# IDEAL characterization of vacuum pp-waves

Igor Khavkine, David McNutt, Lode Wylleman

###### Abstract

An IDEAL characterization of a particular spacetime metric, $g_{0}$, consists
of a set of tensorial equations $T[g]=0$ arising from expressions constructed
from the metric, $g$, its curvature tensor and its covariant derivatives and
which are satisfied if and only if $g$ is locally isometric to the original
metric $g_{0}$. Earlier applications of the IDEAL classification of spacetimes
relied on the construction of particular scalar polynomial curvature
invariants as an important step in the procedure.

In this paper we investigate the well-known class of vacuum pp-wave
spacetimes, where all scalar polynomial curvature invariants vanish, and
determine the applicability of an IDEAL classification for these spacetimes.
We consider a modification of the IDEAL approach which permits a corresponding
extension of the Stewart-Walker lemma. With this change, we are able to
construct invariants and IDEAL-ly classify all of the vacuum pp-wave solutions
which admit a two- or higher-dimensional isometry group, with the exception of
one case.

##  1 Introduction

An old topic in General Relativity [33, Ch.9] [27] is the question of how to
decide when two explicit solutions of Einstein’s equations are actually the
same solution up to a change of coordinates, or conversely are distinct even
allowing for a change of coordinates. These questions are special cases of
_invariant equivalence_ and _characterization_ within the broader scope of
Riemannian, pseudo-Riemannian, and more general kinds of geometries [31]. One
traditional approach to such questions is the so-called _Cartan-Karlhede
algorithm_ [22] [33, §9.2], which assigns a normalized frame, specially
adapted to the algebraic structures of the Weyl curvature tensor and Ricci
tensor, to the spacetime. An alternative, which has gained some attention
recently [12, 13, 11, 14, 15, 19, 18, 20, 26, 6, 25], is the so-called _IDEAL_
111The acronym, explained in [14] (footnote, p.2), stands for Intrinsic,
Deductive, Explicit and ALgorithmic. _characterization_ approach, based on
algebraic identities between (possibly tensorial) curvature invariants of a
spacetime in question, which does not require auxiliary geometric structures
other than the metric itself. An advantage of the Cartan-Karlhede approach is
its generality and breadth of applicability. On the other hand, IDEAL
characterizations have the advantage of being intimately connected to
linearized gauge-invariant observables [23, 17], intrinsic characterization of
a spacetime by its initial data [18], as well as quantitative estimates of
convergence to a target spacetime [3, 30].

Naively, the easiest way to compare two metrics $(g_{1},M_{1})$ and
$(g_{2},M_{2})$ is through their _scalar curvature invariants_. In fact,
picking a certain number of scalars
$\mathcal{R}[g]=(\mathcal{R}^{1}[g],\ldots,\mathcal{R}^{N}[g])$ constructed by
total contractions of copies of the metric $g$, its curvature tensor
$R_{abcd}[g]$, and its covariant $\nabla_{e}$-derivatives, one can immediately
conclude that the metrics on domains $U_{1}\subset M_{1}$ and $U_{2}\subset
M_{2}$ are not isometric if the images of the maps $\mathcal{R}[g_{1}]\colon
U_{1}\to\mathbb{R}^{N}$ and $\mathcal{R}[g_{2}]\colon U_{2}\to\mathbb{R}^{N}$
do not coincide. For Riemannian metrics, under mild uniformity assumptions,
one can conversely conclude that the equality of these images implies the
existence of an isometry between $(g_{1}|_{U_{1}},U_{1})$ and
$(g_{2}|_{U_{2}},U_{2})$, for special choices of $N$ and $\mathcal{R}[g]$ [8,
9].

In the Lorentzian case, there exist large families of spacetime metrics that
cannot be distinguished by any scalar curvature invariants, while containing
non-isometric members [8, 9]. It is for this reason that the Cartan-Karlhede
algorithms supplements scalar cuvature invariants with invariants with respect
to a normalized frame. Analogously, the IDEAL approach makes additional use of
tensorial curvature invariants. The algorithmic aspects of the Cartan-Karlhede
approach have been extensively studied and its scope of applicability and
limitations are well-understood [33, §9.2]. No systematic approach has yet
been developed for IDEAL characterizations and they have been so far found
only by ad-hoc methods for special families of spacetime geometries.

_Vacuum pp-waves_ [10] [33, §24.5] are a family of non-flat spacetimes with
special algebraic and symmetry properties. All possible scalar curvature
invariants of a vacuum pp-wave vanish, yet the family contains mutually non-
isometric subfamilies that are parametrized by free continuous parameters and
even functions. These spacetimes belong to the larger class of VSI spacetimes
for which all scalar polynomial curvature invariants vanish [32].

This property along with the pp-waves interpretation as a toy-model for
gravitational waves has lead to the study of these solutions in string theory
and alternative gravity theories [21]. The vacuum pp-waves provide a simple
illustration of the significant difference in the approaches to the local
characterization of Riemannian metrics and Lorentzian metrics using invariants
constructed from the metric, the curvature tensor and its covariant
derivatives.

Vacuum pp-waves have been extensively studied in the Cartan-Karlhede approach,
including a complete classification [28, 29, 7]. To date, no IDEAL
characterization of vacuum pp-waves has been carried out. In this work, we
partially rectify this omission and give an as complete as possible IDEAL
characterization of isometry classes within the $4$-dimensional vacuum pp-wave
subfamily with more than the minimum number of Killing vectors (_highly
symmetric_ pp-waves). See Theorem 8 for the precise statement, including the
description of the characterized subfamily and the small list of exceptions.
We continue to use ad-hoc methods in this case, made difficult by the
impossibility to use any non-trivial scalar curvature invariants. We hope that
the experience gained in this work will help in formulating a systematic
approach to IDEAL characterizations.

In Section 2, we recall the precise notion of an IDEAL characterization and
introduce some ways that we were forced to generalize it. Section 3 contains
relevant background geometric information about $4$-dimensional vacuum pp-wave
spacetimes. In Section 4 we present our main results including the
classification of isometry classes within the highly symmetric subfamily
(Theorem 7, Table LABEL:tab:pp-wave-isom-classes, Figure 1), and the IDEAL
characterization of thse isometry classes (Theorem 8, Table LABEL:tab:pp-wave-
ideal, Figure 2. Section 5 finishes with a discussion and an outlook toward
future work.

##  2 IDEAL characterization

As was alluded to in the Introduction, an IDEAL characterization of a
spacetime is a local invariant condition that uses only the metric tensor and
quantities that can be covariantly obtained from it. More specifically, an
ideal characterization of a geometry $(M,g_{0})$ is a list of tensorial
equations

$\mathbf{T}_{k}[g]=0,\quad k=1,2,\ldots,N,$ (1)

constructed covariantly out of a metric $g$ and its derivatives (concomitants
of the Riemann tensor) that are satisfied on a small neigborhood of a point if
and only if that neighborhood is isomorphic to a neighborhood of a point of
$(M,g_{0})$. In this case, we say that the tensor identities (1) characterize
the local isometry class of $(M,g_{0})$. We also allow the possibility that
$g_{0}$ is a family of (possibly non-isometric) metrics and the tensor
identities (1) imply that $g$ is locally isometric to some member of that
family. In such cases, we speak of an IDEAL characterization of the whole
family.

The most classical example is of spaces of locally constant curvature,
characterized by the identity

$R_{abcd}=\alpha\left(g_{ac}g_{bd}-g_{ad}g_{bc}\right),$ (2)

where $\alpha$ is a constant; in relativity it is re-expressed via the
cosmological constant $\Lambda=\frac{(n-1)(n-2)}{2}\alpha$, in dimensions
$n>2$. Of course $\alpha=0$ corresponds to the locally flat case. The
_locally_ attribute is important because (2) as any IDEAL characterization (1)
is a fundamentally local condition, which is insensitive to the global
topology of the spacetime. Other notable geometries that have been given IDEAL
characterizations include Schwarzschild [12] (including in higher dimensions
[25]) and Kerr [13] black holes, as well as FLRW [6] cosmological solutions. A
growing list of other examples can be gleaned from the references in the
Introduction.

For the sake of terminology, let _IDEAL tensors_ refer to the left-hand sides
of (1). If $v$ is a vector field and $\mathbf{T}[g]$ is tensor covariantly
constructed from the metric, then the identity (sometimes known as the
_Stewart-Walker lemma_ [34])
$\dot{\mathbf{T}}_{g}[\mathcal{L}_{v}g]=\mathcal{L}_{v}\mathbf{T}[g]$, where
$\mathbf{T}[g+h]=\mathbf{T}[g]+\dot{\mathbf{T}}_{g}[h]+O(h^{2})$, immediately
links an IDEAL characterization to linear gauge invariant observables. Namely,
if $\mathbf{T}[g]$ is one of the IDEAL tensors for $g$, then
$\dot{\mathbf{T}}_{g}[\mathcal{L}_{v}g]=0$ implies that
$\dot{\mathbf{T}}_{g}[h]$ is invariant under local linearized gauge
transformations (diffeomorphisms) of linearized gravity about the background
$g$. In fact, linearizing all the IDEAL tensors for a given $g$ yields a good
candidate for a complete set of linear gauge invariants on that background
[23, 17, 24]. We consider this connection to linear gauge invariants through
linearization to be one of the important applications of IDEAL
characterizations.

Later, we will find that the limitations on how IDEAL tensors $\mathbf{T}[g]$
are to be constructed to be slightly restrictive. For instance,
$\mathbf{A}[g]$ and $\mathbf{B}[g]$ may be two such tensors, of equal valence
say, that become proportional, $\mathbf{A}[g]=X[g]\mathbf{B}[g]$, for some
subclass of metrics. Taking derivatives and tensor products of $\mathbf{A}[g]$
and $\mathbf{B}[g]$, it might be possible to impose some additional limited
set of polynomial conditions on the scalar $X[g]$ and its derivatives, but
non-polynomial conditions would certainly be impossible. That is a significant
limitation while trying to characterize all isometry classes of pp-wave
spacetimes. However, supposing that the proportionality factor $X[g]$ exists,
and using it in further non-polynomial conditions seems to be sufficient to do
the job. A natural question then: is this step in the spirit of IDEAL
characterizations?

The E in the acronym stands for Explicit, so with $X[g]$ being only implicitly
defined, the strict answer seems to be No. However, if $X[g]$ exists, it is
defined uniquely. Moreover, as we shall prove below, when $X[g]$ is defined,
it still satisfies the Stewart-Walker lemma, thus preserving the connection
with linear gauge invariants that we described above. Let us make the last
statement more precise with the following lemma, where a scalar $X[g]$ is just
a special case of a tensor field $\mathbf{X}[g]$.

######  Lemma 1 (extended Stewart-Walker).

Let $\mathbf{A}$ and $\mathbf{B}$ be tensors covariantly constructed from the
metric. Suppose the background metric $g_{ab}$ is chosen so that
$\mathbf{B}\neq 0$ and the relation $\mathbf{A}=\mathbf{X}\otimes\mathbf{B}$
holds for some tensor $\mathbf{X}$. Then there exists a differential operator
$\dot{\mathbf{X}}[h]$ acting on metric perturbations $h_{ab}$ such that
$\mathcal{L}_{v}\mathbf{X}=\dot{\mathbf{X}}[\mathcal{L}_{v}g]$ for any vector
field $v^{a}$.

###### Proof.

Let us denote the linearization by a dot, as in
$\mathbf{A}_{g+h}=\mathbf{A}_{g}+\dot{\mathbf{A}}_{g}[h]+O(h^{2})$. In
general, $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{X}$ may all have different
numbers of indices (including no indices), say $\mathbf{A}_{c_{1}\cdots
c_{k}:a_{1}\cdots a_{j}}$ and $\mathbf{B}_{b_{1}\cdots b_{j}}$, with
$\mathbf{X}_{c_{1}\cdots c_{k}}$, and where $:$ is used only to visually
separate groups of indices. Let us define the operation

$(\mathbf{A}\wedge\mathbf{B})_{c_{1}\cdots c_{k}:a_{1}\cdots a_{j}:b_{1}\cdots
b_{j}}=\mathbf{A}_{c_{1}\cdots c_{k}:a_{1}\cdots a_{j}}\mathbf{B}_{b_{1}\cdots
b_{j}}-\mathbf{A}_{c_{1}\cdots c_{k}:b_{1}\cdots b_{j}}\mathbf{B}_{a_{1}\cdots
a_{j}}.$ (3)

By elementary linear algebra, the relation
$\mathbf{A}=\mathbf{X}\otimes\mathbf{B}$ between $\mathbf{A}$ and $\mathbf{B}$
holds iff $\mathbf{A}\wedge\mathbf{B}$ = 0. It is also convenient to note the
antisymmetry identity
$(\mathbf{X}\otimes\mathbf{B})\wedge\mathbf{C}=-(\mathbf{X}\otimes\mathbf{C})\wedge\mathbf{B}$.
Linearizing the identity $\mathbf{A}\wedge\mathbf{B}=0$ about the chosen
background metric $g_{ab}$, we find (dropping all $g$ subscripts)

$\dot{\mathbf{A}}\wedge\mathbf{B}+\mathbf{A}\wedge\dot{\mathbf{B}}=0.$ (4)

Using antisymmetry and bilinearity of $\wedge$, as well as the background
relation $\mathbf{A}=\mathbf{X}\otimes\mathbf{B}$, we arrive at

$(\dot{\mathbf{A}}-\mathbf{X}\otimes\dot{\mathbf{B}})\wedge\mathbf{B}=0,$ (5)

which implies that there must exist an operator $\dot{\mathbf{X}}[h]$ such
that
$\dot{\mathbf{A}}-\mathbf{X}\otimes\dot{\mathbf{B}}=\dot{\mathbf{X}}\otimes\mathbf{B}$.
The notation $\dot{\mathbf{X}}$ is just formal, since $\mathbf{X}$ itself is
not defined for arbitrary background metrics.

Next, we fix a vector field $v^{a}$ and recall the Stewart-Walker lemma,
$\mathcal{L}_{v}\mathbf{A}=\dot{\mathbf{A}}[\mathcal{L}_{v}g]$ and
$\mathcal{L}_{v}\mathbf{B}=\dot{\mathbf{B}}[\mathcal{L}_{v}g]$. Taking the Lie
derivative of the relation between $\mathbf{A}$ and $\mathbf{B}$ obtains the
relation

$0=\mathcal{L}_{v}(\mathbf{A}-\mathbf{X}\otimes\mathbf{B})\\\
=(\dot{\mathbf{A}}[\mathcal{L}_{v}g]-\mathbf{X}\otimes\dot{\mathbf{B}}[\mathcal{L}_{v}g])-(\mathcal{L}_{v}\mathbf{X})\otimes\mathbf{B}=(\dot{\mathbf{X}}[\mathcal{L}_{v}g]-\mathcal{L}_{v}\mathbf{X})\otimes\mathbf{B},$
(6)

which implies that
$\mathcal{L}_{v}\mathbf{X}=\dot{\mathbf{X}}[\mathcal{L}_{v}g]$, since
$\mathbf{B}\neq 0$. ∎

When $\mathbf{X}[g]$ is defined implicitly as in Lemma 1, we will nevertheless
refer to the corresponding $\dot{\mathbf{X}}_{g}[h]$ as the _linearization_ of
$\mathbf{X}[g]$ about the metric $g_{ab}$. Thus, since the connection to
linear gauge invariants through linearization is important, for the purposes
of this work a characterization of a spacetime geometry will still be
considered to be in the spirit of IDEAL characterization if it is IDEAL except
for explicitly using an implicitly defined tensor like $\mathbf{X}[g]$.

In light of the extended Stewart-Walker lemma, we will refer to a tensors
defined implicitly from two or more IDEAL tensors as

######  Definition 1 (conditional invariants).

Given two tensors ${\bf A}$ and ${\bf B}$ which are covariantly constructed
from the metric, if there is a tensor of lower rank, ${\bf X}$, such that
${\bf A}={\bf X}\otimes{\bf B}$, then ${\bf X}$ is a _conditional invariant
(tensor)_. If the rank is zero, then $X$ is a _conditional invariant
(scalar)_.

Conditional invariants, tensors ${\bf X}$ or scalars $X$, will be determined
from tensors constructed from the curvature tensor and its covariant
derivatives.

##  3 Vacuum pp-waves

Plane-fronted gravitational waves with parallel rays, in short _pp-waves_ ,
can be defined as follows.

######  Definition 2.

A pp-wave is a spacetime that admits a covariantly constant null vector field,
$\ell_{a}$:

$\nabla_{a}\ell_{b}=0,\quad\ell^{a}\ell_{a}=0.$ (7)

These spacetimes made their first appearance in the work of Brinkmann [5].
Subsequently, they have been widely studied [33, §24.5]. For the rest of this
section, we restrict to $4$ spacetime dimensions. In $4$ dimensions, an
Einstein vacuum pp-wave has the line element [33, §24.5]

$\mathrm{d}{s}^{2}=2\mathrm{d}\zeta\mathrm{d}\bar{\zeta}-2\mathrm{d}{u}\mathrm{d}{v}-2H\mathrm{d}{u}^{2},\quad
H\equiv H(u,\zeta,\bar{\zeta})=f(\zeta,u)+\bar{f}(\bar{\zeta},u),$ (8)

where $u,v$ are two real coordinates and $\zeta=x+iy$ is a convenient complex
coordinate, while $H(u,\zeta,\bar{\zeta})$ is real harmonic in $\zeta$ and
jointly smooth, justifying its presentation in terms of the complex
holomorphic in $\zeta$ and jointly smooth $f(\zeta,u)$, $f_{,\bar{\zeta}}=0$.
The $1$-forms

$\bm{\ell}=\mathrm{d}{u},\quad\mathbf{n}=\mathrm{d}{v}+H\mathrm{d}{u},\quad\mathbf{m}=\mathrm{d}\zeta,\quad\bar{\mathbf{m}}=\mathrm{d}\bar{\zeta}$
(9)

constitute a complex null coframe dual to the compex null frame of vector
fields

$\ell=-\partial_{v},\quad n=-\partial_{u}+H\partial_{v},\quad
m=\partial_{\bar{\zeta}},\quad\bar{m}=\partial_{\zeta},$ (10)

normalized as $\ell^{a}n_{a}=n^{a}\ell_{a}=-1$,
$m^{a}\bar{m}_{a}=\bar{m}^{a}m_{a}=1$, and all other contractions vanishing.

After fixing the form (8) of the metric, the residual gauge freedom (coming
from coordinate transformations or underdeterminedness of the parameters)
consists of the continuous transformations

$\displaystyle\zeta^{\prime}=e^{i\beta}(\zeta+h(u)),\quad
u^{\prime}=(u+u_{0})/a,\quad
v^{\prime}=a\,[v+\partial_{u}(h(u)\bar{\zeta}+\bar{h}(u)\zeta)+g(u)],$
$\displaystyle
f^{\prime}(\zeta^{\prime},u^{\prime})=a^{2}\,\left[f(\zeta,u)-\partial_{u}^{2}\bar{h}(u)\zeta+\tfrac{1}{2}\left|\partial_{u}\bar{h}(u)\right|^{2}-\tfrac{1}{2}\partial_{u}g(u)+ic(u)\right],$
(11a) where $\beta$, $a$, $u_{0}$ are real constants, $c(u)$ and $g(u)$ are
smooth real functions, and $h(u)$ is a smooth complex function, and the
discrete transformation $\zeta^{\prime}=\bar{\zeta},\quad v^{\prime}=v,\quad
u^{\prime}=u,\quad f^{\prime}=\bar{f},$ (11b)

where $\bar{f}(\zeta,u)=\overline{f(\bar{\zeta},u)}$. The free choice of
$h(u)$ or $c(u)$ and $g(u)$, respectively, allows us to adjust both
$\partial_{u}^{2}\bar{h}(u)$ and
$\frac{1}{2}\left|\partial_{u}\bar{h}(u)\right|^{2}-\frac{1}{2}\partial_{u}g(u)+ic(u)$
to be arbitrary smooth complex functions of $u$.

The above line element (8) is consistent with $\ell^{a}$ being covariantly
constant, hence in particular a Killing vector. Thus, the spacetime admits at
least a $G_{1}$ group of isometries [33], which is the maximal isometry group
for generic $f$.

###  3.1 Curvature invariants

Denote by $R_{ab}$ the Ricci tensor, by $C_{abcd}$ the Weyl tensor, and by
$\eta_{abcd}$ the Levi-Civita $4$-form. In addition, let us define the
_complex self-dual Weyl tensor_

$\mathbf{C}^{\dagger}=\mathbf{C}-i{}^{*}\mathbf{C},\quad\text{where}\quad{}^{*}C_{abcd}=\frac{1}{2}\eta_{abef}C^{ef}{}_{cd},$
(12)

and the _Bel-Robinson tensor_ [1, 2]

$T_{abcd}=C_{aecf}C_{b}{}^{e}{}_{d}{}^{f}+{{}^{*}}C_{aecf}{{}^{*}}C_{b}{}^{e}{}_{d}{}^{f}=C^{\dagger}_{aecf}\bar{C}^{\dagger}{}_{b}{}^{e}{}_{d}{}^{f},$
(13)

which is fully symmetric $T_{abcd}=T_{(abcd)}$. The spacetime is called
_complex recurrent_ when the complex self-dual Weyl tensor satisfies

$\nabla\mathbf{C}^{\dagger}=\mathbf{K}\otimes\mathbf{C}^{\dagger}$ (14)

for some (unique) $1$-form $\mathbf{K}$. A null vector field $\ell^{a}$ is
itself called _recurrent_ when

$\nabla\bm{\ell}=\mathbf{p}\otimes\bm{\ell}\quad\text{or
equivalently}\quad\ell_{[a}\ell_{b];c}=0,$ (15)

for some (unique) $1$-form $\mathbf{p}$.

