# Non-compact inaudibility of weak symmetry and commutativity via generalized
Heisenberg groups

Teresa Arias-Marco and José Manuel Fernández-Barroso
Department of Mathematics,
University of Extremadura,
Badajoz, Spain
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Two Riemannian manifolds are said to be isospectral if there exists a unitary
operator which intertwines their Laplace-Beltrami operator. In this paper, we
prove in the non-compact setting the inaudibility of the weak symmetry
property and the commutative property using an isospectral pair of 23
dimensional generalized Heisenberg groups.

Keywords: Laplace operator; Isospectral Riemannian manifolds; Weakly-symmetric
spaces; Commutative spaces; g.o. spaces

MSC2020: 58J53; 53C25; 58J50.

##  1 Introduction

Locally symmetric spaces are Riemannian manifolds whose local geodesic
symmetries are local isometries. Symmetric-like spaces are Riemannian
manifolds with a geometric or algebraic property that locally symmetric spaces
satisfy. Weakly symmetric and commutative spaces are examples of symmetric-
like spaces. Selberg in [15] introduced the notion of weakly symmetric spaces
as those on which any two points can be interchanged by an isometry. In the
same paper, Selberg proved that the algebra of all invariant differential
operators is commutative in every weakly symmetric space, these spaces are
known as commutative spaces.

Kaplan, in [10], developed examples of non locally symmetric Riemannian
manifolds, namely generalized Heisenberg groups. The study of symmetric-like
properties in generalized Heisenberg groups has been studied by many authors
(e.g. [14],[13],[5],[4] ).

Two Riemannian manifolds, $M$ and $M^{\prime}$ are said to be isospectral if
there exists a unitary operator $T:L^{2}(M^{\prime})\to L^{2}(M)$ which
intertwines their Laplacians, $\Delta$ and $\Delta^{\prime}$, respectively,
this is such that $T\circ\Delta^{\prime}=\Delta\circ T$. If $M$ and
$M^{\prime}$ are compact Riemannian manifolds, the definition of being
isospectral is equivalent to the fact that both manifolds have the same
spectrum. In particular, Gordon and Wilson in [9] studied the isospectrality
between pairs of generalized Heisenberg groups. Szabó in [17] introduced a new
example of isospectral closed Riemannian manifolds which can be seen as
compact submanifolds of generalized Heisenberg groups. Szabó’s examples were
also introduced independently by Gordon, Gornet, Schueth, Webb and Wilson in
[8], using a different approach.

A geometric property is said to be inaudible, or it cannot be heard, when one
can find isospectral Riemannian manifolds such that one of them satisfies that
property and the other does not. In [1], the authors proved the inaudibility
of some symmetric-like properties using Szabó’s examples. In this paper, we
rely on the work done in [9], and we use a pair of isospectral generalized
Heisenberg groups of dimension 23 to develop our main result:

The property of being weakly symmetric and the property of being commutative
are inaudible properties on non-compact Riemannian manifolds.

In Section 2, we treat the isospectrality between generalized Heisenberg
groups. For that, we first study the isospectrality between compact 2-step
nilmanifolds (i.e. quotients of 2-step nilpotent Lie groups) whose coverings
are generalized Heisenberg groups. Finally, in Section 3, we remember the main
results concerning the study of symmetric-like properties on generalized
Heisenberg groups, and we prove our main result.

##  2 Isospectrality on generalized Heisenberg groups

A well studied kind of manifolds are generalized Heisenberg groups, which are
particular examples of 2-step nilpotent Lie groups. These manifolds were
introduced by Kaplan in [10] (for a more detailed information, consult [5]).
In this section we show how they are constructed, and how they are classified
according to their dimension.

Let consider $\mathfrak{v}$ and $\mathfrak{z}$ real vector spaces which are
orthogonal with respect to an inner product $g$, and a linear map
$j:\mathfrak{z}\to\mathfrak{so(v)}$. Let denote by $\mathfrak{n}$ the real
vector space $\mathfrak{v}\oplus\mathfrak{z}$. Using this $j$ map, one can
define a Lie bracket $[\cdot,\cdot]$ on $\mathfrak{n}$ by

$g\left([X,Y],Z\right)=g\left(j_{Z}X,Y\right),$ (1)

for each $X,Y\in\mathfrak{v}$ and each $Z\in\mathfrak{z}$. This Lie bracket
makes $\mathfrak{n}$ a 2-step nilpotent Lie algebra, this is
$[\mathfrak{n},\mathfrak{n}]\subset\mathfrak{z}$ and
$[\mathfrak{n},\mathfrak{z}]=0$. Thus, $\mathfrak{z}$ is also known as the
center of $\mathfrak{n}$. If we need to distinguish the Lie algebra or the Lie
bracket depending on the $j$ map, we will denote it by $\mathfrak{n}(j)$ and
$[\cdot,\cdot]^{j}$, respectively.

$N(j)$ is the 2-step nilpotent Lie group whose Lie algebra is
$\mathfrak{n}(j)$, and together with the metric induced by the inner product
$g$, that we denote it again by $g$, the pair $(N(j),g)$ is a Riemannian
manifold. The usual Lie group exponential map $\exp^{j}:\mathfrak{n}(j)\to
N(j)$ which satisfies the Cambell–Baker–Hausdorff formula

$\exp^{j}(X)\cdot\exp^{j}(Y)=\exp^{j}\left(X+Y+\frac{1}{2}[X,Y]^{j}\right),$

is a diffeomorphism due to $N(j)$ is simply-connected and 2-step nilpotent. We
will denote $\exp$ instead of $\exp^{j}$, if there is no confusion with $j$.

A 2-step nilpotent Lie algebra, $\mathfrak{n}$, is said to be a generalized
Heisenberg algebra when, additionally, $j$ satisfies the following relation

$j_{Z}^{2}=-\|Z\|^{2}\textup{Id}_{\mathfrak{v}},$ (2)

for every $Z\in\mathfrak{z}$. In that case, its connected, simply-connected
2-step nilpotent Lie group, $N(j)$, is a generalized Heisenberg group, and $j$
is a map of Heisenberg type.

Generalized Heisenberg algebras,
$\mathfrak{n}=\mathfrak{v}\oplus\mathfrak{z}$, can be classified using the
classification of representations of Clifford algebras over the real vector
space $\mathfrak{z}$ attached to a negative definite quadratic form
$-\|\mathfrak{z}\|^{2}$. The classification can be found in [2], but we
include it here:

  1. i)

If $\dim\mathfrak{z}\not\equiv 3\mod 4$, then the Clifford algebra
$Cl(\mathfrak{z})$ has a unique irreducible Clifford module $\mathfrak{v}_{0}$
up to equivalence, and the generalized Heisenberg algebra is obtained up to
isomorphism, taking the direct sum of $p$ times the irreducible Clifford
module, this is, $\mathfrak{v}\cong(\mathfrak{v}_{0})^{p},p\geq 1.$

  2. ii)

If $\dim\mathfrak{z}\equiv 3\mod 4$, then the Clifford algebra
$Cl(\mathfrak{z})$ has two non-equivalent irreducible Clifford modules
$\mathfrak{v}_{1}$ and $\mathfrak{v}_{2}$, up to equivalence. The generalized
Heisenberg algebras are obtained up to isomorphism, taking
$\mathfrak{v}\cong(\mathfrak{v}_{1})^{p}\oplus(\mathfrak{v}_{2})^{q}$, with
$p\geq 0,q\geq 0,p+q\geq 1$, which is isomorphic to
$\mathfrak{v}\cong(\mathfrak{v}_{1})^{q}\oplus(\mathfrak{v}_{2})^{p}$ due to
$\mathfrak{v}_{1}$ and $\mathfrak{v}_{2}$ have the same dimension. In this
case, we name the generalized Heisenberg algebras as $\mathfrak{n}_{p,q}$.

