# Double field theory, twistors, and integrability in 4-manifolds

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

(June 3, 2021)

###### Abstract

The search for a geometrical understanding of dualities in string theory, in
particular T-duality, has led to the development of modern T-duality covariant
frameworks such as Double Field Theory, whose mathematical structure can be
understood in terms of generalized geometry and, more recently, para-Hermitian
geometry. In this work we apply techniques associated to this doubled geometry
to four-dimensional manifolds, and we show that they are particularly well-
suited to the analysis of integrability in special spacetimes, especially in
connection with Penrose’s twistor theory and its applications to general
relativity. This shows a close relationship between some of the geometrical
structures in the para-Hermitian approach to double field theory and those in
algebraically special solutions to the Einstein equations. Particular results
include the classification of four-dimensional, possibly complex-valued,
(para-)Hermitian structures in different signatures, the Lie and Courant
algebroid structures of special spacetimes, and the analysis of deformations
of (para-)complex structures. We also discuss a notion of “weighted
algebroids” in relation to a natural gauge freedom in the framework. Finally,
we analyse the connection with two- and three-dimensional (real and complex)
twistor spaces, and how the former can be understood in terms of the latter,
in particular in terms of twistor families.

###### Contents

  1. 1 Introduction
  2. 2 Background
    1. 2.1 Para-Hermitian and related structures
    2. 2.2 Generalized geometry and algebroids
      1. 2.2.1 Elementary notions
      2. 2.2.2 Lie and Courant algebroids
    3. 2.3 Conformal geometry and the Lee form
  3. 3 Almost para-Hermitian structures in four dimensions
    1. 3.1 Self-dual forms
    2. 3.2 Spaces of almost para-Hermitian structures
      1. 3.2.1 Riemannian signature
      2. 3.2.2 Split signature
      3. 3.2.3 Lorentz signature
    3. 3.3 Gauge freedom and a covariant formalism
  4. 4 Integrable structures
    1. 4.1 Special spinor fields
    2. 4.2 Lie and Courant algebroids
    3. 4.3 Weighted algebroids
  5. 5 Twistors
    1. 5.1 Two-dimensional twistor spaces
    2. 5.2 Three-dimensional twistor spaces
      1. 5.2.1 Connections with other twistor constructions
    3. 5.3 Deformations of the para-complex structure
  6. 6 Summary and conclusions
  7. A Spinors in four dimensions
  8. B Further details on the covariant formalism
  9. C Curvature of $\mathcal{C}_{a}$

##  1 Introduction

Generalized geometry [1, 2] is a branch of differential geometry that unifies
symplectic and complex geometry and in which, in particular, vector fields and
1-forms on a manifold $M$ are treated on an equal footing, as sections of a
“doubled” or generalized tangent bundle, $\mathbb{T}M=TM\oplus T^{*}M$. The
generalization to $\mathbb{T}M$ of the differential structure encoded in the
Lie bracket is captured by the notion of Courant algebroids, which involve the
so-called Dorfman bracket (or its antisymmetric version, the Courant bracket).
During the last years these structures have appeared naturally in theoretical
high-energy physics, as a central tool for the understanding of the geometry
of string theory, related in particular to the geometrical interpretation of
T-duality (e.g. [3]). A prominent line of developments in this respect is
Double Field Theory (see below), where it has recently been shown [4, 5, 6, 7]
that also para-Hermitian geometry is particularly well-suited to the
understanding of its mathematical structure. In this work we show that doubled
geometry is also natural in general relativity, where it provides a new
perspective on ‘integrability issues’ in special spacetimes and their
connection with twistor theory.

As an initial motivation, let us give a rough argument to introduce para-
Hermitian and generalized geometry in string theory, and their connections
with T-duality. When considering string toroidal compactifications, a
distinctive feature of strings is that their extended nature allows them to
wrap non-contractible cycles around the compact dimensions. As a consequence,
strings are not only characterized by momentum modes but also by winding
modes, which describe how the strings wind around the tori. Roughly speaking,
T-duality is a symmetry of string theory that establishes that momentum and
winding states should be considered on an equal footing. As a simple example,
suppose we have only one compact dimension, a circle $S^{1}$ with radius $R$.
The momentum and winding modes are quantized; they are characterized by two
quantum numbers $n,m$. From the mass spectrum for closed strings one can see
(as a standard reference cf. [8, Section 8.3]) that, when $R\to\infty$,
winding modes become very heavy, while momentum modes become light and start
to form a continuum, as corresponds to a non-compact dimension. Similarly,
when $R\to 0$, momentum modes become infinitely massive but winding modes
become light, and the spectrum again approaches a continuum. Therefore, the
$R\to\infty$ and $R\to 0$ limits appear to be physically equivalent, a
remarkable feature that is exclusive to string theory since a field theory of
point-like particles has no winding states (so no spacetime dimension ‘opens
up’ when $R\to 0$ in that case). This equivalence comes from the fact that the
string spectrum is invariant under the simultaneous change
$R\leftrightarrow\alpha^{\prime}/R$, $n\leftrightarrow m$ (where
$\alpha^{\prime}$ is the inverse of the string tension), a symmetry which
extends to any observable in the theory. This is the simplest manifestation of
T-duality; more generally, T-duality relates the physics of strings
propagating on backgrounds with very different geometries.

Since momentum modes are conjugate or ‘dual’ to spacetime coordinates $x^{i}$,
the consideration of momentum and winding modes at the same level suggests the
introduction of a new set of coordinates $\tilde{x}_{i}$ ‘dual’ to winding.
This doubling of coordinates allows to formulate a field theory that
incorporates the degrees of freedom associated to winding without necessarily
representing strings, and such that T-duality is a manifest symmetry of the
theory. This scheme is called Double Field Theory (DFT) since the pioneering
work [9] (see also e.g. [10] for a review), and represents a T-duality
covariant formulation of supergravity that incorporates stringy aspects not
present in the usual treatment of supergravity. Although the doubling of space
coordinates should only affect the compact dimensions, it is useful in
practice to duplicate all space coordinates; in this way one ends up with a
doubled space, which in the simplest case is a product $T\times\tilde{T}$ and
has global coordinates $(x^{i},\tilde{x}_{i})$, where $T$ and $\tilde{T}$ are
$d$-dimensional flat tori that are said to be dual from each other. This
‘extended spacetime’ is intended to be the target space of the fundamental
string, and the physical fields222In the simplest formulation of DFT, the
physical fields correspond to the massless sector of closed strings, namely a
metric tensor, a 2-form field (the Kalb-Ramond field) and a scalar field (the
dilaton). are organized in representations of the T-duality group ${\rm
O}(d,d)$. Furthermore, from the field content of the theory one deduces a
notion of generalized diffeomorphisms, that are infinitesimally generated by a
generalized Lie derivative or ‘D-bracket’. Consistency conditions (i.e.
closure of the algebra of generalized diffeomorphisms) impose a restriction on
the fields. A particular solution to this restriction is the so-called
‘section condition’ or ‘strong constraint’, where the fields are forced to
depend on only half of the doubled coordinates. This half is determined as a
subspace that is maximally isotropic (or totally null) with respect to the
${\rm O}(d,d)$ metric $\eta={\rm d}x^{i}\otimes{\rm d}\tilde{x}_{i}+{\rm
d}\tilde{x}_{i}\otimes{\rm d}x^{i}$ [11], i.e. a $d$-dimensional subspace $S$
such that333The metric $\eta$ involved in (1.1) does not represent the
gravitational field; it is part of the ‘kinematics’ of the theory. The
gravitational field is unified with the 2-form field into a ‘generalized
metric’, which codifies the dynamics; this will not play a role in the present
work.

$\eta|_{S}=0.$ (1.1)

T-duality acts by transforming any such isotropic subspace into another.

One might already anticipate some parallelism of the DFT framework with
generalized geometry, in the sense that both formalisms are based on a doubled
space. However, an important difference is that in the former, what is doubled
is the base manifold, while in the latter is the tangent bundle. This
difference is important if one wants to understand T-duality in the
generalized geometry setting, since, as argued above, the very essence of
T-duality is to exchange the manifolds $T$ and $\tilde{T}$. Upon restriction
to fields satisfying the strong constraint, the D-bracket reduces to the
Dorfman bracket and one can argue that the DFT setting reduces locally to
generalized geometry (see e.g. [12]). A possible formalization of all this
(that is also intended to provide a global formulation of DFT) originates in
the work of Vaisman in [4, 5], who shows, in the first place, that the usual
extended spacetime of DFT is a flat para-Kähler manifold. To see this,
consider first an almost para-Hermitian structure on a $2d$-dimensional
manifold $M$: a pair $(\eta,K)$ where $K\in{\rm End}(TM)$ and $\eta$ is a
metric such that

$K^{2}=\mathbb{I},\qquad\eta(KX,KY)=-\eta(X,Y)$ (1.2)

for any vector fields $X,Y$. The map $K$ is called almost para-complex
structure, and it produces a splitting of the tangent bundle as
$TM=L\oplus\tilde{L}$, where $L,\tilde{L}$ are the $(\pm 1)$-eigenbundles of
$K$. These eigenbundles are maximally isotropic with respect to $\eta$, i.e.
they are $d$-dimensional and satisfy (1.1), and this implies that
$\tilde{L}\cong L^{*}$ (see Section 2 for details). If $L$ and $\tilde{L}$ are
involutive under the Lie bracket then $K$ is said to be integrable, and there
exist local coordinates $(x^{i},\tilde{x}_{i})$, $i=1,...,d$, such that,
denoting $\partial_{i}=\partial/\partial x^{i}$ and
$\tilde{\partial}^{i}=\partial/\partial\tilde{x}_{i}$, one has $L={\rm
span}(\partial_{i})$ and $\tilde{L}={\rm span}(\tilde{\partial}^{i})$. If
$\eta$ is flat, then it can be expressed in these coordinates as $\eta={\rm
d}x^{i}\odot{\rm d}\tilde{x}_{i}$. So there are two complementary foliations
$F,\tilde{F}$ such that $L=TF$ and $\tilde{L}=T\tilde{F}$ 444This notation
means that if $\Sigma$ is a leaf in $F$, then $T\Sigma=L|_{\Sigma}$, etc., and
$\tilde{L}\cong L^{*}$ implies that $T\tilde{F}\cong T^{*}F$, thus $TM$ has
actually the structure of a generalized tangent bundle: $TM\cong TF\oplus
T^{*}F$. Furthermore, Vaisman shows [4] that there is a natural notion of
D-bracket on $TM$, and that it reduces to the Dorfman bracket for fields with
dependence only on $F$, so in this way one recovers generalized geometry.
Finally, different T-dual spacetimes correspond to different choices of para-
complex structures.

The study of DFT from the perspective of para-Hermitian geometry, in settings
more general than the flat para-Kähler case, was further developed in [5] and
[6, 7, 13, 14], and also in several subsequent works, being currently an
active research area. (Early considerations of the relationship between para-
Hermitian structures and T-duality can be found in [15].) A particularly
important difference between Hermitian and para-Hermitian geometry is that the
eigenbundles of an almost-complex structure are complex conjugates of each
other, while those of an almost para-complex structure are independent. As a
consequence, the integrability properties of a para-Hermitian structure can be
split into separate questions about the involutivity of each eigenbundle, so
one can talk about “half-integrability”. We will see that this is especially
important in general relativity and its connection with twistor structures.

A key condition in the analysis above is equation (1.2), which implies (1.1)
for the eigenbundles of $K$. For real $K$, this means that the signature of
$\eta$ must be $(d,d)$; so in four dimensions this is $(++--)$, which is
different from the usual Lorentz signature $(+---)$ of general relativity.
However, if we allow $K$ to be a map on the complexified tangent bundle, then
the metric is allowed to have any signature (or to be complex). Our interest
in complex-valued maps arises from the power of using complex methods in
relativity, which date back to remarkable programmes such as Penrose’s twistor
theory, Newman’s H-space, and Plebański’s hyper-heavenly (HH) construction.
(One can argue that, in a sense, all these constructions can be understood as
parts of the twistor programme.) This is one of our main motivations in this
work: submanifolds where the condition (1.1) holds are actually the basic
object of twistor theory, but they arise from a completely different
perspective, namely the essential premise of the twistor programme that light-
rays are more fundamental than spacetime points. A related motivation comes
from the fact that a “complex-valued almost-complex structure” has some
significance in developments in mathematical relativity and their relation to
twistor structures, see [16, 17, 18, 19, 20]. As in ordinary para-Hermitian
geometry, the phenomenon of “half-integrability” is shared by such complex-
valued maps. The present work can be regarded as the application of some of
the modern DFT-like techniques mentioned above to 4-manifolds, with the extra
ingredient that $K$ can be complex-valued so that one can make contact with
general relativity and twistor theory.

### Main results

The fact that we consider complex-valued “para-Hermitian” structures implies
that, unlike the usual literature in DFT and related matters, we can analyse
different signatures of the metric, by imposing different reality conditions.
Since we work in a 4-manifold $M$ (which we assume to be orientable), this
means that we will deal with Lorentz, Riemannian (or Euclidean) and split (or
neutral) signature. The Lorentzian case is relevant, of course, for general
relativity (and in particular for the so-called hyper-heavenly spaces [21]);
the Riemannian case is relevant because e.g. of the Atiyah-Hitchin-Singer
approach to twistor theory [22]; and the split case can be related, since the
work of Ooguri and Vafa [23], to the geometry of strings with $N=2$
supersymmetry. (It is also related to the LeBrun-Mason twistor construction in
split signature [24].)

Since an arbitrary 4-manifold with a metric does not naturally come equipped
with a (possibly complex-valued) almost para-Hermitian structure, we first
focus on classifying all possible such structures. We show that regardless of
signature, an almost para-Hermitian structure is essentially equivalent to an
(anti-)self-dual 2-form (in the sense of Hodge duality). Then we use this
result to describe the space $P_{x}$ of para-Hermitian structures at any point
$x\in M$: we show that $P_{x}$ has two connected components and, imposing
appropriate reality conditions whenever possible, each component can be
described as: a complex projective line $\mathbb{CP}^{1}$ in Riemannian
signature555This turns out to be analogous to the result on almost-complex
structures, which is already known in the literature [22]., the space
$(\mathbb{RP}^{1}\times\mathbb{RP}^{1})\backslash\mathbb{RP}^{1}$ in split
signature, and the space
$(\mathbb{CP}^{1}\times\mathbb{CP}^{1})\backslash\mathbb{CP}^{1}$ in Lorentz
signature (and in complex 4-manifolds). We show that, topologically, these
spaces are respectively a real 2-sphere $S^{2}$, a hyperboloid of one sheet
$H^{1}$, and a complex 2-sphere $\mathbb{C}S^{2}$. For Hermitian structures,
the corresponding spaces are $\mathbb{CP}^{1}$,
$\mathbb{CP}^{1}\backslash\mathbb{RP}^{1}$ (a hyperboloid of two sheets) and
$(\mathbb{CP}^{1}\times\mathbb{CP}^{1})\backslash\mathbb{CP}^{1}$,
respectively. We also find that there is a natural “gauge freedom” associated
to the representation of a (para-)Hermitian structure, and we develop a
correspondingly covariant formalism.

We then analyse the integrability conditions for almost (para-)Hermitian
structures, and we show that involutivity of one of the eigenbundles of $K$
(or “half-integrability”) is equivalent to the existence of certain special
spinors, that in relativity language give origin to what are called shear-free
null geodesic congruences. This allows us to show that there are natural
algebroid structures associated to any four-dimensional manifold with a shear-
free congruence, namely Lie and Courant algebroids. This gives explicit
examples of Courant algebroids in four dimensions, that include e.g. all
algebraically special Einstein manifolds (such as the Kerr and Schwarzschild
black hole solutions, but also all vacuum Petrov type II solutions), so it
connects the structures studied in generalized geometry to spacetimes of
interest in classical relativity. Furthermore, the Lie algebroid structure has
naturally associated a cochain complex that can be used to prove the existence
of potentials in problems of interest in relativity. We are naturally led to
the question of generalizing the algebroid structure to fields with general
transformation properties under the gauge freedom mentioned above, and we
discuss the possible construction of such an object (“weighted algebroid”) and
the associated differential complex.

Finally, we focus on connections with twistor theory, showing that: any
particular half-integrable para-Hermitian structure defines a 2-dimensional
(2D) twistor space, and if all para-Hermitian structures are half-integrable,
then there is a 3-dimensional (3D) twistor space, which is a one-parameter
family of 2D twistor spaces parametrized by projective spinor fields. This 3D
space is the total space of a fibration of 2D twistor spaces over projective
spinors if and only if the vacuum Einstein equations are satisfied. We also
discuss the relationships with other twistor constructions in the literature.
Finally we define deformations of para-complex structures in $M$ and analyse
their integrability, and we show that small, half-integrable deformations
exist if and only if the Weyl tensor is half-algebraically special, and in
this case they define sections of a line bundle over a 2D twistor space. As in
the rest of the paper, our results here are valid for any metric signature.

### Conventions and overview

Our conventions for different kinds of indices, curvature, etc. follow Penrose
and Rindler [25, 26]. The organization of this work is as follows. We start in
Section 2 with some background material: we review basic notions on para-
Hermitian geometry and related structures (keeping always in mind that we
allow complex-valued maps), generalized geometry and Lie and Courant
algebroids, and conformal geometry and the Lee form. Then in Section 3 we
study almost para-Hermitian structures in 4-manifolds equipped with a metric
of different signatures, and we describe the spaces of such structures in each
case. In Section 4 we study integrability issues: half-integrability of para-
Hermitian structures, the Lie and Courant algebroids associated to special
spacetimes, and a possible notion of “algebroids” for the treatment of
weighted fields. In Section 5 we study some connections with twistor theory:
two- and three-dimensional twistor spaces, twistor families, and integrability
of small deformations of para-complex structures. In Section 6 we make a
summary of this work and comment on some possible future directions. We
include three appendices: Appendix A with a brief review of spinors in 4
dimensions (in any signature), Appendix B with additional details about the
“gauge freedom” and the associated covariant formalism, and Appendix C with
some properties of the curvature of the natural connection in the above
formalism.

##  2 Background

###  2.1 Para-Hermitian and related structures

In the following, $M$ denotes a $d$-dimensional real, smooth manifold. The
definitions still apply if $M$ is complex.

######  Definition 2.1.

Let $E\to M$ be a vector bundle with even rank. Let $g$ be a non-degenerate
symmetric bilinear form on the fibers. Given a map $K\in{\rm End}(E)$, we say
that $(g,K)$ is a para-Hermitian structure on $E$ if it holds:

  1. 1.

$K^{2}=+\mathbb{I}$

  2. 2.

The $\pm 1$ eigenbundles of $K$ have the same rank.

  3. 3.

$g(KX,KY)=-g(X,Y)$ for all $X,Y\in E$.

If we require only the first two conditions, $K$ is said to be a para-complex
structure.

Consider a para-Hermitian structure $(g,K)$. The map $K$ gives a decomposition
$E=L\oplus\tilde{L}$, where $L$ is the $(+1)$-eigenbundle of $K$ and
$\tilde{L}$ the $(-1)$-eigenbundle. Any element $X\in E$ can be written as

$X=x+\tilde{x}$ (2.1)

where $x\in L$ and $\tilde{x}\in\tilde{L}$. In the following, we will
extensively use this notational convention. Note that if $x,y\in L$, i.e.
$Kx=x$ and $Ky=y$, then $g(x,y)=g(Kx,Ky)=-g(x,y)$ so $g(x,y)\equiv 0$ for all
$x,y\in L$, and similarly $g(\tilde{x},\tilde{y})\equiv 0$ for all
$\tilde{x},\tilde{y}\in\tilde{L}$. We then say that $L$ and $\tilde{L}$ are
isotropic with respect to $g$, i.e. $g|_{L}=0=g|_{\tilde{L}}$. The following
is a standard result:

######  Proposition 2.2.

We have the isomorphisms $\tilde{L}\cong L^{*}$ and $L\cong\tilde{L}^{*}$.

