# On Frobenius structures in symmetric cones

Noemie Combe

###### Abstract.

We prove that in any strictly convex symmetric cone $\Omega$ there exists a
non empty locus where the WDVV equation is satisfied (i.e. there exists a
hyperplane being a Frobenius manifold). This result holds over any real
division algebra (with a restriction to the rank 3 case if we consider the
field $\mathbb{O}$) but also on their linear combinations. This theorem holds
as well in the case of pseudo-Riemannian geometry, in particular for a Lorentz
symmetric cone of Anti-de-Sitter type. Our statement can be considered as a
generalisation of a result by Ferapontov–Kruglikov–Novikov and Mokhov. Our
construction is achieved by merging two different approaches: an
algebraic/geometric one and the analytic approach given by Calabi in his
investigations on the Monge–Ampère equation for the case of affine
hyperspheres.

###### Contents

  1. 1 Introduction
  2. 2 Strictly convex symmetric cones
  3. 3 Hessian structures
  4. 4 Algebraic structures
  5. 5 Frobenius structures
  6. 6 The WDVV equation and symmetric cones
  7. 7 Conclusion and future work

##  1\. Introduction

###  1.1. The main problem and result

Consider a strictly convex symmetric cone in the Euclidean space. Does there
exist a relation between strictly convex symmetric cones and the
Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) highly nonlinear PDE equation (also
known as forming Frobenius manifolds)? The answer to this question is yes. In
this paper, we prove that in a strictly convex symmetric cone there exists a
non empty locus (a hyperplane) where the WDVV equation is satisfied. This
result holds over any division algebra (as well as their linear combinations).
In addition, this holds also for symmetric Lorentz cones of an Anti-de-Sitter
type. We separate the latter from the first class of cones due to the
difference in the geometries that they carry. The construction of the proof
allows a deep exposition on the geometrical and algebraic aspects occurring
around those types of manifolds and answers questions raised by Yu. I. Manin,
related to our joint works.

Our statement can be considered as a generalisation of a construction by
Ferapontov–Kruglikov–Novikov [FKN21] and Mokhov [Mok95, Mok09] proving the
existence of connections between the symplectic Monge–Ampère equations of
Hirota type and the WDVV equation. The construction is achieved by merging two
different approaches: an algebraic/geometric one and the analytic approach
given by Calabi in his investigations on the Monge–Ampère equation for the
case of affine hyperspheres.

For a documentation on the importance of the WDVV equation in algebraic
geometry we refer to [Du96, KM94, KKP08, Man98, Man99, Man05, Sa07]. Note that
a manifold satisfying the WDVV equation corresponds to a Frobenius manifold
for algebraic geometers. For a more differential geometry flavoured approach,
relating the WDVV equation and integrable systems see for instance [FM96,
Mok08, Mok09].

###  1.2. Motivation

The reason for our question takes root in a series of differential geometry
problems. In particular discussions with M. Kontsevich [Kont22] at the IHES
during my stay Nov-Dec (2022) and discussions with Yu. I. Manin that have
followed from our joint works [CM20, CMM22A, CMM22B] have inspired this paper.
One of the differential geometry problems related to what we consider is the
problem of classifying flat Lagrangian submanifolds in $\mathbb{R}^{2n}$ with
a given pseudo-Riemannian metric. From the Hessian geometry perspective,
having flat Hessian metrics corresponds exactly to satisfying the WDVV
equation (see [Kito99]). Finally, this problem is also connected to problems
gravitating around manifolds of constant curvature ([To04]).

###  1.3. The realm of strictly convex symmetric cones: 1935–now

Strictly convex symmetric cones play a central role in many different domains.
They appeared at first in the works of Cartan [Car35], Koszul [Kos59, Kos61,
Kos62, Kos65, Kos68A, Kos68B] and Vinberg [Vi60, Vi63, Vi65]. From the works
of Minkowski, Siegel [Sie35, Sie44], Maas [Ma71], Piateski–Shapiro [PS61] (and
many others) those cones are important in number theory. In algebraic
geometry, symmetric convex cones appear under the shape of ”cone of Kähler
classes” for the case of an $n$-dimensional complex torus [Wil04, To04].
Throughout the works of Wishart [Wi28], Constantine [Co63, Co66], James
[Ja61], Muirhead [CM72, CM76], symmetric cones shine in statistics and in
harmonic analysis [FK94]. More recent developments show links towards
information geometry [AW04, CM20, CMM22A, CMM22B]. Their importance is
revealed also on a more applied side of mathematics, such as in optimization
via convex programming and machine learning.

###  1.4. The rich geometry and algebra of symmetric cones

Behind the notion of strictly convex symmetric cones hides a rich geometric
and algebraic world. Our proof is the fruit of merging two different
languages, which intersect at the notion of those strictly convex symmetric
cones. First we use a differential geometry aspect. This comes from the
relations between those cones and the Monge–Ampère equation (showed in Lem.
2). The more algebraic approach is inherited from Cartan’s classification of
symmetric spaces.

Recall the bridge between strictly convex symmetric cones and a special case
of the Monge–Ampere equation. For $x\in\mathbb{R}^{n}$ and $\phi(x)$ a smooth
function, this equation is:

(1) $\det\mathrm{Hess}(\phi)=k,$

where $k$ is a constant and $\mathrm{Hess}(\phi)$ is the Hessian of $\phi$.

Following [Cal58], Eq. 1 has at most one convex solution in a bounded strictly
convex domain, if $\phi$ has prescribed boundary values. Interest is given to
tensors defined by the second and third derivatives of $\phi$. Given $\phi$ a
smooth function on an open subset of a real vector space, it is possible to
define an associated Hessian metric. This metric is obtained by taking the
second derivatives of $\phi$. Hessian metrics are a natural way to construct
Riemannian or pseudo-Riemannian metrics.

Using Cartan’s classification of symmetric spaces those cones are expressed
using a more algebraic flavour. It turns out that those strictly convex cones
fall into two main classes. The first class corresponds to so-called
Lagrangian Grassmannians having nonpositive sectional curvature. The second
class are formed by the Lorentz cones (of anti-de-Sitter type).

###  1.5. A proof giving a new perspective

By merging the geometric and algebraic viewpoints discussed above, we create a
method which allows to prove our result. This new method gives additionally a
very detailed description of the geometric and algebraic landscape occurring
in this topic. For instance, using the language of Lie groups, we reformulate
one part of our statement by saying that: in a noncompact
Lagrangian–Grassmanian symmetric space there exist hypersurfaces satisfying
the axioms of a Frobenius manifold. Those spaces are defined over a division
algebra $\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$ (or a linear combination
of those algebra). Reciprocally, there exist hypersurfaces satisfying the
axioms of a Frobenius manifold within a Lorentz symmetric cone (of anti-de-
Sitter type).

###  1.6. Connections to other works

[FKN21, Mok95] shows that in the real case, using a reformulation of the
dispersionless Hirota type of equation (in the context of the symplectic
Monge–Ampere equation) one can prove the existence of a hyperplane such that
the WDVV equation is satisfied. In the real case, this statement is linked to
our result since it is a confirmation that our statement is true. Since our
result holds over any division algebra (and their linear combinations) as well
as for Lorentz Anti-de-Sitter (AdS) symmetric cones, our result gives a
general statement. Finally, we point out that the method chosen to prove our
result pictures the rich geometric and algebraic patterns, hiding within those
symmetric cones and bridges two different ways of seeing them.

###  1.7. Organisation of the paper

This paper is organised as follows: in the three first sections we expose the
geometric and algebraic state of the art concerning the strictly convex
symmetric cones. It is necessary to recall some known results for the proof.
The the last two sections are devoted to the construction the proof of the
main statement.

Acknowledgments I would like to thank M. Kontsevich for many discussions and
comments during my stays (spring and winter 2022) at the IHES. Both Max Planck
Institutes for Mathematics in Bonn (MPIM) and in Leipzig (MPIMIS) are very
much acknowledged for having supported my research. As well, I am grateful to
the grant Polonez-bis 3 for supporting my research.

##  2\. Strictly convex symmetric cones

This part serves, as a first focus, on what is known of the geometric aspect
of those strictly symmetric convex cones. The second part will bring a
(complementary) algebraic version.

###  2.1. Strictly convex cones

In the following parts of this article we always consider strictly convex
cones. Note that for brevity we simply refer to them as convex cones.

Let us recall some elementary notions on strictly convex cones (see [FK94] for
further information).

######  Definition 1.

Let $V$ be a finite dimensional real vector space. Let $\langle-,-\rangle$ be
a non-singular symmetric bilinear form on $V$. A subset $\Omega\subset V$ is a
convex cone if and only if $x,y\in\Omega$ and $\lambda,\mu>0$ imply $\lambda
x+\mu y\in\Omega$.

###  2.2. Homogeneous cones

The automorphism group $G(\Omega)$ of an open convex cone $\Omega$ is defined
by

$G(\Omega)=\\{g\in GL(V)\,|\,g\Omega=\Omega\\}$

An element $g\in GL(V)$ belongs to $G(\Omega)$ iff
$g\overline{\Omega}=\overline{\Omega}$ [Fauraut Koranyi] So, $G(\Omega)$ is a
closed subgroup of $GL(V)$ and forms a Lie group. The cone $\Omega$ is said to
be homogeneous if $G(\Omega)$ acts transitively upon $\Omega$.

###  2.3. Symmetric cones

From homogeneous cones one can construct symmetric convex cones. Let us
introduce the definition of an open dual cone. An open dual cone $\Omega^{*}$
of an open convex cone is defined by $\Omega^{*}=\\{y\in V\,|\,\langle
x,y\rangle>0,\,\forall\,x\in\overline{\Omega}\setminus 0\\}$. A homogeneous
convex cone $\Omega$ is symmetric if $\Omega$ is self-dual i.e.
$\Omega^{*}=\Omega$. Note that if $\Omega$ is homogeneous then so is
$\Omega^{*}$. A symmetric homogeneous cone is called a Vinberg cone.

######  Remark 1.

If $\Omega$ is a symmetric open cone in $V$, then $\Omega$ is a symmetric
Riemann space.

###  2.4. Automorphism group

Let us go back to the automorphism group of $\Omega$. This discussion relies
on Prop I.1.8 and Prop. I.1.9 in [FK94].

Let $\Omega$ be a symmetric cone in $V$. For any point $a\in\Omega$ the
stabilizer of $a$ in $G(\Omega)$ is given by

$G_{a}=\\{g\in G(\Omega)\,|\,ga=a\\}.$

By [Prop I.1.8 [FK94] ], if $\Omega$ is a proper open homogeneous convex cone
then for any $a$ in $\Omega$, $G_{a}$ is compact. Now, if $H$ is a compact
subgroup of $G$ then $H\subset G_{a}$ for some $a$ in $\Omega$. This means
that the groups $G_{a}$ are all maximal compact subgroups of $G$ and that if
$\Omega$ is homogeneous then all these subgroups are isomorphic.

