\section{Introduction}

In this paper we relate two remarkable constructions associated with a finite Coxeter group $W$. The first one is the Frobenius manifold structure on the space of orbits of $W$ acting in its reflection representation $V$~\cite{Dubrovin_lect}. The key ingredient here is the Saito flat metric on the orbit space $V/W$~\cite{SYS}. This metric is defined as a Lie derivative of the standard contravariant (Arnold) metric. The flat coordinates form a distinguished basis in the ring of invariant polynomials $S(V^*)^W$. This basis is now known explicitly for all irreducible groups $W$. All the cases except $W$ of type $E_7$, $E_8$ were covered in the original paper~\cite{SYS}. The flat coordinates in the latter two cases were found recently both in~\cite{Abriani} and in~\cite{Talamini}.

The other famous construction associated with the group $W$ is the rational Cherednik algebra $H_c(W)$~\cite{EG}. It depends on the $W$-invariant function $c$ on the set of reflections of $W$ which we assume to be constant. The key ingredient here is the Dunkl operator $\nabla_\zeta$, $\zeta \in V$, which acts in the ring of polynomials as a differential-reflection operator~\cite{Dunkl}. For particular values of $c$ the polynomial representation $S(V^*)$ has non-trivial submodules $M$.  These values were completely determined by Dunkl, de Jeu and Opdam  in~\cite{DJO} where it was shown that non-trivial submodules exist if and only if $c$ is a non-integer number of the form $c=l/d$ where $d$ is one of the degrees of the Coxeter group $W$ and $l \in \Z_{>0}$. The lowest homogeneous component $M_0$
of $M$ consists of so-called singular polynomials~\cite{DJO} which are annihilated by Dunkl operators $\nabla_\zeta$ for any $\zeta \in V$. All singular polynomials were found by Dunkl when $W$ has type $A$~\cite{DunklAsing1, DunklAsing2}. Further, it was established in~\cite{ESG} that in this case any submodule $M$ is generated by its lowest homogeneous component $M_0$. In general the structure of submodules of $S(V^*)$ and the corresponding singular polynomials are not known.  Some singular polynomials for the classical groups $W$ and for the icosahedral group were determined in~\cite{EtChm} and~\cite{Du2} (see also~\cite{Balag}) respectively, while dihedral case was fully studied in~\cite{DJO} (see also~\cite{Chm}).

In the paper we study singular polynomials that belong to the isotypic component of the reflection representation $V$ of the Coxeter group $W$. The existence of such singular polynomials is known for the Weyl groups  when $c=r/h$ where $h$ is the Coxeter number of $W$ and $r$ is a positive integer  coprime with $h$ \cite{GG}. It appears that in general the corresponding parameter values have to be $c=(d-1)/h+m$, where $d$ is one of the degrees of $W$  and $m \in \Z_{\ge 0}$.  We explain how to construct all the singular  polynomials in the isotypic component of $V$ in terms of the Saito polynomials that are flat coordinates of the Saito metric. We use theory of Frobenius manifolds and especially Dubrovin's almost duality~\cite{Dubrovin_dual}. We show that singular polynomials under consideration correspond to the $W$-invariant polynomial twisted periods of the Frobenius manifold $V/W$, and we determine all such twisted periods.

 Firstly we prove that the first order derivatives of the Saito polynomials are singular polynomials at appropriate values of the parameter $c=(d-1)/h$ (Corollary~\ref{corSaitoPol}). Then we explain how to construct further singular polynomials with parameter $c$ shifted by an integer (Theorem~\ref{mainth}).
Then we show  in Corollary~\ref{all-sing}
that this construction provides all the singular polynomials in the isotypic component of the reflection representation.

In the last section we present residue formulas for all the polynomial invariant twisted periods in the case of classical Coxeter groups $W$. Then we generalize them to get some singular polynomials for the complex reflection group  $W=S_n\ltimes(\mathbb Z/\ell\mathbb Z)^n$.

\section{Frobenius structures on the orbit spaces}

Let $V=\C^n$ with the standard constant metric $g$ given by $g(e_i,e_j)=(e_i,e_j)=\delta_{ij}$ where $e_i$, $i=1,\ldots,n$, is the standard basis in $V$. Let $(x^1,\ldots, x^n)$ be the corresponding orthogonal coordinates. Let $W$ be an irreducible finite Coxeter group of rank $n$ which acts in $V$ by orthogonal transformations such that $V$ is the complexified reflection representation of $W$. Let $\mathcal R \subset V$ be the Coxeter root system with the group $W$~\cite{Humphreys}. Let $y^1(x),\ldots, y^n(x)$ be a homogeneous  basis in the ring of invariant polynomials $S(V^*)^W=\C[x^1,\ldots,x^n]^W=\C[x]^W$. Let $d_\alpha$ be the corresponding degrees $d_\alpha = \deg y^\alpha(x)$, $\alpha=1,\ldots,n$. We assume the polynomials are ordered so that $d_1\ge\ldots\ge d_n$; $d_1=h$ is the Coxeter number of the group $W$. The polynomials $y^1,\ldots,y^n$ are coordinates on the orbit space ${\mathcal M}=V/W$. The Euclidean coordinates $x^1,\ldots, x^n$ can also be viewed as local coordinates on $\mathcal M\backslash\Sigma$, where $\Sigma$ is the discriminant set. Denote by $\Sympl=\{x\in V | (\g,x)=0 \mbox{ for some } \g \in {\mathcal R} \}$  the preimage of $\Sigma$ in the space $V$.

The metric $g$ is defined on ${\mathcal M}\setminus \Sigma$ due to its $W$-invariance. Let $g^{\alpha\beta}$ be the corresponding contravariant metric. Consider its Lie derivative
$\eta^{\alpha\beta}(y) = \partial_{y^1} g^{\alpha\beta}(y)$.
The metric $\eta$ is called the {\it Saito metric}. It is correctly defined (up to proportionality), and it is flat. There exist homogeneous coordinates $t^\a \in \C[x]^W$ ($1 \le \a \le n$), $\deg t^\a = d_\a$, such that $\eta$ is constant and anti-diagonal:
\begin{align*}
 &\eta^{\alpha\beta}=\partial_{t^1}g^{\alpha\beta}(t)=\delta^{\alpha+\beta}_{n+1}, &&1 \le \a,\b \le n,
\end{align*}
where $\delta^i_j=\delta_{ij}$ is the Kronecker symbol\footnote{We distinguish between the upper and lower indices as we will use the standard differential-geometrical convention that summation over the repeated upper and lower indices is assumed.}.
 Such coordinates are called {\it Saito polynomials}.

The pair of metrics $g, \eta$ forms a flat pencil which defines a Frobenius manifold~\cite{Dubrovin_lect}. We will be mainly concerned with the almost dual Frobenius structure~\cite{Dubrovin_dual}. It is defined by the prepotential
$$
F(x)= \frac{1}{2} \sum_{\g \in \mathcal R_+} (\g,x)^2 \log(\g,x),
$$
where summation is over the set of positive roots, and the roots are normalized so that $(\g,\g)=2$. The prepotential is quasi-homogeneous that is the Lie derivative
$$
{\mathcal L}_E F = \frac{2}{h} F + \mbox{quadratic terms in}  \,\,  x,
$$
where $E$ is the Euler vector field
$$
 E= \frac{1}{h} x^i\frac{\partial}{\partial x^i}= E^\alpha \frac{\partial}{\partial t^\alpha}
 $$
 with $E^\alpha = \frac{d_\alpha}{h}t^\alpha$.

Define the tensor
\begin{align}
 \Cstar_{i j k} = \frac{\partial^3 F}{\partial x^i \partial x^j\partial x^k}=\sum_{\g\in\mathcal R_+}\frac{\g_i\g_j\g_k}{(\g,x)}, \label{Cstar}
\end{align}
where $x \in V$ and $\gamma_i=(\gamma,e_i)$. Let $\Cstar^i_{jk}=g^{i l} \Cstar_{jkl}$. Then for any $x\in V\setminus \Sympl$ the tensor $\Cstar^i_{jk}=\Cstar^i_{jk}(x)$ forms the structure constants of an associative $n$-dimensional algebra~\cite{Dubrovin_dual}.

Define another tensor $C^{ij}_k= \Cstar^{ij}_k=g^{il} \Cstar^{j}_{kl}$ and consider the corresponding tensor with two low indices $C^\alpha_{\beta\lambda} = \eta_{\lambda\varepsilon} C^{\alpha\varepsilon}_\beta$ in the flat coordinates $t^\alpha$. The tensor $C^{\alpha}_{\beta\lambda}$ defines associative multiplication of tangent vectors at any point of the orbit space; the vector field $\p_{t^1}$ is the unity of this multiplication.
Let $C_\alpha$ be the corresponding $n\times n$ matrix with the entries $(C_\alpha)^\beta_\lambda = C^\beta_{\alpha\lambda}$. Let $U$ be the matrix  $U^\alpha_\beta=g^{\alpha\lambda}\eta_{\lambda\beta}$. The following result plays a key role.

\begin{theorem} \cite[Proposition 3.3]{Dubrovin_dual} \label{two-systems}
A function $p(t^1,\ldots,t^n)$ satisfies the system of equations
\begin{align}\label{second-derivatives-p}
\qquad\qquad \frac{\partial^2p}{\partial x^i \partial x^j} &= \nu\, \Cstar_{i j}^k \frac{\partial p}{\partial x^k}, &\qquad\qquad1\le i,j \le n
\end{align}
if and only if the following equations hold
\begin{align}
 \xi_{\a}(t)&=\partial_{t^\alpha} p(t), &&1\le \a\le n,\label{xiDef}  \\
\partial_{t^\alpha}\xi(t)\, U
 &= \xi(t)\big(\nu +\Lambda\big) C_\a, &&1\le \a\le n, \label{ptt}
\end{align}
where $\xi(t)=\big(\xi_1(t),\ldots, \xi_n(t)\big)$ and $\Lambda$ is the diagonal matrix $$\Lambda=-\frac1{h}\Diag(d_1-1,\ldots,d_n-1).$$
\end{theorem}

Functions $p$ satisfying the system (\ref{second-derivatives-p}) are called {\it twisted periods} of the Frobenius manifold~\cite{Dubrovin_dual}.
We will be dealing with the system~\eqref{xiDef},~\eqref{ptt} so we note the following statement.

\begin{lemma} \label{lemFxiTp}
 For any polynomial solution $\xi(t)$ of the system~\eqref{ptt} there exists a polynomial $p(t)$ satisfying~\eqref{xiDef}.
\end{lemma}

\begin{proof}
 It is sufficient to check the compatibility $\partial_{t^\alpha}\xi_\beta=\partial_{t^\beta}\xi_\alpha$. Taking into account invertibility of the matrix $U$ on $\mathcal M\backslash\Sigma$ and using the equation~\eqref{ptt} we rearrange this equality as
\begin{align*}
\xi_\lambda\big(\nu +\Lambda\big)^\lambda_\varepsilon C^\varepsilon_{\beta\rho} (U^{-1})^\rho_\alpha=
\xi_\lambda\big(\nu +\Lambda\big)^\lambda_\varepsilon C^\varepsilon_{\alpha\rho} (U^{-1})^\rho_\beta.
\end{align*}
Note that it is necessary to check that
\begin{align} \label{CUinvCUinv}
 C^\varepsilon_{\beta\rho} (U^{-1})^\rho_\alpha=C^\varepsilon_{\alpha\rho} (U^{-1})^\rho_\beta.
\end{align}
The equality~\eqref{CUinvCUinv} is equivalent to
\begin{align} \label{CUCUeq}
 C^\varepsilon_{\lambda\rho} U^\lambda_\delta=C^\varepsilon_{\lambda\delta} U^\lambda_\rho.
\end{align}
Indeed, multiplying~\eqref{CUCUeq} by $(U^{-1})^\rho_\alpha (U^{-1})^\delta_\beta$ and summing by repeating indexes we obtain~\eqref{CUinvCUinv}. The relation~\eqref{CUCUeq}, in its turn, follows from the property $U^\a_\b = E^\varepsilon C^\a_{\varepsilon \beta}$ (see~\cite{Dubrovin_lect}) and the associativity conditions $C^\varepsilon_{\lambda\rho} C^\lambda_{\alpha\delta}=C^\varepsilon_{\lambda\delta} C^\lambda_{\alpha\rho}$.
\end{proof}

We observe that the Saito polynomials themselves satisfy the equations from Theorem~\ref{two-systems}. More exactly we have the following statement.

\begin{lemma} \label{lemptbeta}
For each $\beta=1,\ldots,n$ the function $p(t)=t^\beta$ is a solution to the system~\eqref{xiDef},~\eqref{ptt} with the parameter
\begin{align*}
 \nu=\frac{d_\beta-1}{h}.
\end{align*}
\end{lemma}

\begin{proof}
By substituting $\xi_\alpha=\delta^\beta_\alpha$ into~\eqref{ptt} we obtain the equations $0=e^\beta( \nu+\Lambda)C_\alpha$, where $e^\beta=(0,0,\ldots,1,\ldots,0)$ with $1$ in the site $\beta$. These equations are satisfied if $\nu-\frac{d_\beta-1}{h}=0$.
\end{proof}

