text:
\section{Transfer of beating effects: Proof of Theorem \ref{thm:traveling_periodic_beating}}\label{sec:transfer}
We devote this section to prove Theorem \ref{thm:traveling_periodic_beating}. First, in Section \ref{sec:MultiSplitting} we prove the transversality of the stable and unstable invariant manifolds of different periodic orbits of the Hamiltonian \eqref{redhamN}. As a consequence of this transversality, we construct orbits which shadow these invariant manifolds for infinite time. Then, in Section \ref{sec:ApplicationMultiple} we prove that the resonant models associated to the Wave, Beam and Hartree equations that we consider fit into the framework of Section \ref{sec:MultiSplitting} and we complete the proof of Theorem \ref{thm:traveling_periodic_beating}.

\subsection{Heteroclinic connections between periodic orbits and their shadowing}\label{sec:MultiSplitting}

Reasoning as in Proposition \ref{prop:SplittingPO2Rect}, the Hamiltonian $\mathcal{H}$ in \eqref{redhamN}  possesses hyperbolic periodic orbits
$\mathtt{P}_{\varepsilon,h,k}^\pm$ at the energy level $h$ whose time parameterization is of the form
\begin{equation*}

\gamma_{\varepsilon,h,k}^{\pm,p}(\tau_k) = (\Psi_{\pm,\varepsilon,1}^*, \dots,\Psi_{\pm,\varepsilon,k-1}^*, \Psi_k^{(h)}(\tau_k),\Psi_{\pm,\varepsilon,k+1}^*, \dots,
\Psi_{\pm,\varepsilon,N}^*, 0, \dots,0, K_k^{(h)}(\tau_k),0, \dots,0)
\end{equation*}
where
\[
 \Psi_{\pm,\varepsilon,1}^*=\pm\Psi_\ast+\OO(\eps)
\]
(see \eqref{def:psi}) and $( \Psi_k^{(h)}, K_k^{(h)})$ is $\eps$-close to the periodic orbit $\mathtt{P}_h$ (see \eqref{def:Gammah}).

When $\varepsilon = 0$, the  invariant manifolds $W^u (\mathtt{P}_{0,h,k}^+)$ and  $W^s(\mathtt{P}_{0,h,k}^-)$ coincide.

\begin{prop}\label{prop:SplittingMulti}
Take any $i,j=1,\ldots, N$, $i\neq j$. Assume that the condition  \eqref{cond:HessD} is satisfied (see \eqref{def:MatrixD},  \eqref{coeffN}). Then, there exists $\varepsilon_0>0$ such that for
$\varepsilon\in(0, \varepsilon_0)$ and $h_0>0$
such that for any $h \in(0, h_0)$,

the manifolds
$W^u(\mathtt{P}_{\varepsilon, h, i}^-)$ and
$W^s(\mathtt{P}_{\varepsilon, h, j}^+)$ intersect transversally
within the energy level.
\end{prop}

The transversality of the invariant manifolds allows to construct orbits which shadow them. Note that in the coordinates introduced in \eqref{def:cartesian}, the periodic orbits $\mathtt{P}_{0,h,k}^+$ and $\mathtt{P}_{\eps,h,k}^-$ blow down to the same periodic orbit, which we denote by $\mathtt{P}_{0,h,k}$. In the coordinates  \eqref{def:cartesian}, Proposition \ref{prop:SplittingMulti} can be restated as that the  manifolds $W^u(\mathtt{P}_{\varepsilon, h, i})$ and $W^s(\mathtt{P}_{\varepsilon, h, j})$ intersect  transversally along an orbit
within the energy level.

\begin{defi}
\label{def:transition_chain} We will say that a family of hyperbolic periodic
orbits $\{\mathtt{P}_{\ell}\}_{\ell \in \N}$ of a system of differential equations,  is a \emph{transition
chain} if  $W^u(\mathtt{P}_{\ell}) \pitchfork
W^s(\mathtt{P}_{{\ell+1}})$, for all $\ell \in \N$.
\end{defi}
Note that Proposition \ref{prop:SplittingMulti} gives full transversality between the invariant manifolds on the energy level. Thus, recalling that  $\mathcal{H}(\mathtt{P}_{i_\ell})=h$, from now on, we restrict the flow to this energy level, which is a regular manifold.

\begin{cor}
\label{cor:transition_chain}
Let $(i_\ell)_{\ell \in \N}$, with $i_\ell \in \{1,\dots,N\}$, be any sequence.
Then, if $\varepsilon>0$ is small enough, there exists $h_0$ such
 that for any $0<h<h_0$, $\{\mathtt{P}_{\eps,h,i_\ell}\}_{\ell \in \N}$ is a transition chain
 of Hamiltonian $\mathcal{H}$ on the manifold $\mathcal{H}=h$.
\end{cor}

\begin{prop}
\label{prop:diffusion_orbits}
Let $(i_\ell)_{\ell \in \N}$, with $i_\ell \in \{1,\dots,N\}$, be any sequence. Assume that
$\varepsilon>0$ is small enough such that $h_0$ in Corollary~\ref{cor:transition_chain} exists.
Let $\{\mathtt{P}_{\eps,h,i_\ell}\}_{\ell \in \N}$ be a transition chain
 of Hamiltonian $\mathcal{H}$. Let $(\nu_\ell)_{\ell \in \N}$, with $\nu_\ell >0$, be an arbitrary sequence.
Let $N_{\ell} := \{z\mid d(z,\mathtt{P}_{\eps,h,i_\ell})<\nu_{\ell}\}$. Then, there exists a trajectory $\gamma(t)$ of  Hamiltonian $\mathcal{H}$ in \eqref{redhamN}
and an increasing sequence  $(t_\ell)_{\ell \in \N}$ of times such that $\gamma(t_\ell) \in N_{\ell}$, for all $\ell\in\N$.
\end{prop}

\begin{proof}
Since $\{\mathtt{P}_{\eps,h,i_\ell}\}_{\ell \in \N}$ is a transition chain

 then the Inclination Lemma in~\cite{FontichM01} (Theorem $4.5$) ensures that $W^s_\varepsilon( \mathtt{P}_{\eps,h,i_{\ell}})\subseteq \overline{ \cup_{t\leq 0} \Phi^t_{\mathcal{H}}(W_{\varepsilon}^{s} (\mathtt{P}_{i_{\ell}+1}))}$ for all $\ell\in\mathbb{N}$ \footnote{More precisely we apply Theorem 4.5 in~\cite{FontichM01}to the flow map $f:=\Phi^{\tau}_{\mathcal{H}}$, where $\tau>0$ is chosen to be not a multiple of any frequency of the periodic orbits $\mathtt{P}_{{\eps,h,i_\ell}}$. Note that the Inclination Lemma stated in~\cite{FontichM01} is stated for the unstable manifold; in order to deduce the statement for the stable manifold it suffices to replace $f$ by $f^{-1}$.} .

  Let $x\in W^s_\varepsilon(\mathtt{P}_{i_0})$. We can find a closed ball $B_0$ centered at $x$ such that
  \begin{equation}\label{eq:inclusion}
  \Phi_{\mathcal{H}}^{t_0}(B_0)\subset N_0
  \end{equation}
  for some $t_0>0$. By the inclination Lemma we have that $W_{\varepsilon}^s(\mathtt{P}_{i_1})\cap B_0\neq \emptyset$. Hence we can find a closed ball $B_1$ centered at a point in $W_{\varepsilon}^s(\mathtt{P}_{i_1})\cap B_0$ such that, besides satisfying \eqref{eq:inclusion},
 \begin{equation*}
  \Phi_{\mathcal{H}}^{t_1}(B_1)\subset N_1
 \end{equation*}
 for some $t_1>t_0$. Proceeding by induction we can construct a sequence of closed nested balls $B_{{i+1}}\subset B_{i} \subset \dots\subset B_0$ and times $t_{i+1}>t_i>\ldots >t_0$ such that
 \[
 \Phi_{\mathcal{H}}^{t_j} (B_i)\subset N_{j}, \qquad i\ge j.
 \]
 Since the balls are compact, the intersection $\cap_{n\geq 0} B_n$ is non-empty, and we can consider $\gamma(t)$ as an orbit with initial datum in that set.

\end{proof}

\subsubsection{Proof of Proposition \ref{prop:SplittingMulti}}

We proceed as in Section \ref{sec:HomoMelnikov} by considering an auxiliary parameter $\de$ and the Hamiltonian $\mathcal{H} = \sum_{j=1}^N \mathbf{H}_0^{(j)}
+ \varepsilon \mathbf{H}_1$ defined in \eqref{def:hamN}. The Hamiltonian \eqref{def:hamN} has two saddle points,
\[
\mathfrak{e}^{(0)}_{\pm,\varepsilon}=  (\Psi_{\pm, \varepsilon,1}^*,\dots, \Psi_{\pm, \varepsilon,N}^*, 0,\dots,0),
\]
which, for $\varepsilon = 0$ are
$\mathfrak{e}^{(0)}_{\pm}$ (see~\eqref{saddlesN}). For $\varepsilon = 0$ and any $\delta>0$ small, they
 are connected by the homoclinic manifolds
\begin{equation}
\label{def:homoclinic_manifold_ndim}
\gamma_0(\vec{\tau} ) = (\Psi_1^{(0)}(\tau_1), \ldots, \Psi_N^{(0)}(\tau_N), K_1^{(0)}(\tau_1), \ldots, K_N^{(0)}(\tau_N)), \quad \vec{\tau}:=(\tau_1, \dots, \tau_N),
\end{equation}
where $\Psi_j^{(0)}$, $K_j^{(0)}$, $j=1,\dots, N$, have been introduced in~\eqref{gammazero}.
This parametrization of the homoclinic manifold satisfies ${\Phi_{\mathcal{H}}^t}_{\mid \varepsilon = 0} \gamma_0(\vec{\tau} )
= \gamma_0(\tau_1+t, \dots,\tau_n+t)$.
Fix $1\le k \le N$. The set
\[
\pi_k = \{(\Psi_1,\dots, \Psi_N, K_1,\dots,K_N): \;K_\ell = 0, \; \ell \neq k\}
\]
is invariant by the flow of $\mathcal{H}$ for any $\varepsilon$ and $\delta$ (this is properly seen in the coordinates \eqref{def:cartesian}, since then $\pi_k$ corresponds to $x_\ell=y_\ell=0$, $\ell\neq k$, see \eqref{redhamNCartesian}).

The dynamics on the  $\pi_k$ plane is integrable and is given by the $1$-d.o.f. Hamiltonian
$\mathbf{H}_0^{(k)} + \varepsilon {\mathbf{H}_1}_{\mid \pi_k}$. This Hamiltonian has two saddles $(\Psi_{\pm,\varepsilon,\ell}^*,0)$
 $\varepsilon$-close to $(\pm \Psi_*,0)=(\pm 2\pi/3,0)$, at the zero energy level. For $h>0$ small, the set $\{\mathbf{H}_0^{(k)} + \varepsilon {\mathbf{H}_1}_{\mid \pi_k}=h\}$ is a periodic orbit, whose period tends to infinity when $h$ goes to $0$. Let $(\Psi_k^{(h)}(\tau_k), K_k^{(h)}(\tau_k))$ be a time parametrization of this periodic orbit satisfying
\begin{equation}
\label{eq:choice_param_periodic_orbit}
\lim_{h\to 0} (\Psi_k^{(h)}(0), K_k^{(h)}(0)) =
(\Psi_k^{(0)}(0), K_k^{(0)}(0)),
\end{equation}
where $(\Psi_k^{(0)}, K_k^{(0)})$ are components of the homoclinic manifold introduced in \eqref{def:homoclinic_manifold_ndim}.

