# Singularities and clusters.

V.V.Fock  Université de Strasbourg  Dedicated to Norbert A’Campo on occasion
of his 80-th birthday.

###### Abstract.

The aim of this note is to describe a geometric relation between simple plane
curve singularities classified by simply laced Cartan matrices and cluster
varieties of finite type also classified by the simply laced Cartan matrices.
We construct certain varieties of configurations of flags out of Dynkin
diagrams and out of singularities and show that they coincide if the Dynkin
diagram corresponds to the singularity.

###### 2000 Mathematics Subject Classification:

14H20, 13F60, 34M40

The correspondence between singularities and cluster varieties was first
observed by A.Fomin, P.Pilyavsky, D.Thurston, and E.Shustin in their
remarkable paper [2]. Starting from certain real forms of deformations of the
singularities introduced by N.A’Campo [1] and S.Gussein-Zade [3] they produced
a cluster variety and showed that different resolutions of the same
singularity give the same cluster variety. In this note we try to make this
correspondence more geometrical and less mysterious. In particular we show
that there is a map from the base of a versal deformation of the singularity
to the corresponding cluster variety. For this purpose we generalize a
construction by R.Nevanlinna [4] brought to our attention by B.Shapiro [5].
R.Nevanlinna studied the map from the space of the differential operators of
the form $D=\partial^{2}/\partial z^{2}+P(x)$, where $P$ is a polynomial, to
the collections of points in a projective line $P^{1}$. These points just
correspond to solutions of the equation $D\psi=0$ vanishing at infinity along
different rays on the complex plane and viewed as lines in the two-dimensional
space of global solutions. On the other hand, the symbol of this operator
$p^{2}+P(x)$ is just an equation for the versal deformation of a plane curve
singularity of type $A$.

Our construction is in a sense a generalization of this one for singularities
of other types. Namely, a versal family of a planar singularity can be
considered as a family of Lagrangian curves in the plane
$(\mathbb{C}^{2},dp\,dx)$. Observe that this family can be locally
parameterized by the cohomology class of the form $pdx$. On the other hand the
family of equations defining these curves can be transformed into a family of
differential operators having the equations as their symbol. The space of
Stokes data at infinity of the differential operator is the corresponding
cluster variety.

The correspondence between symbols and operators is of course not canonical.
First of all, we can change change a representation of the symplectic plane
$(\mathbb{C}^{2},dp\,dx)$ as a cotangent bundle to a complex line $\mathbb{C}$
in different ways. All these ways give different Stokes data, but, as one can
verify in the examples, they are equivalent as cluster varieties. Sometimes
the birational equivalence of the corresponding configuration space is not so
trivial and can be considered as a generalization of the Gale duality.

On the other hand the correspondence between the space of differential
operators and its symbol is not canonical either. We conjecture that this map
becomes canonical in the tropical limit and the cluster coordinates in this
limit are periods of the Lagrangian curves given by the versal family of the
singular one.

I am strongly indebted to F.Chapoton for reading paper and making very crucial
remarks.

##  1\. Recall: Configurations corresponding to planar bipartite graphs

Let $a$ be a collection of points of a projective space $P(V)$ vector space
$V$. Denote by $\langle a\rangle$ a projective subspace generated by the
points from $a$. Recall that a collection $a$ of $k$ points is called free if
$\dim\,\langle a\rangle=k-1$ and a circuit if $\dim\,\langle a\rangle=k-2$.
For example, two points form a circuit if they coincide, three points form a
circuit if they are colinear four points form a circuit if they are coplanar
etc.

Let $\Gamma$ be a bipartite graph with the set of white vertices $W$, black
vertices $B$ and edges $E$. For simplicity we assume here that any two
vertices are connected by no more than one edge. We say that an association of
a point $p_{w}$ in a projective space to every white vertex $w\in W$
corresponds to $\Gamma$ if for every black vertex $b\in B$ the points
corresponding to its white neighbors form a circuit. We also require that the
collection of points does not belong to a proper projective subspace.

Denote by $\mathcal{X}_{\Gamma}$ the set of configurations corresponding to
$\Gamma$ considered up to the action of the projective group $PGL(V)$. We call
the dimension of the configuration the dimension of the projective space
$P(V)$. We say that the graph $\Gamma$ is minimal if removal of any black
vertex increases the dimension of the corresponding configuration.

A discrete connection on a graph $\Gamma$ is an association of one dimensional
vector spaces to vertices and association to edges of isomorphisms of the
spaces corresponding to its endpoints. Given a basis in each of the one-
dimensional subspaces, the discrete connection becomes an association of
nonzero numbers to edges. These numbers can be organized into a matrix
$M_{b}^{w}$ with columns and rows enumerated by the black and white vertices,
respectively and with zeroes for pairs of vertices not connected by an edge.
Changing the bases amounts to the multiplication of this matrix by invertible
diagonal matrices from the left and from the right. Given a closed path of the
graph, the monodromy of the connection around this graph is a composition of
the maps corresponding to the edges. In terms of the connection matrix if the
path passes consecutively through the vertices
$b_{0},w_{0},\ldots,b_{k},w_{k},b_{0}$ then the monodromy is given by a
Laurent monomial in the matrix entries
$M_{b_{1}}^{w_{1}}(M_{b_{2}}^{w_{1}})^{-1}\cdots(M_{b_{0}}^{w_{k}})^{-1}$. If
the graph is planar a graph connection is uniquely determined by the
monodromies around its faces.

The set of graph connections on a graph $\Gamma$ is in a bijection with the
set of configurations corresponding to $\Gamma$. Indeed, given the matrix
$M_{b}^{w}$ representing the connection we can consider it as a map
$M:\mathbb{C}^{B}\to\mathbb{C}^{W}$. The image of the standard basis of
$\mathbb{C}^{W}$ in the projectivized cokernel $P(\mathbb{C}^{W}/\textrm{Im
}M)$ form the desired configuration. Conversely, given a configuration of
points in a projective space $P(V)$, for every white vertex $w$ choose a
vector $\tilde{p}_{w}\in V$ representing $p_{w}$. Given a black vertex $b$ the
chosen vectors corresponding to its white neighbors satisfy a nontrivial
linear relation $\sum M_{b}^{w}\tilde{p}_{w}=0$. The matrix $M_{b}^{w}$ forms
the desired graph connection.

Call two configurations equivalent if one can construct one from another by
adding and removing points which can be constructed from the remaining points.
The graphs corresponding to equivalent configurations are also called
equivalent.

There are four equivalences generating many (and conjecturally all) others.
Namely

  1. 1.

$\vdots$$\vdots$$\longleftrightarrow$$\vdots$$\vdots$

Two-valent black vertex means that two points corresponding to its white
neighbors coincide, and of course one of them can be removed.

  2. $1^{\prime}$.

$\vdots$$\vdots$$\longleftrightarrow$$\vdots$$\vdots$

Two-valent white vertex means that the circuits corresponding to its black
neighbors have one common point, and obviously their union (with one of the
points de-doubled) is also a circuit.

  3. 2.

$\vdots$$\vdots$$\cdots$$\cdots$$d$$b$$c$$a$$X$$Y$$\longleftrightarrow$$\vdots$$\vdots$$\cdots$$\cdots$$d^{\prime}$$b^{\prime}$$c^{\prime}$$a^{\prime}$$Y^{\prime}$$X^{\prime}$

Let $a$,$b$,$c$,$d$ be the circuits corresponding to black vertices and $X$
and $Y$ points corresponding to the white ones. Let $Y^{\prime}=\langle
b\setminus\\{X,Y\\}\rangle\cap(XY)$. The intersection is a point since
obviously the line $(XY)$ belongs to the subspace $\langle b\rangle$ and the
subspace $\langle b\setminus\\{X,Y\\}\rangle$ is of codimension 1 in it.
Similarly let $X^{\prime}=\langle d\setminus\\{X,Y\\}\rangle\cap(XY)$. We can
now define $a^{\prime}=a\cup\\{X^{\prime},Y^{\prime}\\}\setminus\\{X\\}$,
$b^{\prime}=b\cup\\{Y^{\prime}\\}\setminus\\{X,Y\\}$,
$c^{\prime}=c\cup\\{X^{\prime},Y^{\prime}\\}\setminus\\{Y\\}$,
$d^{\prime}=d\cup\\{X^{\prime}\\}\setminus\\{X,Y\\}$.

  4. $2^{\prime}$.

$\vdots$$\vdots$$\cdots$$\cdots$$\longleftrightarrow$$\vdots$$\vdots$$\cdots$$\cdots$

This fragment of the graph corresponds to four collinear points. Any triple of
such point is a circuit. Any two of such circuits define such configurations
and can be exchanged by the remaining two preserving planarity of the graph.

##  2\. From Dynkin diagram to bipartite graph.

For any planar graph (in our examples they will be Dynkin diagrams) one can
associate a planar bipartite graph dual to the original graph (See example
shown on Fig. 1a, 2a and 3a.) Namely, surround every vertex of valency $k>2$
by a $2k$-gon in a way that its every second side crosses the respective edge
of the original graph. For a 1-valent vertex surround it with a square. Then
identify the sides of the polygons crossing the same edges of the original
graph. Finally color vertices in black and white and add one edge with a white
vertex to the every black 2-valent vertex drawn around vertices of valency 1.

##  3\. Stokes data.

Let $D(z,\frac{\partial}{\partial z})$ be a differential operator with
polynomial coefficients of order $n$. Let $D(x,p)$ be its symbol and $\Delta$
be its Newton polygon, namely the convex hull of the points with coordinates
$(a,b)\in\mathbb{Z}^{2}$ for each monomial term $x^{a}p^{b}$ of the polynomial
$D(x,p)$.

Consider the algebraic curve $C=\\{(x,p)\in(\mathbb{C}^{*})^{2}|D(x,p)=0\\}$.
This curve has the genus equal to the number of integer points strictly inside
the polygon $\Delta$ and can be compactified by adding points, called
compactification points, corresponding to sides of the polygon. By a side we
mean a segment of the boundary of $\Delta$ between two adjacent points of
$\mathbb{Z}^{2}$. If we orient the boundary of the polygon counterclockwise
each side $s$ correspond to indivisible vector $(a,b)\in\mathbb{Z}^{2}$. The
sum of such vectors obviously vanishes. At the compactification points
corresponding to $s$ the functions $x$ and $p$ have a zero of order $-b_{s}$
and $a_{s}$, respectively. Therefore in the vicinity of such point we have
$x^{a_{s}}p^{b_{s}}=O(1)$ and hence $p\sim A_{s}x^{-a_{s}/b_{s}}$ for some
constants $A_{s}$.

One should however realize that the correspondence between sides and
compactification points is not canonical but is defined up to permutations of
sides corresponding to equal vectors.