Recall now the Dunkl operators associated with the Coxeter group $W$. We fix a parameter $c \in \C$. The Dunkl operator in the direction $e_i$ ($i=1,\ldots,n$) is given by
\begin{align}\label{dunkl-operator}
 \nabla_i=\partial_{x^i}-c\sum_{\gamma\in\mathcal R_+}\frac{\gamma_i}{(\gamma,x)}(1-s_\gamma),
\end{align}
where $s_\gamma$ denotes the orthogonal reflection with respect to the hyperplane $(\gamma,x)=0$.
The key property of  Dunkl operators is their commutativity~\cite{Dunkl}:
\begin{align*}
 [\nabla_i,\nabla_j]&=0, &&1\le i,j\le n.
\end{align*}

\begin{proposition}\label{thm1}
 Suppose a $W$-invariant polynomial $p(x^1,\ldots,x^n)$ satisfies the system~\eqref{second-derivatives-p}. Then for any $j=1,\ldots,n$ the polynomial $v_j(x)=\partial_{x^j}p(x)$ satisfies the equations
\begin{align}
\nabla_i v_j&=0, &&i=1,\ldots,n, \label{Dbva}
\end{align}
where $\nabla_i$ is the Dunkl operator~\eqref{dunkl-operator} with the parameter $c=\nu$.
\end{proposition}

\begin{proof}
 By using the $W$-invariance of $p(x)$ we rearrange the left-hand side of the equation~\eqref{Dbva} as
\begin{equation*}
\nabla_i v_j =\partial_{x^i}\partial_{x^j} p(x)-\nu\sum_{\gamma\in\mathcal R_+}\frac{\gamma_i}{(\gamma,x)}\big(\partial_{x^j} p(x)-\frac{\partial}{\partial(s_\gamma e_j)}p(x)\big)=
\end{equation*}
\begin{equation*}=\partial_{x^i}\partial_{x^j} p(x)-\nu\sum_{\gamma\in\mathcal R_+}\frac{2\gamma_j\gamma_i}{(\gamma,\gamma)(\gamma,x)}\Big(\gamma,\frac{\partial}{\partial x}\Big)p(x).
\end{equation*}
By using $\big(\gamma,\frac{\partial}{\partial x}\big)=\sum_i\gamma_i\partial_{x^i}$ and the formula~\eqref{Cstar} we obtain
\begin{align}
\nabla_i v_j&=\partial_{x^i}\partial_{x^j} p(x)-\nu\Cstar^k_{ij}\partial_{x^k} p(x).
\end{align}
Thus the property~\eqref{Dbva} follows from~\eqref{second-derivatives-p}.
\end{proof}

\begin{corollary} \label{corSaitoPol}
 Consider the Saito polynomial $t^\beta=t^\beta(x)$ for some $\beta=1,\ldots,n$.  Then the derivatives $v_j=\partial_{x^j} t^\beta(x)$ satisfy the relations~\eqref{Dbva}, that is
\begin{align}
 \nabla_i\nabla_j t^\beta(x)=\nabla_i\partial_{x^j} t^\beta(x)&=0, &&i,j=1,\ldots,n,\label{Dbatbeta}
\end{align}
if the Dunkl operators have parameter $c=\frac{d_\beta-1}{h}$.
\end{corollary}

\begin{definition} \cite{DJO}
A polynomial $q(x)$ is called {\it singular} if $\nabla_i q(x)=0$ for all $i=1,\ldots,n$.
\end{definition}
Thus Corollary~\ref{corSaitoPol} deals with the singular polynomials $v_j$. The $W$-module  $\langle v_1,\ldots, v_n\rangle$ is isomorphic to the reflection representation of the Coxeter group $W$.

\section{Shifting}

In the previous Section we established that derivatives of the Saito polynomials $t^\beta$ are singular polynomials for the appropriate values of the parameter $c=c_\beta=(d_\beta-1)/h$. In this Section we explain how to generate further singular polynomials starting with Saito polynomials. The corresponding parameters $c$ differ from $c_\beta$ by integers.

We start with a known property of the solutions of the system~\eqref{ptt}.

\begin{lemma} \cite[Lemma 3.6]{Dubrovin_dual} \label{lempt1}
 If $\xi(t)$ is a solution of the system~\eqref{ptt} then $\wt\xi(t)=\partial_{t^1}\xi(t)$ is a solution of the same system with $\nu$ replaced by $\nu-1$:
\begin{align*}
\partial_{t^\alpha}\wt\xi(t)\, U
 &=\wt\xi(t)\, \big(\nu-1+ \Lambda\big) C_\alpha, &&1\le \alpha\le n.
\end{align*}
\end{lemma}

Note also that  if a function $p(t)$ is a solution of the system~\eqref{xiDef},~\eqref{ptt} then $\partial_{t^1} p(t)$ is a solution of the same system with $\nu$ replaced by $\nu-1$. Indeed, the partial derivatives of the function $\partial_{t^1} p(t)$ are $\partial_{t^\alpha}\partial_{t^1} p(t)=\partial_{t^1} \xi_\alpha(t)$, so they satisfy  the system~\eqref{ptt} with $\nu-1$.

The new solution $\wt\xi(t)$ given in Lemma~\ref{lempt1} can be represented as $\wt\xi=\xi \big(\nu+\Lambda\big)U^{-1}$. If $\nu\ne\dfrac{d_\alpha-1}h$ for all $\alpha=1,\ldots,n$ then the matrix $\nu+ \Lambda$ is invertible and we can rewrite this relation as $\xi=\wt\xi\, U\big(\nu+\Lambda\big)^{-1}$. This suggests how to invert the Lemma~\ref{lempt1}  in order to generate solutions with increased value of $\nu$.

\begin{lemma} \label{lempt1conv}
Let $\xi(t)$ be a solution of the system~\eqref{ptt}. Assume that $\nu\ne\dfrac{d_\alpha-1}h-1$ for all $\alpha=1,\ldots,n$. Then
\begin{align} \label{whxixi}
 \wh\xi(t)=\xi(t) U\big(\nu+1+ \Lambda\big)^{-1}
\end{align}
is a solution of the system~\eqref{ptt} with $\nu$ replaced by $\nu+1$:
\begin{align}
\partial_{t^\alpha}\wh\xi(t)\, U
 &=\wh\xi(t)\, \big(\nu+1+ \Lambda\big) C_\alpha, &&1\le \alpha\le n. \label{pttHat}
\end{align}
\end{lemma}

\begin{proof}
 Let $t_0$ be a generic point in $\mathcal M$ and let  $\xi(t_0)$ be the value of $\xi(t)$ at this point. Then the value of $\wh\xi(t)$ at this point is $\wh\xi(t_0)=\xi(t_0) U(t_0)\big(\nu+1+\Lambda \big)^{-1}$. There exists a solution $\wh\zeta(t)$ of the system~\eqref{pttHat} such that $\wh\zeta(t_0)=\wh\xi(t_0)$. By Lemma~\ref{lempt1} the covector $\zeta(t)=\wh\zeta(t)\big(\nu+1+\Lambda\big)U^{-1}$ is a solution of~\eqref{ptt}. Note that there exists a unique solution of the system~\eqref{ptt} with a given value in the point $t_0$. So taking into account $\zeta(t_0)=\wh\xi(t_0)\big(\nu+1+\Lambda\big)U^{-1}(t_0)=\xi(t_0)$ one yields $\zeta(t)=\xi(t)$. Therefore $\widehat\zeta(t)=\zeta(t)U\big(\nu+1+\Lambda\big)^{-1}=\xi(t)U\big(\nu+1+\Lambda\big)^{-1}=\wh\xi(t)$ and $\wh\xi(t)$ satisfies the equation~\eqref{pttHat}.
\end{proof}

\begin{remark}\label{rem-increase-p}
Suppose $\xi(t)$ in Lemma~\ref{lempt1conv} is polynomial. Then by Lemma~\ref{lemFxiTp}
there exists a polynomial $\wh p(t)$ such that $\wh\xi_\alpha(t)=\partial_{t^\alpha}\wh p(t)$. Thus the covector $\wh\xi(t)$ satisfies the whole system~\eqref{xiDef},~\eqref{ptt} (with $\nu$ replaced by $\nu+1$)  with some polynomial $\wh p(t)$.
\end{remark}

By applying Lemma~\ref{lempt1conv} to the first order derivatives of the Saito polynomials we get the following result.

\begin{proposition} \label{propwhxibeta}
 The covector $\wh\xi$ with the components
\begin{align} \label{whxibeta}
 \wh\xi_\alpha=\frac{U^\beta_{\alpha}}{d_\beta-d_\alpha+h},   \,\,\,\, \a=1,\ldots,n,
\end{align}
satisfies the equations
\begin{equation}\label{system-hat}
\partial_{t^\alpha}\widehat\xi(t)\, U
 = \widehat\xi(t)\big(\widehat\nu +\Lambda\big) C_\alpha, \,\,\,\,\, 1\le \a\le n,
\end{equation}
with $\widehat\nu=\dfrac{d_\beta-1}h+1$.
\end{proposition}

\begin{proof}
 Let $\nu=\dfrac{d_\beta-1}h$ and consider the solution $p(t)=t^\beta$ of the system~\eqref{xiDef},~\eqref{ptt} and the corresponding covector $\xi$ with the components $\xi_\alpha=\delta^\beta_\alpha$ (see Lemma~\ref{lemptbeta}).

 By Lemma~\ref{lempt1conv} the  covector $\wh\xi$ given by the formula~\eqref{whxixi} is a solution of~\eqref{system-hat} for $\wh\nu=\nu+1$. By substituting $\xi_\alpha=\delta^\beta_\alpha$ into~\eqref{whxixi} we obtain the components~\eqref{whxibeta}.
\end{proof}

This leads us to the following result.
\begin{theorem}
For any $\zeta \in V$, $\beta=1,\ldots,n$
the polynomial
\begin{align} \label{qtheor}
q(x)=\sum_{i,\a=1}^n  \frac{1}{d_\beta - d_\a+h} \frac{\partial t^\beta}{\partial x^i} \frac{\partial t^{n+1-\a}}{\partial x^i}{\partial_\zeta t^\a}
\end{align}
is a singular polynomial for the Dunkl operators with the parameter $c=\dfrac{d_\beta-1}h+1$.
\end{theorem}

\begin{proof}
First we rearrange
\begin{align} \label{U-expand}
U^\beta_\alpha = g^{\beta \lambda} \eta_{\lambda \alpha}=g^{\beta, n+1-\alpha}=
\sum_{a=1}^n\frac{\partial t^\beta}{\partial x^a}\frac{\partial t^{n+1-\alpha}}{\partial x^a}.
\end{align}
It follows from Proposition~\ref{propwhxibeta} and Lemma~\ref{lemFxiTp} that there exists a $W$-invariant polynomial $p(x)$ such that
$$
\partial_{t^\alpha} p =  \frac{1}{d_\beta-d_\alpha+h}
\sum_{a=1}^n \frac{\partial t^\beta}{\partial x^a}\frac{\partial t^{n+1-\alpha}}{\partial x^a}.
$$
By Proposition~\ref{thm1} the derivative $q(x)=\partial_\zeta p(x)$ is a singular polynomial. It has the required form as $\partial_\zeta p = \partial_{t^\a}p  \partial_{\zeta} t^\a$.
\end{proof}

As an example consider the case $\beta=n$. The corresponding Saito polynomial $t^n$ is proportional to $(x^{1})^2+(x^{2})^2+\ldots+(x^{n})^2$. The right-hand side of the equality~\eqref{qtheor} is then proportional to
$$
\sum_{\a=1}^n t^{n+1-\a} \partial_\zeta t^\a
$$
as the polynomial $t^{n+1-\a}$ is homogeneous of degree $d_{n+1-\a}$ and $d_\a+d_{n+1-\a}=h+2$.
We arrive at the following corollary.
\begin{proposition}\label{prophhp1}
For any $\zeta \in V$
the polynomial
\begin{align} \label{qcor}
q(x)=\partial_\zeta \sum_{\alpha=1}^n t^\alpha t^{n+1-\alpha}
\end{align}
is a singular polynomial for the Dunkl operators with the parameter $c=(h+1)/h$.
\end{proposition}

\begin{remark}
For $c=1/h+m, m \in \Z_{\ge 0}$ the existence of the singular polynomials in the isotypic component of the  reflection representation is known from~\cite{BEG} (see also~\cite{Go}). Further, in the case of Weyl groups $W$ and $c=r/h$ where $r$ is a positive integer coprime with $h$,  the existence of singular polynomials in the isotypic component of the reflection representation is known from \cite{GG}.
\end{remark}

\begin{example}
Let ${\mathcal R}={\mathcal A}_{n} \subset V \subset \C^{n+1}$ be given in its standard embedding so that $V\subset \C^{n+1}$ is defined by $\sum_{i=1}^{n+1} z_i=0$ where $z_i$ are standard coordinates in $\C^{n+1}$. Define $s^\alpha = \mathop{\mathrm{Res}}\nolimits_{z=\infty} \prod_{i=1}^{n+1} (z-z_i)^\frac{n+1-\alpha}{n+1} d z$ for $\alpha=1,\ldots,n$. Then Saito coordinate $t^\alpha = \frac{1}{n-\a+1}s^\alpha|_V$ (see~\cite{Dubrovin_lect},~\cite{SYS}). The polynomials $s^\alpha$ satisfy  $\sum_{i=1}^{n+1} \frac{\partial s^\alpha}{\partial z_i} =0$ so for any $\zeta \in \C^{n+1}$, $i=1,\ldots,n+1$ Corollary~\ref{corSaitoPol} gives
$\nabla^{\frac{n+1-\alpha}{n+1}}_i \partial_\zeta s^\alpha=0$,
 where
$$
\nabla^c_i =  \frac{\partial}{\partial z_i} - c\sum_{\nad{j=1}{j \ne i}}^{n+1} \frac{1-s_{ij}}{z_i-z_j}
$$
with $s_{ij}$ exchanging $z_i$ and $z_j$ (see also~\cite[Proposition 11.14]{Et},~\cite{Du} where this fact was established). Further, Proposition~\ref{prophhp1} gives that
the polynomial $q(z_1,\ldots,z_{n+1})=\partial_\zeta \sum_{\alpha=1}^n t^\alpha t^{n+1-\alpha}$ satisfies $\nabla_i ^{\frac{h+1}{h}}q =0$.
\end{example}