By [Prop. I.1.9, [FK94]], if $\Omega$ is a symmetric cone, there exist points
$e$ in $\Omega$ such that $G(\Omega)\cap O(V)\subset G_{e}$, where $O(V)$ is
the orthogonal group of $V$. For every such $e$ one has $G_{e}=G\cap O(V)$

Suppose $\Omega$ is a convex homogeneous domain in $V$. Assume that

  * —

$G(\Omega)$ is the group of all automorphisms;

  * —

$G_{e}=K(\Omega)$ is the stability subgroup for some point $x_{0}\in\Omega$;

  * —

$T(\Omega)$ is a maximal connected triangular subgroup of $G(\Omega).$

Following [[Vi63] Th 1] we have:

$G(\Omega)=K(\Omega)\cdot T(\Omega),$

where $K(\Omega)\cap T(\Omega)=e$ and the group $T$ acts simply transitively.

This decomposition on the Lie group side leads naturally to its Lie algebra.
Cartan’s decomposition for the Lie algebra tells us that
$\mathfrak{g}=\textgoth{k}\oplus\mathfrak{t},$

where:

  * —

$\mathfrak{t}$ can be identified with the tangent space of $\Omega$ at $e$.

  * —

$\textgoth{k}$ is the Lie algebra associated to $K(\Omega)$

and

$[\mathfrak{t},\mathfrak{t}]\subset\textgoth{k},$
$[\textgoth{k},\mathfrak{t}]\subset\mathfrak{t}.$

From now assume that $G$ is semi-simple. The Killing bilinear form is thus not
degenerate on $\mathfrak{g}$ and the symmetric bilinear form is given by:

$\langle X,Y\rangle=-Tr(adX\,adY)$

where $ad\,X(\xi)=[X,\xi]$ and $ad\,Y(\xi)=[Y,\xi]$.

###  2.5. $\mathbb{K}-$modules

Throughout the paper let $\mathbb{K}$ be a finite dimensional real division
algebra. By the well known Kervaire–Milnor there exist only four such division
algebras which are isomorphic to $\mathbb{R},\mathbb{C},\mathbb{H},$ and
$\mathbb{O}$ of respective dimensions 1,2, 4 and 8.

  * —

If the finite dimensional real division algebra $\mathbb{K}$ is unitary and
commutative then it is isomorphic to $\mathbb{R}$ and $\mathbb{C}$.

  * —

If the real division algebra is noncommutative but associative then
$\mathbb{K}$ is isomorphic to $\mathbb{H}$.

  * —

If $\mathbb{K}$ is non associative but alternative then $\mathbb{K}$ is
isomorphic to $\mathbb{O}$.

We discuss briefly the realisation of the $\mathbb{K}$-modules in terms of
linear spaces. If $\mathbb{K}$ is a field of dimension $n$, where
$n\in\\{1,2,4,8\\}$ the (real) realisation of the $\mathbb{K}$-module defines
a (real) vector space of dimension $nm$ where $m$ is the dimension of the
module [Ro97], Sec. 2.1.1. Via the Cayley–Dickson process and starting from
$\mathbb{R}$, remark that all normed division algebras can be obtained. This
means that from $\mathbb{R}$ we get $\mathbb{C}$; from $\mathbb{C}$ we get
$\mathbb{H}$ and finally from $\mathbb{H}$ we get $\mathbb{O}$. Taking the
realisation of $\mathbb{K}$-modules as real vector spaces $V$ then it leads
to:

  1. (1)

the complexification: $V\to(TV,J)$, where $J^{2}=-Id$.

  2. (2)

the symplectification: $V\to(T^{*}V,\omega),$ where $\omega$ is a non-
degenerate closed form.

See [Sh87, Sh02] for further developments concerning the geometry of the real
realisation of $\mathbb{K}$-modules, namely concerning affinors.

###  2.6. Classification of cones

Any symmetric cone (i.e. homogeneous and self-dual) $\Omega$ is in a unique
way isomorphic to the direct product of irreducible symmetric cones
$\Omega_{i}$ (cf. Prop. III.4.5, [FK94]). According to Vinberg we have that:

######  Proposition 1.

Each irreducible homogeneous self–dual cone belongs to one of the following
classes:

Nb | Symbol | Irreducible symmetric cones
---|---|---
1. | $\Pi_{n}(\mathbb{R})$ |  Cone of $n\times n$ positive definite symmetric real matrices.
2. | $\Pi_{n}(\mathbb{C})$ |  Cone of $n\times n$ positive definite self-adjoint complex matrices.
3. | $\Pi_{n}(\mathbb{H})$ |  Cone of $n\times n$ positive definite self-adjoint quaternionic matrices.
4. | $\Pi_{3}(\mathbb{O})$ |  Cone of $3\times 3$ positive definite self-adjoint octavic matrices.
5. | $\Lambda_{n}$ |  Lorentz cone given by $x_{0}>\sqrt{\sum_{i=1}^{n}x_{i}^{2}}$ (aka spherical cone).
Table 1. Classification of irreducible symmetric cones

######  Remark 2.

We have two remarks. The first is that $\Pi_{3}(\mathbb{O})$ corresponds to
the Cayley algebra. The second is that spherical cone $\Lambda_{n}$
corresponds here to an $n$-dimensional Anti-de-Sitter (AdS) space.

###  2.7. Jordan algebra structures

Recall the tight relations between those cones and algebraic objects, namely
formally real simple Jordan algebras. We introduce some notations:

  * —

$Sym(n,\mathbb{K})$ denotes the space of symmetric matrices of dimension
$n\times n$ defined over the field $\mathbb{K}$.

  * —

$Herm(n,\mathbb{K})$ denotes the space of hermitian matrices of dimension
$n\times n$ defined over the field $\mathbb{K}$.

######  Lemma 1.

Consider the cones given by $\Pi_{n}(\mathbb{K})$, where the field
$\mathbb{K}$ is $\mathbb{R},\mathbb{C},\mathbb{H}$ or $\mathbb{O}$. Then, the
tangent space $T_{e}(\Pi_{n}(\mathbb{K}))$ to $\Pi_{n}(\mathbb{K})$ is the
space of self-adjoint matrices.

###### Proof.

Take the symmetric cone $\Omega$ of symmetric positive definite matrices
$n\times n$ over $\mathbb{R}$. The tangent space to $\Omega$ at $x\in V$ is
the space of symmetric matrices with real entries. The matrix exponential and
logarithm realize a one-to-one mapping, between the space of symmetric
matrices to the space of symmetric positive definite matrices. The same
reasoning can be made for the cones over the other division algebras. ∎

######  Definition 2.

An algebra $(\mathscr{A}^{+},\circ)$ is a Jordan algebra if:

  * —

it is commutative and

  * —

$(x^{2}\circ y)\circ x=x^{2}\circ(y\circ x)$,

where we have $x^{2}=x\circ x$ and $x,y\in\mathscr{A}^{+}.$

A Jordan algebra is called Euclidean (or formally real) if it satisfies the
formal reality axiom:

$x_{1}^{2}+\cdots+x_{n}^{2}=0\quad\text{implies}\quad x_{1}=\cdots=x_{n}=0$

In particular, a formally real Jordan algebra is semi-simple ([Koe99] Cor. 5
p.118). To any Vinberg cone there exists a bijectively corresponding
semisimple formally real Jordan algebra.

######  Example 1.

By Lem. 1 the tangent space to $\Pi_{n}(\mathbb{R})$, is $Sym(n,\mathbb{R})$.
Given $X,Y\in Sym(n,\mathbb{R})$, one gets a new product (giving a Jordan
algebra):

$X\circ Y=\frac{1}{2}(XY+YX),$

where $XY$ is the standard matrix product.

######  Proposition 2.

Every formally real Jordan algebra can be written as a direct sum of simple
ones. In finite dimensions, the simple formally real Jordan algebras come in
four infinite families, together with one exceptional case:

Irreducible symmetric cone | Formally real simpl Jordan algebras
---|---
$\Pi_{n}(\mathbb{R})$ |  Jordan algebra of $n\times n$ self-adjoint real matrices.
$\Pi_{n}(\mathbb{C})$ |  Jordan algebra of $n\times n$ self-adjoint complex matrices.
$\Pi_{n}(\mathbb{H})$ |  Jordan algebra of $n\times n$ self-adjoint quaternionic matrices.
$\Pi_{3}(\mathbb{O})$ |  Jordan algebra of $3\times 3$ self-adjoint octonionic matrices:
| Albert algebra.
$\Lambda_{n}$ |  Spin factor algebra $JSpin^{+}$ on the space $\mathbb{R}1\oplus\mathbb{R}^{n}$ for $n\geq 2$.
Table 2. Classification of symmetric cones using their Jordan algebras

######  Remark 3.

Note that one can consider the algebra $JSpin_{n}^{+}$ as a Jordan subalgbera
of a full algebra of hermitian matrices $2^{n}\times 2^{n}$ real matrices.

###  2.8. Cartan’s symmetric spaces

Vinberg cones are Cartan symmetric spaces. In Sec. LABEL:S:HessianMfd we study
their properties as Riemannian symmetric spaces. Recall that a
(pseudo-)Riemannian manifold is globally symmetric if one can assign to every
point $p\in\mathcal{M}$ an isometry $s_{p}$ of $\mathcal{M}$ such that
$s^{2}_{p}=id$ and $p$ is an isolated fixed point of $s_{p}$.

A Riemannian symmetric space $\mathcal{M}$ is diffeomorphic to a homogeneous
space $G/K$, where $G$ is a connected Lie group with an involutive
automorphism whose fixed point set is essentially the compact subgroup
$K\subset G.$ Consider the pair $(G,K)$. We call $(G,K)$ a symmetric pair
provided that there exists an involution $s\in G$, such that
$(K_{s})_{0}\subset K\subset K_{s}$, where $K_{s}$ is the set of fixed points
of $s$ and $(K_{s})_{0}$ is the identity component of $K_{s}$. Riemannian
symmetric spaces may be classified in terms of symmetric Lie algebras. Every
noncompact symmetric space has a compact dual (and reciprocally).

Any simply connected Riemannian symmetric space is a Riemannian product of
irreducible ones. Irreducible, simply connected Riemannian symmetric spaces
are classified as follows:

  * —

Type of irreducible symmetric space: Euclidean. The curvature is 0. It is
therefore isometric to a Euclidean space.

  * —

Type of irreducible symmetric space: Compact. The sectional curvature is
nonnegative (but not identically zero).

  * —

Type of irreducible symmetric space: non-compact.The sectional curvature is
nonpositive (but not identically zero).

######  Definition 3.

  1. (1)

The following non-compact symmetric spaces are called spacelike Lagrangian
Grassmannians non-compact symmetric spaces:

$\mathbb{A}/\mathbb{B}$ | $\mathbb{R}$ | $\mathbb{C}$ | $\mathbb{H}$ | $\mathbb{O}$
---|---|---|---|---
$\mathbb{C}$ | $\frac{SL_{n}(\mathbb{R})}{SO(n)}$ | $\frac{SL_{n}(\mathbb{C})}{SU(n)}$ | $\frac{SL_{n}(\mathbb{H})}{Sp(n)}$ | $\frac{E_{6(-26})}{F_{4}}$
Table 3. Spacelike Lagrangian Grassmannian symmetric spaces

  2. (2)

We call a space of de Sitter type a submanifold of the Minkowski space
$\mathbb{R}^{n,1}$, with metric $ds^{2}=dx_{0}^{2}-\sum_{i=1}^{n-1}dx_{i}^{2}$
such that the isometry group is a Lorentz group $O(1,n)$. The scalar curvature
is positive.

  3. (3)

A space of Anti de Sitter type is a submanifold as in (2) with negative scalar
curvature .

