# Global rigidity of the period mapping

Benson Farb Supported in part by National Science Foundation Grant No.
DMS-181772, the Eckhardt Faculty Fund and the Jump Trading Mathlab Research
Fund.

##  1 Introduction

Let $\mathcal{M}_{g}$ denote the moduli space of smooth, genus $g\geq 1$
curves and let $\mathcal{A}_{h}$ denote the moduli space of $h$-dimensional,
principally polarized abelian varieties. Let

$\operatorname{J}:\mathcal{M}_{g}\to\mathcal{A}_{g}$

denote the period mapping, assigning to a Riemann surface
$X\in\mathcal{M}_{g}$ its Jacobian variety together with the prinicipal
polarization induced from the intersection form on $H_{1}(X;\mathbb{Z})$. The
map $\operatorname{J}$ is an injective (by the Torelli Theorem) morphism of
quasiprojective varieties.

Given the more than 150 years of intensive study of the period mapping, and
the fundamental role it plays in the theory of Riemann surfaces, it is natural
to ask: are there other ways to attach an $h$-dimensional principally
polarized abelian variety to a smooth, genus $g$ Riemann surface in an
algebraically (or even holomorphically) varying manner, not necessarily in a
one-to-one fashion? The main theorem of this paper states that, when $h\leq
g$, the period mapping is the unique nontrivial way to do this, even if one is
allowed the extra data of a finite set of marked points on the surface.

Let $\mathcal{M}_{g,n}$ denote the moduli space of pairs
$(X,(z_{1},\ldots,z_{n}))$ with $X\in\mathcal{M}_{g}=\mathcal{M}_{g,0}$ and
$z_{i}\neq z_{j}\in X$ when $i\neq j$. In this case the period mapping
$\operatorname{J}:\mathcal{M}_{g,n}\to\mathcal{A}_{g}$ factors through the map
$\mathcal{M}_{g,n}\to\mathcal{M}_{g}$ given by
$(X,(z_{1},\ldots,z_{n}))\mapsto X$.

######  Theorem 1.1 (Global rigidity of the period mapping).

Let $g\geq 3,n\geq 0$ and assume that $h\leq g$. Let
$F:\mathcal{M}_{g,n}\to\mathcal{A}_{h}$ be any nonconstant holomorphic map of
complex orbifolds. Then $h=g$ and $F=\operatorname{J}$.

######  Remark 1.2.

Both $\mathcal{M}_{g,n}$ and $\mathcal{A}_{g}$ are complex orbifolds: they are
quotients of contractible complex manifolds by a group of biholomorphic
automorphisms acting properly discontinuously and virtually freely. As such,
in this paper holomorphic maps between them are always taken to be in the
category of orbifolds. The standard examples of maps between orbifold moduli
spaces (including the period mapping $\operatorname{J}$) are holomorphic in
this sense. See §2 for more details.

I do not know if the statement of Theorem 1.1 holds for $g=2$. Some upper
bound on $h$ in terms of $g$ is necessary for Theorem 1.1 to hold: lifting
complex structures to certain characteristic covers (e.g. finite homology
covers) gives various nonconstant holomorphic maps
$\mathcal{M}_{g}\to\mathcal{A}_{h}$ with $h>g$. The following example shows
that if $\mathcal{M}_{g}$ is replaced by a finite cover then the conclusion of
Theorem 1.1 no longer holds.

######  Example 1.3 (Prym map).

Let

$\mathcal{R}_{g}:=\\{(X,\theta):X\in\mathcal{M}_{g},\ 0\neq\theta\in
H^{1}(X;\mathbb{F}_{2})\\},$

a $(2^{2g}-1)$-sheeted cover of $\mathcal{M}_{g}$. Then, in addition to the
composition
$\mathcal{R}_{g}\to\mathcal{M}_{g}\stackrel{{\scriptstyle\operatorname{J}}}{{\to}}\mathcal{A}_{g}$,
there is for $g\geq 2$ a nontrivial morphism of varieties

${\rm Prym}:\mathcal{R}_{g}\to\mathcal{A}_{g-1}$

called the Prym map; see for example the survey [F] by G. Farkas. There are
many other such examples.

I believe that, with the methods of this paper, it should be possible to prove
that if $F$ is any nonconstant holomorphic map
$\mathcal{R}_{g}\to\mathcal{A}_{h}$ with $h\leq g-1$ then $h=g-1$ and $F={\rm
Prym}$.111Since an earlier version of this paper appeared, C. Servan [Se] has
proven this result using the methods of this paper. As Hain observed to me,
post-composing the period mapping with Hecke correspondences on
$\mathcal{A}_{g}$ gives a huge number of distinct holomorphic maps from
various finite covers $X$ of $\mathcal{M}_{g}$ to $\mathcal{A}_{g}$. A
classification of all such maps $X\to\mathcal{A}_{g}$ seems a worthwhile
challenge.

Proof outline. The proof of Theorem 1.1 is divided into six steps, outlined as
follows. As far as I know, this approach to proving uniqueness of holomorphic
maps in a homotopy class - by using a convexity argument along Jacobi fields
to convert a homotopy to an algebraic deformation, and play this off against
rigidity for certain families - is novel.

  1. 1.

A theorem of Korkmaz [K] and the Congruence Subgroup Property classify
representations $\pi^{\rm orb}_{1}(\mathcal{M}_{g,n})\to\pi^{\rm
orb}_{1}(\mathcal{A}_{h})$. The fact that $\mathcal{A}_{h}$ is aspherical
quickly reduces the theorem to the case $h=g$ and $F$ homotopic to
$\operatorname{J}$.

  2. 2.

Since $\mathcal{A}_{g}$ is covered by a bounded symmetric domain in
$\mathbb{C}^{\binom{g+1}{2}}$, a theorem of Borel-Narasimhan [BN] gives
$F=\operatorname{J}$ as long as $F(x)=\operatorname{J}(x)$ for some
$x\in\mathcal{M}_{g}$. The rest of the proof of Theorem 1.1 is devoted to
finding such an $x$.

  3. 3.

For any curve $C\subset\mathcal{M}_{g}$, we replace the homotopy
$\operatorname{J}|_{C}\sim F|_{C}$ by a geodesic homotopy $H_{t}$. We then
follow an argument of Antonakoudis-Aramayona-Souto [AAS]. Using the fact that
$\mathcal{A}_{g}$ has nonpositive sectional curvature, we deduce convexity of
the energy of $H_{t}$ along a Jacobi field. We apply this to a Wirtinger-type
inequality, used as a kind of “holomorphicity detector”, to prove that each
$H_{t}:C\to\mathcal{A}_{g}$ is holomorphic.

  4. 4.

We apply a theorem of Kobayashi-Ochiai [KO] to extend each $H_{t}$ to a map
$\overline{C}\to\overline{\mathcal{A}_{g}}^{\rm S}$ to the Satake
compactification of $\mathcal{A}_{g}$. Chow’s Theorem gives that each $H_{t}$
is a morphism of varieties. This improvement is needed to apply the rigidity
theory of Faltings and Saito for families of abelian varieties.

  5. 5.

We find a curve $C\subset\mathcal{M}_{g}$ that is $\mathcal{A}_{g}$-rigid: it
is an isolated point in the space $\operatorname{Mor}(C,\mathcal{A}_{g})$ of
morphisms $C\to\mathcal{A}_{g}$. The construction is non-explicit, and uses
$g\geq 3$. It produces a family of abelian varieties over $C$ with monodromy
$\operatorname{Sp}(2g,\mathbb{Z})$, and another property, so that a rigidity
criterion of Saito [S] (building on Faltings) can be applied.

  6. 6.

Steps 4 and 5 give that the path $t\mapsto H_{t}$ in
$\operatorname{Mor}(C,\mathcal{A}_{g})$ is constant, from which it follows
that $F(x)=\operatorname{J}(x)$ for all $x\in C$. We have thus found the
required (by Step 2) $x$, completing the proof of the theorem.

An alternative approach to proving Step 3, due to R. Hain, is given in §3.

Related work. There are various powerful theorems of Faltings, Noguchi and
others 222For a recent example, see Theorem 1.7 of Javanpeykar-Litt [JL].
giving the finiteness of various spaces of holomorphic maps between (typically
compact) complex spaces and certain Kobayashi hyperbolic spaces such as
$\mathcal{A}_{g}$; see for example the survey [No] and [JL]. This contrasts
with Theorem 1.1 above, where this finite set is shown to have a single
element. This is similar to the difference between local rigidity for
cocompact, irreducible lattices in semisimple Lie groups, proved by Calabi-
Vasenti-Weil, and the global rigidity later proved by Mostow. It would be
interesting to see if any of the methods in the present paper could be used to
address any of the conjectures in [No].

Acknowledgements. I would like to thank Juan Souto for patiently answering my
questions about [AAS]; Sam Grushevsky for suggesting Example 1.3; Nir Gadish,
Ariyan Javapeykar, Eduard Looijenga and Dan Margalit for useful comments and
corrections on an earlier draft; Richard Hain for suggesting the comment above
as well an alternative proof of Step 3; and Curtis McMullen for various
inspiring conversations, and for useful comments on an earlier draft of this
paper. Finally, it is a pleasure to thank the anonymous referee for several
useful suggestions.

