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\section{Introduction} \label{sec:Intro}
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Computing volumes of moduli spaces has significance in many fields. For instance, the Weil-Petersson volumes of moduli spaces of Riemann surfaces can be written as intersection numbers of tautological classes due to the work of Wolpert (\cite{wolpert85}) and of Mirzakhani for hyperbolic bordered surfaces with geodesic boundaries (\cite{MirzWP}). In this paper we establish similar results for the Mazur-Veech volumes of moduli spaces of Abelian differentials. 
\par 
Denote by $\omoduli[g,n](\mu)$ the moduli spaces (or strata) of Abelian differentials (or flat surfaces) with labeled zeros of type $\mu = (m_1, \ldots, m_n)$, where $m_i\geq 0$ and where $\sum_{i=1}^n m_i = 2g-2$. Masur (\cite{masur82}) and Veech (\cite{veech82}) showed that the hypersurface of flat surfaces of area one in $\omoduli[g,n](\mu)$ has finite volume, called the Masur-Veech volume, and we denote it by ${\rm vol}\,(\omoduli[{g,n}](\mu))$. The starting point of this paper is the following expression of Masur-Veech volumes in terms of intersection numbers on the incidence variety compactification
$\proj\oOmM_{g,n}(\mu)$ described in \cite{strata}. For $1\leq i\leq n$ we define 
\be \label{eq:beta}
\beta_i \= \frac{1}{m_i+1} \,\xi^{2g-2} \, \prod_{j\neq i} \psi_{j}
\quad \in H^{2(2g-3+n)}(\proj\oOmM_{g,n}(\mu),\Q)\,
\ee
where $\xi$ is the universal line bundle class of the projectivized Hodge bundle and 
$\psi_j$ is the vertical cotangent line bundle class associated to the $j$-th marked point
(see Section~\ref{sec:RelMZInt} for a more precise definition of these tautological classes). 
\par
\begin{Thm} \label{intro:IntFormula} The Masur-Veech volumes
can be computed as intersection numbers
\begin{eqnarray} \label{eq:MV/int}
{\rm vol}\,(\omoduli[{g,n}](m_1,\ldots,m_n)) 
&\=& - \frac{2(2i\pi)^{2g}}{(2g-3+n)!}
\int_{\proj \obarmoduli[g,n](\mu)} \!\!\!\!\!\!\!\!  \xi^{2g-2} \cdot \prod_{i=1}^{n} \psi_i \\
\label{for:intMV1} &\=& \frac{2(2i\pi)^{2g}}{(2g-3+n)!}
\int_{\proj \obarmoduli[g,n](\mu)} \!\!\!\!\!\!\!\! \beta_i\cdot  \xi\,
\end{eqnarray}
for each $1\leq i\leq n$. 
\end{Thm}
\par
The equality of the two expressions on the right-hand side is a non-trivial
claim about intersection numbers on $\proj \obarmoduli[g,n](\mu)$. Note that
we follow the volume normalization in \cite{emz} that differs slightly
from the one in \cite{eo} (see \cite[Section~19]{cmz} for the conversion).
\par
Theorem~\ref{intro:IntFormula} is the interpolation and generalization of~\cite[Proposition 1.3]{SauvagetMinimal} (for the minimal strata) and 
\cite[Theorem~4.3]{cmz} (for the Hurwitz spaces of torus covers). In order to prove it, we show that both sides of equation~\eqref{eq:MV/int} satisfy the same recursion formula. On the volume side, the recursion formula
is expressed via an operator acting on Bloch and Okounkov's algebra of
shifted symmetric functions (see~Section~\ref{sec:D2rec}). The recursion for intersection numbers is first proved at the numerical level using the techniques developed in~\cite{SauvagetClass}
to compute the classes of $\proj \obarmoduli[g,n](\mu)$ (see Sections~\ref{sec:hproj} and~\ref{sec:RelMZInt}). Then we 
formally lift this relation to the algebra of shifted symmetric functions and show that it is equivalent to the previous one (see Section~\ref{sec:D2VR}).
\par
In particular, the recursion arising from intersection calculations provides the following useful formula.  We define the rescaled volume 
\ba\label{eq:vol-rescale}
v(\mu) = (m_1+1)\cdots (m_n+1){\rm vol}\,(\omoduli[{g,n}](m_1,\ldots,m_n))\,.