Then,

the Hamiltonian $\mathcal{H}$  possesses two hyperbolic periodic orbits
$\mathtt{P}_{\varepsilon,h,k}^\pm$ at the energy level $h$, whose time parametrization is given by
\begin{equation}\label{def:timeparapo}
\gamma_{\varepsilon,h,k}^{\pm,p}(\tau_k) = (\Psi_{\pm,\varepsilon,1}^*, \dots,\Psi_{\pm,\varepsilon,k-1}^*, \Psi_k^{(h)}(\tau_k),\Psi_{\pm,\varepsilon,k+1}^*, \dots,
\Psi_{\pm,\varepsilon,N}^*, 0, \dots,0, K_k^{(h)}(\tau_k),0, \dots,0).
\end{equation}

When $\varepsilon = 0$, the  invariant manifolds $W^u (\mathtt{P}_{0,h,k}^+)$ and  $W^s(\mathtt{P}_{0,h,k}^-)$ coincide. This homoclinic manifold can be parameterized as
\begin{equation}\label{def:omoclinicpo}
\gamma_{h,k}(\vec{\tau} ) = (\Psi_1^{(0)}(\tau_1), \ldots, \Psi_k^{(h)}(\tau_k), \dots,  \Psi_N^{(0)}(\tau_N), K_1^{(0)}(\tau_1), \ldots, K_k^{(h)}(\tau_k), \dots, K_N^{(0)}(\tau_N)),
\end{equation}
where $(\Psi_k^{(0)}, K_k^{(0)})$ are components of the homoclinic manifold introduced in \eqref{def:homoclinic_manifold_ndim}.

Now, fix $i,j \in \{1,\dots, N\}$.

For small $\varepsilon>0$, the periodic orbits  $\mathtt{P}_{\varepsilon,h,i}^-$, $\mathtt{P}_{\varepsilon,h,j}^+$ and their invariant manifolds,
$W^s(\mathtt{P}_{\varepsilon,h,i}^+)$ and $W^u(\mathtt{P}_{\varepsilon,h,j}^-)$ persist slightly deformed. We show now that the perturbation allows them to intersect.

In order to analyze the possible intersection,

we introduce a $N$-dimensional section in the following way.

We define, taking into account~\eqref{def:hamN},
\begin{equation}\label{def:hamtrick1}
\widetilde{\mathbf{H}}_0^{(k)} = \mathbf{H}_0^{(k)} + \varepsilon \widehat{\mathbf{H}}_0^{(k)}, \qquad k = 1,\dots,N,
\end{equation}
where
\begin{equation*}

\widehat{\mathbf{H}}_0^{(k)}(\psi_k,K_k) =
a_{k} K_k + (b_{k}-1) K^2_k +c_k\,K_k (1-K_k)\,\cos(\psi_k)
\end{equation*}
only depends on $(\psi_k,K_k)$. We have that $\widetilde{\mathbf{H}}_0^{(k)}$ is integrable and $\mathcal{H}$ can be also written as
\begin{equation}
\label{def:newdefinitionofmathcalH}
\mathcal{H} = \sum_{k=1}^N \mathbf{H}_0^{(k)} + \varepsilon \mathbf{H}_1 = \sum_{k=1}^N \widetilde{\mathbf{H}}_0^{(k)} + \varepsilon \widetilde{\mathbf{H}}_1
\end{equation}
where
\begin{equation}\label{def:hamtrick3}
\widetilde{\mathbf{H}}_1(K_1,\dots,K_n) = \sum_{k, \ell=1, k<\ell}^N d_{k \ell} K_k\,K_\ell.
\end{equation}
We consider the $N$-dimensional section
\begin{equation}
\label{def:transsecNrectangles}
\Sigma(\vec{\tau})=\left\{\gamma_{0}(\vec{\tau})+\sum_{k=1}^ Nr_k\,{\nabla \widetilde{\mathbf{H}}^{(k)}_0}_{\mid \gamma_0(\vec{\tau})},
r=(r_1,\dots,r_{N})\in (-m, m)^{N}\right\}

\end{equation}
where $\gamma_0$ is the homoclinic manifold introduced in \eqref{def:homoclinic_manifold_ndim}. Observe that $\gamma_{0}(\vec{\tau})$, which is $N$-dimensional, intersects transversally $\Sigma(\vec{\tau})$ at $r=0$. Then, for $h$ small, $\gamma_{h,i}$ and $\gamma_{h,j}$ (see \eqref{def:omoclinicpo}) intersect transversally
$\Sigma(0)$ at points $r_{i}$ and $r_{j}$, close to $\gamma_{0}(\vec{\tau})$. Hence, for $\eps$ small enough, the invariant manifolds  $W^u(\mathtt{P}_{\varepsilon,h,i}^-)$ and $W^s(\mathtt{P}_{\varepsilon,h,j}^+)$  intersect transversally $\Sigma(0)$ at points $r_{\varepsilon,i}$ and $r_{\varepsilon,j}$ close to $r_{i}$ and $r_{j}$, respectively.

Let $\gamma_{\varepsilon,h,i}^u$ and $\gamma_{\varepsilon,h,j}^s$ be parametrizations of the perturbed invariant manifolds $W^u(\mathtt{P}_{\varepsilon,h,i}^-)$ and $W^s(\mathtt{P}_{\varepsilon,h,j}^+)$ such that
$\gamma_{\varepsilon,h,i}^u(0) = r_{\varepsilon,i}$, $\gamma_{\varepsilon,h,j}^s(0) = r_{\varepsilon,j}$
and

$\Phi^t_{\mathcal{H}} \gamma(\tau_1,\dots, \tau_n) = \gamma (\tau_1+t,\dots, \tau_N+t)$, for $\gamma = \gamma_{\varepsilon,h,i}^{u},\gamma_{\varepsilon,h,j}^{s}$, where $\Phi^t_{\mathcal{H}}$ is the flow of Hamiltonian $\mathcal{H}$.
Up to a shift in the initial conditions in the periodic orbits, the parameterization of the periodic orbits and the homoclinic manifold satisfy the following property: for any $\tau$ there exists constants $\la, K, M>0$ such that
\begin{equation*}

 \begin{aligned}
\|\gamma_{\varepsilon,h,k}^{u}(\tau_1+t,\ldots,\tau_N+t) - \gamma_{\varepsilon,h,i}^{-,p}(\tau_i+t)\| \le &K e^{\lambda t}\qquad &\text{ for }\quad t\leq M\\
\|\gamma_{\varepsilon,h,k}^{s}(\tau_1+t,\ldots,\tau_N+t) - \gamma_{\varepsilon,h,j}^{+,p}(\tau_j+t)\| \le &K e^{- \lambda t}\qquad &\text{ for }\quad t\geq M.
  \end{aligned}
\end{equation*}

Let us remark that $\mathcal{H}_{\mid W^u (\mathtt{P}_{\varepsilon,h,i}^-)}=\mathcal{H}_{\mid W^s (\mathtt{P}_{\varepsilon,h,j}^+)}=h$. Therefore, to analyze their intersections it is enough to measure their distance along $(N-1)$-directions of those defining the section $\Sigma$ in \eqref{def:transsecNrectangles}. That is, the manifolds $W^u (\mathtt{P}_{\varepsilon,h,i}^-)$ and $W^s (\mathtt{P}_{\varepsilon,h,j}^+)$
 intersect transversally along an orbit at the non-degenerate zeros of the vector function (see \eqref{def:hamtrick1})
\begin{equation}
\label{def:distance_between_manifolds_of_periodic_orbits}
d_{\varepsilon,h}(\vec{\tau}) =
\begin{pmatrix}
\widetilde{\mathbf{H}}_0^{(1)}(\gamma_{\varepsilon,h,i}^u(\vec{\tau}))-
\widetilde{\mathbf{H}}_0^{(1)}(\gamma_{\varepsilon,h,j}^s(\vec{\tau})) \\
\vdots \\
\widetilde{\mathbf{H}}_0^{(N-1)}(\gamma_{\varepsilon,h,i}^u(\vec{\tau}))-
\widetilde{\mathbf{H}}_0^{(N-1)}(\gamma_{\varepsilon,h,j}^s(\vec{\tau}))
\end{pmatrix}.
\end{equation}

\begin{lem}
The function $d_{\varepsilon,h}$ in~\eqref{def:distance_between_manifolds_of_periodic_orbits} can be written as
\begin{equation}
\label{eq:difMelnikov}
d_{\varepsilon,h}(\vec{\tau}) = d_{0,h}+ \varepsilon \mathcal{M}_h(\vec{\tau}) + \OO\left(\varepsilon^2\right),
\end{equation}
where  the vector $d_{0,h}=(d_{0,h}^1, \ldots d_{0,h}^{N-1})^\top$ is of the form
\[
 d_{0,h}^i=h,\quad d_{0,h}^j=-h\quad \text{ and }\quad d_{0,h}^k=0\quad\text{ for }\quad k\neq i,j
\]

and
$\mathcal{M}_h(\vec{\tau}) = ( \mathcal{M}_{h}^1(\vec{\tau}) ,\dots,
\mathcal{M}_{h}^{N-1}(\vec{\tau}))^\top$, with
\[
\mathcal{M}_{h}^k(\vec{\tau}) :=\int_{-\infty}^{0} \{ \mathbf{H}_0^{(k)}, \widetilde{\mathbf{H}}_1\} \, \circ \, \Phi^t_{\mathbf{H}_0}(\gamma_{h,i}(\vec{\tau}))\,dt+\int_0^{\infty} \{ \mathbf{H}_0^{(k)}, \widetilde{\mathbf{H}}_1\}\, \circ \,\Phi^t_{\mathbf{H}_0}(\gamma_{h,j}(\vec{\tau}))\,dt.
\]
\end{lem}
\begin{proof}
We will compute the formula for $\widetilde{\mathbf{H}}_0^{(k)}(\gamma_{\varepsilon,h,i}^u(\vec{\tau}))$, $k=1,\dots, N-1$,
being the derivation for the one of $\widetilde{\mathbf{H}}_0^{(k)}(\gamma_{\varepsilon,h,j}^s(\vec{\tau}))$ analogous.