###### Proof.

If $X\in E=L\oplus\tilde{L}$, in general we have $g(X,\cdot)\in
L^{*}\oplus\tilde{L}^{*}$. But if $\tilde{x}\in\tilde{L}$, then
$g(\tilde{x},\tilde{y})=0$ for all $\tilde{y}\in\tilde{L}$ since $\tilde{L}$
is isotropic, therefore $g(\tilde{x},\cdot)\in L^{*}$. Thus we have a map
$\tilde{L}\to L^{*}$ given by

$\tilde{L}\ni\tilde{x}\mapsto\tilde{x}_{\flat}:=g(\tilde{x},\cdot)\in L^{*},$
(2.2)

and since $g$ is non-degenerate, the map is an isomorphism. The proof of
$L\cong\tilde{L}^{*}$ is analogous. ∎

Since here we are not using the abstract index notation, we use the musical
isomorphisms $\flat$ and $\sharp$. From (2.2) it follows that the map

$\varphi:E=L\oplus\tilde{L}\to L\oplus L^{*},\qquad
X=x+\tilde{x}\mapsto\varphi(X)=(x,\tilde{x}_{\flat})$ (2.3)

is also an isomorphism.

######  Definition 2.3.

We say that $(g,K)$ is an almost para-Hermitian structure on a manifold $M$ if
$(g,K)$ is a para-Hermitian structure on $TM\otimes\mathbb{C}$.

######  Remark 2.4.

Notice that the map $K$ in Def. 2.3 satisfies the conditions of Def. 2.1 and
is allowed to be complex-valued. From this perspective, we can equally well
refer to the map $J=iK$ as a “Hermitian structure” in the sense that it
satisfies $J^{2}=-\mathbb{I}$, its eigenbundles have equal rank, and
$g(J\cdot,J\cdot)=g(\cdot,\cdot)$. We choose the terminology ‘para-Hermitian’
because it suggests that the eigenbundles are not related to each other
(unlike in the usual real-valued Hermitian structures), see Remark 2.9 below.
See also Remark 3.3.

######  Remark 2.5.

If $(g,K)$ is an almost para-Hermitian structure on $M$, and $\Omega$ is a
nowhere vanishing scalar field, then $(\Omega^{2}g,K)$ is also an almost para-
Hermitian structure on $M$, since
$(\Omega^{2}g)(K\cdot,K\cdot)=-(\Omega^{2}g)(\cdot,\cdot)$. This means that we
can interpret $K$ as an object associated not to a particular metric $g$ but
to the conformal class $[g]=\\{\Omega^{2}g\;|\;\Omega\in
C^{\infty}(M),\Omega>0\\}$, in the sense that the conditions of definitions
2.3 and 2.1 are satisfied for any $g\in[g]$. Therefore, conformal invariance
will play an important role in our study. We will refer to $([g],K)$ as an
almost para-Hermitian conformal structure on $M$.

A para-Hermitian structure $(g,K)$ in $M$ can also be thought of as an
“almost-symplectic structure”, in the sense that it automatically defines a
non-degenerate 2-form $\omega(X,Y):=g(KX,Y)$; this is usually called the
fundamental 2-form. The relationship of this with conformal geometry is
particularly interesting, see Section 2.3 below. The eigenbundles of $K$ are
isotropic with respect to $\omega$, that is $\omega|_{L}=0$ and
$\omega|_{\tilde{L}}=0$.

######  Definition 2.6.

An almost para-Kähler structure on a manifold $M$ is an almost para-Hermitian
structure such that the fundamental 2-form is closed, ${\rm d}\omega=0$.

Although our primary interest is in the case where there is only one para-
Hermitian structure, in later sections it will appear naturally the case in
which one has more than one para-complex or complex structure. Because of
this, it is useful to recall now the algebras of quaternions $\mathbb{H}$ and
of para-quaternions (or split-quaternions) $\mathbb{H}^{\prime}$:

$\displaystyle\mathbb{H}={}$
$\displaystyle\\{q=a+bi+cj+dk\;|\;i^{2}=j^{2}=k^{2}=-1,\;k=ij=-ji\\},$ (2.4)
$\displaystyle\mathbb{H^{\prime}}={}$
$\displaystyle\\{q^{\prime}=a+bi+cs+dt\;|\;i^{2}=-1,\;s^{2}=t^{2}=1,\;t=is=-si\\}$
(2.5)

where $a,b,c,d\in\mathbb{R}$. The quaternion algebra $\mathbb{H}$ is
associated to a positive definite quadratic form666The conjugate of a
quaternion $q$ is $\bar{q}=a-bi-cj-dk$, and the conjugate of a para-quaternion
$q^{\prime}$ is $\bar{q}^{\prime}=a-bi-cs-dt$.
$q\bar{q}=a^{2}+b^{2}+c^{2}+d^{2}$, while $\mathbb{H}^{\prime}$ is associated
to a quadratic form with split signature
$q^{\prime}\bar{q}^{\prime}=a^{2}+b^{2}-c^{2}-d^{2}$. Based on these algebras,
we define:

######  Definition 2.7.

Let $M$ be a real manifold, and let $I_{1},I_{2},I_{3}$ be three anticommuting
endomorphisms of $TM$. Then:

  1. 1.

$(I_{1},I_{2},I_{3})$ is an almost-hypercomplex (or almost-quaternionic)
structure if $I_{1},I_{2},I_{3}$ satisfy the quaternion algebra (2.4). If $M$
is equipped with a metric $g$, then $(g,I_{1},I_{2},I_{3})$ is an almost-
hyperhermitian structure on $M$ if it is almost-hypercomplex and
$g(I_{i}X,I_{i}Y)=g(X,Y)$ for all $i=1,2,3$ and for all vectors $X,Y$.

  2. 2.

$(I_{1},I_{2},I_{3})$ is an almost-para-hypercomplex (or almost-para-
quaternionic) structure if $I_{1},I_{2},I_{3}$ satisfy the para-quaternion
algebra (2.5). If $M$ is equipped with a metric $g$, then
$(g,I_{1},I_{2},I_{3})$ is an almost-para-hyperhermitian structure on $M$ if
it is almost-para-hypercomplex and $g(I_{1}X,I_{1}Y)=g(X,Y)$ and
$g(I_{i}X,I_{i}Y)=-g(X,Y)$ for $i=2,3$.

The word ‘almost’ in all the definitions above can be removed by introducing a
notion of integrability. To do this, it is instructive to first briefly recall
the situation in complex geometry. An almost-complex structure in a real
manifold $M$ is a real map $J$ in $TM$ such that $J^{2}=-\mathbb{I}$, and
whose $\pm i$ eigenbundles, denoted $T^{\pm}M$, have the same rank. One has
the decomposition $TM\otimes\mathbb{C}=T^{+}M\oplus T^{-}M$ (note that the
elements in $T^{\pm}M$ are necessarily complex). An almost-complex structure
is said to be integrable if $T^{+}M$ is an involutive distribution in
$TM\otimes\mathbb{C}$, that is
$[\Gamma(T^{+}M),\Gamma(T^{+}M)]\subset\Gamma(T^{+}M)$, where $[\cdot,\cdot]$
is the Lie bracket of vector fields, and $\Gamma(E)$ denotes the space of
sections of a vector bundle $E$. A real manifold equipped with an integrable
almost-complex structure is a complex manifold. The integrability condition
for $J$ is equivalent to the vanishing of its associated Nijenhuis tensor,
where, if $A:\Gamma(TM)\to\Gamma(TM)$ is a linear map, the Nijenhuis tensor
associated to $A$ is the map $N_{A}:\Gamma(TM)\times\Gamma(TM)\to\Gamma(TM)$
given by

$N_{A}(X,Y)=\frac{1}{4}\left(A^{2}[X,Y]+[AX,AY]-A([AX,Y]+[X,AY])\right),$
(2.6)

for all vector fields $X,Y$.

Mimicking the above definitions for complex structures, we define

######  Definition 2.8.

A (para-)Hermitian structure on a manifold $M$ is an almost (para-)Hermitian
structure $(g,K)$ such that the Nijenhuis tensor (2.6) associated to $K$
vanishes:

$N_{K}\equiv 0.$ (2.7)

Similarly, a (para-)hyperhermitian structure on $M$ is an
almost-(para-)hyperhermitian structure $(g,I_{1},I_{2},I_{3})$ such that
$N_{I_{i}}\equiv 0$ for all $i=1,2,3$.

A crucial difference between integrability of ordinary Hermitian and para-
Hermitian structures is the following:

######  Remark 2.9.

The fact that an ordinary almost-complex structure $J$ is a real tensor
implies that its eigenbundles are complex conjugates of each other, so $J$ is
integrable if and only if both distributions $T^{+}M$ and $T^{-}M$ are
involutive. In the para-Hermitian case, the two eigenbundles of a real-valued
$K$ are not complex conjugates of each other, so one of the distributions may
be involutive while the other one is not. This also applies if $K$ is complex-
valued as in Def. 2.3.

Recalling the splitting $L\oplus\tilde{L}$ induced by $K$, one can then
separate integrability properties associated to the two eigenbundles $L$ and
$\tilde{L}$. This gives origin to a notion of “half-integrability”:

######  Definition 2.10.

We say that an almost para-Hermitian structure $(g,K)$ on $M$ is half-
integrable if one of the eigenbundles of $K$ is involutive. If both
eigenbundles are involutive, then $K$ is integrable and $(g,K)$ is a para-
Hermitian structure on $M$.

######  Remark 2.11.

In [6], half-integrable almost-para Hermitian structures on a manifold $M$
(with real-valued $K$) are called $L$-para Hermitian or $\tilde{L}$-para
Hermitian manifolds, depending on whether the $(+1)$\- or the
$(-1)$-eigenbundle is integrable, respectively.

It is useful to introduce the projectors to $L$ and $\tilde{L}$; respectively:

$P:=\tfrac{1}{2}(\mathbb{I}+K),\qquad\tilde{P}:=\tfrac{1}{2}(\mathbb{I}-K).$
(2.8)

Using $K^{2}=\mathbb{I}$, one can easily check that these operators satisfy
$P^{2}=P$, $\tilde{P}^{2}=\tilde{P}$ and $P\tilde{P}=0=\tilde{P}P$. The
Nijenhuis tensor (2.6) for $A\equiv K$ can be rewritten in terms of these
projectors as [6, Eq. (3.13)]

$N_{K}(X,Y)\equiv N_{P}(X,Y)+N_{\tilde{P}}(X,Y),$ (2.9)

where

$N_{P}(X,Y)=\tilde{P}[PX,PY],\qquad
N_{\tilde{P}}(X,Y)=P[\tilde{P}X,\tilde{P}Y]$ (2.10)

Note that if $L$ is integrable, then $[PX,PY]\in\Gamma(L)$ and therefore
$\tilde{P}[PX,PY]=0$, i.e. $N_{P}\equiv 0$. Hence $N_{P}$ and $N_{\tilde{P}}$
govern the involutivity of $L$ and $\tilde{L}$ respectively.

######  Remark 2.12.

By Frobenius theorem (see e.g. [27, Theorem 19.21]), if $M$ is a real manifold
and one has an involutive distribution $L\subset TM$ then the collection of
all integral manifolds777An integral manifold of a distribution $L\subset TM$
is an immersed submanifold $N\subset M$ such that $T_{p}N=L|_{p}$ for all
$p\in N$. A generic distribution does not admit integral manifolds. of $L$
forms a foliation of $M$. But if the involutive distribution is complex, i.e.
$L\subset TM\otimes\mathbb{C}$, then the integral manifolds are complex
submanifolds living in the complexification of $M$ 888Note that, given a real
manifold $M$, the complex manifold resulting from an integrable almost-complex
structure, and the complexification of $M$, are two different concepts.. Since
complexification requires real-analyticity of $M$ (which is a restrictive
property from the point of view of relativity), in general we will not assume
the existence of a foliation. The involutivity of the distribution, however,
is well-defined, and is what we mostly need in this work.

######  Remark 2.13.

Since the eigenbundles $L$ and $\tilde{L}$ of $K$ are isotropic with respect
to the almost-symplectic form $\omega$, we can refer to them as Lagrangian
subbundles. If, say, $L$ is integrable and gives origin to a foliation $F$, we
can refer to $F$ as a Lagrangian foliation.

The integrability of almost-para-Hermitian and related structures is a
conformally invariant property. Other properties that are not conformally
invariant are also of interest, in particular the Kähler and related
conditions:

######  Definition 2.14.

Let $M$ be a real manifold.

  1. 1.

An almost-para-Hermitian structure $(g,K)$ on $M$ is para-Kähler if it is
para-Hermitian and almost-para-Kähler, that is, $K$ is integrable and ${\rm
d}\omega=0$.

  2. 2.

An almost-para-hyperhermitian structure $(g,I_{1},I_{2},I_{3})$ on $M$ is
para-hyperkähler if it is para-hyperhermitian and ${\rm d}\omega_{i}=0$, where
$\omega_{i}(\cdot,\cdot)=g(I_{i}\cdot,\cdot)$, $i=1,2,3$.

One can show the following:

######  Proposition 2.15.

Let $(g,K)$ be an almost-para-Hermitian structure on a real manifold $M$. Then
$(g,K)$ is para-Kähler if and only if $K$ is parallel with respect to the
Levi-Civita connection of $g$. Likewise, an almost-para-hyperhermitian
structure $(g,I_{1},I_{2},I_{3})$ on $M$ is para-hyperkähler if and only if
$I_{i}$ is parallel w.r.t. the Levi-Civita connection of $g$ for all
$i=1,2,3$.

###  2.2 Generalized geometry and algebroids

####  2.2.1 Elementary notions

Let $V$ be a $d$-dimensional vector space and $V^{*}$ its dual, and consider
the space $V\oplus V^{*}$. We will denote elements of $V\oplus V^{*}$ by
$(X,\alpha),(Y,\beta)$, etc. There is a natural inner product
$\langle\cdot,\cdot\rangle$ on $V\oplus V^{*}$ given by
$\langle(X,\alpha),(Y,\beta)\rangle=\alpha(Y)+\beta(X)$ 999A factor of $1/2$
is often included in the right hand side of this equation, but for our
purposes this is not important.. Any orthogonal endomorphism, i.e. any elemeny
of $\mathfrak{so}(V\oplus V^{*})$, can be written as

$\left(\begin{matrix}A&\beta\\\ B&-A^{*}\end{matrix}\right)$

where $A\in{\rm End}(V)$, $B$ can be viewed as a 2-form in $\wedge^{2}V^{*}$
and $\beta$ as a bivector in $\wedge^{2}V$. Exponentiating, one gets elements
in ${\rm SO}(V\oplus V^{*})$, that can be separated into

$\exp(B)=\left(\begin{matrix}1&0\\\
B&1\end{matrix}\right),\quad\exp(\beta)=\left(\begin{matrix}1&\beta\\\
0&1\end{matrix}\right),\quad\exp(A)=\left(\begin{matrix}\exp A&0\\\ 0&(\exp
A^{*})^{-1}\end{matrix}\right).$ (2.11)

######  Remark 2.16.

The matrix $\exp(B)$ in (2.11) is called a B-transformation, and it can be
thought of as a shear transformation in the sense that it fixes projections to
$V$ while shearing in the direction of $V^{*}$; see [2, Chapter 2]. This
interpretation and terminology will be particularly well-suited to our
purposes in later sections. Analogous considerations apply to the
$\beta$-transformation given by the matrix $\exp(\beta)$.

While the above is just linear algebra and can be done for any vector space,
in Generalized Geometry one applies this to the case where $V$ is the tangent
space to a point in a $d$-dimensional manifold $M$, and the generalized
tangent bundle is defined as $\mathbb{T}M:=TM\oplus T^{*}M$. The construction
above then describes the pointwise structure of $\mathbb{T}M$. The
differential structure is described by generalizing the notion of Lie bracket
$[\cdot,\cdot]$ of vector fields to a bracket operation on sections of
$\mathbb{T}M$. There are two different bracket operations considered in the
literature, the Courant bracket and the Dorfman bracket. Here we choose the
Dorfman bracket and denote it by $\llbracket\cdot,\cdot\rrbracket$; it is
defined by

$\llbracket(X,\alpha),(Y,\beta)\rrbracket=([X,Y],\pounds_{X}\beta-\pounds_{Y}\alpha+{\rm
d}(i_{Y}\alpha))$

where $\pounds$ denotes the Lie derivative and $i_{Y}$ the interior product.
The Courant bracket is the skew-symmetrization of the Dorfman bracket. While
the ordinary Lie bracket is skew-symmetric and satisfies the Jacobi identity,
the Dorfman bracket is not skew-symmetric, but it satisfies the Jacobi
identity. In turn, the Courant bracket is skew-symmetric, but it does not
satisfy the Jacobi identity.

####  2.2.2 Lie and Courant algebroids

In the literature one frequently encounters real Lie algebroids, but in this
work we need the complexified versions (although the manifold is still real).
This is also used in e.g. [2], see Chapter 3 therein.

######  Definition 2.17.

A (complex) Lie algebroid is a triple $(L,[\cdot,\cdot]_{L},\rho)$ where $L\to
M$ is a complex vector bundle,
$[\cdot,\cdot]_{L}:\Gamma(L)\times\Gamma(L)\to\Gamma(L)$ is a bilinear map,
and $\rho:L\to TM\otimes\mathbb{C}$ is a bundle map called anchor, such that,
for all $X,Y,Z\in\Gamma(L)$ and $f\in C^{\infty}(M)$, the following four
conditions are satisfied101010The anchor $\rho$ extends to a map between
sections $\Gamma(L)\to\Gamma(TM\otimes\mathbb{C})$ that we also denote by
$\rho$.:

  1. 1.

$[X,Y]_{L}=-[Y,X]_{L}$.

  2. 2.

${\rm
Jac}_{L}(X,Y,Z):=[X,[Y,Z]_{L}]_{L}+[Z,[X,Y]_{L}]_{L}+[Y,[Z,X]_{L}]_{L}=0$.

  3. 3.

$\rho([X,Y]_{L})=[\rho(X),\rho(Y)]$, where $[\cdot,\cdot]$ is the Lie bracket
of vector fields.

  4. 4.

$[X,fY]_{L}=(\rho(X)f)Y+f[X,Y]_{L}$.

From the first item we see that $[\cdot,\cdot]_{L}$ is skew-symmetric, and the
second means that the Jacobiator for $[\cdot,\cdot]_{L}$ vanishes, or, in
other words, the Jacobi identity for $[\cdot,\cdot]_{L}$ is satisfied.
Together, these two items imply that $[\cdot,\cdot]_{L}$ is a Lie bracket. The
third item means that $\rho$ is a morphism, and from the fourth item we see
that the anchor and the bracket are subject to the Leibniz rule.

The following are standard examples of Lie algebroids, taken from [2, Chapter
3]:

######  Example 2.18.

The tangent bundle $TM$ gives origin to the tangent Lie algebroid
$(TM,[\cdot,\cdot],\mathbb{I})$, where the bracket is the Lie bracket of
vector fields and the anchor is the identity map.

######  Example 2.19.

If $L\subset TM$ is an involutive distribution, then
$(L,[\cdot,\cdot]_{L},\mathbb{I}_{L})$ is a Lie algebroid, with
$[\cdot,\cdot]_{L}$ and $\mathbb{I}_{L}$ the restrictions of, respectively,
the Lie bracket and the identity map of $TM$ to the subbundle $L$.

We will need some objects that can be naturally constructed from the
structures in a Lie algebroid:

######  Definition 2.20 (Def. 3.7 and 3.8 in [2]).

Let $(L,[\cdot,\cdot]_{L},\rho)$ be a Lie algebroid, let
$\Lambda^{k}=\wedge^{k}L^{*}$ for $k=0,1,2,...$, and let
$\omega\in\Gamma(\Lambda^{k})$ and $X_{0},...,X_{k}\in\Gamma(L)$.

  1. 1.