######  Proposition 3.

— Let $\mathbb{K}$ be a real division algebra. The irreducible Vinberg cones
$\Pi_{n}(\mathbb{K})$ are noncompact symmetric spaces, which are of spacelike
Lagrangian Grassmannian type.

— The irreducible Vinberg cone $\Lambda_{n}$ is a pseudo-Riemannian noncompact
symmetric space, which is a Lorentz manifold (of Anti-De-Sitter type).

###### Proof.

1\. Thm. 1, implies that the first four Vinberg cones are identified to spaces
of symmetric positive definite matrices $n\times n$ over $\mathbb{K}$, where
$\mathbb{K}$ is $\mathbb{R},\mathbb{C},\mathbb{H}$ or $\mathbb{O}$ (concerning
$\mathbb{O}$ the cone holds only for $n=3$). Those cones are symmetric spaces,
so diffeomorphic to a quotient of Lie groups $G/K$ such that:

— $G$ is a connected Lie group

— $K$ is a Lie subgroup which is (a connected component of) the invariant
group of an involution of $G$.

For the class of Vinberg cones $\Omega=\Pi_{n}(\mathbb{K})$, one has the
following associated Lie groups:

  * —

$G$ is identified to $GL_{n}(\mathbb{K})$, where $\mathbb{K}$ is $\mathbb{R}$
(resp. $\mathbb{C},\mathbb{H},\mathbb{O}$).

  * —

The subgroup $K$ is identified to $SO_{n}$ (resp. $SU_{n}$, $Sp_{n}$, where
$Sp_{n}=SU_{n}(\mathbb{H})$, and the exceptional Lie group $F_{4}$, whenever
$\mathbb{K}$ is equal to $\mathbb{O}$).

2\. There exists a short exact sequence:

$1\to SL_{n}(\mathbb{K})\to
GL_{n}(\mathbb{K})\xrightarrow[]{det}\mathbb{K}^{\times}\to 1,$

implying that we have the following decomposition
$GL_{n}(\mathbb{K})=SL_{n}(\mathbb{K})\rtimes\mathbb{K}^{\times}$, where
$\mathbb{K}^{\times}$ is the multiplicative group. One may thus consider
$SL_{n}(\mathbb{K})/K\rtimes\mathbb{K}^{\times}$ and for simplicity we focus
on the submanifold $SL_{n}(\mathbb{K})/K$.

Those submanifolds are classified as follows:

  * —

$\Pi_{n}(\mathbb{C})$, with $\det=1$ is associated to
$SL_{n}(\mathbb{C})/SU_{n}$;

  * —

$\Pi_{n}(\mathbb{H})$, with $\det=1$ is associated to
$SL_{n}(\mathbb{H})/Sp_{n};$

  * —

$\Pi_{3}(\mathbb{O})$, with $\det=1$ is associated to
$SL_{3}(\mathbb{O})/F_{4}$.

Concerning the latter, $\Pi_{3}(\mathbb{O})$ note that $Herm(3,\mathbb{O})$ is
a closed subgroup of the orthogonal group $O(27)$ and thus forms a compact Lie
group. It is referred to as $F_{4}$.

3\. The spherical cone $\Lambda_{n}$ is an anti-de-Sitter space. In terms of
Lorentzian symmetric spaces it is associated to:
$O(1,n-1)/O(n-1)\oplus\mathbb{R}$ (see [FK94] p.7 for a detailed proof).

4\. The full classification of Vinberg cones in terms of Lie groups and Jordan
algebras is presented in the table below (see for details [FK94], Ch.V, p.97
Sec.3). Te Jordan algebras are listed in the leftmost part of the table.

$\mathcal{J}$ | $\Omega$ | $\mathfrak{g}$ | $\mathfrak{k}$ | $dim\mathcal{J}$ | rank $\mathcal{J}$ | $d$
---|---|---|---|---|---|---
$Sym(n,\mathbb{R})$ | $\Pi_{n}(\mathbb{R})$ | $\mathfrak{sl}(n,\mathbb{R})\oplus\mathbb{R}$ |  $\mathfrak{o}$(n) | $\frac{1}{2}n(n+1)$ | $n$ | 1
$Herm(n,\mathbb{C})$ | $\Pi_{n}(\mathbb{C})$ | $\mathfrak{sl}(n,\mathbb{C})\oplus\mathbb{R}$ |  $\mathfrak{su}$(n) | $n^{2}$ | $n$ | 2
$Herm(n,\mathbb{H})$ | $\Pi_{n}(\mathbb{H})$ | $\mathfrak{sl}(m,\mathbb{H})\oplus\mathbb{R}$ |  $\mathfrak{su}(n,\mathbb{H}$) | $n(2n-1)$ | $n$ | 4
$\mathbb{R}\times\mathbb{R}^{n-1}$ | $\Lambda_{n}$ | $\mathfrak{o}(1,n-1)\oplus\mathbb{R}$ |  $\mathfrak{o}$(n-1) | $n$ | 2 | $n-2$
$Herm(3,\mathbb{O})$ | $\Pi_{3}(\mathbb{O})$ | $\mathfrak{e}_{(-26)}\oplus\mathbb{R}$ | $\mathfrak{f}_{4}$ | 27 | 3 | 8

5\. By Nomizu [No54], an irreducible symmetric space $G/H$ is either flat,
compact or noncompact. It is easy to check that the symmetric spaces
considered above have non-positive scalar curvature. To sumarize:

  * —

Cones given by $GL_{n}(\mathbb{K})/K$ are noncompact symmetric spaces.

  * —

The spherical cone $\Lambda_{n}$ corresponds to
$O(1,n-1)/O(n-1)\oplus\mathbb{R}$ and belongs also to the class of noncompact
symmetric space.

To conclude, Vinberg cones (1)–(4) correspond to noncompact symmetric spaces
of spacelike Lagrangian–Grassmanian type; Vinberg cones of type (5) are
Lorentz cones of Anti-de-Sitter type. ∎

######  Proposition 4.

Vinberg cones of space-like Lagrangian–Grassmannian type come equipped with a
$G$-invariant metric and with a symmetric bilinear form given by

$\langle X,Y\rangle=\Re\,Tr(XY),$

where $X,Y\in T_{e}(Gl_{n}(\mathbb{K})/K)\cong\mathfrak{t}\subset\mathfrak{g}$
and where $Tr(\cdot)$ stands for the trace (linear) operator .

###### Proof.

To the Lie group $Sym(n,\mathbb{R})$ the corresponding Lie algebra is given by
the set of symmetric matrices of trace 0, denoted $\mathfrak{s}ym_{0}(n)$.This
statement follows from the Killing form. ∎

##  3\. Hessian structures

We proceed to a differential geometric approach of cones. This allows to
inherit a rich panel of tools, important for our construction.

###  3.1. The Monge–Ampere equation

Consider the non-linear PDE equation Monge–Ampere equation.

(2) $\det\mathrm{Hess}(\phi)=f(x),$

where $\mathrm{Hess}(\phi)$ denotes the Hessian matrix attached to the smooth
function $\phi$ and $f(x)$ is a given positive-valued function with
$x=(x_{1},\cdots,x_{n})\in\mathbb{R}^{n}$.

A special case of the Equ. 2 can be given by:

(3) $\det\mathrm{Hess}(\phi)=1.$

Solutions to this nonlinear PDE equation (such that the hessian matrix is
definite at each point) are so-called elliptic. Note that $\phi$ is locally
either convex or concave.

######  Lemma 2.

In a strictly convex symmetric cone $\Omega$, the equation Eq. 2 has at most
one convex solution $\phi$ with arbitrary smooth boundary values.

###### Proof.

It has been shown that Eq.2 has at most one convex solution $\phi$ (with
arbitrary smooth boundary values) in any strictly convex bounded domain
[Cal58]. Given that the strictly convex symmetric cone is also a strictly
convex bounded domain. This concludes the statement. ∎

Eq. 1 is invariant under (unimodular) linear transformation of the independent
variables $x=(x_{1},\cdots,x_{n})\in\mathbb{R}^{n}$. If we allow only linear
transformations of $x$, the array of all partial derivatives of $\phi$ of any
given order $k$ can be interpreted as the components of a covariant tensor of
valence $k$ which is symmetric in all pairs of indices.

Our interest goes to tensors defined by the second and third derivatives.
Since we assume that $\phi$ is convex and satisfies 2, then the symmetric
tensor $g_{ij}$ is positive definite. It thus definesa Riemannian and also
Hessian metric, in the domain of definition of $\phi$ .

###  3.2. The potential function

We introduce a function $\chi(x):\overline{\Omega}\to\mathbb{R}$ such that:

  1. (1)

$\chi(x)$ is real analytic and positive on $\Omega$

  2. (2)

$\chi(x)$ is continuous on $\overline{\Omega}$ and vanishes on the boundary
$\partial\Omega$.

  3. (3)

$\chi(\lambda x)=\lambda^{n}\chi(x)$ for $\lambda>0$, $x\in\Omega$ and where
$n$ is the dimension of $\Omega$.

  4. (4)

the bilinear form on $\Omega$ is non singular for every $y\in\Omega$. Fixing a
point $c\in\Omega$ we have:

$\langle u,v\rangle=-\partial_{u}\partial_{v}\log\chi(y)|_{y=c}$

######  Definition 4 (Koszul–Vinberg (KV) characteristic function).

Let $\Omega\subset V$ be a strictly convex homogeneous cone. For any vector
$x\in\Omega$, define the KV-characteristic function:

(4) $\chi(x)=\int_{\Omega^{*}}\exp{\\{-\langle x,a^{*}\rangle\\}}da^{*}$

where $da^{*}$ is a volume form invariant under translations in $\Omega^{*}$.

Some data about the KV-function is expressed below.

  * —

The KV-function tends to infinity on the boundary of the cone
$\overline{\Omega}$. By Hölder’s inequality, we have that $\Phi=\ln\chi$ is
strictly convex.

  * —

From the definition of $\chi$ we have that for any $x\in\Omega$ and any $g\in
G(\Omega),$ one has $\chi(gx)=|\det g|^{-1}\chi(x).$ The differential form,
$\alpha=d\chi/\chi$ is invariant under $G(\Omega)$.

  * —

Let $\Omega$ be homogeneous cone. Then the complex tube $T_{V}$—whose real
part is $\Omega$—has a transitive group of holomorphic mappings given by $z\to
gz+\imath t,\,g\in G(\Omega)$. The Bergman kernel is for $T_{V}$ up to
constant factor: $\chi^{2}(z+\overline{w})$.

###  3.3. Affine flat structures

Let $\Omega$ be a strictly convex homogeneous cone. Let $\mathcal{T}_{\Omega}$
be the tangent sheaf. Let $\Omega^{1}_{\Omega}$ be the sheaf of holomorphic
1-forms on $\Omega$. Let us recall the general definition of an affine flat
structure on $\Omega$. It is given by any of the following equivalent data:

  1. (1)

An atlas on $\Omega$ whose transition functions are affine linear.

  2. (2)

A torsionless flat connection
$\nabla_{0}:\mathcal{T}_{\Omega}\to\Omega^{1}_{\Omega}\otimes_{v}\mathcal{T}_{\Omega}$.

  3. (3)

A local system $\mathcal{T}_{\Omega}^{f}\subset\mathcal{T}_{\Omega}$ of flat
vector fields, which forms a sheaf of commutative Lie algebras of rank
$n(=dim\Omega)$ such that
$\mathcal{T}_{\Omega}=\mathcal{O}_{\Omega}\otimes_{k}\mathcal{T}_{\Omega}^{f}$.

