If $X\in{\mathcal{N}}\vee J{\mathcal{N}}J$ is such that
$L:=\operatorname*{w-lim}\limits_{t\to-\infty}\Delta^{it}X\Delta^{-it}$
exists, then $L\in\mathcal{A}_{\infty}$.

We now apply this technique to the half-sided inclusions
$\mathcal{L}_{T}(K)\subset\mathcal{L}_{T}(H)$, $K:=U(1)H\subset H$, defined by
Prop. 4.6. For $T=F$, the relative commutant $\mathcal{C}_{F}(K,H)$ is known
to be non-trivial for $K^{\prime}\cap H\neq\\{0\\}$, and for various symmetric
twists it is expected to be non-trivial.

We begin with a preparatory lemma which does not rely on the half-sided
structure. The lemma says that an operator $A$ commuting with left and right
fields is necessarily quite complicated in the sense that $A\Omega$ cannot be
a finite particle number vector. Here we only need to assume that $T$ is
strict (Def. 2.1).

######  Lemma 4.8.

Let $H,K\subset\mathcal{H}$ be standard subspaces with
$\operatorname{dim}K\geq 2$ and let $T\in{\mathcal{T}}_{>}(H)$ be a strict
twist. Let $A\in\mathcal{L}_{T}(H)^{\prime}\cap\mathcal{R}_{T}(K)^{\prime}$,
$A\neq 0$. If

$\displaystyle n(A):=\sup\\{n\in\mathbb{N}_{0}\,\colon\,[A\Omega]_{n}\neq
0\\}<\infty,$ (4.14)

then $A\in\mathbb{C}1$.

###### Proof.

The supremum $n(A)$ exists because $A\neq 0$ and $\Omega$ separates the
intersection $\mathcal{L}_{T}(H)^{\prime}\cap\mathcal{R}_{T}(K)^{\prime}$ (it
even separates $\mathcal{L}_{T}(H)^{\prime}$ because $H$ is standard, see
Lemma 2.9), hence $A\Omega\neq 0$. We have to show $n(A)=0$ and proceed by
contradiction, assuming $n(A)>0$.

By assumption, $A$ commutes with $\phi^{H}_{L,T}(\xi)$ and
$\phi^{K}_{R,T}(\eta)$ for all $\xi\in H+iH$ and all $\eta\in K+iK$, i.e.

$\displaystyle A\xi$
$\displaystyle=A\phi^{H}_{L,T}(\xi)\Omega=\phi^{H}_{L,T}(\xi)A\Omega=a_{L,T}^{\star}(\xi)A\Omega+a_{L,T}(S_{H}\xi)A\Omega,$
$\displaystyle A\eta$
$\displaystyle=A\phi^{K}_{R,T}(\eta)\Omega=\phi^{K}_{R,T}(\eta)A\Omega=a_{R,T}^{\star}(\eta)A\Omega+a_{R,T}(S_{K}\eta)A\Omega.$

Since $[A\Omega]_{n(A)+2}=0$, projecting these equations onto
$\mathcal{H}_{T,n(A)+1}$ gives (note that $\operatorname{ker}P_{T,m}=0$ for
all $m$ because $T\in{\mathcal{T}}_{>}$, hence we don’t have to project onto a
quotient)

$\displaystyle(A\xi)_{n(A)+1}$
$\displaystyle=a_{L,T}^{\star}(\xi)(A\Omega)_{n(A)}+a_{L,T}(S_{H}\xi)(A\Omega)_{n(A)+2}=\xi\otimes(A\Omega)_{n(A)},$
$\displaystyle(A\eta)_{n(A)+1}$
$\displaystyle=a_{R,T}^{\star}(\eta)(A\Omega)_{n(A)}+a_{R,T}(S_{K}\eta)(A\Omega)_{n(A)+2}=(A\Omega)_{n(A)}\otimes\eta.$

As $H$ is standard, for each $\eta\in K+iK$ we can find a sequence
$(\xi^{\eta}_{l})_{l\in\mathbb{N}}\subset H+iH$ such that
$\|\xi^{\eta}_{l}-\eta\|_{\mathcal{H}}\to 0$ as $l\to\infty$. By continuity
and the first of the two equations above, we then have

$\displaystyle 0$
$\displaystyle=(A\xi^{\eta}_{l})_{n(A)+1}-\xi^{\eta}_{l}\otimes(A\Omega)_{n(A)}\longrightarrow(A\eta)_{n(A)+1}-\eta\otimes(A\Omega)_{n(A)}$

for $l\to\infty$. Comparing this equation with the second equation above
yields

$\displaystyle\eta\otimes(A\Omega)_{n(A)}=(A\Omega)_{n(A)}\otimes\eta\quad\text{
for all }\,\eta\in K+iK,$

which is impossible because $(A\Omega)_{n(A)}\neq 0$ and
$\operatorname{dim}K>1$. ∎

######  Theorem 4.9.

Let $H\subset\mathcal{H}$ be a standard subspace and
$T\in{\mathcal{T}}_{\geq}(H)$ a braided crossing-symmetric twist with
$\|T\|<1$. Let $U$ be a translation group compatible with $H$ and $T$, and set
$K:=U(1)H\subset H$.

Then the half-sided modular inclusion
$\mathcal{L}_{T}(K)\subset\mathcal{L}_{T}(H)$ has trivial relative commutant.

###### Proof.

The idea is to use Prop. 4.7, namely to establish that the vacuum projection
$P_{\Omega}$ lies in the algebra at infinity. The easiest way to achieve this
would be to realize it as a weak limit of
$\Delta^{it}\phi_{L,T}(k)\phi_{R,T}(k^{\prime})\Delta^{-it}$ as $t\to-\infty$
(with $k\in K$, $k^{\prime}\in J_{H}K$), see Prop. 4.7 b).

Hence we take $h_{1},h_{2}\in H$ such that $k:=U(1)h_{1}\in K$ and
$k^{\prime}:=U(-1)J_{H}h_{2}$ satisfy $\langle k,k^{\prime}\rangle=1$ (which
is possible because $H$ is cyclic), and consider the field operators

$\displaystyle\phi_{L,T}(k)$
$\displaystyle\in\mathcal{L}_{T}(K)=\Gamma_{T}(U(1))\mathcal{L}_{T}(H)\Gamma_{T}(U(1))^{\star},$
$\displaystyle\phi_{R,T}(k^{\prime})$
$\displaystyle\in\mathcal{R}_{T}(J_{H}U(1)H)=J(\Gamma_{T}(U(1))\mathcal{L}_{T}(H)\Gamma_{T}(U(1))^{\star})J;$

these are bounded operators because $\|T\|<1$.

In view of the form of the modular unitaries $\Delta^{it}$ of
$(\mathcal{L}_{T}(H),\Omega)$ (Prop. 3.25), we have

$\displaystyle\Delta^{it}\phi_{L,T}(k)\phi_{R,T}(k^{\prime})\Delta^{-it}$
$\displaystyle=\phi_{L,T}(\Delta_{H}^{it}k)\phi_{R,T}(\Delta_{H}^{it}k^{\prime})$
$\displaystyle=a_{L,T}^{\star}(\Delta_{H}^{it}k)a_{R,T}^{\star}(\Delta_{H}^{it}k^{\prime})+a_{L,T}^{\star}(\Delta_{H}^{it}k)a_{R,T}(\Delta_{H}^{it}k^{\prime})$
$\displaystyle\qquad+a_{L,T}(\Delta_{H}^{it}k)a_{R,T}(\Delta_{H}^{it}k^{\prime})+a_{L,T}(\Delta_{H}^{it}k)a_{R,T}^{\star}(\Delta_{H}^{it}k^{\prime}).$

To investigate the limit $t\to-\infty$, we first recall that $\Delta_{H}$ has
purely absolutely continuous spectrum and hence $\Delta_{H}^{it}\to 0$ weakly
as $t\to-\infty$ by the Riemann-Lebesgue–Lemma. This is a consequence of the
commutation relations (4.10) and $\operatorname{ker}{\mathrm{P}}=\\{0\\}$,
which imply that $\log P$ and $\log\Delta_{H}$ satisfy canonical commutation
relations and are hence a multiple of the Schrödinger representation [Lon08,
Prop. 1.7.1].

This implies that for any $\xi\in\mathcal{H}$, we have
$a_{L,T}(\Delta_{H}^{it}\xi)\to 0$ and $a_{R,T}(\Delta_{H}^{it}\xi)\to 0$ in
the strong operator topology as $t\to-\infty$ (evident on vectors of the form
$\psi_{1}\otimes\ldots\otimes\psi_{n}$, then apply uniform boundedness – note
that our assumptions imply that $T$ is strict, i.e. $\mathcal{H}^{\otimes
n}=\mathcal{H}_{T,n}$).

Taking into account the uniform bound (2.18), we see that the first three
summands converge weakly to zero as $t\to-\infty$. With Lemma 3.20, the last
term may be rewritten on a vector $\Psi_{n}\in\mathcal{H}^{\otimes n}$ as

$\displaystyle a_{L,T}(\Delta_{H}^{it}k)a_{R,T}^{\star}($
$\displaystyle\Delta_{H}^{it}k^{\prime})\Psi_{n}=a_{R,T}^{\star}(\Delta_{H}^{it}k^{\prime})a_{L,T}(\Delta_{H}^{it}k)\Psi_{n}$
$\displaystyle\;+\langle\Delta_{H}^{it}k,\Delta_{H}^{it}k^{\prime}\rangle
E_{0}\Psi_{n}+a_{L}(\Delta_{H}^{it}k)T_{1}\cdots
T_{n}(E_{0}^{\perp}\Psi_{n}\otimes\Delta_{H}^{it}k^{\prime}).$

Here the first term $a^{\star}a$ converges weakly to zero as $t\to-\infty$,
and the second term equals $E_{0}\Psi_{n}$ for all $t$.

Since $\phi_{L,T}(k)\phi_{R,T}(k^{\prime})$ is a bounded operator and the unit
ball in ${\mathcal{B}}(\mathcal{F}_{T}(\mathcal{H}))$ is compact in the weak
operator topology,
$\Delta^{it}\phi_{L,T}(k)\phi_{R,T}(k^{\prime})\Delta^{-it}$ has a weak limit
for $t\to-\infty$ along a suitably chosen net $(t_{j})_{j}$ going to
$-\infty$. By construction, the limit lies in the algebra at infinity
$\mathcal{A}_{\infty}$ (4.13).

Hence we conclude that

$\displaystyle
W:=\operatorname*{w-lim}_{j\to\infty}a_{L}(\Delta_{H}^{it_{j}}k)\bigoplus_{n=1}^{\infty}T_{1}\cdots
T_{n}\,a_{R}^{\star}(\Delta_{H}^{it_{j}}k^{\prime})E_{0}^{\perp}$

exists as a bounded operator and $Q:=E_{0}+W$ lies in $\mathcal{A}_{\infty}$.
Moreover, $Q$ is selfadjoint because
$\Delta^{it}\phi_{L,T}(k)\phi_{R,T}(k^{\prime})\Delta^{-it}$ is selfadjoint
for any $t$. Clearly, $\Omega$ is an eigenvector of $Q$, with eigenvalue $1$.
Hence the spectral projection
$\Pi^{Q}_{1}\in{\mathcal{B}}(\mathcal{F}_{T}(\mathcal{H}))$ of $Q$ for
eigenvalue $1$ also lies in $\mathcal{A}_{\infty}$ and satisfies
$\Pi^{Q}_{1}\geq E_{0}$.

If we even have $\Pi^{Q}_{1}=E_{0}$, Prop. 4.7 a) immediately concludes the
proof of the theorem. But in general, the equality $\Pi^{Q}_{1}=E_{0}$ is not
clear and we have to use Lemma 4.8. In order to be able to do so, we now show
that there exists $n_{T}\in\mathbb{N}$ such that $\Pi^{Q}_{1}$ satisfies

$\displaystyle\Pi^{Q}_{1}\mathcal{H}_{T,n}=\\{0\\},\qquad n\geq n_{T}.$ (4.15)

To prove this claim, it will be advantageous to work with the $T$-independent
tensor product norm $\|\cdot\|$ instead of $\|\cdot\|_{T}$. Recall that
$\mathcal{H}^{\otimes n}=\mathcal{H}_{T,n}$ as vector spaces because
$\mathcal{H}^{\otimes n}\subset\mathcal{H}_{T,n}$ is dense in the norm
$\|\cdot\|_{T}$ and the two norms $\|\cdot\|$, $\|\cdot\|_{T}$ are equivalent
because of our assumptions on $T$.

As the operators

$\displaystyle W_{n}(t):=a_{L}(\Delta_{H}^{it}k)T_{1}\cdots
T_{n}\,a_{R}^{*}(\Delta_{H}^{it}k^{\prime})|_{\mathcal{H}^{\otimes n}}$ (4.16)

are well defined operators on $\mathcal{H}^{\otimes n}$, $n\in\mathbb{N}$,
with operator norm (w.r.t. the norm $\|\cdot\|$) bounded by
$\|W_{n}(t)\|\leq\|k\|\|k^{\prime}\|\|T\|^{n}$, the limit
$W_{n}:=\operatorname*{w-lim}_{j\to\infty}W_{n}(t_{j})$ also exists in the
weak operator topology given by the norm $\|\cdot\|$, i.e. $W_{n}$ satisfies
the same bound on its $\|\cdot\|$-operator norm.