With this notation, $\mathfrak{v}$ is said to be isotypic if its Clifford
module structure is irreducible. When $\dim\mathfrak{z}\not\equiv 3\mod 4$,
$\mathfrak{v}$ is trivially isotypic. If $\dim\mathfrak{z}\equiv 3\mod 4$,
$\mathfrak{v}$ is isotypic when $p=0$ or $q=0$, with $p+q\geq 1$.

Note that, in the case when $\dim\mathfrak{z}\equiv 3\mod 4$, if we fix non-
negative integers $(p_{1},q_{1})$ and $(p_{2},q_{2})$ such that
$p_{1}+q_{1}=p_{2}+q_{2}$, then we obtain generalized Heisenberg algebras of
the same dimension, $\mathfrak{n}_{p_{1},q_{1}}$ and
$\mathfrak{n}_{p_{2},q_{2}}$. These algebras are isomorphic if and only if
$(p_{2},q_{2})=\\{(p_{1},q_{1}),(q_{1},p_{1})\\}$.

Now, we want to find a pair of isospectral and non-isometric 2-step
nilmanifolds whose Riemannian coverings are generalized Heisenberg groups. We
also show the isospectrality between those Riemannian coverings which are non-
compact.

Following Wilson [18] and Gordon [6], we say that a Riemannian manifold is a
2-step nilmanifold if it is a quotient of a simply-connected 2-step nilpotent
Lie group $N(j)$ by a possible trivial discrete subgroup $\Gamma(j)$, together
with a left invariant Riemannian metric $g$. We denote it by
$N^{j}=(N(j)/\Gamma(j),g)$ where $\Gamma(j)=\exp^{j}(\mathcal{M}+\mathcal{L})$
with $\mathcal{M}$ and $\mathcal{L}$ cocompact lattices in $\mathfrak{v}$ and
$\mathfrak{z}$, respectively.

Wilson [18] proved that two simply-connected Riemannian 2-step nilmanifolds
are isometric if and only if the associated metric Lie algebras are
isomorphic. According to the classification of the generalized Heisenberg
algebras in terms of the attached Clifford algebra, there are up to
isomorphisms one generalized Heisenberg algebra in each dimension when
$\dim\mathfrak{z}\not\equiv 3\mod 4$, and there are two Lie algebras when
$\dim\mathfrak{z}\equiv 3\mod 4$. We will center our attention in the case
when $\dim\mathfrak{z}\equiv 3\mod 4$.

Gordon introduced in [6] the following map in order to work with the isotypic
and the non isotypic representation of $\mathfrak{v}$ at the same time. Let
$A$ denotes the quaternion numbers $\mathbb{H}$ or the octonion numbers
$\mathbb{O}$. Suppose that $\mathfrak{v}=(A)^{p}\oplus(A)^{q},p,q\geq
0,p+q\geq 1,p,q\in\mathbb{N}$, and the center is the pure elements of $A$,
$\mathfrak{z}=A^{*}$. The map $j^{(p,q)}:\mathfrak{z}\to\mathfrak{so(v)}$ is
given by

$j^{(p,q)}_{Z}(X_{1},\dots,X_{p},X_{p+1},\dots,X_{p+q})=(Z\cdot
X_{1},\dots,Z\cdot X_{p},X_{p+1}\cdot Z,\dots,X_{p+q}\cdot Z),$ (3)

where $Z\in\mathfrak{z},X_{i}\in A,i=1,\dots,p+q$, and $\cdot$ is the usual
multiplication in $A$. This map is always of Heisenberg type independently of
the values of $p$ and $q$.

We denote by $\mathfrak{n}(p,q)$ the generalized Heisenberg algebra
$\mathfrak{n}_{p,q}(j^{(p,q)})$, by $N(p,q)$ the generalized Heisenberg group
$\exp(\mathfrak{n}(p,q))$, and by $\mathcal{M}_{p,q}+\mathcal{L}$ the lattice
in $\mathfrak{n}(p,q)$ spanned by its integer standard basis. The 2-step
nilmanifolds of Heisenberg type obtained by taking the cocompact quotient of
$N(p,q)$ by $\Gamma_{p,q}:=\exp(\mathcal{M}_{p,q}+\mathcal{L})$, with its left
invariant Riemannian metric, are denoted by $N^{p,q}$. With this notation,
Gordon proved in [6] the following result.

######  Theorem 2.1.

$N^{p,q}$ and $N^{p^{\prime},q^{\prime}}$ are isospectral if
$p+q=p^{\prime}+q^{\prime}$. Moreover, they are locally isometric if and only
if $(p^{\prime},q^{\prime})=\\{(p,q),(q,p)\\}$.

######  Remark 2.2.

This result allows us to find pairs of isospectral 2-step nilmanifolds
constructed using generalized Heisenberg groups. One possible situation in the
case $\dim\mathfrak{z}\equiv 3\mod 4$ is

$\textstyle{N(p,q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N(p+q,0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N^{p,q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N^{p+q,0}}$

where $N(p,q)$ and $N(p+q,0)$ with $p\geq 0,q\geq 0,p+q\geq 1$, are the
Riemannian covering of $N^{p,q}$ and $N^{p+q,0}$, respectively, and
$\textstyle{N^{p,q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N^{p+q,0}}$
means that $N^{p,q}$ and $N^{p+q,0}$ are isospectral and not locally isometric
in the compact sense. In particular, if $A=\mathbb{H}$ (resp. $A=\mathbb{O}$),
then $\dim\mathfrak{z}=3$ (resp. $\dim\mathfrak{z}=7$), and $N^{1,1}$ and
$N^{2,0}$ are isospectral. Moreover, they are not locally isometric, due to
their corresponding 11-dimensional (resp. 23-dimensional) generalized
Heisenberg algebras are not isomorphic.

Now, we want to study the isospectrality in the non-compact sense between the
Riemannian coverings $N(p,q)$ and $N(p+q,0)$. Szabó [17], proved their
isospectrality in the case when $A=\mathbb{H}$, by constructing the explicit
unitary operator which intertwines their Laplacians. The purpose of the rest
of the section is to prove their isospectrality also when $A=\mathbb{O}$.