\ea 
For a partition $\mu$, we denote by $n(\mu)$ the cardinality of $\mu$ and by $|\mu|$ the sum of its entries. 
\par
\begin{Thm} \label{intro:VolRec}
The rescaled volumes of the strata satisfy the recursion
\ba \label{eq:volintro}
v(\mu)  \=  \sum_{k \geq 1} 
\sum_{\bfg, \bfmu}
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)}
      {2^{k-1}\, k!\, (2g-3+n)!}\,
\ea
where $\bfg=(g_1,\ldots,g_k)$ is a partition of $g$, 
$\bfmu = (\mu_1, \ldots,\mu_k)$ is a $k$-tuple of multisets with 
$(m_3,\ldots, m_n)= \mu_1\sqcup \cdots \sqcup  \mu_k$, 
and $\bfp = (p_1,\ldots,p_k)$ is defined by $p_i = 2g_i-1-|\mu_i|$
and required to satisfy $p_i>0$.  
Here the Hurwitz number $h_{\PP^1}((m_1,m_2),\bfp)$ is defined for any~$\bfp$ 
by
\be \label{eq:defhP1}
h_{\PP^1}((m_1,m_2),\bfp)= (k-1)! [t_1^{m_1+1}t_2^{m_2+1}] \left( \prod_{i=1}^k
t_1t_2  \frac{(t_2^{p_i}-t_1^{p_i})}{t_2-t_1}\right)\,.
\ee
\end{Thm}
The  relevant Hurwitz spaces of $\PP^1$ covers will be introduced in
Section~\ref{sec:hproj}.  Note that $h_{\PP^1}((m_1,m_2),\bfp) \neq 0 $ only
if $\sum_{i=1}^k (p_i+1) = m_1 + m_2+2$. This implies that $k \leq \min(m_1+1,m_2+1)$
in the summation of the theorem.
\par
For special $\mu$, the strata $\omoduli[g,n](\mu)$ can be disconnected. 
There are up to three connected components altogether, at most one of
which is hyperelliptic, classified by Kontsevich and Zorich (\cite{kz03}).
We show the refinements of Theorem~\ref{intro:IntFormula} and Theorem~\ref{intro:VolRec} for the spin and hyperelliptic components respectively, given as Theorem~\ref{thm:refinedINT} and Theorem~\ref{thm:hypint}
(conditional on a technical Assumption~\ref{asu} in Section~\ref{sec:spin},
which can be deduced from \cite{LMS}).\footnote{This assumption was later verified in~\cite{goodmetric}.}
\par
Equation~\eqref{eq:volintro} has a similar form compared to the recursion formula obtained by Eskin, Masur and Zorich (\cite{emz}) for computing saddle connection Siegel-Veech constants (joining two distinct zeros).  Consider a generic flat surface with~$n$ labeled zeros of orders $\mu=(m_1,\ldots, m_n)$. The growth rate of the number of saddle connections of length at most~$L$ joining, say,
the first two zeros is quadratic in $L$ and the leading term of the asymptotics
(up to a factor of $\pi$ to ensure rationality) is called the saddle connection Siegel-Veech constant.
Intuitively, the saddle connection Siegel-Veech constant should be proportional
to the cone angles around the two concerned zeros. For quadratic differentials this is not correct as shown by Athreya,
Eskin and Zorich (\cite{aez}). Nevertheless as an application of our formulas, we show that for Abelian
differentials the intuitive expectation indeed holds, if we use a minor modification~$c^\ho_{1\lra 2}(\mu)$ of the Siegel-Veech
constant counting homologous saddle connections only once. An
overview about the variants of Siegel-Veech constants is given in Section~\ref{sec:SV}.
\par
\begin{Thm} \label{intro:SVproduct}
The saddle connection Siegel-Veech constant $c^\ho_{1\lra 2}(\mu)$  joining
the first and the second zeros on a generic flat surface of type~$\mu$ is given by
\be \label{eq:SVmain}
c^\ho_{1\lra 2}(\mu) \= (m_1 +1)(m_2 + 1)\,.
\ee
\end{Thm}
\par
When the stratum is disconnected, we also show that the theorem holds for each
connected component under Assumption~\ref{asu}. We remark that as an asymptotic
equality as~$g$ tends to infinity, the formula~\eqref{eq:SVmain} for the
entire stratum was previously shown in the appendix by Zorich
to~\cite{aggarwal} for saddle connections of multiplicity one and
by Aggarwal~\cite{Agg2} for all multiplicities.