Now any $\Psi_{n}\in{\rm Ran}(\Pi^{Q}_{1})\cap\mathcal{H}^{\otimes n}$
satisfies

$\displaystyle\|\Psi_{n}\|=\|Q\Psi_{n}\|=\|W_{n}\Psi_{n}\|\leq\|k\|\|\|k^{\prime}\|\|T\|^{n}\|\Psi_{n}\|.$
(4.17)

As $\|T\|^{n}\to 0$ for $n\to\infty$, this is only possible for
$\|\Psi_{n}\|=0$ (and hence $\|\Psi_{n}\|_{T}=0$) for sufficiently large $n$.
This proves the existence of $n_{T}\in\mathbb{N}$ such that (4.15) holds.

To finish the proof of the theorem, let
$A\in\mathcal{R}_{T}(U(1)H^{\prime})\cap\mathcal{L}_{T}(H)$ be an element of
the relative commutant of the half-sided modular inclusion
$\mathcal{L}_{T}(K)\subset\mathcal{L}_{T}(H)$. Then $A$ commutes with any
element of the algebra at infinity and in particular with $\Pi^{Q}_{1}\geq
E_{0}$. So we obtain

$\displaystyle\Pi^{Q}_{1}A\Omega=A\Pi^{Q}_{1}\Omega=A\Omega.$

But the vector $\Pi^{Q}_{1}A\Omega$ is contained in
$\bigoplus_{n=0}^{n_{T}}\mathcal{H}_{T,n}$, whereas Lemma 4.8 tells us that
$A\Omega$ is not an element of this space unless $A\in\mathbb{C}1$. ∎

The significance of this result in quantum field theory will be explained in
Section 5. Here we note that the following instability result: If we take
$T=qF$, then Theorem 4.9 shows that the half-sided inclusion
$\mathcal{L}_{qF}(K)\subset\mathcal{L}_{qF}(H)$ has trivial relative commutant
for any $-1<q<1$, whereas the relative commutant equals
$\mathcal{L}_{F}(K^{\prime}\cap H)$ (which has $\Omega$ as a standard vector)
for $q=1$ because of the familiar structure of second quantization factors.
Such instability phenomena of inclusions under deformations are in line with
the observations in [BLS11, Tan12, LS22].

###  4.2. $L^{2}$-nuclearity and quasi-split inclusions

So far we have obtained results about (half-sided) inclusions
$\mathcal{L}_{T}(K)\subset\mathcal{L}_{T}(H)$ of twisted Araki-Woods algebras
having trivial relative commutant $\mathcal{R}_{L,T}(H,K)=\mathbb{C}$. We now
add a complementary result, showing that other inclusions of the form
$\mathcal{L}_{T}(K)\subset\mathcal{L}_{T}(H)$ can also have large relative
commutants.

The mechanism behind this discussion is the $L^{2}$-nuclearity condition of
Buchholz, D’Antoni, and Longo [BDL07] and the theory of (quasi-)split
inclusions [DL84, Fid01].

######  Definition 4.10.

An inclusion ${\mathcal{N}}\subset\mathcal{M}$ of von Neumann algebras with
joint cyclic separating vector and corresponding modular operators
$\Delta_{\mathcal{N}}$, $\Delta_{\mathcal{M}}$ is said to satisfy
$L^{2}$-nuclearity if $\Delta_{\mathcal{M}}^{1/4}\Delta_{\mathcal{N}}^{-1/4}$
is of trace class.

If ${\mathcal{N}}\subset\mathcal{M}$ satisfies $L^{2}$-nuclearity, then it is
quasi-split, namely there exist type I factors $\mathcal{I}$,
$\tilde{\mathcal{I}}$ such that
${\mathcal{N}}\otimes\mathbb{C}\subset\mathcal{I}\subset\mathcal{M}\otimes\tilde{\mathcal{I}}$.
If ${\mathcal{N}}$ or $\mathcal{M}$ is a factor, we even have the split
property, namely ${\mathcal{N}}\subset\mathcal{I}\subset\mathcal{M}$ with some
type I factor $\mathcal{I}$.

A consequence of ${\mathcal{N}}\subset\mathcal{M}$ being quasi-split is that
whenever $\mathcal{M}$ is of type III, then also the relative commutant
${\mathcal{N}}^{\prime}\cap\mathcal{M}$ is of type III. These results are
generalizations of [DL84] to the non-factor case [Fid01], see [BM22, Cor.
3.11].

We now state our non-triviality result for the relative commutant
$\mathcal{C}_{T}(K,H)$ under quite strong assumptions. This result is closely
related to the work of D’Antoni, Longo, and Radulescu [DLR01] who considered
the case $T=0$.

######  Proposition 4.11.

Let $K\subset H$ be an inclusion of standard subspaces and $T$ a braided twist
such that

  1. a)

$L^{2}$-nuclearity holds on the standard subspace level, i.e.
$\|\Delta_{H}^{1/4}\Delta_{K}^{-1/4}\|_{1}<1$, where $\|\cdot\|_{1}$ is the
trace norm on $\mathcal{H}$,

  2. b)

$T$ is crossing-symmetric w.r.t. $H$,

  3. c)

$T$ is compatible with $K$ and $H$, and has norm $\|T\|<1$.

Then $\mathcal{L}_{T}(K)\subset\mathcal{L}_{T}(H)$ satisfies
$L^{2}$-nuclearity and is quasi-split. If $\mathcal{L}_{T}(H)$ is of type III,
also the relative commutant $\mathcal{C}_{T}(K,H)$ is of type III.

###### Proof.

We first note that in view of the assumptions, $\Omega$ is cyclic and
separating for $\mathcal{L}_{T}(H)$ and hence also separating for the
subalgebra $\mathcal{L}_{T}(K)$. It is also cyclic for the subalgebra because
$K$ is a standard subspace, so we have $\Omega$ as a joint cyclic separating
vector.

Thanks to the compatibility assumption for $K$ and $H$, the modular operators
$\hat{\Delta}_{K}$ and $\hat{\Delta}_{H}$ of $\mathcal{L}_{T}(K)$ and
$\mathcal{L}_{T}(H)$ are of second quantized form (Prop. 3.25). Hence the
trace norm of $\hat{\Delta}_{H}^{1/4}\hat{\Delta}_{K}^{-1/4}$ is

$\displaystyle\|\hat{\Delta}_{H}^{1/4}\hat{\Delta}_{K}^{-1/4}\|_{T,1}=\sum_{n\geq
0}\|(\Delta_{H}^{1/4}\Delta_{K}^{-1/4})^{\otimes n}\|_{T,1};$ (4.18)

where $\|\cdot\|_{T,1}$ denotes the trace norm of the trace class in
${\mathcal{B}}(\mathcal{H}_{T,n})$.

As $\|T\|<1$ is a strict twist, the operator
$P_{T,n}^{1/2}:\mathcal{H}^{\otimes n}\to\mathcal{H}_{T,n}$ is unitary. Hence
$\|(\Delta_{H}^{1/4}\Delta_{K}^{-1/4})^{\otimes
n}\|_{T,1}=\|P_{T,n}^{-1/2}(\Delta_{H}^{1/4}\Delta_{K}^{-1/4})^{\otimes
n}P_{T,n}^{1/2}\|_{1}$, where $\|\cdot\|_{1}$ denotes the trace norm of the
trace class in ${\mathcal{B}}(\mathcal{H}^{\otimes n})$. But as $T$ is
compatible with $K$ and $H$, the operators $P_{T,n}^{\pm 1/2}$ commute with
$(\Delta_{H}^{1/4}\Delta_{K}^{-1/4})^{\otimes n}$ and thus drop out. Now we
may use the tensor structure of $\|\cdot\|_{1}$ to conclude

$\displaystyle\|\hat{\Delta}_{H}^{1/4}\hat{\Delta}_{K}^{-1/4}\|_{T,1}=\sum_{n=0}^{\infty}\|\Delta_{H}^{1/4}\Delta_{K}^{-1/4}\|_{1}^{n}=\frac{1}{1-\|\Delta_{H}^{1/4}\Delta_{K}^{-1/4}\|_{1}}<\infty.$
(4.19)

∎

Let us comment on the assumptions entering into this proposition. The
compatibility condition for $T$ with $H$ and $K$ is quite strong in general,
but automatically satisfied for $T=qF$, $-1<q<1$. In this case, it is also
known from Hiai’s work that $\mathcal{L}_{T}(H)$ is a type III1 factor if
$\Delta_{H}$ does not have eigenvalue $1$ [Hia01, Prop. 3.2].

The $L^{2}$-nuclearity assumption on the level of the standard subspaces is
also a strong assumption, but it is known to hold (including the norm
inequality $\|\Delta_{H}^{1/4}\Delta_{K}^{-1/4}\|_{1}<1$) for certain
inclusions of standard subspaces arising QFT (see Section 5).

##  5\. Applications to quantum field theory

In this section we explain some applications of our constructions in algebraic
quantum field theory [Haa96, BDFY15], where the main object of interest are
families of von Neumann algebras labeled by open regions in some spacetime
manifold, subject to several physically motivated conditions.

Since we will consider quantum field theories on a variety of different
spacetimes, we use an abstract formulation. From the structure of spacetime as
a globally hyperbolic Lorentzian manifold ${\mathscr{M}}$ we only need that
${\mathscr{M}}$ has a family ${\mathscr{O}}$ of “good” subsets (to be thought
of as the causally complete convex regions), a notion of causal complement,
mapping subsets $\mathcal{O}\in{\mathscr{O}}$ to their complements
$\mathcal{O}^{\prime}\in{\mathscr{O}}$ such that
$(\mathcal{O}^{\prime})^{\prime}=\mathcal{O}$ for all $\mathcal{O}$.
Furthermore, we require that there exists a Lie group $G$ acting on
${\mathscr{M}}$ such that $g\mathcal{O}\in{\mathscr{O}}$ for
$\mathcal{O}\in{\mathscr{O}}$ and preserving complementation,
$g\mathcal{O}^{\prime}=(g\mathcal{O})^{\prime}$ for all $g\in G$,
$\mathcal{O}\in{\mathscr{O}}$.

In this general setting, we require the existence of a reference region with
the following properties.

(reference wedge):

There exists $W_{0}\in{\mathscr{O}}$ such that there is a one-parameter group
$\lambda_{0}(t)\in G$, $t\in\mathbb{R}$, and an involution $j_{0}\in G$ such
that $\lambda_{0}(t)W_{0}=W_{0}$ for all $t\in\mathbb{R}$, and
$j_{0}W_{0}=W_{0}^{\prime}$ as well as
$j_{0}\lambda_{0}(t)=\lambda_{0}(t)j_{0}$.

We will refer to the reference region $W_{0}$ as “reference wedge” although
its geometrical shape can be quite different (see examples below), and to
elements of the $G$-orbit ${\mathscr{W}}:=GW_{0}\subset{\mathscr{O}}$ as
“wedges”.

(wedge separation property):

For any $\mathcal{O}_{1},\mathcal{O}_{2}\in{\mathscr{O}}$ with
$\mathcal{O}_{1}\subset\mathcal{O}_{2}^{\prime}$, there exists a wedge
$W\in{\mathscr{W}}$ such that $\mathcal{O}_{1}\subset
W\subset\mathcal{O}_{2}^{\prime}$.

(wedge intersection property):

For any $\mathcal{O}\in{\mathscr{O}}$, there holds

$\displaystyle\mathcal{O}=\bigcap_{W\in{\mathscr{W}}:W\supset\mathcal{O}}W.$
(5.1)

Two examples of this setting are the following.

  1. a)

Minkowski spacetime ${\mathscr{M}}=\mathbb{R}^{1+s}$, $s\geq 1$, with
${\mathscr{O}}$ the set of all convex causally complete regions,
$\mathcal{O}^{\prime}$ the causal complement, $G$ the Poincaré group, and
$W_{0}=\\{x\in\mathbb{R}^{1+s}\,:\,x_{1}>|x_{0}|\\}$ the Rindler wedge. Here
$\lambda_{0}(t)$ are the Lorentz boosts in $x_{1}$-direction with parameter
$t$, and $j_{0}$ is the reflection at the edge of $W_{0}$.

  2. b)

The lightray ${\mathscr{M}}=\mathbb{R}$ with ${\mathscr{O}}$ the set of all
open intervals and half-lines, $\mathcal{O}^{\prime}$ the set-theoretic
complement, $G$ the affine group of $\mathbb{R}$, and $W_{0}=\mathbb{R}_{+}$.
Here $\lambda_{0}(t)$ are the dilations $x\mapsto e^{t}x$ and $j_{0}$ is the
reflection $x\mapsto-x$.