Affine flat structures on a homogeneous cone imply the existence of a pre-Lie
algebra (aka Vinberg algebra, or Left Symmetric Algebra). Consider $X,Y$ two
vector fields in $\mathcal{T}_{\Omega}$ (or $\mathfrak{t}$). If $\Omega$ is
provided with an affine structure then:

(5) $\nabla_{X}(Y)-\nabla_{Y}(X)-[X,Y]=0,$

Also, written as:

$\nabla_{X}\nabla_{Y}-\nabla_{Y}\nabla_{X}-\nabla_{[X,Y]}=0,$

where $X,Y\in\mathcal{T}_{\Omega}$.

A multiplication operation $``\circ"$ given by $X\circ Y=\nabla_{X}(Y)$
defines a commutative algebra $(\mathfrak{t},\circ)$ satisfying the relation:

$a\circ(b\circ c)-b\circ(a\circ c)=(a\circ b)\circ c-(b\circ a)\circ c,$

for $a,b,c\in\mathfrak{t}.$

This forms a pre-Lie algebra structure, on the set of connections. This is
also known as a Lie-admissible algebra. Each Lie algebra with affine structure
is derived from a Lie-admissible algebra.

######  Lemma 3 ([Bu06]).

There is a one-to-one correspondence of $n$-dimensional convex homogeneous
cones $\Omega$ and $n$-dimensional pre-Lie algebras.

We can thus now state that:

######  Corollary 1.

Convex homogeneous cones and Vinberg cones are domains equipped with an affine
flat structure.

###### Proof.

The proof follows direclty from the lemma 3 and the discussion above. ∎

###  3.4. On Hessian structures

Suppose that $\Omega$ is an affine flat manifold. Then, it comes equipped with
an affine flat connection $\nabla_{0}$ and there exists a class of Riemannian
metrics compatible with $\nabla_{0}$. A Riemannian metric $g$ on $\mathcal{M}$
is said to be a Hessian metric if $g$ is given by $g=\nabla_{0}^{2}\Phi$,
where $\Phi$ is a local smooth function.

######  Definition 5.

The pair $(\nabla_{0},g)$ is called a Hessian structure on $\Omega$. The
triple given by $(\Omega,\nabla_{0},g)$ is called a Hessian manifold.

Let $\Omega$ be a differentiable manifold with a locally flat linear
connection $\nabla_{0}$. Let $p$ be a point of the differentiable manifold
$\Omega$ and let $U$ be an open neighborhood of $p$. Then, for any point $p$
in $U$, there exists a local coordinate system $(x_{1},...,x_{n})$ called an
affine local coordinate system, such that $\nabla_{0}(dx_{i})=0$. A Riemannian
metric $g$ on the differential manifold $\Omega$ is said to be locally Hessian
with respect to $\nabla_{0}$ if there exists for each point $p\in\Omega$, a
real-valued function $\Phi$ of class $C^{\infty}$ on $U$ such that

$g=\frac{\partial^{2}\Phi}{\partial x_{i}\partial x_{j}}dx_{i}dx_{j},$

where $(x_{1},...,x_{n})$ is an affine local coordinate system around $p$.
Then, $(\nabla_{0},g)$ is called a locally Hessian structure on $\Omega$.

######  Remark 4.

A variant of Hessian metrics can be used to define a canonical Riemannian
metric on any convex domain, using the solution of a Monge–Ampère equation by
Loewner–Nirenberg and Cheng–Yau.

Therefore, applying the knowledge above, we can conclude that:

######  Lemma 4.

The convex homogeneous cone $\Omega$ has a Hessian structure.

###### Proof.

We use the KV-function to define a Hessian structure on $\Omega$. Indeed, the
canonical Riemannian metric attached to the cone $\Omega$ is
$g_{V}=g=-\mathrm{Hess}(\ln\chi(x))$. It is invariant under $G(\Omega).$

Let us continue the discussion in local coordinates. Assume
$(x_{1},\cdots,x_{n})$ forms an affine coordinate system on $\Omega$. The
convex homogeneous domain $\Omega$ admits an invariant volume element defined
as $\Phi dx_{1}\wedge\cdots\wedge dx_{n}$.

The canonical bilinear form is:

(6)
$g={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}}\sum\frac{\partial^{2}\ln\chi}{\partial
x_{i}\partial x_{j}}dx_{i}dx_{j},$

where $\Phi=\ln\chi(x)$ is a potential function. The canonical bilinear form
$g$ is positively definite. This gives the Riemannian metric on $\Omega$ and
defines the Hessian structure. ∎

This marks the beginning of the emergence of Frobenius manifold patterns. As
we will see, $\Phi=\ln\chi(x)$ is the potential function in the Frobenius
manifold (see Sec. 5).

##  4\. Algebraic structures

Let $\Omega$ be a Riemannian homogeneous convex cone (cf. Sec.3). We
investigate its relations to algebras.

###  4.1. Frobenius algebras

Consider a finite $n$-dimensional commutative associative algebra
$\mathscr{A}$ over $\mathbb{K}$ possibly with unit ${\bf 1}_{\mathscr{A}}$,
with basis $\\{e_{i}\\}_{i=1}^{n}$ and structural constants $C^{i}_{jk}$,
being components of the (1,2)-tensor
$\circ:\mathscr{A}\times\mathscr{A}\to\mathscr{A}$ such that:

$e_{i}\circ e_{j}=C_{ij}^{s}e_{s},\quad i,j,s\in\\{1,\cdots,n\\}.$

Associativity can be written as follows :

$C_{ab}^{c}C_{cd}^{f}=C_{ad}^{c}C_{cb}^{f}.$

Commutativity implies that

$C^{s}_{ab}=C^{s}_{ba}.$

######  Definition 6.

A unital, commutative, associative algebra $(\mathscr{A},\circ)$ equipped with
a bilinear symmetric form $\langle-,-\rangle$ satisfying

$\langle x\circ y,z\rangle=\langle x,y\circ z\rangle,\quad
x,y,z\in\mathscr{A}$

is a Frobenius algebra.

###  4.2. Connection algebras

######  Definition 7 (Connection algebra).

A connection algebra is a unital, commutative algebra $\mathscr{A}^{+}$ with a
basis $e_{1},\cdots,e_{n}$ and multiplication $\circ:V\times V\to V$ given by

$e_{i}\circ e_{j}=\Gamma_{ij}^{k}e_{k},$

where $\Gamma_{ij}^{k}$ are the structure constants and being equipped with a
symmetric bilinear form $\langle-,-\rangle$, induced from the Riemannian
metric $g$.

######  Definition-Proposition 1.

Let $\Omega$ be a convex (pseudo-)Riemannian homogeneous cone; let
$p\in\Omega$ be a point. Consider a system of linear coordinates
$(x_{1},x_{2},\cdots,x_{n})\in\Omega$.

Then, there exists a connection algebra $\mathscr{A}^{+}$ on the tangent
bundle of $\Omega$ with multiplication operation $\circ$ given for every pair
$X,Y$ in $V$ by:

(7) $(X\circ Y)^{i}=-\sum_{j,k}\Gamma^{i}_{jk}X^{j}Y^{k}\quad 1\leq i\leq n.$

Structure constants are given by
$\Gamma^{i}_{jk}=\frac{1}{2}\partial_{jkl}\Phi g^{li}(p)$ and
$\partial_{jkl}\Phi=C(\partial_{j},\partial_{k},\partial_{l})$ forms a rank
three symmetric tensor.

The construction of this algebra follows from [Vi63].

######  Remark 5.

Note that the symmetry of $C$ comes from the fact that $\chi(e)$ is a real-
analytic function.

###  4.3. Towards Jordan algebras

The connection algebra $\mathscr{A}^{+}$ forms a Jordan algebra if the
Riemannian space is a symmetric space (i.e. is a Vinberg cone). We define the
powers of the elements $u$ in $\mathscr{A}^{+}$ as follows, which leads to the
Jordan algebra:

$X^{1}=X,\quad X^{m+1}:=X\circ X^{m},\quad m\geq 1.$

######  Corollary 2.

If $\mathscr{A}^{+}$ is a Jordan algebra, then $X^{r}\circ X^{s}=X^{r+s}$
holds for all $X\in\mathscr{A}^{+}$, where $r\geq 1$ and $s\geq 1$ and it is
then called power-associative.

The Jordan algebras associated to the Vinberg cones are finite-dimensional
formally real Jordan algebras, being a direct sum of a finite number of simple
ideals. The five basic types of simple building blocks are listed in Sec.2.7

######  Lemma 5.

The pre-Lie algebra $\nabla_{X}(Y)=X\circ Y$ generates the connection algebra
$\mathscr{A}^{+}$.

###### Proof.

The pre-Lie algebra defined above is given by a multiplication operation
$X\circ Y=\nabla_{X}(Y)$ where $X,Y$ lie in the space of vector fields of
$\mathfrak{t}$. The algebra $\mathscr{A}^{+}$ is defined on the vector space
$V$ which identifies with $\mathfrak{t}$.

Let us construct a canonical linear isomorphism between $\mathfrak{t}$ and
$V$, via Vinberg’s $T$-algebras [Vi63]. Every element $a\in\mathfrak{t}$
corresponds to the sum of $a+\tilde{a}\in V=T_{e}(V)$, where $\tilde{a}$ is
the involutive anti-automorphism equipping the $T$-algebra. This determines
the isomorphism:

$\nu:\mathfrak{t}\to V=T_{e}(V)$ $\,a\mapsto\,a+\tilde{a}.$

The multiplication operation on $\mathscr{A}^{+}$ coincides with $X\circ
Y=\nabla_{X}(Y)$. So, the pre-Lie algebra $\nabla_{X}(Y)=X\circ Y$ generates
the connection algebra $\mathscr{A}^{+}$. ∎

The algebra is defined throughout a non-singular symmetric bilinear form on
$V$. Consider a trilinear form $C$ on $V$. Fixing one argument we get a
bilinear form in the remaining two arguments. Therefore, for given $X$ the
form $C(X,Y,Z)$ is a bilinear form in $Y$ and $Z$ and there exists a linear
transformation $\nabla_{X}$ such that

$C(X,Y,Z)=g(\nabla_{X}(Y),Z).$

If the trilinear form is symmetric in $X,Y,Z$ then we have that the algebra is
commutative and that

$g(X\circ Y,Z)=g(X,Y\circ Z).$

######  Lemma 6.

Let $\mathscr{A}^{+}$ be the connection algebra. The symmetric bilinear form
satisfies:

$\langle x\circ y,z\rangle=\langle x,y\circ z\rangle,$

for all $x,z,y\in\mathscr{A}^{+}$.

###### Proof.

The symmetry of the rank 3 tensor $C(x,y,z)$ implies the relation:

$\langle x\circ y,z\rangle=\langle x,y\circ z\rangle.$

∎

######  Corollary 3 ([Koe99], Thm. 12, p.118.).

Let $\mathscr{A}^{+}$ be a real Jordan algebra associated to a Vinberg cone
$\Omega$. Then the following statements are equivalent:

  * •

$\mathscr{A}^{+}$ is formally real;

  * •

there exists a positive definite bilinear form $\langle-,-\rangle$ satisfying

$\langle x,y\circ z\rangle=\langle x\circ y,z\rangle.$

###### Proof.

This follows from [Koe99], Thm. 12, p.118. In fact, in a simple Euclidean
Jordan algebra, every associative symmetric bilinear form is a scalar multiple
of $Tr(xy)$. ∎

######  Lemma 7.

Consider an associative, commutative unital subalgebra
$\mathscr{A}\subset\mathscr{A}^{+}$. Then, this subalgebra forms a Frobenius
algebra.