######  Proposition 2.3.

Let consider $A=\mathbb{O}$. Then, $N(p+q,0)$ and $N(p,q)$, $p\geq 0,q\geq
0,p+q\geq 1$, are isospectral in the non-compact sense. Equivalently, there
exists an operator, $T:L^{2}(N(p,q))\to L^{2}(N(p+q,0))$, intertwining the
Laplacians $\Delta^{(p,q)}$ and $\Delta^{(p+q,0)}$, this is

$T\circ\Delta^{(p,q)}=\Delta^{(p+q,0)}\circ T.$ (4)

###### Proof.

The generalized Heisenberg groups with 7-dimensional center $N(p,q)$ and
$N(p+q,0)$ are diffeomorphic to $\mathbb{R}^{8(p+q)}\times\mathbb{R}^{7}$.
Then, it is enough to prove the existence of an operator
$T:L^{2}(\mathbb{R}^{8(p+q)}\times\mathbb{R}^{7})\to
L^{2}(\mathbb{R}^{8(p+q)}\times\mathbb{R}^{7})$ satisfying (4). From now on,
we write $L^{2}(N(p,q))$ or $L^{2}(N(p+q,0))$ instead of
$L^{2}(\mathbb{R}^{8(p+q)}\times\mathbb{R}^{7})$ to distinguish the space of
functions where the operator is working on. In order to do that, we first
study the structure of $\Delta^{(p,q)}$.

Let consider the following lattice

$\mathcal{L}=\left\\{\sum_{i=1}^{7}\alpha_{i}\mathbf{e}_{i}:\alpha=(\alpha_{1},\dots,\alpha_{7})\in\mathbb{Z}^{7}\right\\},$

where $\\{\mathbf{e}_{i}\\}_{i=1}^{7}$ is the standard basis of
$\mathbb{R}^{7}$. The quotient
$\mathbb{R}^{8(p+q)}\times(\mathbb{R}^{7}/\mathcal{L})$ can be seeing as
$\mathbb{R}^{8(p+q)}\times\mathbb{T}^{7}$, where $\mathbb{T}^{7}$ is the
7-dimensional torus whose volume is 1. Thus,
$\mathbb{R}^{8(p+q)}\times\mathbb{T}^{7}$ is diffeomorphic to $N(p,q)/\Gamma$,
where $\Gamma=\exp(\mathcal{L})$. We denote by $\tilde{\Delta}^{(p,q)}$ the
Laplace-Beltrami operator associated with the Riemannian manifold
$N(p,q)/\Gamma$.

As before, we write $L^{2}(N(p,q)/\Gamma)$ or $L^{2}(N(p+q,0)/\Gamma)$ instead
of $L^{2}(\mathbb{R}^{8(p+q)}\times\mathbb{T}^{7})$ in order to understand
where the operators are acting. To relate $\Delta^{(p,q)}$ with
$\tilde{\Delta}^{(p,q)}$, we need to establish a relation between functions in
$L^{2}(N(p,q)/\Gamma)$ and functions in $L^{2}(N(p,q))$. We shall use the
Fourier transform given by

$\mathcal{F}(f)(X,Z)=\int f(X,\xi)e^{-i\langle\xi,Z\rangle}d\xi,\quad
X\in\mathbb{R}^{8(p+q)},\;Z\in\mathbb{R}^{7},\;\xi\in\mathbb{T}^{7},$

where $f\in(L^{1}\cap L^{2})(N(p,q)/\Gamma)$. It is well defined almost
everywhere and it has a unique extension onto $L^{2}(N(p,q)/\Gamma)$. Indeed,
for each $f\in L^{2}(N(p,q)/\Gamma)$, it is satisfied

$\Delta^{(p,q)}\mathcal{F}(f)=\mathcal{F}(\tilde{\Delta}^{(p,q)}f).$ (5)

Therefore, we only need to study the structure of $\tilde{\Delta}^{(p,q)}$.
Let fix an element
$Z^{\alpha}=\sum_{i=1}^{7}\alpha_{i}\mathbf{e}_{i}\in\mathcal{L}$, with
$\alpha=(\alpha_{1},\dots,\alpha_{7})\in\mathbb{Z}^{7}$, then, the function

$E^{\alpha}(Z)=e^{2\pi i\cdot\langle Z^{\alpha},Z\rangle},\
Z\in\mathbb{R}^{7},$

is periodic in $\mathcal{L}$. Let define the functions in
$L^{2}_{\mathbb{C}}(N(p,q))$ by

$\Phi^{\alpha}(X,Z)=\varphi(X)E^{\alpha}(Z)$ (6)

with $\varphi$ in the complex function space
$L^{2}_{\mathbb{C}}(\mathbb{R}^{8(p+q)})$. In addition, they are also well
defined on $N(p,q)/\Gamma$. The complex function space of $N(p,q)/\Gamma$ can
be decomposed in the following way

$L^{2}_{\mathbb{C}}(N(p,q)/\Gamma)=\bigoplus_{\alpha\in\mathbb{Z}^{7}}L^{2}_{\mathbb{C}^{\alpha}}(N(p,q)/\Gamma),$
(7)

where $L^{2}_{\mathbb{C}^{\alpha}}(N(p,q)/\Gamma)$ is the function space
spanned by the functions of the form (6). Moreover, the Laplace-Beltrami
operator acts separately on each subspace
$L^{2}_{\mathbb{C}^{\alpha}}(N(p,q)/\Gamma)$, and they are invariant by its
action, thus, one can consider the eigenvalue problem separately on each
subspace. Szabó in [17] computed the Laplace-Beltrami operator on these
subspaces, it is given by 111Note that the equations (8) and (9) differ from
the ones presented in [17] in the coefficient $\frac{\dim\mathfrak{v}}{4}$.

$(\tilde{\Delta}_{\alpha}^{(p,q)}\Phi^{\alpha})(X,Z)=\left(\Delta_{\mathfrak{v}}+2\pi
i\cdot\left(j^{(p,q)}_{Z^{\alpha}}X\right)\bullet-4\pi^{2}\|Z^{\alpha}\|^{2}\cdot\left(1+\frac{\dim\mathfrak{v}}{4}\|X\|^{2}\right)\right)(\varphi(X))E^{\alpha}(Z),$
(8)

where $\Delta_{\mathfrak{v}}$ is the Laplace-Beltrami operator in the
Euclidean space $\mathfrak{v}\cong\mathbb{R}^{8(p+q)}$, and
$\left(j^{(p,q)}_{Z}X\right)\bullet\varphi$ acts as a derivation for every
$\varphi\in L^{2}_{\mathbb{C}}(\mathbb{R}^{8(p+q)})$, $X\in\mathfrak{v}$,
$Z\in\mathfrak{z}\cong\mathbb{R}^{7}$, with respect to the vector field
$j^{(p,q)}_{Z}X$, this is