Further examples include the circle ${\mathscr{M}}=S^{1}$ with Möbius group
symmetry or de Sitter spacetime with Lorentz group symmetry. These examples
are well known, and recently a general group-theoretic formulation has been
developed which includes these and further cases [MN21, MNO22].

The first step to a quantum field theory on ${\mathscr{M}}$ consists in
describing the one-particle localization structure, which we shall do by
modular localization. As explained in the work of Brunetti, Guido and Longo
[BGL02] and later generalizations [Mun03, MN21], we consider an (anti)unitary
representation $U$ of $G$ on a Hilbert space $\mathcal{H}$, namely a strongly
continuous group homomorphism $U$ from $G$ to the group of (anti)unitary
operators on $\mathcal{H}$ such that $U(j_{0})$ is antiunitary. Denoting the
connected component of the identity by $G^{\uparrow}\subset G$, we then have
$G=G^{\uparrow}\rtimes{\mathbb{Z}}_{2}$, where ${\mathbb{Z}}_{2}$ acts by
conjugation with $j_{0}$. We also write $G^{\downarrow}:=G\backslash
G^{\uparrow}=G^{\uparrow}j_{0}$.

Then $t\mapsto U(\lambda_{0}(t))=e^{itB}$ is a unitary one-parameter group
with some selfadjoint generator $B$, and $U(j_{0})$ is an antiunitary
involution. Setting

$\displaystyle
H_{0}:=\operatorname{ker}(U(j_{0})e^{\frac{1}{2}B}-1),\qquad\Delta_{H_{0}}^{it}=U(\lambda_{0}(t)),\qquad
J_{H_{0}}=U(j_{0}),$ (5.2)

we obtain a standard subspace $H_{0}\subset\mathcal{H}$ with the modular data
specified above.

We shall assume that $H_{0}$ is localized in $W_{0}$ in the sense that

$\displaystyle g\in G,\;gW_{0}\subset W_{0}\;\Rightarrow\;U(g)H_{0}\subset
H_{0}.$ (5.3)

This inclusion property is known to be closely linked to positive energy
properties of the representation $U$: It holds if $U$ is a positive energy
representation of the Poincaré group (in the Minkowski space example a))
[BGL02], or if if the generator of translations in the affine group is
positive (lightray example b)). See also [MN21] for a general spectral
condition implying (5.3) for $g\in G^{\uparrow}$.

Once (5.3) holds, we may consistently interpret the vectors in $H_{0}$ as
being localized in $W_{0}$, and obtain a net of standard subspaces for wedges,
namely

$\displaystyle gW_{0}\mapsto H(gW_{0}):=U(g)H_{0},\qquad g\in G.$ (5.4)

Note that this map is well-defined thanks to the inclusion property (5.3)
which implies that the stabilizer group of $W_{0}$ fixes $H_{0}$ in the
representation $U$. The standard subspace net has the duality property

$\displaystyle H(W^{\prime})=H(W)^{\prime},\qquad W\in{\mathscr{W}}.$ (5.5)

For $W=W_{0}$ this follows by choosing $g=j_{0}$ in (5.4) and observing
$U(j_{0})H_{0}=J_{H_{0}}H_{0}=H_{0}^{\prime}$, and for general wedges by
covariance.

To connect to the twisted Araki-Woods algebras, we need to amplify $U$ to a
representation on $\mathcal{F}_{T}(\mathcal{H})$ for suitable twists $T$.
Since we also need $T$ to be braided and crossing symmetric for our
standardness results to hold, we add these assumptions here.

######  Definition 5.1.

Let $U$ be an (anti)unitary representation of $G$ such that (5.3) holds. A
twist $T\in{\mathcal{T}}_{\geq}$ is called admissible if it is braided,
$G$-invariant in the sense that

$\displaystyle[U(g)\otimes U(g),T]=0,\quad g\in
G^{\uparrow},\qquad[F(J_{H_{0}}\otimes J_{H_{0}}),T]=0,$ (5.6)

and crossing-symmetric w.r.t. $H_{0}$.

The $G$-invariance condition amounts to $T$ respecting the symmetries of the
spacetime under consideration. It is trivially satisfied if $T=qF$, $-1\leq
q\leq 1$. Note that an admissible twist is automatically compatible with
$H_{0}$ (and hence any $H(W)$, $W\in{\mathscr{W}}$), in the sense of Def. 3.1.

For admissible $T$, the operators

$\displaystyle U_{T}(g):=\Gamma_{T}(U_{1}(g)),\quad g\in G^{\uparrow},\qquad
U_{T}(j_{0}):=\Gamma_{T}^{Y}(J_{H}),$ (5.7)

are well-defined (anti)unitary operators on $\mathcal{F}_{T}(\mathcal{H})$
(Lemma 2.7 and Lemma 3.16). It is clear that $U_{T}$ is a (anti)unitary
representation that leaves $\Omega$ invariant.

Given a representation $U$ and admissible $T$, we thus obtain a representation
$U_{T}$ of $G$ on $\mathcal{F}_{T}(\mathcal{H})$ and two maps

$\displaystyle W\mapsto\mathcal{L}_{T}(H(W)),\qquad
W\mapsto\mathcal{R}_{T}(H(W))$ (5.8)

from wedges $W\in{\mathscr{W}}$ to von Neumann algebras on
$\mathcal{F}_{T}(\mathcal{H})$.

######  Proposition 5.2.

Let $U$ be a (anti)unitary $G$-representation with the inclusion property
(5.3), and $T$ and admissible twist. Then the nets (5.8) satisfy

  1. a)

Isotony: $\mathcal{L}_{T}(H(W_{1}))\subset\mathcal{L}_{T}(H(W_{2}))$ for
wedges $W_{1}\subset W_{2}$, and analogously for $\mathcal{R}_{T}$,

  2. b)

Covariance: For any $W\in{\mathscr{W}}$,

$\displaystyle
U_{T}(g)\mathcal{L}_{T}(H(W))U_{T}(g)^{-1}=\begin{cases}\mathcal{L}_{T}(H(gW))&g\in
G^{\uparrow}\\\ \mathcal{R}_{T}(H(gW))&g\in G^{\downarrow}\end{cases},$ (5.9)

and analogously with $\mathcal{L}_{T}$ and $\mathcal{R}_{T}$ exchanged.

  3. c)

Reeh-Schlieder property: $\Omega$ is cyclic and separating for
$\mathcal{L}_{T}(W)$ and $\mathcal{R}_{T}(W)$, $W\in{\mathscr{W}}$.

  4. d)

Relative locality: For wedges $W_{1}\subset W_{2}^{\prime}$, we have

$\displaystyle\mathcal{L}_{T}(H(W_{1}))\subset\mathcal{R}_{T}(H(W_{2}))^{\prime},$
(5.10)

and analogously with $\mathcal{L}_{T}$ and $\mathcal{R}_{T}$ exchanged.

###### Proof.

a) follows from (5.3) and Prop. 4.1 a). Item b) is a consequence of Prop. 4.1
c), taking into account $U_{T}(j_{0})=J_{H_{0}}$. Item c) is contained in Thm.
3.22, and d) follows from

$\displaystyle\mathcal{L}_{T}(H(W_{1}))\subset\mathcal{L}_{T}(H(W_{2}^{\prime}))=\mathcal{L}_{T}(H(W_{2})^{\prime})=\mathcal{R}_{T}(H(W_{2}))^{\prime}.$

∎

We may interpret this structure as a quantum field theory in two different
ways. In the first version, we consider $G^{\uparrow}$ as our symmetry group.
Then both nets $W\mapsto\mathcal{L}_{T}(H(W))$ and
$W\mapsto\mathcal{R}_{T}(H(W))$ are $G^{\uparrow}$-covariant and relatively
local in the sense of item d) above. However, for $T\neq F$ typically neither
$\mathcal{L}_{T}$ nor $\mathcal{R}_{T}$ is local (see Lemma 4.2).

In the second version, we consider $G$ as our symmetry group. Then the
transformation behavior from item b) above imposes an additional constraint in
case there exists $g_{0}\in G^{\uparrow}$ with $g_{0}W_{0}=W_{0}^{\prime}$.
Namely, in that case we must have
$\mathcal{L}_{T}(H_{0}^{\prime})=\mathcal{R}_{T}(H_{0}^{\prime})$ for
$W\mapsto\mathcal{L}_{T}(H(W))$ to be $G$-covariant. Apart from $T=F$, this is
typically not the case (see Lemma 4.2). However, this conflict does not occur
in several situations of interest, such as two-dimensional Minkowski space,
the lightray example, or higher-dimensional Minkowski space with a restricted
symmetry group (see [BS07] for similar considerations). If
$W_{0}^{\prime}\not\in G^{\uparrow}W_{0}$, we may take

$\displaystyle
gW_{0}\mapsto\tilde{\mathcal{L}}_{T}(H(gW_{0})):=U_{T}(g)\mathcal{L}_{T}(H_{0})U_{T}(g^{-1}),\qquad
g\in G,$ (5.11)

as the definition of our net, which is then $G$-covariant and local as a
consequence of items b) and d) above. Analogously, we could work with
$\mathcal{R}_{T}$ instead of $\mathcal{L}_{T}$.

We will adopt this second point of view because it leads to local nets
$\tilde{\mathcal{L}}_{T}$, and refer to [Lec12] for a scheme (for special
twists) in which the restriction to low-dimensional spacetime or smaller
symmetry groups can be overcome.

The net $\tilde{\mathcal{L}}_{T}$ is so far defined on the set of wedges
${\mathscr{W}}$, but naturally extends to the larger set ${\mathscr{O}}$ of
localization regions. Here the assumptions of wedge separation and wedge
intersection enter.

######  Proposition 5.3.

Let $U$ be a representation with the inclusion property (5.3), $T$ an
admissible twist, and assume that $G^{\uparrow}W_{0}$ does not contain
$W_{0}^{\prime}$. Then the map

$\displaystyle{\mathscr{O}}\ni\mathcal{O}\longmapsto\mathcal{A}_{U,T}(\mathcal{O}):=\bigcap_{W\in{\mathscr{W}}:W\supset\mathcal{O}}\tilde{\mathcal{L}}_{T}(H(W))$
(5.12)

is well-defined,

  1. a)

$G$-covariant, namely
$U_{T}(g)\mathcal{A}_{U,T}(\mathcal{O})U_{T}(g^{-1})=\mathcal{A}_{U,T}(g\mathcal{O})$,
$g\in G$, $\mathcal{O}\in{\mathscr{O}}$,

  2. b)

isotonous, namely
$\mathcal{A}_{U,T}(\mathcal{O}_{1})\subset\mathcal{A}_{U,T}(\mathcal{O}_{2})$
for $\mathcal{O}_{1}\subset\mathcal{O}_{2}$,

  3. c)

local, namely
$\mathcal{A}_{U,T}(\mathcal{O}_{1})\subset\mathcal{A}_{U,T}(\mathcal{O}_{2})^{\prime}$
for $\mathcal{O}_{1}\subset\mathcal{O}_{2}^{\prime}$.

###### Proof.

The map is well-defined because any $\mathcal{O}\in{\mathscr{O}}$ is contained
in a wedge by the wedge intersection property. a) and b) are obvious. For c),
note that for $\mathcal{O}_{1}\subset\mathcal{O}_{2}^{\prime}$ there exists a
wedge $W$ such that $\mathcal{O}_{1}\subset W$ and $\mathcal{O}_{2}\subset
W^{\prime}$ (wedge separation property). Hence either $W$ or $W^{\prime}$ lies
in the $G^{\uparrow}$-orbit of $W_{0}$. In case $W=gW_{0}$, $g\in
G^{\uparrow}$, we have
$\mathcal{A}_{U,T}(\mathcal{O}_{1})\subset\mathcal{L}_{T}(H(W))$ and

$\mathcal{A}_{U,T}(\mathcal{O}_{2})\subset\mathcal{L}_{T}(H(W^{\prime}))=\mathcal{L}_{T}(H(W)^{\prime})=\mathcal{R}_{T}(H(W))^{\prime},$

which implies the claim. The case $W^{\prime}=gW_{0}$, $g\in G^{\uparrow}$, is
analogous. ∎

Given a representation $U$ and a twist $T$ satisfying our assumptions, we
therefore get a net $\mathcal{A}_{U,T}$ satisfying the basic properties of a
quantum field theory. In particular, this construction recovers the free field
for $T=F$.

The representation $U$ entering the definition of $\mathcal{A}_{U,T}$ encodes
the single particle structure (such as particle masses and spins), and the
twist $T$ is expected to be characteristic of the interaction in the
underlying theory. This is well understood in case $T$ comes from a Yang-
Baxter solution with spectral parameter (Example 2.4), in which case it
precisely encodes the two-particle scattering operator and leads to a scheme
for the operator-algebraic construction of integrable QFTs [Lec08, AL17]. In
this case, our notion of crossing symmetry also coincides with the scattering
theory definition of crossing symmetry (Remark 3.15).