By iterating the previous arguments we get the following statement.

\begin{theorem}\label{mainth}
Let $m\in \Z_{\ge0}$, fix $\beta$,  $1\le \beta \le n$. Define the covector $\xi^{(m)}=(\xi^{(m)}_1,\ldots,\xi^{(m)}_n)$ by
\begin{align} \label{xim}
 \xi^{(m)}= \xi^{(0)} \mathop{\overrightarrow\prod}_{1\le j\le m} U\, \big( \Lambda + \frac{d_\beta-1}{h}+j \big)^{-1},
\end{align}
where $\xi^{(0)}$ has components $\xi^{(0)}_\alpha = \delta^{\beta}_\alpha$, $\alpha=1,\ldots,n$ and the factors are ordered as $\mathop{\overrightarrow\prod}\limits_{1\le j\le m}A_j=A_1A_2\cdots A_m$.
Then for any $i=1,\ldots,n$ the polynomials
\begin{align} \label{qi}
q_i(x) =q_{\beta,i}(x) =
\sum_{\a=1}^n \xi^{(m)}_\a \frac{\partial t^\a}{\partial x^i}
\end{align}
are singular polynomials for the Dunkl operators with the parameter $c=\frac{d_\beta-1}{h}+m$. These polynomials are homogeneous of degree $d_\b-1+h m$.
\end{theorem}

\section{Singular polynomials in the reflection representation}

We are going to show that polynomials (\ref{qi}) generate all singular polynomials in the isotypic component of the reflection representation of $W$. Firstly we note that each copy of the reflection representation spanned by the singular polynomials is governed by a single $W$-invariant polynomial.

\begin{proposition}\label{Qexists}
Let a subspace $M_0\subset\mathbb C[x]$ be spanned by the singular polynomials and suppose that $M_0\cong V$ as $W$-modules. Choose a basis $\{P_1,\ldots, P_n\}$ in $M_0$ such that each polynomial $P_i$ is mapped to the basis vector $e_i \in V$ under the isomorphism. Then there exists $Q \in \C[x^1,\ldots,x^n]^W$ such that $\frac{\partial Q}{\partial x^i}= P_i$ for all $i=1,\ldots, n$.
\end{proposition}
\begin{proof}
We have $\nabla_i P_j = \nabla_j P_i =0$ for all $i,j=1,\ldots,n$. Hence
\begin{align} \label{crosspartial}
\partial_{x^i} P_j - \partial_{x^j} P_i - c \sum_{\gamma\in {\mathcal R}_+} \frac{(\gamma,e_i)}{(\gamma,x)}(1-s_\gamma)P_j +  c \sum_{\gamma\in {\mathcal R}_+} \frac{(\gamma,e_j)}{(\gamma,x)}(1-s_\gamma)P_i=0.
\end{align}
Since $(1-s_\gamma)e_i = \frac{2(\gamma,e_i)}{(\gamma,\gamma)}\gamma$ we get
$$
(\gamma,e_i)(1-s_\gamma)P_j = (\gamma,e_j)(1-s_\gamma)P_i
$$
for any $\gamma \in {\mathcal R}_+$.
Thus it follows from the relation~\eqref{crosspartial} that  $\partial_{x^i} P_j = \partial_{x^j} P_i$ so there exists $Q\in \C[x^1,\ldots,x^n]$ such that $\frac{\partial Q}{\partial x^i}=P_i$ for all $i=1,\ldots,n$. Let us also check that $Q$ is $W$-invariant. Fix $\gamma\in {\mathcal R}_+$. Then for any $i=1,\ldots,n$ we have $s_\gamma P_i = \partial_{s_\gamma e_i} (s_\gamma Q)$, and on the other hand $s_\gamma P_i = \partial_{s_\gamma e_i} Q$. Thus $\partial_{s_\gamma e_i}(Q-s_\gamma Q)=0$, so $Q=s_\gamma Q$ as required.
\end{proof}

\begin{corollary}\label{corSaitoPolgeneral}
The singular polynomials~\eqref{qi} can be represented as
\begin{align} \label{qiqi}
q_i = \frac{\partial Q}{\partial x^i},
\end{align}
where
\begin{align} \label{QQ}
Q=Q_\beta = \frac{1}{d_\b+h m}
\sum_{\a=1}^n d_\a \xi^{(m)}_\a  t^\a,
\end{align}
$\,i=1,\ldots,n$, and we keep notations of Theorem \ref{mainth}.
\end{corollary}
\begin{proof}
By Proposition~\ref{Qexists} we have the relation~\eqref{qiqi} for some invariant polynomial $Q$ of degree $d_\b +h m$. Hence $(d_\b+h m) Q=\sum_{i=1}^n x^i q_i$
\end{proof}

\begin{remark}
It is recently explained in~\cite{KL} how ${\mathcal N}=4$ multi-particle mechanical system with $D(2,1;\a)$ superconformal symmetry can be constructed based on a solution of the WDVV equations and a particular twisted period. It follows from Theorem~\ref{mainth} and Corollary~\ref{corSaitoPolgeneral} that the polynomials $q_i, Q$  given by~\eqref{qi},~\eqref{QQ} define a superconformal mechanical system with the bosonic potential proportional to
$
Q^{-2} \sum_{i=1}^n q_i^2
$
at the parameter value $\a=-\frac12 (d_\b + h m)$.
\end{remark}

\begin{remark}
Let $g(x^1,\ldots,x^n)$ be a homogeneous  $W$-invariant polynomial of positive degree. Let $L_g$ be the differential operator which acts on the $W$-invariant functions by $g(\nabla_1,\ldots,\nabla_n)$. The operators $L_g$ commute with each other and include the corresponding {\it Calogero-Moser operator}~\cite{Et}.  It follows from Corollary~\ref{corSaitoPolgeneral} that if $c=\frac{d_\beta-1}{h}+m$ then  $L_g Q_\beta=0$ so the polynomial $Q_\beta$ belongs to the joint kernel of the Calogero-Moser operator and its quantum integrals. In particular, the Saito polynomial $t^\beta$ satisfies $L_g t^\beta=0$ if $c=\frac{d_\beta-1}{h}$.
\end{remark}

Now we move to the main statement of this Section on possible polynomial twisted periods of the Frobenius manifold $\mathcal M$.

\begin{theorem}\label{mthsect4uniqueness}
Let $L$ be the linear space of solutions $p(x)$ to the system~\eqref{second-derivatives-p} such that $p \in \C[x]^W$. Then $\dim L =1$ unless
$\nu=\dfrac{d_\beta-1}{h}+m$ for some degree $d_\beta$, and $m\in\mathbb Z_{\ge0}$. In this case $\dim L=2$ unless $W=D_n$ where $n$ is even and $d_\b=n$ when $\dim L=3$.
\end{theorem}

\begin{proof}
Suppose $p\in \C[x]^W$ is a homogeneous solution of the system~\eqref{second-derivatives-p} such that $D=\deg p >0$. By Proposition~\ref{thm1} the polynomials $v_i(x)= \frac{\partial p(x)}{\partial x^i}$ are singular at the parameter $c=\nu$. It follows from the relation
\begin{align*}
 \sum_{i=1}^n x^i\nabla_i=\sum_{i=1}^n x^i\partial_{x^i}-\nu\sum_{\gamma\in \mathcal R_+}(1-s_\gamma)
\end{align*}
that $\nu= \frac{D-1}{h}\ge0$
(c.f.~\cite{DJO}).
Consider firstly the case $0\le\nu\le1$.
Equation~\eqref{ptt} at $\a=1$ takes the form $0=\xi(t)(\nu+\Lambda)$ since $C_1=Id$ and $\deg\xi_\lambda=D-d_\lambda<h$ for any $1 \le \lambda \le n$.
Hence the matrix $\nu+\Lambda$ is degenerate
so $\nu= \frac{d_\beta-1}{h}$ for some $\beta$, and $D=d_\b$. Moreover $p(x)$ as a polynomial of the Saito coordinates has to be a linear combination of $t^\lambda$ with $d_\lambda=d_\b$.

Now let $\nu>1$. The polynomial $\p_{t^1}p(t)$ is non-constant as the matrix $\nu+\Lambda$ is non-degenerate. By Lemma~\ref{lempt1} the polynomial $\p_{t^1}p(t)$ is a solution of the system~\eqref{second-derivatives-p} with $\nu$ replaced by $\nu-1$. It follows by Lemma~\ref{lempt1conv} and Remark~\ref{rem-increase-p} that the spaces of positive degree homogeneous invariant solutions of~\eqref{second-derivatives-p} for the parameter $\nu$ and $\nu$ replaced with $\nu-1$ are isomorphic.
\end{proof}

The arguments from the proof of Proposition~\ref{thm1} can be reversed so it follows that the polynomial~\eqref{QQ} is a twisted period with $\nu=(d_\b-1+h m)/h$ of the Frobenius manifold $\mathcal M$. Thus Theorem~\ref{mthsect4uniqueness} implies the following statement.

\begin{corollary} \label{all-twist}
Let $p \in \C[x]^W$ be a non-constant  twisted period of the Frobenius manifold $\mathcal M$. Then $\nu =(d_\b-1+h m)/h$ for some degree $d_\b$ and $m \in \Z_{\ge 0}$. Further, $p= \lambda Q_{\beta}$ unless $W=D_{2r}$ and $d_\b=2r$ for some $r \in \N$ in which case $p=\lambda Q_{r}+\mu Q_{r+1}$, where $Q_{\beta}$ are given by~\eqref{QQ} and $\lambda,\mu\in \C$.
\end{corollary}

Consider now a linear space $M_0$ of singular polynomials such that $M_0 \cong V$ as $W$-modules. Then $Q\in\mathbb C[x]^W$ defined by Proposition~\ref{Qexists} is a twisted period and hence Corollary~\ref{all-twist} allows to describe all such $W$-modules $M_0$.

\begin{corollary} \label{all-sing}
Let $q$ be a homogeneous singular polynomial. Suppose the linear space spanned by the polynomials $wq$, $w\in W$, is isomorphic to $V$ as $W$-module. Then $\deg q = d_\b -1 + h m$ for some degree $d_\b$ of $W$,  $m \in \Z_{\ge 0}$, and $c=(d_\b-1+h m)/h$.

Further, $q=\sum_{i=1}^n \eta_i q_{\beta,i}$ unless $W=D_{2r}$ and $d_\b=2r$ for some $r \in \N$ in which case $q=\sum_{i=1}^n  \eta_i (\lambda q_{r,i}+ \mu  q_{r+1,i})$, where $q_{\beta,i}$ are given by~\eqref{qi},~\eqref{xim} and $\lambda,\mu, \eta_i \in \C$.