$\left(j^{(p,q)}_{Z}X\right)\bullet\varphi(X)=\sum_{i=1}^{8(p+q)}\frac{\partial\varphi}{\partial
x_{i}}(X)\cdot g\left(j^{(p,q)}_{Z}X,\mathbf{u}_{i}\right),$

where $\\{x_{i}\\}_{i=1}^{8(p+q)}$ are the coordinate functions on
$\mathfrak{v}$, $\\{\mathbf{u}_{i}\\}_{i=1}^{8(p+q)}$ is the standard basis of
$\mathfrak{v}$, and $g$ is the inner product defined on $\mathfrak{n}(p,q)$
with 7-dimensional center. This derivation is also well defined on functions
$f\in L^{2}_{\mathbb{C}^{\alpha}}(N(p,q)/\Gamma)$, by

$\left(j^{(p,q)}_{Z}X\right)\bullet f(X,V)=\sum_{i=1}^{8(p+q)}\frac{\partial
f}{\partial x_{i}}(X,V)\cdot g\left(j^{(p,q)}_{Z}X,\mathbf{u}_{i}\right).$

We denote both of them with the same notation if there is not risk of
confusion. Thus, using the orthogonal decomposition of
$L^{2}_{\mathbb{C}^{\alpha}}(N(p,q)/\Gamma)$ given in (7) and the expresion of
the Laplace-Beltrami given on each subspace by (8), Szabó [17] established the
Laplace-Beltrami operator associated with $N(p,q)/\Gamma$

$\tilde{\Delta}^{(p,q)}f(X,\xi)=\left(\Delta_{\mathfrak{v}}-i\cdot\left(j^{(p,q)}_{\xi}X\right)\bullet-\left(1+\frac{\dim\mathfrak{v}}{4}\|X\|^{2}\right)\|\xi\|^{2}\right)f(X,\xi),$
(9)

where $f$ is an arbitrary function in $L^{2}_{\mathbb{C}}(N(p,q)/\Gamma)$.

In order to find $T$ satisfying (4), it is important to note that
$\tilde{\Delta}^{(p,q)}$ and $\tilde{\Delta}^{(p+q,0)}$ given by (9) only
differ in the term depending on the $j$ map,
$\left(j^{(p,q)}_{\xi}(X)\right)\bullet$ and
$\left(j^{(p+q,0)}_{\xi}(X)\right)\bullet$. Moreover, due to (7) and (8), it
is enough to obtain an operator intertwining
$\left(j^{(p,q)}_{Z^{\alpha}}(X)\right)\bullet$ and
$\left(j^{(p+q,0)}_{Z^{\alpha}}X\right)\bullet$ on each
$L^{2}_{\mathbb{C}^{\alpha}}(N(p,q)/\Gamma)$.

Let consider $\iota:\mathbb{R}^{7}\to\mathbb{O}^{*}$ the natural embedding of
$\mathbb{R}^{7}$ into the pure octonion space, $\bar{X}$ is the conjugate of
$X\in\mathbb{O}$, and $Z^{\alpha}$ a fixed unitary vector in $\mathcal{L}$.
Let take $\nu^{\alpha}\in\mathbb{R}^{7}$ an unit vector orthogonal to
$Z^{\alpha}$, this means that $\|\nu^{\alpha}\|=1$ and
$\langle\nu^{\alpha},Z^{\alpha}\rangle=0$, with the usual scalar product in
$\mathbb{R}^{7}$, equivalently
$\textup{Re}(\iota(\nu^{\alpha})\cdot\iota(Z^{\alpha}))=0$. Thus, it is
important to note that $\iota(\nu^{\alpha})\cdot\iota(Z^{\alpha})$ remains
inside $\mathbb{O}^{*}$, and $\bar{X}=-X$, for every $X\in\mathbb{O}^{*}$.
Additionally, for any two orthogonal pure octonions, $X$ and $Y$, then $X\cdot
Y=-Y\cdot X$. With that setting, we can define

$\sigma_{(p,q)}^{\alpha}:\mathbb{O}^{p+q}\longrightarrow\mathbb{O}^{p+q},$

by

$\small\sigma_{(p,q)}^{\alpha}(X_{1},\dots,X_{p},X_{p+1},\dots,X_{p+q}):=(X_{1},\dots,X_{p},\overline{X_{p+1}\cdot\iota(\nu^{\alpha})},\dots,\overline{X_{p+q}\cdot\iota(\nu^{\alpha})}).$
(10)

Note that

$\displaystyle\overline{Y\cdot\iota(Z^{\alpha})\cdot\iota(\nu^{\alpha})}$
$\displaystyle=\iota(\nu^{\alpha})\cdot\iota(Z^{\alpha})\cdot\overline{Y}$
$\displaystyle=\iota(Z^{\alpha})\cdot(-\iota(\nu^{\alpha}))\cdot\overline{Y}$
$\displaystyle=\iota(Z^{\alpha})\cdot\overline{Y\cdot\iota(\nu^{\alpha})},$

for every $Y\in\mathbb{O}$. Thus, for some $X\in\mathbb{O}^{p+q}$,

$\displaystyle\sigma_{(p,q)}^{\alpha}$
$\displaystyle(j^{(p,q)}_{\iota(Z^{\alpha})}(X))=\sigma_{(p,q)}^{\alpha}(\iota(Z^{\alpha})\cdot
X_{1},\dots,\iota(Z^{\alpha})\cdot
X_{p},X_{p+1}\cdot\iota(Z^{\alpha}),\dots,X_{p+q}\cdot\iota(Z^{\alpha}))$
$\displaystyle=(\iota(Z^{\alpha})\cdot X_{1},\dots,\iota(Z^{\alpha})\cdot
X_{p},\overline{X_{p+1}\cdot\iota(Z^{\alpha})\cdot\iota(\nu^{\alpha})},\dots,\overline{X_{p+q}\cdot\iota(Z^{\alpha})\cdot\iota(\nu^{\alpha})})$
$\displaystyle=(\iota(Z^{\alpha})\cdot X_{1},\dots,\iota(Z^{\alpha})\cdot
X_{p},\iota(Z^{\alpha})\cdot\overline{X_{p+1}\cdot\iota(\nu^{\alpha})},\dots,\iota(Z^{\alpha})\cdot\overline{X_{p+q}\cdot\iota(\nu^{\alpha})})$
$\displaystyle=j^{(p+q,0)}_{\iota(Z^{\alpha})}(\sigma_{(p,q)}^{\alpha}(X)).$

This is

$\sigma_{(p,q)}^{\alpha}\circ
j^{(p,q)}_{\iota(Z^{\alpha})}=j^{(p+q,0)}_{\iota(Z^{\alpha})}\circ\sigma_{(p,q)}^{\alpha}.$

Therefore, the induced map

$\sigma_{(p,q)*}^{\alpha}:L^{2}_{\mathbb{C}^{\alpha}}(N(p,q)/\Gamma)\to
L^{2}_{\mathbb{C}^{\alpha}}(N(p+q,0)/\Gamma)$

is defined, for each $X_{1},\dots,X_{p+q}\in\mathbb{O}$ and
$Z\in\mathbb{T}^{7}$, by

$\sigma_{(p,q)*}^{\alpha}f(X_{1},\dots,X_{p+q},Z):=f(\sigma_{(p,q)}^{\alpha}(X_{1},\dots,X_{p+q}),Z).$