The complex self-dual Weyl tensor of vacuum pp-waves takes the form

$\mathbf{C}^{\dagger}=8f_{,\zeta\zeta}(\mathrm{d}{u}\wedge\mathrm{d}\zeta)\otimes(\mathrm{d}{u}\wedge\mathrm{d}\zeta)=8f_{,\zeta\zeta}(\bm{\ell}\wedge\mathbf{m})\otimes(\bm{\ell}\wedge\mathbf{m}).$
(16)

The special case $f_{,\zeta\zeta}=0$ corresponds to Minkowski space. We
exclude this case from further consideration, meaning that we assume from now
on assume that $f$ is not a linear polynomial in $\zeta$. Hence
$f_{,\zeta\zeta}\neq 0$ almost everywhere, and henceforth we implicitly
consider the spacetime points where this inequality holds. At those points the
Weyl tensor is of Petrov type N, $\bm{\ell}$ spanning the unique quadruple
_principal null direction (PND)_ [33, $\S$ 4.3]. The Bel-Robinson tensor takes
the form [4]

$T_{abcd}=\beta\ell_{a}\ell_{b}\ell_{c}\ell_{d},\quad\beta=4\left|f_{,\zeta\zeta}\right|^{2}.$
(17)

Direct calculation shows that $\mathbf{C}^{\dagger}$ is complex recurrent,
with recurrence $1$-form

$\mathbf{K}=\mathrm{d}(\ln f_{,\zeta\zeta})=\frac{(f_{,\zeta\zeta
u}\bm{\ell}+f_{,\zeta\zeta\zeta}\mathbf{m})}{f_{,\zeta\zeta}},$ (18)

which clearly satisfies $K^{a}\ell_{a}=0$. For reference, it is useful to
compute

$\nabla{\mathbf{K}}=\left((\ln
f_{,\zeta\zeta})_{,uu}-\bar{f}_{,\bar{\zeta}}(\ln
f_{,\zeta\zeta})_{,\zeta}\right)\bm{\ell}\otimes\bm{\ell}\\\ +(\ln
f_{,\zeta\zeta})_{,\zeta
u}(\bm{\ell}\otimes\mathbf{m}+\mathbf{m}\otimes\bm{\ell})+(\ln
f_{,\zeta\zeta})_{,\zeta\zeta}\mathbf{m}\otimes\mathbf{m},$ (19)
$\nabla\nabla{\mathbf{K}}=\left((\ln
f_{,\zeta\zeta})_{,uuu}-3\bar{f}_{,\bar{\zeta}}(\ln f_{,\zeta\zeta})_{,\zeta
u}\right)\bm{\ell}\bm{\ell}\bm{\ell}+(\ln f_{,\zeta\zeta})_{,\zeta
uu}(\mathbf{m}\bm{\ell}\bm{\ell}+\bm{\ell}\bm{\ell}\mathbf{m}+\bm{\ell}\mathbf{m}\bm{\ell})\\\
+(\ln f_{,\zeta\zeta})_{,\zeta\zeta
u}(\mathbf{m}\bm{\ell}\mathbf{m}+\mathbf{m}\mathbf{m}\bm{\ell}+\bm{\ell}\mathbf{m}\mathbf{m})+(\ln
f_{,\zeta\zeta\zeta})_{,\zeta\zeta}\mathbf{m}\mathbf{m}\mathbf{m}\\\
-\bar{f}_{,\bar{\zeta}}(\ln
f_{,\zeta\zeta})_{,\zeta\zeta}(\mathbf{m}\bm{\ell}\bm{\ell}+\bm{\ell}\bm{\ell}\mathbf{m}+\bm{\ell}\mathbf{m}\bm{\ell})-(\ln
f_{,\zeta\zeta})_{,\zeta}(\bar{f}_{,\bar{\zeta}u}\bm{\ell}\bm{\ell}\bm{\ell}+\bar{f}_{,\bar{\zeta}\bar{\zeta}}\overline{\mathbf{m}}\bm{\ell}\bm{\ell})$
(20)

The vector $\mathbf{K}$ is a conditional invariant tensor and is the main
geometric quantity that we will use to build further invariants that will be
used in later sections. We introduce these derived quantities here. To do
this, it will be necessary at times to contract the vector $\mathbf{K}$ with
various tensors and this will be denoted as

$\mathbf{K}\cdot{\bf A}=A_{a_{1}...a_{k}}K^{a_{k}}.$ (21)

In general the conditional invariants will be rational functions, so they will
be well-defined only when their denominators are non-vanishing.

The simplest invariant is just

$|\mathbf{K}|^{2}:=\overline{\mathbf{K}}\cdot{\mathbf{K}}=\left|(\ln
f_{,\zeta\zeta})_{,\zeta}\right|^{2}.$ (22)

From (17), (18) and (3) we have that

$|\mathbf{K}|^{2}=0\quad\Leftrightarrow\quad
f_{,\zeta\zeta\zeta}=0\quad\Leftrightarrow\quad\mathbf{T}\wedge\mathbf{K}=0,$
(23)

which implies $\mathbf{T}\wedge(\nabla\mathbf{K})=0$. In this case we define
the following conditional invariants:

$\displaystyle\left(I_{\mathrm{a}}^{(2)}\right)^{4}\mathbf{T}$
$\displaystyle:=16\mathbf{K}^{\otimes 4},$ $\displaystyle
I_{\mathrm{a}}^{(2)}$
$\displaystyle=\frac{2^{\frac{1}{2}}}{|f_{,\zeta\zeta}|^{\frac{1}{2}}}(\ln
f_{,\zeta\zeta})_{,u},$ (24)
$\displaystyle\left(I_{\mathrm{b}}^{(2)}\right)^{2}\mathbf{T}$
$\displaystyle:=16(\nabla\mathbf{K})^{\otimes 2},$ $\displaystyle
I_{\mathrm{b}}^{(2)}$ $\displaystyle=\frac{2}{|f_{,\zeta\zeta}|}(\ln
f_{,\zeta\zeta})_{,uu}.$ (25)

The meaning of the superscripts will become clear later. Strictly speaking,
these formulas only define the invariant scalars $I_{\mathrm{a}}^{(2)}$ and
$I_{\mathrm{b}}^{(2)}$ as raised to the powers
$\left(I_{\mathrm{a}}^{(2)}\right)^{4}$ and
$\left(I_{\mathrm{b}}^{(2)}\right)^{2}$. We must choose a branch of the square
and quartic roots to get rid of those powers. In principle, a branch choice
could be made separately at each point of $M$. However, once a branch choice
is made at a point $x\in M$, requiring continuity is sufficient to ensures a
unique branch choice on a sufficiently small contractible neighborhood of $x$,
provided of course that the invariant does not vanish at $x$ and on this
neighborhood. For later convenience in the proof of Theorem 8, we choose the
square root branch at $x$ so that $\operatorname{Im}I_{\mathrm{b}}^{(2)}\geq
0$, with a cut on the positive real axis and $I_{\mathrm{b}}^{(2)}<0$ on the
cut (namely $(+1)^{1/2}=-1$); we choose the quartic root branch so that
$\operatorname{Im}I_{\mathrm{a}}^{(2)}\geq 0$ and
$\operatorname{Re}I_{\mathrm{a}}^{(2)}\leq 0$, with a cut on the positive real
axis and $I_{\mathrm{a}}^{(2)}<0$ on the cut (namely $(+1)^{1/4}=-1$).

The next definitions for conditional invariant tensors and scalars will
require $|\mathbf{K}|\neq 0$:

$\displaystyle D_{ac}$
$\displaystyle:=\frac{\overline{K}^{b}\overline{K}^{d}C^{\dagger}_{abcd}}{|\mathbf{K}|^{4}}=2\frac{f_{,\zeta\zeta}^{3}}{f_{,\zeta\zeta\zeta}^{2}}\ell_{a}\ell_{c},$
(26) $\displaystyle J\mathbf{T}$
$\displaystyle:=|\mathbf{K}|^{4}(\mathbf{D})^{\otimes 2},\quad
J=\frac{f_{,\zeta\zeta}^{3}}{\bar{f}_{,\bar{\zeta}\bar{\zeta}}^{3}}\frac{\bar{f}_{,\bar{\zeta}\bar{\zeta}\bar{\zeta}}^{2}}{f_{,\zeta\zeta\zeta}^{2}},$
(27) $\displaystyle I_{\mathrm{c}}$
$\displaystyle:=\frac{\overline{\mathbf{K}}\cdot(\overline{\bf
K}\cdot\nabla{\bf K})}{|\mathbf{K}|^{4}}=-\left(\frac{1}{(\ln
f_{,\zeta\zeta})_{,\zeta}}\right)_{,\zeta},$ (28) $\displaystyle\mathfrak{L}$
$\displaystyle:=\frac{\overline{\mathbf{K}}\cdot\nabla\mathbf{K}}{|\mathbf{K}|^{2}}-I_{\mathrm{c}}\mathbf{K}=\left(\frac{(\ln
f_{,\zeta\zeta})_{,u}}{(\ln
f_{,\zeta\zeta})_{,\zeta}}\right)_{,\zeta}\bm{\ell},$ (29)
$\displaystyle\left(I_{\mathrm{a}}^{(0)}\right)^{4}\mathbf{T}$
$\displaystyle:=4|\mathbf{K}|^{4}\mathfrak{L}^{\otimes 4},\quad
I_{\mathrm{a}}^{(0)}=\left|\frac{f_{,\zeta\zeta\zeta}^{2}}{f_{,\zeta\zeta}^{3}}\right|^{\frac{1}{4}}\left(\frac{(\ln
f_{,\zeta\zeta})_{,u}}{(\ln f_{,\zeta\zeta})_{,\zeta}}\right)_{,\zeta},$ (30)
$\displaystyle\left(I_{\mathrm{b}}^{(0)}\right)^{2}\mathbf{T}$
$\displaystyle:=|\mathbf{K}|^{4}\left(\nabla\mathfrak{L}-\frac{\overline{\mathbf{K}}\cdot\nabla\mathfrak{L}}{|\mathbf{K}|^{2}}\mathbf{K}\right)^{\otimes
2},$ $\displaystyle I_{\mathrm{b}}^{(0)}$
$\displaystyle=\frac{1}{2}\left|\frac{f_{,\zeta\zeta\zeta}^{2}}{f_{,\zeta\zeta}^{3}}\right|^{\frac{1}{2}}\left[\left(\frac{(\ln
f_{,\zeta\zeta})_{,u}}{(\ln f_{,\zeta\zeta})_{,\zeta}}\right)_{,\zeta
u}-\left(\frac{(\ln f_{,\zeta\zeta})_{,u}}{(\ln
f_{,\zeta\zeta})_{,\zeta}}\right)\left(\frac{(\ln f_{,\zeta\zeta})_{,u}}{(\ln
f_{,\zeta\zeta})_{,\zeta}}\right)_{,\zeta\zeta}\right],$ (31)
$\displaystyle\overline{\mathbf{N}}$
$\displaystyle=\nabla\frac{\overline{\mathbf{K}}\cdot\nabla\mathbf{K}}{|\mathbf{K}|^{2}}-\left(\frac{\overline{\mathbf{K}}\cdot\nabla\mathbf{K}}{|\mathbf{K}|^{2}}\right)^{\otimes
2}+\frac{|\mathbf{K}|^{2}}{2}\overline{\mathbf{D}}.$ (32)

The branch choices for $I_{\mathrm{a}}^{(0)}$ and $I_{\mathrm{b}}^{(0)}$ are
the same as for $I_{\mathrm{a}}^{(2)}$ in (24) and $I_{\mathrm{b}}^{(2)}$ in
(25). When $\mathbf{N}\wedge\mathbf{D}=0$, we can also define the scalar

$N\mathbf{D}:=\mathbf{N}.$ (33)

We will need the coordinate expression for $N$ only in some special pp-wave
subcases. We note that $\mathfrak{L}$ is proportional to the covariantly
constant null vector $\bm{\ell}$ and so $\mathfrak{L}$ will be a recurrent
null vector field.

The next batch of definitions will require $|\mathbf{K}|\neq 0$ and also
$k(I_{\mathrm{c}}-\tfrac{1}{2})-I_{\mathrm{c}}\neq 0$ for some fixed constant
$k\in\mathbb{C}$.

$\displaystyle\mathbf{L}^{(k)}$
$\displaystyle:=\frac{\mathfrak{L}}{k(I_{\mathrm{c}}-\tfrac{1}{2})-I_{\mathrm{c}}}=-2\Xi^{(k)}\bm{\ell},$
(34) $\displaystyle\Xi^{(k)}$
$\displaystyle:=\frac{1}{2}\frac{\left(\frac{(\ln f_{,\zeta\zeta})_{,u}}{(\ln
f_{,\zeta\zeta})_{,\zeta}}\right)_{,\zeta}}{k\left[\left(\frac{1}{(\ln
f_{,\zeta\zeta})_{,\zeta}}\right)_{,\zeta}+\frac{1}{2}\right]-\left(\frac{1}{(\ln
f_{,\zeta\zeta})_{,\zeta}}\right)_{,\zeta}},$ (35) $\displaystyle
I_{\mathrm{d}}^{(k)}\mathbf{D}$
$\displaystyle:=-\frac{1}{2}\left(\mathbf{L}^{(k)}\right)^{\otimes 2},\quad
I_{\mathrm{d}}^{(k)}=-\left(\Xi^{(k)}\right)^{2}\frac{f_{,\zeta\zeta\zeta}^{2}}{f_{,\zeta\zeta}^{3}},$
(36)
$\displaystyle-\frac{1}{2}\overline{I_{\mathrm{e}}^{(k)}}|\mathbf{K}|^{2}\overline{\mathbf{D}}$
$\displaystyle:=\nabla\mathbf{K}-(\mathbf{K}\mathfrak{L}+\mathfrak{L}\mathbf{K})-I_{\mathrm{c}}\mathbf{K}\mathbf{K}$
$\displaystyle\quad{}-(k-1)\left[k(I_{\mathrm{c}}-1)-(I_{\mathrm{c}}+\tfrac{1}{2})\right]\mathbf{L}^{(k)}\mathbf{L}^{(k)},$
(37) $\displaystyle\overline{\mathbf{M}^{(k)}}$
$\displaystyle:=\left(\nabla-\frac{\mathbf{K}}{|\mathbf{K}|^{2}}\overline{\mathbf{K}}\cdot\nabla-\frac{\overline{\mathbf{K}}}{|\mathbf{K}|^{2}}\mathbf{K}\cdot\nabla\right)(\overline{I_{\mathrm{e}}^{(k)}}|\mathbf{K}|^{2}\overline{\mathbf{D}})$
$\displaystyle\quad{}-\mathbf{L}^{(k)}\left[(k-1)I_{\mathrm{c}}+1\right]\overline{I_{\mathrm{e}}^{(k)}}|\mathbf{K}|^{2}\overline{\mathbf{D}}$
$\displaystyle\quad{}-\frac{(k+1)}{2}\mathbf{L}^{(k)}\left[-2|\mathbf{K}|^{2}\overline{\mathbf{D}}+k(k-1)\mathbf{L}^{(k)}\mathbf{L}^{(k)}\right].$
(38)

For any pp-wave, the right hand sides of the last three formulas will be
proportional to a product of $\bm{\ell}$’s. All of the above formulas are
well-defined when $k=0$. On the other hand, most of them diverge or go to zero
as $k\to\infty$. It will be useful for us to define certain rescaled limits of
these formulas when $I_{\mathrm{c}}\neq\frac{1}{2}$:

$\displaystyle\mathbf{L}^{(\infty)}$
$\displaystyle:=\lim_{k\to\infty}k\mathbf{L}^{(k)}=\frac{\mathfrak{L}}{I_{\mathrm{c}}-\frac{1}{2}}=-2\Xi^{(\infty)}\bm{\ell},$
(39) $\displaystyle\Xi^{(\infty)}$
$\displaystyle:=\lim_{k\to\infty}k\Xi^{(k)}=\frac{1}{2}\frac{\left(\frac{(\ln
f_{,\zeta\zeta})_{,u}}{(\ln
f_{,\zeta\zeta})_{,\zeta}}\right)_{,\zeta}}{\left(\frac{1}{(\ln
f_{,\zeta\zeta})_{,\zeta}}\right)_{,\zeta}+\frac{1}{2}},$ (40) $\displaystyle
I_{\mathrm{d}}^{(\infty)}\mathbf{D}$
$\displaystyle:=-\frac{1}{2}\left(\mathbf{L}^{(\infty)}\right)^{\otimes
2},\quad
I_{\mathrm{d}}^{(\infty)}=\lim_{k\to\infty}k^{2}I_{\mathrm{d}}^{(k)}=-\left(\Xi^{(\infty)}\right)^{2}\frac{f_{,\zeta\zeta\zeta}^{2}}{f_{,\zeta\zeta}^{3}},$
(41)
$\displaystyle-\frac{1}{2}\overline{I_{\mathrm{e}}^{(\infty)}}|\mathbf{K}|^{2}\overline{\mathbf{D}}$
$\displaystyle:=\lim_{k\to\infty}-\frac{1}{2}\overline{I_{\mathrm{e}}^{(k)}}|\mathbf{K}|^{2}\overline{\mathbf{D}}$
$\displaystyle=\nabla\mathbf{K}-(\mathbf{K}\mathfrak{L}+\mathfrak{L}\mathbf{K})-I_{\mathrm{c}}\mathbf{K}\mathbf{K}-(I_{\mathrm{c}}-1)\mathbf{L}^{(\infty)}\mathbf{L}^{(\infty)},$
(42) $\displaystyle\overline{\mathbf{M}^{(\infty)}}$
$\displaystyle:=\lim_{\operatorname{Re}k=0,k\to\infty}\overline{\mathbf{M}^{(k)}}$
$\displaystyle:=\left(\nabla-\frac{\mathbf{K}}{|\mathbf{K}|^{2}}\overline{\mathbf{K}}\cdot\nabla-\frac{\overline{\mathbf{K}}}{|\mathbf{K}|^{2}}\mathbf{K}\cdot\nabla\right)(\overline{I_{\mathrm{e}}^{(\infty)}}|\mathbf{K}|^{2}\overline{\mathbf{D}})$
$\displaystyle\quad{}-I_{\mathrm{c}}\mathbf{L}^{(\infty)}\overline{I_{\mathrm{e}}^{(\infty)}}|\mathbf{K}|^{2}\overline{\mathbf{D}}-\frac{1}{2}\mathbf{L}^{(\infty)}\left[-2|\mathbf{K}|^{2}\overline{\mathbf{D}}+\mathbf{L}^{(\infty)}\mathbf{L}^{(\infty)}\right].$
(43)

In a similar way to $\mathbf{L}$, the definitions of $\mathbf{L}^{(k)}$ and
$\mathbf{L}^{(\infty)}$ are proportional to the covariantly constant null
vector field $\bm{\ell}$ and so they are both recurrent null vector fields,
when they are defined.

###  3.2 Specialization to subclasses

The exhaustive classification of the pp-wave classes listed in Proposition 3
was first carried out in the classic work [10, §2–5.6], where Ehlers & Kundt
gave a complete classification of vacuum pp-waves by their Lie algebras of
_infinitesimal local isometries_ 222At each point $x\in M$, the _infinitesimal
local isometries_ of $(M,g)$ are the Killing vector fields that can be defined
on at least some neighborhood of $x$. They obviously have some common domain
of definition on which they have the structure of a Lie algebra, via the Lie
bracket of vector fields. For each class, they gave a representative
functional form of $f(\zeta,u)$ depending on constant or functional
parameters, which we summarize in the following

######  Proposition 3 (pp-wave classes).

Let $(M,g)$ be a non-flat 4-dimensional vacuum pp-wave (Definition 2), where
$M$ is connected. Then, assuming the Lie algebra of infinitesimal local
isometries does not change at any point of $M$, the line element $g$ can be
locally put into the form (8), with the function $f(\zeta,u)$ belonging to one
of the following canonical forms

$f(\zeta,u)=\begin{cases}4\alpha u^{2i\kappa-2}\zeta^{2}&G_{6a}\\\
e^{2i\lambda u}\zeta^{2}&G_{6b}\\\ A(u)\zeta^{2}&G_{5}\\\ 4\alpha
e^{i\gamma}u^{-2}\ln\zeta&G_{3a}\\\ e^{i\gamma}\ln\zeta&G_{3b}\\\
e^{2\lambda\zeta}&G_{3c}\\\ e^{i\gamma}\zeta^{2i\kappa}&G_{3d}\\\
u^{-2}\mathrm{f}(\zeta u^{i\kappa})&G_{2a}\\\ \mathrm{f}(\zeta e^{i\lambda
u})&G_{2b}\\\ A(u)\ln\zeta&G_{2c}\\\ f(\zeta,u)&G_{1}\end{cases},$ (44)

where $\alpha$, $\gamma$, $\lambda$ and $\kappa$ are real constants, $A(u)$ is
an arbitrary smooth complex function and $f(\zeta)$ is an arbitrary complex
holomorphic function. Moreover, these constant and functional parameters are
the same in each adapted chart on $M$, up to form-preserving transformations
allowed by (11a).

We note that in the $G_{3a}$ and $G_{3b}$ cases, the functional form differs
by a phase factor which was not found in the original analysis by Ehler’s and
Kundt but was pointed out in Lemmas 8.9.1 and 8.9.4 of [28]. Our results will
cover the $G_{\text{2--6}}$ subclasses, excluding the generic component of the
$G_{1}$ class. We will refer to these geometries as

######  Definition 4 (highly-symmetric pp-waves).

Any pp-wave geometry covered by the classification of Proposition 3 that has
at least two independent Killing vectors at each point, or equivalently
belonging to one of the $G_{\text{2--6}}$ subclasses, will be called _highly
symmetric_.

Furthermore, we will impose an additional condition on the geometries studied
here, in order to restrict the form of the functional form on the entire
manifold.