###### Proof.

A Frobenius algebra is a unital, commutative, associative algebra (finite
dimensional) equipped with a symmetric bilinear form $\langle-,-\rangle$
where: $\langle x\circ y,z\rangle=\langle x,y\circ z\rangle$, forall $x,y,z$
lying in the Forbenius algebra.

The subalgebra $\mathscr{A}$ is a commutative, associative, unital subalgebra
of $\mathscr{A}^{+}$, equipped with a symmetric bilinear form satisfying
associativity condition (by Lem.6). So, $\mathscr{A}$ is a Frobenius algebra.
∎

##  5\. Frobenius structures

We set out some notions on Frobenius manifolds/ structures and conclude by
showing that this is equivalent to satisfying the WDVV equation.

###  5.1. Compatible flat affine structures with a multiplication

Recall the notion of flat structures introduced in Sec. 3.3. For brevity, let
$\mathcal{M}$ be a smooth manifold. Assume $\mathcal{T}_{\mathcal{M}}$ is
endowed with $\circ$ a $\mathcal{O}_{\mathcal{M}}$-bilinear commutative and
associative multiplication operation, eventually with unit $e$.

Then, according to [Man05] [Def.2.2.1], a flat structure
$\mathcal{T}_{\mathcal{M}}^{f}$ on $\mathcal{M}$ is called compatible with
$\circ$, if in a neighbourhood of any point there exists a vector field
$\mathscr{C}$ such that for any arbitrary local flat vector fields we have

(8) $X\circ Y=[X,[Y,\mathscr{C}]],$

$\mathscr{C}$ is a local vector potential for $\circ$.

Moreover, $\mathcal{T}_{\mathcal{M}}^{f}$ is called compatible with
$(\circ,{\bf 1}_{\mathscr{A}})$ if the equation (8) holds and if $e$ is flat.

By [Man05] Prop 2.2.2: If $\circ$ admits a compatible flat structure then it
satisfies the identity: For all local vector fields $X,Y,Z,W$:

(9) $P_{X\circ Y}(Z,W)=X\circ P_{Y}(Z,W)+(-1)^{XY}Y\circ P_{X}(Z,W),$

where $P_{X}(Z,W):=[X,Z\circ W]-[X,Z]\circ W-(-1)^{XZ}Z\circ[X,W]$.

A manifold $(\mathcal{M},\circ)$, where $\circ$ is an
$\mathcal{O}_{\mathcal{M}}$-bilinear commutative and associative
multiplication operation satisfying the identity 9 is called an $F$-manifold
(see Def 5.1 [Man99].

###  5.2. Pencils of flat connections

Let us consider the following input data:

  * •

a flat structure
$\nabla_{0}:\mathcal{T}_{\mathcal{M}}\to\Omega^{1}_{\mathcal{M}}\otimes_{\mathcal{M}}\mathcal{T}_{\mathcal{M}}$
on $\mathcal{M}$, where $\Omega_{\mathcal{M}}^{1}$ is the sheaf of holomorphic
1-forms on $\mathcal{M}$.

  * •

An odd global section
$\mathcal{A}\in\Omega^{1}_{\mathcal{M}}\otimes_{\mathcal{O}_{\mathcal{M}}}End(\mathcal{T}_{\mathcal{M}}).$

Then, one can produce from it the following datum:

  * •

A pencil of connections
$\nabla_{\lambda}^{\mathcal{A}}=\nabla_{0}+\lambda\mathcal{A}$.

  * •

An $\mathcal{O}_{\mathcal{M}}$-bilinear composition law $\circ$ on
$\mathcal{T}_{\mathcal{M}}$ given by:

$X\circ Y:=i_{X}(\mathcal{A})(Y),$

where for $G\in End(\mathcal{T}_{\mathcal{M}})$ and
$df\in\Omega_{\mathcal{M}}^{1}$ (for $f\in\mathcal{O}_{\mathcal{M}}$) the
following is defined: $i_{X}(df\otimes G):=Xf\cdot G.$

######  Proposition 5 ([Man05], Prop 2.3.1.).

$(\mathcal{M},\circ,\nabla^{\mathcal{A}}_{0})$ is an $F$-manifold with
compatible flat structure iff $\nabla_{\lambda}^{\mathcal{A}}$ is a pencil of
torsionless flat connections.

In that case, $(\mathcal{M},\circ,\nabla_{\lambda}^{\mathcal{A}})$ is an
$F$-manifold with compatible flat structure for any $\lambda$ as well.

###  5.3. (pre-)Frobenius manifolds

Consider the following family of data:

(10)
$(\mathcal{M};\,\circ:\mathcal{T}_{\mathcal{M}}\otimes\mathcal{T}_{\mathcal{M}}\to\mathcal{T}_{\mathcal{M}};\,\mathcal{T}_{\mathcal{M}}^{f}\subset\mathcal{T}_{\mathcal{M}};\
g:S^{2}(\mathcal{T}_{\mathcal{M}})\to\mathcal{O}_{\mathcal{M}}),$

where:

  * •

$\mathcal{M}$ is a manifold;

  * •

$\circ:\mathcal{T}_{\mathcal{M}}\otimes\mathcal{T}_{\mathcal{M}}\to\mathcal{T}_{\mathcal{M}}$
is a multiplication operation on the tangent sheaf;

  * •

$\mathcal{T}_{\mathcal{M}}^{f}\subset\mathcal{T}_{\mathcal{M}}$ is the
subsheaf of flat vector fields;

  * •

$g:S^{2}(\mathcal{T}_{\mathcal{M}})\to\mathcal{O}_{\mathcal{M}}$ is the non-
degenerate symmetric quadratic form with $\mathcal{O}_{\mathcal{M}}$ a sheaf
of holomorphic functions on $\mathcal{M}$.

The main additional structure gluing together this data is given by a family
of (local) potentials $\Phi,$ being sections of $\mathcal{O}_{\mathcal{M}}$,
such that for any local tangent fields $X,Y,Z$:

(11) $g(X\circ Y,Z)=g(X,Y\circ Z)=(XYZ)\Phi.$

If such a structure exists on
$(\mathcal{M};\,\circ:\mathcal{T}_{\mathcal{M}}\otimes\mathcal{T}_{\mathcal{M}}\to\mathcal{T}_{\mathcal{M}};\,\mathcal{T}_{\mathcal{M}}^{f}\subset\mathcal{T}_{\mathcal{M}};\
g:S^{2}(\mathcal{T}_{\mathcal{M}})\to\mathcal{O}_{\mathcal{M}})$ then it forms
a pre-Frobenius manifold.

######  Definition 8.

A pre-Frobenius manifold is called associative if the multiplication $\circ$
is associative.

A pre-Frobenius manifold is called potential if $C$ admits everywhere locally
admits a potential.

A pre-Frobenius manifold is Frobenius if it is simultaneously potential and
associative.

######  Theorem 1 (Th.1.5, [Man99]).

The pre-Frobenius manifold $(\mathcal{M},\circ,\nabla_{0})$ is a Frobenius
manifold with compatible flat structure if and only if
$\nabla_{\lambda}^{\mathcal{A}}$ is a pencil of torsionless, flat connections.

###  5.4. Associativity equations

Consider an $F$-manifold $(\mathcal{M},\circ,\nabla_{0})$ endowed with a
compatible flat structure $\nabla_{0}.$

Consider the algebra $(\mathscr{A},\circ)$ with basis $\\{e_{i}\\}_{i=1}^{n}$
and its structural constants $C^{s}_{ij}$ satisfying the relation

$e_{i}\circ e_{j}=C^{s}_{ij}e_{s}$

for indexes $i,j,s\in\\{1,\cdots,n\\}$. This algebra is defined at each point
$p\in\mathcal{M}$ on the tangent space to $T_{\mathcal{M}}$.

If we choose a local flat coordinate system $(x_{a})$ on $\mathcal{M}$ and
write the local vector potential $C$ as $C=\sum_{c}C^{c}\partial_{c}$,
$X=\partial_{a}$ and $Y=\partial_{b}$ then

$\partial_{a}\circ\partial_{b}=\sum C_{ab}^{c}\partial_{c},\quad
C^{c}_{ab}=\partial_{a}\partial_{b}C^{c}.$

The choice of a flat invariant metric $g$ allows to define $C^{a}$ as
$C^{a}=\sum_{b}g^{ba}\partial_{b}\Phi$.

Let us consider:

(12)
$(\partial_{a}\circ\partial_{b})\circ\partial_{c}=\Bigg{(}\sum_{e}C^{e}_{ab}\partial_{e}\Bigg{)}\circ\partial_{c}=\sum_{ef}C_{ab}^{e}C^{f}_{ec}\partial_{f}.$

(13)
$\partial_{a}\circ(\partial_{b}\circ\partial_{c})=\partial_{a}\circ\sum_{f}C^{f}_{bc}\partial_{f}=(-1)^{a(b+c+e)}\sum_{ef}C_{bc}^{f}C^{e}_{ae}\partial_{e}.$

If $\mathscr{A}$ is associative then

$C_{ab}^{e}C_{ec}^{f}=C_{ae}^{e}C_{cb}^{f},$

which amounts to
$\partial_{a}\circ(\partial_{b}\circ\partial_{c})=(\partial_{a}\circ\partial_{b})\circ\partial_{c}$
i.e. Eq.12=Eq.13.

Given that $C^{c}_{ab}:=\sum_{e}C_{abe}g^{ec}=\sum_{e}\Phi_{abe}g^{ec}$, where
$(g^{ec}):=(g_{ec})^{-1}$ and
$\Phi_{abe}=\partial_{a}\partial_{b}\partial_{e}\Phi$ we can rewrite the
associativity relation for $\Phi$ as

$\forall
a,b,c,d:\quad\sum_{ef}\Phi_{abe}g^{ef}\Phi_{fcd}=(-1)^{a(b+c)}\sum\Phi_{bce}g^{ef}\Phi_{fad},$

which corresponds to the WDVV equation. Given the flat identity denoted
$e=\partial_{0}$, then the equation reduces to $\Phi_{0ab}=g_{ab}$.

##  6\. The WDVV equation and symmetric cones

We proceed as follows.

  1. (1)

We prove the existence of totally geodesic submanifolds in Vinberg cones.

  2. (2)

We prove that those totally geodesic submanifolds carry Frobenius structures
(i.e. satisfy the WDVV equation).

###  6.1. A first insight

To give an first intuition to the reader of why should totally geodesic
submanifolds in Vinberg cones exist, it is enough to go back to a theorem of
Rothaus [Rot66]. According to [Rot66]: if $\Omega$ is a cone of rank $k$ and
of dimension $n$ then every cone $\Omega$ is birationally biregularly
equivalent to the direct product of $k$ half lines and an Euclidean space. The
number of half lines appearing coincides with the rank of the Vinberg cone
$\Omega$.