The Lie algebroid exterior derivative is the map ${\rm
d}^{L}:\Gamma(\Lambda^{k})\to\Gamma(\Lambda^{k+1})$ defined by

$\displaystyle({\rm d}^{L}\omega)(X_{0},...,X_{k})={}$
$\displaystyle\sum_{i}(-1)^{i}\rho(X_{i})\omega(X_{0},...,\hat{X}_{i},...,X_{k})$
$\displaystyle+\sum_{i<j}(-1)^{i+j}\omega([X_{i},X_{j}]_{L},X_{0},...,\hat{X}_{i},...,\hat{X}_{j},...,X_{k})$
(2.12)

where the notation $\hat{X}_{i}$ means that the vector $X_{i}$ is omitted.

  2. 2.

The generalized Lie derivative $\pounds^{L}$ of $\omega$ along $X\in\Gamma(L)$
is the operator defined in terms of ${\rm d}^{L}$ by Cartan’s formula

$\pounds^{L}_{X}\omega=i_{X}{\rm d}^{L}\omega+{\rm d}^{L}i_{X}\omega$ (2.13)

where $i_{X}$ denotes interior product, $i_{X}\omega=\omega(X,...)$.

As examples of the exterior derivative (2.12), if $f\in\Gamma(\Lambda^{0})$
and $\omega\in\Gamma(\Lambda^{1})$ then

$\displaystyle({\rm d}^{L}f)(X)={}$ $\displaystyle\rho(X)f,$ (2.14)
$\displaystyle({\rm d}^{L}\omega)(X,Y)={}$
$\displaystyle\rho(X)\omega(Y)-\rho(Y)\omega(X)-\omega([X,Y]_{L}).$ (2.15)

######  Remark 2.21 (de Rham complexes).

The fact that the bracket of a Lie algebroid satisfies the Jacobi identity,
together with the morphism property of the anchor, imply that the Lie
algebroid exterior derivative ${\rm d}^{L}$ satisfies ${\rm d}^{L}\circ{\rm
d}^{L}=0$. Therefore, $(\Gamma(\Lambda^{\bullet}),{\rm d}^{L})$ is a cochain
complex (see e.g. [2, Chapter 3]). This will be particularly important in
sections 4.2, 4.3.

######  Definition 2.22 (Def. 3.1 in [5]).

A Courant algebroid is a quadruple
$(E,\langle\cdot,\cdot\rangle,\rho,\llbracket\cdot,\cdot\rrbracket)$ where:
$E\to M$ is a vector bundle,
$\langle\cdot,\cdot\rangle:\Gamma(E)\times\Gamma(E)\to C^{\infty}(M)$ is a
non-degenerate symmetric bilinear form, $\rho:E\to TM$ is a bundle map called
anchor, and
$\llbracket\cdot,\cdot\rrbracket:\Gamma(E)\times\Gamma(E)\to\Gamma(E)$ is a
bilinear operation called Dorfman bracket, subject to the following axioms for
all $X,Y,Z\in\Gamma(E)$:

  1. 1.

$\rho(X)\langle Y,Z\rangle=\langle\llbracket X,Y\rrbracket,Z\rangle+\langle
Y,\llbracket X,Z\rrbracket\rangle$

  2. 2.

$\langle\llbracket X,X\rrbracket,Y\rangle=\tfrac{1}{2}\rho(Y)\langle
X,X\rangle$

  3. 3.

$\llbracket X,\llbracket Y,Z\rrbracket\rrbracket=\llbracket\llbracket
X,Y\rrbracket,Z\rrbracket+\llbracket Y,\llbracket X,Z\rrbracket\rrbracket$

In terms of the Dorfman bracket, the axioms in Definition 2.22 have the
following meaning: the first axiom means invariance of the inner product with
respect to the bracket; from the second axiom we see that the bracket is not
skew-symmetric; and the third axiom means that the bracket satisfies the
Jacobi identity.

The definition of a Courant algebroid can be given using two different bracket
operations: the Courant bracket $\llbracket\cdot,\cdot\rrbracket_{\rm Cour}$
or the Dorfman bracket $\llbracket\cdot,\cdot\rrbracket$. The difference is
that one can require the bracket either to be skew-symmetric (Courant) or to
satisfy the Jacobi identity (Dorfman), but not both. The original definition,
introduced in [28], is in terms of the Courant bracket. The two brackets are
related by $\llbracket X,Y\rrbracket_{\rm Cour}=\tfrac{1}{2}(\llbracket
X,Y\rrbracket-\llbracket Y,X\rrbracket)$.

######  Remark 2.23.

If only the first and second axioms in Definition 2.22 are required, the
bracket is said to be metric-compatible. The quadruple
$(E,\langle\cdot,\cdot\rangle,\rho,\llbracket\cdot,\cdot\rrbracket)$ is then
called metric algebroid [4].

From the axioms in Definition 2.22 one can deduce two additional identities:

$\displaystyle\llbracket X,fY\rrbracket={}$
$\displaystyle(\rho(X)f)Y+f\llbracket X,Y\rrbracket,$ (2.16)
$\displaystyle\rho(\llbracket X,Y\rrbracket)={}$
$\displaystyle[\rho(X),\rho(Y)]$ (2.17)

for any $f\in C^{\infty}(M)$. The first identity means that the anchor and the
bracket satisfy the Leibniz rule; the second identity means that $\rho$ is a
morphism of bundles. These identities are sometimes included as part of the
definition of a Courant algebroid, but it is known that they can be obtained
from the axioms: to prove (2.16), compute $\langle\llbracket
X,fY\rrbracket,Z\rangle$ for $Z\in\Gamma(E)$ arbitrary and use the first
axiom; to prove (2.17), compute both sides independently applied to $\langle
Z,W\rangle$ for arbitrary $Z,W\in\Gamma(E)$, and use the first and third
axioms.

As shown in [28], a natural example of a Courant algebroid can be obtained
from a Lie bialgebroid. A related example, which is key for our purposes, is
the following:

######  Proposition 2.24 (See e.g. Example 2.6 in [7]).

Let $(L,[\cdot,\cdot]_{L},\rho_{L})$ be a Lie algebroid, and consider the
vector bundle $L\oplus L^{*}$. Denote sections of $L\oplus L^{*}$ by
$(X,\alpha),(Y,\beta)$ etc., and introduce the following maps:

$\displaystyle\pi_{L\oplus L^{*}}((X,\alpha))=X,$ (2.18)
$\displaystyle\langle(X,\alpha),(Y,\beta)\rangle_{L\oplus
L^{*}}=\alpha(Y)+\beta(X),$ (2.19)
$\displaystyle\llbracket(X,\alpha),(Y,\beta)\rrbracket_{L\oplus
L^{*}}=([X,Y]_{L},\pounds^{L}_{X}\beta-\pounds^{L}_{Y}\alpha+{\rm
d}^{L}(i_{Y}\alpha))$ (2.20)

where ${\rm d}^{L}$ and $\pounds^{L}$ were defined in Definition 2.20. Then
the quadruple

$(L\oplus L^{*},\langle\cdot,\cdot\rangle_{L\oplus L^{*}},\pi_{L\oplus
L^{*}},\llbracket\cdot,\cdot\rrbracket_{L\oplus L^{*}})$ (2.21)

is a Courant algebroid, with inner product $\langle\cdot,\cdot\rangle_{L\oplus
L^{*}}$, anchor $\pi_{L\oplus L^{*}}$ and Dorfman bracket
$\llbracket\cdot,\cdot\rrbracket_{L\oplus L^{*}}$.

This procedure for constructing Courant algebroids from Lie algebroids is
particularly natural in manifolds with a foliation, as described by Vaisman in
[5, Section 3]. The following result is essentially Proposition 3.1 in [5]
(see also [7, Section 3.4]):

######  Proposition 2.25.

Let $(g,K)$ be an almost-para Hermitian structure on a $d$-dimensional real
manifold $M$ (Def. 2.3). Let $L$ and $\tilde{L}$ be the two eigenbundles of
$K$, and assume that $L$ is involutive. Then the quadruple

$(TM\otimes\mathbb{C},g,P,\llbracket\cdot,\cdot\rrbracket)$

is a Courant algebroid, where the inner product $g$ is the metric, the anchor
$P$ is the projector (2.8) to $L$, and the Dorfman bracket is

$\llbracket
X,Y\rrbracket=\varphi^{-1}(\llbracket(x,\tilde{x}_{\flat}),(y,\tilde{y}_{\flat})\rrbracket_{L\oplus
L^{*}}).$

Here, $\varphi$ and $\flat$ are the isomorphisms (2.3) and (2.2),
$\llbracket\cdot,\cdot\rrbracket_{L\oplus L^{*}}$ is the Dorfman bracket
(2.20), and $x=PX$, $\tilde{x}=\tilde{P}X$, $y=PY$, $\tilde{y}=\tilde{P}Y$ are
the decompositions of vectors induced by $K$.

###### Proof.

The tangent bundle splits as $TM\otimes\mathbb{C}=L\oplus\tilde{L}$, where $L$
and $\tilde{L}$ are the eigenbundles of $K$ and $L$ is involutive. From
Example 2.19, the distribution $L$ defines a Lie algebroid
$(L,[\cdot,\cdot]_{L},\mathbb{I}_{L})$. Therefore, applying Proposition 2.24,
the quadruple

$(L\oplus L^{*},\langle\cdot,\cdot\rangle_{L\oplus L^{*}},\pi_{L\oplus
L^{*}},\llbracket\cdot,\cdot\rrbracket_{L\oplus L^{*}})$ (2.22)

is a Courant algebroid, where the anchor, inner product and Dorfman bracket
are given by (2.18), (2.19) and (2.20). Now one simply has to transfer this
structure to $TM\otimes\mathbb{C}$ using the isomorphisms $\varphi$ and
$\flat$ (eqs. (2.3), (2.2)). Any elements $X,Y\in\Gamma(TM\otimes\mathbb{C})$
can be written as $X=x+\tilde{x}$ and $Y=y+\tilde{y}$, where
$x,y,\in\Gamma(L)$ and $\tilde{x},\tilde{y}\in\Gamma(\tilde{L})$. Using these
decompositions, and the fact that $g|_{L}=0=g|_{\tilde{L}}$, a brief
calculation shows that

$g(X,Y)=\langle(x,\tilde{x}_{\flat}),(y,\tilde{y}_{\flat})\rangle_{L\oplus
L^{*}},$

so the metric $g$ plays the role of the inner product in Def. 2.22.
Furthermore, recalling the projector $P$ given in (2.8) and using again
$X=x+\tilde{x}$, we have

$P(x+\tilde{x})=x=\pi_{L\oplus L^{*}}((x,\tilde{x}_{\flat}))$

so $P$ plays the role of the anchor. Finally, let us see the Dorfman bracket.
Since any elements $\alpha,\beta\in\Gamma(L^{*})$ can be written as
$\alpha=\tilde{x}_{\flat}$, $\beta=\tilde{y}_{\flat}$ for some
$\tilde{x},\tilde{y}\in\Gamma(\tilde{L})$, the Dorfman bracket (2.20) is

$\llbracket(x,\tilde{x}_{\flat}),(y,\tilde{y}_{\flat})\rrbracket_{L\oplus
L^{*}}=([x,y]_{L},\pounds^{L}_{x}\tilde{y}_{\flat}-\pounds^{L}_{y}\tilde{x}_{\flat}+{\rm
d}^{L}g(\tilde{x},y))$ (2.23)

where we recall that by definition, $\tilde{x}_{\flat}(y)\equiv
g(\tilde{x},y)$. The right hand side of (2.23) is, of course, an element of
$L\oplus L^{*}$. In order to map this to $L\oplus\tilde{L}$, one uses the
inverse isomorphisms $\sharp:L^{*}\to\tilde{L}$ and $\varphi^{-1}:L\oplus
L^{*}\to L\oplus\tilde{L}$, which map $L\oplus
L^{*}\ni(z,\gamma)\mapsto\varphi^{-1}((z,\gamma))=z+\gamma^{\sharp}\in
L\oplus\tilde{L}$. Replacing $(z,\gamma)$ by the right hand side of (2.23), we
get

$\displaystyle\llbracket x+\tilde{x},y+\tilde{y}\rrbracket\equiv{}$
$\displaystyle\varphi^{-1}(\llbracket(x,\tilde{x}_{\flat}),(y,\tilde{y}_{\flat})\rrbracket_{L\oplus
L^{*}})$ $\displaystyle={}$
$\displaystyle[x,y]+\left(\pounds^{L}_{x}\tilde{y}_{\flat}-\pounds^{L}_{y}\tilde{x}_{\flat}+{\rm
d}^{L}g(\tilde{x},y)\right)^{\sharp}.$ (2.24)

(This is the unnumbered equation above eq. (20) in [7].) ∎

###  2.3 Conformal geometry and the Lee form

As noticed in Remark 2.5, conformal invariance plays an important role in our
work. Recall that, given a manifold $M$ and a metric $g$ on it, the conformal
class of $g$ is defined as $[g]=\\{\Omega^{2}g\;|\;\Omega\in
C^{\infty}(M),\Omega>0\\}$. We refer to the pair $(M,[g])$ as a conformal
structure. The elements in $[g]$ are called conformal representatives.

######  Definition 2.26.

Let $(M,[g])$ be a conformal structure. A Weyl connection is a linear,
torsion-free connection ${}^{\rm w}\nabla$ such that for any conformal
representative $g\in[g]$, it holds ${}^{\rm w}\nabla g=-2f\otimes g$ for some
1-form $f$. We call $f$ the Weyl 1-form.

Under a change of conformal representative $g\to\hat{g}=\Omega^{2}g$, by
definition we must have ${}^{\rm w}\nabla\hat{g}=-2\hat{f}\otimes\hat{g}$ for
some 1-form $\hat{f}$. Replacing $\hat{g}=\Omega^{2}g$ on the left hand side,
we see that $f$ and $\hat{f}$ are related by (from now on we will frequently
use the abstract index notation)

$\hat{f}_{a}=f_{a}-({\rm d}\log\Omega)_{a}.$ (2.25)

If we choose a metric $g\in[g]$ with Levi-Civita connection $\nabla$, the
relationship between ${}^{\rm w}\nabla$ and $\nabla$ is given by a tensor
field $Q:\Gamma(TM)\times\Gamma(TM)\to\Gamma(TM)$. Explicitly, one has

$g({}^{\rm w}\nabla_{X}Y,Z)=g(\nabla_{X}Y,Z)+g(Q(X,Y),Z),$

where

$g(Q(X,Y),Z)=f(X)g(Y,Z)+f(Y)g(X,Z)-f(Z)g(X,Y).$

In index notation this can be expressed as

${}^{\rm w}\nabla_{a}Y^{b}=\nabla_{a}Y^{b}+Q_{ac}{}^{b}Y^{c},$ (2.26)

where

$Q_{ac}{}^{b}=\delta^{b}_{c}f_{a}+\delta^{b}_{a}f_{c}-g^{bd}g_{ac}f_{d}.$
(2.27)

There are interesting relations between the 1-form $f$ associated to a Weyl
connection and certain properties of almost para-Hermitian structures. This
dates back to the original work of H. C. Lee in [29] on almost symplectic
manifolds $(M,a_{ab})$: $M$ is a $d$-dimensional manifold (with $d$ even) and
$a_{ab}=a_{[ab]}$ is a non-degenerate 2-form. The inverse of $a_{ab}$ is
$(a^{-1})^{ab}$, so that $(a^{-1})^{ac}a_{cb}=\delta^{a}{}_{b}$. Lee defines
the “curvature tensor” of $a_{ab}$ as $({\rm d}a)_{abc}$, and the “curvature
vector” as $(a^{-1})^{bc}({\rm d}a)_{abc}$. Two such spaces $(M,a_{ab})$ and
$(\hat{M},\hat{a}_{ab})$ are said to be “conformal” to each other if there
exists a scalar field $\phi$ such that

$\hat{a}_{ab}=\phi a_{ab}.$ (2.28)

Assuming $d>2$ and defining the 1-form

$k_{a}:=\tfrac{1}{(d-2)}(a^{-1})^{bc}({\rm d}a)_{abc},$

it is observed in [29] that $\hat{k}_{a}$ and $k_{a}$ are related by

$\hat{k}_{a}=k_{a}-({\rm d}\log\phi)_{a}$ (2.29)

It follows from this that $({\rm d}k)_{ab}=({\rm d}\hat{k})_{ab}$, so $({\rm
d}k)_{ab}$ is called the “first conformal curvature tensor” by Lee. There is
also a “second conformal curvature tensor”: a 3-form $c_{abc}$ given by

$c_{abc}:=({\rm d}a)_{abc}+(k\wedge a)_{abc}.$ (2.30)

The tensors $c_{abc}$ and $\hat{c}_{abc}$ are related by $\hat{c}_{abc}=\phi
c_{abc}$.

Now, as already noticed, a manifold with an almost-para-Hermitian structure
$(g,K)$ comes automatically with an almost-symplectic form
$\omega_{ab}=g_{cb}K^{c}{}_{a}$. Therefore, we may take
$a_{ab}\equiv\omega_{ab}$ in Lee’s construction. Since
$K^{a}{}_{c}K^{c}{}_{b}=\delta^{a}{}_{b}$, we have
$(\omega^{-1})^{ab}=g^{ac}g^{bd}\omega_{cd}\equiv\omega^{ab}$. Under a
conformal transformation of the metric,
$g_{ab}\to\hat{g}_{ab}=\Omega^{2}g_{ab}$, the new almost-symplectic 2-form is
$\hat{\omega}_{ab}=\Omega^{2}\omega_{ab}$. Thus, we take $\phi=\Omega^{2}$ in
(2.28) and subsequent formulas, and we define:

######  Definition 2.27.

Let $([g],K)$ be an almost (para-)Hermitian conformal structure on a
$d$-dimensional manifold $M$, with $d>2$. For a choice $g\in[g]$, let $\omega$
be the associated almost-symplectic 2-form. We define the Lee form as

$\theta_{a}=\tfrac{1}{2(d-2)}\omega^{bc}({\rm d}\omega)_{abc}.$ (2.31)

If $\nabla_{a}$ is the Levi-Civita connection of $g_{ab}$, a short calculation
using $\omega^{ab}\omega_{ab}=-d$ shows that (2.31) can be written as

$\theta_{a}=\tfrac{1}{(d-2)}K^{b}{}_{c}\nabla_{b}K^{c}{}_{a}.$

This is equivalent to
$\theta_{a}=-\frac{1}{(d-2)}K^{c}{}_{a}\nabla_{b}K^{b}{}_{c}$, or in index-
free notation:

$\theta(X)=-\tfrac{1}{(d-2)}(\delta\omega)(KX)$ (2.32)

for any vector field $X$, where $\delta$ is the codifferential.

From (2.29) and (2.31) we see that the Lee form $\theta_{a}$ has exactly the
transformation property of a Weyl 1-form under conformal transformations of
the metric $g_{ab}\to\hat{g}_{ab}=\Omega^{2}g_{ab}$ (i.e.
$\theta_{a}\to\hat{\theta}_{a}=\theta_{a}-({\rm d}\log\Omega)_{a}$).
Therefore, from the discussion above, we deduce:

######  Proposition 2.28.

An almost (para-)Hermitian structure $(g,K)$ on $M$ induces a natural Weyl
connection in the conformal structure $(M,[g])$, by taking the Lee form as the
Weyl 1-form, $\theta_{a}\equiv f_{a}$.

This motivates the following definition:

######  Definition 2.29.

Let $([g],K)$ be an almost (para-)Hermitian conformal structure on $M$, and
let ${}^{\rm w}\nabla$ be a Weyl connection. We say that ${}^{\rm w}\nabla$
and $K$ are compatible if the Weyl 1-form associated to ${}^{\rm w}\nabla$ and
the Lee form associated to $K$ coincide (in other words, ${}^{\rm w}\nabla$ is
induced by $K$).