######  Definition 5 (regular pp-wave).

A vacuum pp-wave geometry $(M,g)$ is _regular_ at $x\in M$ when the Lie
algebra of infinitesimal local isometries at each point of an open
neighborhood of $x$ is isomorphic to the one at $x$, and it satisfies the
following conditions, broken down by the symmetry classes listed in (44).
Inequalities are interpreted as holding at $x$ and equalities on at least some
open neighborhood of $x$ (note that $\wedge$ and $\vee$ respectively stand for
logical _and_ and _or_):

$G_{5}$: | $(B=\ln A)$
---|---
| $A\neq 0\wedge\dot{B}\neq 0\wedge\left(\ddot{B}\neq 0\vee\ddot{B}=0\right)$
|
${}\wedge\biggl{(}\partial_{u}(\operatorname{Re}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B})\neq
0\vee\Bigl{[}\partial_{u}(\operatorname{Re}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B})=0$
|
$\qquad{}\wedge\left(\partial_{u}(\operatorname{Im}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B})\neq
0\vee\partial_{u}(\operatorname{Im}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B})=0\right)\Bigr{]}\biggr{)}$
$G_{2a}$: | $\mathrm{f}^{\prime\prime}(z)\neq 0\wedge\left(\partial_{z}\left(\frac{\mathrm{f}^{\prime\prime\prime}(z)^{2}}{\mathrm{f}^{\prime\prime}(z)^{3}}\right)=0\vee\partial_{z}\left(\frac{\mathrm{f}^{\prime\prime\prime}(z)^{2}}{\mathrm{f}^{\prime\prime}(z)^{3}}\right)\neq 0\right)$
$G_{2b}$: | $\mathrm{f}^{\prime\prime}(z)\neq 0$
| ${}\wedge\biggl{(}\left[\lambda\neq
0\wedge\Bigl{(}\partial_{z}\left(\frac{\mathrm{f}^{\prime\prime\prime}(z)^{2}}{\mathrm{f}^{\prime\prime}(z)^{3}}\right)\neq
0\vee\partial_{z}\left(\frac{\mathrm{f}^{\prime\prime\prime}(z)^{2}}{\mathrm{f}^{\prime\prime}(z)^{3}}\right)=0\Bigr{)}\right]$
|
$\qquad{}\vee\left[\lambda=0\wedge\Bigl{(}\partial_{z}\left(\frac{\mathrm{f}^{\prime\prime\prime}(z)\mathrm{f}^{\prime}(z)}{\mathrm{f}^{\prime\prime}(z)^{2}}\right)\neq
0\vee\partial_{z}\left(\frac{\mathrm{f}^{\prime\prime\prime}(z)\mathrm{f}^{\prime}(z)}{\mathrm{f}^{\prime\prime}(z)^{2}}\right)=0\Bigr{)}\right]\biggr{)}$
$G_{2c}$: | $(B=\ln A)$
| $A\neq 0\wedge\dot{B}\neq 0\wedge\left(\ddot{B}\neq 0\vee\ddot{B}=0\right)$
|
${}\wedge\biggl{(}\partial_{u}(\operatorname{Re}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B})\neq
0\vee\Bigl{[}\partial_{u}(\operatorname{Re}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B})=0$
| $\qquad{}\wedge\left(\operatorname{Im}\dot{B}\neq
0\vee\operatorname{Im}\dot{B}=0\right)\Bigr{]}\biggr{)}$

If not explicitly listed above, the corresponding class is not restricted. A
_regular_ vacuum pp-wave geometry $(M,g)$ is one that is highly-symmetric
(Definition 4) and _regular_ at each point $x\in M$.

The logic behind the inequalities in Definition 5 is as follows. Some of them
prevent the point $x\in M$ from falling onto a branch point of the implicitly
defined invariants $I_{\mathrm{a}}^{(0,2)}$ and $I_{\mathrm{b}}^{(0,2)}$,
while others simplify the use of the implicit function theorem to find a
relation between certain invariants in the proof of Theorem 7.

In addition to the invariant quantities listed above, we will require some
specialized invariants that appear only appear in particular subclasses. In
the following we will consider the $G_{5}$ subclass along with the $G_{2a}$,
$G_{2b}$ and $G_{2c}$ subclasses and their respective generalizations arising
from including a $\zeta$-linear term and a $\zeta$-independent term to the
original function.

In the $G_{5}$ subclass, $f(\zeta,u)=A(u)\zeta^{2}$, let $A(u)=e^{B(u)}$. Then
we evaluate

$(I_{\mathrm{a}}^{(2)})^{4}=\dot{B}^{4}e^{-2\operatorname{Re}B},\quad(I_{\mathrm{b}}^{(2)})^{2}=\ddot{B}^{2}e^{-2\operatorname{Re}B}.$
(45)

Similarly, in the $G_{2c}$ subclass, $f(\zeta,u)=A(u)\ln\zeta$, let
$A(u)=e^{B(u)}$. Then we evaluate

$(I_{\mathrm{a}}^{(0)})^{4}=\dot{B}^{4}e^{-2\operatorname{Re}B},\quad(I_{\mathrm{b}}^{(0)})^{2}=\ddot{B}^{2}e^{-2\operatorname{Re}B},\quad\frac{(I_{\mathrm{a}}^{(0)})^{2}}{I_{\mathrm{d}}^{(0)}}=e^{i\operatorname{Im}B}.$
(46)

In the slightly more general class
$f(\zeta,u)=A(u)\ln\zeta+A_{1}(u)\zeta+A_{0}(u)$, we evaluate

$|\mathbf{K}|^{2}=\frac{4}{|\zeta|^{2}},\quad\mathbf{D}=-\frac{A}{2}\bm{\ell}\bm{\ell},\quad\mathbf{N}=\frac{A_{1}}{\bar{\zeta}}\bm{\ell}\bm{\ell},\quad\text{and}\quad
N=-\frac{2}{\bar{\zeta}}\frac{A_{1}}{A}.$ (47)

In the $G_{2a}$ class, $f(\zeta,u)=u^{-2}\mathrm{f}(\zeta u^{i\kappa})$.
Letting $z=\zeta u^{i\kappa}$, we then evaluate

$|\mathbf{K}|^{2}=\left|\frac{\mathrm{f}^{\prime\prime\prime}(z)}{\mathrm{f}^{\prime\prime}(z)}\right|^{2},\quad
I_{\mathrm{d}}^{(i\kappa)}=-\frac{\mathrm{f}^{\prime\prime\prime}(z)^{2}}{\mathrm{f}^{\prime\prime}(z)^{3}},\quad
I_{\mathrm{e}}^{(i\kappa)}=(\mathrm{f}^{\prime}(z)+i\kappa(i\kappa-1)\bar{z})\frac{\mathrm{f}^{\prime\prime\prime}(z)}{\mathrm{f}^{\prime\prime}(z)^{2}}.$
(48)

We also evaluate the following invariant quantities in the generalization of
$G_{2a}$, $f(\zeta,u)=u^{-2}\mathrm{f}(\zeta
u^{i\kappa})+A_{1}(u)\zeta+A_{0}(u)$:

$\mathbf{L}^{(i\kappa)}=-\frac{2}{u}\bm{\ell},\quad\mathbf{M}^{(i\kappa)}=2\frac{\overline{\mathrm{f}^{\prime\prime\prime}(z)}}{\overline{\mathrm{f}^{\prime\prime}(z)}}u^{-i\kappa-1}(u\dot{A}_{1}-(i\kappa-2)A_{1})\bm{\ell}\bm{\ell}\bm{\ell}.$
(49)

In the $G_{2b}$ class, $f(\zeta,u)=\mathrm{f}(\zeta e^{i\lambda u})$. Letting
$z=\zeta e^{i\lambda u}$, we then have the following coordinate expressions:

$|\mathbf{K}|^{2}=\left|\frac{\mathrm{f}^{\prime\prime\prime}(z)}{\mathrm{f}^{\prime\prime}(z)}\right|^{2},\quad
I_{\mathrm{d}}^{(\infty)}=\lambda^{2}\frac{\mathrm{f}^{\prime\prime\prime}(z)^{2}}{\mathrm{f}^{\prime\prime}(z)^{3}},\quad
I_{\mathrm{e}}^{(\infty)}=(\mathrm{f}^{\prime}(z)-\lambda^{2}\bar{z})\frac{\mathrm{f}^{\prime\prime\prime}(z)}{\mathrm{f}^{\prime\prime}(z)^{2}}.$
(50)

By considering the slight generalization, $f(\zeta,u)=\mathrm{f}(\zeta
e^{i\lambda u})+A_{1}(u)\zeta+A_{0}(u)$, we find the following

$\mathbf{L}^{(\infty)}=-2i\lambda\bm{\ell},\quad\mathbf{M}^{(\infty)}=2\frac{\overline{\mathrm{f}^{\prime\prime\prime}(z)}}{\overline{\mathrm{f}^{\prime\prime}(z)}}e^{-i\lambda
u}(\dot{A}_{1}-i\lambda A_{1})\bm{\ell}\bm{\ell}\bm{\ell}.$ (51)

It will be helpful to consider a more specific case of this new function,
$f(\zeta,u)=\frac{4\alpha}{u^{2}}e^{i\gamma}\ln\zeta+A_{1}(u)\zeta+A_{0}(u)$
in order to compute a particular conditional invariant

$\frac{1}{I_{\mathrm{d}}^{(k)}}=\alpha e^{i\gamma}.$ (52)

We note that the dependence on $k$ disappears from this expression.

In addition to the above quantities for their respective subclasses or
generalizations, it will be useful to evaluate the $J$ invariant for several
subclasses. Namely, for
$f(\zeta,u)=e^{i\gamma}\ln\zeta+A_{1}(u)\zeta+A_{0}(u)$ or
$f(\zeta,u)=\frac{4\alpha}{u^{2}}e^{i\gamma}\ln\zeta+A_{1}(u)\zeta+A_{0}(u)$,

$J=e^{2i\gamma},$ (53)

for $f(\zeta,u)=e^{i\gamma}\zeta^{2i\kappa}$,

$|\mathbf{K}|=\left|\frac{2(i\kappa-1)}{\zeta}\right|,\quad
J=\frac{(2i\kappa-1)(i\kappa+1)^{2}}{(2i\kappa+1)(i\kappa-1)^{2}}|\zeta|^{4i\kappa}e^{2i\gamma},$
(54)

for $f(\zeta,u)=\mathrm{f}(\zeta)$,

$J=\frac{\mathrm{f}^{\prime\prime}(\zeta)^{3}}{\mathrm{f}^{\prime\prime\prime}(\zeta)^{2}}\frac{\overline{\mathrm{f}^{\prime\prime\prime}(\zeta)^{2}}}{\overline{\mathrm{f}^{\prime\prime}(\zeta)^{3}}}.$
(55)

###  3.3 Notation summary

For convenience, we summarize the invariants that we have introduced above by
their differential order and the non-vanishing conditions needed for their
definition

| $\partial^{2}f(\zeta,u)$ | $\partial^{3}f(\zeta,u)$ | $\partial^{4}f(\zeta,u)$ | $\partial^{5}f(\zeta,u)$
---|---|---|---|---
|  $\mathbf{C}^{\dagger}$, $\mathbf{T}$ |  $\mathbf{K}$, $|\mathbf{K}|$ |  $I_{\mathrm{a}}^{(2)}$, $I_{\mathrm{b}}^{(2)}$ |
$|\mathbf{K}|\neq 0$ |  |  $\mathbf{D}$, $J$ |  $I_{\mathrm{c}}$, $\mathfrak{L}$, $I_{\mathrm{a}}^{(0)}$, $I_{\mathrm{b}}^{(0)}$ |  $N$, $\mathbf{N}$
$\textstyle I_{\mathrm{c}}-\frac{1}{2}\neq 0$, $|\mathbf{K}|\neq 0$ |  |  |  $\mathbf{L}^{(\infty)}$, $I_{\mathrm{d}}^{(\infty)}$, $I_{\mathrm{e}}^{(\infty)}$ | $\mathbf{M}^{(\infty)}$
$\textstyle k(I_{\mathrm{c}}-\frac{1}{2})\neq I_{\mathrm{c}}$, $|\mathbf{K}|\neq 0$ |  |  |  $\mathbf{L}^{(k)}$, $I_{\mathrm{d}}^{(k)}$, $I_{\mathrm{e}}^{(k)}$ | $\mathbf{M}^{(k)}$

For the $G_{5}$ and $G_{2c}$ subclasses:

| $B(u)$ | $\dot{B}(u)$ | $\ddot{B}(u)$
---|---|---|---
|  | $I_{\mathrm{a}}^{(2)}$ | $I_{\mathrm{b}}^{(2)}$
$|\mathbf{K}|\neq 0$ | $J$ | $I_{\mathrm{a}}^{(0)}$ | $I_{\mathrm{b}}^{(0)}$

For the $G_{2a}$ and $G_{2b}$ subclasses:

| $\mathrm{f}^{\prime\prime}(z)$ | $\mathrm{f}^{\prime\prime\prime}(z)$
---|---|---
|  | $|\mathbf{K}|$
$\textstyle I_{\mathrm{c}}-\frac{1}{2}\neq 0$, $|\mathbf{K}|\neq 0$ |  $|\mathbf{K}|^{2}/|I_{\mathrm{d}}^{(\infty)}|$, $(I_{\mathrm{e}}^{(\infty)})^{2}/I_{\mathrm{d}}^{(\infty)}$ |  $I_{\mathrm{d}}^{(\infty)}$, $I_{\mathrm{e}}^{(\infty)}$
$\textstyle k(I_{\mathrm{c}}-\frac{1}{2})\neq I_{\mathrm{c}}$, $|\mathbf{K}|\neq 0$ |  $|\mathbf{K}|^{2}/|I_{\mathrm{d}}^{(k)}|$, $(I_{\mathrm{e}}^{(k)})^{2}/I_{\mathrm{d}}^{(k)}$ |  $I_{\mathrm{d}}^{(k)}$, $I_{\mathrm{e}}^{(k)}$

##  4 Main results

In 4 dimensions, vacuum pp-waves are traditionally classified by the dimension
and isomorphism type of the Lie algebra of infinitesimal local isometries,
ranging through $G_{1}$, $G_{2}$, $G_{3}$, $G_{5}$ and $G_{6}$, with various
subclasses [33, Tbl.24.2]. The next increase in the dimension of the symmetry
algebra already gives $G_{10}$, corresponding to the maximally symmetric
locally flat case, which we exclude from consideration.

The various symmetry classes can be further refined into subclasses by a
careful analysis of associated differential equations with their formfactors.
The result of this is a complete list of formfactors which are uniquely
characterized by invariant constant parameters and signature functions,
whenever an arbitrary function is involved in the formfactor. This analysis is
summarized in Figure 1 and all of the possible formfactors are listed in Table
LABEL:tab:pp-wave-isom-classes.

To connect the various formfactors of the pp-waves to an IDEAL classification,
we begin with the most general case and determine a chain of conditions on
tensor quantities. The binary decision of whether the condition is satisfied
or not satisfied will either lead to a new tensor condition or an end node for
which the sequence of conditions leading to the node fully characterize a
particular subclass of the highly-symmetric pp-waves. This is summarized in
Figure 2.

###  4.1 Four dimensional vacuum pp-waves

The vacuum pp-waves in four dimensions (4D) are characterized by the following

######  Proposition 6.

A 4-dimensional spacetime is a vacuum pp-wave (Definition 2) if and only if
the following conditions hold at every point:

  1. (i)

$R_{ab}=0$,

  2. (ii)

$C^{\dagger}_{ab}{}^{cd}C^{\dagger}_{cdef}=0$,

  3. (iii)

$T_{abc[d}T_{e]fgh;i}=0$.

###### Proof.

Condition (i) is the vacuum condition, while condition (ii) expresses that the
Weyl tensor is of Petrov type N (see, e.g., [16, 35] or [33, §4.1]). Let
$\ell^{a}$ span the unique PND. Then the Bel-Robinson tensor is of the form
[1, 2]

$T_{abcd}=\beta\ell_{a}\ell_{b}\ell_{c}\ell_{d},\quad\beta>0.$ (56)

Given this form we easily obtain

$T_{abc[d}T_{e]fgh;i}=\beta\ell_{a}\ell_{b}\ell_{c}\ell_{f}\ell_{g}\ell_{h}(\ell_{[d}\ell_{e];i}).$

Thus condition (iii) is equivalent to $\ell_{[d}\ell_{e];i}=0$, which by
Definition (15) means that $\ell$ is recurrent. But for 4-dimensional type N
spacetimes, the recurrence of the Weyl PND $\ell$ is equivalent to the complex
recurrence of the self-dual Weyl tensor $\mathbf{C}^{\dagger}$ itself [33,
§35.2]. In turn, the complex recurrent type N vacua are precisely the vacuum
pp-waves [10]. ∎

###  4.2 Classification and characterization

We have now defined sufficiently many invariants to construct an IDEAL
characterization of the individual isometry classes of highly symmetric pp-
wave geometries. Our main results split into two parts. First, in Theorem 7,
we refine the the classification of highly symmetric vacuum pp-wave geometries
from Proposition 3 down to individual isometry classes. Each isometry class is
labelled by a number of invariant parameters and possibly a _signature
function_ $F$. The signature function appears as part of a functional relation
between some scalar invariants, which ultimately translate to an ODE on the
function $A(u)$ or $\mathrm{f}(z)$. These ODEs must have the property that the
integration constants in a general solution must exhaust the residual gauge
freedom (cf. (169), (170), (171) and (172)) in the corresponding isometry
class. The differential order and size of the signature ODE will vary
according to the number of integration constants it needs to produce. The
isometry classes are conveniently listed in Table LABEL:tab:pp-wave-isom-
classes and the logic of the proof follows the corresponding flowchart in
Figure 1, which illustrates how different classes reduce to each other by
specializing parameters.

Second, in Theorem 8, for each isometry class, we give the IDEAL equations
characterizing it. The characterizations are conveniently listed in Table
LABEL:tab:pp-wave-ideal and the logic of the proof is reflected in the
classification flowchart in Figure 2.