###  6.2. Totally geodesic submanifolds and Lie triple systems

We will proceed to an algebraic investigation of the existence of totally
geodesic submanifolds in $\Omega$. Using our method, it is possible to compute
explicitly all the properties of those manifolds.

######  Definition 9.

Let $\Omega$ be a noncompact symmetric space. A totally geodesic
$r$-dimensional submanifold of $\Omega$ isometric to $\mathbb{R}^{r}$ is
called an $r$-flat. If $r$ is the maximal natural number $r$ for which an
$r$-flat exists then it is a maximal flat.

Let $F$ denote an $r$-flat submanifold in an $n$-dimensional Vinberg cone
$\Omega$. We first outline an algebraic description of $F$.

######  Theorem 2 (Thm IV.4.2. and Thm IV.7.2 of [Hel78]).

  1. (1)

The curvature tensor $R$ evaluated at $T_{e}\Omega$ is given by

$R(X,Y)Z=-[[X,Y],Z],\,\text{for}\quad X,Y,Z\,\in\,T_{e}\Omega.$

  2. (2)

The totally geodesic space $F\subset\Omega$ has the form

$F=\exp{\mathfrak{a}}\cdot e,$

where $\mathfrak{a}\subseteq\mathfrak{t}$ is a Lie triple system i.e.
$[[\mathfrak{a},\mathfrak{a}],\mathfrak{a}]\subseteq\mathfrak{a}$.

  3. (3)

Totally geodesic submanifolds through $e$ are of the form
$\exp{\mathfrak{a}}\cdot e$ where $\mathfrak{a}\subseteq\mathfrak{t}$ is a Lie
triple system.

By construction, we know that the sectional curvature restricted to $F$ equals
zero. If $\mathfrak{a}\subseteq\mathfrak{t}$ is a maximal abelian subspace of
dimension $r$ then, $F=\exp{\mathfrak{a}}\cdot e$ is a maximal flat in
$\Omega$.

######  Remark 6.

Given that $\mathfrak{g}$ are semisimple Lie algebras a possible class of Lie
subalgebras $\mathfrak{a}$ that satisfy the necessary requirements for
producing a maximal flat are the Cartan subalgebras of $\mathfrak{g}$. They
are maximal abelian subalgebras of $\mathfrak{g}$ and form a Lie triple
system.

######  Lemma 8.

Suppose that there exists a totally geodesic $F$ submanifold immersed in
$\Omega$. Then, the Lie subalgebra $\mathfrak{a}$ of $\mathfrak{t}$ attached
to the tangent space $T_{e}F$ to $F$ is Lie associative.

###### Proof.

By Thm IV 4.2 in [Hel78], the curvature tensor $R_{0}$ evaluated at
$T_{e}\Omega$ is given by

(14) $R_{0}(X,Y)Z=-[[X,Y],Z],\quad X,Y,Z\in T_{e}\Omega.$

$F$ being totally geodesic it is flat. So, restricting attention onto $F$, the
left hand side of Eq.14 is 0. Thus, we have $0=-[[X,Y],Z]$, where $X,Y,Z\in
T_{e}F$. Now, by Jacobi’s identity:

$[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0.$

Rewriting it one gets $[[Y,Z],X]+[[Z,X],Y]=-[[X,Y],Z]$. So, by Eq.14 for
$X,Y,Z\in T_{e}F$ one has $[[Y,Z],X]=[Y,[Z,X]]$. Thus, the Lie associativity
is satisfied on $F$. ∎

###  6.3. The case of space-like Lagrangian–Grassmanians cones

We prove algebraically that in $n$-dimensional space-like
Lagrangian–Grassmanians cones there exist totally geodesic immersed
submanifolds being maximal flats in $\Omega$.

According to Thm. 6.6 totally geodesic submanifold are found if one has a Lie
triple system. So, in particular we can take the example of Cartan subalgebras
which form a Lie triple sytem. A generalisation is mentioned as Cor. 4.

######  Proposition 6.

Consider the $n$-dimensional irreducible Vinberg cones of
Lagrangian–Grassmanian type. Then, for each of those cones there exists an
$n-1$-dimensional totally geodesic submanifold given by

$F=\exp{\tilde{\mathfrak{a}}}\cdot e,$

such that ${\tilde{\mathfrak{a}}}$ is a Cartan subalgebra of
$\mathfrak{gl}_{n}(\mathbb{K})$ given by ${\tilde{\mathfrak{a}}}=\lambda
I_{n}\oplus\mathfrak{a}$, $\lambda\in\mathbb{K}$ where $\mathfrak{a}$ is
formed by diagonal matrices of null trace.

  1. (1)

If $\mathbb{K}=\mathbb{R}$, $\mathfrak{a}$ is given by all diagonal matrices
with real diagonal entries and such that the trace is 0.

  2. (2)

If $\mathbb{K}=\mathbb{C}$, $\mathfrak{a}$ is given by all diagonal matrices
with diagonal entries $a+\imath b$ and such that the trace is 0.

  3. (3)

If $\mathbb{K}=\mathbb{H}$, $\mathfrak{a}$ is given by all diagonal matrices
of $\bigg{\\{}\begin{pmatrix}X&-\overline{Y}\\\
Y&\overline{X}\end{pmatrix}\,|\,\Re TrX=0.\bigg{\\}}$

  4. (4)

If $\mathbb{K}=\mathbb{O}$, $\mathfrak{a}$ is given by all diagonal $(3\times
3)$ matrices with diagonal entries $a+\imath b$ and of the diagonal matrices
of a Cartan subalgebra of $\mathfrak{g}_{2}$.

###### Proof.

The noncompact symmetric spaces considered are of the type:
$SL_{n}(\mathbb{K})/K\rtimes\mathbb{K}^{*}$, where $K$ is the maximal compact
subgroup at $e$. Thm IV.7.2 of [Hel78] implies that totally geodesic
submanifolds through $e$ are of the form $exp{\mathfrak{a}}\cdot e$, where
$\mathfrak{a}\subset\mathfrak{t}$ is a Lie triple system. Given that we are
working with semi-simple Lie algebras we can investigate in particular the
existence of their Cartan subalgebras which here will be maximal abelian
subalgebras. The Cartan involution is given by $X\mapsto-X^{t}$. One can prove
that for the real division algebras $\mathbb{K}$, the Cartan subalgebras of
$\mathfrak{sl}_{n}(\mathbb{K})$ are diagonal matrices with null trace. We
discuss this statement in detail, according to $\mathbb{K}$.

  1. (1)

If $\mathbb{K}=\mathbb{R}$, then Lie algebra attached to the symmetric space
has the Cartan decomposition
$\mathfrak{sl}_{n}(\mathbb{R})=\mathfrak{so}_{n}\oplus sym_{0}(n)$ where
$\mathfrak{s}ym_{0}(n)$ denotes the set of symmetric matrices of trace 0 with
entries in $\mathbb{R}$. One can check that the maximal abelian subspace
$\mathfrak{a}$ of $sym_{0}(n)$ is given by the set of diagonal matrices of
null trace. The maximal flat is thus given by

$F=\exp{\mathfrak{a}}\cdot
e=\\{Diag(\lambda_{1},\cdots,\lambda_{n}):\,\lambda_{i}=exp(t_{i})\in\,\mathbb{R},\,\prod_{i=1}^{n}\lambda_{i}=1\\}.$

  2. (2)

If $\mathbb{K}=\mathbb{C}$, $\mathfrak{a}$ is given by the diagonal matrices
with complex entries. The maximal flat is

$F=\exp{\mathfrak{a}}\cdot
e=\\{Diag(\lambda_{1},\cdots,\lambda_{n}):\,\lambda_{i}=\exp{a_{i}+\imath
b_{i}}\in\,\mathbb{C},\,\prod_{i=1}^{n}\lambda_{i}=1.\\}$

  3. (3)

If $\mathbb{K}=\mathbb{H}$, $\mathfrak{sl}_{n}(\mathbb{H})$ is given by
diagonal matrices in $\bigg{\\{}\begin{pmatrix}X&-\overline{Y}\\\
Y&\overline{X}\end{pmatrix}\,|\,\Re TrX=0\bigg{\\}}.$ To define
$\mathfrak{a}$, one takes only the diagonal matrices in that set.

  4. (4)

If $\mathbb{K}=\mathbb{O}$, $\mathfrak{a}$ is given by all diagonal $(3\times
3)$ matrices with diagonal entries $a+\imath b$ and of the diagonal matrices
of a Cartan subalgebra of $\mathfrak{g}_{2}$.

∎

######  Proposition 7.

Consider a Vinberg cone of space-like Lagrangian–Grassmanian type. Then, at
any point of the maximal flat, the tangent space to it carries the structure
of a Frobenius algebra.

###### Proof.

Let us discuss the case of Vinberg cones of Lagrangian–Grassmanian type.
Consider the maximal flat $F$, given by $F=\exp{\tilde{\mathfrak{a}}}\cdot e$
where ${\tilde{\mathfrak{a}}}=\lambda I_{n}\oplus\mathfrak{a}$,
$\lambda\in\mathbb{K}$ is a Cartan subalgebra of
$\mathfrak{gl}_{n}(\mathbb{K})$ and where $\mathfrak{a}$ are diagonal matrices
of null trace. The set of diagonal matrices form an associative, commutative
and unital algebra. Therefore, we have a Frobenius algebra structure on the
tangent space to the $n-1$-flat. The tangent space to $F$ is given by
$\tilde{\mathfrak{a}}\subseteq\mathfrak{t}$. On $\mathfrak{t}$ there exists a
bilinear symmetric form given by the Killing form. In particular, we have
$\langle X,Y\rangle=tr(XY)$, where $X,Y\in T_{e}(G/K)$, which satisfies
associativity by Lem. 6. The Cartan subalgebra inherits this bilinear form.
Thus, we have a Frobenius algebra. ∎

######  Lemma 9.

The $n$-dimensional Vinberg cones (1)–(3) are associated with the Weyl
chambers of type $A_{n}$.

###### Proof.

For $SL_{n}(\mathbb{R})/SO_{n}$ consider $H\in Diag(t_{1},\cdots
t_{n})\in\mathfrak{a}$. One gets that

$(adH)E_{ij}=[H,E_{ij}]=(t_{i}-t_{j})E_{ij}.$

So, one has $n(n-1)$ non-zero roots. In particular, we have the following
splitting

$\mathfrak{sl}_{n}(\mathbb{R})=\mathfrak{a}+\sum_{i\neq j}\mathbb{R}\cdot
E_{ij}.$

For $\mathbb{C}$, a similar situation occurs. More abstractly we can write:

$\mathfrak{sl}_{n}(\mathbb{C})=\mathfrak{a}+\oplus\bigg{(}\bigoplus_{i\neq
j}\mathfrak{g}_{\lambda_{i}-\lambda_{j}},)$

where $\mathfrak{g}_{\lambda_{i}-\lambda_{j}}=Spac_{\mathbb{C}}(e_{ij})$ and
$e_{ij}$ represents the basis vector in the $i$-th row and $j$-th column. For
$\mathbb{H}$ and $\mathbb{O}$ a similar argument can be carried out. ∎

######  Lemma 10.

Let $\Omega$ be Vinberg cones of type (1)–(4). A Weyl chamber is isomorphic to
$\exp{\mathfrak{a}^{+}}\in V$, where

(15)
$\mathfrak{a}^{+}:=\\{Diag(t_{1},\cdots,t_{n}):\,\sum_{i=1}^{n}t_{i}=0,\,t_{1}<\cdots<t_{n}\\}.$

###### Proof.

A Weyl chamber is isomorphic to an open Euclidean cone in $\mathfrak{a}$. This
chamber is given by Eq.15. So, in the Vinberg cone $\Omega$ a Weyl chamber is
isomorphic to $\exp{\mathfrak{a}^{+}}\in V$. ∎

###  6.4. The case of Lorentzian cones

######  Proposition 8.

Consider the irreducible Lorentzian cone $\Lambda_{n}$. Then, there exists a
totally geodesic submanifold $\mathscr{H}$ in $\Lambda_{n}$. This totally
geodesic space is given by the maximal flat $\exp{\mathfrak{a}}\cdot e$, where
$\mathfrak{a}$ is a maximal abelian subalgebra of the Lie algebra
$\mathfrak{o}(1,n-1)$.

