation over $U_{0}$, we can write
$\Psi(x,1,\zeta)=(\lambda_{0})^{-k}\Psi(x,\lambda_{0},\lambda_{1})$, which
then gives $\Psi(x,\lambda_{0},\lambda_{1})=(\lambda_{0})^{k}f(x,\zeta)$,
where $f(x,\zeta):=\Psi(x,1,\zeta)$. So $\Psi$ is represented by $f$ over
$U_{0}$, and similarly it is represented by $\tilde{f}=\zeta^{-k}f$ over
$U_{1}$. Noticing that $U_{0}\cap U_{1}$ is an annular region in
$\mathbb{CP}^{1}$, and that the function $f$ is holomorphic there, it follows
that $f$ admits a Laurent series expansion in $\zeta$:

$\displaystyle
f(x,\zeta)=\sum_{j=-\infty}^{\infty}\psi_{j}(x)\zeta^{j},\qquad\psi_{j}(x)=\frac{1}{2\pi{\rm
i}}\oint_{\Gamma}\frac{f(x,\zeta){\rm d}\zeta}{\zeta^{j+1}},$ (3.2)

where the expression for $\psi_{j}$ is the standard formula for the
coefficients of a Laurent series, see e.g. [15], and $\Gamma$ is any contour
in $\mathbb{CP}^{1}$ separating the points $\zeta=0$ and $\zeta=\infty$.
(Since $f$ is holomorphic inside the annulus $U_{0}\cap U_{1}$, the integral
is insensitive to continuous deformations of the contour.) So any
$\Psi\in\Gamma(U_{0}\cap U_{1},\mathcal{O}(k))$ can be expressed (in the
trivialisation over $U_{0}$) as

$\displaystyle\Psi(x,\lambda_{0},\lambda_{1})=(\lambda_{0})^{k}\sum_{j=-\infty}^{\infty}\psi_{j}(x)\frac{(\lambda_{1})^{j}}{(\lambda_{0})^{j}}.$
(3.3)

######  Remark 5.

The coefficients $\psi_{j}$ are weighted scalar fields on $\mathcal{M}$: since
$\Psi$ must have zero weights, using (2.20) and (3.3) we deduce that the
weights of $\psi_{j}$ are

$\displaystyle w(\psi_{j})=k/2,\qquad p(\psi_{j})=2j-k.$ (3.4)

The main result of this subsection is the following:

######  Proposition 6.

Let $(\mathcal{M},g_{ab},J^{a}{}_{b})$ be a Hermitian manifold, let
$\lambda^{A}L_{AA^{\prime}}$ be the twistor distribution (2.21), and let
$\mathscr{C}_{AA^{\prime}}$ be the connection defined in (2.6). Then for any
section $\Psi\in\Gamma(U_{0}\cap U_{1},\mathcal{O}(k))$, with Laurent
expansion (3.3), the following identity holds:

$\displaystyle\lambda^{A}L_{AA^{\prime}}\Psi=(\lambda_{0})^{k}\sum_{j=-\infty}^{\infty}\frac{(\lambda_{1})^{j}}{(\lambda_{0})^{j}}\lambda^{A}\mathscr{C}_{AA^{\prime}}\psi_{j}.$
(3.5)

###### Proof.

We have

$\displaystyle\lambda^{A}L_{AA^{\prime}}\Psi=\sum_{j=-\infty}^{\infty}\left[(\lambda_{0})^{k-j}(\lambda_{1})^{j}\lambda^{A}\partial_{AA^{\prime}}\psi_{j}-\psi_{j}\Gamma_{AA^{\prime}B}{}^{C}\lambda^{A}\lambda^{B}\tfrac{\partial}{\partial\lambda^{C}}\left((\lambda_{0})^{k-j}(\lambda_{1})^{j}\right)\right].$

From (2.19) we deduce
$\frac{\partial\lambda_{0}}{\partial\lambda^{C}}=-o_{C}$,
$\frac{\partial\lambda_{1}}{\partial\lambda^{C}}=-\iota_{C}$, so we find that
$\lambda^{A}L_{AA^{\prime}}\Psi$ is equal to

$\displaystyle\sum_{j=-\infty}^{\infty}(\lambda_{0})^{k-j}(\lambda_{1})^{j}\left[\lambda^{A}\partial_{AA^{\prime}}+\left(\frac{(k-j)}{\lambda_{0}}\Gamma_{AA^{\prime}B}{}^{C}\lambda^{A}\lambda^{B}o_{C}+\frac{j}{\lambda_{1}}\Gamma_{AA^{\prime}B}{}^{C}\lambda^{A}\lambda^{B}\iota_{C}\right)\right]\psi_{j}.$

To compute the connection terms, we express the spin connection in terms of
the objects
$f_{AA^{\prime}},\omega_{AA^{\prime}},\sigma^{0}_{A^{\prime}},\sigma^{1}_{A^{\prime}}$.
Recalling (2.22), a computation using the definitions (2.7), (2.8) gives

$\displaystyle\Gamma_{AA^{\prime}B}{}^{C}o_{C}$
$\displaystyle=o_{B}\omega_{AA^{\prime}}+o_{A}\iota_{B}o^{C}f_{CA^{\prime}}-\iota_{A}\iota_{B}\sigma^{0}_{A^{\prime}},$
(3.6a) $\displaystyle\Gamma_{AA^{\prime}B}{}^{C}\iota_{C}$
$\displaystyle=-\iota_{B}\omega_{AA^{\prime}}-\iota_{A}o_{B}\iota^{C}f_{CA^{\prime}}-o_{A}o_{B}\sigma^{1}_{A^{\prime}}.$
(3.6b)

Therefore:

$\displaystyle\Gamma_{AA^{\prime}B}{}^{C}\lambda^{A}\lambda^{B}o_{C}$
$\displaystyle=-(\lambda_{1})^{2}\sigma^{0}_{A^{\prime}}-\lambda_{0}\lambda_{1}o^{A}(\omega_{AA^{\prime}}-f_{AA^{\prime}})+(\lambda_{0})^{2}\iota^{A}\omega_{AA^{\prime}},$
$\displaystyle\Gamma_{AA^{\prime}B}{}^{C}\lambda^{A}\lambda^{B}\iota_{C}$
$\displaystyle=(\lambda_{1})^{2}o^{A}\omega_{AA^{\prime}}-\lambda_{0}\lambda_{1}\iota^{A}(\omega_{AA^{\prime}}+f_{AA^{\prime}})-(\lambda_{0})^{2}\sigma^{1}_{A^{\prime}}.$

So after a straightforward calculation we arrive at

$\displaystyle\lambda^{A}L_{AA^{\prime}}\Psi=\sum_{j=-\infty}^{\infty}$
$\displaystyle(\lambda_{0})^{k-j}(\lambda_{1})^{j}\left\\{\lambda^{A}\partial_{AA^{\prime}}+\lambda_{1}o^{A}[(2j-k)\omega_{AA^{\prime}}+(k-j)f_{AA^{\prime}}]\right.$
$\displaystyle\left.-\lambda_{0}\iota^{A}[(2j-k)\omega_{AA^{\prime}}+jf_{AA^{\prime}}]+\tfrac{(j-k)(\lambda_{1})^{2}}{\lambda_{0}}\sigma^{0}_{A^{\prime}}-\tfrac{j(\lambda_{0})^{2}}{\lambda_{1}}\sigma^{1}_{A^{\prime}}\right\\}\psi_{j}.$

For an arbitrary scalar field $\psi$ with weights $(w,p)$, using (2.19),
(2.6a) and (2.7) we get

$\displaystyle\lambda^{A}\mathscr{C}_{AA^{\prime}}\psi=\\{\lambda^{A}\partial_{AA^{\prime}}+\lambda_{1}o^{A}[p\omega_{AA^{\prime}}+(w-\tfrac{p}{2})f_{AA^{\prime}}]-\lambda_{0}\iota^{A}[p\omega_{AA^{\prime}}+(w+\tfrac{p}{2})f_{AA^{\prime}}]\\}\psi.$

From remark 2, on a Hermitian manifold we have
$\sigma^{0}_{A^{\prime}}=\sigma^{1}_{A^{\prime}}=0$. Recalling now that
$\psi_{j}$ has weights as in (3.4), the result follows. ∎

###  3.2 Twistor functions and recursion relations

We now assume that $(\mathcal{M},g_{ab})$ is conformally self-dual, eq. (1.3).
From section 2.3, the twistor distribution (2.21) is integrable, and there is
a twistor space $\mathcal{PT}$ (2.23). Similarly to $\mathcal{F}$, this can be
covered by two open sets $V_{0},V_{1}$, which we may take to be the images of
$U_{0},U_{1}$ under the projection $\mathcal{F}\to\mathcal{PT}$. The analogue
of the bundles $\mathcal{O}(k)$ over $\mathcal{F}$ but now over $\mathcal{PT}$
will also be denoted by $\mathcal{O}(k)$ (those over $\mathcal{F}$ are the
pull-back of those over $\mathcal{PT}$). A local section
$\Psi\in\Gamma(V_{0}\cap V_{1},\mathcal{O}(k))$ can be thought of as a local
section555We slightly abuse notation and use the same letter $\Psi$ for both
spaces. $\Psi\in\Gamma(U_{0}\cap U_{1},\mathcal{O}(k))$ which is constant
along the twistor distribution:

$\displaystyle\lambda^{A}L_{AA^{\prime}}\Psi=0.$ (3.7)

We call such functions twistor functions.

