%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Spin and hyperelliptic components} \label{sec:spin}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


In this section we prove a refinement of Theorem~\ref{intro:VolRec}
and Theorem~\ref{intro:IntFormula} with spin structures 
taken into account. We also prove the corresponding refinement
for hyperelliptic components in Section~\ref{sec:VRhyp}.
Along the way we revisit the counting problem for
torus covers with sign given by the spin parity and complete the
proof of Eskin, Okounkov and Pandharipande~(\cite{eop}) that the generating function
is a quasimodular form of the expected weight. We then show that the
$D_2$-recursion has a perfect analog when counting with spin parity
and use the techniques of Section~\ref{sec:D2VR} to convert this into the
recursion for intersection numbers.
\par
In this section we assume that all entries of $\mu=(m_1,\ldots,m_n)$ are even.
The {\em spin parity} of a flat surface $(X,x_1,\ldots, x_n,\omega) \in
\omoduli[{g,n}](\mu)$ is defined as  
$$ \phi(X,\omega) \= h^0\Bigl(X, \sum_{i=1}^n \frac{m_i}{2} x_i\Bigr) \mod 2\,.
$$
The parity is constant in a connected family of flat surfaces
by~\cite{Mumfordtheta}. We will denote by $\Om\M_{g,n}(\mu)^{\bullet}$
with $\bullet \in\{ {\rm odd},{\rm even}\}$ the moduli spaces of flat
surfaces with a fixed odd or even spin parity and simply call them
{\em spin (sub)spaces} of the corresponding strata.
Moreover, we will denote
by $\obarmoduli[{g,n}](\mu)^{\bullet}$ their incidence variety compactification
and will similarly use this symbol, e.g.\ in the form $v(\mu)^\bullet$,
$c_{1\leftrightarrow 2}(\mu,\mathcal{C})^\bullet$, and $c_{\area}(\mu)^{\bullet}$
for volumes and Siegel-Veech constants. 
\par
We remark that with our choice of notation the spin spaces
$\obarmoduli[{g,n}](\mu)^{\bullet}$ are not necessarily connected. Indeed,
for $\mu = (2g-2)$ with $g\geq 4$ and $\mu = (g-1, g-1)$ with $g \geq 5$ odd, one of the two spin spaces is disconnected, since
it contains an extra hyperelliptic component (see \cite[Theorem~2]{kz03}).
Nevertheless, the hyperelliptic components will be treated separately in
Section~\ref{sec:VRhyp}. Taking the difference of the volumes thus gives a
formula for the volume of each connected component. 
\par
To state the refined version of the volume recursion we need a generalization
of the spin parity. Let $(\Gamma,\ell,\bfp)$ be a backbone graph.
A {\em spin assignment} is a function
$$
\phi\colon \{ v \in V(\Gamma), \ell(v)=0\} \to \{ 0, 1\}.
$$
The {\em parity of the spin assignment} is defined as 
\be \label{eq:spin-parity}
\phi(\Gamma) \,:=\, \!\!\! \sum_{v \in V(\Gamma), \ell(v)=0}
\!\!\! \phi(v)\,\,\,\,\,\, {\rm mod} \,\, 2.
\ee
Our goal is the following refinement of Theorems~\ref{intro:VolRec}
and~\ref{intro:IntFormula} under a mild assumption. Recall that the
tautological line bundle $\mathcal{O}(-1)$ over $\proj\omoduli[g,n](\mu)$
has a natural hermitian metric given by the area form
$h(X,\omega)=\frac{i}{2} \int_X \omega\wedge \overline{\omega}$.
\par
\begin{asu} \label{asu}
There exists a desingularization $f\colon Y\to \proj\obarmoduli[g,1](2g-2)$
such that $f^{*}h$ extends to a good hermitian metric on $f^{*}\mathcal{O}(-1)$.
\end{asu}
\par
This assumption was already present in~\cite{SauvagetMinimal} and can
be deduced from the smooth strata compactification constructed in \cite{LMS}.\footnote{Assumption~\ref{asu} was recently verified in~\cite{goodmetric}, to which we refer the reader for more details about the area form being a good metric. See also~\cite{CMSprincipal} for a similar application to the volumes of strata of quadratic differentials with odd zeros.}
Note that we do not need this assumption for Theorem~\ref{intro:VolRec}, 
as it is stated for the entire stratum whose cohomology class 
was computed recursively in~\cite{SauvagetClass}. However, currently we do 
not know the cohomology class of each individual spin subspace. 
\par
\begin{Thm} \label{thm:refinedVR}
If $n\geq 2$, then the rescaled volumes satisfy the recursion  
\bas \label{eq:volrefrec}
v(\mu)^{\rm odd}  \=
\sum_{k \geq 1}
\sum_{\bfg, \bfmu, \, \phi \, {\rm odd} }
  h_{\PP^1}((m_1,m_2),\bfp)
\cdot  \frac{\prod_{i=1}^k (2g_i-1+n(\mu_i))!v(\mu_i,  p_i-1)^{\phi(i)}}
      {{2^{k-1}k!(2g-3+n)!}}\,,
\eas
where the summation conventions for $\bfg$, $\bfmu$ and $\bfp$
are as in Theorem~\ref{intro:VolRec} and the superscript $\phi(i)$
indicates the corresponding spin subspace. 
\end{Thm}
\par
We remark that the same formula holds when replacing ``odd'' by ``even'' in the theorem, which follows simply by subtracting the formula in Theorem~\ref{thm:refinedVR} from that in Theorem~\ref{intro:VolRec}. 
\par
\begin{Thm} \label{thm:refinedINT}
Let $\proj\Om\M_{g,n}(\mu)^\bullet$ with $\bullet \in\{ {\rm odd},{\rm even}\}$
be the subspaces of $\proj\Om\M_{g,n}(\mu)$ with a fixed spin
parity. Then the volume can be computed as an intersection number 
$$ {\rm vol}(\omoduli[g,n](\mu)^\bullet)
\= -\frac{2(2i\pi)^{2g}}{(2g-3+n)!}
\int_{\proj \obarmoduli[g,n](\mu)^\bullet} \xi^{2g-2}\cdot \prod_{i=1}^{n} \psi_i\,.
$$
\end{Thm}
\par
We first show in Section~\ref{sec:intspin} that the intersection numbers on the
right-hand side of Theorem~\ref{thm:refinedINT} satisfy a recursion as in
Theorem~\ref{thm:refinedVR}. This is parallel to Section~\ref{sec:RelMZInt}.