######  Remark 2.30.

In the literature, see e.g. [30, Section 4] (also [18]), the usual definition
of compatibility of a Weyl connection ${}^{\rm w}\nabla$ with a Hermitian
structure $K$ is that they must satisfy

${}^{\rm w}\nabla_{a}K^{a}{}_{b}=0.$ (2.33)

Using (2.26)-(2.27), it follows easily that this is true if and only if the
condition of Definition 2.29 holds, namely the Weyl 1-form is equal to the Lee
form. Thus, the two definitions coincide. However, we have chosen Def. 2.29
since it is an immediate consequence of Lee’s construction, while the
geometric meaning of requiring (2.33) as a compatibility condition is not
clear to us.

Finally, from Lee’s results [29] and the definitions given in Section 2.1 we
have:

######  Proposition 2.31 (Lee [29]).

Let $(g,K)$ be an almost (para-)Hermitian structure on a $d$-dimensional
manifold $M$. Then:

  1. 1.

$(g,K)$ is almost (para-)Kähler if and only if the Lee form (2.31) vanishes.

  2. 2.

For $d=4$, $(g,K)$ is locally conformally almost (para-)Kähler if and only if
the Lee form is closed.

  3. 3.

For $d>4$, $(g,K)$ is locally conformally almost (para-)Kähler if and only if
$c_{abc}\equiv 0$ (where $c_{abc}$ is defined in (2.30)).

###### Proof.

The first item follows from the definitions of almost (para-)Kähler (Def. 2.6,
${\rm d}\omega=0$) and the Lee form (2.31). The second and third items are
Theorem 5 in [29]. ∎

##  3 Almost para-Hermitian structures in four dimensions

###  3.1 Self-dual forms

Consider a 4-dimensional, orientable, real manifold $M$, equipped with a
metric. Let $\Lambda^{k}(M)=\wedge^{k}T^{*}M$ be the space of $k$-forms. The
Hodge star operator $*$ can be seen as a map
$*:\Lambda^{2}(M)\to\Lambda^{2}(M)$ satisfying $*^{2}=(-1)^{s}$, where $s$ is
the number of $(-1)$’s appearing in the signature of the metric. Thus, in
Riemannian ($s=0$) and split ($s=2$) signature, $*$ always defines a para-
complex structure in $\Lambda^{2}(M)$, whereas in Lorentz signature ($s=3$) it
defines a complex structure in $\Lambda^{2}(M)$. The eigenvalues of $*$ are
$\pm\sqrt{(-1)^{s}}$. The space of 2-forms is then decomposed as

$\Lambda^{2}(M)=\Lambda^{2}_{+}(M)\oplus\Lambda^{2}_{-}(M)$ (3.1)

where $\Lambda^{2}_{+}(M)$ is the eigenspace corresponding to the eigenvalue
$+\sqrt{(-1)^{s}}$, and $\Lambda^{2}_{-}(M)$ corresponds to
$-\sqrt{(-1)^{s}}$. Elements of $\Lambda^{2}_{+}(M)$ are called self-dual (SD)
2-forms, and elements of $\Lambda^{2}_{-}(M)$ are anti-self-dual (ASD)
2-forms.

######  Theorem 3.1.

Let $M$ be a real, 4-dimensional, orientable manifold with a metric $g$, let
$A\in{\rm Aut}(TM\otimes\mathbb{C})$ and let $W$ be a bilinear map defined by

$W(X,Y)=g(AX,Y)$ (3.2)

for any vectors $X,Y$. Furthermore, let $a$ be a non-vanishing scalar111111For
the case $a=0$, see Remark 3.7 below.. Then $A$ satisfies the conditions

$\displaystyle A^{2}={}$ $\displaystyle a\mathbb{I},$ (3.3) $\displaystyle
g(AX,AY)={}$ $\displaystyle-ag(X,Y)$ (3.4)

if and only if the map $W$ is either a self-dual or an anti-self-dual 2-form.

###### Proof.

We will use the abstract index notation, and the metric and its inverse to
raise and lower indices as convenient. Notice that since $A$ is an
automorphism, the map $W$ in (3.2) is non-degenerate. Appendix B in [31] will
be useful in the following. Since $M$ is orientable, it has a volume 4-form
$\varepsilon_{abcd}$, and the metric allows to normalize it as
$\varepsilon^{abcd}\varepsilon_{abcd}=(-1)^{s}4!$.

Assume first that $A\in{\rm Aut}(TM\otimes\mathbb{C})$ satisfies (3.3) and
(3.4). In index notation this is

$\displaystyle A^{a}{}_{c}A^{c}{}_{b}={}$ $\displaystyle a\delta^{a}{}_{b},$
(3.5) $\displaystyle g_{cd}A^{c}{}_{a}A^{d}{}_{b}={}$ $\displaystyle-ag_{ab}$
(3.6)

with $a\neq 0$. The bilinear map (3.2) is $W_{ab}=g_{bc}A^{c}{}_{a}$.
Multiplying by $A^{b}{}_{d}$ and using (3.6):

$W_{ab}A^{b}{}_{d}=g_{bc}A^{c}{}_{a}A^{b}{}_{d}=-ag_{ad}.$

Contracting now with $A^{d}{}_{c}$ and using (3.5):

$W_{ab}A^{b}{}_{d}A^{d}{}_{c}=aW_{ac}=-ag_{ad}A^{d}{}_{c}=-aW_{ca},$

which shows that $W_{ac}=-W_{ca}$ and therefore $W_{ab}$ is a 2-form. Now we
want to show that it is (A)SD. We can write the volume form as
$\varepsilon_{abcd}=NW_{[ab}W_{cd]}$, where $N$ is determined by the
normalization of $\varepsilon_{abcd}$. Expanding the skew-symmetrization:

$\varepsilon_{abcd}=\tfrac{N}{3}(W_{ab}W_{cd}+W_{ac}W_{db}+W_{ad}W_{bc}).$

Contracting with $W^{cd}$ and using that $W_{ab}=-W_{ba}$:

$\varepsilon_{abcd}W^{cd}=\tfrac{N}{3}(W_{ab}W_{cd}W^{cd}+2W_{ac}W_{db}W^{cd}).$

Now, using that $W_{db}=g_{eb}A^{e}{}_{d}$ and $W^{cd}=g^{ce}g^{df}W_{ef}$, we
have

$W_{db}W^{cd}=A^{c}{}_{d}A^{d}{}_{b}=a\delta^{c}{}_{b}$

from which it also follows that $W_{cd}W^{cd}=-4a$. Therefore

$\varepsilon_{abcd}W^{cd}=-\tfrac{2}{3}NaW_{ab}.$ (3.7)

Using now the normalization of $\varepsilon_{abcd}$:

$(-1)^{s}4!=\varepsilon^{abcd}\varepsilon_{abcd}=N\varepsilon^{abcd}W_{ab}W_{cd}=\tfrac{8}{3}N^{2}a^{2},$

thus

$Na=\pm 3\sqrt{(-1)^{s}}.$

Replacing in (3.7) and using the definition
${}^{*}W_{ab}=\frac{1}{2}\varepsilon_{abcd}W^{cd}$, we get

${}^{*}W_{ab}=\mp\sqrt{(-1)^{s}}\;W_{ab},$

which shows that $W_{ab}$ must be (A)SD.

Suppose now that the map $W$ defined in (3.2) is a 2-form. Writing the volume
form again as $\varepsilon_{abcd}=NW_{[ab}W_{cd]}$, and using the
normalization $\varepsilon^{abcd}\varepsilon_{abcd}=(-1)^{s}4!$, a short
calculation reveals that $(-1)^{s}4!=2N{}^{*}W^{ab}W_{ab}$, thus

$N=12(-1)^{s}({}^{*}W^{ab}W_{ab})^{-1}.$ (3.8)

Now, from formula (B.2.13) in [31] we have

$(-1)^{s}3!\delta^{a}{}_{b}=\varepsilon^{cdea}\varepsilon_{cdeb}$

Replacing $\varepsilon_{cdeb}=NW_{[cd}W_{eb]}$ and the expression for $N$, we
get the general identity

$\delta^{a}{}_{b}=-4({}^{*}W^{de}W_{de})^{-1}\;{}^{*}W^{ac}W_{cb}.$

Assume now that $W_{ab}$ is (A)SD:

${}^{*}W_{ab}=\epsilon W_{ab},$

where $\epsilon=\pm\sqrt{(-1)^{s}}$, depending on the signature of $g$ and on
whether we consider SD or ASD forms. In any case, the above identity becomes

$\delta^{a}{}_{b}=-4(W^{de}W_{de})^{-1}\;W^{ac}W_{cb},$ (3.9)

which shows that $A^{a}{}_{b}=g^{ac}W_{bc}$ satisfies (3.5), with
$a=-\frac{1}{4}W_{ef}W^{ef}$. Finally, to see that (3.4) holds, we compute

$g_{cd}A^{c}{}_{a}A^{d}{}_{b}=-W_{da}A^{d}{}_{b}=-ag_{ab}$

where the second equality is deduced form (3.5) by contracting with $g_{da}$.
∎

######  Corollary 3.2.

Let $A\in{\rm Aut}(TM\otimes\mathbb{C})$ and let $W$ be defined by (3.2).
Define a scalar field $\phi$, a bilinear map $\omega$, and a map $K\in{\rm
Aut}(TM\otimes\mathbb{C})$ by, respectively,

$\displaystyle\phi={}$ $\displaystyle(-\tfrac{1}{4}W_{ab}W^{ab})^{1/2},$
(3.10) $\displaystyle\omega={}$ $\displaystyle\phi^{-1}W,$ (3.11)
$\displaystyle g(KX,Y)={}$ $\displaystyle\omega(X,Y)$ (3.12)

for any vectors $X,Y$. Then $K$ satisfies the conditions

$\displaystyle K^{2}={}$ $\displaystyle\mathbb{I},$ (3.13) $\displaystyle
g(KX,KY)={}$ $\displaystyle-g(X,Y)$ (3.14)

if and only if $\omega$ is either a self-dual or an anti-self-dual 2-form.

######  Remark 3.3.

Suppose that $\omega$ in (3.11) is an (A)SD 2-form. Then the map $K$ defined
in (3.12) satisfies equations (3.13) and (3.14), and its eigenvalues are
$+1,+1,-1,-1$ so the two eigenbundles of $K$ have the same rank. According to
definitions 2.1 and 2.3, we say that $(g,K)$ is an almost para-Hermitian
structure. Notice that $K$ is not necessarily real-valued, this is the reason
why we use $TM\otimes\mathbb{C}$ instead of $TM$ in Def. 2.3. The reality of
$K$ depends on the signature of $g$:

  1. 1.

Riemannian signature $(++++)$: (A)SD forms can be chosen to be real, but
$\phi$ in (3.10) is always purely imaginary, so $K$ is purely imaginary or
complex-valued.

  2. 2.

Split signature $(++--)$: (A)SD forms can be chosen to be real, but $\phi$ can
be real or complex, so $K$ can be real- or complex-valued.

  3. 3.

Lorentz signature $(+---)$: (A)SD forms are necessarily complex, so $K$ is
complex-valued.

###  3.2 Spaces of almost para-Hermitian structures

The results of Theorem 3.1 and its Corollary 3.2 show us that we can think of
maps satisfying (3.13) and (3.14) equivalently in terms of non-degenerate
(A)SD 2-forms. Here we will give a convenient parametrization of the space of
such maps at a point $x\in M$, that is of the space

$P_{x}:=\\{K\in{\rm
Aut}(T_{x}M\otimes\mathbb{C})\;|\;K^{2}=\mathbb{I}\;\;\text{and}\;\;g(K\cdot,K\cdot)=-g(\cdot,\cdot)\\}.$
(3.15)

As emphasized in Remark 3.3, we consider $K\in{\rm
Aut}(T_{x}M\otimes\mathbb{C})$ since a tensor satisfying (3.13)-(3.14) is not
necessarily real. Imposing reality conditions changes the structure of
$P_{x}$, and this will depend on the metric signature. Our results in this
subsection can be summarized in the following form:

######  Theorem 3.4.

Let $M$ be a real, orientable 4-manifold with a metric $g$, and let $x\in M$.
The space (3.15) has two connected components, and depending on the signature
of $g$, each component can be parametrized as follows:

  1. 1.

Riemannian signature: the elements in the set (3.15) are not real but can be
chosen to be purely imaginary, and each of the two components in the space of
such maps is

$P^{(R)}_{x}\cong\mathbb{CP}^{1}\cong S^{2}$ (3.16)

where $\mathbb{CP}^{1}$ is the complex projective line, and $S^{2}$ is the
unit 2-sphere.

  2. 2.

Split signature: the elements in the set (3.15) can be chosen to be real, and
each of the two components in the space of such maps is

$P^{(S)}_{x}\cong(\mathbb{RP}^{1}\times\mathbb{RP}^{1})\backslash\mathbb{RP}^{1}\cong
H^{1}$ (3.17)

where $\mathbb{RP}^{1}$ is the real projective line, and $H^{1}$ is a
hyperboloid of one sheet.

  3. 3.

Lorentz signature: the elements in the set (3.15) are necessarily complex, and
each of the two components in the space of such maps is

$P^{(L)}_{x}\cong(\mathbb{CP}^{1}\times\mathbb{CP}^{1})\backslash\mathbb{CP}^{1}\cong\mathbb{C}S^{2}$
(3.18)

where $\mathbb{C}S^{2}$ is the complexified 2-sphere.

######  Remark 3.5.

The result (3.16) is analogous to the result about almost-complex structures
in Riemannian geometry, which is already known in the literature (see Remark
3.8 below).

We will first analyse the general structure of (3.15) and then study the
different signatures separately.

First of all, the two connected components in $P_{x}$ refer to the fact that,
if $K\in P_{x}$, then the 2-form $\omega(\cdot,\cdot)=g(K\cdot,\cdot)$ can be
self-dual or anti-self-dual. We will focus on only one of the components of
$P_{x}$; the analysis for the other is analogous.

The parametrization of $P_{x}$ involves the use of spinors. We refer to
Appendix A for a brief review of spinors in four dimensions, as well as for
notation and conventions (see also Appendix B for spinor fields). In
particular, we raise and lower spinor indices with the symplectic forms
$\epsilon_{AB}$, $\epsilon_{A^{\prime}B^{\prime}}$ and their inverses. In any
signature, the spaces of SD and ASD 2-forms have the spinor decomposition

$\Lambda^{2}_{+}(M)\cong\mathbb{S}^{\prime*}\odot\mathbb{S}^{\prime*},\qquad\Lambda^{2}_{-}(M)\cong\mathbb{S}^{*}\odot\mathbb{S}^{*},$
(3.19)

where $\mathbb{S}^{\prime*}$ and $\mathbb{S}^{*}$ are the (dual) primed and
unprimed spinor bundles (see the beginning of Appendix B). In indices, this
means that if $W^{+}_{ab}$ is SD, and $W^{-}_{ab}$ is ASD, then there exist
symmetric spinors $\varphi_{AB}=\varphi_{(AB)}$ and
$\psi_{A^{\prime}B^{\prime}}=\psi_{(A^{\prime}B^{\prime})}$ such that

$\displaystyle W^{+}_{ab}={}$
$\displaystyle\psi_{A^{\prime}B^{\prime}}\epsilon_{AB},$ (3.20) $\displaystyle
W^{-}_{ab}={}$ $\displaystyle\varphi_{AB}\epsilon_{A^{\prime}B^{\prime}}.$
(3.21)

These 2-forms are non-degenerate if and only if
$\psi_{A^{\prime}B^{\prime}}\psi^{A^{\prime}B^{\prime}}\neq 0$ and
$\varphi_{AB}\varphi^{AB}\neq 0$ respectively. In the following, since we will
focus on only one of the two components of $P_{x}$, we are free to choose to
work with either SD or ASD forms. For concreteness, from now on we will focus
on the ASD case (the SD case being entirely analogous), that is on forms like
(3.21), and we omit the superscript “$-$” since it will not be needed.

######  Remark 3.6 (Convention).

If a tensor/spinor depends non-trivially only on unprimed spinors, we will
often refer to it as having “negative chirality”, while if it depends only on
primed spinors we will say that it has “positive chirality”. For example, we
say that $\varphi_{A}$ or
$T^{a}{}_{b}=\tau^{A}{}_{B}\delta^{A^{\prime}}{}_{B^{\prime}}$ have negative
chirality, and that $U_{aB}=\sigma_{A^{\prime}}\epsilon_{AB}$ or
$V_{ab}=\rho_{A^{\prime}B^{\prime}}\epsilon_{AB}$ have positive chirality.
More formally, the notions of negative and positive chirality refer to the two
kinds of representations of the spin group (in the notation of appendix A,
negative chirality corresponds to representations $(n,0)$, and positive
chirality to $(0,m)$).

######  Remark 3.7 (Null-Kähler).

We notice that the proof of Theorem 3.1 is particularly simple if one uses the
isomorphisms (3.1), (3.19) and the explicit expressions (3.20) and (3.21).
Actually, these decompositions also allow to deal with the case $a=0$ in
(3.3). Suppose that $N\in{\rm End}(TN\otimes\mathbb{C})$ satisfies $N^{2}=0$
and $g(NX,Y)+g(X,NY)=0$ for all $X,Y$. Then using (3.1), (3.19)–(3.21), it is
straightforward to show that any $N$ satisfying these conditions must be of
the form

$N^{a}{}_{b}=\alpha^{A}\alpha_{B}\delta^{A^{\prime}}{}_{B^{\prime}}\qquad\text{or
}\qquad N^{a}{}_{b}=\mu^{A^{\prime}}\mu_{B^{\prime}}\delta^{A}{}_{B}$ (3.22)

for some spinors $\alpha^{A}$, $\mu^{A^{\prime}}$, depending on whether the
2-form $g(N\cdot,\cdot)$ is SD or ASD. We may refer to the pair $(g,N)$ as an
‘almost-null Kähler structure’. It follows that the space of such maps at a
point $x\in M$ is simply the space of spinors at $x$. If the spinor
$\alpha^{A}$ (or $\mu^{A^{\prime}}$) is parallel under the Levi-Civita
connection of $g$, then this is called a Null-Kähler structure, see [32] and
[33, Section 10.2.3]. (If $N$ is real, then the signature of $g$ must be
split, however here we allow $N$ to be complex.)

Let $W_{ab}$ be an ASD 2-form at $x\in M$, with the spinor representation
(3.21), and let $(\xi_{A},\eta_{A})$ be a basis of $\mathbb{S}^{*}|_{x}$,
where $\xi_{A}\eta^{A}=\chi\neq 0$. Then $\varphi_{AB}$ can expanded as (see
Appendix A)

$\varphi_{AB}=\chi^{-2}[\varphi_{2}\xi_{A}\xi_{B}-\varphi_{1}(\xi_{A}\eta_{B}+\eta_{A}\xi_{B})+\varphi_{0}\eta_{A}\eta_{B}],$
(3.23)

where $\varphi_{0}=\varphi_{AB}\xi^{A}\xi^{B}$,
$\varphi_{1}=\varphi_{AB}\xi^{A}\eta^{B}$ and
$\varphi_{2}=\varphi_{AB}\eta^{A}\eta^{B}$. The scalar (3.10) is then
$\phi=(-\frac{1}{2}\varphi_{AB}\varphi^{AB})^{1/2}=\chi^{-1}(\varphi^{2}_{1}-\varphi_{0}\varphi_{2})^{1/2}$.
Thus, the tensor $K^{a}{}_{b}=\phi^{-1}g^{ac}W_{bc}$ satisfies (3.13) and
(3.14) and is explicitly

$K^{a}{}_{b}=\frac{1}{\chi\sqrt{\varphi^{2}_{1}-\varphi_{0}\varphi_{2}}}\left[\varphi_{2}\xi^{A}\xi_{B}-\varphi_{1}(\xi^{A}\eta_{B}+\eta^{A}\xi_{B})+\varphi_{0}\eta^{A}\eta_{B}\right]\delta^{A^{\prime}}{}_{B^{\prime}}.$
(3.24)

This can be written in a more convenient form by expressing $\varphi_{AB}$ in
terms of its principal spinors, see Appendix A: since
$\varphi_{AB}\varphi^{AB}\neq 0$, there exist two non-proportional spinors
$\alpha_{A},\beta_{A}$ such that

$\varphi_{AB}=\alpha_{A}\beta_{B}+\alpha_{B}\beta_{A}.$ (3.25)

Independently of signature, the spinors $\alpha_{A}$ and $\beta_{A}$ are in
general complex, since they are obtained by finding the roots of a second
order polynomial; see around eq. (A.10). However, whereas in Lorentzian and
Riemmanian signature they are necessarily complex, in split signature the
combination in the right hand side of (3.25) is real.