######  Proposition 7.

Let $\Psi\in\Gamma(V_{0}\cap V_{1},\mathcal{O}(k))$ be a twistor function.
Then the coefficients $\psi_{j}$ in the associated Laurent series (3.3) solve
the Teukolsky equation for weights $(\frac{k}{2},2j-k)$, eq. (2.18). The
coefficients can be recovered from the twistor function $\Psi$ by a contour
integral:

$\displaystyle\psi_{j}(x)=\frac{1}{2\pi{\rm
i}}\oint_{\Gamma}\frac{\Psi(x,\lambda)\lambda_{A}{\rm
d}\lambda^{A}}{(o_{B}\lambda^{B})^{k-j+1}(\iota_{B}\lambda^{B})^{j+1}}.$ (3.8)

###### Proof.

From proposition 6, we see that if $\Psi$ satisfies (3.7) then the right hand
side of (3.5) must vanish. Using (2.19), it follows that
$o^{A}\mathscr{C}_{AA^{\prime}}\psi_{j}-\iota^{A}\mathscr{C}_{AA^{\prime}}\psi_{j+1}=0$
for all $j$, or equivalently using the operators (2.9):

$\displaystyle\tilde{\mathscr{C}}_{A^{\prime}}\psi_{j}=\mathscr{C}_{A^{\prime}}\psi_{j+1}\qquad\forall\,j.$
(3.9)

Applying $\mathscr{C}^{A^{\prime}}$ to this equation and using (2.10), we see
that $\mathscr{C}^{A^{\prime}}\tilde{\mathscr{C}}_{A^{\prime}}\psi_{j}=0$.
Recalling now (2.17), we get equation (2.18). The integral (3.8) follows from
the expression for the coefficients in (3.2), noticing that
$f(x,\zeta)=(\lambda_{0})^{-k}\Psi(x,\lambda_{0},\lambda_{1})$ and
$\lambda_{A}{\rm d}\lambda^{A}=(\lambda_{0})^{2}{\rm
d}(\frac{\lambda_{1}}{\lambda_{0}})$. ∎

Following the terminology of the twistor literature [13, 6], we call (3.9) the
recursion relations. They are connected to a symmetry operator for the
Teukolsky equations, which we call the recursion operator:

######  Proposition 8.

Let $\mathcal{T}_{(w,p)}$ be the space of solutions to the Teukolsky equation
for weights $(w,p)$. Then there is an operator
$\mathcal{R}:\mathcal{T}_{(w,p)}\to\mathcal{T}_{(w,p+2)}$ defined, for any
$\chi\in\mathcal{T}_{(w,p)}$, by

$\displaystyle\tilde{\mathscr{C}}_{A^{\prime}}\chi=\mathscr{C}_{A^{\prime}}\mathcal{R}\chi.$
(3.10)

###### Proof.

Let $\chi$ be a solution to the Teukolsky equation $\Box^{\mathscr{C}}\chi=0$
for weights $(w,p)$, so $\chi\in\mathcal{T}_{(w,p)}$. Using the first identity
in (2.17), this can be written as
$\mathscr{C}^{A^{\prime}}\tilde{\mathscr{C}}_{A^{\prime}}\chi=0$. From item 3
in remark 2, it follows that there exists a scalar field $\tilde{\chi}$ such
that
$\tilde{\mathscr{C}}_{A^{\prime}}\chi=\mathscr{C}_{A^{\prime}}\tilde{\chi}$.
The weights of $\tilde{\chi}$ are $(w,p+2)$. Applying
$\tilde{\mathscr{C}}^{A^{\prime}}$ and using (2.10), we get
$\tilde{\mathscr{C}}^{A^{\prime}}\mathscr{C}_{A^{\prime}}\tilde{\chi}=0$.
Using the second and first identities in (2.17), we have
$\Box^{\mathscr{C}}\tilde{\chi}=0$, so $\tilde{\chi}\in\mathcal{T}_{(w,p+2)}$.
We then set $\tilde{\chi}\equiv\mathcal{R}\chi$. ∎

With the help of the recursion operator $\mathcal{R}$, we have a converse to
proposition 7:

######  Proposition 9.

Let $\chi$ be a solution to the Teukolsky equation for weights $(w,p)$, eq.
(2.18). Then $\chi$ generates a twistor function $\Psi\in\Gamma(V_{0}\cap
V_{1},\mathcal{O}(2w))$, and we have

$\displaystyle\chi(x)=\frac{1}{2\pi{\rm
i}}\oint_{\Gamma}\frac{\Psi(x,\lambda)\lambda_{A}{\rm
d}\lambda^{A}}{(o_{B}\lambda^{B})^{w+1-\frac{p}{2}}(\iota_{B}\lambda^{B})^{w+1+\frac{p}{2}}}.$
(3.11)

###### Proof.

Given $\chi$, using the recursion operator (3.10) we get a solution
$\chi^{(1)}=\mathcal{R}\chi$ to the Teukolsky equation for weights $(w,p+2)$.
Since $\tilde{\mathscr{C}}^{A^{\prime}}\mathscr{C}_{A^{\prime}}\chi^{(1)}=0$
and the operators commute, we can write
$\mathscr{C}^{A^{\prime}}\tilde{\mathscr{C}}_{A^{\prime}}\chi^{(1)}=0$.
Applying (2.12) again, there exists $\chi^{(2)}$ such that
$\tilde{\mathscr{C}}_{A^{\prime}}\chi^{(1)}=\mathscr{C}_{A^{\prime}}\chi^{(2)}$,
and we have
$\chi^{(2)}=\mathcal{R}(\mathcal{R}\chi)\equiv\mathcal{R}^{2}\chi$. The
weights of $\chi^{(2)}$ are $(w,p+4)$. Iterating this process we get a set
$\\{\mathcal{R}^{m}\chi\\}_{m=0}^{\infty}$, where the weights of
$\mathcal{R}^{m}\chi$ are $(w,p+2m)$. To get negative $m$’s, we start again
from the Teukolsky equation for $\chi$ but now we write it as
$\tilde{\mathscr{C}}^{A^{\prime}}\mathscr{C}_{A^{\prime}}\chi=0$, so there
exists $\chi^{(-1)}$ such that
$\mathscr{C}_{A^{\prime}}\chi=\tilde{\mathscr{C}}_{A^{\prime}}\chi^{(-1)}$.
Comparing to (3.10), we can write $\chi=\mathcal{R}\chi^{(-1)}$, so formally
$\chi^{(-1)}=\mathcal{R}^{-1}\chi$. Repeating the process, we end up with an
infinite family, $\\{\mathcal{R}^{m}\chi\\}_{m=-\infty}^{\infty}$, of
solutions to the Teukolsky equations, connected by the recursion operator
(3.10) or, equivalently, by the recursion relations
$\tilde{\mathscr{C}}_{A^{\prime}}\mathcal{R}^{m}\chi=\mathscr{C}_{A^{\prime}}\mathcal{R}^{m+1}\chi$.
We can then define the power series

$\displaystyle\Psi(x,\lambda_{0},\lambda_{1}):=\sum_{m=-\infty}^{\infty}(\mathcal{R}^{m}\chi)(x)(\lambda_{0})^{w-\frac{p}{2}-m}(\lambda_{1})^{w+\frac{p}{2}+m}.$
(3.12)

The function $\Psi$ has weights $(0,0)$, it is homogeneous of degree $2w$ in
$(\lambda_{0},\lambda_{1})$, and, since the coefficients $\mathcal{R}^{m}\chi$
satisfy the recursion relations, we have $\lambda^{A}L_{AA^{\prime}}\Psi=0$.
Thus $\Psi$ is a twistor function in $\Gamma(V_{0}\cap
V_{1},\mathcal{O}(2w))$. The integral (3.11) is simply (3.8) for $k=2w$,
$j=w+\frac{p}{2}$. ∎

Proposition 9 tells us that any solution to the Teukolsky equation generates a
twistor function, and proposition 8 tells us that, given a solution $\chi$ to
the Teukolsky equation for weights $(w,p)$, the recursion operator produces a
solution $\tilde{\chi}=\mathcal{R}\chi$ to the equation for weights $(w,p+2)$.
It is then natural to ask about the relation between the twistor functions
generated by $\chi$ and by $\mathcal{R}\chi$. We have:

######  Proposition 10.

$\chi$ and $\mathcal{R}\chi$ generate the same twistor function.