We then complete in Section~\ref{sec:SBHur} properties of the strict brackets
introduced by \cite{eop}. The volume recursion in Section~\ref{sec:spinD2}
is parallel to Section~\ref{sec:D2rec} and allows efficient computations
of volume differences of the spin subspaces. We do not need to
prove an analog of Section~\ref{sec:D2VR} but can rather apply the results,
since the structures of
the two recursions are exactly the same as before. Only in Section~\ref{subsec:spin-conclusion} we need Assumption~\ref{asu} to prove the initial case of Theorem~\ref{thm:refinedVR}, i.e. the case of the minimal strata. 
\par

%%%%%%%%%%%%%%%%%%
\subsection{Intersection theory on spin subspaces and
hyperelliptic components}
\label{sec:intspin}
%%%%%%%%%%%%%%%%%%

With a view toward Section~\ref{sec:VRhyp} for the hyperelliptic components, 
we allow here also the profile $\mu=(g-1,g-1)$ (with $g-1$ not necessarily even)
and $\bullet \in\{ {\rm odd},{\rm even}, {\rm hyp}\}$, and study the
corresponding union of connected components $\proj\omoduli[g,n](\mu)^\bullet$.
\par 
Let $(\Gamma,\ell,\bfp)$ be a twisted bi-colored graph and $D$ be an irreducible
component of the boundary $\proj\oOmM_{\Gamma,\ell}^{\bfp}$. We recall from
Section~\ref{sec:RelMZInt} that $\zeta_{\Gamma,\ell}^{\#}(D)$ is a divisor
of $\proj\obarmoduli[g,n](\mu)$ if and only if $\dim(\proj\oOmM_{\Gamma,\ell}^{\bfp})
=\dim(\proj\obarmoduli[g,n](\mu))-1$. Hence in this case we define
$\alpha(D)=\zeta_{\Gamma,\ell*}^{\#}(D)\in H^*(\proj\oOmM_{g,n})$, and define $\alpha(D)=0$ otherwise.  
\par
We will denote by $\proj\oOmM_{\Gamma,\ell}^{\bfp,\bullet}$ the union of the
irreducible components of $\proj\oOmM_{\Gamma,\ell}^{\bfp}$ that are mapped to
$\proj\obarmoduli[g,n](\mu)^\bullet$. 
\par
\begin{Prop} \label{eq:comprecursion}
For $1\leq i\leq n$ and each irreducible component~$D$ of
$\proj\oOmM_{\Gamma,\ell}^{\bfp,\bullet}$,
there exist constants $m_i^\bullet(D)\in \QQ$ such that
$$
(\xi+(m_i+1)\psi_i) [\proj\obarmoduli[g,n](\mu)^\bullet]
\= \sum_{\begin{smallmatrix} (\Gamma,\ell,\bfp)\\ i\mapsto v, \ell(v)=-1 \end{smallmatrix}}
\sum_{D\subset \proj\oOmM_{\Gamma,\ell}^{\bfp,\bullet}} m_i^\bullet(D) \, \alpha(D)\,,
$$
where the sum is over all twisted bi-colored graphs $(\Gamma,\ell,\bfp)$ with
the $i$-th marking in the lower level.  Moreover, if
$D\subset \proj\oOmM_{\Gamma,\ell}^{\bfp}$  and $(\Gamma,\ell,\bfp)$ is a backbone
graph, then $m_i^\bullet(D)=m(\bfp)$ is the multiplicity defined in~\eqref{eq:multtwist}.
\end{Prop}
\par
\begin{proof} We follow the same strategy as in~\cite[Theorem~5]{SauvagetClass}.
We consider the line bundle $\cOO(1)\otimes \cLL_i^{\otimes (m_i+1)}$ 
on $\proj \obarmoduli[g,n](\mu)^\bullet$. It has a global section~$s$ defined 
by mapping a differential to its $(i+1)$-st order at the marked point $x_i$. 
The vanishing locus of this section is exactly the union of the boundary 
components $\zeta_{\Gamma,\ell}^{\#}(\proj\oOmM_{\Gamma,\ell}^{\bfp})$,
thus proving the first part of the proposition.
\par
If $(\Gamma,\ell,\bfp)$ is a backbone graph, then each irreducible component  $D$ of $\proj\oOmM_{\Gamma,\ell}^{\bfp,\bullet}$
is contained in the boundary of exactly one connected component of the stratum (see e.g.~\cite[Corollary 4.4]{chenPB}). 
Thus the neighborhood of a generic point of $D$ in $\proj\omoduli[g,n](\mu)^\bullet$ is given by \cite[Lemma~5.6 and the subsequent formula]{SauvagetClass}.
In particular the multiplicity of $D$ in the vanishing locus of~$s$ is the 
same as that of the entire boundary stratum, which implies that $m_i^\bullet(D)=m(\bfp)$. 
\end{proof}
\par
By the same arguments as in Section~\ref{ssec:reduction} 
(see Proposition~\ref{pr:intbb}) one can show that
$D\cdot \xi^{2g-2} = 0$ unless $D$ is an irreducible component 
of $ \proj\oOmM_{\Gamma,\ell}^{\bfp}$ with $(\Gamma,\ell,\bfp)$ a backbone graph. 
If $(\Gamma,\ell,\bfp)$ is a backbone graph, then we let 
$\alpha_{\Gamma,\ell, \bfp}^{\bullet}=\zeta_{\Gamma,\ell*}^{\#}
[\proj\oOmM_{\Gamma,\ell}^{\bfp,\bullet}]$. Besides,  we let
$a_i(\mu)^{\bullet}= \int_{\proj\obarmoduli[g,n](\mu)^\bullet} \beta_i \cdot \xi$ 
for $1\leq i\leq n$. 