###### Proof.

The Lorentzian cone is related to pseudo-orthogonal Lie algebras. It is
described in terms of quotients of Lie groups by $O(p,q)/O(p)\times O(q)$,
with $p\geq q$, where $p=n-1$ and $q=1$.

To the corresponding Lie algebra, there exists a non-empty maximal abelian
subalgebra. Consider the matrix corresponding to an element of
$\mathfrak{o}(p,q)$:

where entries are real and $a,d$ are skew symmetric. The blue numbers
correspond to numbers of rows/columns. We take $\textgoth{k}$ as matrices with
$b=0$; $\mathfrak{t}$ are matrices with $a=0$ and $d=0$. It has been shown
that for $\mathfrak{o}(p,q)$, one has a class of maximal abelian subalgebra
which are formed by orthogonally decomposable matrices (these are the Cartan
subalgebras). All matrices in that set can be simultaneously represented by
block diagonal matrices, having the same decomposition patterns. A second type
of maximal abelian subalgebra occurring has a matrix representation as
follows:

(16) $\begin{pmatrix}0&\alpha&\cdots&0&\\\ \\\ \cdots&&\cdots&-\alpha^{T}\\\
\\\ 0&&\cdots&0\\\ &0&&\cdots&0\\\ \end{pmatrix}_{n\times n}\text{for some
vector}\,\alpha.$

We construct the maximal flats via Thm. 6.6. The flat is a non-empty subspace
given that the maximal abelian subalgebra. So, there exists a non-empty
totally geodesic submanifold $\mathcal{H}$ in $\Lambda_{n}.$ ∎

The Lorentzian cone $\Lambda_{n}$ is of the Anti-de-Sitter type. Anti-de-
Sitter spaces are Lorentzian manifolds with negative constant sectional
curvature.

######  Lemma 11.

Consider $\mathscr{H}$ the totally geodesic submanifold in $\Lambda_{n}$.
Then, the tangent space at at any point of $\mathscr{H}$ carries the structure
of a (non unital) Frobenius algebra.

###### Proof.

We have a maximal abelian subalgebra $\mathfrak{a}\subset\mathfrak{t}$ which
is associated to $\mathscr{H}$, via $\exp{\mathfrak{a}}\cdot e$ . This defines
the “flat”. Taking $\mathfrak{a}$ to be represented by block diagonal matrices
having same decomposition pattern, one can prove that for the standard matrix
multiplication this forms this forms an associative, commutative algebra.
Given that $\mathfrak{a}$ inherits a symmetric bilinear form from
$\mathfrak{t}$ satisfying Lem. 6 it is a Frobenius algebra. Associativity and
commutativity also holds for the class of matrices of type given in formula 16
as well as the symmetric bilinear form satisfying Lem. 6.

∎

###  6.5. Main statements

###  6.6. New sources of $F$-manifolds

We prove first that on $F$, the multiplication $\circ$ on the tangent sheaf
admits a compatible flat structure.

######  Proposition 9.

Let $\Omega$ be a Vinberg cone. Consider $F$ a totally geodesic immersed
submanifold of $\Omega$. Then, for any vector fields $X,Y$ in
$\mathcal{T}_{F}$ there exists a local vector potential $\mathscr{C}$ such
that we have the following compatibility relation

(17) $X\circ Y=[X,[Y,\mathscr{C}]].$

###### Proof.

By [Hel78], Thm IV 4.2 one has the following:

$R_{0}(Y,Z)X=[X,[Y,Z]],$

where $X,Y,Z\in T_{e}\Omega$. The submanifold $F$ being totally geodesic, the
scalar curvature vanishes on $F$. So, the curvature tensor is 0 and the right
hand side of Eq.17 is

$[X,[Y,Z]]=0.$

Consider the left hand side of Eq. 17. The multiplication operation in the
algebra is given by:

$(X\circ Y)^{i}=\Gamma_{jk}^{i}X^{j}Y^{k},$

where $\Gamma_{jk}^{i}=g^{li}\partial_{jkl}\Phi$. The scalar curvature
vanishes if and only if $C_{ijk}=\partial_{ijl}\Phi$ vanishes. So, $X\circ
Y=0$ and thus $X\circ Y=[X,[Y,\mathscr{C}]]$. ∎

######  Theorem 3.

Let $\Omega$ be a Vinberg cone. Consider $F$ a totally geodesic immersed
submanifold of $\Omega$ given by $F=\exp{\mathfrak{a}}\cdot e$, where
$\mathfrak{a}\subset\mathfrak{t}$ is a Lie triple system. Then, the flat
structure on $F$ is compatible with $\circ$ and $(F,\circ)$ is an
$F$-manifold.

###### Proof.

One applies directly the statement from [Hel78] (Theorem ) ensuring that $F$
is totally geodesic iff $\mathfrak{a}\subset\mathfrak{t}$ is a Lie triple
system. We have shown in Prop. 9 that one has a flat structure compatible with
$\circ$ on $F$. Thus, by Manin [Man] if the multiplication $\circ$ on the
tangent sheaf admits a compatible flat structure it forms an $F$-manifold.

∎

######  Corollary 4.

Let $\Omega$ be a Vinberg cone. Then, $F=\exp{\mathfrak{a}}\cdot e$ in
$\Omega$ is an $F$-manifold iff $\mathfrak{a}$ is a Lie triple system.

###### Proof.

Any Vinberg cone is a linear combination of irreducible Vinberg cones. Each of
those irreducible cones contains a non-empty totally geodesic submanifold
(Prop.8 and Prop.6). The Cartesian product of those totally geodesic
submanifolds gives a totally geodesic submanifold. So, our statement on the
existence of a totally geodesic submanifold in a Vinberg cone holds given any
Vinberg cone defined on any algebra (for example a Clifford algebra) combining
the real division algebras.

∎

######  Lemma 12.

Let $F=\exp{\mathfrak{a}}$, where $\mathfrak{a}$ is Lie triple system. Then,
$(F,\circ,\nabla_{0})$ is an $F$-manifold with compatible flat structure iff
$\nabla_{\lambda}^{\mathcal{A}}$ is a pencil of torsionless flat connections.

###### Proof.

This follows from Thm. 3 and Prop.5. ∎

###  6.7. New sources of Frobenius manifolds

######  Proposition 10.

Vinberg cones are equipped with the structure of a potential pre-Frobenius
manifold.

###### Proof.

Let $\Omega$ be a Vinberg cone.

  1. (1)

By Lem. 1, $\Omega$ has an affine flat structure. Given that it is a Hessian
manifold the flat connection is a Levi–Civita connection $\nabla_{0}$.

  2. (2)

Vinberg cones comes equipped with the data $(\Omega,g,C,\circ)$, where:

  * •

$g$ is a compatible Riemannian (Hessian) metric. Locally we have:
$g_{ij}=\partial_{i}\partial_{j}\Phi$, where $\Phi$ is a potential function
given by the Koszul–Vinberg function given in Def.4.

  * •

$C$ is a rank three symmetric tensor. In the present case given that $\Omega$
is being Hessian , $C$ is given by
$C_{ijk}=\partial_{i}\partial_{j}\partial_{k}\Phi$.

  * •

$``\circ"$ is a multiplication operation on the tangent bundle, defined in
Prop. 7. The subalgebra associated to the locus of $(n-1)$-flats is a
Frobenius algebra, by Prop.1.

By Prop.7 and Lem.6, we have $C(X,Y,Z)=g(X\circ Y,Z)=g(X,Y\circ Z)$.

This gives a pre-Frobenius manifold. One can add that we have a potential pre-
Frobenius manifold. Indeed, $C(X,Y,Z)$ admits everywhere locally a potential.
∎

######  Theorem 4.

Consider a Vinberg cone $\Omega$. Then, the manifold
$F=\exp{\mathfrak{a}}\cdot e$, where $\mathfrak{a}\subset\mathfrak{t}$ is Lie
triple system, forms a Frobenius manifold immersed in $\Omega$.

###### Proof.

Let $F$ be maximal flat in $\Omega$; we think of $F$ as embedded in $\Omega$
for the following paragraphs.

1) By Prop. 10 the Vinberg cone is pre-Frobenius. It is equipped with:

  * •

$\nabla_{0}$, an affine flat structure (cf. Lem. 1);

  * •

$g$, a compatible Riemannian metric;

  * •

$C$, a rank three symmetric tensor;

  * •

$``\circ"$, a multiplication operation on the tangent bundle (see Prop. 1.)

Moreover, one has a compatibility condition: $g(X\circ Y,Z)=g(X,Y\circ
Z)=C(X,Y,Z)$, where $X,Y,Z\in\mathcal{T}_{\Omega}$ is satisfied.

Let us now discuss the proper embedding $i:F\hookrightarrow V$. $F$ is
isometric to an Eucldean space.

2) By Lem. 1, $\Omega$ has an affine flat structure. The embedding $i:F\to V$
induces a Levi–Civita connection on $F$. It is the Levi–Civita connection for
the pullback metric $g^{F}:=i^{*}g.$

3) One verifies that the potentiality axiom holds on $F$. Indeed, as for
$\Omega$, the rank 3 symmetric tensor $C$ admits everywhere locally a
potential function. The local potential $\Phi(=\ln\chi)$ is such that for any
flat local tangent fields one has: $\partial_{abc}\Phi=C_{abc}$.

4) By 7 the tangent space to $F$ forms a Frobenius algebra. This is a strong
statement since it proves that the associativity axiom for $\circ$ i.e
$(X\circ Y)\circ Z=X\circ(Y\circ Z)$ for $X,Y,Z\in\mathcal{T}_{V}$.

At this stage of the proof all axioms to have a Forbenius manifold are
satisfied. So, a maximal flat i.e a totally geodesic submanifold of maximal
dimension, in a Vinberg cone froms a Frobenius manifold.

A different argument can be added. In virtue of Th. 1.5 in [Man99] a pre-
Frobenius manifold M is Forbenius iff the pencil of connections
$\nabla_{\lambda,X}(Y)=\nabla_{0,X}(Y)+\lambda(X\circ Y)$ for
$X,Y\in\mathcal{T}_{V}$ and $\lambda\in\mathbb{R}$ is flat. The maximal flat
is totally geodesic. So, the pencil of connections
$\nabla_{\lambda}^{\mathcal{A}}$ is flat. This allows a direct conclusion. ∎

######  Lemma 13.

Every Frobenius submanifold $F$ of $\Omega$ is necessarily a $G$-translate of
$F$, where $G$ acts by isometries on $\Omega$.