We first observe that, since $\mathcal{H}_{\mid \mathtt{P}_{\varepsilon,h,i}^-}= h$ and
$\widetilde{\mathbf{H}}_{1\,\mid \mathtt{P}_{\varepsilon,h,i}^-}= 0$,
 $\widetilde{\mathbf{H}}_0^{(k)}(\gamma_{\varepsilon,h,i}^{-,p}(\vec{\tau})) = \delta_{ik}h$, being $\delta_{ik}$ the Kronecker's delta. Then, taking into account~\eqref{def:newdefinitionofmathcalH}, it is immediate that

\[
\begin{aligned}
\widetilde{\mathbf{H}}_0^{(k)}(\gamma_{\varepsilon,h,i}^u(\vec{\tau})) &
= \widetilde{\mathbf{H}}_0^{(k)}(\gamma_{\varepsilon,h,i}^u(\vec{\tau})) - \widetilde{\mathbf{H}}_0^{(k)}(\gamma_{\varepsilon,h,i}^{-,p}(\vec{\tau})) + \delta_{ik} h \\
& = \int_{-\infty}^0 \frac{d}{dt} \widetilde{\mathbf{H}}_0^{(k)} \circ \Phi_{\mathcal{H}}^t \gamma_{\varepsilon,h,i}^u(\vec{\tau})\, dt + \delta_{ik} h \\
& = \varepsilon \int_{-\infty}^0 \{ \widetilde{\mathbf{H}}_0^{(k)}, \widetilde{\mathbf{H}}_1\} \circ \Phi_{\mathcal{H}}^t( \gamma_{\varepsilon,h,i}^u(\vec{\tau}))\, dt + \delta_{ik} h \\
& = \varepsilon \int_{-\infty}^0 \{ \widetilde{\mathbf{H}}_0^{(k)}, \widetilde{\mathbf{H}}_1\} \circ \Phi_{\mathbf{H}_0}^t ( \gamma_{h,i}(\vec{\tau}))\, dt + \delta_{ik} h + \mathcal{O}(\varepsilon^2) \\
& = \varepsilon \int_{-\infty}^0 \{ \mathbf{H}_0^{(k)}, \widetilde{\mathbf{H}}_1\} \circ \Phi_{\mathbf{H}_0}^t (\gamma_{h,i}(\vec{\tau}))\, dt + \delta_{ik} h +  \mathcal{O}(\varepsilon^2),
\end{aligned}
\]
where $\gamma_{h, i}$ is defined in \eqref{def:omoclinicpo}.
\end{proof}
We observe that, since the components of $d_{0,h}$ are either $0$ or $\pm h$,
if we consider $h\ll \varepsilon$, the main order of the difference in~\eqref{eq:difMelnikov} is given by $\mathcal{M}_h(\vec{\tau})$. Thus we shall prove that this function has a non-degenerate zero, so that we can conclude by the Implicit Function Theorem that the manifolds $W^s_\varepsilon(\mathtt{P}^-_{\varepsilon, h, i})$ and $W^u_\varepsilon(\mathtt{P}^+_{\varepsilon, h, j})$ intersect transversally.

To do so, we introduce
\begin{equation}
\label{def:homoclinic_Melnikov}
\mathcal{M}_{0}(\vec{\tau}) := ( \mathcal{M}_0^1(\vec{\tau}),\dots, \mathcal{M}_0^{N-1}(\vec{\tau}))^\top,
\end{equation}
where
\begin{equation*}

 \mathcal{M}_{0}^k(\vec{\tau}) := \int_{-\infty}^{\infty} \{ \mathbf{H}_0^{(k)}, \widetilde{\mathbf{H}}_1\}\,\circ\,\Phi^t_{\mathbf{H}_0}(\gamma_{0}(\vec{\tau}))\,dt,
\end{equation*}
where $\gamma_{0}$ was introduced in~\eqref{def:homoclinic_manifold_ndim} (see also~\eqref{gammazero}), which is the Melnikov function associated
to the homoclinic between $\mathfrak{e}^{(0)}_{\pm}$. We observe that the derivative of the Melnikov potential \eqref{def:malnikovpotcaseN} with respect to the variable $\tau_k-\tau_N$ coincides with the Melnikov integral $\mathcal{M}_0^k(\vec{\tau})$ in  \eqref{def:homoclinic_Melnikov}: Indeed, recall that the Melnikov Potential integral associated to $(\mathbf{H}_1-\widetilde{\mathbf{H}}_1)$ is constant, and equivalently
\[
\mathcal{M}_{0}^k(\vec{\tau}) = \int_{-\infty}^{\infty} \{ \mathbf{H}_0^{(k)}, \widetilde{\mathbf{H}}_1\}\,\circ\,\Phi^t_{\mathbf{H}_0}(\gamma_{0}(\vec{\tau}))\,dt= \int_{-\infty}^{\infty} \{ \mathbf{H}_0^{(k)}, \mathbf{H}_1\}\,\circ\,\Phi^t_{\mathbf{H}_0}(\gamma_{0}(\vec{\tau}))\,dt.
\]
Then, by Proposition \ref{prop:homointersaddlesN}, if condition \eqref{cond:HessD} is satisfied and $\varepsilon>0$ is small enough, $\mathcal{M}_{0}(\vec{\tau})$ has a non-degenerate zero.

\begin{lem}
\label{lem:Melnikov_between_periodic_orbits} Let $\varepsilon>0$ and assume that the condition \eqref{cond:HessD} in Proposition \ref{prop:homointersaddlesN} is satisfied. Then, there exists $h_0$ such that  for any $0<h<h_0$,
$\mathcal{M}_{h}(\vec{\tau})$ has a non-degenerate zero.
\end{lem}

This lemma implies Proposition \ref{prop:SplittingMulti};

indeed, one can proceed as in Section \ref{sec:HomoMelnikov} by taking $\de=\eps$ and applying Implicit Function Theorem. We devote the rest of the Section to prove Lemma \ref{lem:Melnikov_between_periodic_orbits}.

\begin{proof}[Proof of Lemma \ref{lem:Melnikov_between_periodic_orbits}]
We have that, for any $1\le k \le N-1$,
\begin{align*}
&\int_{-\infty}^0 \{ \mathbf{H}_0^{(k)}, \widetilde{\mathbf{H}}_1\}\,\circ\,\Phi^t_{\mathbf{H}_0}(\gamma_{h,i}(\vec{\tau}))\,dt
+\int_0^{\infty} \{ \mathbf{H}_0^{(k)}, \widetilde{\mathbf{H}}_1\}\,\circ\,\Phi^t_{\mathbf{H}_0}(\gamma_{h,j}(\vec{\tau}))\,dt-\mathcal{M}_0^{k}(\vec{\tau})=\\
&=\int_{-\infty}^0 \Big(\{ \mathbf{H}_0^{(k)}, \widetilde{\mathbf{H}}_1\}\,\circ\,\Phi^t_{\mathbf{H}_0}(\gamma_{h,i}(\vec{\tau}))-\{ {\mathbf{H}}_0^{(k)}, \widetilde{\mathbf{H}}_1\}\,\circ\,\Phi^t_{\mathbf{H}_0}(\gamma_{0}(\vec{\tau}))\,\Big)\,dt\\
&+\int_0^{\infty}\Big(\{ \mathbf{H}_0^{(k)}, \widetilde{\mathbf{H}}_1\}\,\circ\,\Phi^t_{\mathbf{H}_0}(\gamma_{h,j}(\vec{\tau}))-\{ \mathbf{H}_0^{(k)}, \widetilde{\mathbf{H}}_1\}\,\circ\,\Phi^t_{\mathbf{H}_0}(\gamma_{0}(\vec{\tau}))\,\Big)\,dt \\
& = I_{h,i}(\vec{\tau})+I_{h,j}(\vec{\tau}).
\end{align*}

We prove that, for any compact set $\mathbf{K}\subset \R^{N}$, $\|I_{h,k}\|_{C^1(\mathbf{K})}$ tends to $0$ as $h\to 0$, for $k=i,j$.
We give the argument for $I_{h,i}$, being the one for $I_{h,j}$ analogous. The claim follows immediately from this convergence.

Let $\mathbf{K}\subset \R^{N}$ be a compact set. If $h$ is small enough, the parametrization $\gamma_{h,i}$ is well defined;
since the period of the periodic orbit tends to infinity when $h$ goes to $0$, $\gamma_{h,i}$ intersects $\Sigma(\vec{\tau})$ at a point close to $r=0$, for all $\vec{\tau} \in \mathbf{K}$.

For a given $T>0$, we split the integral $I_{h,i}$ as
\begin{equation}
\label{eq:split_of_Ihi}
\begin{aligned}
I_{h,i} (\vec{\tau})
 = & \int_{-T}^0 \Big(\{ \mathbf{H}_0^{(k)}, \widetilde{\mathbf{H}}_1\}\,\circ\,\Phi^t_{\mathbf{H}_0}(\gamma_{h,i}(\vec{\tau}))
 -\{ \mathbf{H}_0^{(k)}, \widetilde{\mathbf{H}}_1\}\,\circ\,\Phi^t_{\mathbf{H}_0}(\gamma_{0}(\vec{\tau}))\,\Big)\,dt \\
& + \int_{-\infty}^{-T} \Big(\{ \mathbf{H}_0^{(k)}, \widetilde{\mathbf{H}}_1\}\,\circ\,\Phi^t_{\mathbf{H}_0}(\gamma_{h,i}(\vec{\tau}))-\{ \mathbf{H}_0^{(k)}, \widetilde{\mathbf{H}}_1\}\,\circ\,\Phi^t_{\mathbf{H}_0}(\gamma_{0}(\vec{\tau}))\,\Big)\,dt.
\end{aligned}
\end{equation}
From~\eqref{def:hamN}, \eqref{def:hamtrick3}