##  2 Proof of Theorem 1.1

The purpose of this section is to prove Theorem 1.1.

######  Remark 2.1 (Maps of orbifolds).

A topological (resp. complex) orbifold is the quotient $X/\Gamma$ of a
manifold (resp. complex manifold) $X$ by a group $\Gamma$ acting properly
discontinuously on $X$ by homeomorphisms (resp. biholomorphic automorphisms).
Let $Y/\Lambda$ be another orbifold, and let $\rho:\Gamma\to\Lambda$ be a
homomorphism. A continuous (resp. holomorphic) map in the category of
orbifolds $F:X/\Gamma\to Y/\Lambda$ is by definition a continuous (resp.
holomorphic) map $\widetilde{F}:X\to Y$ that intertwines $\rho$:

$\widetilde{F}(g\cdot x)=\rho(g)(\widetilde{F}(x))\ \ \ \text{for all\ }x\in
X,g\in\Gamma.$

When this is the case we use the shorthand $F_{*}:\Gamma\to\Lambda$. Note that
this homomorphism is only defined up to postcomposition by an inner
automorphism of $\Lambda$. Henceforth all maps between orbifolds will be
assumed to be maps in the category of orbifolds.

We now prove Theorem 1.1 in 6 steps.

### Step 1: Homotopy classes of maps $\mathcal{M}_{g,n}\to\mathcal{A}_{h}$

The main goal of this step is the following.

######  Proposition 2.2.

Let $g\geq 3,n\geq 0,h\geq 1$. Assume that $h\leq g$. Let
$F:\mathcal{M}_{g,n}\to\mathcal{A}_{h}$ be any continuous map (in the category
of orbifolds - see below). Then either $F$ is homotopically trivial or $h=g$
and $F$ is homotopic to the period mapping $\operatorname{J}$ (and so in
particular if $n>0$ then $F$ factors through the forgetful map
$\mathcal{M}_{g,n}\to\mathcal{M}_{g}$).

The proof of Proposition 2.2 uses in a crucial way a result of Korkmaz ([K],
Theorem 1) classifying low-dimensional representations of
$\operatorname{Mod}(S_{g,n})$.

###### Proof of Proposition 2.2.

For now fix $g\geq 1,n\geq 0$. Let
$\operatorname{Mod}(S_{g,n}):=\pi_{0}(\operatorname{Diff}^{+}(S_{g};z_{1},\ldots,z_{n}))$
be the mapping class group of a smooth, oriented genus $g$ surface $S_{g}$
fixing $n$ distinct points on $S_{g}$, and let
$\operatorname{Sp}(2g,\mathbb{Z})$ denote the integral symplectic group. Let
$\operatorname{Teich}(S_{g,n})$ be the Teichmüller space of isotopy classes of
complex structures on a smooth, genus $g$ surface $S_{g}$ with $n$ marked
points. Let ${\mathfrak{h}}_{g}$ denote the Siegel upper half-space. The
complex manifold $\operatorname{Teich}(S_{g,n})$ (resp. ${\mathfrak{h}}_{g}$)
is known to be a bounded domain in $\mathbb{C}^{N}$ for $N=3g-3+n$ (resp.
$N=\binom{g+1}{2}$).

The group $\operatorname{Mod}(S_{g,n})$ acts on
$\operatorname{Teich}(S_{g,n})$ properly discontinuously by biholomorphic
automorphisms, and the quotient
$\mathcal{M}_{g,n}:=\operatorname{Teich}(S_{g,n})/\operatorname{Mod}(S_{g,n})$
is the moduli space of genus $g$ Riemann surfaces with $n$ marked (and
ordered) points. Similarly, the integral symplectic group
$\operatorname{Sp}(2g,\mathbb{Z})$ acts properly discontinuously by
biholomorphic automorphisms on Siegel upper half-space ${\mathfrak{h}}_{g}$, a
bounded domain in $\mathbb{C}^{\binom{g+1}{2}}$. The quotient
$\mathcal{A}_{g}:={\mathfrak{h}}_{g}/\operatorname{Sp}(2g,\mathbb{Z})$ is the
moduli space of principally polarized abelian varieties.

The moduli spaces $\mathcal{M}_{g,n}$ and $\mathcal{A}_{g}$ are
quasiprojective varieties, and the period mapping
$\operatorname{J}:\mathcal{M}_{g,n}\to\mathcal{A}_{g}$ (as discussed in the
introduction) is a morphism of complex varieties. Let

$\rho:\operatorname{Mod}(S_{g,n})\to\operatorname{Aut}(H_{1}(S_{g};\mathbb{Z}),\hat{i})=\operatorname{Sp}(2g,\mathbb{Z})$

be the symplectic representation; here $\hat{i}$ denotes the algebraic
intersection form on $H_{1}(S_{g};\mathbb{Z})$. The morphism
$\operatorname{J}$ lifts to a $\rho$-equivariant morphism
$\widetilde{\operatorname{J}}:\operatorname{Teich}(S_{g})\to{\mathfrak{h}}_{g}$,
giving a commutative diagram

$\begin{array}[]{ccc}\operatorname{Teich}(S_{g,n})&\stackrel{{\scriptstyle\widetilde{\operatorname{J}}}}{{\longrightarrow}}&{\mathfrak{h}}_{g}\\\
\Big{\downarrow}&&\Big{\downarrow}\\\
\mathcal{M}_{g,n}&\stackrel{{\scriptstyle\operatorname{J}}}{{\longrightarrow}}&\mathcal{A}_{g}\end{array}$

Thus $\operatorname{J}$ is a holomorphic map (indeed morphism) in the category
of complex orbifolds, with $\operatorname{J}_{*}=\rho$.

As explained above, $F$ induces a homomorphism
$F_{*}:\operatorname{Mod}(S_{g,n})\to\operatorname{Sp}(2h,\mathbb{Z})$. Since
$g\geq 3$, Theorem 1 of Korkmaz [K] gives that either $F_{*}=0$ (the trivial
homomorphism) or $h=g$ and $F_{*}(x)=A\rho(x)A^{-1}$ for some
$A\in\operatorname{GL}(2g,\mathbb{C})$. Assume the latter case. As stated in
Step 5 of the proof of Korkmaz’s theorem, any nontrivial homomorphism
$\operatorname{Mod}(S_{g,n})\to\operatorname{Sp}(2g,\mathbb{Z})$ with $g\geq
4$ factors through the standard symplectic representation $\rho$, inducing a
representation
$\tau:\operatorname{Sp}(2g,\mathbb{Z})\to\operatorname{Sp}(2g,\mathbb{Z})$
with $F_{*}=\tau\circ\rho$. By the Congruence Subgroup Property for
$\operatorname{Sp}(2g,\mathbb{Z})$ (see [Me], Corollar 1 on p.128), any
homomorphism $\tau$ with infinite image (as in our case, since
$F_{*}(x)=A\rho(x)A^{-1}$) is an automorphism and is given by conjugation by
some element of $\operatorname{Sp}(2g,\mathbb{Z})$. It follows that
$F_{*}:\operatorname{Mod}(S_{g,n})\to\operatorname{Sp}(2g,\mathbb{Z})$ is,
after conjugation by some $A\in\operatorname{Sp}(2g,\mathbb{Z})$, equal to the
symplectic representation $\rho$.

Since ${\mathfrak{h}}_{h}$ is contractible, there is a bijection between
continuous maps (as always, in the category of orbifolds)
$F:\mathcal{M}_{g,n}\to\mathcal{A}_{h}$ and pairs $([\widetilde{F}],\tau)$
where $\tau:\operatorname{Mod}(S_{g,n})\to\operatorname{Sp}(2h,\mathbb{Z})$ is
a homomorphism and $[\widetilde{F}]$ denotes equivariant homotopy classes of
continuous maps
$\widetilde{F}:\operatorname{Teich}(S_{g,n})\to{\mathfrak{h}}_{h}$
intertwining $\rho$. Thus if $F_{*}=0$ then $F$ is freely homotopic to a
constant map, and if $F_{*}=\rho$ then $F$ is freely homotopic to the period
mapping $\operatorname{J}$. ∎

Before continuing we dispense with the case where $F$ is homotopically
trivial.

######  Lemma 2.3.

Let $g\geq 3,n\geq 0,h\geq 1$. Let $F:\mathcal{M}_{g,n}\to\mathcal{A}_{h}$ be
any holomorphic map of orbifolds. If $F$ is homotopically trivial then $F$ is
constant.

###### Proof.

Since $F$ is homotopically trivial it lifts to a map
$\widetilde{F}:\mathcal{M}_{g,n}\to{\mathfrak{h}}_{h}$, a bounded domain in
$\mathbb{C}^{N},N:=\binom{h+1}{2}$. Let
$\widetilde{F}_{i}:\mathcal{M}_{g,n}\to\mathbb{C}$ be the composition of
$\widetilde{F}$ with the coordinate function $z_{i}$ on $\mathbb{C}^{N}$. So
$\widetilde{F}_{i}$ is a bounded holomorphic function on a smooth,
quasiprojective variety, hence is constant (see, e.g. [BN], §1.2 (a)). Since
this is true for each $i$ it follows that $\widetilde{F}$ is constant, hence
$F$ is constant. ∎

Proposition 2.2 and Lemma 2.3 reduce the proof of Theorem 1.1 to the case when
$h=g$, when $F$ factors through the forgetful morphism
$\mathcal{M}_{g,n}\to\mathcal{M}_{g}$, and the resulting map
$\mathcal{M}_{g}\to\mathcal{A}_{g}$ is homotopic to $\operatorname{J}$. In
particular the statement of the theorem for $\mathcal{M}_{g,n}$ follows from
that for $\mathcal{M}_{g}$. It thus suffices to assume the following.