In the former cases all the homogeneous singular polynomials for $c=(d_\b-1+h m)/h$ in the isotypic component of the reflection representation are described as linear combinations  $\sum_{i=1}^n \eta_i q_{\beta,i}$ while in the latter case the homogeneous singular polynomials in the isotypic component of the reflection representation form the $2n$-dimensional subspace of polynomials $\sum_{i=1}^n \lambda_i q_{r,i}+\mu_i q_{r+1,i}$, where $\lambda_i, \mu_i \in \C$.
\end{corollary}

\section{Further examples for classical series}

While we express singular polynomials in the isotypic component of the reflection representation of $W$ through the Saito polynomials, in certain cases direct formulas exist. We refer to~\cite{DJO},~\cite{Chm} for the case of dihedral groups, and to~\cite{Du},~\cite{EtChm} for the case when $W$ is of classical type.
For instance, it follows from~\cite{Du} and Corollary \ref{all-twist} that {\it all} the polynomial invariant twisted periods for $W=A_{n}$ are given by
$$
Q=\mathop{\mathrm{Res}}\nolimits_{z=\infty}  \prod_{j=1}^{n+1} (z-z_j)^\nu dz  \Big|_{\sum z_j=0},
$$
where $\nu=\frac{s}{n+1}+m$ with $s=1,\ldots,n$ and $m \in \Z_{\ge 0}$ (c.f.~\cite{Dubrovin_dual}).
Further, let
\begin{equation} \label{twistedperiods}
Q=\mathop{\mathrm{Res}}\nolimits_{z=\infty} z^{a}\prod_{j=1}^n\big(z^2-x_j^2\big)^\nu dz,
\end{equation}
where $x_1,\ldots,x_n$ are standard coordinates in $\C^n$.
Then {\it all} the polynomial invariant twisted periods for $W=B_n$ have the form~\eqref{twistedperiods} where $a=0$, $\nu=\frac{2 s -1}{2n}+m$ with $s=1,\ldots,n$ and $m \in \Z_{\ge 0}$.
Similar, \eqref{twistedperiods}~is the twisted period for $W=D_n$ if $a=-2\nu$, $\nu=\frac{2 s -1}{2(n-1)}+m$ with $s=1,\ldots,n-1$ and $m \in \Z_{\ge 0}$. {\it All} the remaining polynomial invariant twisted periods for $W=D_n$ have the form
\begin{equation} \label{twistedperiodsDrest}
Q=\mathop{\mathrm{Res}}\nolimits_{z=0} z^{-2m-1}\prod_{j=1}^n\big(z^2-x_j^2\big)^{m+\frac12} dz,
\end{equation}
where $m \in \Z_{\ge 0}$ (c.f~\cite{Egouchi}).
We note that for even $n$ the polynomial $Q=Q_{m,\infty}$ given by~\eqref{twistedperiods} with $a=-2m-1$, $\nu = m+\frac12$, $m\in \Z_{\ge 0}$ has same degree as the polynomial
 $Q=Q_{m,0}$ given by~\eqref{twistedperiodsDrest} with the same $m$. These polynomials are not proportional as for $m>0$ the polynomial $\p_{t^1} Q_{m,\infty}$ is a nonzero multiple of the polynomial $Q_{m-1,\infty}$, and  $\p_{t^1} Q_{m,0}$ is a nonzero multiple of $Q_{m-1,0}$.

This leads to the following proposition which can also be checked directly.

\begin{proposition} \label{propnum12}
Let polynomial $Q(x)$ be given by~\eqref{twistedperiodsDrest}
with  $m\in \Z_{\ge 0}$. For any $\zeta \in V$ the polynomial $\p_\zeta Q$ is singular for $W=D_n$ with the parameter $c=m+\frac12$.
\end{proposition}

Proposition~\ref{propnum12} can be generalized to the case of complex reflection group $W=S_n\ltimes(\mathbb Z/\ell\mathbb Z)^n$, where $\ell \in \Z$ satisfies $\ell \ge 2$. This group is generated by the reflections $\sigma_{ij}^{(a)}$ and $s_i$  acting on the standard basis as
\begin{align}
 \sigma_{ij}^{(a)}e_i&=\omega^{-a}e_j, &\sigma_{ij}^{(a)}e_j&=\omega^{a}e_i,
 &\sigma_{ij}^{(a)}e_k&=e_k, \\
 s_ie_i&=\omega e_i, &s_i e_k&=e_k,
\end{align}
where $\omega=e^{2\pi i/\ell}$ is the primitive root of unity, $i,j,k=1,\ldots,n$, $i\ne j\ne k\ne i$, $a=0,\ldots,\ell-1$.
The Dunkl operators in this case have the form~\cite{DO}
\begin{align}\label{dunklgmp}
 \nabla_i=\partial_{x_i}-\nu\sum_{a=0}^{\ell-1}\sum_{\nad{j=1}{j\ne i}}^n\frac1{x_i-\omega^ax_j}\big(1-\sigma_{ij}^{(a)}\big)-\sum_{b=1}^{\ell-1}c_b\sum_{a=0}^{\ell-1}\frac{\omega^{-ab}}{x_i}s_i^a,
\end{align}
where  $\nu,c_1,\ldots,c_{\ell-1}\in\mathbb C$ (we do not suppose any more that parameters of the Dunkl operators are equal). Denote $c_0=0$ and $c_{a\ell+b}=c_b$ for $a\in\mathbb Z$, $b=0,1,\ldots,\ell-1$.

The next statement generalizes Proposition~\ref{propnum12}. The form of the singular polynomials is suggested by~\cite{EtChm} where some singular polynomials for the group $W$ were found using the residues at infinity.

\begin{proposition}\label{complex-gropus}
 Let $q\in \Z$ satisfy $1\le q\le\ell-1$. Suppose that $\nu=m+\frac{\ell-q+s}\ell$, $c_{q-s}=0$ and $c_{-s}=\frac{s}\ell$ for some $m,s\in\mathbb Z_{\ge0}$. Then the formulae
\begin{align}
 f_j=\prod_{i=1}^nx_i^{\ell \nu}\mathop{\mathrm{Res}}\nolimits_{z=0} z^{-\ell m-1}\prod_{i=1}^n\bigg(1-\frac{z^\ell}{x_i^\ell}\bigg)^{\nu}\,\frac{x_j^{q}dz}{x_j^\ell-z^\ell}, \label{fi}
\end{align}
define singular polynomials that is $\nabla_i f_j=0$ for any $i, j=1,\ldots,n$, where the operator $\nabla_i$ is given by~\eqref{dunklgmp} (and it is assumed in~\eqref{fi} that $x_i\ne 0$ for all $i$). These polynomials are homogeneous of degree $(n-1)(m\ell+\ell-q)+ns$ and they span an irreducible $n$-dimensional representation of $W$.
\end{proposition}

\begin{proof}
By calculating the residue in~\eqref{fi} explicitly one yields
\begin{align*}
 f_j= (-1)^m\sum_{k_1,\ldots,k_n=0\atop k_1+\ldots+k_n=m}^\infty \binom{\nu-1}{k_j} x_j^{\ell(m-k_j)+s}\prod_{i=1\atop i\ne j}^n\binom{\nu}{k_i} x_i^{\ell(\nu-k_i)},
\end{align*}
where $\binom{\alpha}k=\alpha(\alpha-1)\cdots(\alpha-k+1)/k!$ and $\binom{\alpha}0=1$. Since $\nu>m$ the function $f_j$ is a homogeneous polynomial of $x_1,\ldots,x_n$, it has degree $(n-1)\ell \nu+s=(n-1)(m\ell+\ell-q)+ns$. Note that all the coefficients in this expression do not vanish and hence $f_j\ne0$. The generators of $W$ act on the polynomials~\eqref{fi} by the formulae
\begin{align}
 \sigma_{ij}^{(a)}f_i&=\omega^{aq}f_j, &\sigma_{ij}^{(a)}f_j&=\omega^{-aq}f_i,
 &\sigma_{ij}^{(a)}f_k&=f_k, \label{sigmaf} \\
 s_if_i&=\omega^{-s} f_i, &s_i f_k&=\omega^{q-s} f_k, \label{sif}
\end{align}
where $i,j,k=1,\ldots,n$, $i\ne j\ne k\ne i$, $a=0,\ldots,\ell-1$. It follows that the space spanned by $f_j$, $j=1,\ldots,n$, is $n$-dimensional irreducible representation of $W$.

Now let us show that the polynomial $f_j$ is singular. Let $i\ne j$, then using the formulae~\eqref{sigmaf},~\eqref{sif} we derive
\begin{align}
 \nabla_i f_j=\partial_{x_i}f_j-\nu\sum_{a=0}^{\ell-1}\frac1{x_i-\omega^ax_j}\big(f_j-\omega^{-aq}f_i\big)-\sum_{b=1}^{\ell-1}c_b\sum_{a=0}^{\ell-1}\frac{\omega^{-ab}}{x_i}\omega^{a(q-s)}f_j. \label{nablafj}
\end{align}
The last term in~\eqref{nablafj} equals $-\ell c_{q-s}x_i^{-1}f_j=0$ as $\sum_{a=0}^{\ell-1}\omega^{ab}=0$ for $b\notin\mathbb\ell\mathbb Z$. Thus $\nabla_i f_j=\prod_{i=1}^nx_i^{\ell \nu}\mathop{\mathrm{Res}}\nolimits_{z=0} z^{-\ell m-1}\prod_{i=1}^n\big(1-z^\ell/x_i^\ell\big)^{\nu}F_{ij}dz$
where
\begin{align}
 F_{ij}&=\nu\frac{\ell x_i^{\ell-1}x_j^q}{(x_i^\ell-z^\ell)(x_j^\ell-z^\ell)}-
\nu\sum_{a=0}^{\ell-1}\frac1{x_i-\omega^ax_j}\Big(\frac{x_j^q}{x_j^\ell-z^\ell}-\frac{\omega^{-aq}x_i^q}{x_i^\ell-z^\ell}\Big)= \notag \\
&=-\frac{\nu}{(x_i^\ell-z^\ell)(x_j^\ell-z^\ell)}\Big(-\ell x_i^{\ell-1}x_j^q+ \notag \\
&+\sum_{a=0}^{\ell-1}\frac{z^\ell\omega^{-aq}(x_i^q-\omega^{aq}x_j^q)+x_i^qx_j^q(x_i^{\ell-q}-\omega^{-aq}x_j^{\ell-q})}{x_i-\omega^ax_j}\Big)= \notag \\
&=-\frac{\nu}{(x_i^\ell-z^\ell)(x_j^\ell-z^\ell)}\Big(-\ell x_i^{\ell-1}x_j^q+ \notag \\
&+\sum_{a=0}^{\ell-1}z^\ell\omega^{-aq}\sum_{b=0}^{q-1}x_i^b\omega^{a(q-1-b)}x_j^{q-1-b}+
\sum_{a=0}^{\ell-1}x_i^qx_j^q\sum_{b=0}^{\ell-q-1}x_i^{\ell-q-1-b}\omega^{ab}x_j^b\Big). \notag
\end{align}
Thus the first double sum in $F_{ij}$ vanishes and the last double sum equals $\ell x_i^{\ell-1}x_j^q$. Thus $F_{ij}=0$ and $\nabla_i f_j=0$ for $i\ne j$.

Now to prove that $\nabla_j f_j=0$ it is sufficient to check $\sum_{i=1}^nx_i\nabla_i f_j=0$. Since $f_j$ is homogeneous of order $(n-1)\ell \nu+s$ we have $\sum_{i=1}^nx_i\partial_{x_i} f_j=\big((n-1)\ell \nu+s\big)f_j$. Hence using $c_{q-s}=0$ as previously one yields
\begin{align}
 \sum_{i=1}^nx_i\nabla_i f_j=\big((n-1)\ell \nu+s\big)f_j+\prod_{k=1}^n x_k^{\ell \nu} \mathop{\mathrm{Res}}\nolimits_{z=0} z^{-\ell m-1}\prod_{k=1}^n(1-z^\ell/x_k^\ell)^{\nu}F_jdz, \notag
\end{align}
where
\begin{multline*}
 F_j=-\nu\sum_{a=0}^{\ell-1}\sum_{i=1\atop i\ne j}^n\frac{x_i}{x_i-\omega^ax_j}\Big(\frac{x_j^q}{x_j^\ell-z^\ell}-\frac{\omega^{-aq}x_i^q}{x_i^\ell-z^\ell}\Big)-\\
-\nu\sum_{a=0}^{\ell-1}\sum_{k=1\atop k\ne j}^n\frac{x_j}{x_j-\omega^ax_k}\Big(\frac{x_j^q}{x_j^\ell-z^\ell}-\frac{\omega^{aq}x_k^q}{x_k^\ell-z^\ell}\Big)-\sum_{b=1}^{\ell-1}c_b\sum_{a=0}^{\ell-1}\frac{\omega^{-ab-as}x_j^q}{x_j^\ell-z^\ell}.
\end{multline*}
By taking into account $\frac{x_i}{x_i-\omega^a x_j}+\frac{x_j}{x_j-\omega^{-a} x_i}=1$ we obtain
\begin{multline*}
 F_j=-\nu\sum_{a=0}^{\ell-1}\sum_{i=1\atop i\ne j}^n\Big(\frac{x_j^q}{x_j^\ell-z^\ell}-\frac{\omega^{-aq}x_i^q}{x_i^\ell-z^\ell}\Big)-\ell c_{-s}\frac{x_j^q}{x_j^\ell-z^\ell}= \\
=-\frac{x_j^q}{x_j^\ell-z^\ell}\big((n-1)\ell \nu+s\big),
\end{multline*}
where we used $c_{-s}=\frac{s}\ell$. Hence we derive $\sum_{i=1}^nx_i\nabla_i f_j=0$.
\end{proof}

\begin{remark}
In the case when the parameters satisfy $\ell | (s-q)$ the singular polynomials~\eqref{fi} appeared earlier in \cite[Proposition 4.1]{EtChm} where they were presented using the residue at infinity. In the other cases the space spanned by~\eqref{fi} does not contain the singular polynomials from \cite[Proposition 4.1]{EtChm} except the case $n=2$, $s=0$ and the case $n=1$.
\end{remark}

\vspace{5mm}

 {\bf Acknowledgements.} M.F. is very grateful to Y. Burman for stimulating discussions particularly in the beginning of the work, and to I.~Strachan for useful discussions and for pointing out the paper~\cite{Talamini}. M.F. is grateful to S.~Krivonos, O.~Lechtenfeld, A.~Marshakov and A.~Varchenko for stimulating discussions. Both authors are grateful to C.~Dunkl and S.~Griffeth for useful comments. The work of A.S. was partially supported by the RFBR grant 09-01-00239-a. The work of both authors was supported by  the EPSRC grant EP/F032889/1.