The decomposition (3.25) implies that $\phi=\alpha_{C}\beta^{C}$, therefore

$K^{a}{}_{b}=(\alpha_{C}\beta^{C})^{-1}(\alpha^{A}\beta_{B}+\beta^{A}\alpha_{B})\delta^{A^{\prime}}{}_{B^{\prime}}.$
(3.26)

From Corollary 3.2 we see that any map satisfying conditions (3.13) and (3.14)
in four dimensions has the form (3.26) for some spinors $\alpha_{A},\beta_{A}$
(the only other possibility being choosing positive instead of negative
chirality). However, (3.26) does not uniquely fix these spinors: we see from
this equation that $K$ is invariant under

$\alpha_{A}\to\lambda\alpha_{A},\qquad\beta_{A}\to\mu\beta_{A}$ (3.27)

for any non-vanishing scalars $\lambda$ and $\mu$. Now, the space of spinors
at a point (the fiber of the spin bundle) is $\mathbb{C}^{2}$ in signature
$(+---)$ and $(++++)$, and $\mathbb{R}^{2}$ in signature $(++--)$. Since any
element of the set (3.15) (with negative chirality) can be put in the form
(3.26), and since the spinors are subject to the projective equivalence
(3.27), we deduce that a priori each component of the set $P_{x}$ can be
parametrized by the product of two projective spaces, that is
$\mathbb{CP}^{1}\times\mathbb{CP}^{1}$. (Notice that although in the split
signature case the projective spin space is $\mathbb{RP}^{1}$, if we allow the
rescalings in (3.27) to be complex then we get the complexification of
$\mathbb{RP}^{1}$, which is $\mathbb{CP}^{1}$.) But since the spinors
$\alpha_{A}$ and $\beta_{A}$ are not allowed to be proportional, we must
remove the “diagonal” from $\mathbb{CP}^{1}\times\mathbb{CP}^{1}$, so we get

$P_{x}\cong\\{(\alpha,\beta)\in\mathbb{CP}^{1}\times\mathbb{CP}^{1}\;|\;\alpha\neq\beta\\}\cong(\mathbb{CP}^{1}\times\mathbb{CP}^{1})\backslash\mathbb{CP}^{1}$
(3.28)

where we are now abusing notation and using $P_{x}$ for one of the two
components of (3.15). Requiring reality conditions for (3.26), however,
imposes additional restrictions on the spinors $\alpha_{A},\beta_{A}$. We will
analyse separately the different signatures.

####  3.2.1 Riemannian signature

In this case we can require the ASD 2-form $W_{ab}$ to be real. To analyse
this in more detail, we need to introduce the Euclidean spinor conjugation.
This is an involution $\dagger$ on the spin spaces, i.e.
$\mathbb{S}\to\mathbb{S}$ and $\mathbb{S}^{\prime}\to\mathbb{S}^{\prime}$ (so
it does not interchange the chirality as in the Lorentzian case), and if we
write a spinor $\varphi_{A}$ in components as $\varphi_{A}=(a,b)$, then
$\varphi^{\dagger}_{A}=(-\bar{b},\bar{a})$. It follows that $\dagger^{2}=-1$.
Since
$\epsilon^{\dagger}_{A^{\prime}B^{\prime}}=\epsilon_{A^{\prime}B^{\prime}}$,
we get that $W_{ab}$ is real if and only if $\beta_{A}=i\alpha^{\dagger}_{A}$,
so
$W_{ab}=i(\alpha_{A}\alpha^{\dagger}_{B}+\alpha^{\dagger}_{A}\alpha_{B})\epsilon_{A^{\prime}B^{\prime}}$.
Furthermore, $(\alpha_{C}\beta^{C})^{-1}=-i/\|\alpha\|^{2}$, where
$\|\alpha\|^{2}:=\epsilon_{AB}\alpha^{A}\alpha^{\dagger B}$. Therefore, with
the additional requirement of real ASD 2-forms, the map (3.26) becomes in the
Riemannian case

$K^{a}{}_{b}=\frac{1}{\|\alpha\|^{2}}(\alpha^{A}\alpha^{\dagger}_{B}+\alpha^{\dagger
A}\alpha_{B})\delta^{A^{\prime}}{}_{B^{\prime}},$ (3.29)

so we see explicitly that $K$ is parametrized by only one projective spinor.
Each of the two components of the set of “Riemannian para-Hermitian
structures” at $x\in M$ is therefore

$P^{(R)}_{x}\cong\mathbb{CP}^{1}.$ (3.30)

The complex projective line $\mathbb{CP}^{1}$ is the Riemann sphere, and is
topologically $\mathbb{CP}^{1}\cong S^{2}$. Thus, there is a 2-sphere of
Riemannian para-Hermitian structures at any point.

######  Remark 3.8.

From (3.29) we see immediately that $K$ is purely imaginary, since
$\dagger^{2}=-1$. Therefore, the map

$J:=iK$

is real-valued and satisfies $J^{2}=-\mathbb{I}$, so it defines a complex
structure in each tangent space, and the space of complex (Hermitian)
structures at any point is again parametrized by $\mathbb{CP}^{1}$. In this
way we recover the Riemannian results for almost-complex structures given in
[34, Section 9.1] and [22] (see eq. (9.1.20) in the first reference and page
429 in the second).

The 2-sphere (3.30) of Riemannian para-Hermitian structures can also be seen
by using non-projective spinors as follows. From the general form (3.24),
choosing different values for $\varphi_{0},\varphi_{1},\varphi_{2}$ we deduce
the following particular cases

$\displaystyle(K_{1})^{a}{}_{b}={}$
$\displaystyle\chi^{-1}(\xi^{A}\eta_{B}+\eta^{A}\xi_{B})\delta^{A^{\prime}}{}_{B^{\prime}},$
(3.31) $\displaystyle(K_{2})^{a}{}_{b}={}$ $\displaystyle
i\chi^{-1}(\xi^{A}\xi_{B}+\eta^{A}\eta_{B})\delta^{A^{\prime}}{}_{B^{\prime}},$
(3.32) $\displaystyle(K_{3})^{a}{}_{b}={}$
$\displaystyle\chi^{-1}(\xi^{A}\xi_{B}-\eta^{A}\eta_{B})\delta^{A^{\prime}}{}_{B^{\prime}}$
(3.33)

(where reality conditions imply $\eta_{A}=i\xi^{\dagger}_{A}$). Any $K$ can be
expressed as a linear combination of these tensors121212Note that (3.24) is
equivalent to
$K^{a}{}_{b}=\frac{(-\varphi_{1})}{\sqrt{\varphi_{1}^{2}-\varphi_{0}\varphi_{2}}}(K_{1}){}^{a}{}_{b}+\frac{(-i)(\varphi_{2}+\varphi_{0})}{2\sqrt{\varphi_{1}^{2}-\varphi_{0}\varphi_{2}}}(K_{2}){}^{a}{}_{b}+\frac{(\varphi_{2}-\varphi_{0})}{2\sqrt{\varphi_{1}^{2}-\varphi_{0}\varphi_{2}}}(K_{3}){}^{a}{}_{b}$
(3.34) . It is straightforward to check the following identities:

$(K_{1})^{2}=(K_{2})^{2}=(K_{3})^{2}=\mathbb{I},\qquad
K_{i}K_{j}=-K_{j}K_{i}\;\;\text{for }\;i\neq j.$ (3.35)

Using these properties, a short calculation shows that

$(aK_{1}+bK_{2}+cK_{3})^{2}=(a^{2}+b^{2}+c^{2})\mathbb{I},$ (3.36)

where $a,b,c$ are real numbers. Thus, as long as $(a,b,c)\in S^{2}$, the
combination $\mathcal{K}_{(a,b,c)}=aK_{1}+bK_{2}+cK_{3}$ is a new para-
Hermitian structure parametrized by an element $(a,b,c)$ of $S^{2}$, so there
is a 2-sphere of such structures.

Since the $K_{i}$ are all purely imaginary, we can define the real maps
$J_{1}=iK_{1}$, $J_{2}=iK_{2}$ and $J_{3}=iK_{3}$; then it follows that these
tensors satisfy

$(J_{i})^{2}=-\mathbb{I},\qquad J_{1}J_{2}=J_{3},\quad J_{2}J_{3}=J_{1},\quad
J_{3}J_{1}=J_{2},$ (3.37)

so the triple $(J_{1},J_{2},J_{3})$ defines an almost-hyperhermitian structure
on $M$ (Def. 2.7).

####  3.2.2 Split signature

In signature $(++--)$ spinors can be chosen to be real. Therefore, the reality
of the map (3.24) depends on the sign of
$\varphi^{2}_{1}-\varphi_{0}\varphi_{2}$. This is equivalent to the reality
conditions for the principal spinors $\alpha_{A}$ and $\beta_{A}$ in (3.25):
if $\varphi^{2}_{1}-\varphi_{0}\varphi_{2}>0$ then $\alpha_{A}$ and
$\beta_{A}$ are real, whereas if $\varphi^{2}_{1}-\varphi_{0}\varphi_{2}<0$
they are complex; see around equation (A.10). Thus, requiring (3.24) to be
real, and hence an ordinary para-Hermitian structure, is equivalent to
requiring $\alpha_{A}$ and $\beta_{A}$ in (3.26) to be real. We can thus
restrict to real rescalings in (3.27), so the space of para-Hermitian
structures at $x$ (of a definite chirality) is

$P^{(S)}_{x}\cong\\{(\alpha,\beta)\in\mathbb{RP}^{1}\times\mathbb{RP}^{1}\;|\;\alpha\neq\beta\\}\cong(\mathbb{RP}^{1}\times\mathbb{RP}^{1})\backslash\mathbb{RP}^{1}.$
(3.38)

The real projective line $\mathbb{RP}^{1}$ is topologically
$\mathbb{RP}^{1}\cong S^{1}$, so the space of para-Hermitian structures at any
point in split signature is the torus $S^{1}\times S^{1}$ with the line
$\alpha=\beta$ removed. To visualize this structure, we consider again the
three cases (3.31), (3.32), (3.33), where now $\xi_{A}$ and $\eta_{A}$ are
real spinors. We see that $K_{1}$ and $K_{3}$ are real, but $K_{2}$ is purely
imaginary, so we define the real map $K^{\prime}_{2}=iK_{2}$. These tensors
satisfy the following:

$(K_{1})^{2}=(K_{3})^{2}=\mathbb{I}=-(K^{\prime}_{2})^{2},\qquad
K_{1}K^{\prime}_{2}=-K^{\prime}_{2}K_{1}=K_{3}.$ (3.39)

From these relations and definition 2.7 we see that the triple
$(K_{1},K^{\prime}_{2},K_{3})$ defines an almost-para-hyperhermitian
structure. If $a,b,c$ are real numbers, we now find

$(aK_{1}+bK^{\prime}_{2}+cK_{3})^{2}=(a^{2}-b^{2}+c^{2})\mathbb{I},$ (3.40)

thus the condition for $aK_{1}+bK^{\prime}_{2}+cK_{3}$ to be an almost-para-
Hermitian structure is $a^{2}-b^{2}+c^{2}=1$, which describes a hyperboloid.
Therefore, the space (3.38) is in this case a hyperboloid of one sheet. One
can also convince oneself of this by drawing $(S^{1}\times S^{1})\backslash
S^{1}$ (where we think of the removed $S^{1}$ as a “diagonal” circle —i.e.
non-contractible).

######  Remark 3.9 (Almost-complex structures).

An analysis analogous to the one that leads to (3.28) can be carried out to
find the set of almost complex structures compatible with $g$ in split
signature (i.e. maps satisfying $J^{2}=-\mathbb{I}$ and
$g(J\cdot,J\cdot)=+g(\cdot,\cdot)$). Requiring $J$ to be real, it is
straightforward to see (cf. around eq. (A.10)) that the principal spinors must
be complex. Since they are complex conjugates of each other, i.e.
$\beta_{A}=\bar{\alpha}_{A}$, $J$ is parametrized by only one spinor
$\alpha_{A}$, and one finds

$J^{a}{}_{b}=-\frac{i}{(\alpha_{C}\bar{\alpha}^{C})}(\alpha^{A}\bar{\alpha}_{B}+\bar{\alpha}^{A}\alpha_{B})\delta^{A^{\prime}}{}_{B^{\prime}}$
(3.41)

(note that this is real since
$\overline{(\alpha_{C}\bar{\alpha}^{C})}=-(\alpha_{C}\bar{\alpha}^{C})$). We
have the freedom $\alpha_{A}\to\lambda\alpha_{A}$ with
$\lambda\in\mathbb{C}^{\times}$, and we must exclude the set
$\bar{\alpha}_{A}\propto\alpha_{A}$, so it follows that the set of Hermitian
structures at a point is $\mathbb{CP}^{1}\backslash\mathbb{RP}^{1}$.
Topologically this is $S^{2}\backslash S^{1}$, which is homemorphic to a
hyperboloid of two sheets. This can also be seen by simply requiring
$a^{2}-b^{2}+c^{2}=-1$ in (3.40). We will invoke this result in section 5.2.1
below.

####  3.2.3 Lorentz signature

As observed in Remark 3.3, in this case the elements of the set (3.15) are
necessarily complex-valued, and $P^{(L)}_{x}$ is simply given by (3.28).
Topologically, this is the product $S^{2}\times S^{2}$ of two 2-spheres with
the “diagonal” $\alpha=\beta$ removed. We can obtain an alternative
description of this structure by considering again the three maps (3.31),
(3.32), (3.33). These tensors satisfy (3.35), but now they are complex-valued.
Considering $\mathcal{K}_{(a,b,c)}=aK_{1}+bK_{2}+cK_{3}$, equation (3.36)
still holds, but now $a,b,c$ are complex numbers. The condition for
$\mathcal{K}_{(a,b,c)}$ to be a “Lorentzian para-Hermitian structure” is thus

$a^{2}+b^{2}+c^{2}=1,\qquad a,b,c\in\mathbb{C}$ (3.42)

which is the complexification of the 2-sphere, $\mathbb{C}S^{2}$. (In this
case it is perhaps not so easy to argue intuitively that the set $(S^{2}\times
S^{2})\backslash S^{2}$ is a complex 2-sphere.)

Each of the two components in the set (3.15), without any reality conditions,
is therefore a complexified sphere $\mathbb{C}S^{2}$. Notice that this is true
in any signature, and also for complex 4-manifolds.

###  3.3 Gauge freedom and a covariant formalism

From Remark 2.5 and the results of section 3.2, we see that fixing a
(para-)complex structure $K$ at a point $x\in M$ is equivalent to fixing
(ignoring reality conditions):

  1. 1.

a conformal structure $[g]$, and

  2. 2.

two (different) points on the Riemann sphere at $x$.

Over an open neighbourhood in $M$, instead of considering point objects we
consider fields, i.e. smooth sections of the appropriate bundles. In practice
we will work with representatives, i.e. with some metric $g_{ab}$, or its
spinor equivalents $\epsilon_{AB}$ and $\epsilon_{A^{\prime}B^{\prime}}$, and
with two non-proportional spinors $\alpha_{A},\beta_{A}$ (which in split
signature are real, while in Riemannian signature one has
$\beta_{A}=i\alpha^{\dagger}_{A}$). But since the only given data is $K$, the
framework should not depend on the representatives chosen. Here we give the
essential points of a formalism that takes into account this issue, namely
that is covariant under the transformations associated to this “gauge
freedom”. A more detailed description of this framework is given in Appendix
B.

We have seen that the explicit expression of $K$ in terms of the
representative spinors is (3.26), that is:

$K^{a}{}_{b}=\frac{1}{(\epsilon^{CD}\alpha_{C}\beta_{D})}\epsilon^{AE}(\alpha_{E}\beta_{B}+\beta_{E}\alpha_{B})\delta^{A^{\prime}}{}_{B^{\prime}}.$
(3.43)

We are free to assign conformal weights to the spinors $\alpha_{A},\beta_{A}$,
i.e. we can impose these spinors to change under a conformal transformation of
the metric as

$\alpha_{A}\to\Omega^{w_{0}+1}\alpha_{A},\qquad\beta_{A}\to\Omega^{w_{1}+1}\beta_{A}$
(3.44)

for some real numbers $w_{0},w_{1}$. Recalling the conformal behavior

$\epsilon_{AB}\to\Omega\epsilon_{AB},\qquad\epsilon_{A^{\prime}B^{\prime}}\to\Omega\epsilon_{A^{\prime}B^{\prime}}.$
(3.45)

we see explicitly from (3.43) that $K$ is invariant under conformal
rescalings. (The conformal weights in (3.44) are chosen so that
$\alpha^{A}\to\Omega^{w_{0}}\alpha^{A}$,
$\beta^{A}\to\Omega^{w_{1}}\beta^{A}$.)

The “gauge freedom” is given by conformal (3.45) and projective (3.27)
rescalings. In Lorentzian and Riemannian signature, these transformations
define the group131313$\mathbb{R}^{+}$ denotes the multiplicative group of
positive real numbers, and $\mathbb{C}^{\times}$ ($\mathbb{R}^{\times}$) is
the multiplicative group of complex (real) numbers.
$G_{o}=\mathbb{C}^{\times}\times\mathbb{C}^{\times}\times\mathbb{R}^{+}$,
where $\mathbb{R}^{+}$ corresponds to conformal rescalings (3.45) and
$\mathbb{C}^{\times}\times\mathbb{C}^{\times}$ to projective rescalings
(3.27). In split signature one replaces
$\mathbb{C}^{\times}\times\mathbb{C}^{\times}$ by
$\mathbb{R}^{\times}\times\mathbb{R}^{\times}$.

######  Definition 3.10.

We say that a spinor/tensor/scalar field
$\varphi^{B...B^{\prime}...}_{C...C^{\prime}...}$ is weighted and has type
$(r,r^{\prime};w)$, where $r,r^{\prime},w$ are real numbers, if under the
transformations (3.45)-(3.27) it changes as

$\varphi^{B...B^{\prime}...}_{C...C^{\prime}...}\to\Omega^{w}\lambda^{r}\mu^{r^{\prime}}\varphi^{B...B^{\prime}...}_{C...C^{\prime}...}.$
(3.46)

This transformation law defines a representation of the group $G_{o}$, see
Appendix B. Examples of weighted fields are $\alpha^{A},\beta^{A}$ and
$\phi=\alpha_{A}\beta^{A}$; we give the specific weights for them in table 1.

| $w$ | $r$ | $r^{\prime}$
---|---|---|---
$\beta^{A}$ | $w_{1}$ | $0$ | $1$
$\beta_{A}$ | $w_{1}+1$ | $0$ | $1$
$\alpha^{A}$ | $w_{0}$ | $1$ | $0$
$\alpha_{A}$ | $w_{0}+1$ | $1$ | $0$
$\phi$ | $w_{0}+w_{1}+1$ | $1$ | $1$
Table 1: Weights of the fields associated to an almost-para Hermitian
structure.