We henceforth assume that $h=g,n=0$ and that $F$ is homotopic to
$\operatorname{J}$.

### Step 2: The Borel-Narasimhan Theorem

Homotopic holomorphic maps are not always equal, even for nonpositively curved
Kahler targets. Here is a simple example (there more serious examples, even
with $\mathcal{A}_{g}$ target, due to Faltings and others; see [S].)

######  Example 2.4 (Cautionary example).

Let $X$ and $Y$ be connected, complex manifolds. For each fixed $x\in X$ let
$F_{x}:Y\to X\times Y$ be defined by $F_{x}(y):=(x,y)$. Then each $F_{x}$ is
holomorphic, all the $F_{x}$ are homotopic to each other, and all the $F_{x}$
are distinct from each other. Note that $F_{x_{1}}(Y)\cap
F_{x_{2}}(Y)=\emptyset$ when $x_{1}\neq x_{2}$.

There is a sufficient criterion, proved by Borel-Narasimhan [BN], for
homotopic holomorphic maps to be equal. Recall that a function
$\sigma:X\to[-\infty,\infty)$ on a complex manifold $X$ is plurisubharmonic
(also called “pseudo-convex” in [BN]) if it is upper semi-continuous, and if
for every domain $\Omega\subseteq\mathbb{C}$ and every holomorphic map
$\psi:\Omega\to X$, the function $\sigma\circ\psi$ is subharmonic on $\Omega$.

######  Theorem 2.5 (Borel-Narasimhan [BN], Theorem 3.6).

Let $X$ be a connected complex manifold that carries no non-constant
plurisubharmonic function that is bounded above. Let $a\in X$ and let $Y$ be a
complex manifold covered by a bounded domain in $\mathbb{C}^{N}$. Let
$u,v:X\to Y$ be two holomorphic maps such that $u(a)=v(a)$ and such that
$u_{*}=v_{*}:\pi_{1}(X,a)\to\pi_{1}(M,u(a))$. Then $u=v$.

Since $\mathcal{M}_{g}$ is a complex quasiprojective variety, Proposition 2.1
of [BN] implies that any bounded plurisubharmonic function on
$\mathcal{M}_{g}$ is constant. Also, $\mathcal{A}_{g}$ has universal cover
${\mathfrak{h}}_{g}$, which is a bounded domain in
$\mathbb{C}^{\binom{g+1}{2}}$. We can thus apply Theorem 2.5 with
$X=\mathcal{M}_{g},Y=\mathcal{A}_{g},u=F$ and $v=\operatorname{J}$. Since $F$
and $\operatorname{J}$ are homotopic and holomorphic, we conclude the
following.

######  Lemma 2.6.

To conclude that $F=\operatorname{J}$ it is enough to find some
$x\in\mathcal{M}_{g}$ such that $F(x)=\operatorname{J}(x)$.

The rest of the proof of Theorem 1.1 is devoted to finding such an
$x\in\mathcal{M}_{g}$.

### Step 3: The Wirtinger squeeze

We continue with the running assumption that
$F:\mathcal{M}_{g}\to\mathcal{A}_{g}$ is a holomorphic map homotopic to the
period mapping $\operatorname{J}$. This step is devoted to proving the
following result.

######  Lemma 2.7 (Paths of holomorphic curves).

Let $C\subset\mathcal{M}_{g}$ be any smooth (not necessarily projective)
curve. There exists a homotopy $H:[0,1]\times C\to\mathcal{A}_{g}$ with
$H_{0}=\operatorname{J}|_{C}$ and $H_{1}=F|_{C}$ such that for each
$t\in[0,1]$ the map

$H_{t}:C\to\mathcal{A}_{g}$

is holomorphic.

The proof of Lemma 2.7 follows closely the proof by Antonakoudis-Aramayona-
Souto of the Imayoshi-Shiga Theorem; see §4 of [AAS].

###### Proof of Lemma 2.7.

Let $G:[0,1]\times C\to\mathcal{A}_{g}$ be the restriction of the given
homotopy $F\sim\operatorname{J}$ to the curve $C$. I claim that there is a
homotopy $H:[0,1]\times C\to\mathcal{A}_{g}$ with the property that
$H_{0}=F|_{C},H_{1}=\operatorname{J}|_{C}$ and for each $x\in C$, each path
$\beta_{x}:[0,1]\to\mathcal{A}_{g}$ defined by $\beta_{x}(t):=H_{t}(x)$ is a
geodesic in $\mathcal{A}_{g}$. To see this, for any $x\in C$ consider the path
$\gamma_{x}(t):=G_{t}(x)$. Lift this path to a path
$\widetilde{\gamma}_{x}:[0,1]\to{\mathfrak{h}}_{g}$. Recall that
${\mathfrak{h}}_{g}$ is a bounded symmetric domain whose
$\operatorname{Sp}(2g,\mathbb{R})$-invariant Kahler metric has nonpositive
sectional curvature. Thus there is a unique (not necessarily unit speed)
geodesic $\beta_{x}:[0,1]\to{\mathfrak{h}}_{g}$ with
$\beta_{x}(0)=\gamma_{x}(0)$ and $\beta_{x}(1)=\gamma_{x}(1)$. Further, there
is a canonical homotopy from $\gamma_{x}$ to $\beta_{x}$ given by orthogonal
projection onto a geodesic segment. Since orthogonal projection onto a
geodesic segment varies continuously with its endpoints, we obtain after
composition with the projection ${\mathfrak{h}}_{g}\to\mathcal{A}_{g}$ the
claimed homotopy $H$.

Given any $x\in C$, since the path $t\mapsto H_{t}(x)$ is a geodesic it
follows that for any $v\in T_{x}C$ the vector field $t\mapsto(D_{x}H_{t})(v)$
is a Jacobi field along $\beta_{x}$ in $\mathcal{A}_{g}$. Since
$\mathcal{A}_{g}$ has nonpositive curvature, the function
$t\mapsto||(D_{x}H_{t})(v)||$ is convex; here the norm is taken with respect
to the inner product on $D_{x}H_{t}(T_{x}C)$.

Let $f:X\to Y$ be any smooth map from a Kahler $1$-manifold to a Kahler
manifold $Y$. The energy of $f$ at $x$ is defined to be

$E_{x}(f):=\frac{1}{2}[||D_{x}f(u)||^{2}+||D_{x}f(v)||^{2}]$

where $\\{u,v\\}$ is any orthonormal basis for $T_{x}X$ and where the norms
are those induced by the inner product metric on $T_{f(x)}Y$ given by the
Riemannian metric on $Y$. The number $E_{x}(f)$ does not depend on the choice
of $\\{u,v\\}$. Since $t\mapsto||(D_{x}H_{t})(v)||$ is convex it follows that
$t\mapsto E_{x}(t)$ is convex for each fixed $x$. For any smooth $f:X\to Y$,
the energy of $f$ is defined to be

$E(f):=\int_{X}E_{x}(f)\operatorname{vol}_{X}$

where $\operatorname{vol}_{X}$ is the volume form on $X$.

The following result is a variation of the Wirtinger inequality. Before
stating it we remark that a complex structure on a genus $g\geq 0$ surface $X$
determines an orientation on $X$, and so an isomorphism
$\wedge^{2}T_{x}X\to\mathbb{R}$ for each $x\in X$. For each $x\in X$, the
standard ordering $\leq$ on $\mathbb{R}$ pulls back via this isomorphism to an
ordering on $\wedge^{2}T_{x}$. Two $2$-forms on $X$ can thus be compared
pointwise.

######  Proposition 2.8 (Eells-Sampson).

Let $f:X\to Y$ be a smooth map from a finite area Kahler $1$-manifold $X$ to a
complete Kahler manifold $Y$. Assume that $E(f)<\infty$. Let $\omega_{X}$ and
$\omega_{Y}$ be the Kahler forms on $X$ and $Y$, respectively. Then

$f^{*}\omega_{Y}(x)\leq E_{x}(f)\omega_{X}(x)\ \ \ \text{for all}\ x\in X$
(2.1)

with equality if and only if $f$ is holomorphic at $x$.