\section*{Introduction}
We consider the following system of mass and momentum conservation with non dimensionless parameters and variables,
 which is the 1D model of blood flow in an artery or a vessel with non uniform elasticity (it is rewritten in a
 conservative form compared to what we usually find in litterature)
\begin{equation}
 \left\{\begin{array}{l}
         \partial_t A+\partial_x Q=0\\
	\partial_t Q+\partial_x \left[\dfrac{Q^2}{A}+\dfrac{1}{3\sqrt{\pi}\rho}kA^{3/2}\right]
=\dfrac{A}{\sqrt{\pi}\rho}\left(\partial_x {\cal A}_0-\dfrac{2}{3}\sqrt{A}\partial_x k\right)-C_f\dfrac{Q}{A}
        \end{array} \right.,\label{eq:system1}
\end{equation}
with ${\cal A}_0=k\sqrt{A_0}$ and where $A(x,t)$ is the cross-section area ($A=\pi R^2$ with $R$ the radius of the
 arteria), $Q(x,t)=A(x,t)u(x,t)$ the flow rate or the discharge, $u(t,x)$ the mean flow velocity, $\rho$ the blood
 density, $A_0(x)$ the cross section at rest and $k(x)$ the stiffness of the artery. System \eqref{eq:system1} is into
 the form of the Saint-Venant problem with variable pressure presented in \cite{Bouchut04}. We have to mention that
 arterial pulse wavelengths are long enough to justify the use of a 1D model rather than a 3D model when a global
 simulation of blood flow in the cardiovascular system is needed.

\section{Numerical method}
 Since \cite{Bermudez94,Greenberg96}, it is well known (in the shallow water community) that the scheme should be
 well-balanced for good source term treatment, {\it i.e.} the scheme should preserve at least some steady states.
 For system \eqref{eq:system1}, we should preserve at least the "man at eternal rest" or "dead man equilibrium"
 \cite{Delestre11} (without artifacts such as \cite{Kirkman03}), it writes
\begin{equation}
\left\{\begin{array}{l}
  u=0\\
  \dfrac{1}{\sqrt{\pi} \rho}A^{3/2}k-\dfrac{A}{\sqrt{\pi} \rho}{\cal A}_0=Cst
       \end{array}\right.,
\end{equation}
this means that steady states at rest are preserved (this is the analogous of the "lake at rest" equilibrium). Thus we
 use the scheme proposed in \cite[p.93-94]{Bouchut04} for that kind of model. This is a finite volume scheme with a
 modification of the hydrostatic reconstruction (introduced in \cite{Audusse04c,Bouchut04} for the shallow water
 model).

\subsection{Convective step}
For the homogeneous system
\begin{equation}
\partial_t U+\partial_x F(U,Z)=0,
\end{equation}
which is \eqref{eq:system1} with:
\begin{equation*}
 U=\left(\begin{array}{c}
          A\\
	  Q
         \end{array}\right),\;
Z=\left(\begin{array}{c}
         {\cal A}_0\\
	k
        \end{array}\right)
\;\text{and}\;
F(U,k)=\left(\begin{array}{c}
              Q\\
	    Q^2/A+kA^{3/2}/(3\sqrt{\pi}\rho)
             \end{array}\right),
\end{equation*}
an explicit first order conservative scheme writes
\begin{equation}
 \dfrac{U_i^{n+1}-U_i^n}{\Delta t}+\dfrac{F_{i+1/2}^n-F_{i-1/2}^n}{\Delta x}=0,\label{eq:scheme-hom}
\end{equation}
where $U_i^n$ is an approximation of $U$
\begin{equation*}
 U_i^n\simeq \dfrac{1}{\Delta x} \int_{x_{i-1/2}}^{x_{i+1/2}} U(x,t_n)dx.
\end{equation*}
$i$ refers to the cell $C_i=(x_{i-1/2},x_{i+1/2})=(x_{i-1/2},x_{i-1/2}+\Delta x)$ and $n$ to time $t_n$ with
 $t_{n+1}-t_n=\Delta t$.\\
The two points numerical flux
\begin{equation*}
F_{i+1/2}^n={\cal F}(U_i^n,U_{i+1}^n,k_{i+1/2}^*),
\end{equation*}
 with $k_{i+1/2}^*=\max(k_i,k_{i+1})$, is an approximation of the flux function $F(U,Z)$ at the cell interface $i+1/2$. This
 numerical flux will be detailled in subsection \ref{HLL}.

\subsection{Source terms treatment}
In system \eqref{eq:system1}, the terms $A(\partial_x {\cal A}_0-2\sqrt{A}\partial_x k/3)/(\sqrt{\pi}\rho)$ are
 involved in steady states preservation, they need a well-balanced treatment: the variables are reconstructed locally
 thanks to a variant of the hydrostatic reconstruction \cite[p.93-94]{Bouchut04}
\begin{equation}
 \left\{\begin{array}{l}
         \sqrt{A_{i+1/2L}}=\max(k_i\sqrt{A_i}+\min(\Delta {{\cal A}_0}_{i+1/2},0),0)/k^*_{i+1/2}\\
	  U_{i+1/2L}=(A_{i+1/2L},A_{i+1/2L}.u_i)^t\\
	  \sqrt{A_{i+1/2R}}=\max(k_{i+1}\sqrt{A_{i+1}}-\max(\Delta {{\cal A}_0}_{i+1/2},0),0)/k^*_{i+1/2}\\
	  U_{i+1/2R}=(A_{i+1/2R},A_{i+1/2R}.u_{i+1})^t
        \end{array}\right.,\label{eq:rec-hydro}
\end{equation}
with $\Delta {{\cal A}_0}_{i+1/2}={{\cal A}_0}_{i+1}-{{\cal A}_0}_i=k_{i+1} \sqrt{{A_0}_{i+1}}-k_i \sqrt{{A_0}_i}$ and
 $k_{i+1/2}^*=\max(k_{i},k_{i+1})$. For consistency, the scheme \eqref{eq:scheme-hom} is modified as follows
\begin{equation}
 U_i^{n+1}=U_i^n-\dfrac{\Delta t}{\Delta x}\left(F_{i+1/2L}^n-F_{i+1/2R}^n\right),\label{eq:scheme-source}
\end{equation}
where
\begin{equation*}
 \begin{array}{l}
  F_{i+1/2L}^n=F_{i+1/2}^n+S_{i+1/2L}\\
  F_{i-1/2R}^n=F_{i-1/2}^n+S_{i-1/2R}
 \end{array},
\end{equation*}
with
\begin{equation*}
 \begin{array}{c}
  F_{i+1/2}^n={\cal F}\left(U_{i+1/2L},U_{i+1/2R},k_{i+1/2}^*\right)\\
  S_{i+1/2L}=\left(\begin{array}{c}
    0\\
    {\cal P}(A_i^n,k_i)-{\cal P}(A_{i+1/2L}^n,k_{i+1/2}^*)
  \end{array}\right)\\
 S_{i-1/2R}=\left(\begin{array}{c}
    0\\
    {\cal P}(A_i^n,k_i)-{\cal P}(A_{i-1/2R}^n,k_{i-1/2}^*)
    \end{array}\right)
 \end{array}
\end{equation*}
and ${\cal P}(A,k)=k A^{3/2}/(3\rho \sqrt{\pi})$. Thus the variation of the radius and the varying elasticity are
 treated under a well-balanced way. In system \eqref{eq:system1}, the friction term $-C_f Q/A$ is treated
 semi-implicitly. This treatment is classical in shallow water simulations \cite{Bristeau01,Liang09b} and had shown
 to be efficient in blood flow simulation as well \cite{Delestre11}. This treatment does not break the "dead man"
 equilibrium. It consists in using first \eqref{eq:scheme-source} as a prediction step without friction, {\it i.e.}:
\begin{equation*}
 U_i^*=U_i^n-\dfrac{\Delta t}{\Delta x}\left(F_{i+1/2L}^n-F_{i-1/2R}^n\right),
\end{equation*}
then we apply a semi-implicit friction correction on the predicted values ($U_i^*$):
\begin{equation*}
 A_i^*\left(\dfrac{u_i^{n+1}-u_i^*}{\Delta t}\right)=-C_f u_i^{n+1}.
\end{equation*}
Thus we get the corrected velocity $u_i^{n+1}$ and we have $A_i^{n+1}=A_i^*$.

\subsection{HLL numerical flux}\label{HLL}
As presented in \cite{Delestre11}, several numerical fluxes might be used (Rusanov, HLL, VFRoe-ncv and kinetic fluxes).
 In this work we will use the HLL flux (Harten Lax and van Leer \cite{Harten83}) because it is the best compromise
 between accuracy and CPU time consuming (see \cite[chapter 2]{Delestre10b}). It writes:
\begin{equation*}
 {\cal F}(U_L,U_R,k^*)=\left\{\begin{array}{ll}
                               F(U_L,k^*) & \text{if}\;0\leq c_1\\
			      \dfrac{c_2 F(U_L,k^*)-c_1 F(U_R,k^*)}{c_2-c_1}
+\dfrac{c_1 c_2}{c_2-c_1}(U_R-U_L) & \text{if}\;c_1<0<c_2\\
			      F(U_R,k^*) & \text{if}\;c_2\leq 0
                              \end{array}\right.,
\end{equation*}
with
\[c_{1}={\inf\limits_{U=U_L,U_R}}({\inf\limits_{j\in\{1,2\}}}\lambda_{j}(U,k^*))\;
\text{and}\;c_{2}={\sup\limits_{U=U_L,U_R}}({\sup\limits_{j\in\{1,2\}}}\lambda_{j}(U,k^*)),\label{eq:flux-hll2}\]
where \(\lambda_1(U,k^*)\) and \(\lambda_2(U,k^*)\) are the eigenvalues of the system and $k^*=\max(k_L,k_R)$.

To prevent blow up of the numerical values, we impose the following CFL (Courant, Friedrichs, Levy) condition
\begin{equation*}
 \Delta t\leq n_{CFL}\dfrac{\Delta x}{\max\limits_{i}(|u_i|+c_i)},
\end{equation*}
where $c_i=\sqrt{k_i \sqrt{A_i}/(2\rho\sqrt{\pi})}$ and $n_{CFL}=1$.

\section{Some numerical results}

\subsection{"The stented man at eternal rest"}

\begin{figure}
\begin{tabular}{cc}
  \includegraphics[angle=-90,width = 0.48 \textwidth]{ManAtRestKCol.eps} & \\[-4.2cm]
  &\includegraphics[angle=-90,width = 0.48 \textwidth]{ManAtRestUCol.eps}
\end{tabular}
\caption{No spurious flows (right) are generated by a change of elasticity (left).}
\label{fig:deadman}
\end{figure}

In this test, we consider a configuration with no flow and with a change of artery elasticity $k(x)$, this is
 the case for a dead man with a stented artery (see Figure 1
 left). The section of the artery is constant
 $R_0(x)=4.0\;10^{-3}\text{m}$ and the
 velocity is $u(x,t)=0\text{m/s}$. We use the following numerical values: $J=50$ cells, $C_f=0$, $\rho=1060\text{m}^3$,
 $L=0.14\text{m}$, $T_{end}=5\text{s}$. As initial conditions, we take a fluid at rest $Q(x,0)=0\text{m}^3/\text{s}$
 and
\begin{equation*}
 k(x)=\left\{\begin{array}{ll}
              k_0 &\text{if}\;x\in[0:x_1]\cup [x_4:L] \\
	      k_0+\dfrac{\Delta k}{2}\left(\sin\left(\dfrac{x-x_1}{x_2-x_1}\pi-\pi/2\right)+1\right) & \text{if}\;x\in]x_1:x_2[\\
	      k_0+\Delta k & \text{if}\;x\in[x_2:x_3] \\
	      k_0+\dfrac{\Delta k}{2}\left(\cos\left(\dfrac{x-x_3}{x_4-x_3}\pi\right)+1\right) & \text{if}\;x\in]x_3:x_4[
             \end{array}\right.,
\end{equation*}
with $k_0=1.0\;10^8\text{Pa}/\text{m}$, $\Delta k=6.0\;10^7\text{Pa}/\text{m}$, $x_1=1.0\;10^{-2}\text{m}$,
 $x_2=3.05\;10^{-2}\text{m}$, $x_3=4.95\;10^{-2}\text{m}$ and $x_4=7.0\;10^{-2}\text{m}$.

The steady state at rest is perfectly preserved in time, we do not notice any spurious oscillation (see Figure 1
 right).