The same polygon defines a curve in $\mathbb{C}^{2}$. The sides of the polygon
with $a>0$ and $b\leq 0$ correspond to compactifcation points with both
coordinates tending to a finite constant and thus belonging to the curve in
$\mathbb{C}^{2}$. The remaining compactification points are called points at
infinity.

Consider now the equation $D(x,p)=0$ as a family of equations for the
indeterminate $p$ depending on $x$ as a parameter. Our aim is to determine the
asymptotic behavior of its roots when $x\to\infty$. This behavior of $x$
corresponds to the sides of the polygon with $b_{s}>0$ (and thus going upward
on a picture where the $p$-axis points up). Each side corresponds to $b_{s}$
roots with asymptotic $p\sim A_{s}x^{-a_{s}/b_{s}}$. The total number of roots
is thus equal to the height of the polygon, which is just the degree of the
polynomial $D(x,p)$ with respect to $p$.

Now proceed to the study of the asymptotic behaviour of the differential
equation $D(x,\frac{\partial}{\partial x})\psi=0$. In a simply connected
domain where the coefficient of the $n$-th derivative vanishes nowhere, it has
an $n$-dimensional space of solutions. However in a sufficiently small
vicinity of infinity the differential equation defines an $n$-dimensional
local system. Indeed, fix a number $R$ such that the coefficient of the
highest derivative of $D$ does not vanish for $|x|>R$, a number
$\varepsilon\in]0,\pi/2[$ and an angle $\alpha\in\mathbb{R}/2\pi\mathbb{Z}$.
The domain $\\{x\in\mathbb{C}|\ |\arg x-\alpha|<\varepsilon,\ |x|>R\\}$
satisfies the above conditions. Define $V_{\alpha}$ as the space of solutions
of the differential equation in this domain. The space $V_{\alpha}$ does
depend neither of $\varepsilon$ nor of $R$ and does not depend on $\alpha$
locally. Namely if $|\alpha_{1}-\alpha_{2}|<\varepsilon$ the corresponding
domains intersect, their intersection is simply connected and therefore the
spaces $V_{\alpha_{1}}$ and $V_{\alpha_{2}}$ can be identified. Hence the
family $V_{\alpha}$ form an $n$-dimensional local system over the circle
$\mathbb{R}/2\pi\mathbb{Z}$.

Every solution of the equation has asymptotic behavior $\psi(x)\sim
e^{\int{p(x)dx}}$ for a root $p(x)$ and therefore for generic angle $\alpha$
the space $V_{\alpha}$ is filtered by the rates of growth of the functions
$\Re e\int^{re^{i\alpha}}\\!\\!p(x)dx$ with $r\to+\infty$.

Consider a sufficiently large $R$ and mark the points
$A_{s}(Re^{i\alpha})^{(b_{s}-a_{s})/b_{s}}$ on the complex plane. (For a given
$s$ there are $b_{s}$ such points.) These points are ordered according to
their projection on the real axis and the interval between the $i$-th and the
$i+1$-st points correspond to the subspace of $V_{\alpha}$ of dimension $i$.
As $\alpha$ runs around the circle, the points rotate around the origin with
the angular speed $(b_{s}-a_{s})/b_{s}$ and the order of their projections
changes. We call the collection of points corresponding to generic $\alpha$
with their angular speeds indicated for each point the growth diagram. When
the $i$-th and the $i+1$-th projections pass through each other the $i$-th
dimensional subspace of $V_{\alpha}$ changes.

The local system and the sequence of flags in its fibers constitute the Stokes
data of the differential operator $D$ at infinity. If the coefficient of the
highest derivative in the operator $D$ is constant, the local system is
trivial and the Stokes data at infinity amounts to the collection of flags in
a fixed vector space.

For nontrivial local systems, taking the universal cover of the circle we can
consider a finite sequence of flags in a local system as a infinite quasi-
periodic one in a fixed vector space. Recall that the sequence is called
quasi-periodic if its shift by a period coincides with the action of an
element of $GL(n)$.

Recall that pairs of complete flags in an $n$-dimensional space up to diagonal
action of $GL(n)$ are in bijection with the permutation group
$\mathfrak{S}_{n}$. The standard generators $s_{i}$ of this group correspond
to the pairs of flags different only in the subspaces of dimension $i$.

Therefore a sequence of flags in a local system on a circle such that adjacent
flags differ in one subspace can be encoded by an infinite periodic word in
the same generators $s_{1},\ldots,s_{n-1}$. We will denote such words as
$[w]$, where $w$ is a period of the infinite word. For example a word
$[(s_{2}s_{1})^{m}]$ corresponds to a 3-dimensional local system with $2m$
flags such that the subspaces of dimension $1$ and $2$ change alternatively.
Such sequence is equivalent to the quasiperiodic sequence of 1-dimensional
subspaces since the 2-dimensional subspaces can be restored from the
1-dimensional ones. In the projective space this sequence corresponds to a
quasi-periodic set of points with the period of length $m$.

Observe that if two words are related by braid relations they correspond to
equivalent sequences of flags.

The word $[(s_{2}s_{1}^{2})^{m}]$ corresponds to a sequence of flags where one
dimensional subspace changes twice after each change of the 2-dimensional
subspace. In the projective space it corresponds to a quasi-periodic broken
line with a marked point on each side.

As another example consider a growth diagram given by a regular $n$-gon
rotating around its center with angular velocity $m/n$. We will show that it
corresponds to an $m$-periodic sequence of points in $P^{n-1}$. Indeed, in
this case all even subspaces change at once and then all odd subspaces change
at once. The corresponding word is therefore
$w=[(w_{\textrm{odd}}w_{\textrm{even}})^{m}]$ where
$w_{\textrm{odd}}=s_{1}s_{3}\cdots$ and $w_{\textrm{even}}=s_{2}s_{4}\cdots$
are the products of the odd and even generators, respectively. We claim that
such configuration is equivalent just to the sequence of 1-dimensional
subspaces with quasi-period $m$. Indeed, given a sequence of flags
corresponding to the word $w$ one can construct the sequence of 1-dimensional
subspaces just discarding all subspaces of higher dimension. On the other hand
given a sequence of 1-dimensional subspaces
$\\{\tilde{p}_{i}|i\in\mathbb{Z}/n\mathbb{Z}\\}$, one can construct a sequence
of $2m$ flags

$F_{2i+1}=\\{\tilde{p}_{i}\subset\tilde{p}_{i}+\tilde{p}_{i+1}\subset\tilde{p}_{i-1}+\tilde{p}_{i}+\tilde{p}_{i+1}\subset\tilde{p}_{i-1}+\tilde{p}_{i}+\tilde{p}_{i+1}+\tilde{p}_{i+2}\subset\cdots\\}$

and

$F_{2i}=\\{\tilde{p}_{i}\subset\tilde{p}_{i-1}+\tilde{p}_{i}\subset\tilde{p}_{i-1}+\tilde{p}_{i}+\tilde{p}_{i+1}\subset\tilde{p}_{i-2}+\tilde{p}_{i-1}+\tilde{p}_{i}+\tilde{p}_{i+1}\subset\cdots\\}$

and observe that $F_{2i}$ differ from $F_{2i+1}$ in odd dimensional terms and
from $F_{2i-1}$ in even dimensional terms.

##  4\. Example: $A_{n}$.

The Dynkin diagram of type $A_{n}$ is just a chain of $n$ vertices and the
corresponding bipartite graph shown on Fig.1a consists of $n+3$ white and
$n+1$ black vertices.

a

b

$(n+3)/2$$(n+3)/2$c

Figure 1.

It corresponds just to $n+3$ collinear points.

The versal deformation of a singularity of type $A_{n}$ is represented by the
polynomial

$D(x,p)=p^{2}+x^{n+1}+a_{1}+a_{2}x+\cdots+a_{n}x^{n-1}.$

The corresponding Newton polygon is a right triangle with sides $2$ and $n$
(shown on Fig.1b for $n=4$) with one or two sides for $n$ odd or even,
respectively, going upwards. It corresponds to a curve of genus
$\lfloor(n-1)/2\rfloor$ with one or two points at infinity, respectively with
homology of rank $n$. The growth diagram (shown on Fig.1c) consists of two
points, which are opposite if $n$ is even, rotating with the angular speed
$(n+3)/2$. It corresponds to the word $[s_{1}^{n+3}]$ and thus the
configurations of $n+3$ points in $P^{1}$.

Interchanging $p$ and $x$ we get a differential equation of order $n+1$, the
growth diagram consists of points forming a regular $n+1$-gon rotating about
its center with angular velocity $(n+3)/(n+1)$ which corresponds to the cyclic
word $[(w_{\textrm{odd}}w_{\textrm{even}})^{n+3}]$ and thus to $n+3$ points in
$P^{n}$. These two configuration space are known to be isomorphic since one
can trace a unique rational normal curve through this point thus obtaining
$(n+3)$ points in $P^{1}$.

##  5\. Example: $D_{4}$.

Consider the Dynkin diagram of the type $D_{4}$ and construct a bipartite
graph shown on Fig. 2a. It corresponds to three groups of collinear points
$C,1,2,A$, $A,3,4,B$ and $B,5,6,A$ and the dimension of the configuration is
2. The configuration is obviously equivalent to the configuration of 6 points
$1,2,3,4,5,6\in P^{2}$ as shown on Fig.2b. This configuration of points
corresponds to a cyclic word $[(s_{2}s_{1}^{3})^{3}]$

$6$$2$$4$$B$$A$$C$$1$$3$$5$a$2$$1$$4$$3$$6$$5$$A$$B$$C$b

c

22d

Figure 2.

The versal deformation of the singularity $D_{4}$ reads as

$D(p,x)=p^{3}+x^{2}p+a_{1}+a_{2}p+a_{3}p^{2}+a_{4}x$

with the Newton polygon is a quadrilateral shown on the Fig.2c. There are
three sides of the polygon directed upward: $(-1,1)$ with multiplicity $2$ and
$(1,1)$ with multiplicity $1$. It corresponds to a curve of genus 1 with three
points at infinity, and with homology of rank 4. The growth diagram consist of
two points rotating with angular speed 2 and one point closer to the center
which does not move. It gives a word $[(s_{2}s_{1}^{2})^{4}]$. Therefore the
space Stokes data can be considered as the configuration space of
quadrilaterals with marked points on each side. Such configurations are
equivalent to the configurations of six points — four points on the sides and
two opposite vertices of the quadrilateral. On the other hand one can deduce
this equivalence from the equality of the words $[(s_{2}s_{1}^{2})^{4}]$ and
$[(s_{2}s_{1}^{3})^{3}]$ in the braid group.

Consider another form of the same singularity just with the variables $p$ and
$x$ interchanged.