\begin{equation}
\label{eq:Poisson_bracket_H0H1}
\{ \mathbf{H}_0^{(k)}, \widetilde{\mathbf{H}}_1\} = - 2 K_k(1-K_k) \sin \psi_k \sum_{\ell \neq k} d_{\ell,k} K_\ell.
\end{equation}
In particular,
\[
\{ \mathbf{H}_0^{(k)}, \widetilde{\mathbf{H}}_1\}_{\mid \mathfrak{e}^{(0)}_{-}}
= \{ \mathbf{H}_0^{(k)}, \widetilde{\mathbf{H}}_1\}_{\mid\mathtt{P}^-_{0, h, i}} = 0.
\]
The hyperbolic character of the periodic orbits implies the existence of constants $C, \lambda >0$ such that, for $\ell \neq i$ and for all $\vec{\tau}\in\mathbf{K}$,
\[
|\pi_{K_\ell} \gamma_{h,i}(\vec{\tau}) | \le C e^{-\lambda(\tau_\ell + t)}, \qquad t \ge 0.
\]
Also, with the same $C, \lambda >0$, for any $1 \le \ell \le N$,
\[
|\pi_{K_\ell} \gamma_{0}(\vec{\tau}) | \le C e^{-\lambda(\tau_\ell + t)}, \qquad t \ge 0.
\]
Hence, by~\eqref{eq:Poisson_bracket_H0H1}, for any $\nu >0$,
since $\mathbf{K}$ is compact, there exists $T>0$ such that, for any
$\vec{\tau} \in \mathbf{K}$,
\[
\left| \int_{-\infty}^{-T} \{ \mathbf{H}_0^{(k)},
\widetilde{\mathbf{H}}_1\}\,\circ\,\Phi^t_{\mathbf{H}_0}(\gamma_{h,i}(\vec{\tau}))\,dt
\right|, \, \left| \int_{-\infty}^{-T} \{ \mathbf{H}_0^{(k)},
\widetilde{\mathbf{H}}_1\}\,\circ\,\Phi^t_{\mathbf{H}_0}(\gamma_{0}(\vec{\tau}))\,dt
\right| \le \nu.
\]
To bound the other part of $I_{h,i}$ in~\eqref{eq:split_of_Ihi}, we observe that,
from~\eqref{man},
\[
\gamma_{h,i}(\vec{\tau}) - \gamma_{0}(\vec{\tau})
= (0, \ldots, \Psi_i^{(h)}(\tau_i)-\Psi_i^{(0)}(\tau_i), \dots,  0, 0, \ldots, K_i^{(h)}(\tau_i)-K_i^{(0)}(\tau_i), \dots, 0).
\]
We remark that, as $h$ goes to $0$, the period of the periodic orbit
$\mathtt{P}_{0,h,i}^-$  goes to $\infty$. Then, the choice of the parametrization of the periodic orbit~\eqref{eq:choice_param_periodic_orbit} implies that,
taking $h$ small enough, $\lim_{h\to 0}(\Psi_i^{(h)}(\tau_i+t), K_i^{(h)}(\tau_i+t)) =
(\Psi_i^{(0)}(\tau_i+t), K_i^{(0)}(\tau_i+t))$, for all $(t, \vec{\tau}) \in [-T,0]\times
\mathbf{K}$ and, furthermore, this convergence is the $C^k$-norm on
$[-T,0]\times \mathbf{K}$. In particular, this implies that, if $h$ is small
enough,
\[
\left| \int_{-T}^0 \Big(\{ \mathbf{H}_0^{(k)},
\widetilde{\mathbf{H}}_1\}\,\circ\,\Phi^t_{\mathbf{H}_0}(\gamma_{h,i}(\vec{\tau}))
 -\{ \mathbf{H}_0^{(k)}, \widetilde{\mathbf{H}}_1\}\,\circ\,\Phi^t_{\mathbf{H}_0}(\gamma_{0}(\vec{\tau}))\,\Big)\,dt
 \right| < \nu.
\]
\end{proof}

\subsection{Application to the Wave, Beam and Hartree equations: Proof of Theorem \ref{thm:geometric}--$(ii)$}\label{sec:ApplicationMultiple}
Recall the matrix \eqref{def:MatrixD}.
We now check the condition \eqref{cond:HessD} in Proposition \ref{prop:homointersaddlesN} for the resonant models associated to the equations \eqref{Wave}, \eqref{Beam} and \eqref{Hartree}. For the Wave and Beam equations this corresponds to choosing suitable sets $\Lambda$ (actually a suitable modification of those obtained in Proposition \ref{prop:LambdaSetWaveHom}). For the Hartree equation this corresponds to imposing a non-degeneracy condition to the potential $V$.

\begin{lem}\label{lemma:WaveBeamMelnikovCondN}
Let us consider either the Wave
equation \eqref{Wave} or the Beam equation \eqref{Beam}. Then, there exists a set $\Lambda\subset\Z^2$ satisfying Propositions \eqref{prop:LambdaSetWaveHom} such that the associated Hamiltonian \eqref{calham0} satisfies
condition \eqref{cond:HessD}.

\end{lem}

\begin{lem}\label{lemma:HartreeMelnikovCondN}
Let us consider Hamiltonian \eqref{calham0} associated to the Hartree equation
\eqref{Hartree} with a potential $V$ as in \eqref{def:potential} and to a set $\Lambda$
satisfying Proposition \ref{prop:LambdaSetHartreeHet}
Then, for a generic choice of the $\{\gamma_{n}\}_{n\in\Lambda}$, the condition  \eqref{cond:HessD}  is satisfied.
\end{lem}

These two lemmas allow us to complete the proof of Item $(ii)$ of Theorem  \ref{thm:geometric}

\begin{proof}[Proof of Item $(ii)$ of Theorem  \ref{thm:geometric}]
Lemmas \ref{lemma:WaveBeamMelnikovCondN} and Lemma \ref{lemma:HartreeMelnikovCondN} ensure that the non-degeneracy condition  \eqref{cond:HessD}  of Proposition \ref{prop:SplittingMulti}. Therefore, any pair of periodic orbits $\mathtt{P}_{\varepsilon,h,i}$, $\mathtt{P}_{\varepsilon,h,j}$ have transverse heteroclinic connections. This implies that all infinite sequences of such periodic orbits form a transition chain in the sense of Definition \ref{def:transition_chain}. Then, to complete the proof of Item $(ii)$ of Theorem \ref{thm:geometric} it is enough to apply  Proposition \ref{prop:diffusion_orbits}.
\end{proof}

We devote the rest of this section to prove Lemmas \ref{lemma:WaveBeamMelnikovCondN} and \ref{lemma:HartreeMelnikovCondN}.
 Lemma \ref{lemma:HartreeMelnikovCondN} is proved following the same argument of the proof of Lemma \ref{lemma:HartreeMelnikovCond}. To the prove  Lemma \ref{lemma:WaveBeamMelnikovCondN}, we consider a set $\Lambda_0\subset\Z^2$ satisfying Proposition \ref{prop:LambdaSetWaveHom} and we modify it slightly. By modification, we refer to construct a set $\Lambda\in\Q^2$ arbitrarily close to $\Lambda_0\subset\Z^2$ and then to scale it  so that the set belongs to $\Z^2$.

\begin{proof}[Proof of Lemma \ref{lemma:WaveBeamMelnikovCondN}]
  Let us call $n^{(i)}_k:=n_{4 (i-1)+k}$ for $i=1,\dots, N$, $k=1, \dots, 4$.
Recall the expression of the coefficients $d_{i j}$ in \eqref{coeffN}. By  using \eqref{def:coeffmatrixA} and Lemma \ref{lem:HresLambda} we obtain
\begin{equation}\label{def:d12N}

 d_{i j}=\frac{3}{32 \mathtt{g}} \,\sum_{\substack{1 \leq r, s \leq 4}}
\frac{(-1)^{r+s}}{|n^{(i)}_r|^{\kappa} |n^{(j)}_s|^\kappa},

 \end{equation}
 where $\kappa=1, 2$ corresponds respectively to the Wave and Beam equations.
 We define  (recall formulas \eqref{def:IndexesPiDelta})
\begin{align}
P_r=\Delta_{n^{(r)}_{1}, n^{(r)}_{2}} (|n_1^{(r)}|^{\kappa} |n^{(r)}_2|^{\kappa}-|n^{(r)}_3|^{\kappa} |n^{(r)}_4|^{\kappa}) \prod_r^{1, 2, 3, 4} \quad \text{where}\quad
\prod_r^{1, 2, 3, 4} &:= \frac{1}{ \prod_{i=1}^4 |n_i^{(r)}|^{\kappa} }. \label{def:Pkk}
\end{align}

We remark that by the resonance relations \eqref{def:Abhwt} and \eqref{def:Aw} we have $\Delta_{n^{(r)}_{1}, n^{(r)}_{2}}=\Delta_{n^{(r)}_{4}, n^{(r)}_{3}}$.
Then we can express the right hand side of \eqref{def:d12N} in terms of the $P_r$'s in the following way:
\[
d_{i j}=\frac{3}{32 \mathtt{g}}\, P_{i} P_j.
\]

Then, the determinant  of the matrix $\mathcal{D}$ in \eqref{def:MatrixD} is of the form
\begin{equation*}

\det(\mathcal{D}) = \left(\frac{3}{32\mathtt{g} }\right)^{N-1}\left( \prod_{k=1}^{N-1} P_k \right) \; \text{det} \;
\begin{pmatrix}
P_N + \sum_{j \neq 1}  P_j & -P_2 & \ldots & -P_{N-1} \\
-P_1 & \ddots & \vdots & \vdots \\
\vdots & \vdots & \ddots &  -P_{N-1} \\
-P_1 & \ldots & -P_{N-2} & P_{N} + \sum_{j \neq N-1} P_j
\end{pmatrix}.

\end{equation*}
This determinant can be written as
\begin{equation}\label{detHessianNP2}
\det(\mathcal{D}) = \left(\frac{3}{32\mathtt{g} }\right)^{N-1}\left( \prod_{k=1}^{N} P_k \right) \left(\sum_{k=1}^{N}P_k\right)^{N-2}.
\end{equation}
Indeed, it is enough to modify the matrix in two steps. First replace the last column by the sum of all columns. Then, the last column is the vector with all components equal to $P_N$. Second, subtract the last row to the other rows. Then, it is very easy to obtain \eqref{detHessianNP2}.

Recall  that in the proof of Lemma \ref{lemma:WaveBeamMelnikovCond} we have shown that the sets $\Lambda$ of Proposition \ref{prop:LambdaSetWaveHom} satisfy
\[
 |n_1^{(r)}|^{\kappa} |n^{(r)}_2|^{\kappa}-|n^{(r)}_3|^{\kappa} |n^{(r)}_4|^{\kappa}\neq 0
\]
(see \eqref{def:CondExtraForPk}). Moreover, in Proposition \ref{prop:LambdaSetWaveHom} it shown that  they also satisfy \eqref{eq:Wave}. These three properties imply
\begin{equation}\label{def:Pk0}
 P_k\neq 0\quad \text{ for all}\quad k=1,\ldots, N
\end{equation}

Therefore, by \eqref{detHessianNP2}, to prove $\det(\mathcal{D}) \neq0$, it only remains to check that
\begin{equation}\label{cond:sum}
 \sum_{k=1}^{N}P_k\neq 0.
\end{equation}
If the set $\Lambda$ obtained in Proposition \ref{prop:LambdaSetWaveHom} satisfies this property, the proof is complete. Now, we show  that if the set $\Lambda$ obtained in these propositions  satisfies $\sum_{k=1}^{N}P_k= 0$, one can modify it slightly so that the new one satisfies \eqref{cond:sum}.
Assume thus that  $\Lambda$ satisfies $\sum_{k=1}^{N}P_k= 0$ and  \eqref{def:Pk0}. Then, we  modify the first resonant tuple $(n_1^{(1)},n_2^{(1)},n_3^{(1)},n_4^{(1)})$ to obtain a set $\Lambda\subset\mathbb{Q}^2$ which satisfies \eqref{cond:sum}. We consider the family of resonant tuples in $\mathbb{Q}^2$, given by
\[
 (\lambda n_1^{(1)}, \lambda n_2^{(1)}, \lambda n_3^{(1)}, \lambda n_4^{(1)}), \qquad \lambda\in\mathbb{Q}\setminus\{0\}.
\]
Then, by \eqref{def:Pkk},
\[
 P_1(\lambda n_1^{(1)}, \lambda n_2^{(1)}, \lambda n_3^{(1)}, \lambda n_4^{(1)})=\lambda^{-\kappa} P_1(n_1^{(1)},n_2^{(1)},n_3^{(1)},n_4^{(1)}).
\]
Then, since $P_1\neq 0$,  $P_1$ is strictly decreasing in $\lambda$ and therefore $\sum_{k=1}^{N}P_k= 0$ can only happen for $\lambda=1$. Thus, one can modify the first rectangle by taking $\lambda\in\mathbb{Q}$ arbitrarily close to 1 and then blowing up the $N$ rectangles so that the new rectangles belong to $\mathbb{Z}^2$. It is clear that with this modification (for $\lambda$ close enough to 1) the properties in Proposition \ref{prop:LambdaSetWaveHom} are still satisfied.