Table 1: Isometry classes of highly symmetric vacuum pp-wave geometries. Any new notation is introduced in the proof of Theorem 7 and can looked up by following the corresponding node number in Figure 1. class | $f(\zeta,u)$ | invariant parameters
---|---|---
$G_{6a;\alpha,\kappa}$: | $\frac{4\alpha}{u^{2}}u^{2i\kappa}\zeta^{2}$ | $\alpha>0,\quad\kappa\geq 0$
$G_{6b;\lambda}$: | $e^{2i\lambda u}\zeta^{2}$ | $\lambda\geq 0$
$G_{5^{\prime};\alpha,F}$: | $\frac{4\alpha}{u^{2}}e^{i\operatorname{Im}B(u)}\zeta^{2}$ | $\begin{gathered}\partial_{u}(\operatorname{Im}\dot{B}e^{-\frac{\operatorname{Re}B}{2}})\neq 0,~{}~{}\alpha>0,\\\ \operatorname{Im}\ddot{B}e^{-\operatorname{Re}B}=F(\operatorname{Im}\dot{B}e^{-\frac{\operatorname{Re}B}{2}}),\\\ F(-y)=-F(y),~{}~{}F(y)\neq-\frac{y}{2\sqrt{\alpha}},\\\ F\colon U\subset\mathbb{R}\to\mathbb{R}\end{gathered}$
$G_{5^{\prime};\infty,F}$: | $e^{i\operatorname{Im}B(u)}\zeta^{2}$ | $\begin{gathered}\partial_{u}(\operatorname{Im}\dot{B}e^{-\frac{\operatorname{Re}B}{2}})\neq 0,\\\ \operatorname{Im}\ddot{B}e^{-\operatorname{Re}B}=F(\operatorname{Im}\dot{B}e^{-\frac{\operatorname{Re}B}{2}}),\\\ F(-y)=-F(y),~{}~{}F(y)\neq 0,\\\ F\colon U\subset\mathbb{R}\to\mathbb{R}\end{gathered}$
$G_{5^{\circ};F}$: | $e^{B(u)}\zeta^{2}$ | $\begin{gathered}\partial_{u}(\operatorname{Re}\dot{B}e^{-\frac{\operatorname{Re}B}{2}})\neq 0,\\\ \begin{pmatrix}\operatorname{Re}\ddot{B}e^{-\operatorname{Re}B}\\\ (\operatorname{Im}\dot{B})^{2}e^{-\operatorname{Re}B}\end{pmatrix}=F(\operatorname{Re}\dot{B}e^{-\frac{\operatorname{Re}B}{2}}),\\\ F_{\operatorname{Im}}(y)\geq 0,~{}~{}F_{\operatorname{Re}}(y)\neq\frac{1}{2}y^{2},\\\ F=\begin{pmatrix}F_{\operatorname{Re}}\\\ F_{\operatorname{Im}}\end{pmatrix}\colon U\subset\mathbb{R}\to\mathbb{R}^{2}\end{gathered}$
$G_{3a^{\prime};\pm\alpha}$: | $\pm\frac{4\alpha}{u^{2}}\ln\zeta$ | $\\{\pm\\},\quad\alpha>0$
$G_{3a^{\circ};\alpha,\gamma}$: | $\frac{4\alpha}{u^{2}}e^{i\gamma}\ln\zeta$ | $\gamma\in[0,\pi],\quad\alpha>0$
$G_{3b^{\prime};\pm}$: | $\pm\ln\zeta$ | $\\{\pm\\}$
$G_{3b^{\circ};\gamma}$: | $e^{i\gamma}\ln\zeta$ | $\gamma\in[0,\pi]$
$G_{3c;\lambda}$: | $e^{2\lambda\zeta}$ | $\lambda>0$
$G_{3d;\gamma,\kappa}$: | $e^{i\gamma}\zeta^{2i\kappa}$ | $\gamma\in[0,\pi],\quad\kappa\in\mathbb{R}\setminus\\{0\\}$
$G_{2a^{\circ};\kappa,c,\operatorname{Re}F}$: | $\frac{1}{u^{2}}\mathrm{f}(\zeta u^{i\kappa})$ | $\begin{gathered}\mathrm{f}^{\prime\prime\prime}(z)^{2}/\mathrm{f}^{\prime\prime}(z)^{3}\neq\text{const},\\\ \frac{\mathrm{f}^{\prime\prime\prime}(z)}{\mathrm{f}^{\prime\prime}(z)}=\exp F(\tfrac{\mathrm{f}^{\prime\prime\prime}(z)^{2}}{\mathrm{f}^{\prime\prime}(z)^{3}}),\\\ \mathcal{F}^{\prime\prime-1}(\mathrm{f}^{\prime\prime}(z))=Z_{a},\\\ |\mathrm{f}^{\prime}(z)-\mathcal{F}^{\prime}(Z_{a})|=c\geq 0,\\\ F(y)\neq\frac{1}{2}\ln y+\text{const},\\\ F\colon U\subset\mathbb{C}\to\mathbb{C}\end{gathered}$
$G_{2a^{\prime};\alpha,\gamma,\kappa,c}$: | $\frac{4\alpha}{u^{2}}e^{i\gamma}\left(\ln\zeta+c\zeta u^{i\kappa}\right)$ | $\begin{gathered}\gamma\in[0,\pi],\quad\kappa\in\mathbb{R},\\\ \alpha>0,\quad c>0\end{gathered}$
$G_{2b^{\circ};c,\operatorname{Re}F}$: | $\mathrm{f}(\zeta e^{iu})$ | $\begin{gathered}\mathrm{f}^{\prime\prime\prime}(z)^{2}/\mathrm{f}^{\prime\prime}(z)^{3}\neq\text{const},\\\ \frac{\mathrm{f}^{\prime\prime\prime}(z)}{\mathrm{f}^{\prime\prime}(z)}=\exp F(\tfrac{\mathrm{f}^{\prime\prime\prime}(z)^{2}}{\mathrm{f}^{\prime\prime}(z)^{3}}),\\\ \mathcal{F}^{\prime\prime-1}(\mathrm{f}^{\prime\prime}(z))=Z_{b},\\\ |\mathrm{f}^{\prime}(z)-\mathcal{F}^{\prime}(Z_{b})|=c\geq 0,\\\ F(y)\neq\frac{1}{2}\ln y+\text{const},\\\ F\colon U\subset\mathbb{C}\to\mathbb{C}\end{gathered}$
$G_{2b^{\prime};\gamma,\operatorname{Re}F}$: | $\mathrm{f}(\zeta)$ | $\begin{gathered}\mathrm{f}^{\prime\prime\prime}(z)\mathrm{f}^{\prime}(z)/\mathrm{f}^{\prime\prime}(z)^{2}\neq\text{const},\\\ \mathrm{f}^{\prime\prime\prime}(z)/\mathrm{f}^{\prime\prime}(z)=\exp F(\mathrm{f}^{\prime\prime\prime}(z)\mathrm{f}^{\prime}(z)/\mathrm{f}^{\prime\prime}(z)^{2}),\\\ [\mathcal{F}^{\prime}(-)/\mathcal{F}^{\prime\prime}(-)]^{-1}(\mathrm{f}^{\prime}(z)/\mathrm{f}^{\prime\prime}(z))=Z_{b^{\prime}},\\\ \operatorname{Re}\left[\frac{\mathrm{f}^{\prime\prime}(z)}{\overline{\mathrm{f}^{\prime\prime}(z)}}\frac{\overline{\mathcal{F}^{\prime\prime}(Z_{b^{\prime}})}}{\mathcal{F}^{\prime\prime}(Z_{b^{\prime}})}\right]^{1/2}=\cos\gamma\in[-1,1],\\\ F(y)\neq\ln(\lambda y),\ln\left(\frac{4c}{2-y}\right),\\\ F\colon U\subset\mathbb{C}\to\mathbb{C}\end{gathered}$
$G_{2b^{\prime\prime};\gamma,\lambda,c}$: | $e^{i\gamma}\left(\ln\zeta+c\zeta e^{i\lambda u}\right)$ | $\gamma\in[0,\pi],\quad\lambda\in\mathbb{R},\quad c>0$
$G_{2b^{\prime\prime\prime};k}$: | $\zeta^{k}$ | $k\in\mathbb{C}\setminus\left(i\mathbb{R}\cup\\{1,2\\}\right)$
$G_{2c^{\prime};\alpha,F}$: | $\frac{4\alpha}{u^{2}}e^{i\operatorname{Im}B(u)}\ln\zeta$ | $\begin{gathered}\partial_{u}(\operatorname{Im}\dot{B})\neq 0,~{}~{}\alpha>0,\\\ \operatorname{Im}\dot{B}e^{-\frac{\operatorname{Re}B}{2}}=F\left(\sin(\operatorname{Im}B)\right),\\\ F(-y)=-F(y),~{}~{}F(y)\neq 0,\\\ F\colon U\subset U(1)\to\mathbb{R}\end{gathered}$
$G_{2c^{\prime};\infty,F}$: | $e^{i\operatorname{Im}B(u)}\ln\zeta$ | $\begin{gathered}\partial_{u}(\operatorname{Im}\dot{B})\neq 0,\\\ \operatorname{Im}\dot{B}e^{-\frac{\operatorname{Re}B}{2}}=F\left(\sin(\operatorname{Im}B)\right),\\\ F(-y)=-F(y),~{}~{}F(y)\neq 0,\\\ F\colon U\subset U(1)\to\mathbb{R}\end{gathered}$
$G_{2c^{\circ};F}$: | $e^{B(u)}\ln\zeta$ |  $\begin{gathered}\partial_{u}(\operatorname{Re}\dot{B}e^{-\frac{\operatorname{Re}B}{2}})\neq 0,\\\ \begin{pmatrix}\operatorname{Re}\ddot{B}e^{-\operatorname{Re}B}\\\ \cos(\operatorname{Im}B)\end{pmatrix}=F(\operatorname{Re}\dot{B}e^{-\frac{\operatorname{Re}B}{2}}),\\\ F_{\operatorname{Re}}(y)\neq\frac{1}{2}y^{2},\\\ F=\begin{pmatrix}F_{\operatorname{Re}}\\\ F_{\operatorname{Im}}\end{pmatrix}\colon U\subset\mathbb{R}\to\mathbb{R}\times[-1,1]\end{gathered}$

$\begin{gathered}\kappa,c,u_{0},\beta\to\infty,\\\ \frac{\kappa}{(-u_{0})}\to
1,(-u_{0})e^{\beta/\kappa}\to 1,\\\ c\mapsto\kappa^{2}c,F(y)\mapsto
F(\kappa^{2}y)\end{gathered}$$\begin{gathered}(F^{-1})^{\prime}(x)\to 0\text{
or}\\\ (\mathrm{f}^{\prime\prime\prime 2}/\mathrm{f}^{\prime\prime
3})^{\prime}=0\end{gathered}$$\left(\frac{(\ln f_{,\zeta\zeta})_{,u}}{(\ln
f_{,\zeta\zeta})_{,\zeta}}\right)_{,\zeta}=0$$\begin{gathered}\kappa,\alpha,u_{0}\to\infty,\\\
(-u_{0})e^{\beta/\kappa}\to
1,\frac{\kappa}{(-u_{0})}\to\lambda,\frac{4\alpha}{u_{0}^{2}}\to
1\end{gathered}$$\begin{gathered}(F^{-1})^{\prime}(x)\to 0\text{ or}\\\
(\mathrm{f}^{\prime\prime\prime 2}/\mathrm{f}^{\prime\prime
3})^{\prime}=0\end{gathered}$$c\to
0$$\alpha,u_{0}\to\infty,\frac{4\alpha}{u_{0}^{2}}\to 1$$c\to
0$$F^{\prime}(y)=\frac{1}{2y}$$\begin{gathered}\alpha,\kappa,u_{0},\beta\to\infty,\\\
\frac{4\alpha}{u_{0}^{2}}=1,\kappa\ln
u_{0}-\beta=0,\frac{\kappa}{u_{0}}=-\lambda\end{gathered}$$F^{\prime}(y)=\frac{1}{2y}$$F(y)=-\frac{y}{2\alpha^{1/2}}$$F(y)=0$$\begin{gathered}F_{\mathrm{Re}}(y)=y^{2}/2\text{
or}\\\
\operatorname{Re}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}=-\alpha^{-1/2}\end{gathered}$$\begin{gathered}\gamma,\kappa,a,z_{0}\to\infty,\\\
\gamma-2i\ln a+2\kappa\ln(-z_{0})\to
0,\frac{\kappa}{(-z_{0})}\to-i\lambda\end{gathered}$$\begin{gathered}F(y)=\ln(\lambda
y)\\\ \text{or
}\frac{\mathrm{f}^{\prime\prime\prime}\mathrm{f}^{\prime}}{(\mathrm{f}^{\prime\prime})^{2}}=1\end{gathered}$$k\to
2i\kappa$$\frac{\mathrm{f}^{\prime\prime\prime}\mathrm{f}^{\prime}}{(\mathrm{f}^{\prime\prime})^{2}}=\frac{k-2}{k-1}$$\begin{gathered}F_{\mathrm{Re}}(y)=y/2\text{
or}\\\
\operatorname{Re}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}=-\alpha^{-1/2}\end{gathered}$$\begin{gathered}\alpha,u_{0}\to\infty,\\\
\frac{4\alpha}{u_{0}^{2}}=1\end{gathered}$$F(y)=\ln\left(\frac{4c}{2-y}\right)$$\mathrm{Re}\dot{B}=0$$k\to
2$$F(y)=0$$F(y)=0$$\alpha,u_{0}\to\infty,\frac{4\alpha}{u_{0}^{2}}=1$$\operatorname{Re}\dot{B}=0$$k\to
0$$e^{i\gamma}\to\pm 1$$e^{i\gamma}\to\pm
1$$\alpha,u_{0}\to\infty,\frac{4\alpha}{u_{0}^{2}}\to
1$$\begin{gathered}G_{6b;\lambda}\\\ \lambda\geq
0\end{gathered}$4$\begin{gathered}G_{5^{\circ};F}\\\
\partial_{u}(\mathrm{Re}\dot{B})\neq 0,\\\ F_{\mathrm{Re}}(y)\neq
y^{2}/2\end{gathered}$1$\begin{gathered}G_{6a;\alpha,\kappa}\\\
\alpha>0,\kappa\geq
0\end{gathered}$5$\begin{gathered}G_{5^{\prime};\alpha,F}\\\ \alpha>0,\\\
F(y)\neq-\frac{y}{2\alpha^{1/2}}\end{gathered}$3$\begin{gathered}G_{3b^{\circ};\gamma}\\\
\gamma\in(0,\pi)\end{gathered}$11$\begin{gathered}G_{3a^{\circ};\alpha,\gamma}\\\
\alpha>0,\gamma\in(0,\pi)\end{gathered}$15$\begin{gathered}G_{2c^{\circ};F}\\\
F_{\mathrm{Re}}(y)\neq
y^{2}/2\end{gathered}$6$\begin{gathered}G_{2c^{\prime};\alpha,F}\\\
\alpha>0,\\\ F(y)\neq
0\end{gathered}$8$\begin{gathered}G_{2b^{\prime\prime};\gamma,\lambda,c}\\\
\lambda\in\mathbb{R},c>0,\gamma\in[0,\pi]\end{gathered}$10$G_{1}$$\begin{gathered}G_{2a^{\prime};\kappa,\alpha,\gamma,c}\\\
\kappa\in\mathbb{R},\alpha>0,c>0,\gamma\in[0,\pi]\end{gathered}$14$\begin{gathered}G_{3d;\gamma,\kappa}\\\
\gamma\in[0,\pi],\\\
\kappa\in\mathbb{R}\setminus\\{0\\}\end{gathered}$20$\begin{gathered}G_{2b^{\prime\prime\prime}:k}\\\
k\in\mathbb{C}\setminus(i\mathbb{R}\cup\\{1,2\\})\end{gathered}$19$\begin{gathered}G_{2a^{\circ};\kappa,c,\operatorname{Re}F}\\\
\kappa>0,c>0,2yF^{\prime}(y)\neq
1\end{gathered}$13$\begin{gathered}G_{2b^{\circ};c,\operatorname{Re}F}\\\
\left(\frac{(\ln f_{,\zeta\zeta})_{,u}}{(\ln
f_{,\zeta\zeta})_{,\zeta}}\right)_{,\zeta}\neq 0\\\
(\mathrm{f}^{\prime\prime\prime 2}/\mathrm{f}^{\prime\prime 3})^{\prime}\neq
0\\\ c>0,2yF^{\prime}(y)\neq
1\end{gathered}$9$\begin{gathered}G_{2b^{\prime};\gamma,\operatorname{Re}F}\\\
(\mathrm{f}^{\prime\prime\prime}\mathrm{f}^{\prime}/(\mathrm{f}^{\prime\prime})^{2})^{\prime}\neq
0,\\\ \alpha\in[0,2\pi),\\\ F(y)\neq\ln(\lambda y),\\\
F(y)\neq\ln\left(\frac{4c}{2-y}\right)\end{gathered}$17$\begin{gathered}G_{3c;\lambda}\\\
\lambda>0\end{gathered}$18$\begin{gathered}G_{5^{\prime};\infty,F}\\\ F(y)\neq
0\end{gathered}$2$\begin{gathered}G_{2c^{\prime};\infty,F}\\\ F(y)\neq
0\end{gathered}$7$G_{3a^{\prime};\pm\alpha}$16$G_{3b^{\prime};\pm}$12

Figure 1: Flowchart for relations between pp-wave classes. Arrows indicate
transitions from generic to more special classes by taking limits of invariant
or gauge parameters. Numbers accompany the proof of Theorem 7.

######  Theorem 7 (pp-wave isometry classes).

Consider a point $x\in M$, where $(M,g)$ belongs to one of the highly
symmetric regular (Definition 5) vacuum pp-wave geometries from Proposition 3.
Then there exists a neighborhood of $x$ such that this neighborhood is
isometric to exactly one of the classes listed in Table LABEL:tab:pp-wave-
isom-classes and Figure 1. These classes are identified by invariant
parameters and/or a signature function $F$ defined on a domain $U$. The domain
$U$ needs to be chosen to correspond to the needed neighborhood of $x$.
Limiting values of the invariant parameters allow for transitions between
different classes according to the flowchart in Figure 1.

###### Proof.

We will proceed by going through the flowchart in Figure 1 step by step, in
the order of the numbers indicated next to each node. In the corresponding
numbered part of the proof, we give some invariant conditions that select a
subclass of formfactor functions $f(\zeta,u)$. These subclasses are then
broken up into individual isometry classes under some non-degeneracy
conditions. The isometry classes and their invariant parameter corresponding
to that node in Figure 1 are precisely recorded by representative formfactors
in Table LABEL:tab:pp-wave-isom-classes. Then, to be exhaustive, the ways in
which the formfactors $f(\zeta,u)$ fail to satisfy the non-degeneracy
conditons, if any, are broken up into cases, which correspond to the arrows
leaving the given node in Figure 1 and are discussed separately. This way, we
show that all of the highly symmetric formfactor families listed in (44) are
exhausted by our analysis.

The functional forms of $f(u,\zeta)$ listed in (44) are not all mutually
exclusive and each one may contain multiple individual isometry classes within
it. Besides $G_{1}$, the most generic subclasses are $G_{2a}$, $G_{2b}$,
$G_{2c}$ and $G_{5}$, each containing functional degrees of freedom. The
remaining subclasses will be specializations thereof. Invariant parameters
that label an isometry class will appear in relations among differential
invariants (_classifying_ or _signature_ relations). We know that we have
isolated an individual isometry class when the integration parameters of the
corresponding set of classifying relations can be completely absorbed by
residual gauge freedom. A combination of $f(\zeta,u)$ and its derivatives can
be checked to be invariant under residual gauge freedom by direct inspection
or, as a shortcut, by realizing it as a component of one of the tensor or
conditional curvature invariants that we have defined in Section 3.

To start, note that $f_{,\zeta\zeta}=0$ is equivalent to
$\mathbf{C}^{\dagger}=0$ (16), which together with the vacuum condition
implies that the spacetime is locally Minkowski. Hence we can presume below
that $f_{,\zeta\zeta}\neq 0$.

  1. 1.

The invariant relation

$(\ln
f_{,\zeta\zeta})_{,\zeta}=\frac{f_{,\zeta\zeta\zeta}}{f_{,\zeta\zeta}}=0$ (57)

integrates to $f(\zeta,u)=A(u)\zeta^{2}+A_{1}(u)\zeta+A_{0}(u)$, where we can
set $A_{1}(u)=A_{0}(u)$ by gauge freedom, which lands us in the $G_{5}$
subclass. Rewriting $A(u)=e^{B(u)}$, the combinations
$\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}$ and
$\ddot{B}e^{-\operatorname{Re}B}$ are invariant under the continuous gauge
symmetries. The discrete gauge symmetry $\zeta\mapsto\bar{\zeta}$ changes the
signs of $\operatorname{Im}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}$ and
$\operatorname{Im}\ddot{B}e^{-\operatorname{Re}B}$, while $u\mapsto-u$ changes
the sign of $\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}$. Generically,
$\partial_{u}(\operatorname{Re}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B})\neq
0$, so there exists a signature function
$F=(F_{\operatorname{Re}},F_{\operatorname{Im}})\colon
U\subset\mathbb{R}\to\mathbb{R}^{2}$, with $F_{\operatorname{Im}}(y)\geq 0$,
such that

$\begin{pmatrix}\operatorname{Re}\ddot{B}e^{-\operatorname{Re}B}\\\
(\operatorname{Im}\dot{B})^{2}e^{-\operatorname{Re}B}\end{pmatrix}=F\left(\operatorname{Re}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}\right).$
(58)

Given any particular solution $B_{F}(u)$, symmetries dictate that the general
3-parameter solution is

$B(u)=B_{F}(a(u-u_{0}))+2\ln
a+2i\beta\quad\text{or}\quad\overline{B_{F}(a(u-u_{0}))}+2\ln a-2i\beta,$ (59)

where the integration constants $a,\beta,u_{0}$ and the complex conjugaton can
be absorbed by corresponding gauge transformations. Hence the signature
function $F$ completely describes the corresponding isometry class, which we
label by $G_{5^{\circ};F}$. The choice $F_{\operatorname{Re}}(y)=y^{2}/2$
implies $\operatorname{Re}B=-2\ln|u-u_{0}|+\ln 4\alpha$, which implies
constant
$\operatorname{Re}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}=\pm\alpha^{-1/2}$,
which must be excluded to be consistent with our genericity assumption.

The only possibilities left to consider are those with constant
$\operatorname{Re}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}\neq 0$ and
$\operatorname{Re}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}=0$.

  2. 2.

The condition

$\operatorname{Re}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}=0$ (60)

integrates to a constant $\operatorname{Re}B(u)=2\ln a$. Generically then,
$\partial_{u}(\operatorname{Im}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B})\neq
0$, so there exists a signature function $F\colon
U\subset\mathbb{R}\to\mathbb{R}$, with $F(-y)=-F(y)$ because both
$\operatorname{Im}B$ and $-\operatorname{Im}B$ must be solutions, such that

$\operatorname{Im}\ddot{B}e^{-\operatorname{Re}B}=F\left(\operatorname{Im}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}\right).$
(61)

Given any particular solution $\operatorname{Im}B_{F}(u)$, symmetries dictate
that the general 2-parameter solution is

$B(u)=i\operatorname{Im}B_{F}(a(u-u_{0}))+2\ln
a+2i\beta\quad\text{or}\quad-i\operatorname{Im}B_{F}(a(u-u_{0}))+2\ln
a-2i\beta.$ (62)

All the integration constants $a,\beta,u_{0}$ and the complex conjugation can
be absorbed by corresponding gauge transformations. Hence, the signature
function $F$ completely describes the corresponding isometry class, which we
label by $G_{5^{\prime};\infty,F}$. The choice $F(y)=0$ implies
$\operatorname{Im}B=2a\lambda(u-u_{0})$, which implies constant
$\operatorname{Im}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}=2\lambda$, which
must be excluded to be consistent with our genericity assumption.

The only possibility left to consider is that with constant
$\operatorname{Im}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}$.

  3. 3.

Let us parametrize

$\operatorname{Re}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}=\pm\alpha^{-1/2}$
(63)

with constant $\alpha>0$, which integrates to
$\operatorname{Re}B(u)=-2\ln|u-u_{0}|+\ln 4\alpha$ with integration constant
$u_{0}$, with the $+$ and $-$ signs corresponding to $u<u_{0}$ and $u>u_{0}$
respectively. Generically then,
$\partial_{u}(\operatorname{Im}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B})\neq
0$, so there exists a signature function $F\colon
U\subset\mathbb{R}\to\mathbb{R}$, with $F(-y)=-F(y)$ because both
$\operatorname{Im}B$ and $-\operatorname{Im}B$ must be solutions, such that

$\operatorname{Im}\ddot{B}e^{-\operatorname{Re}B}=F\left(\operatorname{Im}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}\right).$
(64)

Given any particular solution $\operatorname{Im}B_{F}(u-u_{0})$, symmetries
dictate that the general 2-parameter solution is

$B(u)=i\operatorname{Im}B_{F}(a(u-u_{0}))-2\ln|u-u_{0}|+\ln 4\alpha+2i\beta\\\
\text{or}\quad-i\operatorname{Im}B_{F}(a(u-u_{0}))-2\ln|u-u_{0}|+\ln
4\alpha-2i\beta.$ (65)

All the integration constants $a,\beta,u_{0}$ and the complex conjugation can
be absorbed by corresponding gauge transformations. Hence, the signature
function $F$ completely describes the corresponding isometry class, which we
label by $G_{5^{\prime};\alpha,F}$. The choice $F(y)=-y/(2\alpha^{1/2})$
implies $\operatorname{Im}B=2\kappa\ln|u-u_{0}|$, which implies constant
$\operatorname{Im}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}=\kappa\alpha^{-1/2}$,
which must be excluded to be consistent with our genericity assumption.

The only possibility left to consider is that with constant
$\operatorname{Im}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}$.

Taking the invariant parameter limit $\alpha\to\infty$, accompanied by the
gauge parameter limit $u_{0}\to\infty$, such that $4\alpha/u_{0}^{2}\to 1$, we
recover the $G_{5^{\prime};\infty,F}$ subclass.