$D(p,x)=x^{3}+xp^{2}+a_{1}+a_{2}x+a_{3}x^{2}+a_{4}p$

The Newton polygon is just the reflection of the original one, but the
corresponding differential equation is of order $2$ and the local system is
nontrivial since the coefficient at the highest derivative vanish at the
origin. The growth diagram consists of two points rotating with the angular
speed $2$ and thus the Stokes data amounts to the configuration of four lines
in a two-dimensional local system on a circle.

Remarkably these two configuration space turns out to be birationally
isomorphic. The only isomorphism I know is given by describing both as cluster
varieties, and I don’t know any geometric way to describe it.

##  6\. Example: $E_{8}$.

Consider the Dynkin diagram of the type $E_{8}$ and construct a bipartite
graph shown on Fig. 3a. On the same picture we show the corresponding
bipartite graph. This diagram corresponds to configurations of 13 points
corresponding to white vertices with black vertices corresponding to collinear
triples of points. It implies that there are three groups of collinear points
$C,1,2,3,4,5,A$, $A,6,7,8,B$ and $B,9,10,C$. Such configuration can be
realized in two dimensional projective space $P^{2}$ as a triangle with 2,3
and 5 points on its sides, respectively as shown on Fig. 3b. Observe that this
configuration space is birationally isomorphic to the space of unrestricted
8-tuples of points $1,2,7,8,9,10,X,Y$ in $P_{2}$. Indeed , as it is clear from
the picture the points $A,B,C,3,4,5,6,7$ can be reconstructed out of
$1,2,7,8,9,10,X,Y$ and vise versa.

$10$$7$$3$$B$$A$$C$$1$$2$$4$$5$$6$$8$$9$a$5$$4$$3$$2$$1$$8$$7$$6$$10$$9$$A$$B$$C$$X$$Y$b

c

8/38/38/3d

Figure 3.

The versal deformation of the singularity $E_{8}$ is

$D(x,p)=x^{5}+p^{3}+a_{1}+a_{2}x+a_{3}x^{2}+a_{4}x^{3}+a_{5}p+a_{6}xp+a_{7}x^{2}p+a_{8}x^{3}p$

the Newton polygon is shown of Fig.3c. It has one side $(-5,3)$ directed
upward with multiplicity $1$. Thus the curve has genus 4 with one point at
infinity with the rank of homology group 8. The growth diagram shown on Fig.3d
consists of three points with angle $2\pi/3$ between them rotating with the
angular speed $8/3$. It corresponds to the periodic word $[(s_{1}s_{2})^{8}]$
and thus the Stokes data amounts to a configuration of $8$ points in $P^{2}$.

Exchanging $x$ and $p$ we get on the growth diagram $5$ points in a vertices
of a regular pentagon rotating about its center with the angular speed $8/5$.
It corresponds to a sequence of 8 points in $P^{4}$. The two configuration
space are birationally isomorphic via Gale duality.

##  7\. Other cases.

We leave the detailed consideration of singularities of other types as an
exercise.

  1. $D_{n}$:

The Dynkin diagram corresponds to a configuration of triangles in the
projective plane $P^{2}$ with 2, 2 and $n-2$ points on its respective sides.
The corresponding singularity $xp^{2}+x^{n+1}$ corresponds to a configuration
of $n$ lines in a two-dimensional local system on a circle. The differential
operator corresponding to $p^{n+1}+x^{2}p$ corresponds to configurations of
flags in $P^{n}$.

  2. $E_{6}$:

The Dynkin diagram as well as the singularity $p^{3}+x^{4}$ correspond to a
configuration of triangles in $P^{2}$ with 2,3 and 3 points on the sides,
respectively. The singularity $p^{3}+x^{4}$ corresponds to the word
$[(s_{1}s_{2})^{7}]$, i.e., to configurations of 7-tuples of points of
$P^{2}$. It is easy to see that the two configuration spaces are equivalent.
The singularity $p^{4}+x^{3}$ corresponds to configurations of 7-tuples points
in $P^{3}$.

  3. $E_{7}$:

The Dynkin diagram corresponds to a configuration of triangles in $P^{2}$ with
2,3 and 4 points on the sides, respectively. The singularity $p^{3}+x^{3}p$
corresponds to configurations of pentagons in $P^{2}$ with one marked point on
each side, which is equivalent to the space of configurations given by the
Dynkin diagram. The singularity $xp^{3}+x^{3}$ corresponds to a configuration
space of 21-periodic sequences flags in $P^{3}$ which is too complicated to be
described here.

## References

  * [1] N.A’Campo, Le groupe de monodromie du déploiement des singularités isolées de courbes planes I, Math. Ann. 213(1975), 1–32.
  * [2] S.Fomin, P.Pylyavskyy, E.Shustin, D.Thurston. Morsifications and mutations. J.Lond.Math.Soc. 105(2022)4 pp
  * [3] S.Gusein-Zade, Dynkin diagrams of the singularities of functions of two variables, Funkcional. Anal. i Prilozhen. 8(1974), no. 4, 23–30.
  * [4] R.Nevanlinna, Über Riemannsche Flächen mit endlich vielen Windungspunkten. Acta Math.58,295–373 (1932)
  * [5] B.Shapiro, On Evgrafov–Fedoryuk’s theory and quadratic differentials, Anal. Math. Phys. 5(2015),171–181.


# The generality of closed $\mathrm{G}_{2}$ solitons

Robert L. Bryant  Duke University Mathematics Department
PO Box 90320
Durham, NC 27708-0320<EMAIL_ADDRESS>http://www.math.duke.edu/~bryant

(Date: April 30, 2022)

###### Abstract.

The local generality of the space of solitons for the Laplacian flow of closed
$\mathrm{G}_{2}$-structures is analyzed, and it is shown that the germs of
such structures depend, up to diffeomorphism, on 16 functions of 6 variables
(in the sense of É. Cartan). The method is to construct a natural exterior
differential system whose integral manifolds describe such solitons and to
show that it is involutive in Cartan’s sense, so that Cartan-Kähler theory can
be applied.

Meanwhile, it turns out that, for the more special case of _gradient_
solitons, the natural exterior differential system is _not_ involutive, and
the generality of these structures remains a mystery.

###### Key words and phrases:

$G_{2}$-structures, solitons

###### 1991 Mathematics Subject Classification:

53E99, 53C29

Thanks to the Simons Foundation for its support via the Simons Collaboration
Grant “Special Holonomy in Geometry, Analysis, and Physics”.

_Dedicated, with much admiration and gratitude for his many years of advice,
encouragement, and friendship, to H. Blaine Lawson, Jr. on the occasion of his
80th birthday._

###### Contents

  1. 1 Introduction
    1. 1.1 $\mathrm{G}_{2}$-structures
    2. 1.2 Relation with holonomy
    3. 1.3 Solitons
  2. 2 Basic Identities
    1. 2.1 Expression in differential forms
  3. 3 Formulation as an exterior differential system
    1. 3.1 A differential ideal
    2. 3.2 Involutivity
  4. 4 The gradient case
    1. 4.1 Formulation
    2. 4.2 Non-involutivity

##  1\. Introduction

For the necessary background on $\mathrm{G}_{2}$-structures and the notation
and conventions that this article uses, the reader may consult [2] and,
especially, [3].

###  1.1. $\mathrm{G}_{2}$-structures

A (smooth) $\mathrm{G}_{2}$-structure on a $7$-manifold $M$ is a (smooth)
$3$-form $\phi\in\Omega^{3}(M)$ that is _definite_ in the sense that, for any
nonzero tangent vector $v\in T_{x}M$, the $7$-form $(v\mathbin{\hbox{\vrule
height=1.4pt,width=4.0pt,depth=-1.0pt\vrule
height=4.0pt,width=0.4pt,depth=-1.0pt}}\phi){\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}(v\mathbin{\hbox{\vrule
height=1.4pt,width=4.0pt,depth=-1.0pt\vrule
height=4.0pt,width=0.4pt,depth=-1.0pt}}\phi){\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\phi\in\Lambda^{7}(T^{*}_{x}M)$
is also nonzero. The space of definite $3$-forms on $M^{7}$ will be denoted
$\Omega^{3}_{+}(M)$.

Given a $\sigma\in\Omega^{3}_{+}(M)$, there is a unique metric $g_{\sigma}$
and orientation $\ast_{\sigma}$ (i.e., the Hodge star operator defined by
$g_{\sigma}$ and the orientation) such that

(1.1) $(X\mathbin{\hbox{\vrule height=1.4pt,width=4.0pt,depth=-1.0pt\vrule
height=4.0pt,width=0.4pt,depth=-1.0pt}}\sigma){\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}(Y\mathbin{\hbox{\vrule
height=1.4pt,width=4.0pt,depth=-1.0pt\vrule
height=4.0pt,width=0.4pt,depth=-1.0pt}}\sigma){\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\sigma=6\langle
X,Y\rangle_{g_{\sigma}}\,{\ast_{\sigma}}1$

for all vector fields $X$ and $Y$ on $M$.

It is not difficult to prove the first-order $6$-form identity [2]

(1.2)
$\bigl{(}{\ast_{\sigma}}{\mathrm{d}}({\ast_{\sigma}}\sigma)\bigr{)}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}{\ast_{\sigma}}\sigma+({\ast_{\sigma}}\mathrm{d}\sigma){\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\sigma=0.$

In particular, the equations
$\mathrm{d}\sigma=\mathrm{d}({\ast_{\sigma}}\sigma)=0$ represent only
$35+21-7=49$ first-order equations on a $\mathrm{G}_{2}$-structure $\sigma$.
(It is easy to see that these $49$ equations are independent, i.e., there are
no further algebraic relations between $\sigma$, $\mathrm{d}\sigma$ and
${\ast_{\sigma}}{\mathrm{d}}({\ast_{\sigma}}\sigma)$ beyond those implied by
(1.2).)

###  1.2. Relation with holonomy

Fernández and Gray [7] proved that, if $\sigma$ and ${\ast_{\sigma}}\sigma$
are both closed forms, then $\sigma$ is $g_{\sigma}$-parallel. This, in
particular, implies that the holonomy of $g_{\sigma}$ is isomorphic to a
subgroup of $\mathrm{G}_{2}\subset\operatorname{SO}(7)$. Conversely, if the
holonomy of a metric $g$ on $M^{7}$ is contained in $\mathrm{G}_{2}$, then
there exists a definite $3$-form $\sigma$ (unique up to replacement by
$-\sigma$ if the holonomy of $g$ is isomorphic to $\mathrm{G}_{2}$) satisfying
$\mathrm{d}\sigma=\mathrm{d}({\ast_{\sigma}}\sigma)=0$ and such that
$g=g_{\sigma}$. This characterization has been essential in all constructions
of metrics on $7$-manifolds with holonomy $\mathrm{G}_{2}$, indeed, in even
proving their existence.