Note that $\sigma_{(p,q)*}^{\alpha}(f\cdot
g)=\sigma_{(p,q)*}^{\alpha}f\cdot\sigma_{(p,q)*}^{\alpha}g$, for every $f,g\in
L^{2}_{\mathbb{C}^{\alpha}}(N(p,q)/\Gamma)$. Then, $\sigma_{(p,q)*}^{\alpha}$
intertwines $\left(j^{(p,q)}_{\iota(Z^{\alpha})}X\right)\bullet$ and
$\left(j^{(p+q,0)}_{\iota(Z^{\alpha})}X\right)\bullet$, this is

$\displaystyle\left(j^{(p+q,0)}_{\iota(Z^{\alpha})}X\right)\bullet\sigma_{(p,q)*}^{\alpha}(f(X,Z))$
$\displaystyle=\left(j^{(p+q,0)}_{\iota(Z^{\alpha})}X\right)\bullet\left(\sigma_{(p,q)*}^{\alpha}f(X,Z)\right)$
$\displaystyle=\left(j^{(p+q,0)}_{\iota(Z^{\alpha})}X\right)\bullet
f(\sigma_{(p,q)}^{\alpha}(X),Z))$ $\displaystyle=\frac{\partial f}{\partial
X}(\sigma_{(p,q)}^{\alpha}(X),Z)\cdot
f\left(j^{(p+q,0)}_{\iota(Z^{\alpha})}\left(\sigma_{(p,q)}^{\alpha}(X),Z\right)\right)$
$\displaystyle=\sigma_{(p,q)*}^{\alpha}\left(\frac{\partial f}{\partial
X}(X,Z)\right)\cdot
f\left(\sigma_{(p,q)}^{\alpha}\left(j^{(p,q)}_{\iota(Z^{\alpha})}X\right),Z\right)$
$\displaystyle=\sigma_{(p,q)*}^{\alpha}\left(\frac{\partial f}{\partial
X}(X,Z)\right)\cdot\sigma_{(p,q)*}^{\alpha}f\left(j^{(p,q)}_{\iota(Z^{\alpha})}X,Z\right)$
$\displaystyle=\sigma_{(p,q)*}^{\alpha}\left(\frac{\partial f}{\partial
X}(X,Z)\cdot f\left(j^{(p,q)}_{\iota(Z^{\alpha})}X,Z\right)\right)$
$\displaystyle=\sigma_{(p,q)*}^{\alpha}\left(\left(j^{(p,q)}_{\iota(Z^{\alpha})}X\right)\bullet
f(X,Z)\right),$

for every $f(X,Z)\in L^{2}_{\mathbb{C}^{\alpha}}(N(p,q)/\Gamma)$.

Now, due to $\sigma_{(p,q)*}^{\alpha}$ acts separately for each $\alpha$, it
leaves each subspace $L^{2}_{\mathbb{C}^{\alpha}}(N(p,q)/\Gamma)$ invariant.
Thus, we can define

$\sigma_{(p,q)*}:=\bigoplus_{\alpha}\sigma_{(p,q)*}^{\alpha}:L^{2}_{\mathbb{C}}(N(p,q)/\Gamma)\to
L^{2}_{\mathbb{C}}(N(p+q,0)/\Gamma),$

such that

$\sigma_{(p,q)*}\circ\left(j^{(p,q)}_{\xi}(X)\right)\bullet=\left(j^{(p+q,0)}_{\xi}(X)\right)\bullet\circ\sigma_{(p,q)*}.$

Therefore, $\sigma_{(p,q)*}$ intertwines the Laplacians of $N(p,q)/\Gamma$ and
$N(p+q,0)/\Gamma$, this is

$\sigma_{(p,q)*}\circ\tilde{\Delta}^{(p,q)}=\tilde{\Delta}^{(p+q,0)}\circ\sigma_{(p,q)*}.$

Let consider $T:L^{2}(N(p,q))\to L^{2}(N(p+q,0))$ given by

$T=\mathcal{F}\circ\sigma_{(p,q)*}\circ\mathcal{F}^{-1}.$

Then, for each $f\in L^{2}(N(p,q)/\Gamma)$, it is satisfied

$\displaystyle T\circ\Delta^{(p,q)}(\mathcal{F}(f))$
$\displaystyle=\mathcal{F}\circ\sigma_{(p,q)*}\circ\mathcal{F}^{-1}(\Delta^{(p,q)}(\mathcal{F}(f)))=\mathcal{F}\circ\sigma_{(p,q)*}\circ\mathcal{F}^{-1}(\mathcal{F}(\tilde{\Delta}^{(p,q)}f))$
$\displaystyle=\mathcal{F}\circ\sigma_{(p,q)*}(\tilde{\Delta}^{(p,q)}f)=\mathcal{F}(\tilde{\Delta}^{(p+q,0)}\circ\sigma_{(p,q)*}(f))$
$\displaystyle=\Delta^{(p+q,0)}(\mathcal{F}(\sigma_{(p,q)*}(f)))=\Delta^{(p+q,0)}(\mathcal{F}\circ\sigma_{(p,q)*}\circ\mathcal{F}^{-1}(\mathcal{F}(f)))$
$\displaystyle=\Delta^{(p+q,0)}(\mathcal{F}\circ\sigma_{(p,q)*}\circ\mathcal{F}^{-1})(\mathcal{F}(f))=\Delta^{(p+q,0)}\circ
T(\mathcal{F}(f)).$

Equivalently,

$T\circ\Delta^{(p,q)}=\Delta^{(p+q,0)}\circ T.$

∎

##  3 Main results and conclusions

Generalized Heisenberg groups provide examples of Riemannian manifolds with
different geometric properties depending on their dimension.

Let $(M,g)$ be a Riemannian manifold, $\nabla$ its Levi–Civita connection and
the curvature tensor of $M$,

$R(X,Y)=[\nabla_{X},\nabla_{Y}]-\nabla_{[X,Y]},\quad X,Y\in\mathfrak{X}(M).$

Fixing a point $p\in M$ and a vector $v\in T_{p}M$, let denote by $\gamma_{v}$
the geodesic starting in $p$ with initial direction $v$. Then, a local
geodesic symmetry is the diffeomorphism $s_{p}$ such that it fixes the point
$p$ and involves the vector of the geodesics in a neighborhood of the point,
this is $s_{p}\circ\gamma_{v}=\gamma_{-v}$.

A Riemannian manifold is said to be locally symmetric if the local geodesic
symmetries are in addition isometries. Moreover, Selberg in [15] introduced
the notion of weak symmetry. These spaces are those on which any two points
can be interchanged by an isometry. Berndt, Prüfer and Vanhecke proved in [3]
that weak symmetry is equivalent to ray symmetry introduced by Szabó in [16].

Selberg proved in [15] that the algebra of all invariant differential
operators of a weakly symmetric space is commutative. This fact points out a
new definition, a commutative space is a Riemannian space whose algebra of all
invariant differential operators are commutative. Now, a natural question
arises: Is any commutative Riemannian manifold a weakly symmetric space? This
question was answered negatively by Lauret in [12], where he found examples of
commutative spaces which are not weakly symmetric.