  4. 4.

The condition

$\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}=2i\lambda\iff\operatorname{Re}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}=0,\quad\operatorname{Im}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}=2\lambda$
(66)

integrates to $B(u)=2\ln a+2ia\lambda|u-u_{0}|$ with integration constants
$a$, $u_{0}$, which can be absorbed by the corresponding continuous gauge
freedom. Under $\zeta\mapsto\bar{\zeta}$, $\lambda\mapsto-\lambda$, so we can
choose to represent the isometry class with $\lambda\geq 0$. Hence, the above
condition already completely describes the corresponding isometry class, which
we label by $G_{6b;\lambda}$.

  5. 5.

The condition

$\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}=(i\kappa\pm 1)\alpha^{-1/2}\\\
\iff\operatorname{Re}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}=\pm\alpha^{-1/2},\quad\operatorname{Im}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}=\kappa\alpha^{-1/2},$
(67)

with $\alpha>0$, integrates to

$B(u)=i\beta+\ln 4\alpha+2(i\kappa\operatorname{sgn}(u-u_{0})-1)\ln|u-u_{0}|$
(68)

with integration constants $\beta$, $u_{0}$, which can be absorbed by the
corresponding continuous gauge freedom. The $+$ and $-$ signs correspond to
$u<u_{0}$ and $u>u_{0}$ respectively, with the differrence between the two
cases absorbed by the $u\mapsto-u$ residual gauge transformation, after fixing
$u_{0}=0$. Without loss of generality, we can then assume that $u>0$. Under
$\zeta\mapsto\bar{\zeta}$, $\kappa\mapsto-\kappa$, so we can choose to
represent the isometry class with $\kappa\geq 0$. Hence, the above condition
already completely describes the corresponding isometry class, which we label
by $G_{6a;\alpha,\kappa}$.

Taking the invariant parameter limit $\alpha,\kappa\to\infty$, accompanied by
the gauge parameter limit $\beta,u_{0}\to\infty$, such that
$\frac{\kappa}{2\alpha^{1/2}}=\lambda$, $4\alpha/u_{0}^{2}\to 1$, and
$\kappa\ln u_{0}-\beta=0$, we recover the $G_{6b;\lambda}$ subclass.

  6. 6.

The invariant relation

$-\left(\frac{1}{(\ln f_{,\zeta\zeta})_{,\zeta}}\right)_{,\zeta}=\frac{1}{2},$
(69)

where the invariance of the left-hand side is checked by noting that it is
equal to the $I_{\mathrm{c}}$ conditional curvature invariant in (28),
integrates to $f(\zeta,u)=A(u)\ln(\zeta-h(u))+A_{1}(u)\zeta+A_{0}(u)$, where
we can set $h(u)=A_{0}(u)=0$ by gauge freedom. For this form of $f(\zeta,u)$,
using the invariant $N$ (32) and its evaluation in (47), the invariant
condition $N=0$ is equivalent to

$A_{1}(u)=0,$ (70)

which lands us in the $G_{2c}$ subclass.

Once again, rewriting $A(u)=e^{B(u)}$, the combinations
$\operatorname{Re}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}$,
$\operatorname{Re}\ddot{B}e^{-\operatorname{Re}B}$ and $\operatorname{Im}B$
are invariant under the continuous gauge symmetries. The discrete gauge
symmetry $u\mapsto-u$ changes the signs of
$\operatorname{Re}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}$ and
$\operatorname{Im}\ddot{B}e^{-\operatorname{Re}B}$, while
$\zeta\mapsto\bar{\zeta}$ changes the sign of $\operatorname{Im}B$.
Generically,
$\partial_{u}(\operatorname{Re}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B})\neq
0$, so there exists a signature function
$F=(F_{\operatorname{Re}},F_{\operatorname{Im}})\colon
U\subset\mathbb{R}\to\mathbb{R}\times[-1,1]$, such that

$\begin{pmatrix}\operatorname{Re}\ddot{B}e^{-\operatorname{Re}B}\\\
\cos(\operatorname{Im}B)\end{pmatrix}=F\left(\operatorname{Re}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}\right).$
(71)

Given any particular solution $B_{F}(u)$, symmetries dictate that the general
2-parameter solution is

$B(u)=B_{F}(a(u-u_{0}))+2\ln
a\quad\text{or}\quad\overline{B_{F}(a(u-u_{0}))}+2\ln a,$ (72)

where the integration constants $a,u_{0}$ and the complex conjugaton can be
absorbed by corresponding gauge transformations. Hence the signature function
$F$ completely describes the corresponding isometry class, which we label by
$G_{2^{\circ};F}$. The choice $F_{\operatorname{Re}}(y)=y^{2}/2$ implies
$\operatorname{Re}B=-2\ln|u-u_{0}|+\ln 4\alpha$, which implies constant
$\operatorname{Re}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}=\pm\alpha^{-1/2}$,
which must be excluded to be consistent with our genericity assumption.

The only possibilities left to consider are those with constant
$\operatorname{Re}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}\neq 0$ and
$\operatorname{Re}\dot{B}=0$.

  7. 7.

The condition

$\operatorname{Re}\dot{B}=0$ (73)

integrates to a constant $\operatorname{Re}B(u)=2\ln a$. Generically then,
$\partial_{u}(\operatorname{Im}B)\neq 0$, so there exists a signature function
$F\colon U\subset[-1,1]\to\mathbb{R}$, because both $\operatorname{Im}B$ and
$-\operatorname{Im}B$ must be solutions, with $F(-y)=-F(y)$ such that

$\operatorname{Im}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}=F\left(\sin(\operatorname{Im}B)\right).$
(74)

Given any particular solution $\operatorname{Im}B_{F}(u)$, symmetries dictate
that the general 2-parameter solution is

$B(u)=i\operatorname{Im}B_{F}(a(u-u_{0}))+2\ln
a\quad\text{or}\quad-i\operatorname{Im}B_{F}(a(u-u_{0}))+2\ln a,$ (75)

where the integration constants $a,u_{0}$ and the complex conjugation can be
absorbed by corresponding gauge transformations. Hence, the signature function
$F$ completely describes the corresponding isometry class, which we label by
$G_{2c^{\prime};\infty,F}$. The choice $F(y)=0$ implies constant
$\operatorname{Im}B(u)=\gamma$, which must be excluded to be consistent with
our genericity assumption.

The only possibility left to consider is constant $\operatorname{Im}B$.

  8. 8.

Let us parametrize

$\operatorname{Re}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}=\pm\alpha^{-1/2}$
(76)

with constant $\alpha>0$, which integrates to
$\operatorname{Re}B(u)=-2\ln|u-u_{0}|+\ln 4\alpha$ with integration constant
$u_{0}$, with the $+$ and $-$ signs corresponding to $u<u_{0}$ and $u>u_{0}$
respectively. Generically then, $\partial_{u}(\operatorname{Im}B)\neq 0$, so
there exists a signature function $F\colon U\subset\mathbb{R}\to\mathbb{R}$,
with $F(-y)=-F(y)$ because both $\operatorname{Im}B$ and $-\operatorname{Im}B$
must be solutions, such that

$\operatorname{Im}\dot{B}e^{-\operatorname{Re}B}=F\left(\sin(\operatorname{Im}B)\right).$
(77)

Given any particular solution $\operatorname{Im}B_{F}(u-u_{0})$, symmetries
dictate that the general 2-parameter solution is

$B(u)=i\operatorname{Im}B_{F}(a(u-u_{0}))-2\ln|u-u_{0}|+\ln 4\alpha\\\
\text{or}\quad-i\operatorname{Im}B_{F}(a(u-u_{0}))-2\ln|u-u_{0}|+\ln 4\alpha,$
(78)

where the integration constants $a,u_{0}$ and the complex conjugation can be
absorbed by corresponding gauge transformations. The difference between
$u<u_{0}$ and $u>u_{0}$ can be absorbed by the $u\mapsto-u$ residual gauge
transformation after fixing $u_{0}=0$. Hence, the signature function $F$
completely describes the corresponding isometry class, which we label by
$G_{2c^{\prime};\alpha,F}$. The choice $F(y)=0$ implies constant
$\operatorname{Im}B=\gamma$, which must be excluded to be consistent with our
genericity assumption.

The only possibility left to consider is that with constant
$\operatorname{Im}B$.

Taking the invariant parameter limit $\alpha\to\infty$, accompanied by the
gauge parameter limit $u_{0}\to\infty$, such that $4\alpha/u_{0}^{2}\to 1$, we
recover the $G_{2c^{\prime};\infty,F}$ subclass.

  9. 9.

Consider the equation

$\left(\frac{(\ln f_{,\zeta\zeta})_{,u}}{(\ln
f_{,\zeta\zeta})_{,\zeta}}\right)_{\zeta}=i\lambda\left[\left(\frac{2}{(\ln
f_{,\zeta\zeta})_{,\zeta}}\right)_{\zeta}+1\right].$ (79)

The bracketed expression on the right-hand side is an invariant, while the
left-hand side is invariant up to changing by a factor of $a$ under $u\mapsto
u/a$. So to make the equation invariant, the parameter $\lambda$ must also
transform as $\lambda\mapsto a\lambda$. Thus, $\lambda\in\mathbb{R}$ is not an
invariant parameter, but the property $\lambda=0$ or $\lambda\neq 0$ is
invariant. The first integral of the above equation is

$\frac{(\ln f_{,\zeta\zeta})_{,u}}{(\ln
f_{,\zeta\zeta})_{,\zeta}}-\frac{2i\lambda}{(\ln
f_{,\zeta\zeta})_{,\zeta}}=i\lambda\zeta-e^{-i\lambda u}h^{\prime}(u)\\\ \iff
e^{i\lambda u}(\ln e^{-2i\lambda u}f_{,\zeta\zeta})_{,u}=(\zeta i\lambda
e^{i\lambda u}-h^{\prime}(u))(\ln e^{-2i\lambda u}f_{,\zeta\zeta})_{,\zeta}\\\
\iff\left[(\zeta e^{i\lambda u}-h(u))_{,\zeta}\partial_{u}-(\zeta e^{i\lambda
u}-h(u))_{,u}\partial_{\zeta}\right]\ln e^{-2i\lambda u}f_{,\zeta\zeta}=0,$
(80)

with implicitly defined integration parameter $h(u)$. The general solution to
the last equation is clearly

$\ln e^{-2i\lambda u}f_{,\zeta\zeta}=\ln\mathrm{f}^{\prime\prime}(\zeta
e^{i\lambda u}-h(u))\\\ \iff f(\zeta,u)=\mathrm{f}(\zeta e^{i\lambda
u}-h(u))+A_{1}(u)\zeta+A_{0}(u),$ (81)

for some holomorphic function $\mathrm{f}(z)$ and integration parameters
$A_{0}(u)$, $A_{1}(u)$. We can set $h(u)=A_{0}(u)=0$ by gauge freedom. For
this form of $f(\zeta,u)$, the value of the conditional curvature invariant
$\mathbf{M}^{(\infty)}$ is given in (51). So setting $\mathbf{M}^{(\infty)}=0$
forces $\mathrm{f}^{\prime\prime\prime}(z)=0$ or $A_{1}(u)\zeta=c\zeta
e^{i\lambda u}$. The former possibility results in a formfactor corresponding
to the $G_{5}$ subclass, while in the latter possibility the term
$A_{1}(u)\zeta$ can be absorbed into the definition of $\mathrm{f}(z)$, which
lands us in the $G_{2b}$ subclass.

In this branch of the classification, we assume now that
$\mathrm{f}^{\prime\prime\prime}(z)\neq 0$, and the above conditions hold with
$\lambda\in\mathbb{R}$, so that we are in the $G_{2b}$ subclass.

Suppose that

$\left(\frac{(\ln f_{,\zeta\zeta})_{,u}}{(\ln
f_{,\zeta\zeta})_{,\zeta}}\right)_{\zeta}\neq 0.$ (82)

We consider separately the subfamily where instead the above holds as an
equality, which corresponds to either $\lambda=0$ or $-\left(1/(\ln
f_{,\zeta\zeta})_{,\zeta}\right)=\frac{1}{2}$.

Comparing (79) to the definition of the invariant $\Xi^{(\infty)}$ in (40), we
see that it implies that the invariant $\Xi^{(\infty)}=i\lambda$, while the
inequality (82) implies $\lambda\neq 0$, hence we can set $\lambda=1$ by
setting the gauge parameter $a=1/\lambda$, which we assume from now on. The
residual gauge transformations that preserve the formfactor
$f(\zeta,u)=\mathrm{f}(\zeta e^{iu})$ are $\zeta\mapsto e^{i\beta}\zeta$,
which induces $\mathrm{f}(z)\mapsto\mathrm{f}(e^{-i\beta}z)$,
$\zeta\mapsto\zeta+z_{0}e^{-iu}$, which induces
$\mathrm{f}(z)\mapsto\mathrm{f}(z-z_{0})-\overline{z_{0}}z+\frac{1}{2}|z_{0}|^{2}$,
$v\mapsto v-2u\operatorname{Re}w_{0}$ with $c(u)=\operatorname{Im}w_{0}$,
which induces $\mathrm{f}(z)\mapsto\mathrm{f}(z)+w_{0}$, and
$\zeta\mapsto\bar{\zeta}$, which induces
$\mathrm{f}(z)\mapsto\bar{\mathrm{f}}(z)$.

Generically,
$\partial_{z}(\mathrm{f}^{\prime\prime\prime}(z)^{2}/\mathrm{f}^{\prime\prime}(z)^{3})\neq
0$, so then there exists a holomorphic function $F\colon
U\subset\mathbb{C}\to\mathbb{C}$ such that

$\ln\frac{\mathrm{f}^{\prime\prime\prime}(z)}{\mathrm{f}^{\prime\prime}(z)}=F\left(\frac{\mathrm{f}^{\prime\prime\prime}(z)^{2}}{\mathrm{f}^{\prime\prime}(z)^{3}}\right).$
(83)

The argument of $F(-)$ is an invariant because it coincides with the value of
the invariant $I_{\mathrm{d}}^{(\infty)}$ (41), which was evaluated in (50).
Neither the left-hand side in (83) nor $F(y)$ itself are entirely invariant,
as $\mathrm{f}(z)\mapsto\mathrm{f}(e^{-i\beta}z)$ induces $F(y)\mapsto
F(y)-i\beta$ and $\mathrm{f}(z)\mapsto\bar{\mathrm{f}}(z)$ induces
$F(y)\mapsto\bar{F}(y)$. However, $\operatorname{Re}F(y)$ is invariant under
these transformations and also uniquely fixes $F(y)$ up to these ambiguities.
Thus, we consider $\operatorname{Re}F\colon U\subset\mathbb{C}\to\mathbb{R}$
as our signature function.

If $F^{\prime}(y)=1/(2y)$ or equivalently $F(y)=\frac{1}{2}\ln(2Ay)$, then the
ODE (83) no longer depends on $\mathrm{f}^{\prime\prime\prime}(z)$ and reduces
to $\mathrm{f}^{\prime\prime}(z)=2A$, and hence
$\mathrm{f}^{\prime\prime\prime}(z)=0$, which contradicts our earlier
hypothesis. Otherwise, the solution $\mathrm{f}(z)=Az^{2}+A_{1}z+A_{0}$
recovers the $G_{6b}$ subclass.

Provided $F^{\prime}(y)\neq 1/(2y)$, the ODE (83) can be solved for
$\mathrm{f}^{\prime\prime\prime}(z)$, so the usual existence and uniqueness
theory holds for it. If $\mathrm{f}(z)=\mathcal{F}(z)$ is a particular
solution, then symmetries dictate that the general 3-parameter solution is

$\mathrm{f}(z)=\mathcal{F}(z-z_{0})+A_{1,0}(z-z_{0})+A_{0,0}.$ (84)

While the $z_{0}$ and $A_{0,0}$ integration constants can be absorbed by the
residual gauge freedom, the parameter $A_{1,0}$ cannot be, and it does contain
some invariant information. To be concrete, we uniquely fix333There may be
other ways to uniquely fixing the solution $\mathcal{F}(z)$, for instance by
different initial conditions at $z=0$ or by asyptotic conditions at
$z=\infty$. $\mathcal{F}(z)$ from $F(y)$ by requiring that
$\mathcal{F}(z)=\frac{1}{2}z^{2}+O(z^{3})$. The gauge transformation
$F(y)\mapsto F(y)-i\beta$ induces
$\mathcal{F}(z)\mapsto\mathcal{F}(e^{-i\beta}z)$, $A_{1,0}\mapsto
A_{1,0}e^{-i\beta}$, and $F(y)\mapsto\bar{F}(y)$ induces
$\mathcal{F}(z)\mapsto\overline{\mathcal{F}}(z)$,
$A_{1,0}\mapsto\overline{A_{1,0}}$. So only $|A_{1,0}|$ is invariant in
$A_{1,0}$. Using the ODE (83), and computing the invariants
$I_{\mathrm{d}}^{(\infty)}$ and $I_{\mathrm{e}}^{(\infty)}$, from (41) and
(3.1), as well as the evaluation in (50), we find

$\displaystyle e^{2F(I_{\mathrm{d}}^{(\infty)})}/I_{\mathrm{d}}^{(\infty)}$
$\displaystyle=\mathrm{f}^{\prime\prime}(z)=\mathcal{F}^{\prime\prime}(z-z_{0}),$
(85)
$\displaystyle\frac{I_{\mathrm{e}}^{(\infty)}}{I_{\mathrm{d}}^{(\infty)}}e^{F(I_{\mathrm{d}}^{(\infty)})}$
$\displaystyle=-\overline{(z-z_{0})}+\mathrm{f}^{\prime}(z)$
$\displaystyle=-\overline{(z-z_{0})}+A_{1,0}+\mathcal{F}^{\prime}(z-z_{0}),$
(86)

where $z=\zeta e^{i\lambda u}$. Combining the two formulas, we can extract the
constant

$\displaystyle c=|A_{1,0}|$
$\displaystyle=\left|\frac{I_{\mathrm{e}}^{(\infty)}}{I_{\mathrm{d}}^{(\infty)}}e^{F(I_{\mathrm{d}}^{(\infty)})}+\overline{Z_{b}}-\mathcal{F}^{\prime}(Z_{b})\right|=:\mathcal{A}_{\infty,\operatorname{Re}F}(I_{\mathrm{d}}^{(\infty)},I_{\mathrm{e}}^{(\infty)}),$
(87) $\displaystyle\text{where}\quad Z_{b}$
$\displaystyle=(\mathcal{F}^{\prime\prime})^{-1}\left(e^{2F(I_{\mathrm{d}}^{(\infty)})}/I_{\mathrm{d}}^{(\infty)}\right).$
(88)

While the individual terms entering formula
$\mathcal{A}_{\infty,\operatorname{Re}F}$ that extracts $c=|A_{1,0}|$ are not
invariant, since they depend on the choice of $F(y)$ for given
$\operatorname{Re}F(y)$, it is straightforward to check that the whole formula
is independent of those choices and so defines an invariant. Still, for given
$\operatorname{Re}F(y)$, to make the formula really explicit, we need to pick
a compatible $F(y)$ and determine the unique solution $\mathcal{F}(z)$ from
it.

The only possibilities left to consider are those with, separately,
$\partial_{z}(\mathrm{f}^{\prime\prime\prime}(z)^{2}/\mathrm{f}^{\prime\prime}(z)^{3})=0$
and $\lambda=0$.

  10. 10.

Within the $f(\zeta,u)=\mathrm{f}(\zeta e^{i\lambda u})$ formfactor family
under the condition
$\partial_{z}(\mathrm{f}^{\prime\prime\prime}(z)^{2}/(\mathrm{f}^{\prime\prime}(z)^{3})=0$,
let us parametrize

$\frac{\mathrm{f}^{\prime\prime\prime}(z)^{2}}{\mathrm{f}^{\prime\prime}(z)^{3}}=-4a^{-2}e^{-i\gamma},$
(89)

with constant $\alpha>0$ and $\gamma\in[-\pi,\pi]$, which integrates to

$\mathrm{f}(z)=a^{2}e^{i\gamma}\ln z+A_{1,0}z+A_{0,0}.$ (90)

We can set $A_{0,0}=0$ and $a=1$ by residual gauge freedom, which then makes
the resulting value of $\lambda$ invariant. Further, we can eliminate the
$\mathrm{f}(z)\mapsto\bar{\mathrm{f}}(z)$ gauge freedom by choosing
$\gamma\in[0,\pi]$, and also eliminate the $\zeta\mapsto e^{i\beta}\zeta$
gauge freedom by choosing $A_{1,0}=c\geq 0$. The resulting formfactor takes
the form

$f(\zeta,u)=e^{i\gamma}\ln\zeta e^{i\lambda u}+c\zeta e^{i\lambda u},$ (91)

or after applying another remaining gauge transformation

$f(\zeta,u)=e^{i\gamma}\ln\zeta+c\zeta e^{i\lambda},$ (92)

with $\gamma\in[0,\pi]$, $\lambda\in\mathbb{R}$ and $c\geq 0$ labelling the
corresponding isometry class. When $c\neq 0$, we label this isometry class as
$G_{2b^{\prime\prime};\gamma,\lambda,c}$.

The excluded value $c=|A_{1,0}|=0$ lands us in the $G_{3b}$ subclass.

  11. 11.

Within the formfactor subfamily

$f(\zeta,u)=e^{i\gamma}\ln\zeta+c\zeta e^{i\lambda},$ (93)

where $c=0$, the $\lambda$ parameter drops out and the formfactor reduces to

$f(\zeta,u)=e^{i\gamma}\ln\zeta.$ (94)

We can use the $\zeta\mapsto\bar{\zeta}$ transformation to restrict
$\gamma\in[0,\pi]$, which then becomes invariant. When $e^{i\gamma}\neq\pm 1$,
we label this isometry class as $G_{3b^{\circ};\gamma}$.

The excluded value of $e^{i\gamma}=\pm 1$ lands us in the
$G_{3b^{\prime};\pm}$ subclass.

  12. 12.