While it is easy to write down a local formula for the ‘generic’ closed
$\mathrm{G}_{2}$-structure $\sigma$, the full equations
$\mathrm{d}\sigma=\mathrm{d}{\ast_{\sigma}}\sigma=0$ are highly nonlinear (as
well as being overdetermined). In [2], it was shown how to interpret these
equations as an involutive exterior differential system (see [4]) and analyze
their local generality in the sense of Élie Cartan. In particular, it was
shown that the ‘general’ solution, up to diffeomorphism, depends on six
functions of six variables.

One way of expressing this in terms that are, perhaps, more familiar is the
following: Suppose that one considers the space $\mathcal{J}^{k}$ of $k$-jets
of germs of solutions of $\mathrm{d}\sigma=\mathrm{d}{\ast_{\sigma}}\sigma=0$
on a neighborhood of $0\in\mathbb{R}^{7}$ with the property that the standard
coordinates on $\mathbb{R}^{7}$ are geodesic normal coordinates for
$g_{\sigma}$ at $0$ and the value of $\sigma$ at $0$ takes a standard
form111This nearly removes the diffeomorphism ambiguity. In order to
completely remove it, one would have to quotient by the natural action of
$\mathrm{G}_{2}$ on these jet spaces.. Let
$\delta_{k}=\dim\mathcal{J}^{k}-\dim\mathcal{J}^{k-1}$ and consider the
Poincaré series $P(t)=\delta_{0}+\delta_{1}\,t+\delta_{2}\,t^{2}+\cdots$. Then

(1.3)
$P(t)=t^{2}\left(\frac{14}{(1{-}t)^{2}}+\frac{21}{(1{-}t)^{3}}+\frac{21}{(1{-}t)^{4}}+\frac{15}{(1{-}t)^{5}}+\frac{6}{(1{-}t)^{6}}\right).$

In particular, the ‘leading term’ contributing to the growth of $\delta_{k}$,
namely $6t^{2}/(1{-}t)^{6}$, is the same, up to a degree shift, as the
‘leading term’ describing the growth of $k$-jets of $6$ functions of $6$
variables.

###  1.3. Solitons

D. Joyce pioneered the approach to constructing compact examples of metrics
with holonomy $\mathrm{G}_{2}$ by starting with a ‘soft’ construction of a
closed $\mathrm{G}_{2}$-structure $\sigma$ on certain $7$-manifolds with the
property that $\mathrm{d}({\ast_{\sigma}}\sigma)$ had sufficiently small norm
(in comparison with other geometric features of $g_{\sigma}$) and showing
that, in such situations, one could ‘perturb’ the given closed
$\mathrm{G}_{2}$-structure $\sigma$ to a nearby one that was also co-closed.
(See the fundamental reference [12] and the original papers cited therein for
details.)

The many successes of R. Hamilton’s program for using the Ricci flow in
Riemannian geometry inspired the idea of canonically ‘improving’ a closed
$\mathrm{G}_{2}$-structure $\sigma_{0}$ on a compact $7$-manifold $M$ by
considering the so-called ‘Laplacian flow’

(1.4) $\frac{d\,\sigma}{dt}=\Delta_{g_{\sigma}}\,\sigma$

with $\sigma_{0}$ as initial condition. In particular, a fixed point of this
flow, i.e., a closed $\mathrm{G}_{2}$-structure on $M$ that satisfies
$\Delta_{g_{\sigma}}\,\sigma=0$, would satisfy
$\mathrm{d}\sigma=\mathrm{d}{\ast_{\sigma}}\sigma=0$ and hence lead to a
metric $g_{\sigma}$ with holonomy contained in $\mathrm{G}_{2}$. It has been
shown [6] that the closed $\mathrm{G}_{2}$-Laplacian flow on a compact
$7$-manifold has short-time existence and uniqueness.

Meanwhile, Hitchin [10, 11] observed that the above Laplacian flow is (up to a
constant factor) the gradient flow of the _volume functional_

$\mathcal{V}(\sigma)=\int_{M}\sigma{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}{\ast_{\sigma}}\sigma$

on the space $\mathcal{Z}(M,\gamma)\subset\Omega^{3}_{+}(M)$ of closed
$\mathrm{G}_{2}$-structures in a fixed deRham cohomology class $\gamma\in
H^{3}_{dR}(M)$. Hitchin has shown that any critical points of $\mathcal{V}$ in
$\mathcal{Z}(M,\gamma)\subset\Omega^{3}_{+}(M)$ must be local maxima.

However, there need not be any critical points even if the flow exists for all
time, as the example in Remark 18 of [3] shows.

It is not clear just what sort of singularities one might encounter in the
Laplacian flow, and one strategy for gaining some intuition, based on the
strategy that worked in the case of Ricci flow, is to study the so-called
‘soliton’ solutions.

######  Definition 1.

Let $\lambda$ be a fixed real constant. A pair $(\sigma,X)$, where $\sigma$ is
a closed $\mathrm{G}_{2}$-structure on $M^{7}$ and $X$ is a vector field on
$M^{7}$, is a _$\lambda$ -soliton_ for the $\mathrm{G}_{2}$-Laplacian flow if

(1.5) $\Delta_{\sigma}\,\sigma=\lambda\,\sigma+\mathcal{L}_{X}\sigma\,.$

If, in addition, $X=\nabla^{g_{\sigma}}f$ for some function $f$ on $M$, the
pair $(\sigma,X)$ will be said to be a _gradient $\lambda$-soliton_.

It is not difficult to show that, if $(\sigma,X)$ is a $\lambda$-soliton for
the $\mathrm{G}_{2}$-Laplacian flow on $M$ and $\Phi_{\tau}:M\to M$ is the
time-$\tau$ flow of $X$, then the time-dependent $\mathrm{G}_{2}$-structure

(1.6) $\phi(t)=\mathrm{e}^{\lambda t}\Phi^{*}_{f(t,\lambda)}\sigma$

where $f(t,0)=t$ and $f(t,\lambda)=(1-\mathrm{e}^{-\lambda t})/\lambda$ when
$\lambda\not=0$, satisfies the initial condition $\phi(0)=\sigma$ and the
Laplacian flow equation

(1.7) $\frac{d\,\phi}{dt}=\Delta_{g_{\phi}}\,\phi\,.$

Many explicit examples of such $\lambda$-solitons are now known. Most of the
known examples are constructed using some version of ‘dimensional reduction’,
i.e., assuming some sort of symmetry, thereby reducing the problem to
geometric data on a lower dimensional manifold, see [1, 8, 9, 13, 14].

The goal of this article is to investigate the local generality, in É.
Cartan’s sense, of the space of $\lambda$-solitons. A straightforward count of
equations reveals that, locally, this is 70 equations for the 42 unknowns in
$(\sigma,X)$, so the local generality of the solutions of these equations is
not at all clear. Moreover, the system is degenerate (in particular, every
direction is characteristic) because it is evidently invariant under
diffeomorphisms.

##  2\. Basic Identities

In this section, I will assume that $(\sigma,X)$ is a $\lambda$-soliton for
the $\mathrm{G}_{2}$-Laplacian flow equation on $M^{7}$ and explain how to
express this condition naturally in terms of an exterior differential system.

###  2.1. Expression in differential forms

Because of the fundamental identity (1.2), the closedness of $\sigma$ implies
that
$\bigl{(}{\ast_{\sigma}}\mathrm{d}({\ast_{\sigma}}\sigma)\bigr{)}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}{\ast_{\sigma}}\sigma=0$,
so there is a $2$-form $\tau$ that satisfies

(2.1)
$\mathrm{d}({\ast_{\sigma}}\sigma)=\tau{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\sigma=-{\ast_{\sigma}}\tau.$

(See §4.6 of [3].) Moreover, $\tau$ satisfies

(2.2)
$\mathrm{d}\tau=\Delta_{\sigma}\sigma=\lambda\,\sigma+\mathcal{L}_{X}\sigma=\lambda\,\sigma+\mathrm{d}(X\mathbin{\hbox{\vrule
height=1.4pt,width=4.0pt,depth=-1.0pt\vrule
height=4.0pt,width=0.4pt,depth=-1.0pt}}\sigma),$

which can be rewritten as

(2.3) $\mathrm{d}(\tau-X\mathbin{\hbox{\vrule
height=1.4pt,width=4.0pt,depth=-1.0pt\vrule
height=4.0pt,width=0.4pt,depth=-1.0pt}}\sigma)=\lambda\,\sigma.$

As explained in §2.6 of [3], every $2$-form $\beta\in\Omega^{2}(M)$ can be
written uniquely as a sum $\beta=\beta_{7}+\beta_{14}$ where
${\ast_{\sigma}}\beta_{14}=-\beta_{14}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\sigma$
while
${\ast_{\sigma}}\beta_{7}=\tfrac{1}{2}\,\beta_{7}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\sigma$.
In what follows, set $\beta=\tau-X\mathbin{\hbox{\vrule
height=1.4pt,width=4.0pt,depth=-1.0pt\vrule
height=4.0pt,width=0.4pt,depth=-1.0pt}}\sigma$, so that $\beta_{14}=\tau$ and
$\beta_{7}=-X\mathbin{\hbox{\vrule height=1.4pt,width=4.0pt,depth=-1.0pt\vrule
height=4.0pt,width=0.4pt,depth=-1.0pt}}\sigma$. Then (2.3) can be rewritten as
the $3$-form equation

(2.4) $0=\mathrm{d}\beta-\lambda\,\sigma.$

We already have the 4-form equation

(2.5) $0=\mathrm{d}\sigma$

since $\sigma$ is closed. Using the above algebraic identities, (2.1) can be
written as a 5-form equation

(2.6)
$0=\mathrm{d}({\ast_{\sigma}}\sigma)-\tau{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\sigma=\mathrm{d}({\ast_{\sigma}}\sigma)+\tfrac{2}{3}\,{\ast_{\sigma}}\beta-\tfrac{1}{3}\,\beta{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\sigma.$

Finally, taking the exterior derivative of this last equation and using the
equations found so far, we have the 6-form equation

(2.7) $0=\mathrm{d}(\ast_{\sigma}\beta).$

The virtue of the equations (2.4–7) is that they only involve the constant
$\lambda$, the definite $3$-form $\sigma$ (and its algebraically associated
Hodge star operator), and the $2$-form $\beta$. In fact, this is enough to
recover the structure of a $\lambda$-soliton for the closed
$\mathrm{G}_{2}$-Laplacian flow.