Berndt, Ricci and Vanhecke in [4] distinguished between weakly symmetric
spaces in the broad sense, this is the original definition of Selberg, and
weakly symmetric spaces in the narrow sense, where the two points
interchanging isometries must belong to the identity component of the group of
isometries.

Weak symmetry and commutativity were studied in generalized Heisenberg groups
by Bernd, Ricci and Vanhecke in [4], and by Ricci in [13], respectively. These
authors classified the generalized Heisenberg groups satisfying these
properties in terms of their dimension.

######  Theorem 3.1 (Ricci, [13]).

A generalized Heisenberg group is commutative if and only if

  1. 1.

$\dim\mathfrak{z}=1,2$ or 3,

  2. 2.

$\dim\mathfrak{z}=5$ and $\dim\mathfrak{v}=8$,

  3. 3.

$\dim\mathfrak{z}=6$ and $\dim\mathfrak{v}=8$,

  4. 4.

$\dim\mathfrak{z}=7$ and:

    1. i)

$\dim\mathfrak{v}=8$,

    2. ii)

$\dim\mathfrak{v}=16$, with $\mathfrak{v}$ isotypic, this is
$\mathfrak{n}=\mathfrak{n}_{2,0}\cong\mathfrak{n}_{0,2}$.

######  Theorem 3.2 (Berndt, Ricci, Vanhecke, [4]).

A generalized Heisenberg group is weakly symmetric in the broad sense if and
only if it is commutative. Moreover, every weakly symmetric in the broad sense
generalized Heisenberg group is weakly symmetric in the narrow sense except
the case when $\dim\mathfrak{z}=1$.

Following Kowalski and Vanhecke [11], a g.o. space is a Riemannian manifold
$(M,g)$ whose geodesics are orbits of a one-parameter subgroup of the group of
the isometries of $M$. Generalized Heisenberg groups which are g.o. spaces
were studied and classified by Riehm in [14].

######  Theorem 3.3 (Riehm, [14]).

A generalized Heisenberg group is a Riemannian g.o. space if and only if

  1. 1.

$\dim\mathfrak{z}=1,2$ or 3,

  2. 2.

$\dim\mathfrak{z}=5$ and $\dim\mathfrak{v}=8$,

  3. 3.

$\dim\mathfrak{z}=6$ and $\dim\mathfrak{v}=8$,

  4. 4.

$\dim\mathfrak{z}=7$ and:

    1. i)

$\dim\mathfrak{v}=8$,

    2. ii)

$\dim\mathfrak{v}=16$, with $\mathfrak{v}$ isotypic, this is
$\mathfrak{n}=\mathfrak{n}_{2,0}\cong\mathfrak{n}_{0,2}$,

    3. iii)

$\dim\mathfrak{v}=24$, with $\mathfrak{v}$ isotypic, this is
$\mathfrak{n}=\mathfrak{n}_{3,0}\cong\mathfrak{n}_{0,3}$.

Note that, generalized Heisenberg groups provide an example of g.o. space
which is not weakly symmetric nor commutative, namely the generalized
Heisenberg group with $\dim\mathfrak{z}=7$, $\dim\mathfrak{v}=24$ and
$\mathfrak{v}$ isotypic.

In order to prove the main results, we combine Proposition 2.3 with the
classifications of generalized Heisenberg groups presented in Theorem 3.1,
Theorem 3.3, and Theorem 3.2.

######  Proposition 3.4.

One cannot hear if a non-compact Riemannian manifold is commutative.

###### Proof.

By Proposition 2.3, the 23-dimensional non-compact generalized Heisenberg
groups $N(2,0)$ and $N(1,1)$ with 7-dimensional center are isospectral. The
classification of generalized Heisenberg groups which are commutative spaces,
Theorem 3.1, establishes that $N(2,0)$ is a commutative space while $N(1,1)$
is not a commutative space. ∎

The next result was already proved by Gordon in [7] using a result by Gordon
and Wilson in [9]. Summarizing, its proof is as the previous one, but using
Theorem 3.3.

######  Proposition 3.5.

One cannot hear if a non-compact Riemannian manifold is a g.o. space.

Now, using again the same pair of isospectral manifolds together with Theorem
3.2, we obtain the result about the weak symmetric property.

######  Proposition 3.6.

One cannot hear if a non-compact Riemannian manifold is weakly symmetric,
neither in the broad sense nor in the narrow sense.

These three propositions point out the inaudibility of being a g.o. space,
being a commutative space, and being a weakly symmetric space in the non-
compact case using $N(2,0)$ and $N(1,1)$ with 7-dimensional center. However,
we cannot guarantee if their compact quotients, $N^{2,0}$ and $N^{1,1}$,
inherit the same global properties, and it is still open the audibility
problem of these properties in the compact setting. We can only guarantee that
$N^{2,0}$ is weakly locally symmetric while $N^{1,1}$ is not. This fact proves
the inaudibility in the compact case of the weak local symmetry in dimension
23. Nevertheless, this local property was already proved to be inaudible by
the first author and Schueth in [1], using a pair of 10-dimensional closed
isospectral manifolds introduced by Szabó in [17].

The main results proved in this paper add two new properties to the list of
inaudible Riemannian properties, being a weakly symmetric space and being a
commutative space, only in the non-compact case. So, it is natural to ask what
happen in the compact case. This is:

Can one find examples of isospectral closed Riemannian manifolds proving the
inaudibility of being weakly symmetric, commutative or a g.o. space in the
compact case?

Therefore, it is important to find new examples of isospectral non-isometric
pairs of compact Riemannian manifolds.

We would also like to remark that using the above mentioned classifications,
we can only guarantee the inaudibility in the non-compact case of the weakly
symmetric and the commutative properties in dimension 23. Moreover, using
Theorem 3.3 and working in the same way as in the proof of Proposition 3.5,
but using the 31-dimensional non-compact generalized Heisenberg groups
$N(3,0)$ and $N(2,1)$ with 7-dimensional center, we also obtain the
inaudibility in the non-compact case of the g.o. property in dimension 31.
This means, we can only guarantee the inaudibility of the g.o. property in the
non-compact case in dimension 23 and 31. Then, a second question arises:

Can one find examples of isospectral Riemannian manifolds in another
dimensions which prove the inaudibility of the same symmetric-like properties?

We hope to answer these questions in forthcoming papers using new examples of
closed isospectral manifolds developed by the authors.

Author contributions: All authors contributed equally to this research and in
writing the paper.

Funding: The authors are supported by the grants GR21055 and IB18032 funded by
Junta de Extremadura and Fondo Europeo de Desarrollo Regional. The first
author is also partially supported by grant PID2019-10519GA-C22 funded by
AEI/10.13039/501100011033.

Conflicts of Interest: The authors declare no conflict of interest. The
founders had no role in the design of the study; in the collection, analyses,
or interpretation of data; in the writing of the manuscript, or in the
decision to publish the results.