\par
\begin{Prop}\label{pr:refinedintrec}
 If $n \geq 2$, then the values of $a_i(\mu)^{\bullet}$ are the same for all $1\leq i \leq n$, denoted by $a(\mu)^{\bullet}$, and can be computed as
$$
 (m_1+1)(m_2+1) a(\mu)^\bullet \= \!\!\!\!\sum_{(\Gamma,\ell,\bfp)\in {\rm BB}_{1,2}}
 \! \frac{m(\bfp)}{|{\rm Aut}(\Gamma,\ell,\bfp)|}  \int_{\proj\oOmM_{g,n}} \!\!\! \!\!\!\alpha_{\Gamma,\ell, \bfp}^{\bullet}\cdot \xi^{2g-1}\cdot\prod_{i>2} \psi_i\,.
$$
\end{Prop}
\par
\begin{proof} This follows from the same argument as in the proof of Lemma~\ref{lem:indbis}. 
\end{proof}
\par
\begin{Prop} \label{pr:refinedintrecodd}
For $\mu$ of length bigger than one and with even entries, we have 
\bas
(m_1+1) (m_2+1) a(\mu)^\odd & \= \!\!\!
\sum_{\begin{smallmatrix}(\Gamma,\ell,\bfp,\phi),\\ \phi \, {\rm odd} \end{smallmatrix}} \frac{h_{\proj^1}((m_1,m_2),\bfp)}{|{\rm Aut}(\Gamma,\ell,\bfp,\phi)|}\\
& \, \cdot\,\prod_{v\in V(\Gamma),\,\ell(v)=0} p_v(2g_v-1+n(\mu_v)) a(\mu_v, p_v-1)^{\phi(v)}\,,
\eas
where the sum is over all choices of backbone graphs with only the first two
marked points in the lower level component. 
\end{Prop} 
\par
This proposition is a refined combination of Lemma~\ref{lem:inttree} and 
equation~\eqref{eq:ind3}.
Again we remark that the same formula holds when replacing ``odd'' by ``even'' 
in the proposition, which simply follows from subtracting the above from 
the corresponding formula for the entire stratum.  
\par
\begin{proof}
We apply Proposition~\ref{pr:refinedintrec} to
$\proj\oOmM_{g,n}(\mu)^\odd$. The proposition then follows from the description
of the boundary divisors of connected components of $\proj\oOmM_{g,n}(\mu)$. 
\par
Let $(L\to \mathcal X\to \Delta)$ be a one-parameter family of theta
characteristics, i.e.\ 
$L$ is a line bundle such that $L|_X^{\otimes 2}\simeq \omega_{X}$ for every 
fiber curve $X$ parametrized by a complex disk (centered at the origin) 
such that
\begin{itemize}
\item  the restriction of $\mathcal X$ to $\Delta\setminus 0$ is a family 
of smooth curves;
\item  the central fiber $X_0$ is of compact type.
\end{itemize}
The second condition above is due to the fact that the graphs
in ${\rm BB}(g,n)_{1,2}$ are of compact type. We assume that $L$ is odd,
i.e. $L$ restricted to every smooth fiber is an odd theta characteristic.
The restriction of $L$ to each irreducible component of $X_0$ (minus the nodes)
is a theta characteristic of that component. Since $X_0$ is of compact type,
the parity of $L|_{X_0}$ equals the sum 
of the parities over all irreducible components of $X_0$ (see e.g.~\cite[Proposition 4.1]{chenPB}), which implies that the number of components of $X_0$ with an odd theta characteristic is odd. 
\par 
Let $(\Gamma,\ell,\bfp)\in {\rm BB}(g,n)_{1,2}$. From the above description we deduce that $\proj\oOmM_{\Gamma,\ell}^{\bfp, \rm odd}$ can be written as 
$$
\bigcup_{\phi\,\odd} \oM_{-1}\times \prod_{v \in V(\Gamma)}
\proj\oOmM_{g_v,n_v}(p_v-1,\mu_v)^{\phi(v)}\,
$$
where $\oM_{-1}$ and $\proj\oOmM_{g_v,n_v}(p_v-1,\mu_v)$ are defined as in
Section~\ref{ssec:boundary}. The arguments in the proof of 
Lemma~\ref{lem:indbis}  imply that
\bas
(m_1+1) (m_2+1) a(\mu)^\odd &\= \!\!\!\sum_{\begin{smallmatrix}(\Gamma,\ell,\bfp) \in  {\rm BB}_{1,2}\\ \phi \, {\rm odd} \end{smallmatrix}} \frac{h_{\proj^1}((m_1,m_2),\bfp)}{|{\rm Aut}(\Gamma,\ell,\bfp,\phi)|}\\
&\, \cdot\, \prod_{v\in V(\Gamma),\,\ell(v)=0} p_v^2\ a(\mu_v, p_v-1)^{\phi(v)}\,.
\eas
Then by the same line of arguments as in Section~\ref{ssec:treestrata} 
(expansions over rooted trees), we get the desired expression.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%
\subsection{Strict brackets and Hurwitz numbers with spin parity} \label{sec:SBHur}
%%%%%%%%%%%%%%%%%%%%%%

Let $f\colon \SP \to \QQ$ be any function on the set of strict partitions
(i.e.\ partitions with strictly decreasing part lengths). 
The replacement of the $q$-bracket in the context of spin-weighted
counting is the {\em strict bracket} defined by 
\bes
\bstr{f} \=  \frac{1}{(q)_\infty}
\sum_{\lambda \in \SP} (-1)^{\ell(\lambda)} f(\lambda) q^{|\lambda|}\,,
\qquad \bigl((q)_\infty \= \prod_{n \geq 1} (1-q^n)
= \sum_{\lambda \in \SP} (-1)^{\ell(\lambda)} q^{|\lambda|}\bigr) \,.
\ees
The analog of the algebra $\Lambda^*$  is the algebra $\bfLA^* =
\QQ[\bfp_1,\bfp_3,\bfp_5,\ldots]$ of {\em supersymmetric
functions}, where for odd~$\ell$ the functions~$\bfp_\ell$ are defined by
\bes \label{eq:defboldpf}
\bfp_\ell(\lambda) \= \sum_{i=1}^\infty \lambda_i^\ell - \frac{\zeta(-\ell)}{2}\,.
\ees
Note the modification of the constant term and the absence of the shift
in comparison to~\eqref{eq:defpk}. We provide $\bfLA^*$ with the
{\em weight grading} by declaring $\bfp_\ell$ to have weight~$\ell+1$.