######  Proposition 1.

Let $\lambda$ be a constant, and let $\sigma\in\Omega^{3}_{+}(M)$ and
$\beta\in\Omega^{2}(M)$ satisfy equations (2.4–7). Then for the unique vector
field $X$ on $M$ such that $\beta_{7}=-X\mathbin{\hbox{\vrule
height=1.4pt,width=4.0pt,depth=-1.0pt\vrule
height=4.0pt,width=0.4pt,depth=-1.0pt}}\sigma$, the pair $(\sigma,X)$ is a
$\lambda$-soliton for the closed $\mathrm{G}_{2}$-Laplacian flow. Conversely,
if $(\sigma,X)$ is a $\lambda$-soliton for the closed
$\mathrm{G}_{2}$-Laplacian flow, then setting
$\beta=-\ast_{\sigma}\mathrm{d}({\ast_{\sigma}}\sigma)-X\mathbin{\hbox{\vrule
height=1.4pt,width=4.0pt,depth=-1.0pt\vrule
height=4.0pt,width=0.4pt,depth=-1.0pt}}\sigma$ defines a pair $(\sigma,\beta)$
that satisfies (2.4–7).

###### Proof.

This is a matter of unwinding the definitions. We have $\mathrm{d}\sigma=0$,
and then equation (2.6) implies that $\tau=\beta_{14}$ and that
$\mathrm{d}({\ast_{\sigma}}\sigma)=\tau{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\sigma$,
so that $\mathrm{d}\tau=\Delta_{\sigma}\sigma$. Then (2.4) becomes (2.3). The
converse follows in the same way. ∎

######  Remark 1.

The reader may wonder why (2.7) was included, since it is a consequence of the
other three equations just by applying the exterior derivative to (2.6).
Moreover, when $\lambda$ is nonzero, (2.5) follows from (2.4), so it might
seem that (2.5) could be omitted as well, at least when $\lambda$ is nonzero.

The reason is that I want to make explicit _all_ of the first-order equations
satisfied by the pair $(\sigma,\beta)$. The four equations (2.4–7) are the
vanishing of a $3$-form, a $4$-form, a $5$-form, and a $6$-form, which is
potentially

$35+35+21+7=98$

independent first order equations. However, the identity (1.2), shows that
these 98 equations are _not_ independent; they ‘overlap’ by at least $7$
equations.

In fact, the overlap is _exactly_ the $7$ equations of the fundamental
identity (1.2) since it is easy to see that, for a generic
$\sigma\in\Omega^{3}_{+}(M)$ and $\beta\in\Omega^{2}(M)$, the only relation
among the forms

$\mathrm{d}\beta,\ \mathrm{d}\sigma,\ \mathrm{d}(\ast_{\sigma}\sigma),\
\mathrm{d}(\ast_{\sigma}\beta)$

is the fundamental identity relating the middle two. Thus, the equations
(2.4–7) constitute exactly $91$ independent (quasi-linear) first order
equations on the pair $(\sigma,\beta)$, a fact to which I will return.

Indeed, I am going to argue that this system of $91$ equations is _involutive_
in É. Cartan’s sense, so that the Cartan-Kähler Theorem can be applied to
prove local existence of solutions and describe their ‘degree of generality’.
By contrast, the ‘determined’ system of $35+21=56$ first-order equations for
$(\sigma,\beta)$ represented by combining (2.4) and (2.6) is certainly _not_
involutive.

##  3\. Formulation as an exterior differential system

I now want to explain how the above system of 91 first-order partial
differential equations for the pair $(\sigma,\beta)$, which is a section of a
smooth bundle of fiber rank $35+21=56$ over $M^{7}$, can be interpreted as
defining an exterior differential system for such pairs.

###  3.1. A differential ideal

Let $\Lambda^{3}_{+}(T^{*}M)\subset\Lambda^{3}(T^{*}M)$ denote the open subset
of the bundle of $3$-forms on $M^{7}$ whose (local) sections are the (local)
$\mathrm{G}_{2}$-structures on $M$. Similarly, let $\Lambda^{2}(T^{*}M)$
denote the bundle of $2$-forms on $M$.

Let $X\subset\Lambda^{3}(T^{*}M)\oplus\Lambda^{2}(T^{*}M)$ denote the set of
pairs $(s,b)$ with $s\in\Lambda^{3}_{+}(T^{*}_{m}M)$ and
$b\in\Lambda^{2}(T^{*}_{m}M)$ for some $m\in M$. This $X$ is open in
$\Lambda^{3}(T^{*}M)\oplus\Lambda^{2}(T^{*}M)$ and is a smooth fiber bundle
over $M$ with a submersive base projection $\pi:X\to M$, with $\pi(s,b)=m$.
Let $\pi^{*}:\Lambda^{k}(T^{*}M)\to\Lambda^{k}(T^{*}X)$ denote the induced
‘pullback’ mapping.

There exist natural ‘tautological’ forms on $X$ that are defined as follows:
For each $(s,b)\in X$, let $\mathsf{s}_{(s,b)}=\pi^{*}(s)$ and let
$\mathsf{b}_{(s,b)}=\pi^{*}(b)$. Then $\mathsf{s}$ (respectively,
$\mathsf{b}$) is a smooth $3$-form (respectively, $2$-form) on $X$ with the
‘reproducing’ property that, for any section $(\sigma,\beta):M\to X$, there
holds $(\sigma,\beta)^{*}(\mathsf{s})=\sigma$ and
$(\sigma,\beta)^{*}(\mathsf{b})=\beta$. Moreover, because exterior derivative
commutes with pullback, the identities
$(\sigma,\beta)^{*}(\mathrm{d}\mathsf{s})=\mathrm{d}\sigma$ and
$(\sigma,\beta)^{*}(\mathrm{d}\mathsf{b})=\mathrm{d}\beta$ hold as well.

I will need an extension of this construction: Since, for any
$s\in\Lambda^{3}_{+}(T^{*}_{m}M)$, there is a well-defined Hodge star operator
$\ast_{s}:\Lambda^{p}(T^{*}_{m}M)\to\Lambda^{7-p}(T^{*}_{m}M)$, there also
exists a smooth $4$-form $\mathsf{t}$ (respectively, $5$-form, $\mathsf{p}$)
on $X$ with the property that $\mathsf{t}_{(s,b)}=\pi^{*}(\ast_{s}s)$
(respectively, $\mathsf{p}_{(s,b)}=\pi^{*}(\ast_{s}b)$). These two forms have
the ‘reproducing’ property that, for any section $(\sigma,\beta):M\to X$,
there holds $(\sigma,\beta)^{*}(\mathsf{t})={\ast_{\sigma}}\sigma$ and
$(\sigma,\beta)^{*}(\mathsf{p})={\ast_{\sigma}}\beta$.

Finally, there exists a smooth $7$-form $\omega$ on $X$ with the property that
$\omega_{(s,b)}=\pi^{*}(\ast_{s}1)$ that will be used below, so I introduce it
now.

Note that all of the forms $\mathsf{s}$, $\mathsf{t}$, $\mathsf{b}$,
$\mathsf{p}$, and $\omega$ are $\pi$-semibasic, i.e., their interior product
with any $\pi$-vertical vector field on $X$ vanishes identically.

Now, by construction, for any constant $\lambda$ and any section
$(\sigma,\beta):M\to X$, one has

(3.1)
$\displaystyle(\sigma,\beta)^{*}(\mathrm{d}\mathsf{b}-\lambda\,\mathsf{s})$
$\displaystyle=\mathrm{d}\beta-\lambda\,\sigma$
$\displaystyle(\sigma,\beta)^{*}(\mathrm{d}\mathsf{s})$
$\displaystyle=\mathrm{d}\sigma$
$\displaystyle(\sigma,\beta)^{*}(\mathrm{d}\mathsf{t}+\tfrac{2}{3}\,\mathsf{p}-\tfrac{1}{3}\,\mathsf{b}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\mathsf{s})$
$\displaystyle=\mathrm{d}({\ast_{\sigma}}\sigma)+\tfrac{2}{3}\,{\ast_{\sigma}}\beta-\tfrac{1}{3}\,\beta{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\sigma$
$\displaystyle(\sigma,\beta)^{*}(\mathrm{d}\mathsf{p})$
$\displaystyle=\mathrm{d}({\ast_{\sigma}}\beta)$

This motivates defining the differential forms $\Upsilon_{i}\in\Omega^{i}(X)$,

(3.2) $\displaystyle\Upsilon_{3}$
$\displaystyle=\mathrm{d}\mathsf{b}-\lambda\,\mathsf{s},$
$\displaystyle\Upsilon_{4}$ $\displaystyle=\mathrm{d}\mathsf{s},$
$\displaystyle\Upsilon_{5}$
$\displaystyle=\mathrm{d}\mathsf{t}+\tfrac{2}{3}\,\mathsf{p}-\tfrac{1}{3}\,\mathsf{b}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\mathsf{s},$
$\displaystyle\Upsilon_{6}$ $\displaystyle=\mathrm{d}\mathsf{p}.$

Note the identities

(3.3) $\displaystyle\mathrm{d}\Upsilon_{3}$
$\displaystyle=-\lambda\,\Upsilon_{4}\,,$
$\displaystyle\mathrm{d}\Upsilon_{4}$ $\displaystyle=0,$
$\displaystyle\mathrm{d}\Upsilon_{5}$
$\displaystyle=\tfrac{2}{3}\,\Upsilon_{6}-\tfrac{1}{3}\,\Upsilon_{3}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\mathsf{s}-\tfrac{1}{3}\,\mathsf{b}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\Upsilon_{4}\,,$
$\displaystyle\mathrm{d}\Upsilon_{6}$ $\displaystyle=0,$

which imply that the algebraic ideal $\mathcal{I}_{\lambda}$ in
$\Omega^{*}(X)$ generated by
$\Upsilon_{3},\Upsilon_{4},\Upsilon_{5},\Upsilon_{6}$ is differentially
closed. Consequently, $\mathcal{I}_{\lambda}$ is an exterior differential
system on $X$.

The interest in $\mathcal{I}_{\lambda}$ is explained by the following result.

######  Proposition 2.

A section $(\sigma,\beta):M\to X$ is an integral manifold of
$\mathcal{I}_{\lambda}$ if and only if it comes from a $\lambda$-soliton for
the closed $\mathrm{G}_{2}$-Laplacian flow. In particular, a $7$-dimensional
integral manifold of $\mathcal{I}_{\lambda}$ that is transverse to the fibers
of $\pi:X\to M$ is locally the graph of a section $(\sigma,\beta):M\to X$ that
comes from a $\lambda$-soliton for the closed $\mathrm{G}_{2}$-Laplacian flow.