###### Proof.

For compact $X$ the inequality (2.1) with both sides integrated over $X$ is
stated as the second proposition on page 126 of [ES]. However, the proof there
proceeds by proving (2.1). The proof of (2.1) in [ES], as well as the “if and
only if” statement for equality, is by elementary pointwise estimates, as well
as the assumption $E(f)<\infty$, which we are assuming. ∎

The proof of Lemma 4.1 of [AAS] with the target replaced by $\mathcal{A}_{g}$
with its Kahler metric mutatis mutandis gives that $E(H_{t})<\infty$ for each
$t\in[0,1]$. We can thus apply Proposition 2.8 to each map $H_{t},t\in[0,1]$,
giving

$H_{t}^{*}\omega_{\mathcal{A}_{g}}(x)\leq E_{x}(H_{t})\omega_{C}\ \ \
\text{for all}\ x\in C.$ (2.2)

Integrating (2.2) pointwise gives

$E(H_{t})\geq\int_{C}(H_{t}^{*}\omega_{\mathcal{A}_{g}}).$ (2.3)

On the other hand, since $H_{t}$ is homotopic to $H_{s}$ for all $s,t\in[0,1]$
it follows 333There is an issue when $C$ is not compact, but the argument
verbatim on page 226 of [AAS] gives this result. It uses an exhaustion of $C$
by compact sets, together with Stokes’s Theorem. that

$\int_{C}(H_{s}^{*}\omega_{\mathcal{A}_{g}})=\int_{C}(H_{t}^{*}\omega_{\mathcal{A}_{g}})\
\ \ \text{for all}\ s,t\in[0,1].$ (2.4)

Now, $H_{0}:=F$ and $H_{1}:=\operatorname{J}$ are holomorphic, so the equality
statement of Proposition 2.8 together with (2.4) implies that

$E(H_{0})=\int_{C}(H_{0}^{*}\omega_{\mathcal{A}_{g}})=\int_{C}(H_{1}^{*}\omega_{\mathcal{A}_{g}})=E(H_{1}).$
(2.5)

Combining (2.3), (2.4) and (2.5) gives

$E(H_{t})\geq\int_{C}(H_{t}^{*}\omega_{\mathcal{A}_{g}})=\int_{C}(H_{1}^{*}\omega_{\mathcal{A}_{g}})=E(H_{1})=E(H_{0})\
\ \ \text{for each}\ t\in[0,1].$ (2.6)

Since $t\mapsto E(H_{t})$ is convex, (2.6) implies that for each $t\in[0,1]$:

$E(H_{t})=\int_{C}(H_{t}^{*}\omega_{\mathcal{A}_{g}}).$ (2.7)

Since the pointwise estimate (2.2) holds for each $x\in C$, it follows from
(2.7) (and the fact that $E_{x}(H_{t})\geq 0$) that equality holds in (2.2)
for all $x\in C,t\in[0,1]$. Proposition 2.8 then implies that for each fixed
$t\in[0,1]$ the map $H_{t}:X\to\mathcal{A}_{g}$ is holomorphic.

∎

### Step 4: Improvement to a path of morphisms

The goal of this step is to prove that the holomorphic maps $H_{t}$ are in
fact morphisms.

######  Lemma 2.9.

Let $C\subset\mathcal{M}_{g}$ be any smooth algebraic curve. For each
$t\in[0,1]$ the map $H_{t}:C\to\mathcal{A}_{g}$ is a morphism of varieties.

###### Proof.

We need the following.

######  Proposition 2.10 ([KO], Theorem $2^{\prime}$).

Let $X$ be a complex manifold and let $A\subset X$ be a locally closed complex
submanifold. Let $D$ be a bounded symmetric domain in some $\mathbb{C}^{N}$
and let $G$ be the largest connected subgroup of biholomorphic automorphisms
of $D$. (Thus $G$ is a semisimple Lie group of noncompact, Hermitian type.)
Let $\Gamma\subset G$ be a (not necessarily torsion-free) arithmetic subgroup
of $G$, and let $Y:=D/\Gamma$. Let $\overline{Y}^{S}$ be the Satake
compactification of $Y$. Then every holomorphic mapping $X-A\to Y$ extends to
a holomorphic mapping $X\to\overline{Y}^{S}$.

Apply Proposition 2.10 with $X:=\overline{C}$, the projective closure of $C$;
the set $A:=\overline{C}-C$, a (possibly empty) finite set of points;
$D={\mathfrak{h}}_{g}$, the Siegel upper half-space, a bounded domain in
$\mathbb{C}^{\binom{g+1}{2}}$; the group
$G=\operatorname{Aut}({\mathfrak{h}}_{g})=\operatorname{Sp}(2g,\mathbb{R})$;
the arithmetic group $\Gamma:=\operatorname{Sp}(2g,\mathbb{Z})$; the quotient
$Y:=\mathcal{A}_{g}={\mathfrak{h}}_{g}/\operatorname{Sp}(2g,\mathbb{Z})$; and
the holomorphic map $H_{t}:C\to\mathcal{A}_{g}$. Note that the theorem in [KO]
is explicitly stated for the case when $\Gamma$ is not torsion free, as in the
case $\Gamma=\operatorname{Sp}(2g,\mathbb{Z})$.

Thus $H_{t}:C\to\mathcal{A}_{g}$ extends uniquely to a holomorphic map
$\overline{H}_{t}:\overline{C}\to\overline{\mathcal{A}_{g}}^{S}$ where
$\overline{\mathcal{A}_{g}}^{S}$ denotes the Satake compactification. By
Chow’s Theorem applied to each fixed $H_{t},t\in[0,1]$, the map
$\overline{H}_{t}$ is algebraic; that is, it is a morphism of varieties. It
follows that the restriction $H_{t}:C\to\mathcal{A}_{g}$ to the Zariski open
$C\subset\overline{C}$ is a morphism. ∎

### Step 5: Existence of an $\mathcal{A}_{g}$-rigid curve in $\mathcal{M}_{g}$

For complex varieties $X$ and $Y$ let $\operatorname{Hol}(X,Y)$ denote the
space of holomorphic maps $X\to Y$ equipped with the compact-open topology. It
is known that $\operatorname{Hol}(X,Y)$ is a Zariski open subset of a compact
complex space, but we will not need this. The subset
$\operatorname{Mor}(X,Y)\subseteq\operatorname{Hol}(X,Y)$ of morphisms $X\to
Y$ inherits the subspace topology. A morphism $\phi:X\to Y$ is rigid if it is
an isolated point of $\operatorname{Mor}(X,Y)$.

######  Definition 2.11 ($\mathcal{A}_{g}$-rigid curve).

A curve $i:C\to\mathcal{M}_{g}$ is $\mathcal{A}_{g}$-rigid if

$\operatorname{J}\circ i:C\to\mathcal{A}_{g}$

is rigid; in other words, $\operatorname{J}\circ i$ is an isolated point of
$\operatorname{Mor}(C,\mathcal{A}_{g})$.

Faltings, Saito and others constructed many interesting nonrigid curves in
$\mathcal{A}_{g}$; see [S]. Möller [Mo] found curves in $\mathcal{M}_{g}$ that
are not $\mathcal{A}_{g}$-rigid. In fact, while McMullen [Mc] proved that
Teichmüller curves are rigid in $\mathcal{M}_{g}$, they are not
$\mathcal{A}_{g}$-rigid. I do not know any explicit curves in
$\mathcal{M}_{g}$ that are $\mathcal{A}_{g}$-rigid. However, such curves do
exist.

######  Proposition 2.12.

For all $g\geq 3$ there exists an $\mathcal{A}_{g}$-rigid curve
$i:C\to\mathcal{M}_{g}$.

###### Proof.

In this proof the orbifold nature of $\mathcal{M}_{g}$ will cause a
complication, since the usual (i.e. not orbifold) fundamental group of
$\mathcal{M}_{g}$ is trivial. We will address this by excising the “orbifold
locus” of $\mathcal{M}_{g}$, as follows.

Let $\mathcal{O}\subset\mathcal{M}_{g}$ be the subvariety of smooth, genus $g$
Riemann surfaces with nontrivial automorphism group; this is a union of
irreducible components, one for each topological type of faithful finite group
action on $S_{g}$. The Riemann-Hurwitz Formula implies that the dimension
(over $\mathbb{C})$ of each irreducible component of $\mathcal{O}$ is at most
$2g-1$, with equality precisely for the hyperelliptic locus in
$\mathcal{M}_{g}$. The space $\mathcal{M}_{g}-\mathcal{O}$ is a smooth,
quasiprojective variety.