On the other hand, \cite{eop} used characters of the modified Sergeev
group  $C(d) = S(d) \ltimes \Cliff(d)$ to produce elements in $\bfLA^*$.
Here $\Cliff(d)$ is generated by involutions $\xi_1,\ldots,\xi_d$ and
a central involution~$\ve$ with the relation $\xi_i \xi_j = \ve \xi_j \xi_i$. 
Irreducible representations of $C(d)$ are~$V^\lambda$ indexed by
$\lambda \in \SP$. We denote by $\bff_\mu(\lambda)$ the central character
of the action of a permutation $g_\mu \in S(d) \subset C(d)$ of cycle
type $\mu$ on~$V^\lambda$ by conjugation. The analog of the Burnside
formula is \cite[Theorem~2]{eop} stating that for a fixed profile
$\Pi = (\mu_1,\ldots,\mu_n)$ 
\be \label{eq:spinwtsum}
\sum_p \frac{(-1)^{\phi(p)} q^{\deg(p)}}{|\Aut(p)|}
\= 2^{\sum_{i=1}^n (\ell(\mu_i) - |\mu_i|)/2} \bstr{\bff_{\mu_1} \bff_{\mu_2} \cdots \bff_{\mu_n}}\,,
\ee
where the sum is over all covers $p\colon X \to E$ of a fixed base curve
and profile~$\Pi$. 
\par
\begin{Thm} \label{thm:bfformula}
If we define 
\be \label{eq:bfhldef} 
\bfh_\ell \= \frac{-1}{\ell}  [u^{\ell+1}] \bfP(u)^\ell 
\quad \text{where} \quad
\bfP(u) \= \exp \Bigl( -\sum_{s \geq 1, s\,\odd } u^{s+1} \bfp_s \Bigr)\,,
\ee
then the difference $\bff_\ell - \ell \bfh_\ell$ has weight strictly less
than~$\ell+1$. In particular $\bff_\ell$ belongs to the
subspace $\bfLA^*_{\leq \ell+1}$ of weight less than or equal to $\ell+1$.
More precisely,
\be \label{eq:bffprecise}
\bff_{(\ell)} \= \frac{-1}{2\ell} [t^{k+1}] \Biggl(\prod_{j=1}^{\ell-1} (1-jt)
\exp \Bigl(\sum_{j \,\, \odd} \frac{2\bfp_jt^j}{j}(1-(1-\ell t)^{-j})
\Bigr) \Biggr)\,.
\ee
\end{Thm}
\par
This statement was missing in the proof of the following corollary,
one of the main theorems of \cite{eop}.
\par
\begin{Cor} The strict bracket $\bstr{\bff_{\ell_1}
\bff_{\ell_2} \cdots \bff_{\ell_n}}$ is a quasimodular form of mixed
weight less than or equal to $\sum_{i=1}^n (\ell_i + 1)$.
\end{Cor}
\par
We now prepare for the proof of Theorem~\ref{thm:bfformula} and prove the
corollary along with more precise statements on strict brackets in the next
subsection. From \cite[Definition~6.3 and Proposition~6.4]{IvaGauss} we know
that the central characters are given by 
\be \label{eq:ftoPd}
\bff_\rho \=  \sum_{\mu \in \SP} X^\rho_\mu  P_\mu^\downarrow\,
\ee
where the objects on the right-hand side are defined as follows. 
We define for any partition $\lambda$
the Hall-Littlewood symmetric polynomials
\bes
P_\lambda(x_1,\ldots,x_m; t) \= 
\sum_{\sigma \in S_n}
\sigma\Bigl(\prod_{i=1}^n x_i^{\lambda_i} \prod_{i<j,\, i< \ell(\lambda)}\frac{x_i -tx_j}{x_i -x_j}\Bigr)\,.
\ees
These polynomials have cousins where the
powers are replaced by falling factorials. That is,  writing
$n^{\downarrow k} = n (n-1)(n-2)\cdots (n-k+1)$, we define
\bes
P^\downarrow_\lambda(x_1,\ldots,x_m; t) = 
\sum_{\sigma \in S_n}
\sigma\Bigl(\prod_{i=1}^n x_i^{\downarrow\lambda_i}
\prod_{i<j,\, i< \ell(\lambda)}\frac{x_i -tx_j}{x_i -x_j}\Bigr)\,.
\ees
Next, we define $X^\bullet_\bullet(t)$ to be the base change matrix
from the basis of $\bfp_\rho$ to the basis $P_\lambda(x_1,\ldots,x_m; t)$,
that is, we define them by
\be \label{eq:ptoPwitht}
\bfp_\rho  \= \sum_{\lambda \vdash |\rho|}
X^\rho_\lambda(t) P_\lambda(\,\cdot\,; t)\,.
\ee
The existence and the fact that the $X^\bullet_\bullet(t)$ are polynomials
in~$t$ is shown in \cite[Section~III.7]{mac}. We abbreviate $X^\rho_\lambda=
X^\rho_\lambda(-1)$ and similarly $P_\lambda = P_\lambda(\,\cdot\,; -1)$ and
$P^\downarrow_\lambda = P^\downarrow_\lambda(\,\cdot\,; -1)$.
\par
\begin{proof}[Proof of Theorem~\ref{thm:bfformula}] We need to
prove~\eqref{eq:bffprecise}.
From there one can then derive~\eqref{eq:bfhldef} by expanding the exponential
function (just as in \cite{IvaOlsh} Proposition~3.5 is derived from
Proposition~3.3). We use that for $\rho=(\ell)$ a cycle, the coefficients
$X^\lambda_\rho$ in~\eqref{eq:ftoPd} are supported on~$\lambda$ with at most
two parts. More precisely, by \cite[Example~III.7.2]{mac} we know that
\ba \label{eq:fbasPdown}
\bff_{(\ell)} &\=  P^\downarrow_{(\ell)} + 2 \sum_{i=1}^{\lfloor \ell/2\rfloor}
(-i)^i P^\downarrow_{(\ell-i,i)} \\
&\= \sum_{1 \leq a,b \leq \ell(\lambda) \atop a\neq b} 
\sum_{i=0}^\ell (-1)^i  \lambda_a^{\downarrow \ell-i} \lambda_b^{\downarrow i}
 \frac{\lambda_a +\lambda_b}{\lambda_a -\lambda_b}
\prod_{i \neq a,b}  \frac{\lambda_a +\lambda_i}{\lambda_a -\lambda_i}
 \frac{\lambda_b +\lambda_i}{\lambda_b -\lambda_i} \,.