###### Proof.

It has been shown that a pair $(\sigma,\beta)$ comes from a $\lambda$-soliton
for the closed $\mathrm{G}_{2}$-Laplacian flow if and only if the pair
satisfies the equations (2.4–7). Since this vanishing is exactly the vanishing
of the righthand sides of (3.1), the definitions given in (3.2) and the
definition of $\mathcal{I}_{\lambda}$ show that this vanishing is equivalent
to the section $(\sigma,\beta)$ being a $7$-dimensional integral manifold
(necessarily transverse to the $\pi$-fibers) of $\mathcal{I}_{\lambda}$. The
rest of the statement follows immediately. ∎

###  3.2. Involutivity

I can now state the main result of this article.

######  Theorem 1.

The exterior differential system $\mathcal{I}_{\lambda}$ with independence
condition $\omega$ is involutive, with Cartan characters

$(s_{0},s_{1},s_{2},s_{3},s_{4},s_{5},s_{6},s_{7})=(0,0,1,3,7,15,23,7).$

###### Proof.

The proof will be by Cartan’s Test for involutivity (see, for example, [4]).

The first thing to check for each $(s,b)\in X$ is the dimension of the space
of $7$-dimensional integral elements $E\subset T_{(s,b)}X$ of
$\mathcal{I}_{\lambda}$ on which $\omega$ is non-vanishing. I claim that this
dimension is $301=392-91$.

To see this, choose a splitting $T_{(s,b)}X=\ker\pi^{\prime}(s,b)+W$, where
$\pi^{\prime}(s,b):W\to T_{\pi(s,b)}M$ is an isomorphism. Then an integral
element $E\subset T_{(s,b)}X$ on which $\omega$ is non-vanishing can be
thought of as the graph of a linear map $L:W\to\ker\pi^{\prime}(s,b)$, i.e.,
it can be identified with an element of the vector space
$\operatorname{Hom}(W,\ker\pi^{\prime}(s,b))$. Because the four tautological
forms $\mathsf{s}$, $\mathsf{b}$, $\mathsf{t}$, and $\mathsf{p}$ are
$\pi$-semi-basic, it follows that the induced mapping of
$\operatorname{Hom}(W,\ker\pi^{\prime}(s,b))$ into the 98-dimensional vector
space

$\Lambda^{3}(W^{*})\oplus\Lambda^{4}(W^{*})\oplus\Lambda^{5}(W^{*})\oplus\Lambda^{6}(W^{*})$

induced by evaluating $(\Upsilon_{3},\Upsilon_{4},\Upsilon_{5},\Upsilon_{6})$
on the graph of $L\in\operatorname{Hom}(W,\ker\pi^{\prime}(s,b))$ is an affine
mapping. Moreover, its image lies in the 91-dimensional subspace cut out by
the fundamental identity. Finally, looking at the ‘leading terms’ of
$(\Upsilon_{3},\Upsilon_{4},\Upsilon_{5},\Upsilon_{6})$, i.e.,
$(\mathrm{d}\mathsf{b},\mathrm{d}\mathsf{s},\mathrm{d}\mathsf{t},\mathrm{d}\mathsf{p})$
and considering their interpretation as

$\bigl{(}\mathrm{d}\beta,\mathrm{d}\sigma,\mathrm{d}(\ast_{\sigma}\sigma),\mathrm{d}(\ast_{\sigma}\beta)\bigr{)},$

one sees that the leading order part is a linear mapping that is surjective
onto this 91-dimensional subspace.

Consequently, the set $\mathcal{V}(\mathcal{I}_{\lambda},\omega)$ of
admissible integral elements of $\mathcal{I}_{\lambda}$ at $(s,b)\in X$ is a
(nonempty) affine subspace of
$\operatorname{Hom}\bigl{(}W,\ker\pi^{\prime}(s,b)\bigr{)}$ of dimension
$S=56\times 7-91=392-91=301$. Indeed
$\mathcal{V}(\mathcal{I}_{\lambda},\omega)$ is a smooth submanifold of
$\mathrm{Gr}(7,TX)$, the Grassmann bundle of $7$-dimensional subspaces of the
tangent spaces to $X$.

It remains to compute the Cartan characters and apply Cartan’s Test. Now that
it has been established that the space of admissible integral elements of
$\mathcal{I}_{\lambda}$ at each point is nonempty, the Cartan characters can
be computed by considering just the leading terms listed above.

First, recall that it was established in [2] that the Cartan characters of the
(involutive) equation
$D(\sigma)=\bigl{(}\mathrm{d}\sigma,\mathrm{d}(\ast_{\sigma}\sigma)\bigr{)}=(0,0)$
relative to any flag are

$(s_{0},s_{1},s_{2},s_{3},s_{4},s_{5},s_{6},s_{7})=(0,0,0,1,4,10,13,7),$

Meanwhile, once $\sigma$ is fixed, the characters of the involutive first-
order system
$D(\beta)=\bigl{(}\mathrm{d}\beta,\mathrm{d}(\ast_{\sigma}\beta)\bigr{)}=(0,0)$
for $\beta$ a $2$-form on a manifold with a specified
$\mathrm{G}_{2}$-structure $\sigma$ are

$(s_{0},s_{1},s_{2},s_{3},s_{4},s_{5},s_{6},s_{7})=(0,0,1,2,3,5,10,0).$

It follows that there exists an integral flag of the combined system with
characters

$(s_{0},s_{1},s_{2},s_{3},s_{4},s_{5},s_{6},s_{7})=(0,0,1,3,7,15,23,7).$

It follows that $\sum_{k=0}^{7}k\,s_{k}=301=S$, the dimension of the space of
integral elements of $\mathcal{I}_{\lambda}$ at $(s,b)$. Thus, equality holds
in Cartan’s Test, which implies that such a flag is Cartan-regular.
Consequently, the system is involutive, as was to be shown. ∎

######  Remark 2.

Since $M$ is assumed to be smooth, it also carries a real-analytic structure
(in fact, many, but all of them are equivalent), and $X$ will naturally
inherit a real-analytic structure once one is chosen on $M$. Because of the
nature of their construction, the forms
$\Upsilon_{3},\Upsilon_{4},\Upsilon_{5},\Upsilon_{6}$ will be real-analytic
with respect to the real-analytic structure on $X$. Hence
$\mathcal{I}_{\lambda}$ is real-analytic with respect to any real-analytic
structure on $X$.

######  Corollary 1.

For any $(s,b)\in X$ with $\pi(s,b)=m\in M$, there exists a $\lambda$-soliton
$(\sigma,\beta)$ for the closed $\mathrm{G}_{2}$-Laplacian flow on an
$m$-neighborhood $U\subset M$ such that $(\sigma_{m},\beta_{m})=(s,b)$. In
fact, for any $7$-dimensional integral element $E\subset T_{(s,b)}X$ of
$\mathcal{I}_{\lambda}$ on which $\omega$ is non-vanishing, there exists such
a $(\sigma,\beta):U\to X$ such that $(\sigma,\beta)^{\prime}(s,b)(T_{m}M)=E$.

###### Proof.

In light of Theorem 1 and Remark 2, Corollary 1 follows immediately from the
Cartan-Kähler Theorem. ∎

######  Remark 3 (Generality in Harmonic Coordinates).

Since the last nonzero character of $\mathcal{I}_{\lambda}$ is $s_{7}=7$, the
naïve statement is that local $\lambda$-solitons for the closed
$\mathrm{G}_{2}$-Laplacian flow depend on ‘seven functions of seven variables’
in Cartan’s terminology222The Cartan characters $s_{i}$ indicate how much
essential choice there is in building up a solution to the equations via a
sequence of (possibly underdetermined) Cauchy problems.. However, this
reflects the fact that the equations are invariant under the full
diffeomorphism group in dimension $7$, which itself is described (locally) by
specifying seven functions of seven variables.

There are a number of methods of, at least partially, ‘normalizing’ the local
solutions by choosing special coordinates. The simplest one perhaps, is to ask
how general the local solutions are when presented in harmonic coordinates for
the associated metric $g_{\sigma}$. One way to see this is to specialize to
the case $M=\mathbb{R}^{7}$ with standard coordinates $u=(u^{i})$ and impose
the conditions $\mathrm{d}\bigl{(}\ast_{\sigma}(\mathrm{d}u^{i})\bigr{)}=0$.
This can be added to the exterior differential system by defining the seven
$6$-forms

(3.4)
$\Phi^{i}=\pi^{*}\bigl{(}\ast_{s}(\mathrm{d}u^{i})\bigr{)}\in\Lambda^{6}(T^{*}_{(s,b)}X)$

and then defining the extended ideal $\mathcal{I}_{\lambda}^{+}$ generated by
$\mathcal{I}_{\lambda}$ and the seven closed $7$-forms $\mathrm{d}\Phi^{i}$.
Following the same argument as that used in Theorem 1, one can show that the
system $\mathcal{I}_{\lambda}^{+}$ on $X$ is also involutive, but now with
characters

(3.5) $(s_{0},s_{1},s_{2},s_{3},s_{4},s_{5},s_{6},s_{7})=(0,0,1,3,7,15,30,0).$

One can interpret the local integrals of $\mathcal{I}_{\lambda}^{+}$ as
$\lambda$-solitons for the closed $\mathrm{G}_{2}$-Laplacian flow that are
presented in harmonic coordinates. In Cartan’s terminology, such solitons
depend locally on $s_{6}=30$ functions of six variables. However, the choice
of a local harmonic function, given $\sigma$, depends on 2 functions of six
variables, so a choice of a harmonic coordinate system depends on $2\times
7=14$ functions of six variables. Thus, one can argue that local solitons
$(\sigma,\beta)$ depend on $16=30-14$ functions of six variables, in the sense
that the rate of growth of the dimensions of the $k$-jets of solutions up to
diffeomorphism is the same as the rate of growth of 16 functions of 6
variables.

######  Remark 4 (Analyticity).

Another reason for considering the presentation of $\lambda$-solitons
$(\sigma,\beta)$ in harmonic coordinates is that it makes clear the regularity
of such structures. Because the last nonzero character of
$\mathcal{I}_{\lambda}^{+}$ is $s_{6}=30$, it follows [5, Chapter V] that the
complex characteristic variety
$\Xi_{E}\subset{\mathbb{P}}\bigl{(}(E\otimes{\mathbb{C}})^{*}\bigr{)}\simeq\mathbb{CP}^{6}$
is a subvariety of (complex) dimension $5$ for every integral element $E$.
Because of the $\mathrm{G}_{2}$-invariance of the system and the fact that
$\mathrm{G}_{2}$ acts transitively on (real) $2$-planes in $\mathbb{R}^{7}$,
the only possibility for the support of this characteristic variety for a
given solution $(\sigma,\beta)$ on a neighborhood of a point
$u\in\mathbb{R}^{7}$ is the set of complex co-vectors in
${\mathbb{P}}\bigl{(}(T^{*}_{u}\mathbb{R}^{7}){\otimes}{\mathbb{C}}\bigr{)}$
that are null for $g_{\sigma_{u}}$. In particular, the complex characteristic
variety has no real points, implying that the associated PDE system is
(overdetermined) elliptic. Consequently, $\lambda$-solitons that are
$C^{1,\alpha}$ in harmonic coordinates (for some $\alpha>0$) must be real-
analytic in those coordinates [15].