In general, it is expected that the local algebras (5.12) may fail to have
$\Omega$ as a cyclic vector, or be too small to allow for interesting
observables localized in $\mathcal{O}$. In some cases, we will give proofs of
these expectations below. One should therefore view $\mathcal{A}_{U,T}$ as a
“germ” of a quantum field theory (also called “wedge-local quantum field
theory”). The class of all wedge-local models contains both, “strictly local”
models in which $\Omega$ is cyclic for all $\mathcal{A}_{U,T}(\mathcal{O})$,
$\mathcal{O}\in{\mathscr{O}}$, as well as non-local models in which this
property does not hold.

The challenge to efficiently decide which class a given tuple $U,T$
corresponds to constitutes the QFT part of the motivation for the present
article. We here comment on models with $\|T\|<1$ which have not been
investigated so far.

Typically for a given region $\mathcal{O}\in{\mathscr{O}}$, there will exist
two wedges $W_{1}=g_{1}W_{0}$, $W_{2}=g_{2}W_{0}$, $g_{1},g_{2}\in
G^{\uparrow}$, such that

$\displaystyle W_{1}\subset W_{2},\qquad\mathcal{O}\subset W_{1}^{\prime}\cap
W_{2}.$ (5.13)

This is for instance the case for $\mathcal{O}$ a double cone in Minkowski
space, or an interval on the lightray, where $W_{1}$ is a wedge (half-line)
open to the right, and $W_{2}^{\prime}$ is a wedge (half-line) open to the
left. In this situation,

$\displaystyle\mathcal{A}_{U,T}(\mathcal{O})$
$\displaystyle\subset\tilde{\mathcal{L}}_{T}(H(W_{1}^{\prime}))\cap\tilde{\mathcal{L}}_{T}(H(W_{2}))=\mathcal{R}_{T}(H(W_{1}))\cap\mathcal{L}_{T}(H(W_{2}))$
$\displaystyle=\mathcal{L}_{T}(H(W_{1}))^{\prime}\cap\mathcal{L}_{T}(H(W_{2})),$

namely $\mathcal{A}_{U,T}(\mathcal{O})$ is a subalgebra contained in the
relative commutant of the inclusion
$\mathcal{L}_{T}(H(W_{1}))\subset\mathcal{L}_{T}(H(W_{2}))$. In the cases
mentioned above (two-dimensional Minkowski space or lightray), we actually
have equality between $\mathcal{A}_{U,T}(\mathcal{O})$ and the relative
commutant.

Our results on relative commutants (Theorem 4.9 and Proposition 4.11) then
apply as follows.

  * •

Lightlike inclusions of wedges or lightrays.
Consider $W_{0}+a\subset W_{0}$ a lightlike inclusion of wedges (i.e.
$a=(\lambda,\lambda,0,\ldots,0)$, $\lambda>0$) in the Minkowski example a), or
$\mathbb{R}_{+}+\lambda\subset\mathbb{R}_{+}$, $\lambda>0$, an inclusion of
half lines in the lightray example b). In a positive energy representation, we
are then in the situation of Def. 4.5 and hence the corresponding inclusions
$\mathcal{L}_{T}(H(W_{0}+a))\subset\mathcal{L}_{T}(H(W_{0}))$ and
$\mathcal{L}_{T}(H(\mathbb{R}_{+}+\lambda))\subset\mathcal{L}_{T}(H(\mathbb{R}_{+}))$
are half-sided modular.

Now consider an arbitrary admissible twist with $\|T\|<1$ (for example, $T=qF$
with $|q|<1$, or a scaled Yang-Baxter solution $T=qT_{S}$ as in Example 2.5).
Then Theorem 4.9 implies that the half-sided inclusion has trivial relative
commutant.
In the context of the net $\mathcal{A}_{U,T}$, the interpretation of this fact
is that there are no observables strictly localized in the relative complement
of the inclusion, i.e. on the lightfront
$\overline{W_{0}^{\prime}+a}\cap\overline{W}_{0}$ or the lightlike interval
$(0,\lambda)$. Hence $\mathcal{A}_{U,T}$ is non-local from this point of view.

  * •

$L^{2}$-modular inclusions for massless theories.
Consider Minkowski space $\mathbb{R}^{d+1}$, $d\geq 1$, two double cones
$\mathcal{O}_{1},\mathcal{O}_{2}$ with
$\overline{\mathcal{O}_{1}}\subset\mathcal{O}_{2}$, and the representation $U$
defining the free massless scalar field. Then the corresponding standard
subspaces $H(\mathcal{O}_{1})\subset H(\mathcal{O}_{2})$ satisfy
$L^{2}$-nuclearity including the trace norm bound
$\|\Delta_{H(\mathcal{O}_{2})}^{1/4}\Delta_{H(\mathcal{O}_{1})}^{-1/4}\|_{1}<1$
if the distance between $\mathcal{O}_{1}$ and $\partial\mathcal{O}_{2}$ is
sufficiently large [BDL07].

Now consider an arbitrary admissible twist with $\|T\|<1$; for example, $T=qF$
with $|q|<1$. As the modular operators do not have eigenvalue $1$, in this
case $\mathcal{L}_{T}(H(\mathcal{O}_{i}))$ are type III1 factors and it
follows from Prop 4.11 implies that the relative commutant of the inclusion
$\mathcal{L}_{T}(H(\mathcal{O}_{1}))\subset\mathcal{L}_{T}(H(\mathcal{O}_{2}))$
is of type III as well.

The interpretation of this fact is that there are lots of elements of
$\mathcal{L}_{T}(H(\mathcal{O}_{2}))$ that are localized in
$\mathcal{O}_{1}^{\prime}\cap\mathcal{O}_{2}$ in the sense that they commute
with the subalgebra $\mathcal{L}_{T}(H(\mathcal{O}_{1}))$.

The first result can be understood as a sign that the models with $\|T\|<1$
are strongly non-local. It complements results establishing large relative
commutants for special twists with $\|T\|=1$, and we view such examples as
counterexamples to the construction of strictly local theories. The second
result does not strictly fit to the wedge-local setting because of the
different geometry (double cone inclusions), but does indicate that nets which
appear strongly non-local might contain more local observables than
anticipated.

We expect to learn from both scenarios and other examples how to better
control the local observable content of twisted models in future work on
constructive algebraic quantum field theory.

##  Appendix A Diagram calculus for twisted $n$-point functions

In this section we explain a diagram calculus for the twisted $n$-point
functions $W_{2n}^{\xi_{1},\ldots,\xi_{2n}}(t)$ (3.11).

Our diagrams consist of a disc with $2n$ marked points on its boundary,
labelled $1,2,\ldots,2n$ in clockwise orientation. The points are paired by
directed edges, which are curves $c$ lying in the interior of the disc,
starting at a marked point $s(c)\in\\{1,\ldots,2n\\}$ and ending at a marked
point $t(c)\in\\{1,\ldots,2n\\}$, such that

  * (D1)

every $k\in\\{1,\ldots,2n\\}$ is connected to exactly one curve,

  * (D2)

for every curve $c$, we have $t(c)>s(c)$,

  * (D3)

the curves are drawn in such a way that the number of crossings is minimal and
there are no points where three or more curves meet.

Examples of such diagrams are

$1$$2$$1$$2$$3$$4$$1$$2$$3$$4$$5$$6$

Calling the family of all such diagrams $\mathscr{D}_{2n}$, the idea is that
any $D\in\mathscr{D}_{2n}$ represents a number $\langle
D\rangle^{\xi_{1},\ldots,\xi_{2n}}$ defined in a manner familiar from knot
theory: Every line segment (between two crossings, i.e. an internal line, or
between a crossing and a marked boundary point, i.e. an external line) carries
a vector in $\mathcal{H}$. In case of an external line starting (or ending) at
$k$, this vector is $\xi_{k}$ (or $S_{H}\xi_{k}$), and in case of an internal
line, it is a vector from some orthonormal basis of $\mathcal{H}$. A crossing
represents a matrix element of $T$, namely

$\eta$$\psi$$\zeta$$\gamma$$=\langle\zeta\otimes\eta,T(\psi\otimes\gamma)\rangle,$

and $\langle D\rangle$ is defined by taking the product over all crossings of
$D$ and summing over the orthonormal bases labelling all internal lines. Then

$\displaystyle
W_{2n}^{\xi_{1},\ldots,\xi_{2n}}=\sum_{D\in\mathscr{D}_{2n}}\langle
D\rangle^{\xi_{1},\ldots,\xi_{2n}}.$ (A.1)

For general twists $T$, the rules (D1)–(D3) are however ambiguous and do not
uniquely determine $\langle D\rangle^{\xi_{1},\ldots,\xi_{2n}}$. Only in case
$T$ is braided the above definition of $\langle
D\rangle^{\xi_{1},\ldots,\xi_{2n}}$ is unambiguous. This can for example be
seen by looking at the two diagrams

$1$$2$$3$$4$$5$$6$$1$$2$$3$$4$$5$$6$

which both agree with the diagrammatic rules (D1)–(D3), but only give the same
value $\langle D\rangle$ in case $T$ is braided (invariance under Reidemeister
moves of type III).

For twists that are not braided, this results in somewhat cumbersome diagram
rules which we avoid to spell out in detail. Nonetheless, $\langle D\rangle$
is well defined for diagrams with at most two crossings. For the particular
diagram with three crossings depicted above, the left version is the correct
one for (A.1) to be true for $n=3$ (this can be seen by noting that $P_{T,3}$
contains a term $T_{2}T_{1}T_{2}$ and not $T_{1}T_{2}T_{1}$).

For following the calculations underlying Prop. 3.8 and in particular Lemma
3.10, the diagrammatic rule is very useful. The diagrammatic form of the
expansion of the $6$-point function (3.26) looks as follows:

$\displaystyle w_{0}(t)$
$\displaystyle=\Big{(}\langle\bar{1},2\rangle\langle\bar{3},4\rangle\langle\bar{5},6_{t}\rangle+\langle\bar{2},3\rangle\langle\bar{4},5\rangle\langle\bar{1},6_{t}\rangle\Big{)}$
$\displaystyle\quad+\Big{(}\langle\bar{1},2\rangle\langle\bar{4},5\rangle\langle\bar{3},6_{t}\rangle+\langle\bar{1},4\rangle\langle\bar{2},3\rangle\langle\bar{5},6_{t}\rangle+\langle\bar{2},5\rangle\langle\bar{3},4\rangle\langle\bar{1},6_{t}\rangle\Big{)}$

$1$$2$$3$$4$$5$$6$$1$$2$$3$$4$$5$$6$$1$$2$$3$$4$$5$$6$$1$$2$$3$$4$$5$$6$$1$$2$$3$$4$$5$$6$

$\displaystyle w_{1}(t)$
$\displaystyle=\langle\bar{1},2\rangle\langle\bar{4}\otimes\bar{3},T(5\otimes
6_{t})\rangle+\langle\bar{2},3\rangle\langle\bar{4}\otimes\bar{1},T(5\otimes
6_{t})\rangle$
$\displaystyle\quad+\langle\bar{3},4\rangle\langle\bar{2}\otimes\bar{1},T(5\otimes
6_{t})\rangle+\langle\bar{4},5\rangle\langle\bar{2}\otimes\bar{1},T(3\otimes
6_{t})\rangle$ $\displaystyle\quad+\langle\bar{2}\otimes\bar{1},T(3\otimes
4)\rangle\langle\bar{5},6_{t}\rangle+\langle\bar{3}\otimes\bar{2},T(4\otimes
5)\rangle\langle\bar{1},6_{t}\rangle,$

$1$$2$$3$$4$$5$$6$$1$$2$$3$$4$$5$$6$$1$$2$$3$$4$$5$$6$$1$$2$$3$$4$$5$$6$$1$$2$$3$$4$$5$$6$$1$$2$$3$$4$$5$$6$

$\displaystyle w_{2}(t)$ $\displaystyle=\langle\bar{4}\otimes
a_{3}T(\bar{2}\otimes\bar{1}),T(5\otimes 6_{t})\rangle+\langle
a_{4}T(\bar{3}\otimes\bar{2})\otimes\bar{1},T(5\otimes 6_{t})\rangle$
$\displaystyle\quad+\langle\bar{2}\otimes\bar{1},T(a_{\bar{3}}T(4\otimes
5)\otimes 6_{t})\rangle,$

$1$$2$$3$$4$$5$$6$$1$$2$$3$$4$$5$$6$$1$$2$$3$$4$$5$$6$

$\displaystyle w_{3}(t)$
$\displaystyle=\langle\bar{3}\otimes\bar{2}\otimes\bar{1},T_{2}T_{1}T_{2}(4\otimes
5\otimes 6_{t})\rangle.$

$1$$2$$3$$4$$5$$6$

Here the diagrams are listed in the same order as the terms in the algebraic
expansions of the $w_{k}(t)$.