\subsection{Wave reflection-transmission in a stented artery}

We now observe the reflexion and transmission of a pulse through a sudden change of artery elasticity (from $k_R$
 to $k_L$ with $k_L>k_R$) in an elastic tube of constant radius (see Figure 2 left). We take the following numerical
 values: $J=1500$
 cells, $C_f=0$,
 $k_L=1.6\;10^{8}\text{Pa/m}$,
 $k_R=1.\;10^{8}\text{Pa/m}$, $\Delta k=6.\;10^{7}\text{Pa/m}$,
 $\rho=1060\text{m}^3$, $R_0=4.0\;10^{-3}\text{m}$, $L=0.16\text{m}$, $T_{end}=8.0\;10^{-3}\text{s}$,
 $c_L=\sqrt{k_L R/(2\rho)}\simeq 17.37\text{m/s}$ and $c_R=\sqrt{k_R R/(2\rho)}\simeq 13.74\text{m/s}$. We take
 a decreasing elasticity on a rather small scale:
\begin{equation*}
k(x)=
\left\{\begin{array}{ll}
        k_R+\Delta k & \text{if}\; x\in[0:x_1]\\
	k_R+\dfrac{\Delta k}{2}\left[1+\cos\left(\dfrac{x-x_1}{x_2-x_1}\pi \right)\right] & \text{if}\;x\in]x_1:x_2]\\
	k_R & \text{else}
       \end{array}\right.,
\end{equation*}
with $x_1=19L/40$ and $x_2=L/2$. As initial conditions, we consider a fluid at rest $Q(x,0)=0\text{m}^3/\text{s}$ and
 the following perturbation of radius:
\begin{equation*}
 R(x,0)=\left\{\begin{array}{ll}
		  R_0(x)\left[1+\epsilon \sin\left(\dfrac{100}{20L}\pi\left(x-\dfrac{65L}{100}\right)\right)\right]
 & \text{if}\;x\in[65L/100:85L/100]\\
		  R_0(x) & \text{else}
               \end{array}\right.,
\end{equation*}
with $\epsilon=1.0\;10^{-2}$. The expression for the pressure is
\begin{equation*}
 p(x,t)=p_0(x)+k(x)(R(x,t)-R_0(x)),
\end{equation*}
where $p_0$ is the external pressure.

The numerical results perfectly match with the predictions for a linearized flow. We get the predicted amplitudes both for the
 transmitted and the reflected waves (see Figure 2 right).

\begin{figure}
\begin{tabular}{cc}
  \includegraphics[angle=-90,width = 0.48 \textwidth]{StentedkCol.eps} & \\[-4.2cm]
  &\includegraphics[angle=-90,width = 0.48 \textwidth]{WaveTransReflCol1.eps}
\end{tabular}
\caption{Across a discontinuity of $k$ (left), an initial pulse evolves (right) ; it is transmitted and reflected.}
\label{fig:stented}
\end{figure}

\subsection{Wave "damping"}

In this case, the elasticity is constant in space. We consider the viscous term in the linearized momentum equation.
 A periodic signal is imposed at inflow as a perturbation of a steady state ($R_0=Cst$, $u_0=0$) with a constant
 section at rest. We take $R=R_0+\epsilon R_1$ and $u=0+\epsilon u_1$, where ($R_1$,$u_1$) is the perturbation of the
 steady state.
 Looking for progressive waves ({\it i.e.} under the form $e^{i(\omega t-Kx)}$), we take for the imposed incoming
 discharge
\begin{equation*}
 Q_b(t)=Q(t,x=0)=Q_{amp} \sin(\omega t)\;\text{m}^3/\text{s}.
\end{equation*}
Thus, we have a damping wave in the domain
 \begin{equation*}
  Q(t,x)=\left\{\begin{array}{ll}
                 0 & \text{if}\; k_r x>\omega t\\
		Q_{amp} \sin(\omega t-k_r x)e^{k_i x} &\text{if}\; k_r x \leq \omega t
                \end{array} \right.,
 \end{equation*}
with $\omega=2\pi/T_{pulse}$, $T_{pulse}$ the time length of a pulse and $K=k_r+i k_i$ the wave vector.

We use the following numerical values: $J=750$ cells, $\rho=1060\text{m}^3$, $L=3\text{m}$,
 $k=1.10^{8}\text{Pa/m}$, $R_0=4.10^{-3}\text{m}$ and $T_{end}=5\text{s}$. We consider both  $C_f=0$ and
 $C_f=0.005053$.
 As initial conditions, we take a fluid at rest
 $Q(x,0)=0\text{m}^3/\text{s}$ and as input boundary condition
\begin{equation*}
 Q_b(t)=Q_{amp}\sin(\omega t),
\end{equation*}
with $\omega=2\pi/T_{pulse}=2\pi/0.5\text{s}$ and $Q_{amp}=3.45\;10^{-7}\text{m}^3/\text{s}$. The output is an outgoing wave.

The results are closed to the analytical solution (see Figure 3). We notice a small numerical diffusion for
 $C_f=0.005053$.

\begin{figure}
\begin{tabular}{cc}
  \includegraphics[angle=-90,width = 0.48 \textwidth]{QColCf_0.eps} & \\[-4.2cm]
  &\includegraphics[angle=-90,width = 0.48 \textwidth]{QColCf_0005053.eps}
\end{tabular}
\caption{Response to an oscillating input condition, left with no viscosity and right with viscosity.}
\label{fig::wave}
\end{figure}

\section{Conclusion}

In this work, we have considered the 1D model of flow in an artery with varying elasticity and constant section.
 We have presented a well-balanced finite volume scheme. Thus we get a mass conservative scheme. Moreover, the
 well-balanced property allows to have a good treatment of the source, {\it i.e.} we do not get numerical artifacts.
 This numerical method gave good results on numerical tests. In future works, we will have to add some extra source
 terms in order to get a more realistic model. These extra terms will require to develop a low diffusive high order
 scheme in the spirit of \cite{Delestre10}. Moreover, this will improve the accuracy of the scheme.
 And we will also have to test more complex cases such as bifurcations and networks.
\bibliographystyle{plain}


\section{#1}}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\def\mathacute{\mathaccent"7013}
\def\rm Vol{\rm Vol}
\def\rm constant{\rm constant}
\def\mathbb{N}{\mathbb{N}}
\def\mathbb{R}{\mathbb{R}}
\def\mathbb{C}{\mathbb{C}}
\def{\cal V}{{\cal V}}
\def{\rm i}{{\rm i}}
\def{\rm c.c.}{{\rm c.c.}}
\def{\rm d}{{\rm d}}
\def{\rm SU}(2){{\rm SU}(2)}

\newcommand{\hepth}[1]{\href{http://arxiv.org/abs/hep-th/#1}{arXiv:hep-th/#1}}

\newcommand{\mathph}[1]{\href{http://arxiv.org/abs/math-ph/#1}{arXiv:math-ph/#1}}

\newcommand{\physics}[1]{\href{http://arxiv.org/abs/physics/#1}{arXiv:physics/#1}}

\newcommand{\grqc}[1]{\href{http://arxiv.org/abs/gr-qc/#1}{arXiv:gr-qc/#1}}

\newcommand{\qalg}[1]{\href{http://arxiv.org/abs/q-alg/#1}{arXiv:q-alg/#1}}

\newcommand{\Math}[2]{\href{http://arxiv.org/abs/math.#1/#2}{arXiv:math.#1/#2}}

\newcommand{\MMath}[1]{\href{http://arxiv.org/abs/math/#1}{arXiv:math.#1}}

\newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1/}{arXiv:#1}}

\newcommand{\webpage}[1]{{\color{blue}}\url{#1}{\color{blue}}}

\begin{document}

\title{Emergence of a low spin phase in group field theory condensates}
\author{Steffen Gielen}
\affiliation{Theoretical Physics, Blackett Laboratory, Imperial College London, London SW7 2AZ, U.K. (still EU)}
\email{s.gielen@imperial.ac.uk}

\date{\today}

\begin{abstract}

Recent results have shown how quantum cosmology models can be derived from the effective dynamics of condensate states in group field theory (GFT), where `cosmology is the hydrodynamics of quantum gravity': the classical Friedmann dynamics for homogeneous, isotropic universes, as well as loop quantum cosmology (LQC) corrections to general relativity have been shown to emerge from fundamental quantum gravity. We take one further step towards strengthening the link with LQC and show, in a class of GFT models for gravity coupled to a free massless scalar field and for generic initial conditions, that GFT condensates dynamically reach a low spin phase of many quanta of geometry, in which all but an exponentially small number of quanta are characterised by a single spin $j_0$ (i.e. by a constant volume per quantum). As the low spin regime is reached, GFT condensates expand to exponentially large volumes, and the dynamics of the total volume follows precisely the classical Friedmann equations. This behaviour follows from a single requirement on the couplings in the GFT model under study. We present one particular simple case in which the dominant spin is the lowest one: $j_0=0$ or, if this is excluded, $j_0=1/2$. The type of quantum state usually assumed in the derivation of LQC is hence derived from the quantum dynamics of GFT. These results confirm and extend recent results by Oriti, Sindoni and Wilson-Ewing in the same setting.

\end{abstract}

\keywords{group field theory, hydrodynamics of quantum gravity, loop quantum cosmology, continuum limit in LQG}

\maketitle

\sect{Introduction}

Early universe cosmology is widely seen as the most important arena where quantum gravity could have observational consequences, potentially leading to the validation or exclusion of different quantum gravity approaches as well as providing new input for theoretical cosmology. For practitioners of quantum gravity, this poses the challenge of deriving an effective description at cosmologically relevant scales from the fundamental dynamics defined at the Planck scale. For a fundamental description in terms of discrete excitations of quantum geometry, this process also encompasses the emergence of an effective continuum spacetime from a large number of fundamental quanta \cite{emergence}. This situation is familiar in condensed matter physics: phenomena such as superfluidity can be described macroscopically in terms of a continuum hydrodynamic description, but have a microscopic origin in Bose--Einstein condensation in many-particle quantum physics \cite{BECbook}.

In the context of loop quantum gravity (LQG) \cite{LQG} and its second quantisation formulation \cite{2ndquant} {\em group field theory} (GFT) \cite{GFT}, a line of research has been following this analogy, describing a macroscopic Universe as a {\em condensate of quanta of geometry}, themselves given by quantum excitations of a GFT field \cite{GFCreview}. Rather than following the traditional minisuperspace strategy of quantum cosmology \cite{QCreview} and `quantising' large-scale degrees of freedom such as the scale factor, one aims to derive an effective description at large scales from the `hydrodynamics of quantum gravity'. Being able to obtain such an effective description provides an important self-consistency test for quantum gravity, as well as leading to potentially novel and interesting phenomenology to be explored in cosmology. In the context of LQG and GFT, a particular focus has been on understanding the possible link between loop quantum cosmology (LQC), which is derived using minisuperspace techniques together with input from LQG \cite{LQC}, and a more fundamental description, as in GFT. Indeed, the idea of the universe as a condensate-like configuration of many interacting `patches' can also be understood as a general approach for going beyond the minisuperspace truncation \cite{Martinreview}.

One of the initial aims of work on GFT condensates has been to derive the classical Friedmann equations for homogeneous, isotropic universes in an appropriate large-scale or semiclassical limit, starting with \cite{letter,JHEP} where a WKB approximation was employed to this effect. In particular, \cite{JHEP} introduced a natural way of coupling a free, massless scalar field to gravity in the context of GFT, showing that this leads to the correct matter coupling also at the level of the emergent Friedmann equations, in the same WKB limit, with an effective Newton's constant related to couplings in the GFT action. Considering models of gravity coupled to a massless scalar is of particular interest when comparing with LQC where such a scalar appears as a relational clock variable. A more general class of GFT actions with a similar coupling to a massless scalar field was then recently studied in much more detail in \cite{DLElong,DLEshort}, and several interesting results were found that strengthen the link of GFT condensate dynamics to minisuperspace models of LQC: a systematic way of imposing isotropy on the state led to simpler effective (Gross--Pitaevskii-like) equations for the `condensate wavefunction' (or mean field). In a weak-coupling approximation, these could be translated into {\em relational Friedmann equations} for the volume of the universe $V(\phi)$, seen as a function of `relational time' $\phi$ given by the scalar field\footnote{The scalar field can serve as a clock since its momentum $\pi_\phi$ is conserved and therefore $\phi$ evolves monotonically.}. It was shown that, for a generic choice of mean field and at large volume\footnote{Note that large volume corresponds to small curvature when the Friedmann equation holds, which is what one would expect as the semiclassical limit.}, these reduce precisely to the Friedmann equations of classical general relativity coupled to a massless scalar (where the exact solution is $V(\phi)=V_0\exp(\pm\sqrt{12\pi G}\phi)$), provided an infinite number of parameters in the GFT action are equal (and then identified with $3\pi G$); approximate equality corresponds to classical Friedmann equations with corrections. In the special case of a {\em single-spin} condensate where only quantum geometric excitations of a fixed spin $j=j_o$ (i.e. fundamental quanta with given fixed areas or volumes) are excited, only a single parameter needed to be fixed to recover the Friedmann equations, which then also include high-curvature corrections precisely of the form of the effective equations of LQC, plus further quantum corrections.

The single-spin assumption is particularly relevant for LQC, where in order to derive the quantum-corrected Friedmann equation one uses the picture of a Universe consisting of a large number of LQG quanta with spin $j=1/2$, so that fundamental plaquette areas take the minimum non-zero value in LQG (see, e.g., section IIIB of \cite{BianchiI}); expansion or contraction of the Universe proceeds by creation or annihilation of fundamental quanta. This leads to the idea of {\em lattice refinement} for LQC \cite{latticeref} from which the `improved dynamics' form of holonomy corrections \cite{improdyn} is thought to originate. Some steps towards understanding lattice refinement from the perspective of GFT condensates were taken in \cite{NJP}, leaving the crucial relation $N=N(V)$ between the total number of quanta and the volume of the Universe (or, equivalently, the scale factor) undetermined.