###### Proof.

Let $\chi$ be a solution to the Teukolsky equation for weights $(w,p)$. The
generated twistor function, say $\Psi$, is (3.12). Likewise, the twistor
function corresponding to $\tilde{\chi}=\mathcal{R}\chi$, say $\tilde{\Psi}$,
is given by the same expression (3.12) but replacing $\chi$ by $\tilde{\chi}$
and $p$ by $p+2$, since the weights of $\tilde{\chi}$ are $(w,p+2)$. Thus:

$\displaystyle\tilde{\Psi}(x,\lambda)$
$\displaystyle=\sum_{\ell=-\infty}^{\infty}(\mathcal{R}^{\ell}\tilde{\chi})(x)(\lambda_{0})^{w-\frac{(p+2)}{2}-\ell}(\lambda_{1})^{w+\frac{(p+2)}{2}+\ell}$
$\displaystyle=\sum_{\ell=-\infty}^{\infty}(\mathcal{R}^{\ell+1}\chi)(x)(\lambda_{0})^{w-\frac{p}{2}-\ell-1}(\lambda_{1})^{w+\frac{p}{2}+\ell+1}.$

The second line is simply (3.12) with $m=\ell+1$, so $\tilde{\Psi}=\Psi$. ∎

###  3.3 Twistor cohomology

From prop. 9, any solution $\chi$ to the Teukolsky equation for weights
$(w,p)$ can be expressed as the contour integral of a twistor function $\Psi$
as in (3.11). Suppose we make a transformation

$\displaystyle\Psi\to\Psi^{\prime}=\Psi+h_{0}+h_{1},$ (3.13)

where we assume that $h_{i}$ is a function such that its quotient by the
denominator in (3.11) is holomorphic in $V_{i}$, $i=0,1$. Then the result of
the integral is unchanged. So, from the perspective of the contour integral,
$\chi$ would generate not just a single twistor function but a class of them
related by (3.13), as long as we can find the required functions
$h_{0},h_{1}$.

To investigate whether such functions can be found, we will use sheaf
cohomology, as is usual in twistor constructions. We will follow the Čech
approach, see e.g. [16, 13, 17, 18]. The first Čech cohomology group of
$\mathcal{PT}$ with coefficients in $\mathcal{O}(k)$, denoted
$\check{H}^{1}(\mathcal{PT},\mathcal{O}(k))$, is the set of 1-cocycles modulo
1-coboundaries. In our case we only have two sets in the cover, so the cocycle
condition is trivial, i.e. every cochain is a cocycle. Following [13] we can
then write666Recall that $\Gamma(\mathcal{U},E)$ is the set of holomorphic
sections of a bundle $E$ over an open set $\mathcal{U}$.

$\displaystyle\check{H}^{1}(\mathcal{PT},\mathcal{O}(k))=\frac{\Gamma(V_{0}\cap
V_{1},\mathcal{O}(k))}{\Gamma(V_{0},\mathcal{O}(k))+\Gamma(V_{1},\mathcal{O}(k))}.$
(3.14)

A cohomology class in $\check{H}^{1}(\mathcal{PT},\mathcal{O}(k))$ is thus
represented by a cochain, which we recall is a local holomorphic section of
$\mathcal{O}(k)$ over $V_{0}\cap V_{1}$. So we can say that any such section,
say $F$, is a Čech representative in
$\check{H}^{1}(\mathcal{PT},\mathcal{O}(k))$. The quotient in (3.14) means
that $F$ and $F^{\prime}\equiv F+h_{0}+h_{1}$, where $h_{0}$ is holomorphic in
$V_{0}$ and $h_{1}$ is holomorphic in $V_{1}$, represent the same cohomology
class. Elements of the same class are said to be cohomologous.

With the above terminology, we can replace the term ‘twistor function’ in
propositions 7 and 9 by ‘Čech representative’. We can also investigate the
conditions under which $\chi$ generates not just a Čech representative but a
cohomology class: that is, given $\chi$, we say that it “generates a
cohomology class” if it is invariant under (3.13).

######  Proposition 11.

Let $\chi$ be a solution to the Teukolsky equation for weights $(w,p)$. Then
$\chi$ generates a cohomology class in
$\check{H}^{1}(\mathcal{PT},\mathcal{O}(2w))$ if and only if $w$ and $p$
satisfy

$\displaystyle w\pm\frac{p}{2}\in-\mathbb{N}.$ (3.15)

###### Proof.

Let $h_{i}\in\Gamma(V_{i},\mathcal{O}(2w))$, with $i=0,1$. Put
$\Psi^{\prime}=\Psi+h_{0}+h_{1}$, then $\Psi^{\prime}$ and $\Psi$ are
cohomologous. Letting $\chi^{\prime}$ be the space-time field generated by
inserting $\Psi^{\prime}$ in the integrand in the right hand side of (3.11),
we wish to know whether $\chi^{\prime}$ and $\chi$ are equal for any choice of
$h_{0},h_{1}$. Write $h_{0},h_{1}$ in terms of their Taylor series:

$\displaystyle
h_{0}(x,\lambda)=(\lambda_{0})^{2w}\sum_{j=0}^{\infty}a_{j}(x)\frac{(\lambda_{1})^{j}}{(\lambda_{0})^{j}},\qquad
h_{1}(x,\lambda)=(\lambda_{1})^{2w}\sum_{j=0}^{\infty}b_{j}(x)\frac{(\lambda_{0})^{j}}{(\lambda_{1})^{j}}.$
(3.16)

Replacing in the contour integral (3.11) for $\Psi^{\prime}$, we find

$\displaystyle\chi^{\prime}$ $\displaystyle=\chi+\frac{1}{2\pi{\rm
i}}\oint_{\Gamma}\frac{h_{0}(x,\lambda)\lambda_{A}{\rm
d}\lambda^{A}}{(\lambda_{0})^{w+1-\frac{p}{2}}(\lambda_{1})^{w+1+\frac{p}{2}}}+\frac{1}{2\pi{\rm
i}}\oint_{\Gamma}\frac{h_{1}(x,\lambda)\lambda_{A}{\rm
d}\lambda^{A}}{(\lambda_{0})^{w+1-\frac{p}{2}}(\lambda_{1})^{w+1+\frac{p}{2}}}$
$\displaystyle=\chi+\sum_{j=0}^{\infty}\frac{a_{j}}{2\pi{\rm
i}}\oint_{\Gamma}\zeta^{j-w-1-\frac{p}{2}}{\rm
d}\zeta-\sum_{j=0}^{\infty}\frac{b_{j}}{2\pi{\rm
i}}\oint_{\Gamma}\tilde{\zeta}^{j-w-1+\frac{p}{2}}{\rm d}\tilde{\zeta}$ (3.17)

where in the second line we replaced (3.16) together with $\lambda_{A}{\rm
d}\lambda^{A}=(\lambda_{0})^{2}{\rm d}\zeta=-(\lambda_{1})^{2}{\rm
d}\tilde{\zeta}$ (recall (3.1)). The functions $h_{0},h_{1}$ are arbitrary, so
we see that $\chi^{\prime}$ and $\chi$ will be equal iff the terms with
$a_{j}$ and $b_{j}$ in (3.17) vanish independently. The terms with $a_{j}$
vanish iff $j-w-1-\frac{p}{2}$ is a non-negative integer for all $j$, and the
terms with $b_{j}$ vanish iff $j-w-1+\frac{p}{2}$ is a non-negative integer
for all $j$. Since from (3.16) we have $j\geq 0$, this is equivalent to

$\displaystyle w+1\pm\frac{p}{2}\in-\mathbb{N}\cup 0,$

which can be rewritten as (3.15). ∎

We now wish to give an interpretation of the restrictions (3.15). Notice first
that if $w\geq-\frac{1}{2}$, then there is no $p$ such that (3.15) can be
satisfied. So we must have $w\leq-1$. In this case, suppose that
$\chi_{(w,p)}$ is a solution to the Teukolsky equation for weights $(w,p)$,
where $w\pm\frac{p}{2}\in-\mathbb{N}$. Applying the recursion operator
$\mathcal{R}$ and its inverse, we generate other solutions
$\chi_{(w,p_{j})}=\mathcal{R}^{j}\chi_{(w,p)}$, with $p_{j}=p+2j$,
$j\in\mathbb{Z}$. A certain number of the new solutions will still satisfy the
constraint $w\pm\frac{p_{j}}{2}\in-\mathbb{N}$. To find this number, notice
that the conditions $w\pm\frac{p_{j}}{2}\in-\mathbb{N}$ are equivalent to
$p_{j}=-2w-2n$ and $p_{j}=2w+2\tilde{n}$ for $n,\tilde{n}\in\mathbb{N}$, so
the possible values of $p_{j}$ are $2w+2,2w+4,2w+6$, $...$,
$-2w-6,-2w-4,-2w-2$. The number of different $p$’s is then $-2w-1$. So we have
$-2w-1$ fields $\\{\chi_{(w,p_{j})}\\}_{j=0}^{-2w-2}$, where each
$\chi_{(w,p_{j})}$ solves the Teukolsky equation for weights $(w,p_{j})$, and
since $(w,p_{j})$ satisfy (3.15), each $\chi_{(w,p_{j})}$ generates a
cohomology class in $\check{H}^{1}(\mathcal{PT},\mathcal{O}(2w))$. But since
these fields are related by the recursion operator, proposition 10 implies
that they generate the same twistor function $\Psi$, and so the same
cohomology class $[\Psi]\in\check{H}^{1}(\mathcal{PT},\mathcal{O}(2w))$. In
summary, the restrictions (3.15) mean that a single class in
$\check{H}^{1}(\mathcal{PT},\mathcal{O}(2w))$ is associated to $N$ solutions
to the Teukolsky equations, where $N=0$ if $w>-1$ and $N=-2w-1$ if $w\leq-1$.
But now we see that this is simply the known structure of the cohomology group
$\check{H}^{1}(\mathcal{PT},\mathcal{O}(2w))$: it is well-known (see e.g. [16,
13]) that, on restriction to a twistor line $\mathbb{CP}^{1}$, one has

$\displaystyle\check{H}^{1}(\mathbb{CP}^{1},\mathcal{O}(2w))=\begin{cases}\mathbb{C}^{-2w-1}\quad&\text{if
}\quad w\leq-1\\\ 0\quad&\text{if }\quad w>-1\end{cases}.$ (3.18)

###  3.4 The conformally Kähler case

Suppose that the geometry satisfies (1.3) and is also conformal to a Kähler
metric (which is then a scalar-flat Kähler surface). In our framework this
means that the Lee form is locally exact, $f_{a}=\partial_{a}\log\phi$ for
some $\phi$. The interesting point now is that, by using the recursion
operator $\mathcal{R}$, we can map the Teukolsky equation with any weights
$(w,p)$ to the conformal Laplacian, as long as $p$ is even.