For the analytic continuation in $t$ from $\mathbb{R}$ to $\mathbb{R}-i$,
another diagrammatic rule is helpful: The analytic continuation of the
function $\langle D\rangle^{\xi_{1},\ldots,\Delta_{H}^{it}\xi_{2n}}$ is

$\displaystyle\langle
D\rangle^{\xi_{1},\ldots,\Delta_{H}^{-1}\xi_{2n}}=\langle
D^{\prime}\rangle^{\xi_{2n},\xi_{1},\ldots,\xi_{2n-1}},$ (A.2)

where the diagram $D^{\prime}$ is obtained from $D$ by clockwise rotation by
the angle $\frac{\pi}{n}$ and reversing the orientation of the curve ending in
$2n$. For example:

$1$$2$$3$$4_{t}$$t\leadsto
t-i$$1$$2$$3$$4_{t}$$1$$2$$3$$4$$5$$6_{t}$$t\leadsto
t-i$$1$$2$$3$$4$$5$$6_{t}$

With such a diagram calculus, the calculations in Lemma 3.10 are easily
verified.

We also mention a diagrammatic representation of crossing symmetry. In the
usual graphical notation for tensor product operators [EK98, Chapt. 2.6],
crossing symmetry takes the form (with a subscript indicating the action of
modular unitaries $\Delta_{H}^{-it}$ and a line reversing its up/down
direction indicating the action of a modular conjugation $J_{H}$)

$\langle 2_{t}\otimes 1,T(3\otimes
4_{t})\rangle=$$T$$2_{t}$$1$$3$$4_{t}$$t\leadsto
t+\frac{i}{2}$$T$$2_{t}$$1$$3$$4_{t}$$=\langle 1\otimes
J_{H}4_{t},T(J_{H}2_{t}\otimes 3)\rangle$

It is curious to note that this diagram looks similar to the string Fourier
transforms of Jaffe and Liu [JL17], but we shall not investigate this
connection any further in the present work.

Finally, the diagrammatic view on crossing symmetry gives also a different
perspective on the derivation of the Yang-Baxter equation in Thm. 3.12 b). The
analytic continuation can be seen as a two-fold application of the crossing
symmetry above, which transforms $T_{2}T_{1}T_{2}$ into $T_{1}T_{2}T_{1}$:

$T$$T$$T$$3_{t}$$4$$2$$1$$5$$6_{t}$$t\leadsto
t+\frac{i}{2}$$T$$T$$T$$3_{t}$$4$$2$$1$$5$$6_{t}$$=$$T$$T$$T$$J6_{t}$$5$$1$$2$$4$$J3_{t}$

Here $J=J_{H}$, and in combination with the cyclic permutation underlying the
KMS condition, one obtains the Yang-Baxter equation.

## Acknowledgements

We thank Maximilian Duell for discussions at the beginning of this work,
Roberto Longo for highlighting the possibility of split inclusions in [DLR01],
and Roland Speicher for an exchange about the von Neumann algebras
$\mathcal{L}_{T}(H)$. Support by the German Research Foundation DFG through
the Heisenberg project “Quantum Fields and Operator Algebras” (LE 2222/3-1) is
gratefully acknowledged.

## Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

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# Probing the dark Universe with gravitational waves

Antonio Enea Romano  ICRANet, Piazza della Repubblica 10, I–65122 Pescara,
Italy  Instituto de Física, Universidad de Antioquia, A.A.1226, Medellín,
Colombia<EMAIL_ADDRESS>
###### Abstract

Gravitational waves (GW) are expected to interact with dark energy and dark
matter, affecting their propagation on cosmological scales. In order to model
this interaction, we derive a gauge invariant effective equation and action
valid for all GWs polarizations, based on encoding the effects of the
interaction of GWs at different orders in perturbations, in a polarization,
frequency and time dependent effective speed. The invariance of perturbations
under time dependent conformal transformations and the gauge invariance of the
GWs allow to obtain the unitary gauge effective action in any conformally
related frame, making transparent the relation between Einstein and Jordan
frame. The propagation time and luminosity distance of different GWs
polarizations allow to probe at different frequencies and redshift the dark
Universe, which act as an effective medium, whose physical properties can be
modeled by the GWs effective speed.

###### pacs:

Valid PACS appear here

##  I Introduction

The direct observation of GWs by the Laser Interferometer Gravitational Wave
Observatory (LIGO) and Virgo has started the era of GW astronomy
LIGOScientific:2016aoc . These observations are in good agreement with general
relativity (GR) predictions, but even in GR tensors modes are expected to
acquire a frequency and polarization dependent effective speed Romano:2022jeh
due to the interaction with other fields, an effect which can be tested with
multimessenger observations in different bands, and with measurements of the
luminosity distance of different polarizations modes.

In order to study the effects of the interaction of GWs, we generalize to all
possible GWs polarizations, the effective speed approach Romano:2022jeh ;
Romano:2023ozy ; Romano:2018frb ; Romano:2023uwf , derived for tensor modes,
including gauge invariant scalar and vector modes. We obtain a general
effective propagation equation and action which can be applied for model
independent analysis of the effects of interaction between different
polarizations of GWs, or of GWs with other fields Romano:2023lxf , such as
dark matter or dark energy. In this effective approach, the other fields, and
the other GWs polarizations, act as an effective medium for each GW
polarization, which can consequently propagate with a different frequency and
time dependent effective speed. This effective momentum and polarization
dependent speed is encoding interaction effects which are conceptually
analogous to those encoded in the modified dispersion relation for
electromagnetic waves (EMW) propagating in a plasma. Given the generality of
this effective approach, it is particularly suitable for model independent
phenomenological analysis of observational data, and predicts that the speed
of gravitational waves $c_{ij}$ and the gravitational luminosity distance can
depend on the frequency and polarization of GWs.

##  II Effective speed of arbitrary GW polarization

GWs $h_{ij}$ propagating in the $z$-direction Ezquiaga:2018btd , can be
decomposed in different polarizations as

$h_{ij}=\begin{pmatrix}h_{S}+A_{+}&h_{\times}&h_{V1}\\\
h_{\times}&h_{S}-h_{+}&A_{V2}\\\ h_{V1}&h_{V2}&h_{L}\end{pmatrix}\,,$ (1)

where $h_{+}$ and $h_{\times}$ are the tensor modes, $h_{V1,2}$ the vector
polarizations, and $h_{S}$ and $h_{L}$ the transverse and longitudinal scalar
modes. Note that the gauge has not been fully fixed in the above equation, and
this can lead to GWs modes which depend on the observer frame Bonvin:2022mkw .
We will resolve this ambiguity in the next section, in which we will obtain
the effective action for gauge invariant GWs.

In order to show the general applicability of the effective speed approach
Romano:2022jeh , let’s consider for example this ansatz for the modified GW
propagation equation Nishizawa:2017nef

$\displaystyle
h^{\prime\prime}_{ij}+(2+\nu)\mathcal{H}h^{\prime}_{ij}+(c_{g}^{2}k^{2}+m^{2}a^{2})h_{ij}$
$\displaystyle=$ $\displaystyle\Pi_{ij}(h_{ij},h_{pq},\phi_{m})\,,$ (2)
$\displaystyle L_{m}(\phi_{m},h_{ij})$ $\displaystyle=$ $\displaystyle 0\,,$
(3)

where $\nu$ accounts for a modification of the friction term, $c_{g}$ for an
modified propagation speed, $m$ is an effective mass, $\Pi_{ij}$ is the source
term associated to the self-interaction, interaction with other fields or
other GWs polarizations $h_{pq}$, $\phi_{m}$ denotes abstractly other fields
interacting with GWs, and $L_{m}$ is the differential operator corresponding
to the equation of motion (EOM) of each $\phi_{m}$. After rewriting eq.(2) as

$h^{\prime\prime}_{ij}+2\mathcal{H}h^{\prime}_{ij}+c^{2}\,k^{2}h_{ij}=\Pi_{ij}-\nu\mathcal{H}h^{\prime}_{ij}-\Big{[}(c_{g}^{2}-c^{2})k^{2}+m^{2}a^{2}\Big{]}h_{ij}=\Pi_{ij}^{eff}\,,$
(4)

we obtain the effective equation

$h_{ij}^{\prime\prime}+2\Big{(}\frac{a^{\prime}}{a}-\frac{c^{\prime}_{ij}}{c_{ij}}\Big{)}h^{\prime}_{ij}+k^{2}c_{ij}^{2}h_{ij}=0\,,$
(5)

where we have defined Romano:2022jeh  the effective polarization, momentum and
time dependent speed $c_{ij}$ as

$c_{ij}^{2}(\eta,k)=\Big{(}1-\frac{\hat{g}_{ij}}{\hat{h}^{\prime}_{ij}}\Big{)}^{-1}\quad,\quad\hat{g}_{ij}=\frac{1}{a^{2}}\int
a^{4}\hat{\Pi}^{eff}_{ij}\,d\eta\,,$ (6)

In the above equation a hat denotes quantities obtained by substituting the
solutions $\hat{h}_{ij},\hat{\phi}_{m}$ of the system of coupled differential
equations given in eqs.(2-3), i.e. $\hat{\Pi}^{eff}_{ij}$ is just a function
of space-time coordinates after the substitution, which accounts for the
integrated modified propagation effects. Eq.(6) shows that each polarization
mode $h_{ij}$ can have a different frequency and time dependent effective
speed, depending on the effective anisotropy tensor $\hat{\Pi}^{eff}_{ij}$,
evaluated along the propagation path.

For a set of initial conditions consistent with those used to obtain
$\hat{h}_{ij}$, eq.(5) gives by construction the same solution $\hat{h}_{ij}$.
Solutions of eq.(5) corresponding to different initial conditions are not
physically relevant, such as for example $h_{ij}=0$. Any solution of eq.(2),
or any other ansatz which can be manipulated to put it in the form given in
eq.(4), can always be obtained as a solution of the effective equation 5, with
the effective speed defined in eq.(6). This effective approach is convenient
to relate the modified propagation effects to observations, because the GW-EMW
luminosity distance ratio is related to the effective speed
ratioRomano:2022jeh , making the effective equation useful for model
independent observational data analysis.

##  III Einstein frame definition

In the previous section we have not specified explicitly the frame in which
the ansatz (2) is given, since our goal was to give an example of the general
applicability of the effective speed approach. In order to derive a general
effective action, without assuming any ad hoc ansatz, it is important to
clarify the relation between Jordan and Einstein frame. We defined the Jordan
frame Lagrangian of a modified gravity theory with Jordan frame matter-
coupling (JMC) as

$\mathcal{L}_{\rm JMC}=\sqrt{g_{\rm J}}\Big{[}\Omega^{2}R_{\rm J}+L^{\rm
MG}_{\rm J}+L^{\rm matter}_{\rm J}(g_{\rm J})\Big{]}\,\,,$ (7)

where $L^{\rm MG}$ and $L^{\rm matter}$ are respectively the modified gravity
and matter Lagrangians. After performing a conformal transformation $g_{\rm
E}=\Omega^{2}g_{\rm J}$, in the Einstein frame we have

$\mathcal{L}_{\rm JMC}=\sqrt{g_{\rm E}}\Big{[}R_{\rm E}+L^{\rm MG}_{\rm
E}+L^{\rm matter}_{\rm E}(\Omega^{-2}g_{\rm E})\Big{]}\,.$ (8)

Note that while the tensor modes speed $c_{T}$ is invariant under conformal
transformations, it is not invariant under disformal transformations, allowing
to define a combination of disformal, conformal and coordinate transformations
taking to a frame Creminelli:2014wna  in which $c_{T}=1$. Note nevertheless
that the ratio $c_{T}/c$ between the speed of gravitational and
electromagnetic waves is disformal invariant, implying that, if $c_{T}/c$ is
time dependent, in the $c_{T}=1$ frame the speed of light is time dependent.
For this reason, while the $c_{T}=1$ frame is useful to study the resilience
of inflationary predictionsCreminelli:2014wna , we define the Einstein frame
as that in which the coefficient of the Ricci scalar and the speed of light
are constant, and not the $c_{T}=1$ frame, in agreement with the definition
adopted in Gubitosi:2012hu  for example.

##  IV Gauge invariant gravitational waves

For the purpose of understanding the role played by gauge transformation it is
convenient to interpret the GW polarization tensor defined in eq.(1) in terms
of cosmological perturbations theory CPTLifshitz ; Kodama:1985bj ;
Baumann:2009ds . Since tensor perturbations are gauge invariant at first
order, we will just consider scalar and vector perturbations. Using the
scalar-vector-tensor (SVT) decomposition CPTLifshitz , the scalar and vector
perturbations of the flat FRW line element can be written as

${\rm d}s^{2}=(1+2\Phi){\rm d}t^{2}-2a(t)\Big{(}B_{,i}+S_{i}\Big{)}{\rm
d}x^{i}{\rm
d}t-a^{2}(t)\Big{[}(1-2\Psi)\delta_{ij}+2(E_{,ij}+F_{(i,j)})\Big{]}{\rm
d}x^{i}{\rm d}x^{j}\,,$ (9)

where $S_{i}$ and $F_{i}$ are the vector perturbations, satisfying
$S_{i,i}=F_{i,i}=0$, and $\Phi,B,\Psi,E$ are the scalar perturbations.
Applying the SVT decomposition to an infinitesimal time and space translation,
we obtain

$\displaystyle t$ $\displaystyle\to$ $\displaystyle t+T\ ,$ (10)
$\displaystyle x^{i}$ $\displaystyle\to$ $\displaystyle
x^{i}+\beta^{i}\,+\delta^{ij}\beta_{,j}\,.$ (11)

where $\beta_{i}$ and $\beta$ are respectively the vector and scalar part of
the infinitesimal space translation, satisfying $\beta_{i,i}=0$, and $T$ is
the infinitesimal time translation.