 \subsection{End of the proof of Theorem \ref{thm:traveling_periodic_beating}}
  Lemmas \ref{lemma:WaveBeamMelnikovCondN}, \ref{lemma:HartreeMelnikovCondN} imply that the assumptions required by Proposition \ref{prop:SplittingMulti} hold. Then we can use Proposition \ref{prop:diffusion_orbits} to deduce dynamical results on the resonant models  of equations \eqref{Wave}, \eqref{Beam} and \eqref{Hartree}.

 Let us fix $N\geq 2$, $k\gg 1$ and a sequence $ \omega_1, \dots, \omega_k $ with $\omega_p\in \{1, \dots, N\}$ for $k=1\ldots k$.
 We apply Proposition \ref{prop:diffusion_orbits} choosing $\nu_{\ell}=\varepsilon$ for all $\ell=1, \dots, k$. Then there exist $T>0$ and  an orbit
 \[
 \gamma(t)=(\Psi_1(t), \dots, \Psi_N(t), K_1(t), \dots, K_N (t)), \quad t\in [0, T]
 \]
 of the Hamiltonian $\mathcal{H}$ (see \eqref{redhamN}) which has the following behavior:

There exists  some $\al_p,\beta_p$ satisfying $\al_p<\beta_p<\al_{p+1}$
such that, if one splits the time interval as $[0, T]=\calI_1\cup \calJ_{1, 2}\cup \calI_2\cup \calJ_{2, 3}\cup\dots \cup \calJ_{k-1, k }\cup \calI_k$
with
\[
\calI_p=[\al_p,\beta_p],\quad \calJ_{p,p+1}=[\delta^{-2}\beta_p,\delta^{-2}\al_{p+1}],
\]
the orbit $\gamma(t)$ has two different regimes

\begin{itemize}
 \item \emph{Beating regime:} For $t\in [\alpha_{p}, \beta_{p}]$,  $\gamma(t)$ belongs to an $\varepsilon$-neighborhood of the periodic orbit $\mathtt{P}_{\varepsilon, h, \omega_{p}}$. The orbit $\ga(t)$ spends $\OO(\log\varepsilon)$-time inside this neighborhood and then it leaves it.
  \item \emph{Transition regime:} For $t\in (\beta_{p}, \alpha_{p+1})$,  the orbit $\gamma(t)$ shadows the heteroclinic connection between two hyperbolic periodic orbits $\mathtt{P}_{\varepsilon, h, \omega_{p}}$ and $\mathtt{P}_{\varepsilon, h,  \omega_{p+1}}$.

 \end{itemize}
By  \eqref{def:timeparapo}, the time parameterization of the periodic orbit  $\mathtt{P}_{\varepsilon, h, \omega_{p}}$ satisfies
\[
 K_p(t)=Q(t), \qquad K_i(t)=0\quad \text{for}\quad i\neq p,
\]
where $Q(t)$ is a periodic orbit. Then, the orbit $\gamma(t)$ satisfies that for $t\in [\alpha_{p}, \beta_{p}]$
 \[
|K_p(t)-Q(t-t_p)|\le\eps,\qquad  | K_i(t) |^2\le \varepsilon \qquad \forall  \quad i\neq p,
 \]

for some $t_p>0$.

In the time interval $(\beta_{\ell}, \alpha_{\ell+1})$, the travel along the heteroclinic connection implies that all the actions $|K_i|^2$ experience a change of order $\OO(1)$ (see the proof of Proposition \ref{prop:SplittingMulti}).

By the symplectic reduction performed in Section \ref{sectionsymplecticreduction} there exists $r(t)$ solution of $H_{\Res}$ in \eqref{Hres} with Fourier support $\Lambda$ such that the actions $|r_{n_1^{(\omega_i)}}|^2$ satisfy
\[|r_{n_1^{(\omega_i)}}(t)|^2=| K_{i}(t) |^2\qquad \text{for}\qquad t\in [0,T].\]
This can be seen using Remark \ref{rem:Kcartcoord}, which gives also the behavior of the other actions.

Since the solutions of $H_{\Res}$ are invariant under the scaling \eqref{def:scaling}, we can consider $r^{\delta}(t):=\delta r(\delta^2 t)$. Then, $r^{\delta}(t)$ is also a solution of $H_{\Res}$ for  $t\in [0, \delta^{-2} T]$.

Now it only remains to obtain an orbit for the equations \ref{Wave}, \ref{Beam} and \ref{Hartree} which is close (up to certain changes of coordinates) to $r^{\delta}(t)$. First step is to apply Proposition \ref{approximationargument}. It ensures that there exists $0<\delta_2\ll 1$ such that for all $\delta\in (0, \delta_2)$, there exists a solution $w(t)$ of $H\circ \Gamma\circ \Psi=H_{\Res}+\mathcal{R}'$   such that $w(t)=r^{\delta}(t)+\widetilde{R}(t)$ with $\widetilde{R}(0)=0$, $\| \widetilde{R}(t) \|_{\rho}\lesssim \delta^2$ for $t\in [0, \delta^{-2} T]$. We note that,  by Item $(ii)$ of Proposition \ref{WBNF}, the Birkhoff map $\Gamma$ is $\delta^3$-close to the identity. Finally the transformations \eqref{rotatingcoord1} and \eqref{fase} preserve the modulus of the Fourier coefficients. The last change of coordinates that one has to apply (for the Wave \eqref{Wave} and Beam \eqref{Beam} equations) is  passing from complex coordinates \eqref{def:complexcoordinates} to the original ones. We remark that by \eqref{eq:Wave} if $n_i\in\Lambda$ then $-n_i\notin \Lambda$. Thus
  \[
  u_{n_i}=\frac{1}{\sqrt{2 |j|}} \Psi_{n_i} \qquad n_i\in\Lambda.
  \]

\end{proof}

config: arxiv


text:
\begin{document}
\title[The 2-point function for 3-loop superstrings]{The vanishing of two-point functions for three-loop superstring scattering amplitudes}
\author{Samuel Grushevsky}
\address{Mathematics Department, Princeton University, Fine Hall,
Washington Road, Princeton, NJ 08544, USA. }
\thanks{Research is supported in part by National Science Foundation under the grant DMS-05-55867.}
\email{sam@math.princeton.edu}
\author{Riccardo Salvati Manni}
\address{Dipartimento di Matematica, Universit\`a ``La Sapienza'',
Piazzale A. Moro 2, Roma, I 00185, Italy}
\email{salvati@mat.uniroma1.it}
\date{\today}

\begin{abstract}
In this paper we show that the two-point function for the three-loop chiral superstring measure ansatz proposed by Cacciatori, Dalla Piazza, and van Geemen \cite{CDPvG} vanishes. Our proof uses the reformulation of ansatz given in \cite{G}, theta functions, and specifically the theory of the $\Gamma_{00}$ linear system on Jacobians introduced by van Geemen and van der Geer \cite{vgvdg}.

At the two-loop level, where the amplitudes were computed by D'Hoker and Phong \cite{DHP1,DHPa,DHPb,DHPc,DHPd,DHPe}, we give a new proof of the vanishing of the two-point function (which was proven by them). We also discuss the possible approaches to proving the vanishing of the two-point function for the proposed ansatz in higher genera \cite{G,SM,CDPvG2}.
\end{abstract}
\maketitle

\section{Introduction}
The problem of computing the superstring measure explicitly for arbitrary genus of the worldsheet was begun by the work of
Green and Schwarz \cite{GS}, who performed the integration over supermoduli and gave an explicit formula in genus 1. D'Hoker and Phong in a series of papers \cite{DHP1,DHPa,DHPb,DHPc} introduced a gauge-fixing procedure and computed from first principles the genus 2 superstring measure, verifying  that it satisfied the physical constraints, eg.~the vanishing of the 1,2,3-point functions. They also proposed in \cite{DHPI,DHPII} to search for an ansatz for the superstring measure in arbitrary genus as the product of the bosonic measure and a modular form.

The ansatz for three-loop measure in this form was then proposed by Cacciatori, Dalla Piazza, and van Geemen in \cite{CDPvG}. The genus $g\le 3$ ansatze were reformulated in terms of syzygetic subspaces by the first author in \cite{G}, where an ansatz for general genus was proposed, under the assumption on holomorphicity of certain $2^r$-roots. Cacciatori, Dalla Piazza, and van Geemen in \cite{CDPvG2} give the genus 4 ansatz in terms of quadrics in the theta constants. The second author in \cite{SM} showed that the proposed ansatz is holomorphic in genus 5. Dalla Piazza and van Geemen in \cite{DPvG} proved the uniqueness of the modular form in genus 3 satisfying the factorization constraints. Morozov in \cite{M1} surveyed this work and gave an alternative proof that factorization constraints are satisfied for the ansatz; in \cite{M2} he has also investigated the 1,2,3-point functions of the proposed ansatz, proving under certain non-trivial mathematical assumption that they vanish on the hyperelliptic locus.

\smallskip
In this paper we use the techniques of theta functions, and especially the $\Gamma_{00}$ sublinear system of the linear system $|2\Theta|$ introduced by van Geemen and van der Geer \cite{vgvdg} to prove the vanishing of the 2-point function in genus 3. We also obtain a new proof of the vanishing of the 2-point function in genus 2.

\section{Notations and definitions}
We denote by $\A_g$ the moduli space of complex principally polarized abelian varieties of dimension $g$, and by $\H_g$ the Siegel upper half-space of symmetric complex matrices with positive-definite imaginary part, called period matrices. The space $\H_g$ is the universal cover of $\A_g$, with the deck group $\Sp(2g,\Z)$, so that we have $\A_g=\H_g/\Sp(2g,\Z)$ for a certain action of the symplectic group. A function $f:\H_g\to\C$ is called a (scalar) modular form of weight $k$ with respect to a subgroup $\Gamma\subset\Sp(2g,\Z)$ if
$$
 f(\gamma\circ\tau)=\det(C\tau+D)^kf(\tau)\qquad\forall\gamma\in\Gamma, \forall\tau\in\H_g,
$$
where $C$ and $D$ are the lower blocks if we write $\gamma$ as four $g\times g$ blocks.

For a period matrix $\tau\in\H_g$ the principal polarization $\T_\tau$ on the abelian variety $A_\tau:=\C^g/(\Z^g+\tau\Z^g)$ is the divisor of the theta function
$$
  \theta(\tau,z):=\sum\limits_{n\in\Z^g}\exp(\pi i (n^t\tau n+2n^t
  z)).
$$
Notice that for fixed $\tau$ theta is a function of $z\in\C^g$, and its automorphy properties under the lattice $\Z^g+\tau\Z^g$ define the bundle $\T_\tau$.