######  Proposition 12.

Let $(\mathcal{M},g_{ab})$ be conformal to a scalar-flat Kähler surface. Let
$\chi_{(w,p)}$ be an arbitrary weighted scalar field with weights $(w,p)$,
where $p$ is even. Then we have the identity

$\displaystyle\Box^{\mathscr{C}}\mathcal{R}^{-\frac{p}{2}}\chi_{(w,p)}=\phi^{-(w+1)}\left(\Box+\tfrac{R}{6}\right)\left[\phi^{w+1}\mathcal{R}^{-\frac{p}{2}}\chi_{(w,p)}\right].$
(3.19)

###### Proof.

Recall that if $\chi_{(w,p)}$ has weights $(w,p)$, then
$\mathcal{R}^{m}\chi_{(w,p)}$ has weights $(w,p+2m)$, with $m\in\mathbb{Z}$.
If $p$ is even, it follows that $\mathcal{R}^{-\frac{p}{2}}\chi_{(w,p)}$ has
weights $(w,0)$. Then the identity (2.16) for the Teukolsky operator
$\Box^{\mathscr{C}}$ acting on scalar fields with weights $(w,0)$ applies to
$\mathcal{R}^{-\frac{p}{2}}\chi_{(w,p)}$. But since $\Psi_{2}\equiv 0$, we see
that the operator simply reduces to the conformal Laplacian composed with
conjugation by $\phi^{w+1}$. ∎

If $\chi_{(w,p)}$ is now a solution to the Teukolsky equation for weights
$(w,p)$, then $\mathcal{R}^{-\frac{p}{2}}\chi_{(w,p)}$ is a solution to the
corresponding equation for weights $(w,0)$, since $\mathcal{R}$ maps solutions
to solutions. Then the scalar field
$\phi^{w+1}\mathcal{R}^{-\frac{p}{2}}\chi_{(w,p)}$ has weights $(-1,0)$ and,
in view of (3.19), is a solution to the conformal wave equation. Conversely,
suppose that $\Phi$ has weights $(-1,0)$ and solves the conformal wave
equation $(\Box+\frac{R}{6})\Phi=0$. For $p$ even, the field
$\chi_{(w,p)}:=\phi^{-(w+1)}\mathcal{R}^{\frac{p}{2}}\Phi$ has weights $(w,p)$
and, again as a consequence of (3.19), solves
$\Box^{\mathscr{C}}\chi_{(w,p)}=0$, which is the Teukolsky equation for
weights $(w,p)$.

##  4 Quaternionic-Kähler Metrics

Here we specialise to the case in which the geometry $(\mathcal{M},g_{ab})$
satisfies not only (1.3) but is also Einstein, with non-zero cosmological
constant $\lambda$. In four dimensions, these manifolds are called
quaternionic-Kähler (or self-dual Einstein)777Despite the name, these
manifolds are generically not Kähler.. The curvature satisfies

$\displaystyle\Psi_{ABCD}=0,\qquad R_{ab}=\lambda g_{ab},\qquad\lambda\neq 0.$
(4.1)

###  4.1 Gravitational perturbations

We will focus on metric deformations, that is, we perturb the metric
$g_{ab}\to g_{ab}+\varepsilon h_{ab}$, and we wish to study the linearised
Einstein equations $\mathcal{E}_{ab}[h]=0$, where

$\displaystyle\mathcal{E}_{ab}[h]:=-\tfrac{1}{2}\Box
h_{ab}-\tfrac{1}{2}\nabla_{a}\nabla_{b}{\rm
tr}(h)+\nabla^{c}\nabla_{(a}h_{b)c}-\tfrac{1}{2}g_{ab}[\nabla^{c}\nabla^{d}h_{cd}-\Box{\rm
tr}(h)]+\lambda h_{ab}=0$ (4.2)

and $\Box=g^{ab}\nabla_{a}\nabla_{b}$, ${\rm tr}(h)=g^{ab}h_{ab}$.

######  Proposition 13.

Let $(\mathcal{M},g_{ab})$ be quaternionic-Kähler (4.1). Fix an integrable
complex structure, with principal spinors $o_{A},\iota_{A}$, Lee form $f_{a}$,
and connection $\mathscr{C}_{a}$ as in section 2. Then for an arbitrary
symmetric tensor field $h_{ab}$, the following identities hold:

$\displaystyle-\mathring{\Omega}^{-1}W^{abcd}_{\bf
i}(\nabla_{a}-4f_{a})\nabla_{d}\mathcal{E}_{bc}[h]=\tfrac{1}{4}\Box^{\mathscr{C}}\left(\mathring{\Omega}^{-1}W^{abcd}_{\bf
i}\dot{C}_{abcd}[h]\right),\qquad{\bf i}=0,...,4$ (4.3)

where $\mathring{\Omega}$ is an auxiliary constant conformal factor,
$\mathcal{E}_{ab}[h]$ is the linearised Einstein operator (4.2),
$\dot{C}_{abcd}[h]$ is the linearised Weyl tensor thought of as a linear
functional of $h_{ab}$, and

$\displaystyle W_{\bf i}^{abcd}:=S_{\bf
i}^{ABCD}\epsilon^{A^{\prime}B^{\prime}}\epsilon^{C^{\prime}D^{\prime}},$
(4.4)

with $S_{0}^{ABCD}=o^{(A}o^{B}o^{C}o^{D)}$,
$S_{1}^{ABCD}=o^{(A}o^{B}o^{C}\iota^{D)}$, …,
$S_{4}^{ABCD}=\iota^{(A}\iota^{B}\iota^{C}\iota^{D)}$.

###### Proof.

Suppose first that $(\mathcal{M},g_{ab},J^{a}{}_{b})$ is an arbitrary almost-
Hermitian manifold, with ASD Weyl spinor $\Psi_{ABCD}$. Recalling that
$\Psi_{ABCD}$ has conformal weight zero, let $\mathring{\Omega}$ be an
(auxiliary) constant conformal factor, $\nabla_{a}\mathring{\Omega}=0$, and
define

$\displaystyle\varphi_{ABCD}:=\mathring{\Omega}^{-1}\Psi_{ABCD},$ (4.5)

which has conformal weight $w=-1$ (and $p=0$). Using (2.6b), we have:

$\displaystyle S_{\bf
i}^{ABCD}\mathscr{C}_{AA^{\prime}}\mathscr{C}^{A^{\prime}E}\varphi_{BCDE}$
$\displaystyle=\mathring{\Omega}^{-1}S_{\bf
i}^{ABCD}(\nabla_{AA^{\prime}}-4f_{AA^{\prime}})\nabla^{A^{\prime}E}\Psi_{BCDE}$
$\displaystyle=\mathring{\Omega}^{-1}S_{\bf
i}^{ABCD}(\nabla_{AA^{\prime}}-4f_{AA^{\prime}})\nabla_{B}^{B^{\prime}}\Phi_{CDB^{\prime}}{}^{A^{\prime}}$
$\displaystyle=\mathring{\Omega}^{-1}S_{\bf
i}^{ACBD}\epsilon^{A^{\prime}C^{\prime}}\epsilon^{B^{\prime}D^{\prime}}(\nabla_{AA^{\prime}}-4f_{AA^{\prime}})\nabla_{DD^{\prime}}\Phi_{BCB^{\prime}C^{\prime}}$
$\displaystyle=-\tfrac{1}{2}\mathring{\Omega}^{-1}W^{abcd}_{\bf
i}(\nabla_{a}-4f_{a})\nabla_{d}(G_{bc}+\tfrac{R}{4}g_{bc}),$

where in the second line we used Bianchi identities [2, Eq. (4.10.7)], and in
the last line we used the definition (4.4) and the relation between the Ricci
spinor $\Phi_{ABA^{\prime}B^{\prime}}$ and the Einstein tensor $G_{ab}$ [2,
Eq. (4.6.25)]. We now linearise the above equation around a space that
satisfies (4.1). The ASD linearised Weyl tensor has a spinor decomposition
$\dot{C}^{-}_{abcd}\equiv\dot{\Psi}_{ABCD}\epsilon_{A^{\prime}B^{\prime}}\epsilon_{C^{\prime}D^{\prime}}$,
and correspondingly we have a field $\dot{\varphi}_{ABCD}$ as in (4.5). The
fields $S_{\bf i}^{ABCD}$ and the operator $\mathscr{C}_{AA^{\prime}}$ can now
be taken to be the ones of the background, so we have

$\displaystyle S_{\bf
i}^{ABCD}\mathscr{C}_{AA^{\prime}}\mathscr{C}^{A^{\prime}E}\dot{\varphi}_{BCDE}=\tfrac{1}{2}\Box^{\mathscr{C}}(S_{\bf
i}^{ABCD}\dot{\varphi}_{ABCD}),$

thus (4.3) follows. ∎

We see that the role of the auxiliary constant conformal factor
$\mathring{\Omega}$ is simply to allow us to apply the operator
$\mathscr{C}_{AA^{\prime}}$ to fields with the appropriate conformal weights;
once the final formulae have been obtained, $\mathring{\Omega}$ can simply be
set to one.