Under the above infinitesimal coordinate transformations the metric tensor
transformations implies the gauge transformations for the perturbations
Baumann:2009ds

$\displaystyle S_{i}$ $\displaystyle\to$ $\displaystyle
S_{i}+a\dot{\beta}_{i}\,,$ (12) $\displaystyle F_{i}$ $\displaystyle\to$
$\displaystyle F_{i}-\beta_{i}\,,$ (13) $\displaystyle\Phi$ $\displaystyle\to$
$\displaystyle\Phi-\dot{T}\,,$ (14) $\displaystyle B$ $\displaystyle\to$
$\displaystyle B+a^{-1}T-a\dot{\beta}\,,$ (15) $\displaystyle E$
$\displaystyle\to$ $\displaystyle E-\beta\,,$ (16) $\displaystyle\Psi$
$\displaystyle\to$ $\displaystyle\Psi+HT\,.$ (17)

while scalar fields transform as

$\delta\phi\rightarrow\delta\phi-\dot{\phi}\,T\,.$ (18)

Note that the above gauge transformations have a purely geometrical origin,
i.e. they are independent of the specific theory, since they are simply a
direct consequence of the fact that the metric and the energy-momentum tensor
transform as tensors under a coordinate transformation. This allows to obtain
a gauge invariant definition of GWs polarizations valid in any gravity theory,
since the SVT decomposition of a tensor is also purely geometrical, and is not
based on assuming any symmetry of the action defining the theory. In other
words gauge transformations are just a manifestation of general covariance,
which any theory should satisfy, and corresponds to the Lagrangian not
containing any free tonsorial index, i.e. being a coordinate invariant. By
comparing eq.(1) with the spatial part of the perturbed metric we obtain:

$\displaystyle h_{S}=1-2\Psi\quad$ , $\displaystyle\quad
h_{L}=1-2\Psi+E_{,zz}\,,$ (19) $\displaystyle h_{V1}=-F_{x,z}\quad$ ,
$\displaystyle\quad h_{V2}=-F_{y,z}\,,$ (20)

from which we obtain the gauge transformations of the GWs polarizations

$\displaystyle h_{S}$ $\displaystyle\to$ $\displaystyle h_{S}-2HT\,,$ (21)
$\displaystyle h_{L}$ $\displaystyle\to$ $\displaystyle
h_{L}-2HT-\beta_{,zz}\,,$ (22) $\displaystyle h_{V1}$ $\displaystyle\to$
$\displaystyle h_{V1}+\beta_{x,z}\,,$ (23) $\displaystyle h_{V2}$
$\displaystyle\to$ $\displaystyle h_{V2}+\beta_{y,z}\,.$ (24)

We can now use the above gauge transformations to define gauge invariant GWs
polarizations. It is convenient to fix the Einstein gauge (EG), defined by the
condition

$\nabla^{\mu}\Big{(}h_{\mu\nu}-\frac{1}{2}hg^{0}_{\mu\nu}\Big{)}=0$ (25)

where $g^{0}_{\mu\nu}$ denotes the background metric, $h=h^{ab}g^{0}_{ab}$ is
the trace of $h_{ab}$, and the covariant derivative is w.r.t. the background
metric. The above gauge fixing condition gives a set of four differential
equations which can be solved to obtain the infinitesimal translations
necessary to switch to the EG. Denoting the solutions of the gauge fixing
equations as $\\{T^{E},\beta^{E},\beta_{i}^{E}\\}$, we can define the gauge
invariant GWs polarizations $\bar{h}_{ij}$

$\displaystyle\bar{h}_{S}$ $\displaystyle=$ $\displaystyle h_{S}-2HT^{E}\,,$
(26) $\displaystyle\bar{h}_{L}$ $\displaystyle=$ $\displaystyle
h_{L}-2HT^{E}-\beta^{E}_{,zz}\,,$ (27) $\displaystyle\bar{h}_{V1}$
$\displaystyle=$ $\displaystyle h_{V1}+\beta^{E}_{x,z}\,,$ (28)
$\displaystyle\bar{h}_{V2}$ $\displaystyle=$ $\displaystyle
h_{V2}+\beta^{E}_{y,z}\,,$ (29)

which are gauge invariant by construction, and we will call EGWs. We are
denoting with a bar the gauge invariant quantities EGWs defined above, while
tensor modes are gauge invariant a first order in perturbations, i.e.
$h_{\times}=\bar{h}_{\times},h_{+}=\bar{h}_{+}$. Some residual gauge freedom
is present even after fixing the EG Flanagan:2005yc , but we will not use it,
since our main purpose is to define gauge invariant variables satisfying a
wave equation, and imposing the EG is enough to achieve this, as we will show
in the next section.

##  V Gauge invariant gravitational waves equation

The field equations of a generic JMC modified gravity theory defined by the
action (7) can be written in the Einstein frame as

$G_{\rm E,\mu\nu}=T^{tot}_{\mu\nu}\,,$ (30)

where $T^{tot}_{\mu\nu}$ is the sum of terms associated to the matter fields,
the modified gravity fields, and their interaction with matter due to the non
minimal coupling of matter with the Einstein frame metric. The linearized
perturbed Einstein equations w.r.t. to a curved background Ezquiaga:2018btd
in the Einstein gauge read

$\Box\psi_{\mu\nu}+2R^{{}_{\text{B}}}_{\mu\alpha\nu\beta}\psi^{\alpha\beta}=2\delta
T_{\mu\nu}+2R^{{}_{\text{B}}~{}\alpha}_{~{}(\mu}\psi_{\nu)\alpha}-R^{{}_{\text{B}}}\psi_{\mu\nu}+g^{0}_{\mu\nu}R^{\alpha\beta}_{{}_{\text{B}}}\psi_{\alpha\beta}\,,$
(31)

where $\psi_{\mu\nu}=\bar{h}_{\mu\nu}-\frac{1}{2}\bar{h}g^{0}_{\mu\nu}$. From
the above equation we obtain

$\Box\bar{h}_{\mu\nu}=\delta T^{eff}_{\mu\nu}\,,$ (32)

where the d’Alambert operator is defined w.r.t. the background metric and
$T^{eff}_{\mu\nu}$ is an effective energy-stress (ES) tensor given by the sum
of the matter ES tensor, and other terms involving GWs, the Ricci tensor, and
possible additional fields related to the gravity modification. Note that the
TT gauge cannot be imposed in a generic curved space Flanagan:2005yc , and the
scalar and vector modes cannot be gauged away in a generic modified gravity
theory, since this is only possible in vacuo, but the extra fields associated
to the gravity modification act as an effective source in the r.h.s. of
eq.(32), which in general is not zero even in absence of matter fields. In GR,
assuming a flat background, and in vacuo, the effective source term is zero,
because the Ricci and Riemann tensors are zero. This allows use the residual
gauge freedom to set $\bar{h}=0$, and fix the TT gauge, in which only the two
tensor modes survive. Note that eq.(32) is quite general, since it is valid
for any theory which admits an Einstein frame formulation, for a general
background metric, and as mentioned earlier, the definitions of the gauge
invariant variables is even more general, since it does not assume any form of
the action, and the EG can be fixed even for a background different form the
FRW solution.

Considering a FRW background, following a method similar to the one used in
the previous section, an effective equation for the gauge invariant GWs can be
derived from eq.(32)

$\bar{h}_{ij}^{\prime\prime}+2\Big{(}\frac{a^{\prime}}{a}-\frac{c^{\prime}_{ij}}{c_{ij}}\Big{)}\bar{h}^{\prime}_{ij}+k^{2}c_{ij}^{2}\bar{h}_{ij}=0\,,$
(33)

where

$c_{ij}^{2}(\eta,k)=\Big{(}1-\frac{\hat{g}_{ij}}{\hat{\bar{h}}^{\prime}_{ij}}\Big{)}^{-1}\quad,\quad\hat{g}_{ij}=\frac{1}{a^{2}}\int
a^{4}\delta T^{eff}_{ij}\,d\eta\,,$ (34)

Note that, contrary to the previous section derivation, which was based on
assuming a non gauge invariant ansatz for the GWs propagation equation, the
effective equation above is gauge invariant by construction and is quite
general, since it can be obtained for any theory which admits an Einstein
frame formulation. Formally it can also be applied to the solutions
$\hat{g}_{\mu\nu}$ of theories whose field equations are not of the form given
in eq.(30), by appropriately defining and effective ES tensor

$\displaystyle F[\hat{g}_{\mu\nu}]$ $\displaystyle=$ $\displaystyle
T_{\mu\nu}[\hat{g}_{\mu\nu},\hat{\phi}_{i}]\quad\,,\quad
M_{i}[\hat{\phi}_{i}]=0\,,$ (35) $\displaystyle G_{\mu\nu}[\hat{g}_{\mu\nu}]$
$\displaystyle=$ $\displaystyle\hat{T}^{eff}_{\mu\nu}\,,$ (36)
$\displaystyle\hat{T}^{eff}_{\mu\nu}(x^{\rho})$ $\displaystyle=$
$\displaystyle
T_{\mu\nu}[\hat{g}_{\mu\nu},\hat{\phi}_{i}]-F[\hat{g}_{\mu\nu}]+G_{\mu\nu}[\hat{g}_{\mu\nu}]\,,$
(37)

where $F$ and $M_{i}$ are the differential operator corresponding to the
gravity and matter field equations, $\\{\hat{g}_{\mu\nu},\hat{\phi}_{i}\\}$
are solutions of eq.(35), and the components of $\hat{T}^{eff}_{\mu\nu}$ are
functions of space and time, obtained by substituting into
${T}^{eff}_{\mu\nu}$ the solutions of the coupled matter and gravity field
equations (35).

Note that the above equations should be interpreted as a statement about the
fact that any solution $\hat{g}_{\mu\nu}$ of eq.(35) can also be obtained as a
solution of eq.(36), for an appropriate choice of $\hat{T}^{eff}_{\mu\nu}$ and
boundary conditions, not as a statement about the full equivalence between
general relativity and the generic theory corresponding to eq.(35), which may
indeed involve different differential operators, and be fundamentally
different. Since what we observe are the solutions of the field equations, not
the equations themselves, eq.(36) is enough to obtain an effective description
of the solutions of eq.(35), according to the method given above.

##  VI Generalization to higher order in perturbations

Note that the gauge invariant quantities EGWs and eq.(32) are defined at
linear order in perturbations, and at higher order new gauge invariant
variables ${\bar{h}}^{(i)}_{\mu\nu}$ can be defined Mollerach:2003nq ;
DeLuca:2019ufz ; Chang:2020iji . The expansion of the Einstein equations (30)
will give new equations which can always be put in the canonical form

$\Box\bar{h}^{(i)}_{\mu\nu}=\delta T^{(i)eff}_{\mu\nu}\,,$ (38)

by appropriately defining the effective perturbed ES tensor $\delta
T^{(i)eff}_{\mu\nu}$, even if the d’Alambert operator does not appear
explicitly, by adding it on both sides of the expanded equations. We have
fixed the gauge by imposing the condition

$\nabla^{\mu}\Big{(}\bar{h}^{(i)}_{\mu\nu}-\frac{1}{2}\bar{h}^{(i)}g^{0}_{\mu\nu}\Big{)}=0\,,$
(39)

where $g^{0}_{\mu\nu}$ denotes the background metric,
$\bar{h}^{(i)}=\bar{h}^{(i)}_{\mu\nu}g^{0,\mu\nu}$ is the trace of
$\bar{h}^{(i)}_{\mu\nu}$, and the covariant derivative is w.r.t. the
background metric. This effective equation allows to interpret GWs at any
order in perturbations as the solutions of a wave equation with an
appropriately defined source term. Since the d’Alambert operator is defined
w.r.t. to the same background metric in all these equations, the equations can
be summed to give

$\Box\bar{h}^{(N)}_{\mu\nu}=\delta T^{(N)eff}_{\mu\nu}\,.$ (40)

where we have defined the summed GWs and ES tensor perturbations

$\bar{h}^{(N)}_{\mu\nu}=\sum^{N}_{i=1}\bar{h}^{(i)}_{\mu\nu}\quad,\quad
T^{(N)eff}_{\mu\nu}=\sum^{N}_{i=1}\delta T^{(i)eff}_{\mu\nu}\,.$ (41)

Note that $\bar{h}^{(N)}_{\mu\nu}$ are the physically observable GWs, given by
the sum of the contributions from different orders in perturbations. From
eq.(40), similarly to what shown in the previous section for linear
perturbations, it is possible to define an effective speed, equation, and
action for the summed GWs $\bar{h}^{(N)}_{\mu\nu}$. Note that this effective
speed is not simply the sum of the effective speeds corresponding to
perturbations equations at different orders, due to the coupling between
perturbations at different orders. After solving the system of coupled
differential equations for the perturbations equations at all relevant orders,
the solutions $\hat{\bar{h}}^{(i)}_{\mu\nu}$ can be substituted to obtain
$\hat{T}^{(N)eff}_{\mu\nu}$, from which the effective speed and equation can
be obtained

$\bar{h}^{(N)^{\prime\prime}}_{ij}+2\Big{(}\frac{a^{\prime}}{a}-\frac{c^{\prime}_{ij}}{c_{ij}}\Big{)}\bar{h}^{(N)^{\prime}}_{ij}+k^{2}c_{ij}^{2}\bar{h}^{(N)}_{ij}=0\,,$
(42)

$c_{ij}^{2}(\eta,k)=\Bigg{[}1-\frac{\hat{g}_{ij}}{\hat{\bar{h}}_{ij}^{(N)^{\prime}}}\Bigg{]}^{-1}\quad,\quad\hat{g}_{ij}=\frac{1}{a^{2}}\int
a^{4}\delta\hat{T}^{(N)eff}_{ij}\,d\eta\,.$ (43)

The effective speed is encoding in a single quantity all the interaction
effects between different GWs polarizations, and between GWs and other fields,
up to order $N$ in perturbations.