##  4\. The gradient case

In this final section, I will discuss what is known about the special case
that the $\lambda$-soliton $(\sigma,X)$ is a _gradient_ soliton, i.e.,
$X=\nabla^{g_{\sigma}}f$ for some function $f$. This case is interesting
partly because, in the case of solitons for the Ricci flow, particularly in
dimension $3$, the gradient solitons are the most important.

######  Remark 5 (Ricci solitons in dimension $3$).

Just for comparison, it might be interesting to note that the generality
analysis for general Ricci solitons in dimension $3$, i.e., pairs $(g,X)$ of a
metric $g$ and a vector field $X$ on a manifold $M^{3}$ that satisfy

(4.1) $\mathrm{Ric}(g)=\lambda\,g+\mathcal{L}_{X}g\,,$

shows that, up to local diffeomorphism, the local solutions of this system
depend on $s_{2}=6$ functions of two variables. If one adds the ‘gradient
condition’ that $X=\nabla^{g}f$ for some function $f$, then one sees that, up
to local diffeomorphism, the local solutions satisfying this extra condition
depend on $s_{2}=2$ functions of two variables. In both cases, the associated
exterior differential system is involutive. See [4, §5.5] for the gradient
case.

Many explicit examples of gradient $\lambda$-solitons are known, as can be
seen by looking at the examples constructed in the articles [1, 8, 9, 13, 14].
However, as of this writing, it is not known what their generality is in the
sense of É. Cartan. I will conclude this article by explaining where the
difficulty lies.

###  4.1. Formulation

The gradient $\lambda$-soliton case, where $X=\nabla^{g_{\sigma}}f$ for some
function $f$, is locally equivalent to requiring that $X^{\flat}$, the
$1$-form that is $g_{\sigma}$-dual to the vector field $X$, be closed.

Starting with the algebraic identity of $5$-forms

(4.2) $(X\mathbin{\hbox{\vrule height=1.4pt,width=4.0pt,depth=-1.0pt\vrule
height=4.0pt,width=0.4pt,depth=-1.0pt}}\sigma){\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\sigma=2X^{\flat}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}{\ast_{\sigma}}\sigma\,$

and applying the exterior derivative to the left hand side of (4.2) gives

(4.3) $\mathrm{d}\bigl{(}(X\mathbin{\hbox{\vrule
height=1.4pt,width=4.0pt,depth=-1.0pt\vrule
height=4.0pt,width=0.4pt,depth=-1.0pt}}\sigma){\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\sigma\bigr{)}=\mathrm{d}(X\mathbin{\hbox{\vrule
height=1.4pt,width=4.0pt,depth=-1.0pt\vrule
height=4.0pt,width=0.4pt,depth=-1.0pt}}\sigma){\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\sigma=(\mathrm{d}\tau-\lambda\,\sigma){\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\sigma=0,$

since
$\mathrm{d}({\ast_{\sigma}}\sigma)=\tau{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\sigma$
and $\mathrm{d}\sigma=0$ implies
$\mathrm{d}\tau{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\sigma=0$.
Meanwhile applying the exterior derivative to the right hand side of of (4.2)
(which is closed because of (4.3)) gives

(4.4)
$0=2\,\mathrm{d}(X^{\flat}){\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}{\ast_{\sigma}}\sigma-2\,X^{\flat}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\tau{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\sigma.$

In particular, if $\mathrm{d}(X^{\flat})=0$, which is the condition for a
gradient $\lambda$-soliton, it follows that
$X^{\flat}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\tau{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\sigma=0$,
which is a algebraic relation on $X$ and $\tau$ that is equivalent to

(4.5) $X\mathbin{\hbox{\vrule height=1.4pt,width=4.0pt,depth=-1.0pt\vrule
height=4.0pt,width=0.4pt,depth=-1.0pt}}\tau=0,$

i.e., the vanishing of a $1$-form that is bilinear in $X$ and $\tau$.

###  4.2. Non-involutivity

The condition $\mathrm{d}(X^{\flat})=0$ is expressed in terms of $\beta$ as

(4.6)
$\mathrm{d}\bigl{(}{\ast_{\sigma}}(\beta{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}{{\ast_{\sigma}}\sigma})\bigr{)}=0.$

Thus, it is natural to study the gradient case by augmenting the differential
ideal $\mathcal{I}_{\lambda}$ by adding to it the closed $2$-form
$\mathrm{d}\mathsf{z}$, where $\mathsf{z}$ is the $1$-form on $X$ defined as

$\mathsf{z}_{(s,b)}=\pi^{*}\bigl{(}{\ast_{s}}(b{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}{{\ast_{s}}s})\bigr{)}.$

Unfortunately, as (4.5) shows, this augmented differential ideal does not have
any admissible integral elements at any $(s,b)\in X$ for which the $6$-form
$\bigl{(}{\ast_{s}}(b{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}{{\ast_{s}}s})\bigr{)}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}b{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}s$
is nonzero. Hence, the augmented ideal is not involutive.

The standard ‘next step’ in such a case is to pull back the augmented
differential ideal to the (nonsmooth) subvariety $Z\subset X$ on which
$\mathsf{z}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\mathsf{b}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\mathsf{s}=0$
and check that exterior differential system on $Z$ for involutivity.
Unfortunately, calculation shows that that exterior differential system is not
involutive either, even on the smooth locus of $Z$.

Two successive applications of the standard prolongation procedure to this
system does not yield an involutive exterior differential system, and the
calculations become increasingly complicated. So far, no method has been found
that leads to an involutive system whose solutions correspond to the gradient
$\lambda$-solitons.

As a result, the generality of the gradient $\lambda$-solitons remains unknown
as of this writing. All we have is a collection of the various examples
already mentioned to show that gradient $\lambda$-solitons do, indeed, exist.

## References

  * [1] G. Ball, _Quadratic closed $\mathrm{G}_{2}$-structures_, arXiv:2006.14155
  * [2] R. Bryant, _Metrics with exceptional holonomy_ , Annals of Mathematics (2), 126 (1987), 525–576.
  * [3] by same author, _Some remarks on $\mathrm{G}_{2}$-structures_, in Proceedings of Gökova Geometry-Topology Conference 2005, edited by S. Akbulut, T. Onder, and R.J. Stern. Gökova Geometry/Topology Conference (GGT), Gökova, 2006. See also arXiv:math/0305124
  * [4] by same author, _Notes on exterior differential systems_ , arXiv:1405.3116.
  * [5] R. Bryant, S.-s. Chern, R. Gardner, H. Goldschmidt, and P. Griffiths, _Exterior Differential Systems_ , Mathematical Sciences Research Institute Publications 18 (1991).
  * [6] R. Bryant and F. Xu, _Laplacian flow for closed $\mathrm{G}_{2}$-structures: short time behavior_, arXiv:1101.2004
  * [7] M. Fernández and A. Gray, _Riemannian manifolds with structure group $\mathrm{G}_{2}$_, Ann. Mat. Pura Appl. (4) 132 (1982), 19–45.
  * [8] U. Fowdar, _$S^{1}$ -invariant Laplacian flow_, to appear in the Journal of Geometric Analysis, arXiv:2007.05130
  * [9] M. Haskins and J. Nordström, _Cohomogeneity-one solitons in Laplacian flow: local, smoothly-closing and steady solitons_ , arXiv:2112.09095
  * [10] N. Hitchin, _The geometry of three-forms in six and seven dimensions_ , arXiv:math/0010054.
  * [11] N. Hitchin, _Stable forms and special metrics_ , in Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000), 70–89, Contemp. Math. 288, Amer. Math. Soc., Providence, RI, 2001.
  * [12] D. Joyce, _Compact manifolds with special holonomy_ , Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000. MR 2001k:53093
  * [13] J. Lauret, _Laplacian flow of homogeneous $\mathrm{G}_{2}$-structures and its solitons_, Proc. London Math. Soc. 114 (2017), 527–560.
  * [14] J. Lotay, _Geometric flows of $\mathrm{G}_{2}$-structures_, To appear in a forthcoming volume of the Fields Institute Communications entitled Lectures and Surveys on G2 manifolds and related topics. Also, see arXiv:1810.13417.
  * [15] M. Taylor, _Regularity of solutions to nonlinear overdetermined elliptic PDE_ , https://mtaylor.web.unc.edu/notes, Item 5.15.


# Unraveling the Complexity of Splitting Sequential Data: Tackling Challenges
in Video and Time Series Analysis

Diego Botache  Kristina Dingel  Rico Huhnstock  Arno Ehresmann  Bernhard Sick

###### Abstract

Splitting of sequential data, such as videos and time series, is an essential
step in various data analysis tasks, including object tracking and anomaly
detection. However, splitting sequential data presents a variety of challenges
that can impact the accuracy and reliability of subsequent analyses. This
concept article examines the challenges associated with splitting sequential
data, including data acquisition, data representation, split ratio selection,
setting up quality criteria, and choosing suitable selection strategies. We
explore these challenges through two real-world examples: motor test benches
and particle tracking in liquids.

Machine Learning, ICML, train-test-split, data-centric machine learning,
sequential data, videos, time series, splitting,

##  1 Introduction

Sequential data, such as time series and video data, are crucial for technical
systems and require accurate modelling of time dependencies to predict future
behaviour and identify meaningful patterns. Understanding how variables change
over time can gain insights into underlying mechanisms driving the dynamic
behaviour of a particular system. Analysis of sequential data can furthermore
assist in detecting anomalies or identifying causal relationships. Although
one may use basic modelling strategies such as differential equations and
stochastic processes, computational limitations arise when dealing with high-
dimensional data. While modelling and analyzing sequential data is crucial in
multiple application areas (Barra et al., 2020; Martínez et al., 2022; Masini
et al., 2023), there is currently no default approach concerning splitting
sequential information for data-driven algorithms. Moreover, there is a lack
of research on evaluating the generalization ability of sequential models
trained for specific tasks and assessing the data quality used to train, e.g.,
machine learning (ML) models. The upcoming section will summarise the main
research challenges regarding modelling time dependencies in technical
systems, especially during the training phase of an ML cycle. We will then
frame these challenges in two use cases, namely real-world applications for
time series and video data analysis.