\ea
Using 
\bas
-\sum_{j \in \NN} \frac{1-(-1)^j\bfp_j(\lambda)t^j}{j}(1-\ell t)^{-j}
&\=
\log \Bigl( \prod_{i=1}^{\ell(\lambda)} \frac{1-(\lambda_i+ \ell)t}
{1+(\lambda_i-\ell)t} \Bigr) \\ 
\eas
and the specialization of this formula for $\ell=0$, our goal is to
show that
\bas
\bff_{(\ell)}(\lambda) &\= \frac{-1}{2\ell} [t^{1}] \Biggl(\prod_{j=0}^{\ell-1} (t^{-1}-j)
\prod_{i=1}^{\ell(\lambda)} \frac{t^{-1} +\lambda_i}{t^{-1}-\lambda_i} 
 \frac{t^{-1} -(\lambda_i+\ell)} {t^{-1} +(\lambda_i-\ell)} 
\Biggr) \\
&\= \frac{-1}{2\ell} [z^{-1}] \Biggl(\prod_{j=0}^{\ell-1} (z-j)
\prod_{i=1}^{\ell(\lambda)} \frac{z +\lambda_i}{z -\lambda_i} 
 \frac{z -(\lambda_i+\ell)} {z +(\lambda_i-\ell)} 
\Biggr) \\
 &\= \sum_{a=1}^{\ell(\lambda)} \Bigl( 
\prod_{j=0}^{\ell-1} (\lambda_a-j)
\prod_{i \neq a}  \frac{\lambda_a +\lambda_i}{\lambda_a -\lambda_i}
\frac{\lambda_a -(\lambda_i+\ell)} {\lambda_a +(\lambda_i-\ell)}
\Bigr)\,.
\eas
Using that for $\ell$ odd 
\bes
\sum_{i=0}^\ell (-1)^i \bigl( \lambda_a^{\downarrow \ell-i} \lambda_b^{\downarrow i}
\,-\, \lambda_b^{\downarrow \ell-i} \lambda_a^{\downarrow i}\bigr)
\= \frac{ \lambda_a^{\downarrow \ell}(\lambda_a - \lambda_b -\ell) -   
\lambda_b^{\downarrow \ell} (\lambda_b - \lambda_a -\ell)}
{\lambda_a+\lambda_b-\ell}\,,
\ees
we see that our goal and the known~\eqref{eq:fbasPdown} agree.
\end{proof}


%%%%%%%%%%%%%%%%%%%%%%
\subsection{Volume computations via cumulants for strict brackets}
\label{sec:spinD2}
%%%%%%%%%%%%%%%%%%%%%%

We denote by an upper index~$\Delta$ the difference of the even and
odd spin related quantities, e.g.\ $v(\mu)^\Delta = v(\mu)^\even - v(\mu)^\odd$.
Cumulants for strict brackets are defined by the same formula~\eqref{slash}
as for $q$-brackets. We are interested in cumulants for the same reason
as we were for the case of the strata as in~\eqref{eq:cumutovol}.
\par
\begin{Prop} \label{prop:spinvol}
The difference of the volumes of the even and odd spin subspaces
of $\omoduli[g,n](\mu)$ can be computed in terms of cumulants by
\bes
\vol(\omoduli[g,n](\mu))^\Delta \= \frac{(2\pi i)^{2g}}{(2g-2+n)!}
\bsL{\bff_{(m_1+1)}| \cdots |\bff_{(m_n+1)}}\,,
\ees
and thus, in combination with Theorem~\ref{thm:bfformula} we have 
\bes
v(\mu)^\Delta \= \frac{(2\pi i)^{2g}}{(2g-2+n)!}
\bsL{\bfh_{(m_1+1)}| \cdots |\bfh_{(m_n+1)}}\,.
\ees
Here the subscript~$L$ refers to the leading term
\bas \label{eq:bfdeflead}
\la g_1|\cdots|g_n\ra_{\str,L} &\= [\hslash^{-k-1+n}] \,\la g_1|\cdots|g_n\ra_{\str,\hslash} 
\= \lim_{h\to 0} \hslash^{k+1-n}\,\evh[\langle g_1|\cdots|g_n\rangle_{\str,q}](\hslash)\,
\eas
for $g_i$ homogeneous of weight $k_i$ and $k = \sum_{i=1}^n k_i$.
\end{Prop}
\par
This proposition was certainly the motivation of \cite{eop}, which stops short
of this step. To derive the proposition from~\eqref{eq:spinwtsum}, we need one
more tool, the analog of the degree drop in Proposition~\ref{prop:degdrop}.
We use the fact that for strict brackets we have a closed formula (rather than only
a recursion as for $q$-brackets), proved in~\cite[Section~3.2.2]{eop} and
in more detail in~\cite[Section~13]{blochokounkov}. First, 
\be \label{eq:plstrict}
(-1) \cdot \bstr{\bfp_\ell} \= G_{\ell +1}  \,:=\, \frac{\zeta(-\ell)}{2}
+ \sum_{n \geq 1} \sigma_{\ell}(n) q^n
\ee
and for the more general statement we define the ``oddification'' of the
Eisenstein series to be
\bes
\cGG^\odd(z_1,\ldots,z_n) 
\= - \sum_{r=1}^\infty D_q^{(n-1)}G_{2r} \sum_{s_1+\cdots+s_n = r+n-1}
\frac{z_1^{2s_1-1}\cdots z_n^{2s_n-1}}{(2s_1-1)!\cdots(2s_n-1)!}\,,
\ees
where $D_q = q\partial/\partial q$. 
Then by Proposition~13.3 in loc.\ cit.\ the $n$-point function is given by
\be \label{eq:strnpoint}
\sum_{\ell_i \geq 1, \,\, \ell_i \,\odd} \bstr{\bfp_{\ell_1} \bfp_{\ell_2} \cdots \bfp_{\ell_n}}
\frac{z_1^{\ell_1}\cdots z_n^{\ell_n}}{\ell_1!\cdots \ell_n!} 
\= \sum_{\alpha \in \PPP(n)} \, \prod_{A\in\alpha}\,
\cGG^\odd\Bigl(\{z_a\}_{a \in A}\Bigr)\,.