##  VII Effective Lagrangian and metric

The Lagrangian corresponding to eq.(42) is

$\mathcal{L}^{eff}_{h}=\frac{a^{2}}{c_{ij}^{2}}\Bigg{[}{\Big{(}\bar{h}^{(N)^{\prime}}_{ij}\Big{)}}^{2}+k^{2}c_{ij}^{2}{\Big{(}\bar{h}^{(N)}_{ij}\Big{)}}^{2}\Bigg{]},$
(44)

generalizing to scalar and vector modes the tensor perturbations effective
action Romano:2022jeh . The effective method can also be applied in physical
space, and it can be shown that Romano:2022jeh  the effective Lagrangian can
be obtained from the GR Lagrangian density

$\mathcal{L}_{h}^{GR}=a^{2}\Big{[}h^{\prime 2}_{ij}-c^{2}(\nabla
h_{ij})^{2}\Big{]}=\sqrt{-g}(\partial_{\mu}h_{ij}\partial^{\mu}h_{ij})\,,$
(45)

via the transformation

$a\rightarrow\alpha_{ij}=\frac{a}{\hat{c}_{ij}}\quad,\quad
c\rightarrow\hat{c}_{ij}\quad,\quad h_{ij}\rightarrow\bar{h}^{(N)}_{ij}\,,$
(46)

where we have denoted with $c$ the speed of light, to avoid ambiguity, and the
space effective sound speed $\hat{c}_{ij}(\eta,x^{i})$ Romano:2022jeh , is
defined in terms of the physical space effective ES tensor, not of its Fourier
transform. Note that ${c}_{ij}(\eta,k)$ is not the Fourier transform of
$\hat{c}_{ij}(\eta,x^{i})$ Romano:2022jeh . The physical space effective
action is

$\mathcal{L}^{eff}_{h}=\sqrt{-g^{\rm{eff}}}(\partial_{\mu}h_{ij}\partial^{\mu}h_{ij})\,,$
(47)

where the effective metric is

$ds^{2}_{eff}=g^{\rm{eff}}_{\mu\nu}dx^{\mu}dx^{\nu}=a^{2}\Big{[}\hat{c}_{ij}d\eta^{2}-\frac{\delta_{mm}}{{\hat{c}_{ij}}}dx^{m}dx^{m}\Big{]}\,,$
(48)

from which the EOM are given in terms of the covariant d’Alembert operator

$\square
h_{ij}=\frac{1}{\sqrt{-g^{\rm{eff}}}}\partial_{\mu}\Big{(}\sqrt{-g^{\rm{eff}}}\partial^{\mu}h_{ij}\Big{)}=0\,.$
(49)

In this effective geometrical description, the effects of the interactions of
GWs are encoded in the effective metric, and in the eikonal approximation, the
solutions of the EOM are geodesics in the effective curved space corresponding
to the effective metric.

##  VIII Jordan frame effective action

Cosmological perturbations w.r.t. to a flat FRW background are invariant under
time dependent conformal transformations, since they correspond to a scale
factor redefinition $a=\Omega\,\tilde{a}$, which has no effects on the
perturbations

$ds^{2}=a^{2}\Big{[}(\eta_{\mu\nu}+\delta
g_{\mu\nu})dx^{\mu}dx^{\nu}\Big{]}=\Omega^{2}(\eta)\tilde{a}^{2}\Big{[}(\eta_{\mu\nu}+\delta
g_{\mu\nu})dx^{\mu}dx^{\nu}\Big{]}\,.$ (50)

The gauge invariant GWs do not depend on the coordinate choice by definition,
and can in particular be evaluated in the unitary gauge, defined by
$\delta\phi=0$, in which the conformal factor $\Omega$ becomes a function of
time only. Using the conformal and gauge invariance of the GWs $\bar{h}_{ij}$,
we can obtain the Jordan frame effective action in the unitary gauge

$\mathcal{L}^{eff}_{\bar{h}}=\frac{\Omega^{2}(\eta)\tilde{a}^{2}}{\hat{c}_{ij}^{2}}\Big{[}\bar{h}^{\prime
2}_{ij}-\hat{c}_{ij}^{2}(\nabla\bar{h}_{ij})^{2}\Big{]}=\hat{M}^{2}_{ij}\tilde{a}^{2}\Big{[}\bar{h}^{\prime
2}_{ij}-\hat{c}_{ij}^{2}(\nabla\bar{h}_{ij})^{2}\Big{]}\,.$ (51)

where $\hat{M}_{ij}=\Omega/\hat{c}_{ij}$ plays the role of space and
polarization dependent effective Planck mass.

##  IX Polarization and frequency dependency of the luminosity distance

After defining the effective scale factor

$\alpha_{ij}=\frac{a}{c_{ij}}\,,$ (52)

equation (42) can be re-written as

$\bar{h}^{(N)^{\prime\prime}}_{ij}+2\frac{\alpha_{ij}^{\prime}}{\alpha_{ij}}\bar{h}^{(N)^{\prime}}_{ij}+k^{2}c_{ij}^{2}\bar{h}^{(N)}_{ij}=0\,,$
(53)

from which we obtain

$\chi_{ij}^{\prime\prime}+\Big{(}c_{ij}k^{2}-\frac{\alpha_{ij}^{\prime\prime}}{\alpha_{ij}}\Big{)}\chi_{ij}=0\,,$
(54)

where we have defined $\bar{h}^{(N)}_{ij}=\chi_{ij}/\alpha_{ij}$. In the sub-
horizon limit $\alpha_{ij}^{\prime\prime}/\alpha_{ij}$ is negligible, and the
amplitude of $\bar{h}^{(N)}_{ij}$ is proportional to $1/\alpha_{ij}$, giving
Romano:2022jeh

$\frac{d_{ij}^{GW}}{d_{L}^{EM}}(z)=\frac{a(z)}{\alpha_{ij}(z)}\frac{\alpha_{ij}(0)}{a(0)}=\frac{c_{ij}(z,k)}{c_{ij}(0,k)}\,,$
(55)

where we have used $d_{ij}^{GW}=r\,\alpha_{ij}(0)/\alpha_{ij}(z)$,
$d_{L}^{EM}=r\,a(0)/a(z)$, assuming $(1+z)=a(0)/a(z)$, i.e. that matter is
minimally coupled to the Einstein frame metric.

If matter is minimally coupled to the Jordan frame metric, eq.(53) is still
valid, due to the conformal invariance of GWs, but the relation between the
scale factor and $\alpha$ is modified

$\alpha_{ij}=\frac{a}{c_{ij}}=\frac{\Omega\tilde{a}}{c_{ij}}\,,$ (56)

implying that

$\frac{d_{ij}^{GW}}{d_{L}^{EM}}(z)=\frac{\tilde{a}(z)}{\alpha_{ij}(z)}\frac{\alpha_{ij}(0)}{\tilde{a}(0)}=\frac{c_{ij}(z,k)}{c_{ij}(0,k)}\frac{\Omega(0)}{\Omega(z)}=\frac{M_{ij}(0,k)}{M_{ij}(z,k)}\,,$
(57)

where we have used $d_{ij}^{GW}=r\,\alpha_{ij}(0)/\alpha_{ij}(z)$,
$d_{L}^{EM}=r\,\tilde{a}(0)/\tilde{a}(z)$, assuming
$(1+z)=\tilde{a}(0)/\tilde{a}(z)$, i.e. that matter is minimally coupled to
the Jordan frame metric. The quantity $M_{ij}=\Omega/c_{ij}$ plays the role of
polarization and momentum dependent effective Planck mass.

Note that independently of the type of matter-gravity coupling, the GWs
luminosity distance is predicted to be frequency and polarization dependent.

##  X Observational implications

The polarization and frequency dependency of the effective speed implies that
different polarizations of GWs emitted by the same source at different
frequencies can spend different times to reach the observer. This effect could
be detected by comparing the time delay between the detection of different GWs
polarizations with different observatories, operating at different
frequencies, and can be observed even in absence of an electromagnetic
counterpart.

Another observable effect is the polarization and frequency dependency of the
GWs luminosity distance. For theories with matter minimally coupled to the
Jordan frame metric the GW luminosity distance is related to the effective
Planck mass ratio

$d_{ij}^{GW}(z)=\frac{M_{ij}(0,k)}{M_{ij}(z,k)}d_{L}^{EM}(z)\,,$ (58)

while for theories minimally coupled to the Einstein frame is related to the
effective speed ratio

$d_{ij}^{GW}(z)=\frac{c_{ij}(0,k)}{c_{ij}(z,k)}d_{L}^{EM}(z)\,.$ (59)

Time delay observations allow to constrain at different frequencies the ratio
between the speed of different GWs polarizations

$r^{c}_{ijpq}(k,z)=\frac{c_{ij}(k,z)}{c_{pq}(k,z)}\,,$ (60)

while the GWs waveform, which are inversely proportional to the GW luminosity
distance, allow to constrain the corresponding distance ratio

$r^{d}_{ijpq}(k,z)=\frac{d_{ij}^{GW}(z)}{d_{pq}^{GW}(z)}.$ (61)

For multimessenger events additional constraints can be set on $c_{ij}(k,z)/c$
and $d_{ij}^{GW}(z)/d^{EM}_{L}(z)$.

For GWs propagating in vacuum according to GR, only tensor modes are expected,
and there should not be any redshift, frequency or polarization dependency,
i.e. $c_{\times}(k,z)=c_{+}(k,z)=c$ and
$d_{\times}^{GW}(z)=d_{+}^{GW}(z)=d^{EM}_{L}(z)$ at any frequency. If the
effective ES tensor defined in eq.(41) is not negligible along the GW
propagation, either because of GWs interaction with matter fields, or due to
the effects of gravity modification, other polarizations modes could be
detected and the corresponding speed and luminosity distance could be
constrained observationally using time delay and GWs waveform observations.
These observations can be used to probe the dark Universe by its interaction
with GWs, modeled by the effective GWs speed.

##  XI Conclusions

We have derived a gauge invariant effective equation and action for GWs,
encoding the effects of interaction at different orders in perturbations
theory in a polarization, frequency and time dependent effective speed. The
invariance of perturbations under time dependent conformal transformations and
the gauge invariance of the GWs allow to obtain the unitary gauge effective
action in any conformally related frame. The propagation time and luminosity
distance of different GWs polarizations, emitted at different frequencies and
redshifts by dark or bright sirens, allow to probe the interaction with other
fields, and in particular with the dark Universe.

In the future it will important compute the effective GWs speed for different
dark matter and dark energy theories, to constrain these theories with
luminosity distance and time delay observations.

###### Acknowledgements.

I thank Antonaldo Diaferio, Francesco Pace e Piero Rettegno and for
interesting discussions regarding GWs polarizations. I thank the YITP, and
Osaka University Theoretical Astrophysics group for the kind hospitality
during the preparation of part of this paper.

## References

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# Quotients of Jacobians

Arnaud Beauville  Université Côte d’Azur
CNRS – Laboratoire J.-A. Dieudonné
Parc Valrose
F-06108 Nice cedex 2, France<EMAIL_ADDRESS>
###### Abstract.

Let $C$ be a curve of genus $g$, and let $G$ be a finite group of
automorphisms of $C$. The group $G$ acts on the Jacobian $J$ of $C$; we prove
that for $g\geq 21$ the quotient $J/G$ has canonical singularities and Kodaira
dimension $0$. On the other hand we give examples with $g\leq 4$ for which
$J/G$ is uniruled.