Given a point of order two on $A_\tau$, which can be uniquely
represented as $\frac{\tau\e+\de}{2}$ for $\e,\de\in \F^g$ (where $\F=\lbrace 0,1\rbrace$ is the additive group), the associated theta function with characteristic is
$$
  \tt\e\de(\tau,z):=\sum\limits_{n\in\Z^g}\exp(\pi i ((n+\e)^t\tau
  (n+\e)+ 2(n+\e)^t( z+\de)).
$$
As a function of $z$, $\tt\e\de$ is odd or even depending on whether
the scalar product $\e\cdot\de\in\F$ is equal to 1 or 0, respectively. The theta function with characteristic is the generator of the space of sections of the bundle $\T_\tau+\frac{\tau\e+\de}{2}$ (where we have implicitly identified the principally polarized abelian variety with its dual, and think of points as bundles of degree 0). Thus the square of any theta function with characteristic is a section of $2\T_\tau$, and the basis for the space of sections of this bundle is given by theta functions of the second order
$$
 \T[\e](\tau,z):=\tt\e0(2\tau,2z)
$$
for all $\e\in\F^g$. Riemann's addition formula is an explicit expression of the squares of theta functions with characteristics in this basis:
\begin{equation}\label{riem}
 \tt\e\de(\tau,z)^2=\sum\limits_{\sigma\in\F^g}(-1)^{\de\cdot\sigma} \T[\sigma](\tau,0)\T[\sigma+\e](\tau,z).
\end{equation}

Theta constants are restrictions of theta functions to $z=0$; thus all theta constants with odd characteristics vanish identically in $\tau$, while theta constants with even characteristics and all theta constants of the second order do not vanish identically. All theta constants with characteristics are modular forms of weight one half with respect to a certain normal subgroup of finite index $\Gamma(4,8)\subset\Sp(2g,\Z)$, while all theta constants of the second order are modular forms of weight one half with respect to a bigger normal subgroup $\Gamma(2,4)\supset\Gamma(4,8)$.

\smallskip
Theta constants with characteristics are not algebraically
independent, and satisfy a host of  algebraic identities, some of which follow from Riemann's addition formula. However, the theta constants of the second order are algebraically independent for $g=1,2$, and the only relation among them in genus 3 is of degree 16, and has been known classically. It is discussed in detail in \cite{vgvdg} --- here we give the explicit formula for easy reference. Indeed, a special case of Riemann's quartic addition theorem in genus 3 is the following identity for theta constants (where we suppress the argument $\tau$)
$$
 \tt{0\ 0\ 0}{0\ 0\ 0}\tt{0\ 0\ 0}{1\ 0\ 0}
     \tt{0\ 0\ 0}{0\ 1\ 0}\tt{0\ 0\ 0}{1\ 1\ 0}=
$$
$$
\tt{0\ 0\ 1}{0\ 0\ 0}\tt{0\ 0\ 1}{1\ 0\ 0}
     \tt{0\ 0\ 1}{0\ 1\ 0}\tt{0\ 0\ 1}{1\ 1\ 0}+
\tt{0\ 0\ 0}{0\ 0\ 1}\tt{0\ 0\ 0}{1\ 0\ 1}
     \tt{0\ 0\ 0}{0\ 1\ 1}\tt{0\ 0\ 0}{1\ 1\ 1}.
$$
If we denote the three terms in this relations by $r_i$, so that the relation is $r_1=r_2+r_3$, then multiplying the 4 ``conjugate'' relations $r_1=\pm r_2\pm r_3$ yields the identity
\begin{equation}\label{F}
  F:=r_1^4+r_2^4+r_3^4-2r_1^2r_2^2-2r_2^2r_3^2-2r_3^2r_1^2=0.
\end{equation}
Notice that $F$ is a polynomial of degree 8 in the squares of theta constants with characteristics, and thus by applying Riemann's addition formula (\ref{riem}) $F$ can be rewritten as a polynomial of degree 16 in theta constants of the second order. We refer to \cite{BL,I} for details on theta functions and modular forms, and the current knowledge about the ideal of relations among theta constants of the second order for $g>3$ (which is not known completely even for $g=4$).

\section{The linear system $\Gamma_{00}$}
In this section we review the definition and some facts about the linear system $\Gamma_{00}\subset|2\T|$ introduced and studied in \cite{vgvdg}. We refer to that paper for details, as well as to \cite{vg,G2,Iz} for a review and results on the importance of the linear system $\Gamma_{00}$ for the Schottky problem of characterizing Jacobians.

The linear system $\Gamma_{00}\subset |2\T|$ is defined to consist of all sections vanishing to order at least four at the origin. Since all sections of $2\T$ are even, this is equivalent to the value and the second derivatives $\p_{z_i}\p_{z_j}$ vanishing at zero. These conditions turn out be independent, when $(A_{\tau}, \Theta)$ is an indecomposable ppav (i.e.~not isomorphic to a product of lower-dimensional ppavs). In this case the matrix
$$
 \left(\begin{matrix} \T[\e_1](\tau,0)&\frac{\p \T[\e_1](\tau,0)}{\p\tau_{11}}&\ldots&\frac{\p \T[\e_1](\tau,0)}{\p\tau_{gg}}\\
  \vdots&\vdots&\vdots&\vdots\\
 \T[\e_{2^g}](\tau,0)&\frac{\p \T[\e_{2^g}](\tau,0)}{\p\tau_{11}}&\ldots&\frac{\p \T[\e_{2^g}](\tau,0)}{\p\tau_{gg}}\end{matrix}\right)
$$
has rank $\frac{g(g+1)}{2}+1$, cf\cite{Sas}
and thus (\cite{vgvdg}, proposition 1.1)
\begin{equation}\label{dimgamma00}
 \dim\Gamma_{00}=\dim |2\T|-1-\sum\limits_{1\le i\le j\le g}1=2^g-1-\frac{g(g+1)}{2}.
\end{equation}
Thus the linear system $\Gamma_{00}$ is zero for $g\ge 2$, has dimension 1 for $g=3$, and higher dimension for all other genera.\bigskip

The above description leads to a simple construction of a basis for  the space $\Gamma_{00}$.

\begin{prop}
Let $\tau_0$ be an irreducible point of $\H_g$ (i.e. corresponding to indecomponsable ppav). Denote $N:=1+\frac{g(g+1)}{2}$, and choose $\e_1,\ldots,\e_N\in\F^g$ such that the the modular form
$$
 g_{\e_1,\ldots,\e_N}(\tau):=\det\left(\begin{matrix}\T[\e_1](\tau,0)&\frac{\p \T[\e_1](\tau,0)}{\p\tau_{11}}&\ldots&\frac{\p \T[\e_1](\tau,0)}{\p\tau_{gg}}\\
   \vdots&\vdots&\vdots\\
\T[\e_N](\tau,0)&\frac{\p \T[\e_N](\tau,0)}{\p\tau_{11}}&\ldots&\frac{\p \T[\e_N](\tau,0)}{\p\tau_{gg}}\end{matrix}\right)
$$
does not vanish at $\tau_0$. Then the sections
$$ f_\e(\tau_0,\, z):=
\det\left(\begin{matrix}\T[\e_1](\tau_0,z)&\T[\e_1](\tau_0,0)&\frac{\p \T[\e_1](\tau_0,0)}{\p\tau_{11}}&\ldots&\frac{\p \T[\e_1](\tau_0,0)}{\p\tau_{gg}}\\
  \vdots&\vdots&\vdots&\vdots\\
  \T[\e_N](\tau_0,z)&\T[\e_N](\tau_0,0)&\frac{\p \T[\e_N](\tau_0,0)}{\p\tau_0\tau_{11}}&\ldots&\frac{\p \T[\e_N](\tau_0,0)}{\p\tau_{gg}}\\
 \T[\e](\tau_0,z)&\T[\e](\tau_0,0)&\frac{\p \T[\e](\tau_0,0)}{\p\tau_{11}}&\ldots&\frac{\p \T[\e](\tau_0,0)}{\p\tau_{gg}}
 \end{matrix}\right),
$$
for $\e\in\F^g\setminus\lbrace\e_1,\ldots,\e_N\rbrace$ form a basis of $\Gamma_{00}\subset|2\T_{\tau_0}|$.
\end{prop}
\begin{proof}
The proof is a simple linear algebra argument that we recall  for completeness. First note that  $ f_\e(\tau_0,\,z)$ belongs to $\Gamma_{00}$, as the determinant and all the second $z$-derivatives (equal to the first $\tau$-derivatives by the heat equation) vanish for $z=0$, as two of the columns of the matrix become identical. It thus remains to show that the functions $f_\e$ for various $\e$ are linearly independent. Indeed, recall that theta functions of the second order form a basis of sections of $2\T$, and now note that the basis element $\T[\e](\tau_0, z)$  enters only the expression  of  $f_\e(\tau_0,  z)$, and that with  non-zero coefficient  $g_{\e_1,\ldots,\e_N}(\tau_0)$.
\end{proof}

\begin{rem}
It can be shown that on the open sets $\lbrace g_{\e_1,\ldots,\e_N}(\tau)\neq 0\rbrace$ the coefficients of the basis vectors are in fact modular forms of weight
$g+1+N/2$, see \cite{GSM}. In particular when $g=3$ we have a global expression of the unique section $f(\tau, z)$ of the space $\Gamma_{00}$.
\end{rem}

\begin{rem}
Observe that if the  period matrix $\tau$ is decomposable, then the dimension of $\Gamma_{00} $ increases; however, a basis can still be constructed by using the same method.
\end{rem}

There exists another  method  for constructing elements of $\Gamma_{00}$ --- it is described in \cite{vgvdg}, and is as follows. Suppose $I$ is an algebraic relation among theta constants of the second order (in genus $g$). This is to say, suppose $I\in\C[x_{0\ldots0},\ldots, x_{1\ldots 1}]$ is a polynomial in $2^g$ variables such that for any $\tau\in\H_g$ we have $I(\T[\e](\tau))=0$. Then the function
$$
 f_I(z):=\sum\limits_{\e\in\F^g}\frac{\p I}{\p x_\e}\left(\T[0\ldots0](\tau,0),\ldots, \T[1\ldots1](\tau,0)\right)
 \, \T[\e](\tau,z)
$$
lies in $\Gamma_{00}\subset|2\T_\tau|$. Indeed, since $I$ vanishes identically on $\H_g$, by Euler's formula we have $f_I(0)=0$. Moreover, by the heat equation
$$
 2\pi i(1+\de_{j,k})\frac{\p^2 f_I}{\p z_j\p z_k}|_{z=0} =\sum\limits_{\e\in\F^g}\frac{\p I}{\p x_\e}
 \, \frac{\p \T[\e](\tau)}{\p\tau_{jk}}=\frac{\p I(\T[0\ldots0],\ldots\T[1\ldots1])}{\p\tau_{jk}},
$$
which is zero since $I$ vanishes identically on $\H_g$, and thus its derivative in any direction is also zero. In \cite{vgvdg}, proposition 1.2 it is shown that as $I$ ranges over the ideal of relations among theta constants, the functions $f_I$ generate the linear system $\Gamma_{00}$. Since for $g\ge 4$ the ideal of algebraic relations among theta constants of the second order is not completely known, for $g\ge 4$ this method does not yield a complete description of the basis of $\Gamma_{00}$. However, the geometry of these relations is intriguing, and this methods produces elements of $\Gamma_{00}$ with coefficients algebraic in theta constants, rather than involving their derivatives as well.