## References

  * [1] T. Arias-Marco and D. Schueth.  On inaudible curvature properties of closed Riemannian manifolds.  Ann. Global Anal. Geom., 37(4):339–349, 2010.
  * [2] M. F. Atiyah, R. Bott, and A. Shapiro.  Clifford modules.  Topology, 3(suppl, suppl. 1):3–38, 1964.
  * [3] J. Berndt, F. Prüfer, and L. Vanhecke.  Symmetric-like Riemannian manifolds and geodesic symmetries.  Proc. Roy. Soc. Edinburgh Sect. A, 125(2):265–282, 1995.
  * [4] J. Berndt, F. Ricci, and L. Vanhecke.  Weakly symmetric groups of Heisenberg type.  Differential Geom. Appl., 8(3):275–284, 1998.
  * [5] J. Berndt, F. Tricerri, and L. Vanhecke.  Generalized Heisenberg groups and Damek-Ricci harmonic spaces, volume 1598 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin, 1995.
  * [6] C. S. Gordon.  Isospectral closed Riemannian manifolds which are not locally isometric.  J. Differential Geom., 37(3):639–649, 1993.
  * [7] C. S. Gordon.  Homogeneous Riemannian manifolds whose geodesics are orbits.  In Topics in geometry, volume 20 of Progr. Nonlinear Differential Equations Appl., pages 155–174. Birkhäuser Boston, Boston, MA, 1996.
  * [8] C. S. Gordon, R. Gornet, D. Schueth, D. L. Webb, and E. N. Wilson.  Isospectral deformations of closed Riemannian manifolds with different scalar curvature.  Ann. Inst. Fourier (Grenoble), 48(2):593–607, 1998.
  * [9] C. S. Gordon and E. N. Wilson.  Continuous families of isospectral Riemannian metrics which are not locally isometric.  J. Differential Geom., 47(3):504–529, 1997.
  * [10] A. Kaplan.  Riemannian nilmanifolds attached to Clifford modules.  Geom. Dedicata, 11(2):127–136, 1981.
  * [11] O. Kowalski and L. Vanhecke.  Riemannian manifolds with homogeneous geodesics.  Boll. Un. Mat. Ital. B (7), 5(1):189–246, 1991.
  * [12] J. Lauret.  Commutative spaces which are not weakly symmetric.  Bull. London Math. Soc., 30(1):29–36, 1998.
  * [13] F. Ricci.  Commutative algebras of invariant functions on groups of Heisenberg type.  J. London Math. Soc. (2), 32(2):265–271, 1985.
  * [14] C. Riehm.  Explicit spin representations and Lie algebras of Heisenberg type.  J. London Math. Soc. (2), 29(1):49–62, 1984.
  * [15] A. Selberg.  Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series.  J. Indian Math. Soc. (N.S.), 20:47–87, 1956.
  * [16] Z. I. Szabó.  Spectral theory for operator families on Riemannian manifolds.  In Differential geometry: Riemannian geometry (Los Angeles, CA, 1990), volume 54 of Proc. Sympos. Pure Math., pages 615–665. Amer. Math. Soc., Providence, RI, 1993.
  * [17] Z. I. Szabó.  Locally non-isometric yet super isospectral spaces.  Geom. Funct. Anal., 9(1):185–214, 1999.
  * [18] E. N. Wilson.  Isometry groups on homogeneous nilmanifolds.  Geom. Dedicata, 12(3):337–346, 1982.


# Energy-Frontier Lepton-Hadron Collisions at CERN:
the LHeC and the FCC-eh

Masahiro Kuze  Department of Physics, Tokyo Institute of Technoogy
Tokyo, 152-8551, Japan
<EMAIL_ADDRESS>
###### Abstract

Lepton-hadron colliders that use a proton or nucleus beam of current and
future hadron colliders and let it collide with an electron beam from a newly
built electron accelerator bring attractive physics programs which are strong
and complementary to the hadron collider physics. Machine development for
Energy Recovery LINAC and physics performance studies of such electron-hadron
colliders, specifically the LHeC that uses the existing LHC beam and FCC-eh
that is an option of Future Circular Collider program, are ongoing and
reviewed in this article.

###### keywords:

Deep Inelastic Scattering; LHeC; FCC-eh.

##  1 Introduction

Deep inelastic scattering (DIS) of electrons on protons or nuclei has been
traditionally the best way to probe the inner structure of nucleon and nuclei.
At high energies, in addition to the electromagnetic (photon) exchange, the
electroweak bosons play important roles; the $\gamma/Z$ exchange induces
neutral current (NC) DIS, while the $W$ exchange results in charged current
(CC) DIS, in which the outgoing neutrino is undetected and leaves a missing
energy signature. The scattering is described by two kinematic variables,
$Q^{2}$, the squared momentum transfer between the lepton and hadron, and
Bjorken $x$, the fraction of the nucleon momentum carried by the scattering
parton.

The HERA collider at DESY was the last and highest-energy $ep$ collider which
had a center-of-mass (cms) energy of 318 GeV. It provided the PDF (parton
distribution functions) of the proton up to the scale of $Q^{2}\approx
10^{4}~{}\rm GeV^{2}$ and down to $x\approx 10^{-5}$, which are indispensable
inputs to the physics at the LHC.

It is natural to consider the possibilities of future colliders using the
proton (or heavy-ion) beam of a hadron collider and let it collide with an
electron beam (polarizable) from a newly built electron machine. Two ideas
have been discussed; the LHeC[1] collides a 60 GeV $e$-beam with the 7 TeV
$p$-beam of the LHC, with a center-of-mass (cms) energy of 1.3 TeV, and FCC-eh
collides a 60 GeV $e$-beam with the 50 TeV $p$-beam of the planned FCC (Future
Circular Collider), with a cms energy of 3.5 TeV. Both ideas have an option to
use a beam of nuclei in addition to the proton beam.

Since such a facility uses a beam of the already built hadron collider, it can
be realized at an affordable cost. It can run concurrently with hadron-hadron
collision experiments, and provides much cleaner collision environment than
$h$-$h$ experiments (negligible pile-up), while realizing higher cms energy
than $e^{+}e^{-}$ colliders.

Figure 1: Layout of Energy Recovery LINAC[2].

##  2 Machine and Detector

The proposed electron machine for LHeC/FCC-eh is an energy recovery LINAC
(ERL), which is a horserace-track like ring with two 10 GeV LINACs. After
three turns, the beam is accelerated to 60 GeV. The circumference of the ring
is approximately 9 km (see Fig. 1). A unique characteristic of the ERL is that
the beam after the collision runs in the same LINAC at an opposite phase to
the accelerated beam and is thus decelerated, giving back the power for
acceleration. In this way the RF power is recycled and a lot of wall-plug
power consumption can be saved.

The LINAC has a series of 802 MHz five-cell superconducting cavities with an
accelerating gradient of 18 MV/m. With high current electron beam, the
collider aims at an instantaneous luminosity of $10^{34}~{}{\rm
cm^{-2}s^{-1}}$, providing physics dataset of 100 $\rm fb^{-1}$ per year. A
small-scale ERL demonstrator called PERLE[3] is proposed at LAL, Orsay. It
will have two LINACs with four cavities each, which after three turns give
$\approx$ 400 MeV beam of $\approx$ 15 mA. The main purpose of PERLE is to
probe the ERL operation in multi-Megawatt regime and the multipass mode with a
very high current, but also a low-energy, high-intensity $ep/eA(\gamma
p/\gamma A)$ physics program can be envisaged.