I am indebted to A. Höring for pointing out the reference [K-L].

##  1\. Introduction

Let $C$ be a (smooth, projective) curve of genus $g$, and let $G$ be a finite
group of automorphisms of $C$. The group $G$ acts on the Jacobian $J$ of $C$.
There are two rather different possibilities for the quotient $J/G$ [K-L]:
either it has canonical singularities and Kodaira dimension $0$, or it is
uniruled. The aim of this note is to show that the latter case is rather
exceptional – in fact, it does not occur for $g\geq 21$. This bound is rough,
and can certainly be lowered. On the other hand we give examples of uniruled
$J/G$ for $g\leq 4$. When $g=3$ this is an essential ingredient in our proof
that the cycle $[C]-[(-1)^{*}C]$ on $J$ is torsion modulo algebraic
equivalence [B-S]; in fact this note grew out of the observation that already
in genus 3, the case where $J/G$ is uniruled is quite rare.

##  2\. $J/G$ canonical

Let $C$ be a curve of genus $g$ and $J$ its Jacobian. Let $G$ be a subgroup of
$\operatorname{Aut}(C)$, hence also of $\operatorname{Aut}(J)$. By [K-L,
Theorem 2], there are two possibilities:

$\bullet$ Either $J/G$ has canonical singularities, and Kodaira dimension $0$;

$\bullet$ or $J/G$ is uniruled.

######  Proposition 1.

Assume $g\geq 21$. The quotient variety $J/G$ has canonical singularities, and
Kodaira dimension $0$.

Proof : We first observe that the fixed locus
$\operatorname{Fix}(\sigma)\subset J$ of any element $\sigma\neq 1$ of $G$ has
codimension $\geq 2$. Indeed the dimension of $\operatorname{Fix}(\sigma)$ is
the multiplicity of the eigenvalue $1$ for the action of $\sigma$ on
$H^{0}(C,K_{C})$, that is, the genus of $C/\langle{\sigma}\rangle$, which is
$\leq g-2$ as soon as $g\geq 4$ by the Hurwitz formula.

Let $\sigma$ be an element of order $r$ in $G$, and let $p\in J$ be a fixed
point of $\sigma$. The action of $\sigma$ on the tangent space $T_{p}(J)$ is
isomorphic to the action on $T_{0}(J)=H^{0,1}(C)$. Let $\zeta=e^{\frac{2\pi
i}{r}}$; we write the eigenvalues of $\sigma$ acting on $H^{0,1}(C)$ in the
form $\zeta^{a_{1}},\ldots,\zeta^{a_{g}}$, with $0\leq a_{i}<r$. By Reid’s
criterion [R, Theorem 3.1], $J/G$ has canonical singularities if and only if
$\sum a_{i}\geq r$ for all $\sigma$ in $G$.

The eigenvalues of $\sigma$ acting on $H^{1}(C,\mathbb{C})$ are
$\zeta^{a_{1}},\ldots,\zeta^{a_{g}};{\zeta}^{-a_{1}},\ldots,{\zeta}^{-a_{g}}$.
Thus $\operatorname{Tr}\sigma^{*}_{|H^{1}(C,\mathbb{C})}=\allowbreak
2\sum\cos\dfrac{2\pi a_{i}}{r}\,\cdot$ By the Lefschetz formula, this trace is
equal to $2-f$, where $f$ is the number of fixed points of $\sigma$; in
particular, it is $\leq 2$. Using $\cos x\geq 1-\dfrac{x^{2}}{2}$, we find

$1\geq\sum\cos\dfrac{2\pi a_{i}}{r}\geq g-\dfrac{2\pi^{2}}{r^{2}}\sum
a_{i}^{2}\ ,$

hence $\displaystyle\ (\sum a_{i})^{2}\geq\sum
a_{i}^{2}\geq\dfrac{g-1}{2\pi^{2}}\,r^{2}$. Thus if $g\geq
1+2\pi^{2}=20.739\ldots$, we get $\sum a_{i}>r$.∎

_Remark_ $.-$ The proposition does not extend to the case of an arbitrary
abelian variety: indeed we give below examples where $J/G$ is uniruled; then
if $A$ is any abelian variety, the quotient of $J\times A$ by $G$ acting
trivially on $A$ is again uniruled.

##  3\. $J/G$ uniruled

We will now give examples of (low genus) curves $C$ such that $J/G$ is
uniruled. The genus 2 case is quite particular (and probably well known). We
put $\rho:=e^{\frac{2\pi i}{3}}$.

######  Proposition 2.

If $g(C)=2$, $J/G$ has Kodaira dimension $0$ if and only if
$G=\langle{\sigma}\rangle$ and $\sigma$ is the hyperelliptic involution, or an
automorphism of order $3$ with eigenvalues $(\rho,\rho^{2})$ on
$H^{0}(C,K_{C})$, or an automorphism of order $6$ with eigenvalues
$(-\rho,-\rho^{2})$ on $H^{0}(C,K_{C})$.

Proof : Let $\sigma$ be an element of order $r$ in $G$, and let $\zeta$ be a
primitive $r$-th root of unity. The eigenvalues of $\sigma$ acting on
$H^{0}(C,K_{C})$ are $\zeta^{a},\zeta^{b}$, with $0\leq a,b<r$. Suppose that
$J/G$ has canonical singularities. By Reid’s criterion we must have $a+b\geq
r$; replacing $\zeta$ by $\zeta^{-1}$ gives $2r-a-b\geq r$, hence $a+b=r$.
Since $\sigma$ has order $r$, $\sigma^{k}$ acts non-trivially on
$H^{0}(C,K_{C})$ for $0<k<r$; therefore $a$ and $r$ are coprime. Replacing
$\zeta$ by $\zeta^{a}$ we may assume that the eigenvalues of $\sigma$ are
$\zeta$ and $\zeta^{-1}$.

Put $\zeta=e^{i\alpha}$; the trace of $\sigma$ acting on $H^{1}(C,\mathbb{Z})$
is $2(\zeta+\zeta^{-1})=4\cos\alpha$, and it is an integer. This implies
$\cos\alpha=0$ or $\pm\dfrac{1}{2}$, hence
$\alpha\in\mathbb{Z}\cdot\dfrac{2\pi}{6}$ and $r\in\\{2,3,6\\}$. If $r=2$ we
must have $\zeta=\zeta^{-1}=-1$, hence $\sigma$ is the hyperelliptic
involution; if $r=3$ or $6$ we have
$\\{\zeta,\zeta^{-1}\\}=\\{\rho,\rho^{2}\\}$ or $\\{-\rho,-\rho^{2}\\}$.

Thus $G$ contains elements of order $3$ and $6$, and at most one element of
order $2$. Since the order of $\operatorname{Aut}(C)$ is not divisible by $9$
[B-L, 11.7], $G$ is isomorphic to either $\mathbb{Z}/3$, or a central
extension of $\mathbb{Z}/3$ by $\mathbb{Z}/2$ – that is, $\mathbb{Z}/6$. This
proves the Proposition. ∎

The last two cases are realized by the curves $C$ of the form
$y^{2}=x^{6}+kx^{3}+1$, with the automorphisms $\sigma_{1}:(x,y)\mapsto(\rho
x,y)$ and $\sigma_{2}:(x,y)\mapsto(\rho x,-y)$. The surfaces
$S_{i}:=J/\langle{\sigma_{i}}\rangle$ are (singular) K3 surfaces, each
depending on one parameter. The surface $S_{1}$ is studied in detail in [B-V]:
it has 9 singular points of type $A_{2}$. The surface $S_{2}$ is the quotient
of $S_{1}$ by the involution induced by $(-1)_{J}$; it has one singular point
of type $A_{5}$ (the image of $0\in J$), 4 points of type $A_{2}$
corresponding to pairs of singular points of $S_{1}$, and $5$ points of type
$A_{1}$ corresponding to triples of points of order 2 of $J$.

It is more subtle to give examples with $g\geq 3$. By Reid’s criterion, we
must exhibit an automorphism $\sigma$ of $C$, of order $r$, such that the
eigenvalues of $\sigma$ acting on $H^{0}(C,K_{C})$ are of the form
$\zeta^{a_{1}},\ldots,\zeta^{a_{g}}$, with $\zeta$ a primitive $r$-th root of
unity, $a_{i}\geq 0$ and $\sum a_{i}<r$. In the following tables we give the
curve $C$ (in affine coordinates), the order $r$ of $\sigma$ and $\zeta$, the
expression of $\sigma$, a basis of eigenvectors for $\sigma$ acting on
$H^{0}(C,K_{C})$, the exponents $a_{1},\ldots,a_{g}$; in the last column we
check the inequality $\sum a_{i}<g$, which implies that
$J/\langle{\sigma}\rangle$ is uniruled.

$g=3$

$\begin{array}[]{|c|c|c|c|c|c|}\hline\cr\text{curve}&r&\sigma(x,y)=&\text{basis
of }H^{0}(K_{C})&a_{1},\ldots,a_{3}&\sum a_{i}\\\
\hline\cr\rule[-14.22636pt]{0.0pt}{2.0pt}y^{2}=x^{8}-1&8&(\zeta
x,y)&\dfrac{dx}{y},\dfrac{xdx}{y},\dfrac{x^{2}dx}{y}&1,2,3&6<8\\\
\hline\cr\rule[-14.22636pt]{0.0pt}{2.0pt}y^{2}=x(x^{7}-1)&14&(\zeta^{2}x,\zeta
y)&\dfrac{dx}{y},\dfrac{xdx}{y},\dfrac{x^{2}dx}{y}&1,3,5&9<14\\\
\hline\cr\rule[-14.22636pt]{0.0pt}{2.0pt}y^{3}=x(x^{3}-1)&9&(\zeta^{3}x,\zeta
y)&\dfrac{dx}{y^{2}},\dfrac{xdx}{y^{2}},\dfrac{dx}{y}&1,4,2&7<9\\\
\hline\cr\rule[-14.22636pt]{0.0pt}{2.0pt}y^{3}=x^{4}-1&12&(\zeta^{-3}x,\zeta^{-4}y)&\dfrac{dx}{y^{2}},\dfrac{xdx}{y^{2}},\dfrac{dx}{y}&5,2,1&8<12\\\
\hline\cr\end{array}$

The fact that $J/\langle{\sigma}\rangle$ is uniruled for the third curve $C$
is an important ingredient of the proof in [B-S] that the Ceresa cycle
$[C]-[(-1)^{*}C]$ in $J$ is torsion modulo algebraic equivalence. The same
property for the fourth curve is observed in [L-S].

$g=4$

$\begin{array}[]{|c|c|c|c|c|c|}\hline\cr\text{curve}&r&\sigma(x,y)=&\text{basis
of }H^{0}(K_{C})&a_{1},\ldots,a_{4}&\sum a_{i}\\\
\hline\cr\rule[-14.22636pt]{0.0pt}{2.0pt}y^{2}=x(x^{9}-1)&18&(\zeta^{2}x,\zeta
y)&\dfrac{dx}{y},\dfrac{xdx}{y},\dfrac{x^{2}dx}{y},\dfrac{x^{3}dx}{y}&1,3,5,7&16<18\\\
\hline\cr\rule[-14.22636pt]{0.0pt}{2.0pt}y^{3}=x^{5}-1&15&(\zeta^{-3}x,\zeta^{-5}y)&\dfrac{dx}{y^{2}},\dfrac{xdx}{y^{2}},\dfrac{x^{2}dx}{y^{2}},\dfrac{dx}{y}&7,4,1,2&14<15\\\
\hline\cr\end{array}$

In genus 5 there is no example:

######  Proposition 3.

Assume $g=5$. The quotient variety $J/G$ has canonical singularities, and
Kodaira dimension $0$.

Proof : The paper [K-K] contains the list of possible automorphisms of genus 5
curves, and gives for each automorphism of order $r$ its eigenvalues
$e^{\frac{2\pi ia_{1}}{r}},\ldots,e^{\frac{2\pi ia_{5}}{r}}$ on $H^{0}(K)$,
with $0\leq a_{i}<5$. We want to prove $\sum a_{i}\geq r$.

We first observe that if an eigenvalue and its inverse appear in the list, we
have $\sum a_{i}\geq r$; this eliminates all cases with $r\leq 10$. For $r\geq
11$ the numbers $a_{1},\ldots,a_{5}$ are always distinct and nonzero;
therefore $\sum a_{i}\geq 1+2+\cdots+5\allowbreak=15$. The remaining cases are
$r=20$ and $22$; one checks immediately that for all $k$ coprime to $r$ we
have $\sum b_{i}\geq r$, where $b_{i}\equiv ka_{i}\pmod{r}$ and $0<b_{i}<r$.
This proves the Proposition.∎

I know no example in genus $\geq 6$, and I suspect there are none.

## References

  * [B-L] C. Birkenhake, H. Lange : Complex abelian varieties. Grund. der Math. Wiss. 302. Springer-Verlag, Berlin, 2004.
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