\ee
Consequently, the cumulants are simply given by 
\be \label{eq:strcumulants}
\bstr{\bfp_{\ell_1}| \bfp_{\ell_2}| \cdots |\bfp_{\ell_n}}
\= \Bigl[\frac{z_1^{\ell_1}\cdots z_n^{\ell_n}}{\ell_1!\cdots \ell_n!} \Bigr]
\cGG^\odd(z_1,\ldots,z_n)\,.
\ee
\par
\begin{proof}[Proof of Proposition~\ref{prop:spinvol}]
Recall the evaluation map $\evX$ used in~\cite[Section~8]{cmz}
and in Section~\ref{sec:lift}. We have
$\deg\big(\evX \bstr{\bfp_{\ell_1} \bfp_{\ell_2} \cdots \bfp_{\ell_n}}\big) =
\frac12 \sum_{i=1}^n \ell_i$, the highest term being contributed by the partition
into singletons. From~\eqref{eq:strcumulants} we deduce that
$\deg\big(\evX \bstr{\bfp_{\ell_1}| \bfp_{\ell_2}| \cdots |\bfp_{\ell_n}}\big) =
\frac12 \sum_{i=1}^n \ell_i - (n-1)$, and thus obtain the expected degree drop. The claim
now follows from the usual approximation of Masur-Veech volumes by counting
torus covers (\cite{eo} and~\cite[Proposition~19.1]{cmz}).
\end{proof}
\par
\medskip
While~\eqref{eq:strcumulants} provides an easy and efficient way to
compute cumulants of strict brackets, we show that the more complicated
way via lifting of differential operators to~$\bfLA^*$ and the Key Lemma~\ref{KL:D2recursion}
also works here. The analog of Proposition~\ref{prop:liftviadelta} is the following result.  
\par
\begin{Prop} \label{prop:strhviaDiffOp}
With $\Delta(f) \= \sum_{\ell_1,\,\ell_2\,\ge\,1} (\ell_1+\ell_2) \,
\bfp_{\ell_1+\ell_2-1}\,\frac{\p^2}{\p \bfp_{\ell_1}\,\p \bfp_{\ell_2}}$ we have
\be
\la f \ra_{\str,\hslash} \= \frac{1}{\hslash^k}(e^{\hslash(\Delta -\p/\p \bfp_1)/2} f)\,
(\emptyset)\,,
\ee
where the  evaluation at the empty set is explicitly given by $\bfp_\ell \mapsto
-\frac{\zeta(-\ell)}{2}$.
\end{Prop}
\par
Note that we can regard the differential operator $(\Delta -\p/\p \bfp_1)/2$
appearing in the exponent the same as the operator~$D$ defined
in~\eqref{eq:defD} 
when viewing $\bfLA^*$ as a quotient algebra of $\Lambda^*$ with
all the even~$p_\ell$ set to zero, since the differential operator $\partial$
sending $p_\ell$ to a multiple of $p_{\ell-1}$ is zero on this quotient.
\par
\begin{proof} Using the description~\eqref{eq:defevh} of the $\hslash$-evaluation
we need to show that $\fd \la f \ra_{\str,\hslash} =  \la (\Delta -\p/\p \bfp_1) f
\ra_{\str,\hslash}$. Contrary to the case of $q$-brackets we will actually show the
stronger statement that $\fd \la f \ra_{\str} =  \la (\Delta -\p/\p \bfp_1) f
\ra_{\str}$. It suffices to check this for all the $n$-point functions.
For $n=1$ this can be checked directly from~\eqref{eq:plstrict}. For general $n$, we write
$W(z) = \sum_{s \geq 1} z^{2s-1}/(2s-1)!$. Using~\eqref{eq:strnpoint} and that
the commutator $[\fd,D_q]$ is multiplication by the weight, we compute that
\begin{flalign} 
\fd \la \prod_{i=1}^n W(z_i) \ra_{\str} &\=
\sum_{\alpha \in \PPP(n)} \sum_{A_1 \in \alpha, \atop |A_1| \geq 2}
\Bigl(\fd \cGG^\odd\bigl(\{z_a\}_{a \in A_1}\bigr) \cdot \!\!\!\!
\prod_{A \in \alpha \setminus \{A_1\}} \cGG^\odd\bigl(\{z_a\}_{a \in A}
\bigr)\Bigr)
\nonumber \\
&\phantom{\=} \,-\, \frac12 \sum_{i=1}^n z_i \cdot \Bigl(
\sum_{\alpha \in \PPP(\{1,\ldots, n\} \setminus \{i\})}
\prod_{A \in \alpha} \cGG^\odd\bigl(\{z_a\}_{a \in A}\bigr)\Bigr)
\label{eq:fdFn}
\end{flalign}
where for the factor in the summand with $|A_1| \geq 2$
\be \label{eq:fdGodd}
\fd \cGG^\odd\bigl(\{z_a\}_{a\in A_1}\bigr) \= - \frac12 
\sum_{r \geq 1} 2r D_q^{|A_1|-2} G_{2r} \cdot \sum_{s_a \geq 1, \atop
  \sum s_a = r + |A_1|-1} \prod_{a \in A_1}
\frac{z_a^{2s_a-1}}{(2s_a-1)!}\,,
\ee
and where the summation is over all tuples~$(s_a)_{a \in A_1}$.
Since $\tfrac12 \p/\p \bfp_1 (\prod_{i=1}^n W(z_i)) = \tfrac12
\sum_{i=1}^n z_i \prod_{j\neq i} W(z_j)$, the strict bracket of this expression
is precisely the second line on the right-hand side of~\eqref{eq:fdFn}.
Since
\bes
\frac12 \Delta\Big(\prod_{i=1}^n W(z_i)\Big) = \sum_{1 \leq i\neq j \leq n}
(z_i+z_j) W(z_i+z_j) \prod_{k \in \{1,\ldots,n\} \setminus \{i,j\}} W(z_k)\,,
\ees
its strict bracket matches the first line on the right-hand side of~\eqref{eq:fdFn},
and the part containing the variable for $W(z_i+z_j)$ produces of course the
special factor~\eqref{eq:fdGodd}.