Figure 1: We present a path towards a comprehensive data splitting strategy
that covers all stages, from data acquisition/selection to finding a suitable
representation for machine learning. Our approach involves dividing the data
based on an appropriate split ratio and applying quality criteria tailored to
a specific use case.

##  2 Splitting Sequential Data: Challenges

The common 80% and 20% splitting strategy for non-sequential data to train and
test ML routines, inspired by the Pareto principle, is unsuitable for
modelling time dependencies as it lacks temporal information and dependencies.
One common modification to split sequential data is to use a temporal split,
where the first percentage of data, e.g., 80 %, is used as training data; the
remaining data is a test set. This way, one keeps the temporal order of the
data. This approach, however, may lack generalization of the overall process.
Cross-validation (Bergmeir & Benítez, 2012) can be applied to sequential data,
extracting multiple segments for a predefined number $k$ of groups, e.g.,
folds. This is particularly useful for hyperparameter optimisation considering
the average performance. Still, cross-validation methods assume that each
sequence in the dataset is independent and identically distributed (i.i.d),
which is not valid for correlated time series data. As a result, common cross-
validation methods can lead to overly optimistic estimates of the model’s
performance. The sliding window approach (Keogh et al., 2001) is another
standard method for segmenting sequential data for ML model training. However,
critical factors are selecting an optimal window size that captures relevant
information and determining the overlap between consecutive windows to avoid
missing transitions and patterns. Given the various options, the question of
how to choose the appropriate data selection method for ML training arises.
This article states four key aspects towards this goal (cf. Fig. 1). However,
it is vital to note that selecting a method depends on the specific task and
should be evaluated before addressing the following points. Additionally, we
assume that there are no data limitations and that new data can be obtained
through simulations or additional experiments. Lastly, we assume that the
labeling process has already been completed.

I. Data Acquisition / Selection: Prior to starting the ML process, the
experiment’s use cases, conditions, restrictions, and boundaries need to be
evaluated. The purpose is to establish a sufficient and diverse database
containing all relevant information either from experimentation or simulation.
This involves considering the system’s operational areas and design parameter
constraints in the experiment.

II. Representation: To prepare data for ML training (next to preprocessing, of
course), one must consider the most fitting representation for the desired
experiment. It is essential to evaluate if the parameters of this
representation, e.g., sequence length, features, and periodicity, are
sufficiently taken into account for training the model and accomplishing the
defined task.

III. Split Ratio: In general, all splits should contain all relevant scenarios
in the experiment to avoid overfitting. High data versatility is necessary to
prevent bias during training. Lastly, an isolated validation dataset is
recommended when tuning the model’s hyperparameters to improve generalization
performance.

IV. Quality Criteria: Their definition (in advance) is essential, depending on
the use case. Before placing an instance in a particular data split, tools
such as statistical tests and distance measures can be used to assess the
information gain or similarity of individual pre-selected samples. Addressing
changes in the environment, i.e. domain shifts, during experiments and
identifying new operation areas are necessary for updating the training data.
Task-dependent criteria for change point detection, novelty and anomaly
detection are critical for setting up a database which provides enough
information about specific scenarios. If it is decided that there are not
enough representatives for a scenario, new ones must be sampled/generated
considering further selection strategies, cf. I.

### Video Data Use Case: Particle Tracking in Liquids

Tracking in video data is a common challenge in computer vision tasks where
one wants to track an object through a series of images or, in this case,
extract the centre coordinates $(x,y,z)$ of particles moving in liquids
(Dingel et al., 2021a) to implement lab-on-a-chip technologies (Lim et al.,
2010). Unlike video data captured in the everyday world, additional factors in
microscopic video data can significantly affect tracking, i.e., depth of
focus, motion and blur, noise, artefacts, lighting, video binning, and
contrast. In addition to that, the same issues as with classical object
tracking tasks apply, such as missing labels and the difficulty of
incorporating simulation data due to the large gap between simulation and
reality (Dingel et al., 2021b). Further, it must be noted that video content
can vary significantly from experiment to experiment. But also, consecutive
video frames within one experiment may appear similar due to high frame rates
or long sequences without particle movement. It leads to imbalanced datasets,
and selecting relevant video sequences for training and testing is of utmost
importance (I, IV). Especially in the case of tracking, attention must be paid
to the sequence length (II) considering the possible shortage of storage space
during the recording of an experiment, which results in short video sequences
whose separation would not be ideal. This makes it all the more important to
consider an appropriate data representation (II) and split ratio (III) and
finally establish data quality criteria, which in turn will largely dictate
the selection strategy for extending the data splits. Finally, this particular
use case shows the benefit of having a closed-loop-experimentation station
(Sanchez-Lengeling & Aspuru-Guzik, 2018). It allows the creation of new
training data on the fly by actively (Kottke et al., 2021) asking for missing
or underrepresented data using the pre-defined quality criteria and
requirements (IV).

### Time Series Data Use Case: Motor Test Bench

This use case involves monitoring a motor test bench using deep-learning
techniques (Botache et al., 2021). The data is high-dimensional and
heterogeneous, obtained from multiple sensors capturing factors affecting
motor performance. An ML-supported monitoring strategy must consider
correlations and time dependencies to detect faults early on, improving the
experimental process. When designing fault detection as a binary
classification task, the data should contain faulty instances for each
operating area. Also, acquiring multiple measurements independently from each
other is critical to generate splits containing single experiments. Therefore,
the i.i.d. assumption should hold for each data split (I). Extracting segments
of the same length is the first step for creating the input space for ML
techniques (II). Still, long sequences represent higher model complexity for a
constant sampling rate and shorter sequences, on the other side, may not
capture enough time dependencies in the data. Next, extracting significant
features could reduce the complexity of the task, e.g., by removing
categorical signals and employing representation learning techniques. In a
binary classification task, fault instances are usually rare, challenging to
acquire, and therefore underrepresented in data splits. Oversampling
techniques (Kovács, 2019), active-learning with class-balance selection (Cai,
2022), and synthetic data generation are highly recommended for improving the
split ratio of the datasets (Westmeier et al., 2022) (III). In addition, using
weighted losses (Lin et al., 2017) can improve the learning task of the
algorithms. Finally, the quality assessment of the data split using similarity
measurements is necessary for identifying potentially unexploited areas in the
data distribution (IV).

##  3 Conclusion

Especially with respect to ML training processes, extracting appropriate
training and test splits for time-dependent data is still a major challenge.
In this concept paper, we identified four key research questions and put them
in concrete terms using two use cases for time series and video data, which
now require further investigation.

## Acknowledgements

We gratefully acknowledge the assistance and support of the Joint Laboratory
Artificial Intelligence Methods for Experiment Design (AIM-ED) between
Helmholtzzentrum für Materialien und Energie, Berlin, and the University of
Kassel.

## References

  * Barra et al. (2020) Barra, S., Carta, S. M., Corriga, A., Podda, A. S., and Recupero, D. R.  Deep learning and time series-to-image encoding for financial forecasting.  _IEEE/CAA Journal of Automatica Sinica_ , 7(3):683–692, May 2020.
  * Bergmeir & Benítez (2012) Bergmeir, C. and Benítez, J. M.  On the use of cross-validation for time series predictor evaluation.  _Information Sciences_ , 191:192–213, May 2012.
  * Botache et al. (2021) Botache, D., Bethke, F., Hardieck, M., Bieshaar, M., Brabetz, L., Ayeb, M., Zipf, P., and Sick, B.  Towards Highly Automated Machine-Learning-Empowered Monitoring of Motor Test Stands.  _Proceedings - 2021 IEEE International Conference on Autonomic Computing and Self-Organizing Systems, ACSOS 2021_ , pp. 120–130, 2021.
  * Cai (2022) Cai, X.  Active learning for imbalanced data: The difficulty and proportions of class matter.  _Wireless Communications and Mobile Computing_ , 2022, 2022.
  * Dingel et al. (2021a) Dingel, K., Huhnstock, R., Knie, A., Ehresmann, A., and Sick, B.  AdaPT: Adaptable Particle Tracking for spherical microparticles in lab on chip systems.  _Computer Physics Communications_ , 262:107859, May 2021a.
  * Dingel et al. (2021b) Dingel, K., Liehr, A., Vogel, M., Degener, S., Meier, D., Niendorf, T., Ehresmann, A., and Sick, B.  AI - Based on the Fly Design of Experiments in Physics and Engineering.  _Proceedings - 2021 IEEE International Conference on Autonomic Computing and Self-Organizing Systems Companion, ACSOS-C 2021_ , pp. 150–153, 2021b.
  * Keogh et al. (2001) Keogh, E., Chu, S., Hart, D., and Pazzani, M.  An online algorithm for segmenting time series.  In _Proceedings 2001 IEEE International Conference on Data Mining_ , pp. 289–296, November 2001.
  * Kottke et al. (2021) Kottke, D., Herde, M., Sandrock, C., Huseljic, D., Krempl, G., and Sick, B.  Toward optimal probabilistic active learning using a bayesian approach.  _Machine Learning 2021 110:6_ , 110:1199–1231, May 2021\.
  * Kovács (2019) Kovács, G.  An empirical comparison and evaluation of minority oversampling techniques on a large number of imbalanced datasets.  _Applied Soft Computing_ , 83:105662, 2019.
  * Lim et al. (2010) Lim, Y. C., Kouzani, A. Z., and Duan, W.  Lab-on-a-chip: a component view.  _Microsystem Technologies 2010 16:12_ , 16(12):1995–2015, Sep 2010.
  * Lin et al. (2017) Lin, T.-Y., Goyal, P., Girshick, R., He, K., and Dollár, P.  Focal loss for dense object detection.  In _Proceedings of the IEEE international conference on computer vision_ , pp. 2980–2988, 2017.
  * Martínez et al. (2022) Martínez, F., Frías, M. P., Pérez-Godoy, M. D., and Rivera, A. J.  Time Series Forecasting by Generalized Regression Neural Networks Trained With Multiple Series.  _IEEE Access_ , 10:3275–3283, 2022.
  * Masini et al. (2023) Masini, R. P., Medeiros, M. C., and Mendes, E. F.  Machine learning advances for time series forecasting.  _Journal of Economic Surveys_ , 37(1):76–111, 2023.
  * Sanchez-Lengeling & Aspuru-Guzik (2018) Sanchez-Lengeling, B. and Aspuru-Guzik, A.  Inverse molecular design using machine learning: Generative models for matter engineering.  _Science_ , 361(6400):360–365, Jul 2018.
  * Westmeier et al. (2022) Westmeier, T., Botache, D., Bieshaar, M., and Sick, B.  Generating Synthetic Time Series for Machine-Learning-Empowered Monitoring of Electric Motor Test Benches.  _Proceedings - 2022 IEEE 9th International Conference on Data Science and Advanced Analytics_ , 2022.