Also detector designs are ongoing in the LHeC/FCC-eh working group aiming at
optimization of physics performance. Because of the large asymmetry of the
beam energies, the detector is also asymmetric like the detectors at HERA.
Very low-angle tagging of particles is important so the detector coverage
extends to high repidity. Fig. 2 shows a schematic of a detector design.

Figure 2: Schematic of a future $ep$ detector[4].

##  3 Physics Opportunities at Lepton-Hadron Colliders

A variety of physics programs are discussed for LHeC/FCC-eh, which are unique
and complementary to hadron collider physics programs. Some highlights among
them are briefly discussed in the following.

###  3.1 Ultimate precision measurements of PDF and $\alpha_{S}$

It is clear from LHC experience that precise knowledge of PDFs is vital
information for searches and precision physics at hadron colliders. High-$x$
region is relevant for searches for new, very high mass particles. At very
large collision energy like FCC (cms energy of 100 TeV), small-$x$ region
below $10^{-5}$ becomes relevant even for common physics targets such as
$W/Z$, Higgs or top quarks. Figure 4 shows the uncertainty of gluino pair
production cross section at the LHC (cms energy 14 TeV). Using the current PDF
sets on the market, the uncertainty exceeds 100% above gluino mass of 2 TeV.
With the knowledge of precise PDF from LHeC measurements, this uncertainty
squeezes to below 10%. Also the uncertainty of Higgs production cross section
from PDF shrinks enough using the LHeC PDF so that the cross section
measurement at the LHC becomes sensitive to the Higgs mass value[6].

Figure 3: Uncertainty of gluino pair production cross section at LHC[5]

Figure 4: Invariant mass of $H\to b\bar{b}$ signal on top of backgrounds[8]

From the DIS measurements, also the strong coupling constant $\alpha_{S}$ can
be extracted by a simultaneous fit with PDFs. A precision of 0.3% is expected
at the LHeC, which can further improve to 0.1% when combined with HERA
results[7].

###  3.2 $ep$ collider as a Higgs factory

The production cross section of Higgs bosons in $ep$ collision lies at sub-
picobarn range at LHeC and FCC-eh energies, which makes it very attracting for
precise Higgs studies. The CC channel is particularly interesting due to
suppression of NC DIS background and the possibility of increasing the cross
section with electron polarization ($-80\%$ is assumed in the performance
evaluation).

Figure 4 shows an invariant mass distribution for $H\to b\bar{b}$
reconstruction with a cut-based analysis assuming 10 years (1000 fb-1) of data
taking. In the signal mass window, 3600 signal events are observed on top of
1250 background events. This leads to a $Hbb$ coupling measurement of below 1%
precision (statistical error only).

Using multivariate techniques, much developed in the LHC physics, one can
further improve the precision and could even measure the $Hcc$ coupling which
is deemed very difficult at hadron colliders. Table 3.2 summarizes the
precision of the couplings anticipated at several configurations including the
one using Double-energy LHC (proton beam energy of 14 TeV with stronger
magnets in the existing LHC tunnel).

Higgs coupling precisions expected at future lepton-hadron colliders[9].
$E_{e}$=60 GeV is assumed.  LHeC $e$+DLHC FCC-eh ($E_{p}$=7 TeV, ($E_{p}$=14
TeV, ($E_{p}$=50 TeV, Coupling $\sqrt{s}\approx$1.3 TeV) $\sqrt{s}\approx$1.8
TeV) $\sqrt{s}\approx$3.5 TeV) $\kappa(Hbb)$ 0.5% 0.3% 0.2% $\kappa(Hcc)$ 4%
2.8% 1.8%

###  3.3 Top and electroweak physics

Figure 5: $W$ mass measurement accuracy at HERA and future $ep$ colliders[10].
As a reference, current accuracy from ATLAS[11] is 19 MeV, slightly larger
than the PDG2016 band ($\pm 15$ MeV).

Figure 6: Anticipated improvement on nuclear PDF precision at LHeC (gluon
distribution in lead)[14]

The energy-frontier electron-hadron collider produces also a large number of
top quarks and $W/Z$ bosons, with little background from hadronic QCD events
and pileup events inherent to hadron colliders. Figure 6 shows a prospect of
$W$ mass measurements from CC DIS data from LHeC, FCC-eh and a combination of
them. A very competitive measurement can be made in comparison with the
current accuracy. The possibility of polarizing the electron beam (up to 80%
is expected) brings further opportunities in the electroweak measurements. A
search for single-top production brings a competitive test of FCNC top
couplings, especially with $u$ quarks which are abundant in the proton.

###  3.4 Beyond SM physics

It is fair to say that the highest-energy hadron collider is the front runner
in the discovery of new heavy particles or states, but there are places where
$ep$ collider can make a case. An example is a leptoquark (LQ), a hypothetical
state that couples directly with a lepton and a quark. It is expected that LQs
found in HL-LHC would be also found at $ep$ colliders. Then, $ep$ colliders
can study thoroughly the characteristics of the new particle, by determining
its quantum numbers such as lepton/baryon numbers, spin and generation
indices, thanks to the ability to control the electron beam charge and
polarization[12]. Other topics of interest include compositeness, charged
Higgs, sterile neutrinos, long-lived parties, or anomalous couplings.

###  3.5 Diffractive physics and nuclear PDF

Another interesting area to be probed is the low-$x$ and diffractive physics.
Compared to HERA, the reachable kinematics is much enhanced. At very low-$x$
below $10^{-4}$, there is no data to constrain the gluon distribution, which
is expected to saturate somewhere. Also a lot of diffractive measurements can
be done, using rapidity gap events or installing roman-pot type forward proton
spectrometers[13].

If a beam of nuclei is available in the hadron machine (like Pb in the LHC),
the first measurement of nuclear PDFs using electon-hadron collider can be
made (note that HERA circulated only protons as the hadron beam). Compared to
the past measurements from fixed-target experiments, the gain of kinematics is
four orders of magnitude in $x$ and $Q^{2}$. Figure 6 shows an example of
improvement in the accuracy of nuclear PDF measurement.

##  4 Conclusions

A new electron-hadron collider, using a hadron beam of existing or planned
hadron colliders, is a cost-effective and attractive future program. A design
of an ERL with 60 GeV electron beam is at an advanced state, and a
demonstrator PERLE is proposed. An $ep/eA$ energy frontier machine, with 100
times $Q^{2}$ reach and 1000 times integrated luminosity compared to HERA,
will bring a rich physics program which is complementary to, and strengthens,
the discovery programs at HL-LHC and FCC-hh. It has a different physics
objectives from the Electron Ion Collider[15] (EIC) in US, which is a lower
energy machine and focuses on spin and medium-$x$ structure of nucleon and
nuclei.

## References

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