In the formfactor subfamily

$f(\zeta,u)=\pm\ln\zeta,$ (95)

the only invariant is the sign $\pm$. We label this isometry class as
$G_{3b^{\prime};\pm}$.

  13. 13.

Consider the invariant equation

$\left((u-u_{0})\frac{(\ln f_{,\zeta\zeta})_{,u}}{(\ln
f_{,\zeta\zeta})_{,\zeta}}\right)_{,\zeta}=\left(\frac{2i\kappa-2}{(\ln
f_{,\zeta\zeta})_{,\zeta}}\right)_{,\zeta}+i\kappa,$ (96)

where $\kappa\in\mathbb{R}$ is an invariant parameter. Its first integral is

$(u-u_{0})\frac{(\ln f_{,\zeta\zeta})_{,u}}{(\ln
f_{,\zeta\zeta})_{,\zeta}}-\frac{2i\kappa-2}{(\ln
f_{,\zeta\zeta})_{,\zeta}}=i\kappa\zeta-u^{-i\kappa+1}h^{\prime}(u)\\\
\iff\frac{(u-u_{0})^{i\kappa+1}(\ln(u-u_{0})^{-2i\kappa+2}f_{,\zeta\zeta})_{,u}}{(\ln(u-u_{0})^{-2i\kappa+2}f_{,\zeta\zeta})_{,\zeta}}=(\zeta
i\kappa(u-u_{0})^{i\kappa}-(u-u_{0})h^{\prime}(u))\\\
\iff\left[(\zeta(u-u_{0})^{i\kappa}-h(u))_{,\zeta}(u-u_{0})\partial_{u}\right.\\\
\left.{}-(u-u_{0})(\zeta(u-u_{0})^{i\kappa}-h(u))_{,u}\partial_{\zeta}\right]\ln(u-u_{0})^{-2i\kappa+2}f_{,\zeta\zeta}=0,$
(97)

with implicitly defined integration parameter $h(u)$. The general solution to
the last equation is clearly

$\ln(u-u_{0})^{-2i\kappa+2}f_{,\zeta\zeta}=\ln\mathrm{f}^{\prime\prime}\left(\zeta(u-u_{0})^{i\kappa}-h(u)\right)\\\
\iff
f(\zeta,u)=\frac{1}{(u-u_{0})^{2}}\mathrm{f}\left(\zeta(u-u_{0})^{i\kappa}-h(u)\right)+A_{1}(u)\zeta+A_{0}(u),$
(98)

for some holomorphic function $\mathrm{f}(z)$ and integration parameters
$A_{0}(u)$, $A_{1}(u)$. We can set $h(u)=A_{0}(u)=u_{0}=0$ by gauge freedom.
For this form of $f(\zeta,u)$, the value of the curvature invariant
$\mathbf{M}^{(i\kappa)}$ in (3.1) is given in (49). So setting
$\mathbf{M}^{(i\kappa)}=0$ forces $\mathrm{f}^{\prime\prime\prime}(z)=0$ or
$A_{1}(u)\zeta=A_{1,0}\zeta u^{i\kappa-2}$. The former possibility results in
a formfactor corresponding to $G_{5}$ subclass, while in the latter
possibility the term $A_{1}(u)\zeta$ can be absorbed into the definition of
$\mathrm{f}(z)$, which lands us in the $G_{2a}$ subclass.

In this branch of the classification, we assume now that
$\mathrm{f}^{\prime\prime\prime}(z)\neq 0$, and the above conditions hold with
$\kappa\in\mathbb{R}$, so that we are in the $G_{2a}$ subclass.

Comparing (96) to the definition of the invariant $\Xi^{(i\kappa)}$ in (35),
we see that it implies that the invariant $\Xi^{(i\kappa)}=1/u$. The residual
gauge transformations that preserve the formfactor
$f(\zeta,u)=u^{-2}\mathrm{f}(\zeta u^{i\kappa})$ are $\zeta\mapsto
e^{i\beta}\zeta$, which induces
$\mathrm{f}(z)\mapsto\mathrm{f}(e^{-i\beta}z)$,
$\zeta\mapsto\zeta+z_{0}u^{-i\kappa}$, which induces
$\mathrm{f}(z)\mapsto\mathrm{f}(z-z_{0})+i\kappa(i\kappa-1)\overline{z_{0}}z+\frac{1}{2}|z_{0}|^{2}$,
$v\mapsto v+2u^{-1}\operatorname{Re}w_{0}$ with
$c(u)=u^{-2}\operatorname{Im}w_{0}$, which induces
$\mathrm{f}(z)\mapsto\mathrm{f}(z)+w_{0}$, and $\zeta\mapsto\bar{\zeta}$,
which induces $\mathrm{f}(z)\mapsto\bar{\mathrm{f}}(z)$.

Generically,
$\partial_{z}(\mathrm{f}^{\prime\prime\prime}(z)^{2}/\mathrm{f}^{\prime\prime}(z)^{3})\neq
0$, so then there exists a holomorphic function $F\colon
U\subset\mathbb{C}\to\mathbb{C}$ such that

$\ln\frac{\mathrm{f}^{\prime\prime\prime}(z)}{\mathrm{f}^{\prime\prime}(z)}=F\left(\frac{\mathrm{f}^{\prime\prime\prime}(z)^{2}}{\mathrm{f}^{\prime\prime}(z)^{3}}\right).$
(99)

The argument of $F(-)$ is an invariant because it coincides with the value of
the invariant $I_{\mathrm{d}}^{(i\kappa)}$ in (36), which was evaluated in
(48). Neither the left-hand side in (99) nor $F(y)$ itself are entirely
invariant, as $\mathrm{f}(z)\mapsto\mathrm{f}(e^{-i\beta}z)$ induces
$F(y)\mapsto F(y)-i\beta$ and $\mathrm{f}(z)\mapsto\bar{\mathrm{f}}(z)$
induces $F(y)\mapsto\bar{F}(y)$. However, $\operatorname{Re}F(y)$ is invariant
under these transformations and also uniquely fixes $F(y)$ up to these
ambiguities. Thus, we consider $\operatorname{Re}F\colon
U\subset\mathbb{C}\to\mathbb{R}$ as our signature function.

If $F^{\prime}(y)=1/(2y)$ or equivalently $F(y)=\frac{1}{2}\ln(2Ay)$, then the
ODE (99) no longer depends on $\mathrm{f}^{\prime\prime\prime}(z)$ and reduces
to $\mathrm{f}^{\prime\prime}(z)=2A$, and hence
$\mathrm{f}^{\prime\prime\prime}(z)=0$, which contradicts our earlier
hypothesis. Otherwise, the solution $\mathrm{f}(z)=Az^{2}+A_{1}z+A_{0}$
recovers the $G_{6a}$ subclass.

Provided $F^{\prime}(y)\neq 1/(2y)$, the ODE (99) can be solved for
$\mathrm{f}^{\prime\prime\prime}(z)$, so the usual existence and uniqueness
theory holds for it. If $\mathrm{f}(z)=\mathcal{F}(z)$ is a particular
solution, then symmetries dictate that the general 3-parameter solution is

$\mathrm{f}(z)=\mathcal{F}(z-z_{0})+A_{1,0}(z-z_{0})+A_{0,0}.$ (100)

While the $z_{0}$ and $A_{0,0}$ integration constants can be absorbed by the
residual gauge freedom, the parameter $A_{1,0}$ cannot be and does contain
some invariant information. To be concrete, we uniquely fix444There may be
other ways to uniquely fixing the solution $\mathcal{F}(z)$, for instance by
different initial conditions at $z=0$ or by asyptotic conditions at
$z=\infty$. $\mathcal{F}(z)$ from $F(y)$ by requiring that
$\mathcal{F}(z)=\frac{1}{2}z^{2}+O(z^{3})$. The gauge transformation
$F(y)\mapsto F(y)-i\beta$ induces
$\mathcal{F}(z)\mapsto\mathcal{F}(e^{-i\beta}z)$, $A_{1,0}\mapsto
A_{1,0}e^{-i\beta}$, and $F(y)\mapsto\bar{F}(y)$ induces
$\mathcal{F}(z)\mapsto\overline{\mathcal{F}}(z)$,
$A_{1,0}\mapsto\overline{A_{1,0}}$. So only $c=|A_{1,0}|$ is invariant
contained in $A_{1,0}$. Using the ODE (99), and computing the invariants
$I_{\mathrm{d}}^{(\infty)}$ and $I_{\mathrm{e}}^{(\infty)}$, from (36) and
(3.1), as well as the evaluation in (48), we find

$\displaystyle-e^{2F(I_{\mathrm{d}}^{(i\kappa)})}/I_{\mathrm{d}}^{(i\kappa)}$
$\displaystyle=\mathrm{f}^{\prime\prime}(z)=\mathcal{F}^{\prime\prime}(z-z_{0}),$
(101)
$\displaystyle-\frac{I_{\mathrm{e}}^{(i\kappa)}}{I_{\mathrm{d}}^{(i\kappa)}}e^{F(I_{\mathrm{d}}^{(\infty)})}$
$\displaystyle=i\kappa(i\kappa-1)\overline{(z-z_{0})}+\mathrm{f}^{\prime}(z)$
$\displaystyle=i\kappa(i\kappa-1)\overline{(z-z_{0})}+A_{1,0}+\mathcal{F}^{\prime}(z-z_{0}),$
(102)

where $z=\zeta u^{i\kappa}$. Combining the two formulas, we can extract the
constant

$\displaystyle c=|A_{1,0}|$
$\displaystyle=\left|\frac{I_{\mathrm{e}}^{(i\kappa)}}{I_{\mathrm{d}}^{(i\kappa)}}e^{F(I_{\mathrm{d}}^{(i\kappa)})}-i\kappa(i\kappa-1)\overline{Z_{a}}-\mathcal{F}^{\prime}(Z_{a})\right|$
$\displaystyle\quad=:\mathcal{A}_{\kappa,\operatorname{Re}F}(I_{\mathrm{d}}^{(i\kappa)},I_{\mathrm{e}}^{(i\kappa)}),$
(103) $\displaystyle\text{where}\quad Z_{a}$
$\displaystyle=(\mathcal{F}^{\prime\prime})^{-1}\left(-e^{2F(I_{\mathrm{d}}^{(i\kappa)})}/I_{\mathrm{d}}^{(i\kappa)}\right).$
(104)

While the individual terms entering formula
$\mathcal{A}_{\kappa,\operatorname{Re}F}$ that extracts $c=|A_{1,0}|$ are not
invariant, since they depend on the choice of $F(y)$ for given
$\operatorname{Re}F(y)$, it is straightforward to check that the whole formula
is independent of those choices and so defines an invariant. Still, for given
$\operatorname{Re}F(y)$, to make the formula really explicit, we need to pick
a compatible $F(y)$ and determine the unique solution $\mathcal{F}(z)$ from
it.

The only possibility left to consider is that with
$\partial_{z}(\mathrm{f}^{\prime\prime\prime}(z)^{2}/\mathrm{f}^{\prime\prime}(z)^{3})=0$.

Suppose we reparametrize
$\operatorname{Re}F(y)=\operatorname{Re}\tilde{F}(\kappa^{2}y)$. Then the
solutions of the signature ODE (99) scale as
$\mathrm{f}(z)=\kappa^{2}\tilde{\mathrm{f}}(z)$, where $\tilde{\mathrm{f}}(z)$
solves the signature ODE for $\operatorname{Re}\tilde{F}(y)$. Taking the
invariant parameter limit $\kappa,c\to\infty$, accompanied by the gauge
parameter limits $u_{0},\beta\to-\infty$, such that
$(-u_{0})e^{\beta/\kappa}\to 1$, $\kappa/(-u_{0})\to 1$ and
$c/\kappa^{2}\to\tilde{c}$, we obtain

$\zeta e^{i\beta}(u-u_{0})^{i\kappa}\to\zeta
e^{iu}\quad\text{and}\quad\frac{1}{(u-u_{0})^{2}}\mathrm{f}(z)\to\tilde{\mathrm{f}}(z),$
(105)

and hence we recover the $G_{2b^{\circ};\tilde{c},\operatorname{Re}\tilde{F}}$
subclass.

  14. 14.

Within the $f(\zeta,u)=u^{-2}\mathrm{f}(\zeta u^{i\kappa})$ formfactor family
under the condition
$\partial_{z}(\mathrm{f}^{\prime\prime\prime}(z)^{2}/\mathrm{f}^{\prime\prime}(z)^{3})=0$,
let us parametrize

$\frac{\mathrm{f}^{\prime\prime\prime}(z)^{2}}{\mathrm{f}^{\prime\prime}(z)^{3}}=-\alpha
e^{-i\gamma},$ (106)

with constant $\alpha>0$ and $\gamma\in[-\pi,\pi]$, which integrates to

$\mathrm{f}(z)=4\alpha e^{i\gamma}\ln z+A_{1,0}z+A_{0,0}.$ (107)

We can set $A_{0,0}=0$ by residual gauge freedom. Further, we can eliminate
the $\mathrm{f}(z)\mapsto\bar{\mathrm{f}}(z)$ gauge freedom by choosing
$\gamma\in[0,\pi]$, and also eliminate the $\zeta\mapsto e^{i\beta}\zeta$
gauge freedom by choosing $A_{1,0}=c\geq 0$. The resulting formfactor takes
the form

$f(\zeta,u)=\frac{4\alpha}{u^{2}}e^{i\gamma}\ln\zeta e^{i\lambda u}+c\zeta
u^{i\kappa},$ (108)

or after applying another remaining gauge transformation

$f(\zeta,u)=\frac{4\alpha}{u^{2}}\left(e^{i\gamma}\ln\zeta+c\zeta
u^{i\kappa}\right),$ (109)

with $\alpha>0$, $\gamma\in[0,\pi]$, $\kappa\in\mathbb{R}$ and $c\geq 0$
labelling the corresponding isometry class. When $c\neq 0$, we label this
isometry class as $G_{2a^{\prime};\alpha,\gamma,\kappa,c}$.

The excluded value of $c=|A_{1,0}|=0$ lands us in the $G_{3a}$ subclass.

Taking the invariant parameter limit $\kappa,\alpha\to\infty$, accompanied by
the gauge parameter limits $u_{0},\beta\to-\infty$, such that
$4\alpha/u_{0}^{2}\to 1$, $(-u_{0})e^{\beta/\kappa}\to 1$ and
$\kappa/(-u_{0})\to\lambda$, we recover the
$G_{2b^{\prime\prime};\gamma,\lambda,c}$ subclass.

  15. 15.

Within the formfactor subfamily

$f(\zeta,u)=\frac{4\alpha}{u^{2}}\left(e^{i\gamma}\ln\zeta+c\zeta
u^{i\kappa}\right),$ (110)

where $c=0$, the $\kappa$ parameter drops out and the formfactor reduces to

$f(\zeta,u)=\frac{4\alpha}{u^{2}}e^{i\gamma}\ln\zeta.$ (111)

We can use the $\zeta\mapsto\bar{\zeta}$ transformation to restrict
$\gamma\in[0,\pi]$, which then becomes invariant, along with $\alpha$. When
$e^{i\gamma}\neq\pm 1$, we label this isometry class as
$G_{3a^{\circ};\gamma}$.

The excluded value of $e^{i\gamma}=\pm 1$ lands us in the
$G_{3a^{\prime};\pm\alpha}$ subclass.

Taking the invariant parameter limit $\alpha\to\infty$, accompanied by the
gauge parameter limit $u_{0}\to\infty$, such that $4\alpha/u_{0}^{2}\to 1$, we
recover the $G_{3b^{\circ};\gamma}$ subclass.

  16. 16.

In the formfactor subfamily

$f(\zeta,u)=\pm\frac{4\alpha}{u^{2}}\ln\zeta,$ (112)

the only invariants are the sign $\pm$ and $\alpha>0$. We label this isometry
class as $G_{3a^{\prime};\pm\alpha}$.

Taking the invariant parameter limit $\alpha\to\infty$, accompanied by the
gauge parameter limit $u_{0}\to\infty$, such that $4\alpha/u_{0}^{2}\to 1$, we
recover the $G_{3b^{\prime};\pm}$ subclass.

  17. 17.

Suppose that within the $G_{2b}$ subclass, the following invariant condition
holds

$\left(\frac{(\ln f_{,\zeta\zeta})_{,u}}{(\ln
f_{,\zeta\zeta})_{,\zeta}}\right)_{\zeta}=0.$ (113)

As discussed earlier, it corresponds to either $-(1/(\ln
f_{,\zeta\zeta})_{,\zeta})=\frac{1}{2}$, which lands us in the
$G_{2b^{\prime\prime}}$ subclass, or $\lambda=0$. In this branch of the
classification we consider the latter option, which corresponds to the
formfactor family

$f(\zeta,u)=\mathrm{f}(\zeta).$ (114)

The evaluation (50) of the invariant $I_{\mathrm{d}}^{(\infty)}$ with
$\lambda=0$ simplifies to

$I_{\mathrm{e}}^{(\infty)}=\frac{\mathrm{f}^{\prime\prime\prime}(z)\mathrm{f}^{\prime}(z)}{\mathrm{f}^{\prime\prime}(z)^{2}},$
(115)

where $z=\zeta$. Generically
$\partial_{z}(\mathrm{f}^{\prime\prime\prime}(z)\mathrm{f}^{\prime}(z)/\mathrm{f}^{\prime\prime}(z)^{2})=0$,
so then there exists a holomorphic function $F\colon
U\subset\mathbb{C}\to\mathbb{C}$ such that

$\ln\frac{\mathrm{f}^{\prime\prime\prime}(z)}{\mathrm{f}^{\prime\prime}(z)}=F\left(\frac{\mathrm{f}^{\prime\prime\prime}(z)\mathrm{f}^{\prime}(z)}{\mathrm{f}^{\prime\prime}(z)^{2}}\right).$
(116)

Neither the left-hand side above nor $F(y)$ are entirely invariant, as
$\mathrm{f}(z)\mapsto\mathrm{f}(e^{-i\beta}z)$ induces $F(y)\mapsto
F(y)-i\beta$ and $\mathrm{f}(z)\mapsto\bar{\mathrm{f}}(z)$ induces
$F(y)\mapsto\bar{F}(y)$. However, $\operatorname{Re}F(y)$ is invariant under
these transformations and also uniquely fixes $F(y)$ up to these ambiguities.
Thus, we consider $\operatorname{Re}F\colon U\subset\mathbb{C}\to\mathbb{R}$
as our signature function.

If $F^{\prime}(y)=1/y$ or equivalently $F(y)=\ln(\lambda y)$, then the ODE
(116) no longer depends on $\mathrm{f}^{\prime\prime\prime}(z)$ and reduces to
$\mathrm{f}^{\prime\prime}(z)=\lambda\mathrm{f}^{\prime}(z)$, and hence
$\mathrm{f}^{\prime\prime\prime}(z)\mathrm{f}^{\prime}(z)/\mathrm{f}^{\prime\prime}(z)^{2}=1$,
which contradicts our genericity hypothesis. If $F^{\prime}(y)=1/(2-y)$, or
equivalently $F(y)=\ln\frac{4c}{2-y}$, then the ODE (116) has the general
3-parameter solution

$\mathrm{f}(z)=A(\ln(z-z_{0})+cz+A_{0,0}),$ (117)

which lands us in the $G_{2b^{\prime\prime}}$ subclass, as well as a singular
2-parameter solution

$\mathrm{f}(z)=e^{4c(z-z_{0})}+A_{0,0},$ (118)

which also implies
$\mathrm{f}^{\prime\prime\prime}(z)\mathrm{f}^{\prime}(z)/\mathrm{f}^{\prime\prime}(z)^{2}=1$
and hence contradicts the genericity hypothesis.

Provided $F^{\prime}(y)\neq 1/y$, the ODE (116) can be solved for
$\mathrm{f}^{\prime\prime}(z)$, so the usual existence and uniqueness theory
holds for it. If $\mathrm{f}(z)=\mathcal{F}(z)$ is a particular solution, then
symmetries dictate that the general 3-parameter solution is

$\mathrm{f}(z)=A\mathcal{F}(z-z_{0})+A_{0,0}.$ (119)

Writing $A=a^{2}e^{i\gamma}$ with $a>0$ and $\gamma\in[-\pi,\pi]$, the $a$ and
$A_{0,0}$ integration constants can be absorbed by the residual gauge freedom.
To be concrete, we uniquely fix555There may be other ways to uniquely fixing
the solution $\mathcal{F}(z)$, for instance by different initial conditions at
$z=0$ or by asyptotic conditions at $z=\infty$. $\mathcal{F}(z)$ from $F(y)$
by requiring that $\mathcal{F}(z)=\frac{1}{2}z^{2}+O(z^{3})$. The gauge
transformation $F(y)\mapsto F(y)-i\beta$ induces
$\mathcal{F}(z)\mapsto\mathcal{F}(e^{-i\beta}z)$, and $F(y)\mapsto\bar{F}(y)$
induces $\mathcal{F}(z)\mapsto\overline{\mathcal{F}}(z)$,
$\gamma\mapsto-\gamma$. So only $|\gamma|\in[0,\pi]$ is invariant among the
integration constants. Using the signature ODE (116), and computing the
invariants $J$ and $I_{\mathrm{e}}^{(\infty)}$, from (27) and (3.1), we find

$\displaystyle I_{\mathrm{e}}^{(\infty)}e^{-F(I_{\mathrm{e}}^{(\infty)})}$
$\displaystyle=\frac{\mathrm{f}^{\prime}(z)}{\mathrm{f}^{\prime\prime}(z)}=\frac{\mathcal{F}^{\prime}(z-z_{0})}{\mathcal{F}^{\prime\prime}(z-z_{0})},$
(120) $\displaystyle Je^{4\operatorname{Im}F(I_{\mathrm{e}}^{(\infty)})}$
$\displaystyle=\frac{\mathrm{f}^{\prime\prime}(z)}{\overline{\mathrm{f}^{\prime\prime}(z)}}=e^{2i\gamma}\frac{\mathcal{F}^{\prime\prime}(z-z_{0})}{\overline{\mathcal{F}^{\prime\prime}(z-z_{0})}}.$
(121)

Combining the two formulas, we can extract the constant

$\displaystyle\cos\gamma$
$\displaystyle=\operatorname{Re}\left[Je^{4\operatorname{Im}F(I_{\mathrm{e}}^{(\infty)})}\frac{\overline{\mathcal{F}^{\prime\prime}(Z_{b^{\prime}})}}{\mathcal{F}^{\prime\prime}(Z_{b^{\prime}})}\right]^{1/2}=:\mathcal{G}_{\operatorname{Re}F}(I_{\mathrm{e}}^{(\infty)}),$
(122) $\displaystyle\text{where}\quad Z_{b^{\prime}}$
$\displaystyle=\left[\frac{\mathcal{F}^{\prime}(-)}{\mathcal{F}^{\prime\prime}(-)}\right]^{-1}\left(I_{\mathrm{e}}^{(\infty)}e^{-F(I_{\mathrm{e}}^{(\infty)})}\right).$
(123)

Note that the value of $\cos\gamma$ fixes a unique $\gamma\in[0,\pi]$, which
we combine with the square root prescription
$[e^{2i\gamma}]^{1/2}=e^{i\gamma}$ with $\gamma\in[0,\pi]$. While the
individual terms entering formula $\mathcal{G}_{\operatorname{Re}F}$ that
extracts $\cos\gamma$ are not invariant, since they depend on the choice of
$F(y)$ for given $\operatorname{Re}F(y)$, it is straightforward to check that
the whole formula is independent of those choices and so defines an invariant.
Still, for given $\operatorname{Re}F(y)$, to make the formula really explicit,
we need to pick a compatible $F(y)$ and determine the unique solution
$\mathcal{F}(z)$ from it.