We first consider the case when $g\geq 4$. We claim that in this case
$\pi_{1}(\mathcal{M}_{g}-\mathcal{O})\cong\operatorname{Mod}(S_{g})$. To see
this, let $p:\operatorname{Teich}(S_{g})\to\mathcal{M}_{g}$ be the quotient of
$\operatorname{Teich}(S_{g})$ by the action of $\operatorname{Mod}(S_{g})$.
Let $\widetilde{\mathcal{O}}:=p^{-1}(\mathcal{O})$. The action of
$\operatorname{Mod}(S_{g})$ on $\operatorname{Teich}(S_{g})$ restricts to an
action of $\operatorname{Mod}(S_{g})$ on
$\operatorname{Teich}(S_{g})-\widetilde{\mathcal{O}}$ that is free and
properly discontinuous. Since

${\rm codim}_{\mathbb{C}}(\widetilde{\mathcal{O}})=(3g-3)-(2g-1)=g-2\geq 2\ \
\text{for $g\geq 4$}$

it follows that

$\pi_{1}(\operatorname{Teich}(S_{g})-\widetilde{\mathcal{O}})=\pi_{1}(\operatorname{Teich}(S_{g}))=0,$

so that

$\pi_{1}(\mathcal{M}_{g}-\mathcal{O})=\pi_{1}((\operatorname{Teich}(S_{g})-\widetilde{\mathcal{O}})/\operatorname{Mod}(S_{g}))=\operatorname{Mod}(S_{g}).$

Let $\operatorname{\overline{\mathcal{M}}_{g}^{\rm DM}}$ denote the Deligne-
Mumford compactification of $\mathcal{M}_{g}$. It is a smooth projective
variety $\operatorname{\overline{\mathcal{M}}_{g}^{\rm
DM}}\subset\mathbb{P}^{N}$ with normal crossing divisor. Recall that
$\dim_{\mathbb{C}}\operatorname{\overline{\mathcal{M}}_{g}^{\rm
DM}}=\dim_{\mathbb{C}}\mathcal{M}_{g}=3g-3$ for $g\geq 2$. Bertini’s Theorem
implies that for generic hyperplanes $P_{1},\ldots,P_{3g-4}$ in
$\mathbb{P}^{N}$, the intersection

$C:=(\mathcal{M}_{g}-\mathcal{O})\cap P_{1}\cap\cdots\cap P_{3g-4}$

is a smooth curve in $\mathcal{M}_{g}-\mathcal{O}$. Let
$i:C\to\mathcal{M}_{g}-\mathcal{O}$ denote the inclusion. The Lefschetz
Hyperplane Theorem for quasiprojective varieties (see [HT]) implies that

$i_{*}:\pi_{1}(C)\to\pi_{1}(\mathcal{M}_{g}-\mathcal{O})\cong\operatorname{Mod}(S_{g})$
(2.8)

is a surjection.444Note that, had we not excised $\mathcal{O}$, then the above
argument gives no information since $\pi_{1}(\mathcal{M}_{g})=0$ for $g\geq
1$; here $\pi_{1}$ is the topological (not orbifold) fundamental group.

The codimension $1$ strata of
$\partial\mathcal{M}_{g}:=\operatorname{\overline{\mathcal{M}}_{g}^{\rm
DM}}-\mathcal{M}_{g}$ consist of limits of degenerations of complex curves in
$\mathcal{M}_{g}$ that pinch some (topological type of) simple closed curve
$\beta$ on $S_{g}$ to a point. Let $Z\subset\partial\mathcal{M}_{g}$ denote
the unique codimension-$1$ stratum corresponding to the case where the
topological type of $\beta$ is non-separating. Note that the monodromy of the
universal family over $\mathcal{M}_{g}$ restricted to a small loop in
$\mathcal{M}_{g}$ around $Z$ is the cyclic subgroup generated by a Dehn twist
about a non-separating simple closed curve.

The pullback $(\operatorname{J}\circ i)^{*}$ of the universal principally
polarized abelian variety over $\mathcal{A}_{g}$ gives an algebraic family
$f:E\to C$ of $d$-dimensional, principally polarized abelian varieties over
$C$. The monodromy representation of this family is

$\mu:=(\operatorname{J}\circ
i)_{*}:\pi_{1}(C)\to\operatorname{Sp}(2g,\mathbb{Z})$

where $\operatorname{Sp}(2g,\mathbb{Z})$ acts on
$H_{1}(f^{-1}(c_{0});\mathbb{Z})$ as the standard symplectic representation;
here $c_{0}\in C$ is a basepoint for the family. By (2.8), and since the
action of $\operatorname{Mod}(S_{g};\mathbb{Z})$ on
$H_{1}(f^{-1}(c_{0});\mathbb{Z})$ gives the standard symplectic
representation, it follows that the monodromy
$\mu:\pi_{1}(C)\to\operatorname{Sp}(2g,\mathbb{Z})$ of the family $f:E\to C$
is surjective. In particular

The monodromy of the family $f:E\to C$ is irreducible.  (2.9)

Since
$\dim_{\mathbb{C}}Z=\dim_{\mathbb{C}}\operatorname{\overline{\mathcal{M}}_{g}^{\rm
DM}}-1=3g-4$, it follows from Bezout’s Theorem applied to the Zariski closure
$\overline{C}$ in $\operatorname{\overline{\mathcal{M}}_{g}^{\rm DM}}$ that
$S:=\overline{C}\cap Z\neq\emptyset$. Thus $C$ is a noncompact curve with
punctures $\\{s\in S\\}$. If $\gamma_{s}$ is a small loop around $s\in S$
then, as explained above, $i_{*}([\gamma])\in\operatorname{Mod}(S_{g})$ is a
Dehn twist about a nonseparating curve. Thus the monodromy
$\mu([\gamma])\in\operatorname{Sp}(2g,\mathbb{Z})$ is a symplectic
transvection. In particular

The monodromy $\mu([\gamma])\in\operatorname{Sp}(2g,\mathbb{Z})$ has infinite
order. (2.10)

The properties given in (2.9) and (2.10) are precisely the hypotheses of a
criterion of Saito ([S], Theorem 8.6), which states that if these conditions
hold then the family $f:E\to C$ is rigid; that is, the map $J\circ
i:C\to\mathcal{A}_{g}$ is an isolated point in
$\operatorname{Mor}(C,\mathcal{A}_{g})$, finishing the proof of the $g\geq 4$
case.

We now consider the $g=3$ case (the proof actually works for all $g\leq 7$).
Saito ([S], Corollary 8.4) proved that any abelian scheme with no isotrivial
factors and of relative dimension at most $7$ must be rigid. Let
$i:C\to\mathcal{M}_{3}$ be any curve with the property that there exists $c\in
C$ such that $J\circ i(c)$ is a simple abelian variety; such $C$ are known to
exist for all $g\geq 1$. Applying Saito’s result gives an
$\mathcal{A}_{g}$-rigid curve in $\mathcal{M}_{g}$ for all $g\leq 7$, in
particular for $g=3$. ∎

### Step 6: Finishing the proof

Let $C\subset\mathcal{M}_{g}$ be the $\mathcal{A}_{g}$-rigid curve constructed
in Step 5. By Steps 3 and 4, in particular Lemma 2.9, the homotopy
$G_{t}:\mathcal{M}_{g}\to\mathcal{A}_{g}$ between $J$ and $F$ restricts to a
homotopy $H_{t}:C\to\mathcal{A}_{g}$, giving a path
$\alpha:[0,1]\to\operatorname{Mor}(C,\mathcal{A}_{g})$ defined by
$\alpha(t):=H_{t}$. Since $C$ is $\mathcal{A}_{g}$-rigid the path component of
$\alpha(0)=H_{0}=J|_{C}$ in the space $\operatorname{Mor}(C,\mathcal{A}_{g})$
is a single point, and so the path $\alpha$ is constant. In particular

$J(x)=H_{0}(x)=H_{1}(x)=F(x)\ \ \ \text{for all $x\in C$}.$

By Lemma 2.6, this implies that $J(x)=F(x)$ for all $x\in\mathcal{M}_{g}$,
proving Theorem 1.1.

##  3 An Alternative approach to Theorem 1.1

In this section we give a different approach to proving Theorem 1.1 by
replacing Step 3 with a Hodge theory argument. This proof is due to Richard
Hain, who informed us of it after reading an earlier draft of this paper. For
simplicity we consider only the case $n=0$.

###### Proof.

By Step 1, we are reduced to the case $h=g\geq 3$ and $F$ homotopic to
$\operatorname{J}$. By a slight variation of the argument in Step 5, we can
cover $\mathcal{M}_{g}$ by complete curves with the property that the
inclusion $i:C\to\mathcal{M}_{g}$ induces a surjection
$i_{*}:\pi_{1}(C)\to\operatorname{Mod}(S_{g})$. Let $C$ be a curve in such a
cover.

By the same argument as in Step $4$, the restriction of $F$ to $C$ is in fact
a morphism of varieties. Let $V$ denote the standard polarized variation of
Hodge structure (PVHS) over $A$, where the fiber over the PPAV $A$ is
$H^{1}(A;\mathbb{Z})$. Let $V_{F}$ (resp. $V_{\operatorname{J}}$) be the
pullback of $V$ to $C$ via $F$ (resp. $\operatorname{J}$) restricted to $C$.
Since $\pi_{1}(C)\to\operatorname{Mod}(S_{g})$ is surjective, it follows that
each composition $(\operatorname{J}\circ i)_{*}$ and $(F\circ i)_{*}$ is
surjective on $\pi_{1}$, so that $V_{F}$ and $V_{\operatorname{J}}$ are simple
PVHS. It follows that

$H^{0}(C;\operatorname{Hom}(V_{F},V_{\operatorname{J}}))\cong\mathbb{Z}$

as abelian groups. Since $V_{F}$ and $V_{\operatorname{J}}$ have the same
weight, it follows that this is a Hodge structure of weight $0$ and dimension
$1$. Thus it must be $\mathbb{Z}(0)$. The Theorem of the Fixed Part gives that
for each $x\in C$ that

$H^{0}(X,\operatorname{Hom}(V_{F},V_{\operatorname{J}}))_{x}\to\operatorname{Hom}(V_{F},V_{\operatorname{J}})_{x}$

is a morphism of Hodge structures, so that
$(V_{F})_{x}\cong(V_{\operatorname{J}})_{x}$ is an isomorphism.