##  7\. Conclusion and future work

We have thus proved in this paper an interesting relation between the WDVV PDE
and strictly convex symmetric cones, leading to intriguing questions relating
the 2D (TFT) [Du96] and those cones. This is achieved using geometric and
algebraic methods (using Cartan’s classification of symmetric spaces) on the
one hand side; and Calabi’s investigation on the Monge–Ampère equation. This
allows for instance to generalise a result of [FKN21, Mok95] on symplectic
Monge–Ampere equations of Hirota type. It as well allows an expression of this
idea in the realm of geometry and algebra.

Applications of this result can be achieved for instance in information
geometry [CM20, Ch82]. This is done by using the fact that to any point
$x\in\Omega$ one may attach a probability measure on the dual cone. This forms
a main object in information geometry and statistics, since the KV
characteristic function corresponds to the density of a probability measure:

$\forall x\mapsto p_{x}(x^{*})=\frac{\exp(-\langle x^{*},x\rangle)}{\chi(x)}.$

To conclude, results of this paper reveal an enlargement of the classification
of sources of Frobenius manifolds. In particular, it is shown that there exist
Frobenius manifolds defined over linear combinations of real division algebras
of finite dimension and as well that there exist Frobenius manifolds in the
context of pseudo-Riemannian manifolds: the Lorentz manifolds of Anti-De-
Sitter type.

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1Department of Mechatronics and Robotics, Egypt-Japan University of Science
and Technology, Alexandria, Egypt
(E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
2 Department of Modern Mechanical Engineering, Waseda University, Tokyo, Japan
(E-mail<EMAIL_ADDRESS>

# Towards Spiral Brick Column Building Robots

Yasser Ashraf 1 Ahmed Abdallah1 Abdelhaleem Osman1 Victor Parque2 and Samy
Assal1

###### Abstract

Automation in construction has the potential to expand the technological
landscape of labor intensive tasks, and bring gains in efficiency and
productivity to sustain global competitiveness. In this paper we propose a
task-level approach for assembly of spiral brick columns. Our extensive
computational simulations using the generalized models of spiral brick columns
show the feasibility, the effectiveness and efficiency of our proposed
approach. Our results offer the potential to use robots in automated
construction of spiral brick columns with utmost efficiency.

###### keywords:

spiral brick column, construction robots, task planning, automation.

##  1 Introduction

Figure 1: Basic idea of he general framework in our approach. The manipulator
is attached to a gripper and is tasked with assembling a user-defined spiral
brick column by using incoming bricks from a belt conveyor.

The autonomous assembly of bricks in construction buildings consists in
locating, picking, transporting and assembling brick objects to build pre-
defined structures. Since the manual teaching of such tasks is inefficient,
the model-driven task planning is potential to expand productivity frontiers.

Past research has explored the assembly of wall scenarios in construction by
serial manipulators[1, 2, 3]. Also, mobile robots have been used to evaluate
the assembly task[4], to aid in localization in construction sites[5]: to
automate the generation of robot programs for prefabrication of walls[6] and
to integrate with building information modeling (BIM) to develop plans to
construct walls, stairs, and a pyramid with up to 30 bricks[7]

Nowadays, there already exists a number of commercially available initiatives
for the assembly of buildings in the market. Developed by Construction
Robotics, SAM 100 is the first commercially bricklaying robot for straight-
line mason construction, and MULE 135 (Material Unit Lift Enhancer) is a lift
assisted device for handling and placing heavy materials in construction
tasks[8]. Developed by Fast-brick Robotics, Hadrian X is an automated
bricklaying system able to configure the first multi-room house structure from
a 3D CAD model without human intervention[9].

The full automation of bricklaying in arbitrary building structures is still
unfeasible and requires human supervision. Although the use of robotics in
straight line walls has been proposed, the study of building spiral brick
columns has received little attention. In this paper, we propose a task-level
planning of assembly of spiral brick columns and evaluate its feasibility and
efficiency by using a generalized modeling to render a relevant set of spiral
brick models and simulation environments (as shown by Fig. 1). Our study is
potential to build robot plans being effective for automated construction of
aesthetic spiral brick columns.

##  2 Proposed Approach

###  2.1 Spiral Brick Models

Figure 2: Spiral brick column models considered in our study.

We considered the modeling of spiral brick columns as shown by Fig. 2, which,
according to the geometry of the base of the column and its configuration
parameters, topologically distinct geometries are renderable. When the base of
the spiral column is composed by segments with _parallel_ and _orthogonal_
configurations, we let the formation of $s$ segments each with $B_{i}$ block
units. Then the margin $m$ between two blocks is computed by:

$m=B_{i}.l+\tau.w+\lambda(B_{i}-1)$ (1)

$\tau=\frac{1}{\tan(\frac{\theta}{2})}$ (2)

, where $B_{i}$ is the number of blocks in the $i$th segment of the base
configuration of the spiral column, $l$ is the block length, $w$ is the block
width, $\tau$ is the factor of the angle between segment configurations,
$\theta$ is the angle between two segment configurations, and $\lambda$ is a
user-defined coefficient for margin preference. Thus, to enable a spiral brick
column we let $B_{i}=B_{j}$ for $i,~{}j\in[s],i\neq j$, when the segments are
in parallel configuration ($\theta=\pi$), as portrayed by Fig. 2-(a), and
$B_{i}\neq B_{j}$ for $i,~{}j\in[s],i\neq j$, as portrayed by Fig. 2-(b), when
the segments are in orthogonal configuration ($\theta=\pi/2$). The above-
described configuration is a general mechanism which is a function of the
number of segments and the configuration between segments. The segment base
configuration is rotated around its centroid by an angle $\phi$ to form $L$
consecutive layers of spiral brick column.

When the base of the spiral column is a _regular polygon_ , the number of
segments $s$ in the column base is equal to the number of edges $n$ of the
regular polygon. Thus, the spiral brick column is renderable by the geometry
of the base polygon (the angle $\theta$ is computable from the number of sides
$n$ of the regular polygon), and the configurations in the number of blocks in
each edge of the polygon base. To allow the formation of a spiral column, we
let $B_{i}=B_{j}$, for $i,~{}j\in[n],i\neq j$, and let rotate the base polygon
around its centroid by an angle $\phi$ to form $L$ consecutive layers.
Examples of spiral brick columns wit regular polygonal configuration are
portrayed by Fig. 2(c)-(e) in which the base polygon consists of not only
convex geometry (triangle and square) but also concave geometry (concave
decagon) .

It is also possible to build spiral column configurations by using polynomial
functions. Here, the pose of the bricks are allowed to follow the trajectory
of an arbitrary polynomial function $f$ over a fixed domain and range. Here,
the margin $m$ between two bricks is computed by:

$m=w.\sin\Big{(}\frac{\pi-\theta}{2}\Big{)}+\kappa$ (3)

, where $\theta$ is the angle between two successive bricks, and $\kappa$ is a
user defined coefficient for margin preference. In order to allow the
formation of a spiral column, we let the polynomial function $f$ and $-f$ to
form a closed loop, and let it rotate around its centroid by an angle $\phi$
over the plane to form $L$ consecutive layers.

(a) Depth Map

(b) Point Cloud

(c) Filtered Point Cloud

(d) Estimated Pose

Figure 3: Pose estimation of arbitrary bricks on a conveyor.

(a) Position Error

(b) Orientation Difference

(c) Trajectory Time

(d) Pose Estimation Time

Figure 4: Performance metrics in the assembly of spiral brick columns. Figure
5: Examples of the performance while assembling spiral brick columns.

###  2.2 Pose Estimator

To estimate the pose of incoming bricks from a conveyor, we use a depth camera
close to the robot manipulator (see Fig. 1), and compute the point cloud
$\mathcal{P}$ satisfying the region of interest near the brick by using the
pinhole cameral model and RANSAC with maximum likelihood and consensus
(MLESAC) algorithm, in which errors are modeled by a mixture of Gaussians[10]
instantaneously (Fig. 3-(a-c)).

Furthermore, to estimate the pose of bricks, we find the minimum bounding box
by using the _calipper_ algorithm with bounding box containing edges of the
convex hull of the plane[11]. Fig. 3-(d) shows an instance of pose estimation,
in which the bounding box is in by blue color and the orientation vector is in
red color.

###  2.3 Task Execution

The task planner has the main role of deciding the sequences of manipulator
motion to enable the effective pick and place of bricks.

Given the pose $(\mathbf{p},\alpha$) of the brick in the conveyor and its
target pose $(\mathbf{p}_{b},\alpha_{b}$), the manipulator proceeds as
follows:

  * •

The gripper is aligned to the brick pose $(\mathbf{p},\alpha$) by first
aligning over the $x-y$ plane over the point
$\mathbf{q}=\mathbf{p}+\eta.\mathbf{\hat{k}}$ for $\eta>0$. Then, the brick is
picked up from the midpoints of its largest sides and is moved to
$\mathbf{q}$.

  * •

The planner finds a path that drives the robot from its configuration at
$\mathbf{q}$ to the target pose $(\mathbf{p}_{b},\alpha_{b}$) in the shortest
path in Cartesian space (a straight line).

  * •

Move the gripper into the desired pose $(\mathbf{p}_{b},\alpha_{b}$), open the
gripper and move back to the position and wait for the gripper to grasp the
next brick.

  * •

Continue and repeat all the above steps until all bricks are transported to
their desired poses in the spiral brick columns.

##  3 Computational Experiments

In order to evaluate the effectiveness of our approach we used a simulation
environment which represents the interactions and physics of rigid body
dynamics. We used Matlab for 3D modeling and planning execution tasks and
Coppelia Simulation environment for robot manipulation tasks. simulation
parameters involve bricks with $w=0.5$, $l=0.1$, and height $h=0.025$,
$\lambda=0.01$, $\kappa=0.05$, $L=17$, and $\eta=1.25$. We used the geometries
and configurations of base polygons as shown by Fig. 2. Fine tuning of the
parameters is our of the scope of this paper.

In order to show the performance of our proposed approach, Fig. 4 shows the
errors in position and orientation in the spiral brick column, and the time
used in pose estimation and robot pick and place. Also, Fig. 5 shows a number
of examples of spiral brick columns in our simulation environment. As we can
observe from Fig. 4-(a,b) the position error in terms of euclidean distance is
less than o.025 with a number of spikes at certain bricks in orthogonal
column, and the orientation error in terms of absolute terms coincides with a
mean of about 0.025 degrees. There are spikes at certain bricks which give a
maximum difference at 0.125 degrees.

Also, by observing Fig. 4-(c), the pose estimation is less than half a second
with a mean of about 0.2 seconds, implying the real usefulness for real time
interaction. Furthermore, by observing Fig. 4-(d), the trajectory time is the
time needed for the robot assembly from pick to place in the desired position
and orientation. Fig. 4-(d) sows that the concave decagon model takes larger
time compared to other models such that maximum’s trajectory time per brick is
7 s.. Also, the orthogonal model takes the shortest time compared to other
models such that the minimum time is 3 s.

##  4 Final Notes

In this paper, we have propose an approach for task-level planning of assembly
of spiral brick columns. Our computational studies have shown the feasibility
and efficiency of our approach to render spiral brick columns in simulation
environments. Our study aims at elucidating the robot plans being effective
for automated construction of aesthetic spiral brick columns.

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