A non-trivial issue is how to define a derivative operator that is covariant
under all the transformations involved, since, to begin with, in a conformal
structure one does not have the usual Levi-Civita connection. The first point
is to apply Lee’s construction, seen in Section 2.3, to use the natural Weyl
connection ${}^{\rm w}\nabla_{a}$ induced by an almost-para Hermitian
conformal structure $([g],K)$ (Proposition 2.28). Then, after applying the
appropriate machinery, one arrives at:

######  Definition 3.11.

Let $(M,[g],K)$ be a real, 4-dimensional, almost-para-Hermitian conformal
structure, and let $\alpha_{A},\beta_{A}$ be the spinor fields representing
$K$ as in (3.26), with $\epsilon^{AB}\alpha_{A}\beta_{B}=\phi\neq 0$. Let
${}^{\rm w}\nabla_{AA^{\prime}}$ and $f_{AA^{\prime}}$ be the Weyl connection
and Lee form induced by $K$, and let
$\varphi^{B...B^{\prime}...}_{C...C^{\prime}...}$ be a spinor field of type
$(r,r^{\prime};w)$. We define the covariant derivative
$\mathcal{C}_{AA^{\prime}}$ by

$\mathcal{C}_{AA^{\prime}}\varphi^{B...B^{\prime}...}_{C...C^{\prime}...}={}^{\rm
w}\nabla_{AA^{\prime}}\varphi^{B...B^{\prime}...}_{C...C^{\prime}...}+(wf_{AA^{\prime}}+rL_{AA^{\prime}}+r^{\prime}M_{AA^{\prime}})\varphi^{B...B^{\prime}...}_{C...C^{\prime}...}$
(3.47)

where

$\displaystyle L_{AA^{\prime}}:={}$
$\displaystyle\phi^{-1}\beta_{B}\nabla_{AA^{\prime}}\alpha^{B}+\phi^{-1}\beta_{A}f_{A^{\prime}B}\alpha^{B}-w_{0}f_{AA^{\prime}},$
(3.48) $\displaystyle M_{AA^{\prime}}:={}$
$\displaystyle-\phi^{-1}\alpha_{B}\nabla_{AA^{\prime}}\beta^{B}-\phi^{-1}\alpha_{A}f_{A^{\prime}B}\beta^{B}-w_{1}f_{AA^{\prime}}$
(3.49)

(with $w_{0}$ and $w_{1}$ defined by (3.44)), and where $\nabla_{AA^{\prime}}$
is any Levi-Civita connection in the conformal class.

To understand the construction of (3.47), we refer to the discussion that
leads to eq. (B.9) in Appendix B. Under a gauge transformation (3.46),
$\mathcal{C}_{AA^{\prime}}$ satisfies

$\mathcal{C}_{AA^{\prime}}\varphi^{B...B^{\prime}...}_{C...C^{\prime}...}\to\Omega^{w}\lambda^{r}\mu^{r^{\prime}}\mathcal{C}_{AA^{\prime}}\varphi^{B...B^{\prime}...}_{C...C^{\prime}...}.$
(3.50)

The projection of $\mathcal{C}_{AA^{\prime}}$ into the eigenbundles of $K$
deserves special attention, so we introduce:

######  Definition 3.12.

Let $(M,[g],K)$ be a real, 4-dimensional, almost-para-Hermitian conformal
structure, and let $\alpha_{A},\beta_{A}$ be the spinor fields representing
$K$ as in (3.26). Let $\mathcal{C}_{AA^{\prime}}$ be the associated covariant
derivative, defined by (3.47). We define the operators

$\tilde{\mathcal{C}}_{A^{\prime}}:=\alpha^{A}\mathcal{C}_{AA^{\prime}},\qquad\mathcal{C}_{A^{\prime}}:=\beta^{A}\mathcal{C}_{AA^{\prime}}.$
(3.51)

######  Remark 3.13.

Let $(g,K)$ be an almost-para-Hermitian structure, let $P$ and $\tilde{P}$ be
the associated projectors (2.8) to its eigenbundles, and let
$\alpha^{A},\beta^{A}$ be the spinor fields representing $K$ (eq. (3.26)).
Furthermore, let $\nabla$ be an arbitrary derivative operator, and
$X\in\Gamma(TM\otimes\mathbb{C})$. In the modern para-Hermitian approach to
double field theory, see in particular [6] and [7], an especially important
role is played by the operators $\nabla_{\tilde{P}(X)}$ and $\nabla_{P(X)}$
(in arbitrary dimensions). These operators are also crucial in the present
work, and in our notation we have the equivalence:

$\displaystyle\tilde{\nabla}_{A^{\prime}}:=\alpha^{A}\nabla_{AA^{\prime}}\quad\Leftrightarrow\quad\nabla_{\tilde{P}(X)},$
(3.52a)
$\displaystyle\nabla_{A^{\prime}}:=\beta^{A}\nabla_{AA^{\prime}}\quad\Leftrightarrow\quad\nabla_{P(X)}.$
(3.52b)

In particular, $\tilde{\mathcal{C}}_{A^{\prime}}$ corresponds to
$\mathcal{C}_{\tilde{P}(X)}$, and $\mathcal{C}_{A^{\prime}}$ corresponds to
$\mathcal{C}_{P(X)}$.

##  4 Integrable structures

###  4.1 Special spinor fields

The characterization given by Theorem 3.4 of the space of almost para-
Hermitian structures allows immediately an explicit description of the
eigenbundles: given $K$, there exist projective spinors $\alpha^{A},\beta^{A}$
such that eq. (3.26) (or (3.29) in the Riemmanian case) holds, and from this
expression one can readily check that the $\pm 1$ eigenbundles are
respectively

$\displaystyle L={}$ $\displaystyle\\{u^{a}\in
TM\otimes\mathbb{C}\;|\;u^{a}=\beta^{A}\nu^{A^{\prime}},\;\;\nu^{A^{\prime}}\in\mathbb{S}^{\prime}(0,-1;-w_{1})\\},$
(4.1) $\displaystyle\tilde{L}={}$ $\displaystyle\\{v^{a}\in
TM\otimes\mathbb{C}\;|\;v^{a}=\alpha^{A}\mu^{A^{\prime}},\;\;\mu^{A^{\prime}}\in\mathbb{S}^{\prime}(-1,0;-w_{0})\\},$
(4.2)

where $\beta^{A},\alpha^{A}$ are fixed and $\nu^{A^{\prime}},\mu^{A^{\prime}}$
vary, and where $\mathbb{S}^{\prime}(r,r^{\prime};w)$ represents the weighted
bundles described in Appendix B. We have to include these weighted bundles
because, since $\beta^{A}$ and $\alpha^{A}$ have non-trivial weights (see
table 1), the spinors $\nu^{A^{\prime}},\mu^{A^{\prime}}$ should also have
non-trivial weights so that the elements of $L,\tilde{L}$ are ordinary
vectors.

As discussed in Section 2.1, the integrability properties of an almost para-
Hermitian structure like (3.26) refer to whether the distributions $L$ and
$\tilde{L}$ are involutive. In particular, according to Definition 2.10, $K$
is half-integrable if and only if $N_{P}\equiv 0$ or $N_{\tilde{P}}\equiv 0$,
and integrable if and only if $N_{P}=N_{\tilde{P}}=0$. For the case of (3.26),
the projectors (2.8) to $L$ and $\tilde{L}$ are respectively

$\displaystyle P^{a}{}_{b}={}$
$\displaystyle\tfrac{1}{2}(\delta^{a}{}_{b}+K^{a}{}_{b})=\phi^{-1}\beta^{A}\alpha_{B}\delta^{A^{\prime}}{}_{B^{\prime}},$
(4.3a) $\displaystyle\tilde{P}^{a}{}_{b}={}$
$\displaystyle\tfrac{1}{2}(\delta^{a}{}_{b}-K^{a}{}_{b})=-\phi^{-1}\alpha^{A}\beta_{B}\delta^{A^{\prime}}{}_{B^{\prime}}.$
(4.3b)

where $\phi=\alpha_{A}\beta^{A}$.

######  Lemma 4.1.

Consider an almost (para-)Hermitian structure on a real 4-manifold $M$ (Def.
2.3), and let $\nabla$ be the Levi-Civita connection of $g$. For concreteness
suppose that the (para-)complex structure is of negative chirality (see Remark
3.6). Then the (para-)Hermitian structure is

  1. 1.

half-integrable if and only if there exists a non-trivial unprimed spinor
field $\varphi^{A}$ such that

$\varphi^{A}\varphi^{B}\nabla_{AA^{\prime}}\varphi_{B}=0.$ (4.4)

  2. 2.

integrable if and only if there are two linearly independent solutions to
equation (4.4).

###### Proof.

Denote the (para-)complex structure by $K$, and suppose it has negative
chirality. Then $K$ can be written as (3.26) for some (projective) spinor
fields $\alpha_{A},\beta_{A}$ (where in the Riemannian case we have
$\beta_{A}=i\alpha^{\dagger}_{A}$). Then the projectors $P,\tilde{P}$ take the
form (4.3a)-(4.3b). Now let us compute $N_{P}(X,Y)$ for arbitrary $X,Y$, using
(4.3a)-(4.3b). We have:

$\displaystyle(N_{P}(X,Y))^{AA^{\prime}}={}$
$\displaystyle-\phi^{-1}\alpha^{A}\beta_{B}[PX,PY]^{BA^{\prime}}$
$\displaystyle={}$
$\displaystyle-\phi^{-1}\alpha^{A}\beta_{B}\left(\phi^{-1}\beta^{C}\alpha_{D}X^{DC^{\prime}}\nabla_{CC^{\prime}}(\phi^{-1}\beta^{B}\alpha_{E}Y^{EA^{\prime}})\right.$
$\displaystyle\left.-\phi^{-1}\beta^{C}\alpha_{D}Y^{DC^{\prime}}\nabla_{CC^{\prime}}(\phi^{-1}\beta^{B}\alpha_{E}X^{EA^{\prime}})\right)$
$\displaystyle={}$
$\displaystyle-\phi^{-3}\alpha^{A}\beta_{B}(\beta^{C}\alpha_{D}X^{DC^{\prime}}\alpha_{E}Y^{EA^{\prime}}\nabla_{CC^{\prime}}\beta^{B}-\beta^{C}\alpha_{D}Y^{DC^{\prime}}\alpha_{E}X^{EA^{\prime}}\nabla_{CC^{\prime}}\beta^{B})$
$\displaystyle={}$
$\displaystyle-\phi^{-3}\alpha^{A}\beta_{B}\beta^{C}\alpha_{E}\alpha_{D}(X^{DC^{\prime}}Y^{EA^{\prime}}-X^{DA^{\prime}}Y^{EC^{\prime}})\nabla_{CC^{\prime}}\beta^{B}$
$\displaystyle={}$
$\displaystyle-\phi^{-3}(\alpha_{E}\alpha_{D}X_{D}{}_{E^{\prime}}Y^{EE^{\prime}})\alpha^{A}\beta^{B}\beta^{C}\nabla_{C}{}^{A^{\prime}}\beta_{B},$

therefore we see that $N_{P}\equiv 0$ if and only if $\beta^{A}$ satisfies
equation (4.4), which proves the first item. An analogous calculation applies
to $\alpha^{A}$, and this shows that $N_{\tilde{P}}\equiv 0$ iff $\alpha^{A}$
satisfies (4.4). Therefore, the condition $N_{P}=N_{\tilde{P}}\equiv 0$ is
equivalent to the condition that both spinors $\alpha_{A}$ and $\beta_{B}$
satisfy eq. (4.4); this proves the second item. ∎

Not any 4-manifold admits non-trivial solutions to (4.4), so half-
integrability of $K$ imposes restrictions. We may call equation (4.4) the
“shear-free condition”, since in Lorentzian general relativity its solutions
define what are known as shear-free null geodesic congruences: the real vector
field $\ell^{a}=\varphi^{A}\bar{\varphi}^{A^{\prime}}$ is tangent to a null
geodesic congruence which is shear-free (see e.g. [31, Section 9.2] for the
definition of shear of a geodesic congruence). If, for example, one also asks
the metric to be Einstein, then by the Goldberg-Sachs theorem the Weyl tensor
must be algebraically special, which is a restriction on the curvature. Still,
many physically interesting spacetimes in general relativity satisfy this
property; in particular, stationary black hole solutions such as Kerr or
Schwarzschild (with or without cosmological constant).

In the DFT language, a choice of $K$ means a choice of T-dual ‘spacetimes’,
where the physical spacetime is related to the involutive eigenbundle. If we
change $K$, the new $K$ might not be half-integrable so it would not define
such T-dual spaces. Requiring any $K$ to be “appropriate” in this sense
imposes more severe restrictions:

######  Lemma 4.2.

Let $M$ be a real, 4-dimensional, orientable manifold equipped with a metric
$g$. If all almost-para-Hermitian structures (of a definite chirality) are
half-integrable, then the Weyl curvature of $g$ is (A)SD.

###### Proof.

Let $K$ be an arbitrary almost para-Hermitian structure, and assume for
concreteness that it has negative chirality. Then we know that it is
represented by two projective spinor fields $\alpha^{A},\beta^{A}$. We also
know that $K$ is half-integrable if and only if one of these spinor fields
satisfies eq. (4.4). Let $\varphi^{A}$ denote any of these spinors. From (4.4)
we deduce that
$\varphi^{A}\nabla_{AA^{\prime}}\varphi_{B}=\varphi_{B}\pi_{B^{\prime}}$ for
some spinor field $\pi_{B^{\prime}}$. Applying now
$\varphi^{C}\nabla_{C}{}^{A^{\prime}}$ to equation (4.4), one obtains the
following integrability condition:

$\displaystyle 0={}$
$\displaystyle\varphi^{C}\nabla_{C}{}^{A^{\prime}}(\varphi^{A}\varphi^{B}\nabla_{AA^{\prime}}\varphi_{B})$
$\displaystyle={}$ $\displaystyle
2\pi^{A^{\prime}}\varphi^{A}\varphi^{B}\nabla_{AA^{\prime}}\varphi_{B}-\varphi^{C}\varphi^{A}\varphi^{B}\Box_{AC}\varphi_{B}$
$\displaystyle={}$
$\displaystyle-\Psi_{ABCD}\varphi^{A}\varphi^{B}\varphi^{C}\varphi^{D}$ (4.5)

where $\Box_{AC}=\nabla_{A^{\prime}(A}\nabla_{C)}{}^{A^{\prime}}$, and
$\Psi_{ABCD}$ is the Weyl curvature spinor (see [25] for details). If any
almost para-Hermitian structure $K$ is half-integrable, then (4.5) must be
satisfied by all spinors $\varphi^{A}$, which implies that $\Psi_{ABCD}\equiv
0$ and thus the Weyl curvature of $g$ is SD. (Choosing positive chirality
instead leads to ASD Weyl curvature.) ∎

######  Remark 4.3.

When saying that “any almost-para-Hermitian structure is half-integrable”, it
is convenient to be more specific and say, using the terminology mentioned in
Remark 2.11, that “the manifold is $L$-para Hermitian with respect to any
$K$”, i.e. the $(+1)$-eigenbundle of any $K$ is involutive. From this
perspective, if a manifold is $L$-para Hermitian with respect to any $K$, then
it follows that it is para-Hermitian. To see this: let $K$ be an arbitrary
$L$-para Hermitian structure, i.e. its $(+1)$-eigenbundle is involutive. If
any other $K$ is also $L$-para Hermitian, then in particular this is true for
$K^{\prime}=-K$. The $(+1)$-eigenbundle of $K^{\prime}$ is then involutive,
and it is the $(-1)$-eigenbundle of $K$. So $K$ is both $L$-para Hermitian and
$\tilde{L}$-para Hermitian, i.e., it is para-Hermitian.

The (A)SD restriction in lemma 4.2 is quite strong. In Lorentz signature, the
left- and right-handed Weyl curvature spinors are complex conjugates of each
other, which implies that (A)SD Weyl curvature means vanishing Weyl tensor, so
the manifold must be conformally flat. In Riemannian and split signature, the
two Weyl curvature spinors are independent from each other, so one of them can
vanish while the other one is general. In the Riemannian case, if one also
asks Ricci-flatness (as well as (A)SD Weyl curvature), then it can be shown
that the manifold must be hyperkähler.

In the DFT literature, a change of para-complex structure is referred to as a
change of polarization. As mentioned, an arbitrary change of polarization in
general does not lead to a new pair of T-dual ‘spacetimes’, unless all para-
complex structures are half-integrable. Another possibility is to consider
“small” changes of polarization, by which we mean deformations of the para-
complex structure. These issues will be addressed in section 5, in connection
also with twistor constructions.

###  4.2 Lie and Courant algebroids

We have seen that, given a 4-dimensional real manifold $M$ with a metric $g$,
any $K\in{\rm Aut}(TM\otimes\mathbb{C})$ satisfying conditions (3.13) and
(3.14) is characterized by two projective spinor fields
$[\alpha_{A}],[\beta_{A}]$. The tangent bundle splits as
$TM\otimes\mathbb{C}=L\oplus\tilde{L}$, where $L$ and $\tilde{L}$ are the
eigenbundles of $K$, given explicitly in eqs. (4.1), (4.2). Because of this
splitting, any vector in $TM\otimes\mathbb{C}$ can be decomposed into two
pieces, as originally done in (2.1). We can explicitly describe these pieces
using (4.1), (4.2), the identity $P+\tilde{P}=\mathbb{I}$ and the fact that
the projectors $P$ and $\tilde{P}$ are given by (4.3a), (4.3b). That is, any
$X\in\Gamma(TM\otimes\mathbb{C})$ is decomposed as

$X^{AA^{\prime}}=(PX)^{AA^{\prime}}+(\tilde{P}X)^{AA^{\prime}}\equiv
x^{AA^{\prime}}+\tilde{x}^{AA^{\prime}},$ (4.6)

where

$\displaystyle(PX)^{AA^{\prime}}\equiv
x^{AA^{\prime}}=\beta^{A}x^{A^{\prime}},\qquad
x^{A^{\prime}}=\phi^{-1}\alpha_{B}X^{BA^{\prime}},$ (4.7)
$\displaystyle(\tilde{P}X)^{AA^{\prime}}\equiv\tilde{x}^{AA^{\prime}}=\alpha^{A}\tilde{x}^{A^{\prime}},\qquad\tilde{x}^{A^{\prime}}=-\phi^{-1}\beta_{B}X^{BA^{\prime}}.$
(4.8)

Analogous decompositions and notation apply to any other vector
$Y^{AA^{\prime}}=y^{AA^{\prime}}+\tilde{y}^{AA^{\prime}}$, etc. Similarly, a
1-form $W\in\Gamma(T^{*}M\otimes\mathbb{C})$ is decomposed as

$W_{AA^{\prime}}=\omega_{AA^{\prime}}+\tilde{\omega}_{AA^{\prime}}=\alpha_{A}\omega_{A^{\prime}}+\beta_{A}\tilde{\omega}_{A^{\prime}},$
(4.9)

with

$\displaystyle\omega_{A^{\prime}}={}$
$\displaystyle\phi^{-1}\beta^{B}\omega_{BA^{\prime}},$
$\displaystyle\tilde{\omega}_{A^{\prime}}={}$
$\displaystyle-\phi^{-1}\alpha^{B}\omega_{BA^{\prime}}.$

######  Remark 4.4.

Ordinary tensor fields have trivial $r,r^{\prime},w$ weights (in the notation
of Definition 3.10), but the spinors $\alpha^{A},\beta^{A}$ have non-trivial
weights (given in Table 1). Therefore, when decomposed into pieces like (4.6)
or (4.9), the primed spinor parts are also non-trivially weighted. For
example, the spinors $x^{A^{\prime}}$ and $\tilde{x}^{A^{\prime}}$ in the
decomposition (4.7)-(4.8) have non-trivial weights, which can be deduced from
those of $\phi$ and $\alpha^{A},\beta^{A}$. This is analogous to the fact that
$\nu^{A^{\prime}},\mu^{A^{\prime}}$ in (4.1), (4.2) are also weighted.

We now use the above decomposition to describe the Lie and Courant algebroid
structures of 4-manifolds admitting non-trivial solutions to (4.4). Since this
equation involves only a conformal structure and a projective spinor field,
the covariant formalism of Section 3.3 is here appropriate to give a
formulation invariant under the associated gauge freedom. We recall that the
metric is allowed to have any signature.