The only possibilities left to consider are those with constant
$\mathrm{f}^{\prime\prime\prime}(z)\mathrm{f}^{\prime}(z)/\mathrm{f}^{\prime\prime}(z)^{2}=\frac{k-2}{k-1}$,
separately for $k=\infty$ and $k\neq\infty$.

  18. 18.

Within the $f(\zeta,u)=\mathrm{f}(\zeta)$ formfactor family, consider the
invariant condition

$I_{\mathrm{e}}^{(\infty)}=\frac{\mathrm{f}^{\prime\prime\prime}(z)\mathrm{f}^{\prime}(z)}{\mathrm{f}^{\prime\prime}(z)^{2}}=1,$
(124)

where we have replaced $z=\zeta$, which integrates to the general
$3$-parameter solution

$\mathrm{f}(z)=e^{2\lambda(z-z_{0})}+A_{0,0}.$ (125)

We can set the integration constants $z_{0}=A_{0,0}=0$ by residual gauge
freedom, as well as use it to select $\lambda>0$, whose value then becomes the
only invariant of this isometry class, which we label as $G_{3c;\lambda}$.

  19. 19.

Within the $f(\zeta,u)=\mathrm{f}(\zeta)$ formfactor family, consider the
invariant condition (with $k\neq\infty$)

$I_{\mathrm{e}}^{(\infty)}=\frac{\mathrm{f}^{\prime\prime\prime}(z)\mathrm{f}^{\prime}(z)}{\mathrm{f}^{\prime\prime}(z)^{2}}=\frac{k-2}{k-1},$
(126)

where we have replaced $z=\zeta$, which integrates to the general
$3$-parameter solution

$\mathrm{f}(z)=\begin{cases}a^{2}e^{i\gamma}(z-z_{0})^{k}+A_{0,0}&(k\neq
0,1)\\\ a^{2}e^{i\gamma}\ln(z-z_{0})+A_{0,0}&(k=0)\end{cases}.$ (127)

When $k=0$, we recover the $G_{2b^{\prime\prime\prime}}$ subclass, while the
value $k=1$ is not allowed by the previous formula. When $k=2$ we recover the
$G_{6b}$ subclass. When $k\not\in i\mathbb{R}\cup\\{1,2\\}$, we can set
$a^{2}e^{i\gamma}=1$ and $z_{0}=A_{0,0}=0$ by gauge freedom. We label the
resulting isometry class by $G_{2b^{\prime\prime\prime};k}$.

The only possibility left to consider is that with $k\in i\mathbb{R}$.

  20. 20.

Within the formfactor family

$f(\zeta,u)=a^{2}e^{i\gamma}\zeta^{k},$ (128)

with $k=2i\kappa$, $\kappa\in\mathbb{R}\setminus\\{0\\}$, we can set $a^{2}=1$
by residual gauge freedom, while by exploiting the $\zeta\mapsto\bar{\zeta}$
gauge transformation, we can restrict $\gamma\in[0,\pi]$, which then becomes
an invariant. We label the corresponding isometry class as
$G_{3d;\gamma,\kappa}$.

Taking the invariant parameter limit $\gamma,\kappa\to\infty$, accompanied by
the gauge parameter limit $a,z_{0}\to\infty$, such that $\gamma-2i\ln
a+2\kappa\ln(-z_{0})\to 0$ and $\kappa/(-z_{0})=-i\lambda$, we obtain

$e^{i\gamma}\zeta^{2i\kappa}\to e^{2\lambda\zeta}$ (129)

and hence we recover the $G_{3c;\lambda}$ subclass.

Together with the data in Figure 1 and Table LABEL:tab:pp-wave-isom-classes
this completes the classification of local isometry classes of regular highly
symmetric pp-wave geometries. ∎

Table 2: IDEAL characterization of highly symmetric pp-wave isometry classes to accompany Figure 2 and Theorem 8. The details of the notation and the decision logic are contained in the statement and proof of the theorem. class | parameters | inequalities/equations
---|---|---
$G_{6a;\alpha,\kappa}$: |  $\alpha>0$, $\kappa\geq 0$ |  $|\mathbf{K}|^{2}=0$, $I_{\mathrm{a}}^{(2)}=(i\kappa-1)\alpha^{-1/2}$
$G_{6b;\lambda}$: | $\lambda\geq 0$ | $\begin{gathered}|\mathbf{K}|^{2}=0,\quad I_{\mathrm{a}}^{(2)}=2i\lambda\end{gathered}$
$G_{5^{\prime};\alpha,F}$: |  $\infty\geq\alpha>0$, $F(y)$ | $\nabla\operatorname{Im}I_{\mathrm{a}}^{(2)}\neq 0$
|  | $\begin{gathered}|\mathbf{K}|^{2}=0,\quad\operatorname{Re}I_{\mathrm{a}}^{(2)}=-\alpha^{-1/2},\\\ \operatorname{Im}I_{\mathrm{b}}^{(2)}=F(\operatorname{Im}I_{\mathrm{a}}^{(2)})\end{gathered}$
$G_{5^{\circ};F}$: | $F(y)$ | $\nabla\operatorname{Re}I_{\mathrm{a}}^{(2)}\neq 0$
|  | $\begin{gathered}|\mathbf{K}|^{2}=0,\\\ \begin{pmatrix}\operatorname{Re}I_{\mathrm{b}}^{(2)}\\\ (\operatorname{Im}I_{\mathrm{a}}^{(2)})^{2}\end{pmatrix}=F(\operatorname{Re}I_{\mathrm{a}}^{(2)})\end{gathered}$
$G_{3a^{\prime};\pm\alpha}$: |  $\alpha>0$, $\\{\pm\\}$ | $|\mathbf{K}|^{2}\neq 0$
|  | $\begin{gathered}I_{\mathrm{c}}=1/2,\quad\mathbf{N}=0,\\\ \operatorname{Re}I_{\mathrm{a}}^{(0)}=-\alpha^{-1/2},\\\ J=1,\quad I_{\mathrm{d}}^{(0)}=\pm\alpha^{-1/2}\end{gathered}$
$G_{3a^{\circ};\alpha,\gamma}$: |  $\alpha>0$, $\gamma\in(0,\pi)$ | $|\mathbf{K}|^{2}\neq 0$
|  | $\begin{gathered}I_{\mathrm{c}}=1/2,\quad\mathbf{N}=0,\\\ \operatorname{Re}I_{\mathrm{a}}^{(0)}=-\alpha^{-1/2},\quad J=e^{2i\gamma}\end{gathered}$
$G_{3b^{\prime};\pm}$: | $\\{\pm\\}$ | $|\mathbf{K}|^{2}\neq 0$
|  | $\begin{gathered}I_{\mathrm{c}}=1/2,\quad\mathbf{N}=0,\\\ \operatorname{Re}I_{\mathrm{a}}^{(0)}=0,\quad J=1,\\\ \exists\bm{\ell}:-2\mathbf{D}=\pm\bm{\ell}\bm{\ell}\end{gathered}$
$G_{3b^{\circ};\gamma}$: | $\gamma\in(0,\pi)$ | $|\mathbf{K}|^{2}\neq 0$
|  | $\begin{gathered}I_{\mathrm{c}}=1/2,\quad\mathbf{N}=0,\\\ \operatorname{Re}I_{\mathrm{a}}^{(0)}=0,\quad J=e^{2i\gamma}\end{gathered}$
$G_{3c;\lambda}$: | $\lambda>0$ |  $|\mathbf{K}|^{2}\neq 0$, $I_{\mathrm{c}}\neq 1/2$
|  | $\begin{gathered}\mathfrak{L}=0,\quad\mathbf{M}^{(\infty)}=0,\\\ I_{\mathrm{e}}^{(\infty)}=1,\quad|\mathbf{K}|^{2}=4\lambda^{2}\end{gathered}$
$G_{3d;\gamma;\kappa}$: |  $\gamma\in[0,\pi]$, $\kappa\geq 0$ |  $|\mathbf{K}|^{2}\neq 0$, $I_{\mathrm{c}}\neq 1/2$
|  | $\begin{gathered}\mathfrak{L}=0,\quad\mathbf{M}^{(\infty)}=0,\quad I_{\mathrm{e}}^{(\infty)}=\frac{2(i\kappa-1)}{2i\kappa-1},\\\ J=e^{2i\gamma}\frac{(2i\kappa-1)(i\kappa+1)^{2}}{(2i\kappa+1)(i\kappa-1)^{2}}\left|\frac{\mathbf{K}}{2(i\kappa-1)}\right|^{-4i\kappa}\end{gathered}$
$G_{2b^{\prime\prime\prime};k}$: |  $k\in\mathbb{C}$, $\operatorname{Re}k\neq 0$, $k\neq 1,2$ |  $|\mathbf{K}|^{2}\neq 0$, $I_{\mathrm{c}}\neq 1/2$
| $\begin{gathered}\mathfrak{L}=0,\quad\mathbf{M}^{(\infty)}=0,\quad
I_{\mathrm{e}}^{(\infty)}=\frac{k-2}{k-1}\end{gathered}$
$G_{2b^{\prime\prime};\gamma,\lambda,c}$: |  $\lambda\in\mathbb{R}$, $\gamma\in[0,\pi]$, $c>0$ | $|\mathbf{K}|^{2}\neq 0$
|
$\begin{gathered}I_{\mathrm{d}}=1/2,\quad\mathfrak{L}=0,\quad|N|=c|\mathbf{K}|\\\
\left(\nabla N-\frac{N}{2}\overline{\mathbf{K}}\right)^{\otimes 2}=2\lambda
e^{-i\gamma\operatorname{sgn}\lambda}N^{2}\mathbf{D}^{\dagger},\\\
\begin{cases}\mathbf{D}=2e^{i\gamma}\overline{\mathbf{D}}&\text{if
$e^{i\gamma}\neq\pm 1$}\\\ -2\mathbf{D}=\pm\bm{\ell}\bm{\ell}&\text{if
$e^{i\gamma}=\pm 1$}\end{cases}\end{gathered}$
$G_{2a^{\prime};\alpha,\gamma,\kappa,c}$: |  $\alpha>0$, $\gamma\in[0,\pi]$, $c>0$ |  $|\mathbf{K}|^{2}\neq 0$, $\mathfrak{L}\neq 0$
| $\begin{gathered}I_{\mathrm{d}}=1/2,\quad\mathbf{M}^{(i\kappa)}=0,\\\
|N|=c|\mathbf{K}|\end{gathered}$
$G_{2a^{\circ};\kappa,c,\operatorname{Re}F}$: |  $\kappa>0$, $c>0$, $\operatorname{Re}F(z)$ |  $|\mathbf{K}|^{2}\neq 0$, $I_{\mathrm{c}}\neq 1/2$, $\mathfrak{L}\neq 0$,
|  $\nabla\frac{I_{\mathrm{c}}}{(I_{\mathrm{c}}-\frac{1}{2})}\neq 0$, $\nabla
I_{\mathrm{d}}^{(i\kappa)}\neq 0$
|  | $\begin{gathered}2\nabla\mathbf{L}^{(i\kappa)}=\mathbf{L}^{(i\kappa)}\mathbf{L}^{(i\kappa)},\quad\mathbf{M}^{(i\kappa)}=0,\\\ \ln(|\mathbf{K}|^{2})=2\operatorname{Re}F(-I_{\mathrm{d}}^{(i\kappa)}),\\\ \mathcal{A}_{\kappa,\operatorname{Re}F}(I_{\mathrm{d}}^{(i\kappa)},I_{\mathrm{e}}^{(i\kappa)})=c\end{gathered}$
$G_{2b^{\circ};c,\operatorname{Re}F}$: |  $c>0$, $\operatorname{Re}F(z)$ |  $|\mathbf{K}|^{2}\neq 0$, $I_{\mathrm{c}}\neq 1/2$, $\mathfrak{L}\neq 0$,
|  |  $\nabla\frac{I_{\mathrm{c}}}{(I_{\mathrm{c}}-\frac{1}{2})}\neq 0$, $\nabla I_{\mathrm{d}}^{(\infty)}\neq 0$
|  | $\begin{gathered}\operatorname{Re}\mathbf{L}^{(\infty)}=0,\quad\nabla\mathbf{L}^{(\infty)}=0,\\\ \mathbf{M}^{(\infty)}=0,\\\ \ln(|\mathbf{K}|^{2})=2\operatorname{Re}F(I_{\mathrm{d}}^{(\infty)}),\\\ \mathcal{A}_{\infty,\operatorname{Re}F}(I_{\mathrm{d}}^{(\infty)},I_{\mathrm{e}}^{(\infty)})=c\end{gathered}$
$G_{2b^{\prime};\gamma,\operatorname{Re}F}$: |  $\alpha\in[0,\pi]$, $\operatorname{Re}F(z)$ |  $|\mathbf{K}|^{2}\neq 0$, $I_{\mathrm{c}}\neq 1/2$, $\nabla I_{\mathrm{e}}^{(\infty)}\neq 0$
|  | $\begin{gathered}\mathfrak{L}=0,\quad\mathbf{M}^{(\infty)}=0,\\\ \ln(|\mathbf{K}|^{2})=2\operatorname{Re}F(I_{\mathrm{e}}^{(\infty)}),\\\ \mathcal{G}_{\operatorname{Re}F}(I_{\mathrm{e}}^{(\infty)})=\cos\gamma\end{gathered}$
$G_{2c^{\prime};\alpha,F}$: |  $\infty\geq\alpha>0$, $F(y)$ |  $|\mathbf{K}|^{2}\neq 0$, $\nabla\operatorname{Im}[(I_{\mathrm{a}}^{(0)})^{2}/I_{\mathrm{d}}^{(0)}]\neq 0$
|  | $\begin{gathered}\operatorname{Re}I_{\mathrm{a}}^{(0)}=-\alpha^{-1/2},\\\ \operatorname{Im}I_{\mathrm{a}}^{(0)}=F\left(\operatorname{Im}[(I_{\mathrm{a}}^{(0)})^{2}/I_{\mathrm{d}}^{(0)}]\right)\end{gathered}$
$G_{2c^{\circ};F}$: | $F(y)$ |  $|\mathbf{K}|^{2}\neq 0$, $\nabla\operatorname{Re}I_{\mathrm{a}}^{(0)}\neq 0$
|  | $\begin{gathered}\begin{pmatrix}\operatorname{Re}I_{\mathrm{b}}^{(0)}\\\ \operatorname{Re}[(I_{\mathrm{a}}^{(0)})^{2}/I_{\mathrm{d}}^{(0)}]\end{pmatrix}=F(\operatorname{Re}I_{\mathrm{a}}^{(0)})\end{gathered}$