\par 
Another important kind of Siegel-Veech constants is the area Siegel-Veech constant, which counts cylinders (weighted by the reciprocal of their areas) on flat surfaces and is related to the sum of Lyapunov exponents (\cite{ekz}) (see Section~\ref{sec:SV}
for the definition of area Siegel-Veech constants). We similarly establish an intersection formula for area Siegel-Veech constants.  
\par 
\begin{Thm} \label{intro:IntFormula2} The area Siegel-Veech constants of
the strata can be evaluated as 
\be \label{eq:SV/int}
c_{\rm area}(\mu) 
\= \frac{-1}{4\pi^2} \frac{
\int_{\proj \obarmoduli[g,n](\mu)} \!\beta_i\cdot  \delta_0}
{\int_{\proj \obarmoduli[g,n](\mu)} \! \beta_i\cdot  \xi}
\ee
for each $1\leq i\leq n$, where $\delta_0$ is the divisor class of the 
locus of curves with a non-separating node.  
\end{Thm}
\par 
Theorem~\ref{intro:IntFormula2} completes the investigation of area Siegel-Veech constants
begun in \cite[Section 4]{cmz} (for the principal strata) and \cite[Equation~(2)]{SauvagetMinimal} (for the minimal strata). It also justifies 
a speculation of Kontsevich (\cite[Section 7]{kontsevich}) about the existence of such a $\beta$-class for computing area Siegel-Veech constants and sums of Lyapunov exponents of the strata. 
\par
Another application of the volume recursion is a geometric proof of 
the large genus limit conjecture by Eskin and Zorich (\cite{EZVol})
for the volumes of the strata and area Siegel-Veech constants. A proof using direct 
combinatorial arguments was given by Aggarwal (\cite{Agg2, aggarwal}).
Our proof, in addition, gives a uniform expression for the 
second order term as conjectured in~\cite{SauvagetMinimal} (see 
Section~\ref{sec:asy}).
\par
\begin{Thm} [{\cite[Main Conjectures]{EZVol}}]  \label{cor:EZconj}
Consider the strata $\omoduli[{g,n}](\mu)$ such that all the entries
of $\mu$ are positive.  Then 
%there exist two positive constants $C_1$ and $C_2$ such that
\bas
{v(\mu)} &\= 4 \,-\, \frac{2\pi^2}{3\cdot \sum_{i=1}^n (m_i+1)} \+ 
O(1/g^2) \,, %\epsilon_1(\mu)\,,
%\quad \text{and} 
\\
{c_{\rm area}(\mu)}&\= \frac{1}{2} \,-\, \frac{1}{ 2 \sum_{i=1}^n (m_i+1)} \+
O(1/g^2) \,, %\+ \epsilon_2(\mu)\,
\eas
where the implied constants are independent of $\mu$ and~$g$.
\end{Thm}
\par
Finally we settle another conjecture of Eskin
and Zorich on the asymptotic comparison of spin components. 
\par
\begin{Thm} [{\cite[Conjecture 2]{EZVol}}]  \label{thm:EZconj3} The volumes 
and area Siegel-Veech constants
of odd and even spin components are comparable for large values of $g$. More
precisely, 
\bas
\frac{v(\mu)^{\rm odd}}{v(\mu)^{\rm even}} \= 1 \+ O(1/g)\,, 
\\
\frac{c_{\rm area}(\mu)^{\rm odd}}{c_{\rm area}(\mu)^{\rm even}} \= 1 \+  O(1/g)\,,
\eas
where the implied constants are independent of $\mu$ and~$g$.
\end{Thm}
%\par
%From this theorem one can presumably also deduce that  
%\bes 
%\frac{c_{\rm area}(\mu)^{\rm odd}}{c_{\rm area}(\mu)^{\rm even}} \= 
%1 \+  O(1/g)\,,
%\ees
%by the strategy of Zorich's appendix to \cite{aggarwal} or by repeating
%the strategy of the proof for volumes in the context of strict brackets
%(see Section~\ref{sec:proofareaSV}). 