\section{The proposed ansatz for the superstring measure}
An ansatz for the 3-loop superstring measure was proposed in \cite{CDPvG}. The reformulation of this ansatz in terms of products of theta constants with characteristics in a syzygetic subspace given in \cite{G} is as follows. For any $i=0\ldots g$ define
\begin{equation}\label{defG}
  G_i^{(g)}\tch\e\de(\tau):=\sum\limits_{V\subset\F^{2g};\, \dim V=i} \ \ \prod\limits_{\tch\alpha\beta\in V}\tt{\e+\alpha}{\de+\beta}(\tau)^{2^{4-i}}.
\end{equation}
Notice that since any $i$-dimensional linear subspace contains zero, all products will contain $\tt\e\de$. Since all odd theta constants vanish identically, it is enough to sum over the even cosets of syzygetic $i$-dimensional subspaces containing $[\e,\de]$, see \cite{G,SM}.

To simplify notations, we write $m:=[\e,\de]\in\F^{2g}$ for characteristics and similarly write $\t_m:=\tt\e\de$. Then the proposed ansatz for the superstring measure is the product of the bosonic measure (which is a form on $\M_g$) and, for any even characteristic $m$, the expression
\begin{equation}\label{defXi}
 \Xi_m^{(g)}:=\sum\limits_{i=0}^g(-1)^i2^{\frac{i(i-1)}{2}} G_i^{(g)}[m]
\end{equation}
which is a modular form of weight 8 with respect to a subgroup of $\Sp(2g,\Z)$ conjugate to $\Gamma(1,2)$. In particular for genus 3 we have
$$
  \Xi_m^{(3)}:=G_0^{(3)}[m]-G_1^{(3)}[m]+2G_2^{(3)}[m]-8G_3^{(3)}[m].
$$
In \cite{SM} it is shown that the sum $\sum_m \Xi_m^{(3)}$ is a non-zero multiple of the modular form $F$ given by (\ref{F}), and thus vanishes identically on $\H_3$.

From definition (\ref{defG}) of the summands $G_i^{(g)}[m]$ of the measure $\Xi_m^{(g)}$ it follows that $G_i^{(g)}[m]$ is a polynomial in the squares of theta constants with characteristics for $i\le 3$, divisible by $\t_m^2(\tau)$. Since this is the only kind of summands appearing in the definition of $\Xi_m^{(g)}$ for $g\le 3$, by applying Riemann's addition formula (\ref{riem}) we get
\begin{prop}\label{expressible}
For $g\le 3$ the modular form $\Xi_m^{(g)}$ defined by (\ref{defXi}), and moreover the ratio $\Xi_m^{(g)}/\t_m^2(\tau,0)$ are both polynomials in theta constants of the second order, of degrees 16 and 14, respectively.
\end{prop}

\section{The vanishing of the 2-point function}
We recall (see \cite{DHPa} for explicit formulas) that the vanishing of the cosmological constant reduces to the identity $\sum_m \Xi_m^{(g)}=0$ (proven for the proposed ansatz for $g\le 4$ in \cite{SM}), and this also implies the vanishing of the 1-point function, while as shown in \cite{DHPe}the vanishing of the two-point function is equivalent to the vanishing of
$$
 \sum_m \Xi_m^{(g)} S_m(a,b)^2
$$
for any points $a,b$ on the Riemann surface (thought of as embedded into its Jacobian), where $S_m$ is the Sz\"ego kernel
$$
 S_m(a,b):=\frac{\t_m(a-b)}{\t_m(0)E(a,b)},
$$
with $E$ being the prime form on the Riemann surface.

Since the prime form does not depend on $m$, it is a common factor in all summands above, and thus does not matter for the vanishing of the 2-point function, so the vanishing of the 2-point is equivalent to the vanishing of
$$
 X_2(a,b):=\sum_m \frac{\Xi_m^{(g)}(\tau)}{\t_m^2(\tau,0)} \t_m(\tau,a-b)^2
$$
where $\tau$ is the period matrix of the Jacobian $Jac(C)$ of a Riemann surface $C$, and $a,b\in C\subset Jac(C)$ are arbitrary.
We will now relate the vanishing of the 2-point function and the $\Gamma_{00}$ linear system. Set
\begin{equation}\label{X_2}
 X_2(\tau,z):=\sum_m\frac{\Xi_m(\tau)}{\t_m^2(\tau,0)}\t_m(\tau,z)^2
\end{equation}
and note that this function is a section of $|2\T_\tau|$. The vanishing of the 2-point function is then equivalent to $X_2(z)$ vanishing along the surface $C-C\subset Jac(C)$. By proposition 2.1 in \cite{vgvdg} and the subsequent remark, a section of $|2\T|$ vanishes along the surface $C-C$ if and only if it lies in $\Gamma_{00}$. We thus get
\begin{thm}\label{thm}
The 2-point function for the proposed superstring measure ansatz vanishes (for genus $g$) if and only if for the period matrix $\tau$ of any Jacobian of a Riemann surface of genus $g$ the section $X_2(\tau,z)$ of $2\T_\tau$ defined above lies in $\Gamma_{00}$.
\end{thm}

Since for $g=1,2$ the linear system $\Gamma_{00}$ is zero, the vanishing of the two-point function is equivalent to $X_2(\tau,z)$ vanishing identically in $z$ and $\tau$ for $g\le 2$. If we write out $X_2(\tau,z)$ as a linear combination of the basis for sections of $2\T_\tau$ given by theta functions of the second order
\begin{equation}\label{X2}
 X_2(\tau,z)=\sum c_\e(\tau)\T[\e](\tau,z),
\end{equation}
then $X_2$ vanishes identically if and only if each $c_\e(\tau)$ vanishes identically. This allowes us to recover the result of D'Hoker and Phong in genus 2.
\begin{prop}
The 2-point function for the proposed superstring ansatz vanishes identically for $g\le 2$.
\end{prop}
\begin{proof}
Let us apply Riemann's addition formula (\ref{riem}) to the definition (\ref{X_2}) of the two-point function to rewrite it in terms of theta functions of the second order (notice that the only term depending on $z$ is $\t_m(\tau,z)^2$, and we apply the addition formula to it as well). Notice that by proposition \ref{expressible} the coefficients $c_\e$ in (\ref{X2}) obtained in this way are explicit polynomials of degree 15 in theta constants of the second order, and since there are no algebraic relations among theta constants of the second order for $g\le 2$, one needs to verify that they all polynomials $c_\e$ are zero. This can be done on a computer (we used Maple). Note that the computation can be made easier by noting that since $X_2(\tau,z)$  has a transformation formula  with respect to the entire symplectic group, (i.e.~it is a Jacobi form)  and the coefficients $c_\e(\tau)$ are permuted under the action  a suitable subgroup $\Sp(2g,\Z)$ that acts monomially on the theta constants of the second order, it is enough to check that just one of $c_\e$ is the zero polynomial.
\end{proof}

In the case of $g=3$, recall from (\ref{dimgamma00}) that the space $\Gamma_{00}$ is one-dimensional, and by the results of \cite{vgvdg} we know that it is generated by
$$
 F_2:=\sum\limits_{\e\in\F^3}\frac{\p F}{\p x_\e}\left(\T[000](\tau,0),\ldots,\T[111](\tau,0)\right) \T[\e](\tau,z),
$$
where we recall that $F$, given by  (\ref{F}), is the only polynomial relation of degree 16 among the 8 theta constants of the second order for $g=3$.

\begin{prop}
For any $\tau\in\H_3$ the sections $F_2$ and $X_2$ of $2\T_\tau$ are proportional; more precisely $F_2=-\frac{14}{5}X_2$.
\end{prop}
\begin{proof}
We have explicit expressions for $F_2$ and $X_2$ as linear combinations of the basis of the sections of $2\Theta$ given by the second order theta functions. Thus what we need to verify is that the coefficient in $5F_2+14X_2$ of any $\T[\e](z)$ is equal to zero. This coefficient is a polynomial of degree 15 in theta constants of the second order and can be verified to be zero using Maple (since the only relation among theta constants of the second order is of degree 16, a polynomial in theta constants of the second order of degree 15 vanishes identically only if it is zero). Notice that by modularity it is again enough to verify that the coefficient of $\T[000](z)$ in $5F_2+14X_2$ is equal to zero.
\end{proof}

\begin{thm}
The 2-point function for the proposed ansatz for the 3-loop superstring measure vanishes identically.
\end{thm}
\begin{proof}
By the above proposition we see that for any $\tau\in\H_3$ the function $X_2$, being a constant multiple of $F_2$, lies in the linear system $\Gamma_{00}\subset |2\T_\tau|$. By theorem \ref{thm} this is equivalent to the identical vanishing of the two-point function.
\end{proof}

\begin{rem}
We  note  that the global section $F_2$ is proportional also to the global section
$f(\tau,\, z)$. Really we have that
$$F_2(\tau, z)=c(\tau) f (\tau,\, z)$$
 for any irreducible $\tau \in \H_g$. Moreover $c(\tau)$ results to be a modular function with respect to $\Sp(6,\Z)$  that is  regular on the set of irreducible point, so it is regular everywhere ( modular form) and hence it is a non zero constant. This identity produces eight non trivial identities expressing  each jacobian determinant  $ g_{\e_1,\ldots,\e_7}(\tau)$ as a polynomial of degree 15 in the theta constants
 $\T[\sigma](\tau,0)$

 \end{rem}

\section{Conclusion}
There are two generalizations that it is natural to try to prove.

\smallskip
First, one could ask whether the vanishing of the 2-point function can be obtained for the proposed in \cite{G} ansatz in higher genera. For genus 4 the ansatz is also given in \cite{CDPvG2} and is manifestly holomorphic in either formulation. The holomorphicity of the ansatz in genus 5 was proven in \cite{SM}, and thus it is natural to ask whether the 2-point function vanishes for $g\le 5$. By theorem \ref{thm} we know that this is equivalent to $X_2$ lying in the linear system $\Gamma_{00}$. However, already for genus 4 the geometry of the situation is much more complicated: instead of just one relation $F$ in genus 3 the ideal of relations among theta constants of the second order in genus 4 is unknown.
 and an explicit basis for $\Gamma_{00}$ is unknown for $g=4$ .

Moreover, it could be that here the fact that we are working on the moduli space of curves $\M_4$ rather than $\A_4$ plays a role --- the geometry of $\Gamma_{00}$ depends on this, see \cite{Iz}.

\smallskip
Second, one could try to prove the vanishing of the 3-point function. As shown in \cite{DHPe} for genus 2, this is equivalent to proving that the sum
$$
 \sum\limits_m \Xi_m^{(g)} S_m(a,b)S_m(b,c)S_m(c,a)
$$
vanishes. Using the explicit formula for the Sz\"ego kernel and canceling the $m$-independent factor, this is equivalent to the function
$$
 X_3(a,b,c):= \sum\limits_m \frac{\Xi_m(\tau)}{\t_m^3(\tau,0)} \t_m(a-b)\t_m(b-c)\t_m(c-a)
$$
vanishing identically for $a,b,c\in C$. However, in this case we do not know a natural function on $Jac(C)^{\times n}$ of which $X_3$ is a restriction, and there is no analog of the theory of the $\Gamma_{00}$ for more points. It seems that the identity among the third order theta functions obtained by Krichever in his proof of the trisecant conjecture (\cite{K}, formula (1.18)) may potentially be useful in reducing the vanishing of the 3-point to the vanishing of the 2-point function, but so far we have not been able to find an explicit way to do this.