$\begin{gathered}\kappa,c,u_{0},\beta\to\infty,\\\ \frac{\kappa}{(-u_{0})}\to
1,(-u_{0})e^{\beta/\kappa}\to 1,\\\ c\mapsto\kappa^{2}c,F(y)\mapsto
F(\kappa^{2}y)\end{gathered}$$\begin{gathered}(F^{-1})^{\prime}(x)\to 0\text{
or}\\\ (\mathrm{f}^{\prime\prime\prime 2}/\mathrm{f}^{\prime\prime
3})^{\prime}=0\end{gathered}$$\left(\frac{(\ln f_{,\zeta\zeta})_{,u}}{(\ln
f_{,\zeta\zeta})_{,\zeta}}\right)_{,\zeta}=0$$\begin{gathered}\kappa,\alpha,u_{0}\to\infty,\\\
(-u_{0})e^{\beta/\kappa}\to
1,\frac{\kappa}{(-u_{0})}\to\lambda,\frac{4\alpha}{u_{0}^{2}}\to
1\end{gathered}$$\begin{gathered}(F^{-1})^{\prime}(x)\to 0\text{ or}\\\
(\mathrm{f}^{\prime\prime\prime 2}/\mathrm{f}^{\prime\prime
3})^{\prime}=0\end{gathered}$$c\to
0$$\alpha,u_{0}\to\infty,\frac{4\alpha}{u_{0}^{2}}\to 1$$c\to
0$$F^{\prime}(y)=\frac{1}{2y}$$\begin{gathered}\alpha,\kappa,u_{0},\beta\to\infty,\\\
\frac{4\alpha}{u_{0}^{2}}=1,\kappa\ln
u_{0}-\beta=0,\frac{\kappa}{u_{0}}=-\lambda\end{gathered}$$F^{\prime}(y)=\frac{1}{2y}$$F(y)=-\frac{y}{2\alpha^{1/2}}$$F(y)=0$$\begin{gathered}F_{\mathrm{Re}}(y)=y^{2}/2\text{
or}\\\
\operatorname{Re}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}=-\alpha^{-1/2}\end{gathered}$$\begin{gathered}\gamma,\kappa,a,z_{0}\to\infty,\\\
\gamma-2i\ln a+2\kappa\ln(-z_{0})\to
0,\frac{\kappa}{(-z_{0})}\to-i\lambda\end{gathered}$$\begin{gathered}F(y)=\ln(\lambda
y)\\\ \text{or
}\frac{\mathrm{f}^{\prime\prime\prime}\mathrm{f}^{\prime}}{(\mathrm{f}^{\prime\prime})^{2}}=1\end{gathered}$$k\to
2i\kappa$$\frac{\mathrm{f}^{\prime\prime\prime}\mathrm{f}^{\prime}}{(\mathrm{f}^{\prime\prime})^{2}}=\frac{k-2}{k-1}$$\begin{gathered}F_{\mathrm{Re}}(y)=y/2\text{
or}\\\
\operatorname{Re}\dot{B}e^{-\frac{1}{2}\operatorname{Re}B}=-\alpha^{-1/2}\end{gathered}$$\begin{gathered}\alpha,u_{0}\to\infty,\\\
\frac{4\alpha}{u_{0}^{2}}=1\end{gathered}$$F(y)=\ln\left(\frac{4c}{2-y}\right)$$\mathrm{Re}\dot{B}=0$$k\to
2$$F(y)=0$$F(y)=0$$\alpha,u_{0}\to\infty,\frac{4\alpha}{u_{0}^{2}}=1$$\operatorname{Re}\dot{B}=0$$k\to
0$$e^{i\gamma}\to\pm 1$$e^{i\gamma}\to\pm
1$$\alpha,u_{0}\to\infty,\frac{4\alpha}{u_{0}^{2}}\to
1$ynnyyynynnnynnyyy$\alpha=\infty$$\alpha\neq\infty$$\operatorname{Re}I_{\mathrm{a}}^{(2)}\neq\text{const}$$\operatorname{Im}I_{\mathrm{a}}^{(2)}\neq\text{const}$yy$\;\alpha=\infty$$\alpha\neq\infty\;$$\operatorname{Re}I_{\mathrm{a}}^{(0)}\neq\text{const}$$J\neq\text{const}$ynny$\kappa\in\mathbb{R}$nny$k\neq
0,1,2,\infty$$k=\infty$$I_{\mathrm{e}}^{(\infty)}\neq\text{const}$$\mathfrak{L}\neq
0$$\mathbf{M}^{(\infty)}\neq
0$yynnn$\kappa=\infty$$\kappa\neq\infty$$\begin{gathered}G_{6b;\lambda}\\\
\lambda\geq 0\end{gathered}$4$\begin{gathered}G_{5^{\circ};F}\\\
\partial_{u}(\mathrm{Re}\dot{B})\neq 0,\\\ F_{\mathrm{Re}}(y)\neq
y^{2}/2\end{gathered}$1$\begin{gathered}G_{6a;\alpha,\kappa}\\\
\alpha>0,\kappa\geq
0\end{gathered}$5$\begin{gathered}G_{5^{\prime};\alpha,F}\\\ \alpha>0,\\\
F(y)\neq-\frac{y}{2\alpha^{1/2}}\end{gathered}$3$\begin{gathered}G_{3b^{\circ};\gamma}\\\
\gamma\in(0,\pi)\end{gathered}$11$\begin{gathered}G_{3a^{\circ};\alpha,\gamma}\\\
\alpha>0,\gamma\in(0,\pi)\end{gathered}$15$\begin{gathered}G_{2c^{\circ};F}\\\
F_{\mathrm{Re}}(y)\neq
y^{2}/2\end{gathered}$6$\begin{gathered}G_{2c^{\prime};\alpha,F}\\\
\alpha>0,\\\ F(y)\neq
0\end{gathered}$8$\begin{gathered}G_{2b^{\prime\prime};\gamma,\lambda,c}\\\
\lambda\in\mathbb{R},c>0,\gamma\in[0,\pi]\end{gathered}$10$G_{1}$$\begin{gathered}G_{2a^{\prime};\kappa,\alpha,\gamma,c}\\\
\kappa\in\mathbb{R},\alpha>0,c>0,\gamma\in[0,\pi]\end{gathered}$14$\begin{gathered}G_{3d;\gamma,\kappa}\\\
\gamma\in[0,\pi],\\\
\kappa\in\mathbb{R}\setminus\\{0\\}\end{gathered}$20$\begin{gathered}G_{2b^{\prime\prime\prime}:k}\\\
k\in\mathbb{C}\setminus(i\mathbb{R}\cup\\{1,2\\})\end{gathered}$19$\begin{gathered}G_{2a^{\circ};\kappa,c,\operatorname{Re}F}\\\
\kappa>0,c>0,2yF^{\prime}(y)\neq
1\end{gathered}$13$\begin{gathered}G_{2b^{\circ};c,\operatorname{Re}F}\\\
\left(\frac{(\ln f_{,\zeta\zeta})_{,u}}{(\ln
f_{,\zeta\zeta})_{,\zeta}}\right)_{,\zeta}\neq 0\\\
(\mathrm{f}^{\prime\prime\prime 2}/\mathrm{f}^{\prime\prime 3})^{\prime}\neq
0\\\ c>0,2yF^{\prime}(y)\neq
1\end{gathered}$9$\begin{gathered}G_{2b^{\prime};\gamma,\operatorname{Re}F}\\\
(\mathrm{f}^{\prime\prime\prime}\mathrm{f}^{\prime}/(\mathrm{f}^{\prime\prime})^{2})^{\prime}\neq
0,\\\ \alpha\in[0,2\pi),\\\ F(y)\neq\ln(\lambda y),\\\
F(y)\neq\ln\left(\frac{4c}{2-y}\right)\end{gathered}$17$\begin{gathered}G_{3c;\lambda}\\\
\lambda>0\end{gathered}$18$\begin{gathered}G_{5^{\prime};\infty,F}\\\ F(y)\neq
0\end{gathered}$2$\begin{gathered}G_{2c^{\prime};\infty,F}\\\ F(y)\neq
0\end{gathered}$7$G_{3a^{\prime};\pm\alpha}$16$G_{3b^{\prime};\pm}$12 vacuum
pp-wave
$G_{5^{\circ};F}$$G_{6a;\alpha,\kappa}$$G_{6b;\lambda}$$G_{3b^{\prime};\pm}$$G_{3a^{\prime};\pm\alpha}$$G_{2c^{\circ};F}$$G_{1}$$\overline{\mathbf{K}}\cdot\mathbf{K}=0$2$I_{\mathrm{c}}=\tfrac{1}{2}$5$\operatorname{Re}I_{\mathrm{a}}^{(2)}=-\alpha^{-1/2}$3$I_{\mathrm{a}}^{(2)}=2i\lambda-\alpha^{-1/2}$4$\operatorname{Re}I_{\mathrm{a}}^{(0)}=-\alpha^{-1/2}$7$J=e^{2i\gamma}$8$\mathbf{C}^{\dagger}=0$1$\begin{pmatrix}\operatorname{Re}I_{\mathrm{b}}^{(2)}\\\
(\operatorname{Im}I_{\mathrm{a}}^{(2)})^{2}\end{pmatrix}=F\left(\operatorname{Re}I_{\mathrm{a}}^{(2)}\right)$$\operatorname{Im}I_{\mathrm{b}}^{(2)}=F\left(\operatorname{Im}I_{\mathrm{a}}^{(2)}\right)$$\begin{pmatrix}\operatorname{Re}I_{\mathrm{b}}^{(0)}\\\
\operatorname{Re}[(I_{\mathrm{a}}^{(0)})^{2}/I_{\mathrm{d}}^{(0)}]\end{pmatrix}=F(\operatorname{Re}I_{\mathrm{a}}^{(0)})$$\operatorname{Im}I_{\mathrm{a}}^{(0)}=F\left(\operatorname{Im}[(I_{\mathrm{a}}^{(0)})^{2}/I_{\mathrm{d}}^{(0)}]\right)$Minkowski$\begin{gathered}2\nabla\mathbf{L}^{(i\kappa)}=\mathbf{L}^{(i\kappa)}\mathbf{L}^{(i\kappa)}\\\
\mathbf{M}^{(i\kappa)}=0\end{gathered}$18$\operatorname{Re}\mathbf{L}^{(\infty)}=0$19$G_{2a^{\circ};\kappa,c,\operatorname{Re}F}$$G_{2b^{\circ};c,\operatorname{Re}F}$$G_{2b^{\prime};\gamma,\operatorname{Re}F}$16$G_{2b^{\prime\prime};\gamma,\lambda,c}$$\begin{gathered}\mathfrak{L}=0\\\
\mathbf{M}^{(\infty)}=0\end{gathered}$14$k(I_{\mathrm{c}}-\frac{1}{2})=I_{\mathrm{c}}$17$I_{\mathrm{e}}^{(\infty)}=\tfrac{k-2}{k-1}$15$G_{3c;\lambda}$$G_{3d;\gamma,\kappa}$$\mathbf{N}=0$6$\begin{gathered}\alpha
e^{i\gamma}=1/I_{\mathrm{d}}^{(i\kappa)}\\\
\mathbf{M}^{(i\kappa)}=0\end{gathered}$13$G_{1}$$G_{2a^{\prime};\alpha,\gamma,\kappa,c}$$c=\frac{|N|}{|\mathbf{K}|}$$\lambda=\frac{1}{2}|\mathbf{K}|$$G_{2b^{\prime\prime\prime}:k}$$e^{2i\gamma}=\frac{(2i\kappa+1)(i\kappa-1)^{2}}{(2i\kappa-1)(i\kappa+1)^{2}}\left|\frac{\mathbf{K}}{2(i\kappa-1)}\right|^{4i\kappa}J$$\begin{gathered}\ln(|\mathbf{K}|^{2})=2\operatorname{Re}F(I_{\mathrm{d}}^{(\infty)})\\\
c=\mathcal{A}_{\infty,\operatorname{Re}F}(I_{\mathrm{d}}^{(\infty)},I_{\mathrm{e}}^{(\infty)})\end{gathered}$$\begin{gathered}\ln(|\mathbf{K}|^{2})=2\operatorname{Re}F(I_{\mathrm{e}}^{(\infty)})\\\
\cos\gamma=\mathcal{G}_{\operatorname{Re}F}(I_{\mathrm{e}}^{(\infty)})\end{gathered}$$\begin{gathered}\ln(|\mathbf{K}|^{2})=2\operatorname{Re}F(-I_{\mathrm{d}}^{(i\kappa)})\\\
c=\mathcal{A}_{\kappa,\operatorname{Re}F}(I_{\mathrm{d}}^{(i\kappa)},I_{\mathrm{e}}^{(i\kappa)})\end{gathered}$$k=2i\kappa$$\kappa=2\lambda\alpha^{1/2}$$\mathfrak{L}=0$11$\begin{gathered}\left(\nabla
N-\frac{N}{2}\overline{\mathbf{K}}\right)^{\otimes
2}=2\lambda^{2}e^{-i\gamma\operatorname{sgn}\lambda}N^{2}\mathbf{D}\\\
\mathbf{D}=e^{2i\gamma}\overline{\mathbf{D}}\text{ or
}{-2\mathbf{D}}={\pm\bm{\ell}\bm{\ell}}\end{gathered}$12$e^{2i\gamma}=1$9$e^{2i\gamma}=1$10$G_{3a^{\circ};\alpha,\gamma}$$G_{3b^{\circ};\gamma}$$G_{5^{\prime};\infty,F}$$G_{5^{\prime};\alpha,F}$$\alpha=\infty$$\alpha=\infty$$G_{2c^{\prime};\infty,F}$$G_{2c^{\prime};\alpha,F}$$\pm\alpha=1/I_{\mathrm{d}}^{(0)}$$-2\mathbf{D}={\pm\bm{\ell}\bm{\ell}}$

Figure 2: Flowchart for the IDEAL characterization of vacuum pp-waves of high
symmetry, accompanied by Table LABEL:tab:pp-wave-ideal and Theorem 8.

######  Theorem 8 (IDEAL characterization of pp-waves).

Consider a point $x\in M$, where $(M,g)$ is a Lorentzian metric. Then, locally
at $x$, $(M,g)$ belongs to a given one of the highly symmetric regular pp-wave
isometry types from Theorem 7 if and only if, on a neighborhood of $x$, the
corresponding tensor identities from Table LABEL:tab:pp-wave-ideal hold. These
identities can isolate every individual isometry class with one small
exception (they cannot distinguish between $G_{3d;0,\kappa}$ and
$G_{3d;\pi,\kappa}$). The decision process for recognizing and distinguishing
these isometry classes is given by the flowchart in Figure 2.

###### Proof.

We will proceed by going through the flowchart in Figure 2 step by step, in
the order of the numbers indicated next to each node. In the corresponding
numbered part of the proof, for decision nodes, we decode the significance of
the given tensorial condition by mapping it to the corresponding branch point
in the proof of Theorem 7 (we denote such references by IC.#), and also argue
why the arrows leaving the node exhaust all the possibilities. In some cases,
the latter arguments will rely on the fact that some possibilities may have
already been eliminated by previous branches of the flowchart. For each leaf
node, we decode the tensor identities that extract the invariant parameters of
the isometry class, which are often attached to the branch by a dotted line,
again by mapping them to the relevant point in the proof of Theorem 7. One can
jump directly into any numbered point in the proof below, but in general to
get the full context one may need to retrace to its origin the corresponding
branch in Figure 2.

The entry point is the set of tensorial curvature conditions from Proposition
6 that identify the vacuum pp-wave class. For the rest of the diagram, we will
assume the corresponding line element (8) and will rely on the conditional
curvature invariants defined in Section 3.1. Some of the scalar invariants
defined there required a choice of branch of a square or quartic root, which
was indicated in the definition. We point out when these choices affect the
possible invariant relations that can be encountered in the proof, for
instance when equating an invariant to a constant. We do not need to be so
careful for functional relations between invariants, as different branch
choices can at worst be absorbed into the free functional parameters of the
relation itself.

  1. 1.

Since we are already in vacuum, $R_{ab}=0$, the vanishing
$\mathbf{C}^{\dagger}=0$ of the self-dual Weyl tensor (12) implies that the
entire Riemann tensor $R_{abcd}=0$, which lands us in the trivial case of
Minkowski space. It remains to examine the possibilities where
$\mathbf{C}^{\dagger}\neq 0$, which by the evaluation (16) is equivalent to
$f_{,\zeta\zeta}\neq 0$.

  2. 2.

The condition $\overline{\mathbf{K}}\cdot\mathbf{K}=0$ by (22) is equivalent
to $(\ln f_{,\zeta\zeta})_{,\zeta}=f_{,\zeta\zeta\zeta}/f_{,\zeta\zeta}=0$. As
we saw in IC.1, this condition lands us in the $G_{5}$ subclass,
$f(\zeta,u)=A(u)\zeta^{2}$. It remains to examine both possibilities, that the
equality holds or not.

  3. 3.

The condition $\operatorname{Re}I_{\mathrm{a}}^{(2)}=-\alpha^{-1/2}$, for some
$\alpha>0$666Recall that according to the choice of branch in (24)
$\operatorname{Re}I_{\mathrm{a}}^{(2)}\leq 0$, so $\alpha$ cannot be
negative., results in the three possibilities,
$\operatorname{Re}I_{\mathrm{a}}^{(2)}\neq\text{const}$, $\alpha=\infty$ and
$\alpha\neq\infty$. In the first case, the relation

$\begin{pmatrix}\operatorname{Re}I_{\mathrm{b}}^{(2)}\\\
(\operatorname{Im}I_{\mathrm{a}}^{(2)})^{2}\end{pmatrix}=F\left(\operatorname{Re}I_{\mathrm{a}}^{(2)}\right).$
(130)

combined with the evaluation (45) coincides with the signature equation (58)
and hence by the arguments from IC.1 it isolates the isometry class
$G_{5^{\circ};F}$.

It remains to examine the possibilities where
$\operatorname{Re}I_{\mathrm{a}}^{(2)}$ is constant, including both
$\alpha=\infty$ and $\alpha\neq\infty$.

  4. 4.

The condition $\operatorname{Im}I_{\mathrm{a}}^{(2)}=2\lambda$ is either
satisfied by some constant $\lambda\geq 0$777Recall that according to the
choice of branch in (24) $\operatorname{Im}I_{\mathrm{a}}^{(2)}\geq 0$, so
$\lambda$ cannot be negative., or
$\operatorname{Im}I_{\mathrm{a}}^{(2)}\neq\text{const}$. In the latter case,
the relation

$\operatorname{Im}I_{\mathrm{b}}^{(2)}=F\left(\operatorname{Im}I_{\mathrm{a}}^{(2)}\right).$
(131)

combined with the evaluation (45) coincides with the signature equation (61)
and hence by the arguments from IC.2 or IC.3 it isolates the isometry class
$G_{5^{\prime};\infty,F}$ or $G_{5^{\prime};\alpha,F}$, depending on the value
of $\alpha$.

It remains to examine the possibilities
$I_{\mathrm{a}}^{(2)}=2i\lambda-\alpha^{-1/2}$. Taking the evaluation (45)
into account, the case $\alpha=\infty$ lands us in the $G_{6b;\lambda}$ class
(cf. IC.4), while the case $\alpha\neq\infty$ lands us in the
$G_{6a;\alpha,\kappa}$ subclass with $\kappa=2\lambda\alpha^{1/2}$ (cf. IC.5).

  5. 5.

The condition $I_{\mathrm{c}}=\frac{1}{2}$ combined with the evaluation in
(28) coincides with (69). Hence, by the arguments from IC.6, it lands us in
the $f(\zeta,u)=A(u)\ln\zeta+A_{1}(u)\zeta+A_{0}(u)$ subclass. It remains to
impose further conditions to characterize the invariant content of the
$u$-dependent coefficients, or to examine the possibility that
$I_{\mathrm{c}}\neq\frac{1}{2}$.

  6. 6.

The condition $\mathbf{N}=0$ combined with the evaluation in (47) coincides
with setting $A_{1}(u)=0$ as in (70), which by the arguments of IC.6 lands us
in the $G_{2c}$ subclass, which can be further subclassified based on the
invariant content in the $A(u)$ coefficient. It remains to examine the
possibility that $\mathbf{N}\neq 0$.

  7. 7.

The condition $\operatorname{Re}I_{\mathrm{a}}^{(0)}=-\alpha^{-1/2}$, for some
$\alpha>0$888Recall that according to the choice of branch in (30)
$\operatorname{Re}I_{\mathrm{a}}^{(0)}\leq 0$, so $\alpha$ cannot be
negative., results in the three possibilities
$\operatorname{Re}I_{\mathrm{a}}^{(0)}\neq\text{const}$, $\alpha=\infty$ and
$\alpha\neq\infty$. In the first case, the relation

$\begin{pmatrix}\operatorname{Re}I_{\mathrm{b}}^{(0)}\\\
\operatorname{Re}\left[(I_{\mathrm{a}}^{(0)})^{2}/I_{\mathrm{d}}^{(0)}\right]\end{pmatrix}=F\left(\operatorname{Re}I_{\mathrm{a}}^{(0)}\right)$
(132)

combined with the evaluation (46) coincides with the signature equation (71)
and hence by the arguments from IC.6 it isolates the isometry class
$G_{2c^{\circ};F}$.

  8. 8.

By virtue of its definition (27), the invariant $|J|=1$. The condition
$J=e^{2i\gamma}$ is either satisfied for some unique constant
$\gamma\in[0,\pi)$ or $J\neq\text{const}$. In the latter case, the relation

$\operatorname{Im}I_{\mathrm{a}}^{(0)}=F\left(\operatorname{Im}\left[(I_{\mathrm{a}}^{(0)})^{2}/I_{\mathrm{d}}^{(0)}\right]\right).$
(133)

combined with the evaluation (46) coincides with the signature equation (74)
and hence by the arguments from IC.7 and IC.8 it isolates the isometry class
$G_{2c^{\prime};\infty,F}$ or $G_{2c^{\prime};\alpha,F}$, depending on the
value of $\alpha$.

It remains to examine the possibility that $J$ is constant, which by the
evaluation in (53) restricts to the subclass
$f(\zeta,u)=e^{\operatorname{Re}B}e^{i\gamma}\ln\zeta$, under the two
different branches $\alpha=\infty$ ($\operatorname{Re}B=0$) and
$\alpha\neq\infty$ ($\operatorname{Re}B=\ln 4\alpha-2\ln u$).

  9. 9.

Under the $\alpha=\infty$ condition, we are in the class
$f(\zeta,u)=e^{i\gamma}\ln\zeta$. The condition $e^{2i\gamma}=1$ distinguishes
between the possibilities $\gamma\in\\{0,\pi\\}$ (when satisfied) and
$\gamma\in(0,\pi)$ (when not). The latter case by the arguments in IC.11
isolates the $G_{3b^{\circ};\gamma}$ isometry class. In the remaining
possibility, $e^{i\gamma}=\pm 1$, we can evaluate

$-2\mathbf{D}=\pm\bm{\ell}\bm{\ell}.$ (134)

We have found no further tensor or scalar invariants that can directly
distinguish between the two signs, but they do not give isometric geometries.
However, it seems that an invariant way of distinguishing between the two
signs. Namely, in the $+$ case $-2\mathbf{D}$ is the square a real null vector
$\bm{\ell}$, while in $-$ case $-2\mathbf{D}$ is the negative square of a real
null vector $\bm{\ell}$. Recall that the aligned null vector $\bm{\ell}$ is
not actually known at this stage of our classification, so its existence is a
separate condition. By the arguments from IC.12 These conditions respectively
isolate the $G_{3b^{\prime};\pm}$ isometry classes.

  10. 10.

Under the $\alpha\neq\infty$ condition, we are in the class
$f(\zeta,u)=\frac{4\alpha}{u^{2}}e^{i\gamma}\ln\zeta$. The condition
$e^{2i\gamma}=1$ distinguishes between the possibilities
$\gamma\in\\{0,\pi\\}$ (when satisfied) and $\gamma\in(0,\pi)$ (when not). The
latter case by the arguments in IC.15 isolates the
$G_{3a^{\circ};\alpha,\gamma}$ isometry class. In the remaining possibility,
$e^{i\gamma}=\pm 1$, we recall the evaluation (52),

$\frac{1}{I_{\mathrm{d}}^{(0)}}=\pm\alpha,$ (135)

whose value captures both the $\alpha>0$ and $\pm$ sign invariants. By the
arguments from IC.16 this condition isolates the $G_{3a^{\prime};\pm\alpha}$
isometry class.

  11. 11.

There are only two possibilities to examine, either $\mathfrak{L}=0$ or
$\mathfrak{L}\neq 0$.

  12. 12.

We are in the formfactor class
$f(\zeta,u)=A(u)\ln\zeta+A_{1}(u)\zeta+A_{0}(u)$. Taking into account the
evaluation in (29), the condition $\mathfrak{L}=0$ is equivalent to

$-\frac{1}{2}\frac{\dot{A}}{A}=0$ (136)

or $A(u)=a^{2}e^{i\gamma}$ constant. As we already know, the coefficient
$A_{0}(u)$ is pure gauge, while the coefficient of $\zeta$ must take the form
$A_{1}(u)=A_{1,0}e^{i\lambda u}$ to correspond to one of the highly symmetric
isometry classes. Consider the covariant condition

$\left(\nabla N-\frac{N}{2}\overline{\mathbf{K}}\right)^{\otimes 2}=2\lambda
e^{-i\gamma\operatorname{sgn}\lambda}N^{2}\mathbf{D}^{\dagger}.$ (137)

with external parameters $\lambda\in\mathbb{R}$ and $\gamma\in[0,\pi]$; when
the the condition is satisfied with $\lambda\neq 0$, it uniquely fixes the
values of $\lambda$ and $\gamma$, in the given ranges, for which it holds.
Within our formfactor class, both sides of the equality are proportional to
$\bm{\ell}\bm{\ell}$, while equating the coefficients gives the condition

$\dot{A}_{1}^{2}+\lambda^{2}A_{1}^{2}=0.$ (138)

Either it does not hold and we end up in the generic $G_{1}$ class, or it does
and it is solved by $A_{1}(u)=A_{1,0}e^{\pm i\lambda u+i\gamma}$, where
$A_{1,0}\in\mathbb{C}$ is an integration constant. We can use the
$f\mapsto\bar{f}$, $u\mapsto-u$ and $\zeta\mapsto e^{i\beta}\zeta$ gauge
freedoms to restrict to $\gamma\in[0,\pi]$, $A_{1,0}=c\geq 0$ and the
$e^{+i\lambda u}$ possibility.

When $\lambda=0$, the formula in (137) no longer depends on $\gamma$ and hence
cannot determine its value. In that case, as long as $A(u)=a^{2}e^{i\gamma}$,
the following covariant relation must hold

$\mathbf{D}=e^{2i\gamma}\overline{\mathbf{D}},$ (139)

which uniquely determines the value of $\gamma\in[0,\pi]$ as long as
$e^{2i\gamma}\neq 1$. In the latter case, using the evaluation in (47), we can
differentiate between the values

$-2\mathbf{D}=+(a\bm{\ell})^{\otimes
2}\quad\text{and}\quad{-(a\bm{\ell})^{\otimes 2}}$ (140)

by whether $-2\mathbf{D}$ is the positive ($\gamma=0$) or negative
($\gamma=\pi$) square of a real null vector ($a\bm{\ell}$), respectively.
Finally, again relying on the evaluations in (47), we can use the following
formula to extract the invariant parameter

$c=|A_{1,0}|=\frac{|N|}{|\mathbf{K}|}.$ (141)

In this branch of the classification, we are guaranteed to have $c>0$, because
the $c=0$ case was already handled by the branch containing the $G_{3b}$