\end{proof}
\par
This proposition provides an efficient algorithm to compute the differences
of volumes of the spin subspaces. The definitions below are
completely analogous  to the beginning of Section~\ref{sec:D2rec},
except that objects with even indices
have disappeared and they are written in boldface letters for distinction.
For the substitution, we define
\bes
\bfP_Z(u) \= \exp\Bigl(\sum_{ j \geq 1} \Bigl(\frac{1}{2}\Bigr)^{\frac{j+1}{2}}
\zeta(-j)  \,u^{j+1}\Bigr)
\quad \text{and} \quad
\ual_\ell \= \bigl[u^{\ell}\bigr] \frac{1}{(u/\bfP_Z(u))^{-1}}\,.
\ees
We let $\bfR = \QQ[\bfh_1,\bfh_3,\ldots]$ and define for a finite set
$I=\{i_1,\ldots,i_n\}$ of positive  integers the formal series $\bfcH_I\in
\bfR[[z_{i_1}, \ldots, z_{i_n}]]$ by
\begin{flalign}
&\bfcH_{\{i\}} \= \frac{1}{z_i} + \sum_{\ell\geq 1} \bfh_\ell z_i^\ell\,, 
\quad  
\bfcH_{\{i,j\}} \= \frac{z_i \bfcH'(z_i) - z_j\bfcH'(z_j)}
{\bfcH(z_j) - \bfcH(z_i)} -1\,,   \nonumber  \\
&\bfcH_{I} = \frac{1}{2(n-1)}\sum_{I=I'\sqcup I''}
D_2(\bfcH_{I'},\bfcH_{I''})\,, \label{eq:bfHindform}
\end{flalign}
with
\bes
D_2(f,g) \= \sum_{\ell_1,\ell_2 \geq 1,\,\, \odd} [z_1^{\ell_1} z_2^{\ell_2}] \bfcH_{\{1,2\}}\,
\frac{\partial f}{\partial \bfh_{\ell_1}}\,\frac{\partial g}{\partial \bfh_{\ell_2}} \,.
\ees
We still set $\bfcH_{n}=\bfcH_{[\![1,n]\!]}$ and 
$\bfh_{\ell_1,\ldots,\ell_n} = [z_1^{\ell_1}\cdots z_n^{\ell_n}]\bfcH_{n}$.
\par
\begin{Cor} \label{cor:eodiff}
The even-odd volume differences of the stratum with signature $\mu=(m_1,\ldots,m_n)$ can
be computed as 
\bes
v(\mu)^\Delta \=  \frac{(2\pi i)^{2g}}{(2g-2+n)! } \,
\bigl.\bfh_{m_1+1,\ldots,m_n+1}\bigr|_{\bfh_\ell \mapsto \ual_\ell} 
\ees
using the recursion~\eqref{eq:bfHindform}.
\end{Cor}
\par
\begin{proof} Thanks to Proposition~\ref{prop:strhviaDiffOp} and the subsequent
remark, the proof of Theorem~\ref{thm:D2recursion} can be copied verbatim here.
The extra factor $2^{-\ell/2}$ in the definition of $\bfP_Z$ in comparison to the
constant term $-\zeta(-\ell)/2$ of the evaluation of $\bfp_\ell$ compensates for
the fact that the strict bracket of the $\bff_\ell$ gives the counting function
in~\eqref{eq:spinwtsum} up to a power of two.
\end{proof}


%%%%%%%%%%%%%%%%%%
\subsection{Conclusion of the proofs for spin subspaces}
\label{subsec:spin-conclusion}
%%%%%%%%%%%%%%%%%%

\begin{proof}[Proof of Theorem~\ref{thm:refinedINT} and Theorem~\ref{thm:refinedVR}]
Theorem~\ref{thm:refinedVR} is a consequence of Corrolary~\ref{cor:eodiff}. Indeed the arguments of Section~\ref{sec:D2VR} adapted to the series $\bfcH_n$ show that the recursion in Theorem~\ref{thm:refinedVR} is a consequence of the recursion in~\eqref{eq:bfHindform}. 
\par
To prove Theorem~\ref{thm:refinedINT}, we consider first the case $n=1$, i.e. $\mu = (2g-2)$. Let $\ol{\nu}_{\mu}$ be the 
push-forward of the Masur-Veech volume $\nu_\mu$ form to $\proj\OmM_{g,n}(\mu)$.
Assumption~\ref{asu} implies by the same argument as in
\cite[Lemma~2.1]{SauvagetMinimal}
that $\frac{2(2i\pi)^{2g}}{(2g-1)!}\xi^{2g-1}$  can be represented by a
meromorphic differential form (of Poincar\'e growth at the boundary), 
whose restriction to $\proj \Om\M_{g,1}(2g-2)$ is equal to $\ol{\nu}_{\mu}$. 
This implies that 
$$
{\rm vol}(2g-2)^\bullet \= \frac{2(2i\pi)^{2g}}{(2g-1)!}
\int_{\proj \obarmoduli[g,1](2g-2)^\bullet} \xi^{2g-1}\,.
$$
Now for the case $n \geq 2$ Theorem~\ref{thm:refinedVR} and 
Proposition~\ref{pr:refinedintrecodd} determine $a(\mu)^{\bullet}$ 
and $v(\mu)^{\bullet}$ by the equivalent recursive formulas, and hence 
they coincide up to the obvious normalizing factors.  
\end{proof}
\par
%%%%%%%%%%%%%%%%%%%%%
\subsection{Volume recursion for hyperelliptic components}
\label{sec:VRhyp}
%%%%%%%%%%%%%%%%%%%%%

In this subsection we prove the volume recursion for hyperelliptic
components, which is  analogous to but not quite the same as the recursion
in Theorem~\ref{intro:VolRec}.
It is a consequence of the work of Athreya, Eskin and Zorich (\cite{aez})
on volumes of the strata of quadratic differentials in genus zero. 
\par
Recall that only the strata $\omoduli[g](g-1,g-1)$ and $\omoduli[g](2g-2)$
have hyperelliptic components. For the hyperelliptic components we still
have an interpretation of their volumes as intersection numbers
as well as a volume recursion as follows.  