######  Lemma 4.5.

Let $(M,[g_{ab}])$ be a 4-dimensional, real conformal structure. Let
$g_{ab}\in[g_{ab}]$, with Levi-Civita connection $\nabla_{AA^{\prime}}$.
Suppose that there is a projective spinor field $[\beta_{A}]$, where any
representative $\beta_{A}$ satisfies the equation

$\beta^{A}\beta^{B}\nabla_{AA^{\prime}}\beta_{B}=0.$ (4.10)

Let $\alpha_{A}$ be any spinor field such that
$\epsilon^{AB}\alpha_{A}\beta_{B}\equiv\phi\neq 0$. Let $L\subset
TM\otimes\mathbb{C}$ be defined by (4.1), and let $\mathcal{C}_{A^{\prime}}$
be the operator introduced in (3.51).

  1. 1.

The triple $(L,[\cdot,\cdot]_{L},P)$ is a Lie algebroid, where $P$ is the
projector (4.3a), and the Lie algebroid bracket is

$[X,Y]^{AA^{\prime}}_{L}=\beta^{A}(x^{B^{\prime}}\mathcal{C}_{B^{\prime}}y^{A^{\prime}}-y^{B^{\prime}}\mathcal{C}_{B^{\prime}}x^{A^{\prime}}),$
(4.11)

for any vector fields $X,Y\in\Gamma(L)$.

  2. 2.

Let $\Lambda^{k}=\wedge^{k}L^{*},k=0,1,2,...$. The Lie algebroid exterior
derivative of $(L,[\cdot,\cdot]_{L},P)$ is given by

$\displaystyle({\rm d}^{L}f)_{AA^{\prime}}={}$
$\displaystyle\phi^{-1}\alpha_{A}\mathcal{C}_{A^{\prime}}f,$ (4.12)
$\displaystyle({\rm d}^{L}\omega)_{AA^{\prime}BB^{\prime}}={}$
$\displaystyle\phi^{-1}(\mathcal{C}_{C^{\prime}}\omega^{C^{\prime}})\alpha_{A}\alpha_{B}\epsilon_{A^{\prime}B^{\prime}},$
(4.13)

and ${\rm d}^{L}\omega=0$ for any $\omega\in\Gamma(\Lambda^{k})$ with $k\geq
2$, where $\omega^{C^{\prime}}$ in the right hand side of (4.13) refers to the
decomposition (4.9).

  3. 3.

For a 1-form $\omega\in\Gamma(L^{*})$, the generalized Lie derivative along
$X\in\Gamma(L)$ is

$(\pounds^{L}_{X}\omega)_{AA^{\prime}}=\alpha_{A}[x_{A^{\prime}}(\mathcal{C}_{B^{\prime}}\omega^{B^{\prime}})+\mathcal{C}_{A^{\prime}}(x^{B^{\prime}}\omega_{B^{\prime}})].$
(4.14)

###### Proof.

For the first item, we need to check that the conditions in Definition 2.17
are satisfied. First of all, we know that $L$ is a vector bundle over $M$. We
have also seen that the condition (4.10) implies that sections of $L$ are in
involution with respect to the usual Lie bracket $[\cdot,\cdot]$ of vector
fields, thus its restriction to $L$, $[\cdot,\cdot]_{L}$, indeed maps
$\Gamma(L)\times\Gamma(L)$ to $\Gamma(L)$, is skew-symmetric and satisfies the
Jacobi identity. In addition, the identity in $TM$ restricted to $L$ coincides
with $P$, and it satisfies the conditions to be an anchor. So it only remains
to prove the expression (4.11) for the Lie algebroid bracket. To do this,
notice first that, since ordinary vector fields $X,Y$ have trivial
$w,r,r^{\prime}$ weights (see Remark 4.4), we have for example

$\mathcal{C}_{b}X^{a}=\nabla_{b}X^{a}+Q_{bc}{}^{a}X^{c},$

where $Q_{bc}{}^{a}$ was defined in (2.27). Since
$Q_{bc}{}^{a}=Q_{(bc)}{}^{a}$, we get

$X^{b}\nabla_{b}Y^{a}-Y^{b}\nabla_{b}X^{a}=X^{b}\mathcal{C}_{b}Y^{a}-Y^{b}\mathcal{C}_{b}X^{a}.$

Using this and decompositions like (4.6)-(4.7)-(4.8) for $X,Y\in\Gamma(L)$, we
have:

$\displaystyle[X,Y]^{AA^{\prime}}_{L}={}$ $\displaystyle
X^{BB^{\prime}}\nabla_{BB^{\prime}}Y^{AA^{\prime}}-Y^{BB^{\prime}}\nabla_{BB^{\prime}}X^{AA^{\prime}}$
$\displaystyle={}$ $\displaystyle
x^{B^{\prime}}\beta^{B}\mathcal{C}_{BB^{\prime}}(\beta^{A}y^{A^{\prime}})-y^{B^{\prime}}\beta^{B}\mathcal{C}_{BB^{\prime}}(\beta^{A}x^{A^{\prime}})$
$\displaystyle={}$
$\displaystyle\beta^{A}(x^{B^{\prime}}\mathcal{C}_{B^{\prime}}y^{A^{\prime}}-y^{B^{\prime}}\mathcal{C}_{B^{\prime}}x^{A^{\prime}})$

where in the third line we used (B.11c) and (4.10), together with the
definition (3.51) for $\mathcal{C}_{A^{\prime}}$.

Now let us prove the second item. From Definition 2.20, the Lie algebroid
exterior derivative acting on a 0-form $f\in\Gamma(\Lambda^{0})$ is given by
$({\rm d}^{L}f)(X)=P(X)(f)$ (since here the anchor is the projector $P$).
Since $f$ has trivial weights, we can replace the ordinary
$\partial_{AA^{\prime}}$ implicit in this formula by the covariant derivative
$\mathcal{C}_{AA^{\prime}}$, so we get

$({\rm d}^{L}f)(X)=(PX)^{AA^{\prime}}\mathcal{C}_{AA^{\prime}}f.$

Using (4.7), formula (4.12) follows. Consider now a 1-form
$\omega\in\Gamma(\Lambda^{1})$. The operator ${\rm d}^{L}$ acting on $\omega$
is given by the expression (2.15), with $\rho\equiv P$. Since $\omega,X,Y$
have trivial weights we can, as before, replace $\partial_{AA^{\prime}}$ by
$\mathcal{C}_{AA^{\prime}}$:

$({\rm
d}^{L}\omega)(X,Y)=(PX)^{AA^{\prime}}\mathcal{C}_{AA^{\prime}}(\omega_{BB^{\prime}}Y^{BB^{\prime}})-(PY)^{AA^{\prime}}\mathcal{C}_{AA^{\prime}}(\omega_{BB^{\prime}}X^{BB^{\prime}})-\omega_{AA^{\prime}}[X,Y]^{AA^{\prime}}_{L}.$

Replacing the expressions for $X,Y$ and $[X,Y]_{L}$, this is

$\displaystyle({\rm d}^{L}\omega)(X,Y)={}$ $\displaystyle
x^{B^{\prime}}\mathcal{C}_{B^{\prime}}(\phi\omega_{A^{\prime}}y^{A^{\prime}})-y^{B^{\prime}}\mathcal{C}_{B^{\prime}}(\phi\omega_{A^{\prime}}x^{A^{\prime}})-\phi\omega_{A^{\prime}}(x^{B^{\prime}}\mathcal{C}_{B^{\prime}}y^{A^{\prime}}-y^{B^{\prime}}\mathcal{C}_{B^{\prime}}x^{A^{\prime}})$
$\displaystyle={}$
$\displaystyle\phi\left[x^{B^{\prime}}\mathcal{C}_{B^{\prime}}(\omega_{A^{\prime}}y^{A^{\prime}})-y^{B^{\prime}}\mathcal{C}_{B^{\prime}}(\omega_{A^{\prime}}x^{A^{\prime}})-\omega_{A^{\prime}}x^{B^{\prime}}\mathcal{C}_{B^{\prime}}y^{A^{\prime}}+\omega_{A^{\prime}}y^{B^{\prime}}\mathcal{C}_{B^{\prime}}x^{A^{\prime}}\right]$
$\displaystyle={}$
$\displaystyle\phi(x^{A^{\prime}}y^{B^{\prime}}-y^{A^{\prime}}x^{B^{\prime}})\mathcal{C}_{A^{\prime}}\omega_{B^{\prime}}$
(4.15)

where in the second line we used (B.11d). Now:

$\displaystyle x^{A^{\prime}}y^{B^{\prime}}-y^{A^{\prime}}x^{B^{\prime}}={}$
$\displaystyle(\epsilon_{C^{\prime}D^{\prime}}x^{C^{\prime}}y^{D^{\prime}})\epsilon^{A^{\prime}B^{\prime}}$
$\displaystyle={}$
$\displaystyle(\epsilon_{C^{\prime}D^{\prime}}\phi^{-2}\alpha_{C}X^{CC^{\prime}}\alpha_{D}Y^{DD^{\prime}})\epsilon^{A^{\prime}B^{\prime}}$
$\displaystyle={}$
$\displaystyle(\phi^{-2}\alpha_{C}\alpha_{D}\epsilon_{C^{\prime}D^{\prime}}\epsilon^{A^{\prime}B^{\prime}})X^{CC^{\prime}}Y^{DD^{\prime}}$

where in the second line we replaced the expression (4.7) for
$x^{A^{\prime}}$, and similarly for $y^{A^{\prime}}$. Replacing now in (4.15),
and noticing that

$({\rm d}^{L}\omega)(X,Y)=({\rm
d}^{L}\omega)_{CC^{\prime}DD^{\prime}}X^{CC^{\prime}}Y^{DD^{\prime}},$

formula (4.13) follows. If now $\omega\in\Gamma(\Lambda^{k})$, $k\geq 2$, then
${\rm d}^{L}\omega$ vanishes since it is a $(k+1)$-form with $k\geq 2$ in a
2-dimensional space.

Finally, to show the third item, from the general expression (2.13) we have

$(\pounds^{L}_{X}\omega)_{AA^{\prime}}=X^{BB^{\prime}}({\rm
d}^{L}\omega)_{BB^{\prime}AA^{\prime}}+[{\rm
d}^{L}(X^{BB^{\prime}}\omega_{BB^{\prime}})]_{AA^{\prime}}.$

Using the second item, this is

$\displaystyle(\pounds^{L}_{X}\omega)_{AA^{\prime}}={}$ $\displaystyle
X^{BB^{\prime}}\phi^{-1}\alpha_{B}\alpha_{A}\epsilon_{B^{\prime}A^{\prime}}\mathcal{C}_{C^{\prime}}\omega^{C^{\prime}}+\phi^{-1}\alpha_{A}\mathcal{C}_{A^{\prime}}(\phi
x^{B^{\prime}}\omega_{B^{\prime}})$ $\displaystyle={}$
$\displaystyle\alpha_{A}x_{A^{\prime}}\mathcal{C}_{C^{\prime}}\omega^{C^{\prime}}+\alpha_{A}\mathcal{C}_{A^{\prime}}(x^{B^{\prime}}\omega_{B^{\prime}}),$

which proves (4.14). ∎

######  Lemma 4.6.

Let $(M,[g_{ab}])$ be a 4-dimensional, real conformal structure. Let
$g_{ab}\in[g_{ab}]$, with Levi-Civita connection $\nabla_{AA^{\prime}}$.
Suppose that there is a projective spinor field $[\beta_{A}]$, where any
representative $\beta_{A}$ satisfies the equation

$\beta^{A}\beta^{B}\nabla_{AA^{\prime}}\beta_{B}=0.$ (4.16)

Let $\alpha_{A}$ be any spinor field such that
$\epsilon^{AB}\alpha_{A}\beta_{B}=\phi\neq 0$. Then the quadruple

$(TM\otimes\mathbb{C},g,P,\llbracket\cdot,\cdot\rrbracket)$ (4.17)

is a Courant algebroid, where the anchor $P$ and the Dorfman bracket are
respectively

$\displaystyle P^{AA^{\prime}}{}_{BB^{\prime}}={}$
$\displaystyle\phi^{-1}\beta^{A}\alpha_{B}\delta^{A^{\prime}}{}_{B^{\prime}},$
(4.18) $\displaystyle\llbracket X,Y\rrbracket^{AA^{\prime}}={}$
$\displaystyle\beta^{A}(x^{B^{\prime}}\mathcal{C}_{B^{\prime}}y^{A^{\prime}}-y^{B^{\prime}}\mathcal{C}_{B^{\prime}}x^{A^{\prime}})+\alpha^{A}\left(x^{A^{\prime}}\mathcal{C}_{B^{\prime}}\tilde{y}^{B^{\prime}}-y^{A^{\prime}}\mathcal{C}_{B^{\prime}}\tilde{x}^{B^{\prime}}+\mathcal{C}^{A^{\prime}}(x^{B^{\prime}}\tilde{y}_{B^{\prime}})\right)$
(4.19)

with $\mathcal{C}_{A^{\prime}}$ the operator defined in (3.51).

###### Proof.

Define the bundle $L\subset TM\otimes\mathbb{C}$ by (4.1); then from Lemma
4.5, the triple $(L,[\cdot,\cdot]_{L},P)$ is a Lie algebroid. Thus, applying
Proposition 2.24, the vector bundle $L\oplus L^{*}$ together with the maps
defined by (2.18), (2.19) and (2.20), constitute a Courant algebroid. Using
now the pair of spinors $(\alpha_{A},\beta_{A})$ to define, via (3.26), a
tensor field $K$ satisfying conditions (3.13) and (3.14), the eigenbundles of
$K$ are given by (4.1) and (4.2) and we have the splitting
$TM\otimes\mathbb{C}=L\oplus\tilde{L}$. We can then apply Proposition 2.25, so
the quadruple (4.17) is also a Courant algebroid. The anchor is simply the
projector (4.3a). What remains now is to translate the expression (2.24) for
the Dorfman bracket to the language we are using in this section. Using (4.7),
(4.8), (4.12) and (4.14), we have

$\displaystyle(\tilde{x}_{\flat})_{AA^{\prime}}=\epsilon_{AB}\epsilon_{A^{\prime}B^{\prime}}\tilde{x}^{BB^{\prime}}=\alpha_{A}\tilde{x}_{A^{\prime}},$
(4.20)
$\displaystyle(\tilde{y}_{\flat})_{AA^{\prime}}=\epsilon_{AB}\epsilon_{A^{\prime}B^{\prime}}\tilde{y}^{BB^{\prime}}=\alpha_{A}\tilde{y}_{A^{\prime}},$
(4.21) $\displaystyle
g(\tilde{x},y)=\epsilon_{AB}\epsilon_{A^{\prime}B^{\prime}}\tilde{x}^{AA^{\prime}}y^{BB^{\prime}}=\phi\tilde{x}_{A^{\prime}}y^{A^{\prime}},$
(4.22) $\displaystyle({\rm
d}^{L}g(\tilde{x},y))_{AA^{\prime}}=\alpha_{A}\mathcal{C}_{A^{\prime}}(\tilde{x}_{B^{\prime}}y^{B^{\prime}}),$
(4.23)
$\displaystyle[\pounds^{L}_{x}(\tilde{y}_{\flat})]_{AA^{\prime}}=\alpha_{A}[x_{A^{\prime}}\mathcal{C}_{B^{\prime}}\tilde{y}^{B^{\prime}}+\mathcal{C}_{A^{\prime}}(x^{B^{\prime}}\tilde{y}_{B^{\prime}})].$
(4.24)

The Dorfman bracket (2.24) is

$\llbracket
X,Y\rrbracket^{AA^{\prime}}=[x,y]^{AA^{\prime}}+(\pounds^{L}_{x}(\tilde{y}_{\flat})-\pounds^{L}_{y}(\tilde{x}_{\flat})+{\rm
d}^{L}g(\tilde{x},y))^{AA^{\prime}}.$

Since $x,y$ are sections of $L$, we have
$[x,y]^{AA^{\prime}}=[x,y]^{AA^{\prime}}_{L}$, so we can use formula (4.11).
Replacing also (4.23), (4.24), we get

$\displaystyle\llbracket X,Y\rrbracket^{AA^{\prime}}={}$
$\displaystyle\beta^{A}(x^{B^{\prime}}\mathcal{C}_{B^{\prime}}y^{A^{\prime}}-y^{B^{\prime}}\mathcal{C}_{B^{\prime}}x^{A^{\prime}})+\alpha^{A}[x^{A^{\prime}}\mathcal{C}_{B^{\prime}}\tilde{y}^{B^{\prime}}+\mathcal{C}^{A^{\prime}}(x^{B^{\prime}}\tilde{y}_{B^{\prime}})]$
$\displaystyle-\alpha^{A}[y^{A^{\prime}}\mathcal{C}_{B^{\prime}}\tilde{x}^{B^{\prime}}+\mathcal{C}^{A^{\prime}}(y^{B^{\prime}}\tilde{x}_{B^{\prime}})]+\alpha^{A}\mathcal{C}^{A^{\prime}}(\tilde{x}_{B^{\prime}}y^{B^{\prime}})$
$\displaystyle={}$
$\displaystyle\beta^{A}(x^{B^{\prime}}\mathcal{C}_{B^{\prime}}y^{A^{\prime}}-y^{B^{\prime}}\mathcal{C}_{B^{\prime}}x^{A^{\prime}})+\alpha^{A}\left(x^{A^{\prime}}\mathcal{C}_{B^{\prime}}\tilde{y}^{B^{\prime}}-y^{A^{\prime}}\mathcal{C}_{B^{\prime}}\tilde{x}^{B^{\prime}}+\mathcal{C}^{A^{\prime}}(x^{B^{\prime}}\tilde{y}_{B^{\prime}})\right)$

which is (4.19). ∎

Recall from Remark 2.21 that a Lie algebroid has naturally associated a
cochain complex $(\Gamma(\Lambda^{\bullet}),{\rm d}^{L})$. In our case this is

$0\to\Gamma(\Lambda^{0})\xrightarrow{{\rm
d}^{L}}\Gamma(\Lambda^{1})\xrightarrow{{\rm d}^{L}}\Gamma(\Lambda^{2})\to 0.$
(4.25)

We can think of this as a (twisted) de Rham complex. Since a twisted de Rham
complex is locally exact, if an object $\varphi$ satisfies ${\rm
d}^{L}\varphi=0$, then there exists, locally141414Of course, global existence
of potentials is a different story, where one has to take into account the
cohomology of the complex (4.25)., another field $\psi$ such that
$\varphi={\rm d}^{L}\psi$. Therefore, the Lie algebroid structure leads to a
“potentialization” scheme. For example, suppose
$W_{AA^{\prime}}=\alpha_{A}\omega_{A^{\prime}}\in\Gamma(\Lambda^{1})$
satisfies ${\rm d}^{L}W=0$, then there exists $f\in\Gamma(\Lambda^{0})$ such
that $W={\rm d}^{L}f$. Using (4.12)-(4.13), the equation ${\rm d}^{L}W=0$ is
$\mathcal{C}_{A^{\prime}}\omega^{A^{\prime}}=0$, and the equation $W={\rm
d}^{L}f$ is $W_{AA^{\prime}}=\phi^{-1}\alpha_{A}\mathcal{C}_{A^{\prime}}f$. In
other words:

$\mathcal{C}_{A^{\prime}}\omega^{A^{\prime}}=0\quad\Rightarrow\quad\omega_{A^{\prime}}=\phi^{-1}\mathcal{C}_{A^{\prime}}f$
(4.26)

for some $f$. Such a basic ‘potentialization’ scheme turns out to be very
powerful when applied to problems of interest in relativity (this will be the
subject of a separate work). Notice, however, that the weights of the
potentials here are restricted by the fact that the spaces
$\Gamma(\Lambda^{k})$ in the complex (4.25) are composed of objects with zero
weights. If we want to study whether the complex (4.25) can be generalized to
fields with other weights, we are led to the question of constructing Lie
algebroids for weighted fields. This turns out to be a bit tricky, and will be
analysed in the next subsection.

###  4.3 Weighted algebroids

The algebroids constructed in lemmas 4.5 and 4.6 consist of vectors with
trivial weights (in the sense of def. 3.10). From the discussion at the end of
the previous subsection, we are also interested in analysing whether spaces of
non-trivially weighted objects can be given an algebroid structure. We will
see here that ordinary Lie algebroids do not seem to be compatible with
weighted fields.