The point of the identity (4.3) is the following. Suppose first that $h_{ab}$
is a solution to the linearised Einstein equations, $\mathcal{E}_{ab}[h]=0$.
Define the weighted scalar fields

$\displaystyle\chi_{\bf i}:=\tfrac{1}{4}\mathring{\Omega}^{-1}W^{abcd}_{\bf
i}\dot{C}_{abcd}[h],$ (4.6)

whose weights are $w=-3$, $p=4-2{\bf i}$. Then (4.3) tells us that each
$\chi_{\bf i}$, ${\bf i}=0,...,4$, is a solution to the Teukolsky equation for
weights $(-3,4-2{\bf i})$. From proposition 11, since the weights satisfy
(3.15), each $\chi_{\bf i}$ then generates a cohomology class in
$\check{H}^{1}(\mathcal{PT},\mathcal{O}(-6))$. In addition, the linearised
Bianchi identity $\nabla^{AA^{\prime}}\dot{\Psi}_{ABCD}=0$ implies
$\mathscr{C}^{AA^{\prime}}\dot{\varphi}_{ABCD}=0$, which gives
$\tilde{\mathscr{C}}_{A^{\prime}}(\iota^{A}\dot{\varphi}_{ABCD})=\mathscr{C}_{A^{\prime}}(o^{A}\dot{\varphi}_{ABCD})$.
Contracting with $o^{A}$’s and $\iota^{A}$’s, we see that the $\chi_{\bf i}$’s
satisfy the recursion relations, therefore, from prop. 10, they generate the
same class. In summary, a solution $h_{ab}$ to the linearised Einstein
equations generates a cohomology class in
$\check{H}^{1}(\mathcal{PT},\mathcal{O}(-6))$ via the perturbed curvature.

In the opposite direction, from prop. 7 we can use a representative in
$\check{H}^{1}(\mathcal{PT},\mathcal{O}(-6))$ to generate solutions to the
Teukolsky equations; however, how do we reconstruct a perturbed metric from
them? The identity (4.3) gives a way to do this, but it also suggests that it
is actually the group $\check{H}^{1}(\mathcal{PT},\mathcal{O}(+2))$ which is
naturally involved here:

######  Proposition 14.

Let $\Phi^{\bf i}$ be a solution to the Teukolsky equation with weights $w=1$,
$p=2{\bf i}-4$, for any ${\bf i}$ in $\\{0,...,4\\}$. Then the tensor field

$\displaystyle h_{ab}^{\bf i}:=-\nabla^{d}[(\nabla^{c}+4f^{c})W^{\bf
i}_{c(ab)d}\Phi^{\bf i}]$ (4.7)

(no sum over $\bf i$) is a solution to the linearised Einstein equations,
$\mathcal{E}_{ab}[h^{\bf i}]=0$.

###### Proof.

We use Wald’s adjoint operators method [19] and write (4.3) as the operator
identity $\mathcal{S}_{\bf i}\mathcal{E}[h_{ab}]=\mathcal{O}\mathcal{T}_{\bf
i}[h_{ab}]$ valid for any symmetric $h_{ab}$, where: $\mathcal{E}$ is defined
in (4.2), $\mathcal{O}(\chi):=\Box^{\mathscr{C}}\chi$, $\mathcal{T}_{\bf
i}[h_{ab}]:=\chi_{\bf i}$ (with $\chi_{\bf i}$ as in (4.6)), and
$\mathcal{S}_{\bf i}$ is the operator that is acting on $\mathcal{E}_{bc}[h]$
in the left hand side of (4.3). The adjoint identity is
$\mathcal{E}\mathcal{S}^{\dagger}_{\bf i}[\Phi]=\mathcal{T}^{\dagger}_{\bf
i}\mathcal{O}^{\dagger}[\Phi]$, valid for any scalar field $\Phi$ with the
appropriate weights, where we used that $\mathcal{E}$ is self-adjoint [19]. A
straightforward calculation shows that $\mathcal{S}^{\dagger}_{{\bf
i}\,ab}\equiv h^{\bf i}_{ab}$, with $h^{\bf i}_{ab}$ given by (4.7). It
follows that if $\mathcal{O}^{\dagger}[\Phi^{\bf i}]=0$, then (4.7) is a
solution to the linearised Einstein equations. So it remains to understand the
operator $\mathcal{O}^{\dagger}$. To this end, we recall that the adjoint of
$\mathcal{O}$ is the unique operator $\mathcal{O}^{\dagger}$ defined by

$\displaystyle\Phi^{\bf i}\mathcal{O}[\chi_{\bf i}]=\chi_{\bf
i}\mathcal{O}^{\dagger}[\Phi^{\bf i}]+\nabla_{a}v^{a}_{\bf i}\qquad\text{(no
sum over $\bf i$)}$

for some vector field $v^{a}_{\bf i}$, cf. [19, Eq. (9)]. Since $\chi_{\bf i}$
has weights $(-3,4-2{\bf i})$, we have that $\mathcal{O}[\chi_{\bf
i}]=g^{ab}\mathscr{C}_{a}\mathscr{C}_{b}\chi_{\bf i}$ has weights $(-5,4-2{\bf
i})$. Multiplying by $\Phi^{\bf i}$, we get $\Phi^{\bf i}\mathcal{O}[\chi_{\bf
i}]=\mathscr{C}_{a}v^{a}_{\bf i}+(\Box^{\mathscr{C}}\Phi^{\bf i})\chi_{\bf
i}$, where $v^{a}_{\bf i}=\Phi^{\bf i}\mathscr{C}^{a}\chi_{\bf i}-\chi_{\bf
i}\mathscr{C}^{a}\Phi^{\bf i}$. Using (2.6c), it is straightforward to show
that in order to have $\mathscr{C}_{a}v^{a}_{\bf i}=\nabla_{a}v^{a}_{\bf i}$,
the weights of $\Phi^{\bf i}$ must be $w=+1$, $p=2{\bf i}-4$. So we see that
for $\mathcal{O}=\Box^{\mathscr{C}}$ as defined by (4.3) (that is, acting on
scalars with weights $(-3,4-2{\bf i})$), the adjoint is
$\mathcal{O}^{\dagger}=\Box^{\mathscr{C}}$ but acting on scalars with weights
$(1,2{\bf i}-4)$. ∎

Since in the above proposition $\Phi^{\bf i}$ is a solution to the Teukolsky
equation for weights $(1,2{\bf i}-4)$, applying prop. 9 we see that $\Phi^{\bf
i}$ is generated by a representative, say $\Psi^{\bf i}$, of a cohomology
class in $\check{H}^{1}(\mathcal{PT},\mathcal{O}(2))$:

$\displaystyle\Phi^{\bf i}(x)=\frac{1}{2\pi{\rm
i}}\oint_{\Gamma}\frac{\Psi^{\bf i}(x,\lambda)\lambda_{A}{\rm
d}\lambda^{A}}{(o_{B}\lambda^{B})^{4-{\bf i}}(\iota_{B}\lambda^{B})^{\bf i}}.$
(4.8)

Now, notice that, on restriction to any twistor line $\mathbb{CP}^{1}$, the
group $\check{H}^{1}(\mathbb{CP}^{1},\mathcal{O}(2))$ is trivial (recall
(3.18)). This turns out to be manifested in the second item of the following
proposition:

######  Proposition 15.

Let $(\mathcal{M},g_{ab})$ be quaternionic-Kähler (4.1).

  1. 1.

Let $\Pi_{A^{\prime}ABC}=\Pi_{A^{\prime}(ABC)}$ be a spinor field satisfying

$\displaystyle\nabla^{A^{\prime}}_{(A}\Pi_{BCD)A^{\prime}}=0,$ (4.9)

and let
$\gamma_{CDA^{\prime}B^{\prime}}:=\nabla_{(A^{\prime}}^{A}\Pi_{B^{\prime})ACD}$.
Then the following identity holds:

$\displaystyle\nabla_{(A}^{A^{\prime}}\nabla_{B}^{B^{\prime}}\gamma_{CD)A^{\prime}B^{\prime}}=0.$
(4.10)

  2. 2.

The linearised ASD Weyl spinor of the metric perturbation (4.7) identically
vanishes.

###### Proof.

For the first item, the calculation is tedious but straightforward: one has to
use repeatedly the spinor form of the commutator
$[\nabla_{AA^{\prime}},\nabla_{BB^{\prime}}]$ and related identities as in [2,
Section 4.9], together with the assumptions (4.1) and (4.9).