\par
\begin{Thm} \label{thm:hypint}
For the hyperelliptic components we have 
$$
{\rm vol}(\omoduli[g,1](2g-2)^\hyp) \= \frac{2(2i\pi)^{2g}}{(2g-1)!}
\int_{\proj \obarmoduli[g,1](2g-2)^\hyp} \xi^{2g-1}\,
$$
and
$$
{\rm vol}(\omoduli[g,2](g-1,g-1)^\hyp) \= \frac{2(2i\pi)^{2g}}{g(2g-1)!}
\int_{\proj \obarmoduli[g,2](g-1,g-1)^\hyp} \xi^{2g-1}\psi_2\,,
$$
provided that Assumption~\ref{asu} holds.
\end{Thm}
\par
As before we set
\bas \ 
v(2g-2)^\hyp &\= (2g-1){\rm vol}\,(\omoduli[{g,1}](2g-2)^\hyp)\,, \\
v(g-1,g-1)^\hyp &\= g^2\,{\rm vol}\,(\omoduli[{g,2}](g-1,g-1)^\hyp) \,.
\eas
\par
\begin{Prop} \label{prop:HRvolrec}
The volumes of the hyperelliptic components $\omoduli[g,2](g-1,g-1)$
satisfy the recursion
\bas
v(g\ms 1,g\ms 1)^\hyp = v(2g\ms 2)^\hyp   
+ \sum_{\ell = 1}^{g-1}  \frac{(2\ell\ms1)!v(2\ell\ms2)^\hyp\, (2(g\ms\ell)\ms1)!
v(2g\ms2\ell\ms2)^\hyp}
{4 (2g-1)!}\,.
\eas
\end{Prop}
\par
Note in comparison to Theorem~\ref{intro:VolRec} that only the terms $k=1$
and $k=2$ appear and that the Hurwitz number $h_\PP$ is identically one here.
As a preparation for the proof recall that the canonical double cover construction
provides isomorphisms
\bas
\cQQ_g(2g-3,(-1)^{2g-3}) &\,\cong\, \omoduli[{g,1}](2g-2)^\hyp\,, \\
\cQQ_g(2g-2,(-1)^{2g-2}) &\,\cong\, \omoduli[{g,2}](g-1,g-1)^\hyp
\eas
that preserve the Masur-Veech volume and the $\SL_2(\RR)$-action. Taking into account 
the factorials for labeling zeros and poles the main result of \cite{aez} can
be translated as 
\bas
{\rm vol}\,(\omoduli[{g,1}](2g-2)^\hyp) \= \frac{2}{(2g+1)!} \frac{(2g-3)!!}{(2g-2)!!} \pi^{2g}\,, \\
{\rm vol}\,(\omoduli[{g,2}](g-1,g-1)^\hyp) \= \frac{8}{(2g+2)!} \frac{(2g-2)!!}{(2g-1)!!} \pi^{2g}
\eas
where the double factorial notation means $(2k)!! = 2^k k!$ and $(2k-1)!! = (2k)!/2^kk!$.
\par
\begin{proof} Expanding the definition of the double factorials and including the
summand $v(2g-2)^\hyp$ as the two boundary terms of the sum 
(i.e. $\ell = 0$ and $\ell=g$), we need to show that
\be
\sum_{\ell=0}^{g} \frac{1}{2\ell+1}\binom{2\ell}{\ell}
\frac{1}{2g-2\ell+1} \binom{2(g-\ell)}{g-\ell}
\= 2 \frac{16^g}{(g+1)^2} {\binom{2g+2}{g+1}}^{-1} \,.
\ee
For this purpose it suffices to prove the following two identities of generating series 
\be \label{eq:arctan1}
\sum_{\ell \geq 0}  \frac{1}{2\ell+1}\binom{2\ell}{\ell} x^{2\ell}
\= \frac{1}{2x} \arctan\Biggl(\frac{2x}{\sqrt{1-4x^2}}\Biggr)
\ee
and 
\be \label{eq:arctan2}
2\, \sum_{g \geq 0} \frac{16^g}{g^2} {\binom{2g}{g}}^{-1} x^{2g} 
\= \frac{1}{4x^2} \arctan\Biggl(\frac{2x}{\sqrt{1-4x^2}}\Biggr)^2\,,
\ee
so that we can take the square of the first series and compare the $x^{2g}$-terms. To
prove~\eqref{eq:arctan1} we multiply it by~$x$, differentiate, and are then left with
showing that $\sum_{\ell \geq 0}  \binom{2\ell}{\ell} x^{2\ell} = 1/\sqrt{1-4x^2}$,
which follows from the binomial theorem. To prove~\eqref{eq:arctan2} we
differentiate and are then left with the identity which is already proved in \cite[p.~452, Equation~(9)]{lehmer}.
\end{proof}
\par
The last ingredient is the following straightforward consequence of
Proposition~\ref{pr:refinedintrec} (analogous to the case of spin
subspaces in Proposition~\ref{pr:refinedintrecodd}). 
\par
\begin{Prop}
\label{prop:hyp-int-recursion}
For $\mu=(g-1,g-1)$, we have  
\ba
g^2 a(g-1, g-1)^\hyp & \=  (2g-1)^2 a(2g-2)^\hyp \\
&\+  \frac{1}{2} \sum_{g_1=1}^{g-1} \bigg(
h_{\proj^1}\big((g-1,g-1),(2g_1-1, 2g-2g_1-1)\big)  \\
& \quad 
\cdot (2g_1-1)^2a(2g_1-2)^\hyp(2g-2g_1-1)^2 a(2g-2g_1-2)^\hyp\bigg)\,.
\ea
\end{Prop} 
\par
\begin{proof}[Proof of Theorem~\ref{thm:hypint}]
Since the proof of Theorem~\ref{thm:hypint} for the case $\mu=(2g-2)$ was already given
along with the proof of Theorem~\ref{thm:refinedINT}, it remains to show
that $v(g-1,g-1)^\hyp$ and $a(g-1,g-1)^\hyp = \int_{\proj\obarmoduli[g,2](g-1,g-1)^\hyp} \beta_i \cdot \xi$
satisfy the same recursion. It is elementary to check 
that $h_{\proj^1}\big((g-1,g-1),(2g_1-1, 2g-2g_1-1)\big) = 1$.
Then the desired conclusion thus follows from Propositions~\ref{prop:HRvolrec} and~\ref{prop:hyp-int-recursion}. 
\end{proof}