In this paper, we strengthen and extend the results reported in \cite{DLElong,DLEshort}. Namely, within the class of GFT models considered in \cite{DLElong,DLEshort}, we exhibit a single condition -- that the ratio $B_j/A_j$ of $j$-dependent couplings $A_j$ and $B_j$ takes a positive maximum for some $j>0$ -- for the following to occur: if the condition is satisfied, a phase in which almost all quanta in a GFT condensate have the same spin emerges {\em dynamically} in the mean-field approximation, for generic choices of mean field. In this process, the Universe grows to an exponentially large volume and follows precisely the classical Friedmann dynamics. Compared to \cite{DLElong,DLEshort}, these new results show that a large-volume or small-curvature limit in which the Friedmann equations hold is reached generically, with weaker assumptions on the GFT model than considered before; they show that the single-spin assumption is always satisfied asymptotically; the latter result then also leads directly to the `improved dynamics' scheme in LQC. Furthermore, it was already shown in \cite{DLElong,DLEshort} that the volume has a non-zero minimum leading to a bounce and resolution of singularities, and this remains true here. In order to clarify the link to previous papers, we first show our results in the case of the model proposed in \cite{JHEP}, and then generalise to the wider class of models of \cite{DLElong,DLEshort}. In the model of \cite{JHEP} and when there are no quanta with spin $j=0$, a $j=1/2$ phase emerges, giving further support for the model that adds to its motivation from the theoretical GFT perspective.

An important remark concerns a difference between the Hilbert spaces of GFT and LQG \cite{newGFTreview}: in LQG, excitations with $j=0$ have no physical significance; by cylindrical consistency, a state containing $j=0$ links is equivalent to one in which these are removed \cite{LQG}. This is not true in GFT, where $j=0$ excitations contribute to the total number of quanta but (for the most common choices of area and volume operators) have zero volume and area, so that a match between the LQC and GFT dynamics appears impossible for these $j=0$ quanta. We take this fact into account by only considering GFT condensates with no $j=0$ excitations. Within the approximations taken, this condition is preserved by the dynamics, and is therefore a consistent restriction on the choice of mean field, within the mean-field approximation. It constitutes fine-tuning from the perspective of GFT condensates, but appears necessary for a closer link of GFT condensates to LQG.

\sect{Cosmology from GFT condensates}

In this section we review the most essential aspects of GFT as an approach to quantum gravity, the extraction of an effective dynamics for coherent condensate states, and the connection to cosmology. For more background, we refer to the review \cite{GFCreview} and to the extensive discussions in \cite{JHEP, DLElong}.

In GFT, spacetime geometry emerges from excitations of a (complex) quantum field $\varphi$, defined on a group manifold that is {\em not} physical space or spacetime, but the configuration space of the degrees of freedom of a tetrahedron (with boundary). Concretely, if the local gauge group of gravity is taken to be ${\rm SU}(2)$, the gravitational configuration space is ${\rm SU}(2)^4/{\rm SU}(2)$: there are four parallel transports associated to four links dual to the faces of the tetrahedron, with invariance under diagonal right multiplication by an element of ${\rm SU}(2)$ which corresponds to a discrete gauge transformation at the vertex where these links meet. In the models we are interested in, one also includes a scalar field $\phi$ as an additional degree of freedom on the vertex of the tetrahedron \cite{JHEP,DLElong}. $\varphi$ is then defined as
\begin{equation}
\varphi: {\rm SU}(2)^4\times \mathbb{R} \rightarrow \mathbb{C}\,,\quad \varphi(g_I,\phi)=\varphi(g_Ih,\phi)\quad\forall h\in{\rm SU}(2)\,.
\label{fielddef}
\end{equation}
Here and in the following, the index $I$ runs over the four ${\rm SU}(2)$ arguments of $\varphi$, corresponding to four faces of the tetrahedron. The domain space of the scalar field is $\mathbb{R}$; an alternative choice would be ${\rm U}(1)$, the space of `point holonomies' used in LQG \cite{JHEP}.

Dynamics are now defined through an action
\begin{equation}
S[\varphi,\bar\varphi]=-\int ({\rm d} g)^4\,{\rm d}\phi\;\bar\varphi(g_I,\phi)\mathcal{K}\varphi(g_I,\phi) + \mathcal{V}[\varphi,\bar\varphi]
\label{akshn}
\end{equation}
which we split into kinetic (quadratic) and interaction parts, where the kinetic part is assumed to be local. A trivial kinetic term ($\mathcal{K}=1$) is sufficient to define a class of GFT models whose Feynman amplitudes correspond to the amplitudes of a class of spin foam models \cite{RovelliReis} through the choice of $\mathcal{V}$. This indeed motivated GFT as a field-theoretic description of spin foam dynamics. More general choices of $\mathcal{K}$ are possible and, as usual, renormalisation affects the form of the action and can generate additional terms. In particular, a Laplace--Beltrami operator on ${\rm SU}(2)^4$ seems naturally generated by quantum fluctuations \cite{radiative} (see also e.g. \cite{GFTrenorm} for recent more general results on GFT renormalisation), motivating us to study such more general choices of $\mathcal{K}$.

The theory is now defined either through a covariant path integral (with perturbative expansion in Feynman amplitudes) or in an operator formalism. In the latter, one promotes $\varphi$ and $\bar\varphi$ to field operators with bosonic commutation relations
\begin{equation}
[\hat\varphi(g_I,\phi),\hat\varphi(g'_I,\phi')]=[\hat\varphi^\dagger(g_I,\phi),\hat\varphi^\dagger(g'_I,\phi')]=0\,,\quad [\hat\varphi(g_I,\phi),\hat\varphi^\dagger(g'_I,\phi')]={\bf 1}(g_I,g'_I)\delta(\phi-\phi')\,,
\end{equation}
where ${\bf 1}(g_I,g'_I)=\int {\rm d} h\prod_{I=1}^4 \delta(g_I h (g'_I)^{-1})$ is a delta distribution on ${\rm SU}(2)^4/{\rm SU}(2)$. One can then define a Fock space, starting from a `no-space' vacuum $|0\rangle$ annihilated by $\hat\varphi(g_I,\phi)$;
\begin{equation}
\hat\varphi^\dagger(g_I,\phi)|0\rangle \equiv |g_I,\phi\rangle
\end{equation}
is a one-particle state, representing a single tetrahedron with connection data given by parallel transports $g_I$ and the value $\phi$ for the scalar field. Such a single-particle state is just the smallest perturbation of a vacuum in which no space is present at all; in order to describe a macroscopic continuum Universe one needs to excite a large number of GFT `particles'. A good candidate for this is a GFT {\em condensate} state, whose main property is {\em wavefunction homogeneity}: the state is determined by a single-particle wavefunction.

The simplest choice for such a state is
\begin{equation}
|\sigma\rangle =\mathcal{N}(\sigma) \exp\left(\int ({\rm d} g)^4\,{\rm d}\phi\;\sigma(g_I,\phi)\,\hat\varphi^\dagger(g_I,\phi)\right)|0\rangle\,,
\label{condendef}
\end{equation}
where $\sigma(g_I,\phi)$ is called the `condensate wavefunction' and $\mathcal{N}(\sigma)$ is a normalisation factor. $\sigma(g_I,\phi)$ itself is not normalised, but determines the number of quanta in the state;
\begin{equation}
N(\phi)=\langle\sigma|\hat{N}(\phi)|\sigma\rangle\equiv \int ({\rm d} g)^4 \;\langle\sigma|\hat\varphi^\dagger(g_I,\phi)\hat\varphi(g_I,\phi)|\sigma\rangle = \int ({\rm d} g)^4\;|\sigma(g_I,\phi)|^2
\end{equation}
is the expectation value of the number of quanta $N(\phi)$ at the value $\phi$ of the scalar field \cite{DLElong}. For further discussion of the role of wavefunction homogeneity, arguments for the interpretation of $|\sigma\rangle$ as a macroscopic, homogeneous geometry, and the interpretation of $\sigma$ as a wavefunction, see \cite{GFCreview}. One can define more general condensates satisfying wavefunction homogeneity, containing additional topological structure and defined by a sum over connected graphs of arbitrary complexity \cite{gencond}.

Due to (\ref{fielddef}), $\sigma$ satisfies $\sigma(g_I,\phi)=\sigma(g_Ih,\phi)$ for all $h\in{\rm SU}(2)$; in order for the state to only contain the gauge-invariant degrees of freedom of a tetrahedron, one also imposes left invariance $\sigma(g_I,\phi)=\sigma(kg_I,\phi)$ for all $k\in{\rm SU}(2)$, so that $\sigma$ is a function on ${\rm SU}(2)\backslash{\rm SU}(2)^4/{\rm SU}(2)$, which is isomorphic to the space of connection degrees of freedom of a homogeneous Universe \cite{CQG}.

Equation (\ref{condendef}) is a coherent state, satisfying
\begin{equation}
\hat\varphi(g_I,\phi)|\sigma\rangle = \sigma(g_I,\phi)|\sigma\rangle
\end{equation}
which implements the {\em mean-field approximation}: all normal-ordered expectation values of products of field operators can be evaluated by replacing $\hat\varphi$ by $\sigma$ and $\hat\varphi^\dagger$ by $\bar\sigma$. This approximation is commonly used in condensed matter physics, where it captures well the physics of diluted, weakly interacting Bose--Einstein condensates \cite{BECbook}. It represents a semiclassical limit, in which field operators can be replaced by classical fields. It is thus very well suited for describing the emergence of a semiclassical, macroscopic and nearly homogeneous universe in which spatial gradients are small. Just as in condensed matter physics, one expects the mean-field approximation to break down when interactions get stronger, i.e. in the early Universe in cosmology. We refer to \cite[section 4.2]{DLElong} for a discussion of criteria for the validity of the mean-field approximation in the GFT context.

Within the mean-field approximation, equations encoding the GFT quantum dynamics can be rewritten as conditions on the mean field $\sigma(g_I,\phi)$. As a first approximation to the full dynamics, consider the expectation value of the GFT operator equation of motion,
\begin{equation}
\langle\sigma|\frac{\delta\hat{S}[\hat\varphi,\hat\varphi^\dagger]}{\delta\hat\varphi^\dagger(g_I,\phi)}|\sigma\rangle = \frac{\delta S[\sigma,\bar\sigma]}{\delta\bar\sigma(g_I,\phi)}=-\mathcal{K}\sigma(g_I,\phi) + \frac{\delta \mathcal{V}[\sigma,\bar\sigma]}{\delta\bar\sigma(g_I,\phi)}=0\,.
\label{grosspit}
\end{equation}
In condensed matter physics, this procedure leads to the {\em Gross--Pitaevskii equation} for the condensate wavefunction. For a GFT condensate, (\ref{grosspit}) and corresponding higher order equations define the cosmological dynamics. Indeed, one of the main goals of work on GFT condensates is to show how (\ref{grosspit}) leads to effective cosmological dynamics consistent with general relativity at low curvature, and with interesting quantum corrections.

\sect{Solving the condensate dynamics}

Let us start by revisiting the GFT for gravity coupled to a scalar field introduced in \cite{JHEP}, with
\begin{equation}
\mathcal{K}=\mu+ \sum_I \Delta_{g_I} + \tau\frac{\partial^2}{\partial\phi^2}\,,
\label{kinetic}
\end{equation}
where $\Delta_{g_I}$ is the Laplace--Beltrami operator acting on the $I$th ${\rm SU}(2)$ argument, and we have absorbed a possible coupling in front of it into a normalisation of the action. As we have mentioned, adding a Laplacian in $g_I$ to a constant $\mathcal{K}$ is well motivated by studies of GFT renormalisation, and it is natural to include the same derivative with respect to the matter variable $\phi$. The latter, in turn, has also been derived from independent arguments as a way to couple a free scalar field to gravity in the GFT setting \cite{DLElong,DLEshort}.

Using (\ref{kinetic}) means that one can study a weak-coupling limit in which the contribution of $\mathcal{V}$ in (\ref{grosspit}) is neglected. This approximation has been employed in most studies so far \cite{GFCreview}, and is again motivated by the necessity for the Universe to be nearly homogeneous with small spatial gradients. From the microscopic point of view, interactions have to be weak for the simple coherent states (\ref{condendef}) to be a reliable approximation to the exact physical state.