For the second item, we use (4.4) to write (4.7) in spinor form:

$\displaystyle h^{\bf i}_{CDA^{\prime}B^{\prime}}$
$\displaystyle=\nabla_{(A^{\prime}}^{A}\Pi^{\bf
i}_{B^{\prime})ACD},\qquad\text{where}\qquad\Pi^{\bf
i}_{B^{\prime}ACD}:=(\nabla_{B^{\prime}}^{B}+4f_{B^{\prime}}^{B})S^{\bf
i}_{ABCD}\Phi^{(i)}.$

Using that the weights of $S^{\bf i}_{ABCD}\Phi^{\bf i}$ are $w=3$, $p=0$, a
short calculation gives $\Pi^{\bf
i}_{B^{\prime}ACD}=\mathscr{C}_{B^{\prime}}^{B}[S^{\bf i}_{ABCD}\Phi^{\bf
i}]$. We then get

$\displaystyle\nabla_{(A}^{A^{\prime}}\Pi^{\bf
i}_{BCD)A^{\prime}}=\mathscr{C}_{(A}^{A^{\prime}}\Pi^{\bf
i}_{BCD)A^{\prime}}=S^{\bf
i}_{(ABC}{}^{E}\mathscr{C}_{D)A^{\prime}}\mathscr{C}^{A^{\prime}}_{E}\Phi^{\bf
i}=-\tfrac{1}{2}S^{\bf i}_{ABCD}\Box^{\mathscr{C}}\Phi^{\bf i},$

where in the first equality we used that $\Pi^{\bf i}_{B^{\prime}ACD}$ has
weights $w=2$, $p=0$, and in the last equality we used (2.17). Since
$\Phi^{\bf i}$ is a solution to $\Box^{\mathscr{C}}\Phi^{\bf i}=0$, then
$\Pi^{\bf i}_{B^{\prime}BCD}$ satisfies (4.9). Setting
$\gamma_{CDA^{\prime}B^{\prime}}\equiv h^{\bf i}_{CDA^{\prime}B^{\prime}}$, we
see that eq. (4.10) is satisfied for $h^{\bf i}_{ab}$. But the left hand side
of (4.10) is precisely the formula for the linearised ASD Weyl spinor of a
metric perturbation $\gamma_{ab}$, see [2, Eq. (5.7.15)], thus the result
follows. ∎

###  4.2 Quaternionic deformations

Since the metric perturbations (4.7) have vanishing linearised Ricci and ASD
Weyl, they preserve the conditions (4.1) to linear order. We then say that the
perturbations are quaternionic deformations. This fact allows us to connect
the Teukolsky equations and the approach in the current work with the
treatments of this kind of deformations in [7] and [8]. The latter are based
on Przanowski’s characterisation of any geometry satisfying (4.1): in [20],
Przanowski showed that any solution to (4.1) admits local complex coordinates
$z^{\alpha}=(z^{0},z^{1})$ and a real scalar field
$\mathcal{K}=\mathcal{K}(z^{\alpha},\bar{z}^{\alpha})$ such that the metric
can be written as

$\displaystyle{\rm
d}{s}^{2}=\frac{6}{\lambda}\mathcal{K}_{\alpha\bar{\beta}}{\rm
d}{z}^{\alpha}{\rm d}\bar{z}^{\beta}+\frac{12}{\lambda}e^{\mathcal{K}}{\rm
d}{z}^{1}{\rm d}\bar{z}^{1}$ (4.11)

where
$\mathcal{K}_{\alpha\bar{\beta}}=\partial_{\alpha}\partial_{\bar{\beta}}\mathcal{K}$.
The self-dual Einstein condition (4.1) is satisfied iff $\mathcal{K}$ solves
Przanowski’s equation ${\rm Prz}(\mathcal{K})=0$, where

$\displaystyle{\rm
Prz}(\mathcal{K}):=\mathcal{K}_{0\bar{0}}\mathcal{K}_{1\bar{1}}-\mathcal{K}_{0\bar{1}}\mathcal{K}_{1\bar{0}}+(2\mathcal{K}_{0\bar{0}}-\mathcal{K}_{0}\mathcal{K}_{\bar{0}})e^{\mathcal{K}}.$
(4.12)

A class of quaternionic deformations is obtained by simply perturbing
Przanowski’s potential,
$\mathcal{K}\to\mathcal{K}+\varepsilon\delta\mathcal{K}$, where
$\delta\mathcal{K}$ is required to satisfy ${\rm dPrz}(\delta\mathcal{K})=0$,
with ${\rm dPrz}$ the linearised Przanowski operator:

$\displaystyle{\rm dPrz}(\delta\mathcal{K})={}$
$\displaystyle(\mathcal{K}_{1\bar{1}}+2e^{\mathcal{K}})\delta\mathcal{K}_{0\bar{0}}+\mathcal{K}_{0\bar{0}}\delta\mathcal{K}_{1\bar{1}}-\mathcal{K}_{1\bar{0}}\delta\mathcal{K}_{0\bar{1}}-\mathcal{K}_{0\bar{1}}\delta\mathcal{K}_{1\bar{0}}$
(4.13)
$\displaystyle+e^{\mathcal{K}}[(2\mathcal{K}_{0\bar{0}}-\mathcal{K}_{0}\mathcal{K}_{\bar{0}})\delta\mathcal{K}-\mathcal{K}_{0}\delta\mathcal{K}_{\bar{0}}-\mathcal{K}_{\bar{0}}\delta\mathcal{K}_{0}]$

where again
$\delta\mathcal{K}_{\alpha\bar{\beta}}=\partial_{\alpha}\partial_{\bar{\beta}}\delta\mathcal{K}$.
In [7], Högner constructed a conformally invariant differential operator on
quaternionic-Kähler 4-manifolds, and showed that, when acting on scalar fields
with conformal weight $w=1$, it is proportional to ${\rm dPrz}$. It turns out
that Högner’s operator is essentially the quaternionic-Kähler specialisation
of the Teukolsky operator of section 2.2. Using (2.6a) and (2.6c), one can
indeed check that on scalars with weights $(w,0)$,

$\displaystyle\Box^{\mathscr{C}}=\Box+(w+1)(2f^{a}\nabla_{a}+wf^{a}f_{a})-w\tfrac{R}{6}.$

In terms of the quantities in (4.11), the Lee form is $f_{a}{\rm
d}{x}^{a}=-(\partial\log\mathcal{K}_{\bar{0}}+\bar{\partial}\log\mathcal{K}_{0})$.
Then on any scalar with weights $w=1$, $p=0$, after a computation (and using
$R=4\lambda$) we find

$\displaystyle\Box^{\mathscr{C}}=\frac{6}{\lambda\det
g_{\alpha\bar{\beta}}}{\rm dPrz}$ (4.14)

in agreement with the pioneering work [7]. Furthermore, Högner also gave a
contour integral formula for perturbations of Przanowski’s potential, see [7,
Eq. (5)] (cf. also [8]). This coincides with the case ${\bf i}=2$ in formula
(4.8) (which corresponds to $w=1$, $p=0$).

While perturbations of Przanowski’s potential give one class of quaternionic
deformations, the results of section 4.1 show that there are other such
deformations described by the tensors (4.7) with ${\bf i}\neq 2$. In these
cases, the scalar equation involved is the Teukolsky equation with $p\neq 0$
(instead of (4.13)), and (4.8) gives a contour integral formula for its
solutions. Note, however, that in Euclidean signature the tensor fields (4.7)
with ${\bf i}\neq 2$ are not real.

###  4.3 The case with an isometry

Finally, we comment very briefly on the case in which $(\mathcal{M},g_{ab})$
satisfies (4.1) and has also a Killing vector. In this case, Tod showed in
[21] that the geometry must be (locally) conformally Kähler, so the
construction of section 3.4 applies. In the notation of that section, and
using the expression for the Lee form given before, we deduce
$\phi^{-1}=\mathcal{K}_{0}=\mathcal{K}_{\bar{0}}$. In particular, using (3.19)
and (4.14), for $w=1$ we recover [8, Eq. (6.5)]. More generally, we can map
back and forth between the conformal wave equation and the Teukolsky equation
for any weights $(w,p)$ with $p$ even, as described in section 3.4.

##  5 Final Remarks

We constructed a correspondence between twistor functions and solutions to
Teukolsky-like equations in conformally self-dual spaces, and we gave a
contour integral formula for the latter in terms of essentially free
holomorphic twistor data. Apart from the conformal self-duality assumption
(1.3), the geometries we considered are generic (except in section 4 where we
specialised to the self-dual Einstein case). To obtain more explicit
coordinate descriptions of the space-time solutions, one must specialise to a
specific geometry with its twistor space and use the corresponding incidence
relations that express twistor coordinates in terms of space-time coordinates.
For instance, in [7, Section 5] one can find examples of twistor coordinates
for a number of geometries.

Following the twistor literature [13, 6], there are other structures of
interest that one can explore using the results in this work, such as the
existence of hierarchies of commuting flows, Hamiltonian formulations, hidden
symmetry algebras, the construction of special coordinates for twistor space
$\mathcal{PT}$, the linear deformations of $\mathcal{PT}$ induced by
gravitational perturbations, etc. A point of major interest for us would be to
analyse whether the construction in this work can be used to study
perturbations of non-self-dual black holes: in this case the background space
also has an integrable complex structure and the connection $\mathscr{C}_{a}$
and many of its properties can be used (cf. [22]), but the twistor
distribution has trivial kernel so ordinary twistor functions do not exist. We
leave these questions for future work.