\par
\subsection*{Further directions}
Our work opens an avenue to study a series of related questions. First, we point out an interesting comparison with the proofs by Mirzakhani (\cite{MirzWP}), by Kontsevich (\cite{KontWitten}), and by Okounkov-Pandharipande (\cite{OPWitten}) of Witten's conjecture: the generating function of  $\psi$-class intersections on moduli spaces of curves is a solution of the KdV hierarchy of partial differential equations. Mirzakhani considered the Weil-Petersson volumes of moduli spaces of hyperbolic surfaces and analyzed geodesics that bound pairs of pants, while we consider the Masur-Veech volumes of moduli spaces of flat surfaces and analyze geodesics that join two zeros (i.e. saddle connections). Kontsevich interpreted $\psi$-classes as associated to certain polygon bundles, while we have the interpretation of Abelian differentials as polygons. Okounkov and Pandharipande used Hurwitz spaces of $\proj^1$ covers, while we rely on Hurwitz numbers of torus covers. Therefore, we speculate that generating functions of Masur-Veech volumes and area Siegel-Veech constants should also satisfy a certain interesting hierarchy as in Witten's conjecture. 
\par
In another direction, one can consider saddle connections joining a zero to itself (see~\cite[Part 2]{emz}) or impose other specific configurations to refine the Siegel-Veech counting (see e.g.~the appendix by Zorich to~\cite{aggarwal}). From the viewpoint of intersection theory, such a refinement should pick up the corresponding part of the principal boundary when flat surfaces degenerate along the configuration, hence we expect that the resulting Siegel-Veech constant can be described similarly by a recursion formula involving intersection numbers.  
\par
One can also investigate volumes and Siegel-Veech constants for affine invariant manifolds (i.e. $\SL_2(\mathbb R)$-orbit closures in the strata). It is thus natural to seek intersection theoretic interpretations of these invariants for affine invariant manifolds, e.g.~the strata of quadratic differentials (see~\cite{DGZZ, TRMVV, CMSprincipal} for interesting related results in the case of the principal strata).
We plan to treat these questions in future work.  
\par
\subsection*{Organization of the paper}
In Section~\ref{sec:hproj} we introduce relevant intersection numbers on Hurwitz spaces of $\PP^1$ covers that will appear as coefficients in the volume recursion. In Section~\ref{sec:RelMZInt} we prove that the expression of volumes by intersection numbers satisfies the recursion in~\eqref{eq:volintro}, thus showing the equivalence of Theorems~\ref{intro:IntFormula} and~\ref{intro:VolRec}. In Section~\ref{sec:D2rec} we exhibit another recursion of volumes by using the algebra of shifted symmetric functions and cumulants. In Section~\ref{sec:D2VR} we show that the two recursions are equivalent by interpreting  them as the same summation over certain oriented graphs, thus completing the proof of Theorems~\ref{intro:IntFormula} and~\ref{intro:VolRec}. In Section~\ref{sec:spin} we refine the results for the spin and hyperelliptic components of the strata. In Sections~\ref{sec:SV}, ~\ref{sec:VRandSC} and~\ref{sec:NewHurwitz} we 
respectively review the definitions of various Siegel-Veech constants, prove Theorem~\ref{intro:SVproduct} regarding saddle connection Siegel-Veech constants and interpret the result from the perspective of Hurwitz spaces of torus covers.  In Section~\ref{sec:areaSV} we establish similar intersection and recursion formulas for area Siegel-Veech constants, thus proving Theorem~\ref{intro:IntFormula2}. Finally in Section~\ref{sec:asy} we apply our results to evaluate large genus limits of volumes and area Siegel-Veech constants, proving Theorems~\ref{cor:EZconj} and~\ref{thm:EZconj3}.  
\par
\subsection*{Acknowledgments}
We are grateful to Anton Zorich for valuable suggestions on Siegel-Veech constants and numerical cross-checks.  
Moreover, we would like to thank Alex Eskin and Andrei Okounkov for their help about spin components.  We also thank Amol Aggarwal, Ben Bakker, Qile Chen, Yitwah Cheung, Eleny Ionel, Felix Janda, Xin Jin, Edward Looijenga, Duc-Manh Nguyen, Aaron Pixton, Song Sun, Alex Wright and Dimitri Zvonkine for inspiring discussions on related topics.  We thank the Mathematisches Forschungsinstitut Oberwolfach (MFO), the Max Planck Institute for Mathematics (MPIM Bonn), the l'Institut Fourier (Grenoble), the American Institute of Mathematics (AIM), the Banff International Research Station (BIRS) and the Yau Mathematical Sciences Center (YMSC) for their hospitality during the preparation of this work. Finally we thank the referee for carefully reading the paper and many helpful suggestions.