\section*{Acknowledgements}
We are grateful to Eric D'Hoker and Duong Phong for introducing us to questions about the superstring scattering amplitudes and explanations regarding the conjectured properties of $N$-point functions. The computations for this paper were done using Maplesoft's Maple$^\copyright$ software.


TITLE: Law of total covariance inequality
QUESTION [1 upvotes]: The law of total variance: $$ \text{Var}(X) = \mathbb{E}(\text{Var}(X\mid Y)) + \text{Var}(\mathbb{E}(X\mid Y)).$$
There is also something called the law of total covariance: $$ \text{Cov}(X,Y) = \mathbb{E}(\text{Cov}(X,Y\mid Z)) + \text{Cov}(\mathbb{E}(X\mid Z),\mathbb{E}(Y\mid Z)).$$
The law of total variance gives us the following two inequalities, since the variance is non-negative: $$\text{Var}(X) \geq \mathbb{E}(\text{Var}(X\mid Y))$$ $$\text{Var}(X) \geq \text{Var}(\mathbb{E}(X\mid Y)).$$
Are the following true with the covariances like when dealing with variances?
$$ |\text{Cov}(X,Y)| \geq |\mathbb{E}(\operatorname{Cov}(X,Y\mid Z))|$$ $$ |\operatorname{Cov}(X,Y)| \geq |\operatorname{Cov}(\mathbb{E}(X\mid Z),\mathbb{E}(Y\mid Z))|$$

REPLY [1 votes]: If $\text{Cov}(X,Y)=0$ and the others are non-zero then this would be a counter-example to your covariance conjectures.  For example if there are four equally likely combinations of values:
 x    y    z
+1   -2   -5
-2   -1   -5
-1   +2   +5
+2   +1   +5

these are in fact $(x,y)$ vertices of an oblique  square with $z=x+3y$ but that is not significant
Then

$\text{Cov}(X,Y) = 0$
$\mathbb{E}(\text{Cov}(X,Y\mid Z)) = -\frac34$
$\text{Cov}(\mathbb{E}(X\mid Z),\mathbb{E}(Y\mid Z)) =+\frac34$


TITLE: Type theory for $\infty$-categories: how do we inhabit $\text{hom}$ types?
QUESTION [7 upvotes]: I've started reading Riehl and Shulman's A type theory for synthetic $\infty$-categories, which looks like it develops some beautiful theory, but I want to make sure I'm not misunderstanding some of the initial formalities. In particular, I want to make sure I understand the constructor for function types out of shapes.
For instance, recall $\Delta^1:\equiv\{t:\mathbb{2}\space\vert\space\top\}$ (for definitions and formal deduction rules I refer you to the linked paper). Now, as I understand it, given a type $A$, to define a non-dependent function $\Delta^1\rightarrow A$ (shorthand for $\langle \Delta^1\rightarrow A\space\vert\space^{\bot}_{\text{rec}_\bot}\rangle$), we need a term $a$ (possibly containing $t$ free) such that $\{t:\mathbb{2}\space\vert\space\top\}\vdash a:A$; then $(\lambda t.a):\Delta^1\rightarrow A$. The problem is, I don't see how to define any non-constant functions of this kind (ie functions where $a$ has non-trivial dependence on $t$).
I believe the only non-trivial rule for constructing type elements that depend on shape variables is the given recursion principle for disjunction of topes, which says that if $\{t:I\space\vert\space\varphi\}\vdash a_\varphi:A$, $\{t:I\space\vert\space\psi\}\vdash a_\psi:A$, and $\{t:I\space\vert\space\varphi\wedge\psi\}\vdash a_\varphi\equiv a_\psi$, then $\{t:I\space\vert\space\varphi\vee\psi\}\vdash \text{rec}^{\varphi, \psi}_\vee(a_\varphi, a_\psi):A$, with computation rules given by $\{t:I\space\vert\space\varphi\}\vdash \text{rec}^{\varphi, \psi}_\vee(a_\varphi, a_\psi)\equiv a_\varphi$ and $\{t:I\space\vert\space\psi\}\vdash \text{rec}^{\varphi, \psi}_\vee(a_\varphi, a_\psi)\equiv a_\psi$. So, for instance, defining non-trivial functions out of the boundary $\partial\Delta^1:\equiv \{t:\mathbb{2}\space\vert\space t\equiv 0\vee t\equiv 1\}$ is easy: just choose terms $a_0, a_1:A$, and then $\lambda t.\text{rec}^{t\equiv 0, t\equiv 1}_\vee(a_0, a_1):\partial\Delta^1\rightarrow A$.
However, I cannot see any way of writing $\top$ as a disjunction that would allow this recursion principle to be useful for maps out of $\Delta^1$. In particular, I do not understand how we would inhabit a $\text{hom}$ type with non-trivial elements; recall that given a type $A$ and terms $x, y:A$, we define $\text{hom}_A(x, y):\equiv\langle\Delta^1\rightarrow A\space\vert\space^{\partial\Delta^1}_{\text{rec}^{t\equiv 0, t\equiv 1}_\vee(x, y)}\rangle$. It seems to me that the only way we can inhabit this type (with only the rules given so far) is if $x\equiv y$, and in this case the only inhabitant we can construct is $\lambda t.x$. (Which is imagine is supposed to correspond to the "identity morphism" at $x$?)
As a first question, I would just like someone to confirm for me that this is a correct understanding. As a second question, if that is the case, this situation seems to me comparable to the fact that the only constructor for identity types in undirected HoTT is $\text{refl}$. Indeed, if I remember correctly, it is relatively consistent in Martin-Lof type theory without univalence for every type to be a set. However, once one has univalence, then one can begin constructing non-trivial paths – for instance, the path induced by the non-trivial auto-equivalence of $\mathbf{2}$. So, my question is whether there is anything analogous in Riehl and Shulman's theory. A priori can one exhibit a type with a non-trivial $\text{hom}$ type? And if not, is there a "directed analog" of the univalence axiom that would allow us to construct them?
Bear in mind that I haven't yet read the paper in any detail, so apologies if this is discussed somewhere within it – I want to make sure that I'm understanding these preliminaries before tackling the paper in full. Also, if possible, I would prefer answers that don't depend too heavily on the semantic interpretation of this theory in Segal spaces, as I am not well-versed in the latter. I do know basic definitions though so could (hopefully) follow if that language is necessary to answer the question properly.

REPLY [1 votes]: You're correct that in our type theory without additional axioms, there are no nontrivial closed terms in hom-types.  (Of course, by hypothesizing elements of hom-types, we can produce other elements of hom-types.)  Thus, the theory is only potentially a theory of $(\infty,1)$-categories, in the same way (as you said) that MLTT is potentially a theory of $\infty$-groupoids.  Directed univalence is a subject of current research; Cavallo-Riehl-Sattler have proven a form of it in the bisimplicial model, and Weaver-Licata have a constructive version in a bicubical model, but I don't think any of it has appeared publically yet.


TITLE: Naive questions on the ground states of Kitaev model
QUESTION [4 upvotes]: I got some naive questions on the ground states of honeycomb Kitaev model (with open boundary conditions):
(1) Consider a simple case that $J_x=J_y=0$, then the model reduces to $$H=J_z\sum_{z\text{ }\text{links}}S_i^zS_j^z.$$  It's obvious that $H$ has highly degenerate GSs (degeneracy$=2^N$, where $N$ is the number of unit cells), and in each GS configuration, every two spins connected by a $z$ link are aligned parallel (if $J_z<0$) or antiparallel (if $J_z>0$). Thus, each of these GSs corresponds to a spin configuration and is not a spin liquid (SL).
On the other hand, the Majorana exact solution tells us that the GS of $H$ is a gapped SL.
So, does the exact solution just single out one of the highly degenerate GSs, which happens to be a SL (some certain manner of superposition of the above $2^N$ spin configurations) ??
(2) Following question (1), if the exact solution does single out just one of the highly degenerate GSs of $H$, and how could this be?
According to Kitaev's discussion on P.19 in his paper, the GS is achieved by the vortex-free field conﬁguration, which follows from a theorem proved by Lieb.
But shouldn't an exact solution gives all the $2^N$ GSs ? Why only single out one which strongly contradicts the obviously correct GS degeneracy$=2^N$ ? Is there something wrong with applying the Lieb's theorem here ? I'm confused....

REPLY [2 votes]: I think I can answer my question now.
The Lieb's theorem here is a statement about which flux conﬁguration (or flux conﬁgurations) minimizes the ground state energy of the quadratic Hamiltonians with each corresponding flux conﬁguration/sector. In the case that $J_x=J_y=0$ (or more generic cases that $J_xJ_yJ_z=0$), the corresponding $x,y$ link terms are absence in the quadratic Hamiltonians, and the flux (and hence the flux conﬁguration) through each elementary plaquette is even not well defined. Therefore, the Lieb's theorem does not work in the case that $J_x=J_y=0$ (or more generic cases that $J_xJ_yJ_z=0$) while the theorem itself is still correct. And the Kitaev model at $J_x=J_y=0$ does have highly degenerate GSs.
On the other hand, in the cases that $J_xJ_yJ_z\neq 0$, the Lieb's theorem works, which states that only one flux conﬁguration (vortex-free field conﬁguration here) minimizes the ground state energy. Thus, the GS of Kitaev model is always unique (non-degenerate) as long as $J_xJ_yJ_z\neq 0$. Due to this uniqueness, the GS must preserve all the symmetries of the Kitaev Hamiltonian. And this should be one of the reasons for the name of spin liquid GS of Kitaev model. (And see Prof.Wen's comments there.)
Note that for the Kitaev model defined on a decorated honeycomb lattice, there are two distinct flux conﬁgurations (which are related by time-reversal) that minimize the ground state energy, and hence there are two degenerate chiral spin liquid GSs which breaks time-reversal symmetry while preserves lattice translational and spin symmetries due to the 'uniform' flux conﬁguration instead of the 'uniqueness' of flux conﬁguration in the honeycomb case.
Note: Note that for the honeycomb Kitaev model, as we take the thermodynamic limit,  we have to specify a certain sequence of sizes $\rightarrow \infty $, e.g., let the number of unit cells$\rightarrow \infty $ rather than the number of lattice sites$\rightarrow \infty $ , which ensure that the number of lattice sites is always even. Otherwise, for the odd number of lattice sites, there would be Kramers degeneracy due to the time-reversal symmetry.
BTW: I get another naive question. As we know, if an eigenstate of a Hamiltonian is non-degenerate, then this eigenstate must preserve all the symmetries of the Hamiltonian. While, is there any possibility that this eigenstate has additional symmetry which is absence in the Hamiltonian ? Does someone know any example of this kind?