Let $L$ be the bundle (4.1), where $\beta^{A}$ satisfies (4.4). Consider the
weighted vector bundle
$L_{(r,r^{\prime};w)}:=L\otimes\mathbb{S}(r,r^{\prime};w)$, where
$\mathbb{S}(r,r^{\prime};w)$ is the line bundle defined in (B.4). An element
of $L_{(r,r^{\prime};w)}$ is a vector with weights $(r,r^{\prime};w)$ (so $L$
is simply $L_{(0,0;0)}$):

$X\in L_{(r,r^{\prime};w)}\quad\Leftrightarrow\quad
X^{a}=\beta^{A}x^{A^{\prime}}\quad\text{and }\quad
X\xrightarrow{G_{o}}\lambda^{r}\mu^{r^{\prime}}\Omega^{w}X.$

$L_{(r,r^{\prime};w)}$ is a vector bundle over $M$. In order to construct a
Lie algebroid, we also need an anchor and a Lie bracket satisfying the
conditions given in Def. 2.17.

Let us first focus on the bracket. Given two weighted vector fields
$X\in\Gamma(L_{(r_{1},r^{\prime}_{1};w_{1})})$,
$Y\in\Gamma(L_{(r_{2},r^{\prime}_{2};w_{2})})$, we have a natural candidate
for a bracket operation:

$[X,Y]^{a}_{\mathcal{C}}:=X^{b}\mathcal{C}_{b}Y^{a}-Y^{b}\mathcal{C}_{b}X^{a}.$
(4.27)

This is skew-symmetric, and, since $\mathcal{C}_{a}$ satisfies (3.50) and
$\mathcal{C}_{b}\beta^{A}=0$, the result of $[X,Y]^{a}_{\mathcal{C}}$ is again
a vector of the form $\beta^{A}\pi^{A^{\prime}}$ with well-defined weights.
However, the first problem that arises is that the weights of
$[X,Y]^{a}_{\mathcal{C}}$ are
$(r_{1}+r_{2},r^{\prime}_{1}+r^{\prime}_{2};w_{1}+w_{2})$, so
$[X,Y]^{a}_{\mathcal{C}}$ is neither an element of
$L_{(r_{1},r^{\prime}_{1};w_{1})}$ nor of $L_{(r_{2},r^{\prime}_{2};w_{2})}$,
and consequently, we cannot use $[\cdot,\cdot]_{\mathcal{C}}$ as a Lie bracket
for a particular $L_{(r,r^{\prime};w)}$. More precisely, what we have is

$[\cdot,\cdot]_{\mathcal{C}}:\Gamma(L_{(r_{1},r^{\prime}_{1};w_{1})})\times\Gamma(L_{(r_{2},r^{\prime}_{2};w_{2})})\to\Gamma(L_{(r_{1}+r_{2},r^{\prime}_{1}+r^{\prime}_{2};w_{1}+w_{2})}).$
(4.28)

This suggests that we consider the vector bundle with graded fibers

$\mathbb{L}=\bigoplus_{(r,r^{\prime};w)}L_{(r,r^{\prime};w)},$ (4.29)

where the sum runs over all possible values of $(r,r^{\prime};w)$. We then
have
$[\cdot,\cdot]_{\mathcal{C}}:\Gamma(\mathbb{L})\times\Gamma(\mathbb{L})\to\Gamma(\mathbb{L})$.
So the first item in Def. 2.17 is satisfied, and we can try to give a Lie
algebroid structure to $\mathbb{L}$.

The second item in Def. 2.17 requires the Jacobi identity for
$[\cdot,\cdot]_{\mathcal{C}}$ to hold. Since $\mathcal{C}_{b}$ cannot be
replaced by $\partial_{b}$ in (4.27) (because $X$ and $Y$ now have non-trivial
weights), whether or not the Jacobi identity for $[\cdot,\cdot]_{\mathcal{C}}$
is satisfied is a non-trivial issue. To investigate this, we use the
Jacobiator:

${\rm
Jac}_{\mathcal{C}}(X,Y,Z):=[X,[Y,Z]_{\mathcal{C}}]_{\mathcal{C}}+[Z,[X,Y]_{\mathcal{C}}]_{\mathcal{C}}+[Y,[Z,X]_{\mathcal{C}}]_{\mathcal{C}}.$
(4.30)

The following result shows that in order for this to vanish, we also need to
require (half-)algebraic speciality of the Weyl curvature:

######  Lemma 4.7.

Suppose that the spinor field $\beta^{A}$ is shear-free. Then the Jacobiator
(4.30) vanishes if and only if $\beta^{A}$ is a repeated principal spinor of
the ASD Weyl curvature.

###### Proof.

Take three arbitrary vector fields in $\Gamma(\mathbb{L})$ (with possibly
different weights), $X^{a}=\beta^{A}x^{A^{\prime}}$,
$Y^{a}=\beta^{A}y^{A^{\prime}}$, $Z^{a}=\beta^{A}z^{A^{\prime}}$. A tedious
but straightforward calculation shows that

$({\rm
Jac}_{\mathcal{C}}(X,Y,Z))^{AA^{\prime}}=\beta^{A}[(x_{C^{\prime}}y^{C^{\prime}})(\mathcal{C}_{B^{\prime}}\mathcal{C}^{B^{\prime}}z^{A^{\prime}})-(x_{C^{\prime}}z^{C^{\prime}})(\mathcal{C}_{B^{\prime}}\mathcal{C}^{B^{\prime}}y^{A^{\prime}})+(y_{C^{\prime}}z^{C^{\prime}})(\mathcal{C}_{B^{\prime}}\mathcal{C}^{B^{\prime}}x^{A^{\prime}})].$

Therefore, the Jacobi identity for $[\cdot,\cdot]_{\mathcal{C}}$ is satisfied
if and only if

$\mathcal{C}_{A^{\prime}}\mathcal{C}^{A^{\prime}}=0$ (4.31)

when acting on any weighted primed spinor field. This equation is investigated
in Appendix C, where the proof of Lemma C.5 shows that equation (4.31) is
satisfied if and only if $\Psi_{ABCD}\beta^{B}\beta^{C}\beta^{D}=0$, i.e.,
$\beta^{A}$ is a repeated principal spinor of the ASD Weyl curvature. ∎

Let us now focus on the construction of the anchor. From Def. 2.17, this
should be a map $\rho:\mathbb{L}\to TM\otimes\mathbb{C}$. An element $X\in
L_{(r,r^{\prime};w)}\subset\mathbb{L}$ has non-trivial weights, so we cannot
use the projector $P$ as an anchor as before, since $PX$ has also non-trivial
weights and thus it is not an element of $TM\otimes\mathbb{C}$. Therefore, to
any weighted vector $X$ we must somehow associate a vector $\rho(X)$ with
trivial weights. To this end, consider three scalar fields $A,B,C$ of types
$(1,0;0)$, $(0,1;0)$ and $(0,0;1)$ respectively; that is:

$A\xrightarrow{G_{o}}\lambda A,\qquad B\xrightarrow{G_{o}}\mu B,\qquad
C\xrightarrow{G_{o}}\Omega C.$

Then we can define a map

$\rho:L_{(r,r^{\prime};w)}\to
TM\otimes\mathbb{C},\qquad\rho(X):=A^{-r}B^{-r^{\prime}}C^{-w}X.$ (4.32)

Although this is rather ad hoc since we have not specified anything about the
scalars $A,B,C$, we restrict the possible choices of them by requiring

$\mathcal{C}_{A^{\prime}}A=\mathcal{C}_{A^{\prime}}B=\mathcal{C}_{A^{\prime}}C=0$
(4.33)

where as before $\mathcal{C}_{A^{\prime}}=\beta^{A}\mathcal{C}_{AA^{\prime}}$.
In Lemma C.5 it is shown that, if $\beta^{A}$ is shear-free and a repeated
principal spinor, the integrability condition
$\mathcal{C}^{A^{\prime}}\mathcal{C}_{A^{\prime}}\Phi=0$ is satisfied for any
weighted scalar field $\Phi$, so there are indeed solutions to (4.33). With
the choice (4.33), if $X\in\Gamma(L_{(r_{1},r^{\prime}_{1};w_{1})})$,
$Y\in\Gamma(L_{(r_{2},r^{\prime}_{2};w_{2})})$, we have

$\displaystyle\rho([X,Y]_{\mathcal{C}})^{a}={}$ $\displaystyle
A^{-(r_{1}+r_{2})}B^{-(r^{\prime}_{1}+r^{\prime}_{2})}C^{-(w_{1}+w_{2})}[X,Y]^{a}_{\mathcal{C}}$
$\displaystyle={}$ $\displaystyle
A^{-(r_{1}+r_{2})}B^{-(r^{\prime}_{1}+r^{\prime}_{2})}C^{-(w_{1}+w_{2})}(X^{b}\mathcal{C}_{b}Y^{a}-Y^{b}\mathcal{C}_{b}X^{a})$
$\displaystyle={}$ $\displaystyle
A^{-r_{1}}B^{-r^{\prime}_{1}}C^{-w_{1}}X^{b}\mathcal{C}_{b}(A^{-r_{2}}B^{-r^{\prime}_{2}}C^{-w_{2}}Y^{a})-A^{-r_{2}}B^{-r^{\prime}_{2}}C^{-w_{2}}Y^{b}\mathcal{C}_{b}(A^{-r_{1}}B^{-r^{\prime}_{1}}C^{-w_{1}}X^{a})$
$\displaystyle={}$
$\displaystyle\rho(X)^{b}\mathcal{C}_{b}(\rho(Y)^{a})-\rho(Y)^{b}\mathcal{C}_{b}(\rho(X)^{a})$
$\displaystyle={}$ $\displaystyle[\rho(X),\rho(Y)]^{a}$

where in the third line we used (4.33), in the fourth we used the definition
(4.32), and in the fifth the fact that $\rho(X),\rho(Y)$ have trivial weights.
Therefore, we see that with the choice of the anchor (4.32)-(4.33) and the
bracket (4.27), the map (4.32) is a morphism and the third item in the
definition 2.17 of a Lie algebroid is satisfied.

The last condition that we need to investigate is the fourth item in Def.
2.17, namely the Leibniz rule for the bracket and the anchor:

$[X,fY]_{\mathcal{C}}\stackrel{{\scriptstyle?}}{{=}}(\rho(X)f)Y+f[X,Y]_{\mathcal{C}}$
(4.34)

for any $X\in\Gamma(L_{(r_{1},r^{\prime}_{1};w_{1})})$,
$Y\in\Gamma(L_{(r_{2},r^{\prime}_{2};w_{2})})$ and $f\in C^{\infty}(M)$.
However, the gradation (4.28) of the bracket implies immediately that we
cannot expect this condition to be satisfied: If $f\in C^{\infty}(M)$ is an
ordinary function, then $fY\in\Gamma(L_{(r_{2},r^{\prime}_{2};w_{2})})$, so,
because of (4.28), the left hand side of (4.34) is an element of
$\Gamma(L_{(r_{1}+r_{2},r^{\prime}_{1}+r_{2};w_{1}+w_{2})})$; but in the right
hand side, while the second term is in this space as well, the first term
lives in $\Gamma(L_{(r_{2},r^{\prime}_{2};w_{2})})$, so eq. (4.34) is
generally inconsistent. (There is the possibility however of taking
$\rho\equiv 0$, but this is not something that we wish to consider here; see
(4.35) below.) In other words, asking the condition (4.34) to hold is
incompatible with the gradation (assuming $\rho\neq 0$). The structure we
appear to need is then some sort of “graded Lie algebroid” (if such a thing
can be defined at all). Here we simply observe that computation of the left
hand side of (4.34) gives

$[X,fY]_{\mathcal{C}}=X(f)\;Y+f[X,Y]_{\mathcal{C}},$ (4.35)

which is the Leibniz rule satisfied by the ordinary Lie bracket of vector
fields.

In conclusion, the triple $(\mathbb{L},\rho,[\cdot,\cdot]_{\mathcal{C}})$,
where $\mathbb{L}$, $\rho$ and $[\cdot,\cdot]_{\mathcal{C}}$ are defined
respectively by (4.29), (4.32)-(4.33) and (4.27), satisfies the conditions to
be a Lie algebroid except for the Leibniz rule (4.34), which must be replaced
by the ordinary Leibniz rule (4.35). More precisely, the Leibniz rule (4.34)
cannot be compatible with the gradation of a vector bundle (when $\rho\neq
0$), so it seems that one would need to find an appropriate modification of it
in order to accommodate a situation with graded fibers.

Another possibility is to introduce a notion of “weighted algebroids”, by
simply replacing the tangent bundle in the original definition 2.17 by the
weighted tangent bundle
$TM^{\mathbb{C}}_{(r,r^{\prime};w)}=TM\otimes\mathbb{C}\otimes\mathbb{S}(r,r^{\prime};w)$.
The bracket (4.27) is also defined for any section of
$TM^{\mathbb{C}}_{(r,r^{\prime};w)}$, so the ordinary Lie bracket of the
tangent bundle could be replaced by (4.27). A “weighted Lie algebroid” would
then be given by definition 2.17 and its four items, but using
$TM^{\mathbb{C}}_{(r,r^{\prime};w)}$ instead of the tangent bundle. Then, by
simply taking the inclusion map $L_{(r,r^{\prime};w)}\hookrightarrow
TM^{\mathbb{C}}_{(r,r^{\prime};w)}$ as “anchor” (in other words, $\rho=P$
again), the triple $(\mathbb{L},P,[\cdot,\cdot]_{\mathcal{C}})$ is a “weighted
Lie algebroid” in the above sense.

The above definition is completely ad hoc, and it is not clear if one gains
anything at all by introducing such a structure, apart from perhaps adapting
the Lie algebroid definition to a situation with graded fibers in this
particular case. However, our motivation for analysing this issue was, as
mentioned at the end subsection 4.2, to see if the differential complex (4.25)
can be generalized to objects with arbitrary weights. The fact that (4.25) is
indeed a differential complex is a consequence of: $(i)$ the morphism property
of the anchor, and $(ii)$ the vanishing Jacobiator for the Lie algebroid
bracket. If we now take the inclusion $L_{(r,r^{\prime};w)}\hookrightarrow
TM^{\mathbb{C}}_{(r,r^{\prime};w)}$ as anchor, then it is trivially a
morphism, and the Jacobiator for the bracket (4.27) has been investigated in
Lemma 4.7. Therefore, it still makes sense to try to construct an analogue of
the complex (4.25). To this end, we first introduce weighted differential
forms as sections of the space151515Note that
$\Lambda^{k}_{(r,r^{\prime};w)}\neq\wedge^{k}(L\otimes\mathbb{S}(r,r^{\prime};w))^{*}$.
$\Lambda^{k}_{(r,r^{\prime};w)}:=(\wedge^{k}L^{*})\otimes\mathbb{S}(r,r^{\prime};w)$,
and we define the “weighted Lie algebroid differential”,

${\rm
d}^{\mathbb{L}}:\Gamma(\Lambda^{k}_{(r,r^{\prime};w)})\to\Gamma(\Lambda^{k+1}_{(r,r^{\prime};w)}),$
(4.36)

by its action on weighted 0- and 1-forms, in close analogy to (2.14)-(2.15):

$\displaystyle({\rm d}^{\mathbb{L}}f)(X):={}$ $\displaystyle\mathcal{C}_{X}f,$
(4.37) $\displaystyle({\rm d}^{\mathbb{L}}\omega)(X,Y):={}$
$\displaystyle\mathcal{C}_{X}\omega(Y)-\mathcal{C}_{Y}\omega(X)-\omega([X,Y]_{\mathcal{C}}).$
(4.38)

Explicitly, these expressions are formally equal to the right hand sides of
(4.12)-(4.13), but now $f$ and $\omega$ are allowed to have arbitrary weights.
We have:

######  Lemma 4.8.

Suppose that $\beta^{A}$ is shear-free. Then the following sequence is a
differential complex if and only if $\beta^{A}$ is also a repeated principal
spinor of the ASD Weyl curvature161616A similar complex was obtained in [20],
but not in the context of Lie algebroids.:

$0\to\Gamma(\Lambda^{0}_{(r,r^{\prime};w)})\xrightarrow{{\rm
d}^{\mathbb{L}}}\Gamma(\Lambda^{1}_{(r,r^{\prime};w)})\xrightarrow{{\rm
d}^{\mathbb{L}}}\Gamma(\Lambda^{2}_{(r,r^{\prime};w)})\to 0.$ (4.39)

###### Proof.

We have to check ${\rm d}^{\mathbb{L}}\circ{\rm d}^{\mathbb{L}}=0$. We only
have to see this for (weighted) 0-forms, since for a 1-form $\omega$, ${\rm
d}^{\mathbb{L}}{\rm d}^{\mathbb{L}}\omega$ is a 3-form in a 2-dimensional
space and thus it vanishes automatically. Consider an arbitrary weighted
scalar $f\in\Gamma(\Lambda^{0}_{(r,r^{\prime};w)})$, and two arbitrary
weighted vectors $X^{a}=\beta^{A}x^{A^{\prime}}$,
$Y^{a}=\beta^{A}y^{A^{\prime}}$, then, using (4.37), (4.38):

$\displaystyle({\rm d}^{\mathbb{L}}{\rm d}^{\mathbb{L}}f)(X,Y)={}$
$\displaystyle\mathcal{C}_{X}\mathcal{C}_{Y}f-\mathcal{C}_{Y}\mathcal{C}_{X}f-\mathcal{C}_{[X,Y]_{\mathcal{C}}}f$
$\displaystyle={}$ $\displaystyle
X^{a}Y^{b}(\mathcal{C}_{a}\mathcal{C}_{b}-\mathcal{C}_{b}\mathcal{C}_{a})f$
$\displaystyle={}$ $\displaystyle
x^{A^{\prime}}y^{B^{\prime}}\beta^{A}\beta^{B}[\mathcal{C}_{a},\mathcal{C}_{b}]f$
$\displaystyle={}$ $\displaystyle
x_{B^{\prime}}y^{B^{\prime}}\mathcal{C}_{A^{\prime}}\mathcal{C}^{A^{\prime}}f$

where in the fourth line we used (C.8). Therefore, ${\rm d}^{\mathbb{L}}{\rm
d}^{\mathbb{L}}f=0$ if and only if
$\mathcal{C}_{A^{\prime}}\mathcal{C}^{A^{\prime}}f=0$, which, because of Lemma
C.5, is equivalent to requiring $\beta^{A}$ to be a repeated principal spinor.
∎

Therefore, we see that as long as $\beta^{A}$ is shear-free and a repeated
principal spinor, the “potentialization” (4.26) is also valid for fields with
arbitrary weights.

######  Remark 4.9.

If we are only interested in a particular metric $g_{ab}$, i.e. not in a
conformal structure, then instead of $\mathcal{C}_{AA^{\prime}}$ we can use
the so-called “GHP” covariant derivative $\Theta_{AA^{\prime}}$ (which
consists of only some parts of $\mathcal{C}_{AA^{\prime}}$) and its
projections $\Theta_{A^{\prime}},\tilde{\Theta}_{A^{\prime}}$. One can show
that if $\beta^{A}$ is shear-free and a repeated principal spinor, then
$\Theta_{A^{\prime}}=\beta^{A}\Theta_{AA^{\prime}}$ also satisfies
$\Theta_{A^{\prime}}\Theta^{A^{\prime}}=0$ (on scalars and primed spinor
fields), so we again have a “potentialization” scheme for
$\Theta_{A^{\prime}}$. This is particularly useful for analysing integrability
of the Einstein equations.

######  Remark 4.10.

Although the Leibniz rule is not included in the definition 2.22 of a Courant
algebroid, we saw in eq. (2.16) that it is actually a consequence of the
axioms; thus, the vector bundle $\mathbb{L}$ cannot give an ordinary Courant
algebroid structure either.

##  5 Twistors