Theorem 7.24 of Schmid 555As Schmid remarks, this result with target a locally
symmetric variety - which is the case here - already follows from Borel-
Narasimhan [BN], as in our Step 2. the quotient states that for two variations
of Hodge $V,W$ structures over a base $C$ that is a Zariski open in a compact
analytic space, and if there exists $x\in C$ with $V_{x}$ isomorphic to
$W_{x}$, and if the isomorphism preserves the action of $\pi_{1}(C)$, then the
isomorphism extends to an isomorphism of the variations of Hodge structure.
Applying this to our case gives that

$V_{F}\cong V_{\operatorname{J}}\ \ \text{as PVHS.}$ (3.1)

Now, if $\mathcal{A}_{g}$ were a classifying space for PVHS, then (3.1) would
imply that $f=\operatorname{J}$. However, this is not quite true since
$\mathcal{A}_{g}$ is only a coarse moduli space for PPAV. Thus one must repeat
the above argument for $\mathcal{A}_{g}[3]$, the (fine!) moduli space of PPAV
with level $3$ structure, conclude that the lifts of $F$ and $J$ agree on this
cover, and intertwine the $\operatorname{Sp}(2g,\mathbb{F}_{3})$-actions. We
leave the details to the reader. ∎

## References

  * [AAS] S. Antonakoudis, J. Aramayona and J. Souto, Holomorphic maps between moduli spaces, Annales de L’Institut Fourier, Tome 68, no. 1, 2018, p. 217–228.
  * [BN] A. Borel and R. Narasimhan, Uniqueness conditions for certain holomorphic mappings, Inventiones Math., Vol. 2, 1967, p. 247–255.
  * [ES] J. Eells, Jr. and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. Jour. Math., 86, 1964, p. 109–160.
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  * [HT] H. Hamm and L Trang, Lefschetz theorems on quasi-projective varieties, Bull. Soc. Math. france, 113, 1985, p. 123–142.
  * [JL] A. Javapeykar and D. Litt, Integral points on algebraic subvarieties of period domains: from number fields to finitely generated fields, Ariyan Javanpeykar, Daniel Litt, preprint, May 2021, arXiv:1907.13536v4.
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  * [KO] S. Kobayashi and T. Ochiai, Satake compactification and the great Picard theorem, Jour. Math. Soc. Japan, Vol. 23, No. 2, 1971, p.340–350.
  * [Mo] M. Möller, Variations of Hodge structures of a Teichmüller curve, Journal AMS, Vol. 19, No. 2, 2005, p. 327–344.
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  * [Me] J. Mennicke, Zur Theorie der Siegelschen Modulgruppe, Math. Annalen 159, 1965, p. 115–129.
  * [No] J. Noguchi, Some problems in value distribution and hyperbolic manifolds, in Int. Symp. ”Holomorphic Mappings, Diophantine Geometry and Related Topics” (Kyoto, 1992), Sur. Kok. No. 819 (1993), 66–79.
  * [S] M. Saito, Classification of non-rigid families of abelian varieties, Tohoku Math. J., Vol. 45, 1993, p. 159–189.
  * [Se] C. Serván, On the uniqueness of the Prym map, in preparation.

Dept. of Mathematics, University of Chicago
E-mail<EMAIL_ADDRESS>


# Muon accelerators - Muon lifetime measurements as window to Planck scale
physics

Iarley P. Lobo<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of
Chemistry and Physics, Federal University of Paraíba, Rodovia BR 079 - km 12,
58397-000 Areia-PB, Brazil  Physics Department, Federal University of Lavras,
Caixa Postal 3037, 37200-900 Lavras-MG, Brazil  Christian Pfeifer
<EMAIL_ADDRESS>ZARM, University of Bremen, 28359 Bremen,
Germany.

###### Abstract

A prominent effective description of particles interacting with the quantum
properties of gravity is through modifications of the general relativistic
dispersion relation. Such modified dispersion relations lead to modifications
in the relativistic time dilation. A perfect probe for this effect, which goes
with the particle energy cubed $E^{3}$ over the quantum gravity scale
$E_{\text{QG}}$ and the square of the particle mass $M^{2}$ would be a very
light unstable particle for which one can detect the lifetime in the
laboratory as a function of its energy to very high precision. In this article
we conjecture that a muon collider or accelerator would be a perfect tool to
investigate the existence of an anomalous time dilation, and with it the
fundamental structure of spacetime at the Planck scale.

##  I Introduction

A self consistent theory of quantum gravity is still elusive and the
theoretical predictions, as well as experimental searches for traces of
quantum gravity are part of the fundamental ongoing endeavours in fundamental
physics Addazi:2021xuf .

The main caveat to find quantum gravity effects is that these, if they become
relevant at the Planck scale, are highly suppressed: in terms of length by
$\ell_{\text{Pl}}\sim 10^{-35}\,\text{m}$ or, in terms of energy by
$E_{\text{Pl}}\approx 1.2\times 10^{19}\,\text{GeV}$. Until now, no dedicated
quantum gravity effects have been detected, and the chances are low to detect
them by accident. Therefore, to increase the chances for a detection, mainly
two things are needed: First, a promising, rigorously derived prediction from
a fundamental or phenomenological model of quantum gravity, in order to know
where to look for the effect; second, an amplification mechanism which brings
the predicted effect into the range of nowadays measurement precision.

The most prominent amplifiers in the literature are appear in the context of
cosmic astrophysical or cosmological systems. In the search for quantum
gravity induced time delays of high energetic particles (photons or
neutrinos), possible tiny modified dispersion relation (MDR) effects should
accumulate over the enormous cosmological travel distance of the particles,
and become visible in gamma-ray and neutrino telescopes FermiGBMLAT:2009nfe ;
MAGIC:2020egb ; IceCube:2021tdn . For quantum gravity modified particle
interactions and threshold effects, the amplifier is the power of the energy
of the parent particle. In ultra-high energetic cosmic rays the highest
energetic particles could trigger such modified interactions and which the
become visible in the observation of the rays and the products of the
interaction of the rays with the particles of the atmosphere Jacobson:2002hd ;
HAWC:2019gui .

Despite these high-energy, large distance amplifiers in cosmic systems, there
also exist, maybe a bit unexpectedly, amplifiers, which open up a window to
Planck scale physics in Earth bound, local physical systems. Prominent
examples here are cold atoms experiments Amelino-Camelia:2009wvc , or
manifestations of generalized uncertainty principles and minimal length
scenarios Hossenfelder:2012jw ; Bosso:2023aht .

Recently Lobo:2020qoa , we found an unexpected amplifier in the relativistic
lifetime of elementary particles. We investigated how modified dispersion
relations, lead to a modified time dilation factor between the laboratory
frame and the rest frame of the particle, by applying the clock postulate
rigorously to Planck scale modified dispersion relations. The result was, that
for certain modified dispersion relations, the time dilation contains a
modification which scales with the energy of the particle to the third power,
which would lead to a detectable modification for high energetic particles,
even if the effect is suppressed by the Planck scale. In this letter we
outline how such a modified time dilation factor could in principle be
detected (or excluded) in dedicated accelerator experiments on Earth.

The advantage of Earth, or Earth orbit, based experiments over the observation
of cosmic messengers is that the setup and the initial conditions of the
system under observation can be highly controlled. Moreover, usually the
measurement precision and measurement time is usually higher. The downside is
that the energies which can be achieved on Earth, or Earth orbit, are not as
high as the ones of cosmic messengers.

##  II Time dilation from modified dispersion relations

The time dilation effect in Special Relativity is a consequence of any of two
features:

  1. a)

the Lorentz transformation between frames,

  2. b)

the so called clock postulate, which states that the proper time that an
observer measures between two events on spacetime is given by the length of
its trajectory between the two events.

This second approach has a conceptual advantage in comparison to the former,
as it does not rely on Lorentz transformations as symmetry of spacetime and
can thus be applied to curved spacetimes as well and includes gravitational
time dilations or redshifts.

Moreover, the clock postulate can be extended, even to spacetime geometries
beyond Riemannian spacetime geometry, which makes this approach most suitable
to study time dilations in the absence of a spacetime metric. The only
geometric ingredient needed on spacetime to employ the clock postulate is a
geometric length measure for curves. The most general spacetimes with this
property are so called Finsler spacetimes Pfeifer:2019wus . Hence, to
determine the time dilation induced by Planck scale modified dispersion
relations, we need to derive the corresponding length measure/clock for
massive particle trajectories.