#### Acknowledgements.

The author acknowledges support of the Institut Henri Poincaré (UAR 839 CNRS-
Sorbonne Université) and LabEx CARMIN (ANR-10-LABX-59-01) in Paris, during a
research stay in the spring 2024 in which this work was started.

## References

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  * [2] R. Penrose and W. Rindler,  Spinors and space-time: Volume 1, Two-spinor calculus and relativistic fields, volume 1.  Cambridge University Press, 1984.
  * [3] E. D. Fackerell and J. R. Ipser, Weak electromagnetic fields around a rotating black hole, Phys. Rev. D 5, 2455 (1972).
  * [4] T. Regge and J. A. Wheeler, Stability of a Schwarzschild singularity, Phys. Rev. 108 (1957), 1063-1069
  * [5] F. J. Zerilli, Effective potential for even parity Regge-Wheeler gravitational perturbation equations, Phys. Rev. Lett. 24 (1970), 737-738
  * [6] M. Dunajski and L. J. Mason, HyperKahler hierarchies and their twistor theory, Commun. Math. Phys. 213 (2000), 641-672 [arXiv:math/0001008 [math.DG]].
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  * [8] S. Alexandrov, B. Pioline and S. Vandoren, Self-dual Einstein Spaces, Heavenly Metrics and Twistors, J. Math. Phys. 51 (2010), 073510 [arXiv:0912.3406 [hep-th]].
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# Reply to Comment on “Synchronization dynamics in non-normal networks: the
trade-off for optimality”

Riccardo Muolo1, Timoteo Carletti1, James P. Gleeson2, Malbor Asllani3
1Department of Mathematics and naXys, Namur Institute for Complex Systems,
University of Namur, rue Grafé 2, 5000 Namur, Belgium  2MACSI, Department of
Mathematics and Statistics, University of Limerick, Limerick V94 T9PX, Ireland
3School of Mathematics and Statistics, University College Dublin, Belfield,
Dublin 4, Ireland

###### Abstract

We reply to the recent note “Comment on Synchronization dynamics in non-normal
networks: the trade-off for optimality” comment , showing that the authors
base their claims mainly on general theoretical arguments that do not
necessarily invalidate the adequacy of our previous study ourpaper . In
particular, they do not specifically tackle the correctness of our analysis
but instead limit their discussion on the interpretation of our results and
conclusions, particularly related to the concept of optimality of network
structure related to synchronization dynamics. Nevertheless, their idea of
optimal networks is strongly biased towards their previous work and does not
necessarily correspond to our framework, making their interpretation
subjective and not consistent. We bring here further evidence from the
existing and more recent literature, omitted in the Comment note, that the
synchronized state of oscillators coupled through optimal networks, as
intended by the authors, can indeed be highly fragile to small but finite
perturbations, confirming our original results.

In the Comment note comment , the authors claim we have drawn ambiguous or
incorrect conclusions in our recent paper ourpaper  regarding the lack of
optimality of non-normal networks and the failure of the linear stability
analysis as a prediction tool in such a setting. Here we show that such claims
contradict the recent literature (including previous papers written by authors
of comment ) or omit part of the information already present in our paper.
Despite the fact that we found the Comment comment  harsh and partial, we are
grateful to its authors for having publicly expressed their point of view. In
fact, such reaction has sparked an immediate interest in the scientific
community, in particular the recent studies sorrentino ; fish_bollt  carried
out independently by researchers unrelated to the authors of this Reply
letter. As we will show in the following, not only do such studies confirm our
main results presented in ourpaper , in neat contrast with the Comment note
comment , but they have pushed them further focusing exclusively on the class
of optimal networks as intended by the authors of the Comment comment . In the
following we will respond to the supposed issues raised in comment  aiming
this way to avoid any possible misunderstanding from the interested readers.
However, before we address such claims, we find it necessary to briefly
summarize the emergence of real networks’ non-normal properties and the
consequences in the linear dynamic regime.

A matrix A is defined as non-normal if it does not commute with its self-
adjoint: $\textbf{A}\textbf{A}^{*}\neq\textbf{A}^{*}\textbf{A}$. Such a
property has important consequences for the dynamics of linear systems of the
form $\dot{\textbf{x}}=\textbf{A}\textbf{x}$, as well as for nonlinear ones
with A the linearized operator. As a prominent example, we mention here the
seminal work by Trefethen et al. in hydrodynamic stability trefethen2 . In
this context, they used the pseudo-spectrum of J defined as
$\sigma_{\delta}\left(\textbf{J}\right)=\sigma\left(\textbf{J}+\bm{\Delta}\right)$
for all $||\bm{\Delta}||\leq\delta$, where J represent the linearised
(Jacobian) operator, to determine a lower bound of the transient growth as a
function of the non-normality of J. Such a relation reads $\sup_{t\geq
0}||e^{\textbf{A}t}||\geq\mathcal{K}$ where we make use of the Kreiss constant
$\mathcal{K}=\sup_{\delta\geq
0}\alpha_{\delta}\left(\mathbf{J}\right)\big{/}\delta$ with
$\alpha_{\delta}\left(\mathbf{J}\right)=\sup\Re{\sigma_{\delta}(\mathbf{J})}$.
The latter expression is crucial since it builds a strong link between the
pseudo-spectrum and the non-monotonic evolution of the linear dynamics. Based
on that, Trefethen et al. have shown that when the amount of non-normality
increases beyond some threshold, small but finite perturbations can undergo an
initial growth and the nonlinear system can eventually stabilize in another
state different from the one near which the perturbations originated. As
Trefethen $\&$ Embree have often noted in their book trefethen , although such
outcomes do not contradict the mathematical formulation of the stability
theory, there are still neat evidences where linear methods will practically
fail to predict the final behavior of the system under investigation. The
reason for that is attributed to the fact that an increase in non-normality
will drastically reduce the basin of attraction of the linearly stable state,
which we further stress, is defined on local results only. The latter puts to
the fore a general deficiency of local analysis compared to the global one,
which is built on the promises that such description can capture the general
behavior of the system for initial conditions starting near the stable state.
Such results make local stability techniques in a natural or experimental
setting, where perturbations will be very small but still finite, of little
practical utility to analyze non-normal systems. Recently, a set of systematic
statistical studies has brought to light the fact that most real networks,
including biological, ecological, social, transport, economic, etc.
top_resilience ; malbor_teo ; me ; duccio_stoch ; hierarchy , possess a high
level of non-normality with peculiar structural features. In this regard, it
has been recently shown for networked systems in me  that despite a negative
Master Stability Function (MSF) pecora  indicating a strictly stable fixed
point, the orbits starting near such a state could escape from it, stabilizing
in other stationary equilibrium.

–Adequacy of the Linear Stability Analysis (e.g., MSF) and the finite
perturbations question
Based on the considerations above, in our paper ourpaper , we have studied
synchronization dynamics in non-normal networks based on the MSF formalism.
From the local stability analysis perspective, we have shown that non-normal
networks present spectral features that optimize the stability of the
synchronized state. Still, the evolution of the nonlinear system evades the
linear prediction because of the reduction of the basin of attraction of the
synchronized manifold, in line with the previous results known in the
literature. In their original papers mott_nish1 ; mott_nish2 , the authors
have been limited to a theoretical analysis based exclusively on linear
stability analysis and have not performed numerical simulations. Thus they
have not emphasized the nonlinear behavior and in particular the restriction
of the synchronization basin at variance with the authors of sorrentino ;
fish_bollt  who have carried out systematic numerical investigations and
consequently confirmed our original findings. Furthermore the affirmations
made in comment  are surprisingly contradictory with a very recent
contribution of the first two authors of the Comment note mott_nn  agreeing
with our statements (pg. 1)

“Even when a fixed point is linearly stable in a nonlinear system described by
ordinary differential equations, if the corresponding Jacobian matrix is non-
normal, a small but finite perturbation can transiently grow beyond the
validity of the linear approximation and enter into the nonlinear regime,
preventing the perturbation from decaying to zero.”

which makes clear that our results should not be surprising at all regarding
the validity limitations of linear approximations that, as we clearly show in
the following, extend also to the case of the linearization about a limit
cycle. Before we address further concerns, we will bring to the attention of
the interested reader that similar qualitative results to ours in ourpaper
were independently obtained in Sun . We quote from Sun  in the following (pg.
1)

“Our results show that optimal networks that are more sensitive with respect
to structural perturbations (thus having higher sensitivity index) tend to be
slower in synchronization, and may sometimes not synchronize at all despite
being deemed synchronizable under linear stability analysis.”

Moreover they continue (pg. 7)

“This loss of synchronization cannot be interpreted by a linear theory, since
the equivalent linear system would always converge to the (global) stable
state despite initial transient growth. The actual nonlinear system, however,
can have a non-global basin of synchronization. If the initial transient
growth is too fast, as seen in some of the very sensitive networks, it “kicks”
the state of the system outside of the basin, and thus prevent the system from
returning to eventual synchronization. ”

It is worth mentioning that although the sensitive optimal networks have been
considered in several works prl_exp ; Fish ; PRX , the authors of comment
have omitted this fact in their Comment note. On the other hand, the
observations of Sun  do not diminish the novelty of our results since they do
not relate to the transient growth behavior that drives such evasion from the
linear prediction to the non-normality of such networks. Furthermore, in our
paper and in full analogy with trefethen2 , we have systematically quantified
such relationships using techniques such as the pseudo-spectrum (Fig. 5 $a)$),
grounded on the strong link between the latter and the transient growth
mentioned earlier.