The resulting equation of motion for $\sigma(g_I,\phi)$,
\begin{equation}
-\left(\mu+ \sum_I \Delta_{g_I} + \tau\frac{\partial^2}{\partial\phi^2}\right)\sigma(g_I,\phi) = 0\,,
\label{dyneq}
\end{equation}
now admits an easy direct solution. Using the Peter--Weyl decomposition \cite{PeterWeyl} of functions on ${\rm SU}(2)$ into representations, together with $\sigma(g_I,\phi)=\sigma(g_Ih,\phi)=\sigma(kg_I,\phi)$, gives \cite[equation (52)]{DLElong}
\begin{equation}
\sigma(g_I,\phi)= \sum_{j_I,\iota_l,\iota_r}\sigma^{j_I\iota_l\iota_r}(\phi)\;C^{j_I\iota_l\iota_r}(g_I)
\end{equation}
where $j_I\in\mathbb{N}_0/2$ are representation labels (spins) for ${\rm SU}(2)$, and $C^{j_I\iota_l\iota_r}(g_I)$ are given, up to factors of $(2j_K+1)$, by a product of four Wigner $D$-matrices $D^{j_K}_{m_K n_K}(g_K)$ (for $K=1,\ldots,4$) with their $m$ and $n$ indices contracted by intertwiners $\iota_l$ and $\iota_r$. The Wigner matrices are eigenfunctions of the ${\rm SU}(2)$ Laplacian, so that (\ref{dyneq}) becomes
\begin{equation}
\left(\sum_I j_I(j_I+1) -\mu - \tau\frac{\partial^2}{\partial\phi^2}\right)\sigma^{j_I\iota_l\iota_r}(\phi) = 0\,.
\label{jdyneq}
\end{equation}
As shown in \cite{DLElong}, a natural way to impose isotropy on the condensate is to demand that it consists of equilateral tetrahedra: $\sigma^{j_I\iota_l\iota_r}(\phi)$ is only non-zero if all $j_I$ are equal, and the intertwiners $\iota_l$ and $\iota_r$ are fixed to maximise the tetrahedron volume for given $j$. With these assumptions, the first term in the brackets in (\ref{jdyneq}) becomes $4j(j+1)$, and the general solution is simply
\begin{equation}
\sigma_j(\phi)=\alpha^+_j\exp\left(\sqrt{\frac{4j(j+1)-\mu}{\tau}}\,\phi\right)+\alpha^-_j\exp\left(-\sqrt{\frac{4j(j+1)-\mu}{\tau}}\,\phi\right)\,.
\label{exactsol}
\end{equation}
The behaviour of these solutions clearly depends on whether $(4j(j+1)-\mu)/\tau$ is positive or negative: they either exponentially grow and decay or oscillate. We assume $\tau<0$, which is required for the correct coupling of matter to gravity in the effective Friedmann equation \cite{JHEP}. Then, for $\mu>0$, the behaviour of the different spin components changes dramatically above a critical spin $j_c=\frac{1}{2} \lfloor \sqrt{\mu+1}-1\rfloor$; the $\sigma_j$ with $j>j_c$ oscillate around $\alpha_j^\pm$, whereas $\sigma_j$ with $j\le j_c$ generically (unless some $\alpha^\pm_j$ are zero) grow exponentially as $\phi\rightarrow\pm \infty$, with fastest growth for $j=0$, followed by $j=\frac{1}{2}$, etc. We exclude $\mu<0$, where there are only oscillating solutions which do not lead to a macroscopic universe.

As $|\sigma_j(\phi)|^2$ is the expectation value for the number of quanta in the $j$ representation at `time' $\phi$, the condensate is quickly dominated by $j=0$ quanta, with occupation numbers for higher $j$ exponentially suppressed. As we have said, such quanta do not have a clear interpretation from the perspective of LQG, where one could remove them using cylindrical consistency. In order to facilitate comparison between the effective GFT dynamics and LQC, we hence set $\alpha_0=\beta_0=0$. This is a finely-tuned initial condition in GFT, but self-consistent as all $j$ are decoupled in (\ref{jdyneq}).

The dominant contribution at large $|\phi|$ then comes from $\sigma_{1/2}(\phi)$, and the total volume $V(\phi)$ at the value $\phi$ of the scalar field asymptotes to
\begin{equation}
V(\phi) = \sum_j V_j\,|\sigma_j(\phi)|^2 \;\stackrel{\phi\rightarrow\pm\infty}{\sim}\; V_{1/2} \left|\alpha^\pm_{1/2}\right|^2 \exp\left(\pm 2\sqrt{\frac{\mu-3}{|\tau|}}\,\phi\right)
\label{volumeev}
\end{equation}
with the expression for $V(\phi)$ given in \cite{DLElong}; $V_{1/2}$ is the (Planckian) volume in LQG for an equilateral tetrahedron with $j=1/2$ (and more generally, $V_j\sim j^{3/2}V_{1/2}$). For positive and negative large $\phi$, (\ref{volumeev}) reduces to the solution for the volume of a homogeneous, isotropic Universe in classical general relativity coupled to a massless scalar field
\begin{equation}
V=V_0\exp(\pm\sqrt{12\pi G}\phi)\,,
\end{equation}
if we identify $\frac{\mu-3}{|\tau|}=3\pi G$. In particular, (\ref{volumeev}) in this regime satisfies the classical Friedmann equations. Due to the exponentials in (\ref{exactsol}), this regime is reached quickly as $\phi\gtrsim m_{{\rm Pl}}$, with emergent Planck mass $m_{{\rm Pl}}\sim G^{-1/2}=\sqrt{\frac{3\pi|\tau|}{\mu-3}}$. The full solution $V(\phi)$ interpolates between a contracting and an expanding classical Universe, reminiscent of the bounce in LQC \cite{LQC}.

Note that the last term in (\ref{volumeev}) is proportional to the asymptotic total number of quanta $N(\phi)$ and thus for $\phi\gtrsim m_{{\rm Pl}}$, $N(V)\propto V$, the relation needed for the improved dynamics scheme \cite{improdyn} in LQC. This is merely a different way of saying that asymptotically almost all quanta have spin $j=1/2$, fixing the ratio of total volume to the number of quanta.

Using methods developed in \cite{DLElong,DLEshort} for relational observables in GFT and excluding GFT quanta with $j=0$, we have thus shown that the GFT kinetic term (\ref{kinetic}) with $\mu>0$ and $\tau<0$ gives effective dynamics in which generic initial conditions for $j\neq 0$ lead to a regime where almost all quanta have spin $1/2$, and the total volume follows precisely the classical Friedmann equations. This provides strong evidence that this class of GFT models leads to the correct large-scale, semiclassical limit, adding to the motivation for (\ref{kinetic}) in particular from GFT renormalisation. It also strengthens the results in \cite{JHEP} for the emergence of a Friedmann equation with correct coupling to matter.

These results can be strengthened further by straightforward extension to a more general class of GFT models. In particular, one can study more general dynamical equations
\begin{equation}
\left(-B_j+ A_j\frac{\partial^2}{\partial\phi^2}\right)\sigma_j(\phi) = 0\,,
\label{generaleq}
\end{equation}
as done in \cite{DLElong,DLEshort}, with general solution
\begin{equation}
\sigma_j(\phi)=\alpha^+_j\exp\left(\sqrt{\frac{B_j}{A_j}}\,\phi\right)+\alpha^-_j\exp\left(-\sqrt{\frac{B_j}{A_j}}\,\phi\right)\,.
\label{anothersol}
\end{equation}
This solution was essentially given in \cite{MarcoMairi}, and used to study the bounce for single-spin condensates. Again, its properties depend on the sign of $B_j/A_j$; $B_j/A_j>0$ leads to exponentially expanding and contracting universes, whereas for $B_j/A_j<0$ the number of quanta $|\sigma_j(\phi)|^2$ oscillates around some initial configuration. Furthermore, for {\em all models} for which $B_j/A_j$ has a positive maximum for some $j=j_0$ (again excluding $j=0$)
\begin{equation}
V(\phi) = \sum_j V_j\,|\sigma_j(\phi)|^2 \;\stackrel{\phi\rightarrow\pm\infty}{\sim}\; V_{j_0} \left|\alpha^\pm_{j_0}\right|^2 \exp\left(\pm 2\sqrt{\frac{B_{j_0}}{A_{j_0}}}\,\phi\right)\,;
\end{equation}
in this case, to obtain an exponentially expanding Universe satisfying the classical Friedmann dynamics at large volume, for generic choices of mean field, only a single identification $B_{j_0}/A_{j_0}=3\pi G$ is needed. If the maximum occurs at a low $j_0$, a low spin regime emerges dynamically. These observations confirm and extend the results in \cite{DLElong,DLEshort}, where $B_j/A_j=3\pi G$ for all $j$ was assumed to recover the correct classical limit; this is not necessary.

To facilitate further comparison, we compute the conserved quantities $E_j$ and $Q_j$ identified in \cite{DLElong,DLEshort} for the solution (\ref{anothersol}). For $B_j/A_j>0$, they are
\begin{equation}
E_j=-4\frac{B_j}{A_j}\,{\rm Re}(\alpha_j^+ \overline{\alpha_j^-})\,,\quad Q_j=2\sqrt{\frac{B_j}{A_j}}\,{\rm Im}(\alpha_j^+ \overline{\alpha_j^-})\,,
\label{EQ1}
\end{equation}
whereas for $B_j/A_j<0$
\begin{equation}
E_j=-2\frac{B_j}{A_j}(|\alpha_j^+|^2+|\alpha_j^-|^2)\,,\quad Q_j = \sqrt{\left|\frac{B_j}{A_j}\right|}(|\alpha_j^+|^2-|\alpha_j^-|^2)\,.
\label{EQ2}
\end{equation}
The classical limit assumed in \cite{DLElong,DLEshort} to recover the Friedmann equations is $|\sigma_j|^2\gg \left|\frac{A_j}{B_j}E_j\right|$ and $|\sigma_j|^2\gg \left|\sqrt{\left|\frac{A_j}{B_j}\right|}Q_j\right|$. For $B_j/A_j>0$, this classical limit is reached at $|\phi| \gtrsim \sqrt{\frac{A_j}{B_j}}$ for generic initial conditions, whereas spins with $B_j/A_j<0$ never reach such a regime, as these modes fail to expand sufficiently. The requirement of reaching a classical limit is only a restriction on the values of the couplings in the GFT action, and thus allows distinguishing between different GFT models. For appropriate couplings, the classical limit is reached generically without further assumptions.

The last statement might look puzzling from the perspective of quantum cosmology, where only special states lead to the emergence of a classical Universe; how can it be that this happens generically for GFT condensates? The answer is that the mean-field approximation already is a statement about semiclassicality; failure of semiclassicality is manifest in the breakdown of this approximation, rather than any condition on the `condensate wavefunction', consistent with the conceptual basis of GFT condensate cosmology \cite{CQG,NJP,JHEPalone,GFCreview}. One needs to check whether the assumption of negligible contributions from the GFT interactions that is used in the mean-field approximation is self-consistent \cite{DLElong}.

As a last comment, we repeat the results of \cite{DLElong,DLEshort} regarding the resolution of singularities: as long as one of the $Q_j$ is non-zero, the total volume has a non-zero bound from below. The explicit expressions (\ref{EQ1}) and (\ref{EQ2}) make clear that this condition is satisfied for almost all condensate states, meaning that singularity resolution is generic.

\sect{Conclusions}

Starting from the GFT analogue of the Gross--Pitaevskii equation, defining the effective dynamics of GFT condensates (\ref{condendef}), together with a weak-coupling approximation common in this approach, we have obtained strong results that give further support for the use of condensates for the extraction of cosmological dynamics from GFT. We have confirmed and extended the results of \cite{DLElong,DLEshort} to a wider class of GFT models: if, in (\ref{generaleq}), the ratios $B_j/A_j$ of GFT couplings take a positive maximum for some $j=j_0\neq 0$, consistency with the classical Friedmann equations at large volumes follows directly from setting $B_{j_0}/A_{j_0}=3\pi G$. For such GFT models, a generic condensate will dynamically expand into a configuration in which almost all quanta have spin $j_0$, with other spins exponentially suppressed. This shows explicitly how the requirement on GFT condensates to expand to a large Universe can constrain the possible couplings in the fundamental GFT.

We have shown one example, discussed already in \cite{JHEP}, which is of this form (after we exclude $j=0$ quanta which have no clear interpretation in LQG) and leads to $j_0=1/2$, precisely what is needed in order to derive the intuitive picture used in LQC, and to give a new and convincing derivation of the improved dynamics form \cite{BianchiI,improdyn} of LQC dynamics from a fundamental theory (to be compared with previous derivations from GFT condensates \cite{JHEPalone,Calcagni} and in other LQG settings \cite{AlesciCianfrani}). This result is even more interesting as the kinetic term in this model (\ref{kinetic}) is simple and very natural, given that a Laplace--Beltrami operator on the group appears in GFT renormalisation. The existence of $j=0$ quanta in GFT is indeed one of the main differences to standard LQG \cite{newGFTreview}.

We have not made any statements about the precise form of quantum corrections away from the large-volume regime, where subleading terms corresponding to different $j$ contribute significantly; such corrections depend on the state and on additional couplings that are not fixed by the classical limit. This observation opens the door for a rich phenomenology of GFT condensate cosmology, and the derivation of novel quantum gravity corrections to the classical Friedmann equations, to be explored in future work. Additional assumptions appear necessary to obtain the precise form of the modified Friedmann equations in LQC at high curvature (for example, assuming that only a single spin $j$ is excited throughout \cite{DLElong,DLEshort}), and it needs to be understood better when these hold.

In order to further strengthen the results presented here, one should ultimately go beyond the weak-coupling approximation in (\ref{jdyneq}). Certain forms of the GFT interaction, for example the EPRL \cite{EPRL} interaction considered in \cite{DLElong}, lead to equations that are local in $j$ for isotropic states, so that the decoupling of different $\sigma_j$ modes is maintained. One could hope to adapt methods used in condensed matter physics, such as the Thomas--Fermi approximation for Bose--Einstein condensates \cite{ThomFerm}, to such models, but this is again left for future work.

\begin{acknowledgments}

I would like to thank Daniele Oriti, Edward Wilson-Ewing and an anonymous referee for helpful comments on the manuscript. The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7/2007-2013) under REA grant agreement n$^{{\rm o}}$ 622339.

\end{acknowledgments}