###  II.1 The modified time dilation formula

In Lobo:2020qoa  we performed this derivation in all detail for general
modifications of the general relativistic dispersion relation. Assuming
isotropy, i.e. that the dispersion relation only depends on the norm of the
spatial momentum of the particle, $p=|\vec{p}|$, the MDR can be displayed as

$\displaystyle M^{2}=E^{2}-p^{2}+\frac{1}{E_{\text{QG}}}h(E,p)\,,$ (1)

where $E$ is the energy of the particle and $E_{\text{QG}}$ is the energy
scale from which quantum gravity effects become apparent, which is supposedly
of the order of the Planck energy and suppresses deviations from
special/general relativistic expressions. Introducing the zeroth order,
special relativistic $\gamma=(1-v^{2})^{-1/2}=E/M$ factor the resulting time
dilation becomes

$\displaystyle\tau=\frac{t}{\gamma}\left(1+\frac{1}{E_{QG}}f(\gamma)\right)\,,$
(2)

where $f(\gamma)$ is determined from the function $h(E,p)$ which defines the
MDR, $\tau$ is the proper time in the particle rest frame and $t$ is the
laboratory time. A prominent model is the $\kappa$-Poincaré dispersion
relation in the bicrossproduct basis in which $h(E,p)=-Ep^{2}$. The resulting
time dilation, expressed in terms of the energy $E$ of the particle, contains
a term $E^{3}$

$\displaystyle
t(E,m)=\gamma\tau\left(1-\frac{M}{2E_{QG}}(\gamma^{2}-1)\right)=t_{\text{SR}}\left(1+\frac{M}{2E_{QG}}\left(\frac{M}{E}-2\frac{E}{M}+\frac{E^{3}}{M^{3}}\right)\right)\,,$
(3)

where $t_{\text{SR}}=E\tau/M$ is the result from Special Relativity. Thus,
even if $E_{\text{QG}}$ is the Planck energy, measuring the lifetime of
particles of reasonably high energy $E$ can lead to an observable deviation
from the special relativistic prediction. Such an amplification effect does
not only emerge in this example MDR, but for many models with polynomial
modifications.

Therefore, we conjecture to devise a dedicated experiment to measure $t(E,M)$
for unstable particles, since these measurements are a window to new physics
and capable to constrain, or find evidence for, deviations from local Lorentz
invariance. The special relativistic term, the linear term in $E$ of $t(E,M)$
might just be the first order approximation, to a more complex dependency of
the dilated lifetime of particles with their energy.

###  II.2 The actual observable

In actual experiments like accelerators, one usually does not consider
$t(E,M)$, when one analyses the lifetime of fundamental particles. The
quantity considered is the the lifetime of the particle at rest $\tau$ as a
function of its mass defined by the particle data group Workman:2022ynf
$M_{\text{PDG}}$, its transverse momentum $p_{\text{T}}$ and the decay
distance of the $L_{xy}$ in the laboratory, ATLAS:2012cvl ; ALICE:2023ecf ,

$\displaystyle\tau=\tau(M_{\text{PDG}},L_{xy},p_{\text{T}})\,.$ (4)

We consider a simplifyied situation of motion in 1+1 D, such that for the
modified dispersion relations, we identify $M_{\text{PDG}}=M$ and
$p_{\text{T}}=p=|\vec{p}|$, so that the only variable left to be identified is
$L_{xy}$.

Considering (1) as Hamilton function which determines the motion of particles
$M=H(x,p)$. The Hamilton equations of motion lead to energy and momentum
conservation, since we are considering translation invariant dispersion
relations $H(x,p)=H(p)$,

$\displaystyle\dot{E}=-\partial_{t}H=0\,,\quad\dot{p}=-\partial_{x}H=0\,,$ (5)

and determine the worldlines of the particles, here for the $\kappa$-Poincaré
model $h(E,p)=-Ep^{2}$,

$\displaystyle\dot{t}=\partial_{E}H=2E-\frac{1}{E_{\text{QG}}}p^{2}\,,\quad\dot{x}=\partial_{p}H=-2p\left(1+\frac{1}{E_{\text{QG}}}E\right)\,.$
(6)

Hence for the motion of the particle in the lab frame we find, using the
dispersion relation $M^{2}=E^{2}-p^{2}-\frac{1}{E_{\text{QG}}}Ep^{2}$ to
replace $E$ in terms of $p$ and $M$,

$\displaystyle v$
$\displaystyle=\frac{dx}{dt}=\frac{\dot{x}}{\dot{t}}=-\frac{p}{\sqrt{M^{2}+p^{2}}}-\frac{p}{E_{\text{QG}}}\Rightarrow\quad
x(t)=x(0)-t\left(\frac{1}{\sqrt{M^{2}+p^{2}}}+\frac{p}{E_{\text{QG}}}\right)\,.$
(7)

Fixing the boundary conditions such that the particle gets created at $x(0)$
and decays at $x(t)$, we identify its decay length as $L_{xy}=|x(t)-x(0)|$.
Using (3) to express the decay length as function of the proper lifetime
$\tau$ of the particle and solving for $\tau$ yields

$\displaystyle\tau=\frac{L_{xy}M}{p}\left(1-\frac{1}{E_{\text{QG}}}\frac{\sqrt{M^{2}+p^{2}}(2M^{2}+p^{2})}{2M^{2}}\right)\,.$
(8)

Our prediction is that if this quantity is calculated using data from particle
accelerators with higher and higher energies, discrepancies should eventually
emerge when we compare this result with the lower energies measurements, if
the determination of the lifetime in accelerators is done assuming the special
relativistic expression $L_{xy}M/p$. We conjecture to perform searches for
deviations from Lorentz invariance this way.

###  II.3 Towards detecting anomalous time delays in particle decays

Our discussion in the previous section suggests that the special relativistic
time dilation might just be the first order of a Taylor series of a more
complicated time dilation mechanism, whose coefficients could be determined
measuring $t(E,M)$ to high precision. Any deviation from the linear relation
would indicated a modification of local Lorentz invariance.

The question is, how could one measure the time dilation to the required
precision? What would be good search strategies? In the following we discuss
some ideas, with the aim to start a discussion about a dedicated measurement
of $t(E,M)$.

####  II.3.1 Measuring $t(E,M)$ in different experiments, which are working at
different energy scales.

This strategy is similar, in a certain sense, to the one that has been carried
out recently concerning time delays of photons emitted from gamma ray bursts
(GRBs) Zhang:2014wpb ; Amelino-Camelia:2016ohi ; Bolmont:2022yad , where an
ensemble of astrophysical events is considered in the same analysis and
deviations from a horizontal line (in which there is no time delay) would be a
signature of a deformed kinematics of photons propagating in a quantum
spacetime.

Unfortunately, there are some difficulties for this approach: although we are
capable of reaching controllable energies at the TeV scale with current
experiments, like those performed at the LHC (and possibly for the future the
FCC), this is only valid for the hadron beams. The produced particles have
energies (or transverse momenta) distributed along a range of a few GeV
ALICE:2013cdo  (the issue on the necessary energy for reaching Planck scale
sensitivity will be discussed below). This is known from dedicated analyses of
the distribution of produced particles per energy ALICE:2010syw  in such
experiments, and this range is considered as an input in the likelihood
analysis carried out for the determination of particles’ lifetimes. Therefore,
we do not have the direct information regarding the duo “unstable-particle
dilated lifetime” and “unstable-particle energy” and even if we had it, the
energies/momentum involved are some orders of magnitude below the TeV scale of
the beam.

####  II.3.2 Search for a momentum dependence of $L_{xy}M/p$

A closer look on (8) shows that we can reinterpret the corrections to the
proper time of the particle as a momentum dependence of the quantity
$L_{xy}M/p$. Special Relativity predicts that the ratio of the the particle’s
travel distance until it decays and its momentum always gives a constant
number $\tau$, as can be seen in (8) for $E_{\textrm{QG}}\to\infty$.
Therefore, if one is able to consider ranges with higher energies and lengths,
we should expect a departure of the constancy of this ratio. In fact, if the
effect that we are describing in this paper exists, we predict the momentum
dependence for the $\kappa$-Poincaré dispersion relation to be

$\displaystyle\frac{L_{xy}M}{p}=\tau\left(1+\frac{1}{E_{\text{QG}}}\frac{\sqrt{M^{2}+p^{2}}(2M^{2}+p^{2})}{2M^{2}}\right)\,.$
(9)

Therefore, one should observe a shift in the likelihood fit of such ratio at
higher momentum/length ranges. An actual discrepancy would just be actually
perceivable when the relative uncertainty of measurements of this quantity
matches the dimensionless correction of Eq. (9). We like to point out that for
small masses the effect becomes larger, as the correction term diverges for
$M\to 0$.

Let us discuss the recent measurement of the lifetime of the $\Lambda$ hyperon
performed at ALICE ALICE:2023ecf  in this context. From this we can get an
estimate between the order of magnitude of the momentum of the decaying
particle and the magnitude of the uncertainty in the measurement. This
particle was chosen not just because its lifetime encompasses a very recent
control of uncertainties, but also because it is the lightest hyperon and we
see from (9) that the lighter the particle, the stronger is the effect.

The $\Lambda$ hyperon has a an average PDG mass of
$M_{\Lambda}=1115.683\,\text{MeV}$ Workman:2022ynf  and two two-body decay
channels $\Lambda\rightarrow p+\pi^{-}$ and
$\bar{\Lambda}\rightarrow\bar{p}+\pi^{+}$. Its lifetime has been reconstructed
at ALICE as $\tau_{\Lambda}=[261.07\pm 0.37\,\text{(stat.)}\pm
0.72\,\text{(syst.)}]\,\text{ps}$, and the particles were produced from a Pb-
Pb beam collision with a center of mass energy of
$\sqrt{s_{\text{NN}}}=5.02\,\text{TeV}$.