As mentioned earlier, the recent discussion triggered by our original paper
ourpaper  has drawn the attention of the scientific community sorrentino ;
fish_bollt . In particular, and similarly to Sun , the authors of such recent
studies, focus exclusively on models of optimal networks following the
definition as introduced by the first two authors of the Comment in mott_nish1
; mott_nish2 . Here an excerpt from the conclusions of sorrentino

“Our numerical results confirm some of the claims presented in Refs. ourpaper
; footnote  and indicate that networks with non-normal Laplacian matrix are
typically characterized by a smaller basin of attraction than networks with a
normal Laplacian matrix. Our work also complements some of the early results
in Sun … Finally, our work points out the need for an alternative measure of
network synchronizability which addresses the important issue of the basin of
attraction for the synchronous solution.”

and the conclusions of fish_bollt

“In this work we have examined the stability of the synchronous state of
”optimal” non-normal matrices, specifically of non-normal graph Laplacians.
For such networks, MSF analysis is no longer useful in some circumstances and
we must appeal to other measures to characterize the stability.”

In both cases, the authors agree that infinitesimal perturbations make no
sense in the real scenarios “It is well known that the validity of the MSF
approach is limited to infinitesimal perturbations about the synchronous
solution (local stability.) However, in all applications, one is interested in
the effects of finite perturbations.” in sorrentino  and “Of note in real
world systems as verified by experiment, noise plays an important role and so
finite perturbations cannot be ignored.” in fish_bollt , respectively. Thus,
we are confident that the qualitative reproduction of our study ourpaper
carried out independently by different groups of researchers and unrelated to
the authors of this Reply letter are in line with the best scientific practice
and should be sufficient to dispel any doubt regarding the validity of our
findings. Similar confirmations can be found also in Fish , referring to the
use of MSF approach in non-normal networks for finite perturbations: “In such
cases, initial conditions near the synchronization manifold may suffer large
transients that can expel trajectories from the expected basin of stability
Sun ; me ; malbor_teo ”

–Optimality of non-normal networks and the averaging method
Regarding the spectral optimality of the non-normal networks, we have been
focused on a network model (a directed chain line with reciprocal links
uniformly weighted with $\epsilon$), whose Laplacian always admits a basis of
eigenvectors. Such a model converges to the class of optimal networks
mott_nish1 ; mott_nish2  for $\epsilon\rightarrow 0$, but otherwise, it is
generally different from them. In fact, from the Comment note, one could draw
the wrong conclusion that our work was focused exclusively on the optimal
networks mott_nish1 ; mott_nish2 . In contrast, nowhere in our paper have we
defined our network model based on such a class of optimal networks. Instead,
we have stated that they share similar features related to strong directedness
or Laplacian degeneracy (for $\epsilon=0$), but they are otherwise different
being diagonalizable. Thus our concept of optimality, although based on the
same MSF approach, differs from the one of mott_nish1 ; mott_nish2  without
any prejudice for the latter. On their side, the authors firmly refer to the
optimal networks they have previously introduced, but as we have pointed in
ourpaper  different definitions of optimal networks already exist in
literature skardal1 ; skardal2 . Furthermore, there is a striking difference
between our model of non-normal networks and the class of optimal networks
mott_nish1 ; mott_nish2  which are non-diagonalizable, as also correctly
pointed out in sorrentino . As we have previously noticed in top_resilience
the geometric degenerancy of the spectrum can bring about a transient growth,
as also explained in the Comment note. However, there is a crucial point that
has been ommitted in the Comment comment : that such transient growth is
possible in a larger class of networks that are not necessarily non-
diagonalizable, the non-normal ones. This fact refutes the incorrect assertion
made in the Comment that we haven’t extended the idea of non-normal dynamics
to the synchronization phenomenon. At variance with mott_nish1 ; mott_nish2 ,
we make use of the classical MSF formulation pecora . We so require
diagonalizable networks with varying imaginary part (tuning the $\epsilon$
parameter) of the Laplacian eigenvalues to quantify the spectral optimality.
Such numerical analysis has been presented in Fig. 2 of ourpaper . In our
paper, we emphasize that alternatively analytical insight can be obtained by
approximating the Master Stability Function by averaging the time-dependent
Jacobian over the period $\langle\bm{\mathcal{J}}\rangle_{T}$ (Fig. 3), this
approach aims to bridge our problem of optimality to known results malbor ;
malbor2  where the role of the imaginary part of the Laplacian spectrum has
been proved analytically. Such an approximation is based on the Magnus
expansion, and its validity for studying the stability of limit cycles is
given in chall  where a detailed analysis of the same model (i.e.,
Brusselator) has been carried out. In fact, in our paper, we clearly state
that “the set of model parameters for which we expect a good agreement with
the original model corresponds to the case when higher-order terms are
negligible.”. This alternative approach is suggested to analytically justify
the optimality we obtain numerically in Fig. 2 by varying the $\epsilon$
parameter, but we have not developed such argument further and, contrary to
what is stated in the Comment comment , our claims of optimality are based on
the numerical approach of Fig. 2 rather than the analytical one of Fig. 3. On
the other hand, we agree that the averaging method does not apply for all
models, however because of the latter remark, the counterexample provided in
comment  does not fit in our framework. Regarding the use of the averaging
method, the authors of the Comment note seem to contradict previous statements
made by some of them. In particular the third author of comment  has several
times advised the averaging of the time-dependent Jacobian matrix as a valid
alternative to the MSF pecora1 ; pecora2 , for instance (pg. 3 of pecora1 ):

“We noted that other stability criteria are possible. Each will produce its
own master function over the complex coupling plane. Among them are the
following four (roughly in order of strictness): (1) Calculate an exponent for
an averaged Jacobian [Brown $\&$ Rulkov, 1997a, 1997b],” (the second
reference, here rulk , was already included in our paper when referred to the
averaging of the Laplacian).

–Further minor points
We agree with the authors of the Comment note that we have written “(real part
of the maximum) Lyapunov exponent” which is redundant since the Lyapunov
exponents are real by definition. Nevertheless, such unfortunate oversight
does not compromise the results of our analysis as far as the real part of a
real number is still a real number. Also, with “autonomous version of the MSF”
we intended the MSF of an autonomous linearized system as shown in our paper.
We are confident that such unharmful lapses do not compromise our study, and
the attentive reader will easily grasp the novelty of our contribution, as
was, for instance, the case of the authors of sorrentino ; fish_bollt . Also,
in the Comment note there is a long discussion about the bubbling phenomenon.
Although we do agree in principle with the authors on it, bringing such a
phenomenon as an argument in this discussion can result misleading since non-
normality and bubbling are not necessarily connected. What we are stating in
our paper is that transient growth and the eventual escaping from the linear
stability prediction can be certainly caused by the network non-normality. We
did not observe the bubbling effect in our study, but we are by no means
excluding the possibility that other factors, such as bubbling, can cause
similar behavior. Thus we find either in such comparison confusing rather than
clarifing. As a last note, we did not find neither in the Comment’s
references, nor in the broader literature, any explicit relation between the
non-normality and the robustness of the synchronized regime.

To conclude, we want to emphasize that there is an increasing research field,
among which our paper sits, that clearly points towards the same conclusion we
reached with our numerical study, i.e., when a non-normal matrix governs the
dynamics, a linear stability analysis may fail in predicting the global
stability of a system. Let us stress that, in addition, our results have not
been tested nor reproduced, making a comparison on the same basis difficult.
And last but not least, we want to reemphasize an elementary fact:
perturbation theories (where also the MSF method belong), since they were
first deployed, were aimed to understand the behavior of nonlinear systems by
approximating their dynamical evolution with few terms of series expansions
based on a small but still finite parameter and thus the term _infinitesimal_
, used by the authors of the Comment, is highly misleading. Such an
approximation will be valid to study the linear stability as long as the
perturbations lie inside the attraction basin of the state under
consideration. Unfortunately, even in the most controlled laboratory
conditions, such perturbations cannot be “infinitesimal,” and for the case of
highly non-normal systems, as is the case of many real networks, the basin of
attraction can be drastically small (in our paper, this can be clearly
observed in Fig. 5 panels $c_{1}$ and $c_{2})$), thus making linear methods
fail to achieve their goal. Finally, for consistency with what is stated
above, we quote our paper where the term “finite perturbations” has been
abundantly used strongly contrasting the authors’ claims in the Comment paper:

“In a linear regime, a finite perturbation regarding a stable equilibrium goes
through a transient amplification (blue curve in Figure 4 $d)$) that is
proportional to the level of non-normality before it is eventually reabsorbed
in the steady state trefethen , while, in the full non-linear system, the
finite perturbation could persist indefinitely (red curve in Figure 4 $d)$).”

## Funding

R.M. is supported by a FRIA-FNRS PhD fellowship, Grant FC 33443, funded by the
Walloon region. The work of J.P.G. and M.A. is partly funded by Science
Foundation Ireland (Grants No. 16/IA/4470, No. 16/RC/3918 and No. 12/RC/2289
P2) and co-funded under the European Regional Development Fund. M.A. is also
supported by a OBRSS Research Support Scheme no. R23036.

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