This setup gives a relative uncertainty of the order $\sigma\sim 0.1\%$, as
can be seen if we express the result of the measurement as
$\tau_{\Lambda}=261.07[1\pm 0.14\%\,\text{(stat.)}\pm
0.28\%\,\text{(syst.)}]\,\text{ps}$, and we estimate $\tau_{\Lambda}\sim
261.07[1\pm\sigma]\,\text{ps}$.

In Fig.1, we plotted the dimensionless correction in Eq.(9) as a function of
the momentum, assuming $E_{\text{QG}}=E_{\text{Pl}}=1.2\times
10^{19}\,\text{GeV}$ which the Planck energy. We see that in order to achieve
Planck scale sensitivity by performing experiments of the lifetime with
nowadays uncertainty of $0.1\%$, we would need a momentum range of the order
$300\,\text{TeV}$ (blue, dashed line). Furthermore, even if we somehow managed
to improve the measurement precision in one order of magnitude, going to
$\sigma\sim 0.01\%$ (red, dashed line), or two orders of magnitude, going to
$\sigma\sim 0.001\%$ (purple, dashed line), we would still need a momentum
range of the order $150\,\text{TeV}$ or $\sim 60\,\text{TeV}$, respectively.
This scenario is unachievable in the foreseen future regarding the energy of
the hadron beam (for the first two uncertainties) and even more regarding the
momentum range of the produced $\Lambda$ hyperon.

Figure 1: The dimensionless correction of the lifetime as function of the
momentum (solid, black line) for $E_{\text{QG}}=E_{\text{Pl}}=1.2\times
10^{19}\,\text{GeV}$ (Planck energy). We considered the $\Lambda$ hyperon mass
$M=1115.7\,\text{MeV}$, whose lifetime has been measured in ALICE:2023ecf
with relative uncertainty $\sigma=0.1\%$ (blue, dashed line), where we see the
need for $|p|\sim 300\,\text{TeV}$ to achieve Planck scale sensitivity. We
also described hypothetical relative uncertainties of $\sigma=0.01\%$ (red,
dashed line) and $\sigma=0.001\%$ (purple, dashed line), where we see the need
for $|p|\sim 150\,\text{TeV}$ and $\sim 60\,\text{TeV}$ to achieve Planck
scale sensitivity. We highlighted the precision achieved nowadays for an
actual measurement of the $\Lambda$ hyperon lifetime.

As can be seen in (9), a way to improve this effect is to consider light
particles, however, they cannot be so light that it has a too long dilated
propagation distance such that the result of its decay is produced beyond the
detector. For example, the experiments carried out at the LHC is uncapable of
detecting the product of the decay of the muon, which is a light particle with
an average PDG mass $M_{\pi}=105.658\,\text{MeV}$. If the lifetime of the muon
could be measured at such accelerators, we would be able to reduce the
necessary energy for reaching the Planck scale as can be seen in Fig.2. This
figure shows that for nowadays control of uncertainties (blue, dashed line),
we would need muons with momenta of the order $65\,\text{TeV}$. Each order of
magnitude in improvement in the precision of the measurement reduces in half
the necessary momentum for reaching Planck scale sensitivity, where we reach
LHC-like energies only for a relative uncertainty of the order $10^{-6}$. In
any of these cases, we still have the problem for hadron colliders in which
only the hadrons beam would achieve such energies, and the energy of the
unstable particle itself would be just a fraction of it, which does not help
us in scrutinizing the Planck scale.

Figure 2: The dimensionless correction of the lifetime as function of the
momentum (solid, black line) for $E_{\text{QG}}=E_{\text{Pl}}=1.2\times
10^{19}\,\text{GeV}$ (Planck energy). We considered the muon mass
$M=105.7\,\text{MeV}$. The horizontal lines represent some possible relative
uncertainties: $\sigma=0.1\%$ (blue, dashed line), where one would need for
$|p|\sim 65\,\text{TeV}$; $\sigma=0.01\%$ (red, dashed line), where one would
need for $|p|\sim 30\,\text{TeV}$; $\sigma=0.001\%$ (purple, dashed line),
where one would need for $|p|\sim 14\,\text{TeV}$ to achieve Planck scale
sensitivity. We highlighted nowadays precision in lifetime measurements in
accelerators.

####  II.3.3 Accelerating light unstable particles: muon accelerators

The main lesson from the two previous subsections is that it is difficult to
use hadron colliders to measure the lifetime, or travel distance of unstable
particles to a precision needed to detect Planck-scale induced deviations from
Lorentz invariance.

What is needed are light unstable particles which can be accelerated to
energies which are achievable in hadron accelerators like the LHC or FCC.

_The most natural candidate for such an undertaking which satisfies this
requirement is to study the lifetime of muons and to build a muon
accelerator/collider._

A muon collider Long:2020wfp  is a proposal that has recently gained traction
in the community, including some Snowmass papers dedicated to its
idealization, summarized in Black:2022cth  and the establishment of the
International Muon Collider Collaboration Black:2022cth . The main objective
of these efforts is the construction of a $10+$TeV accelerator capable of
colliding muons, which would have the advantage of allowing the exploration of
a leptonic environment at energies that are higher than those achievable for
electron-positron colliders Black:2022cth . Besides that, it would provide a
cleaner environment for collecting data on for such unstable particle physics
in comparison to hadron colliders. The cleanness of the environment is also
fundamental to compare bounds found for deviations of special relativity
dilated lifetimes from cosmic rays data PierreAuger:2021mve , where the
analysis of particles at extensive air showers heavily suffers from
uncertainties Addazi:2021xuf  such that it becomes hard to find smoking guns
that could point out to new physics from the presence of anomalies in this
scenario, like explaining the recently found muon deficit PierreAuger:2021qsd
.

The idea of having a muon beam produced and accelerated in a ring by a
magnetic field and measuring the result of its decay in order to verify its
dilated lifetime is not new and has been used in the past to verify the time
dilation prediction of Special Relativity Bailey:1977de  at $0.1\%$ relative
uncertainty. In this case, the apparatus that served to verify such dilation
was the same as the one used to measure the anomalous magnetic moment of the
muon at CERN. Although this kind of experiment served well to verify the
standard time dilation, it would not be suitable to test Planck scale
corrections due to the needs for muon $g-2$ measurements, where it is
necessary to have a fixed and low (for our standards) “magic” Lorentz factor
$\gamma\approx 29.3$ in order to remove the contribution of the stabilizing
quadrupole electrostatic field from the muon’s relation between the angular
frequency and the electromagnetic field, named Thomas-Bargmann-Michel-Telegdi
equation jegerlehner2007anomalous ; Lobo:2018zrz . We imagine that if a
similar scenario could be realized, but for a $10+$TeV muon accelerator in the
future, this would lead to fundamental new insights on the validity of Lorentz
invariance as fundamental symmetry of nature at these high energy scales.

A most spectacular outcome of such a measurement would be the evidence that
indeed Lorentz invariance is just the low energy approximation of a more
fundamental symmetry, the less spectacular, but not lesser important outcome
would be to constrain models of quantum gravity.

##  III Conclusion

Recently, it has been suggested that the way accelerated particles lifetimes
is dilated due to relativistic effects could present reminiscent signatures of
an underlying quantum spacetime structure. Such corrections would present an
amplifier of the order of square of the ratio between the energy of the
particle and its mass, which represents a great opportunity for
phenomenological analyses carried out using light and high energetic unstable
particles. Preliminary studies have been performed in the environment of
extensive air showers from cosmic rays, but with a drawback that such
environment is very polluted with uncertainties, making it hard to find a
smoking gun, i.e., an unambiguous signal that could just be explained by
quantum gravity. In order to clean up such environment and aid in the search
for such effect, we suggest that particle accelerators could indeed serve to
scrutinize this scenario, at least at first order perturbation in the
supposedly Planckian scale.

In this paper, we demonstrated some difficulties that one would face when
investigating this effect even at a next generation hadron collider, that
could reach energies $50+$TeV, like the FCC. The main difficulty concerns the
fact that the produced, unstable particles would present just a fraction of
the energy of the primary beam at the order of tens of GeVs, and as we
verified, one would need to reach energies beyond the TeV scale in order to
scrutinize this effect with Planck scale sensitivity.

The most prominent solution to this technological problem is actually the
development of an accelerator capable of accelerating a light unstable
particle to such scale. This candidate very naturally turns out to be a muon
collider or accelerator capable of reaching $10+$TeV, that is currently under
discussion for the next twenty years. Therefore, this paper adds an extra
brick to the set of possible achievements of such apparatus, for the research
on quantum gravity and fundamental spacetime symmetries.

## Acknowledgments

I. P. L. was partially supported by the National Council for Scientific and
Technological Development - CNPq grant 306414/2020-1 and by the grant
3197/2021, Paraíba State Research Foundation (FAPESQ). The authors would like
to acknowledge networking support by the COST Action QGMM (CA18108), supported
by COST (European Cooperation in Science and Technology). C.P. was funded by
the cluster of excellence Quantum Frontiers funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy - EXC-2123 QuantumFrontiers - 390837967.

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