Masur-Veech volumes and intersection theory on moduli spaces of abelian differentials

We show that the Masur-Veech volumes and area Siegel-Veech constants can be obtained by intersection numbers on the strata of Abelian differentials with prescribed orders of zeros. As applications, we evaluate their large genus limits and compute the saddle connection Siegel-Veech constants for all strata. We also show that the same results hold for the spin and hyper-elliptic components of the strata.


Introduction
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 ([Wol85]) and of Mirzakhani for hyperbolic bordered surfaces with geodesic boundaries ( [Mir07]). In this paper we establish similar results for the Mazur-Veech volumes of moduli spaces of Abelian differentials.
Denote by ΩM g,n (µ) the moduli spaces (or strata) of Abelian differentials (or flat surfaces) with labeled zeros of type µ = (m 1 , . . . , m n ), where m i ≥ 0 and where n i=1 m i = 2g − 2. Masur ([Mas82]) and Veech ([Vee82]) showed that the Research of the first author is partially supported by the NSF CAREER grant DMS-1350396. Research of the second author is partially supported by the DFG-project MO 1884/1-1 and by the LOEWE-Schwerpunkt "Uniformisierte Strukturen in Arithmetik und Geometrie". hypersurface of flat surfaces of area one in ΩM g,n (µ) has finite volume, called the Masur-Veech volume, and we denote it by vol (ΩM g,n (µ)). The starting point of this paper is the following expression of Masur-Veech volumes in terms of intersection numbers on the incidence variety compactification PΩM g,n (µ) described in [BCG + 18]. For 1 ≤ i ≤ n we define where ξ is the universal line bundle class of the projectivized Hodge bundle and ψ j is the vertical cotangent line bundle class associated to the j-th marked point (see Section 3 for a more precise definition of these tautological classes).
The equality of the two expressions on the right-hand side is a non-trivial claim about intersection numbers on PΩM g,n (µ). Note that we follow the volume normalization in [EMZ03] that differs slightly from the one in [EO01] (see [CMZ18, Section 19] for the conversion).
Theorem 1.1 is the interpolation and generalization of [Sau18, Proposition 1.3] (for the minimal strata) and [CMZ18,Theorem 4.3] (for the Hurwitz spaces of torus covers). In order to prove it, we show that both sides of equation (2) 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 4). The recursion for intersection numbers is first proved at the numerical level using the techniques developed in [Sau17] to compute the classes of PΩM g,n (µ) (see Sections 2 and 3). 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 5).
For special µ the strata ΩM g,n (µ) can be disconnected, with up to three spin and hyperelliptic connected components, classified by Kontsevich and Zorich ([KZ03]). We show the refinements of Theorem 1.1 and Theorem 1.2 for the spin and hyperelliptic components respectively, given as Theorem 6.3 and Theorem 6.12 (conditional on Assumption 6.1 which will be proved in an appendix).
Equation (5) has a similar form compared to the recursion formula obtained by Eskin, Masur and Zorich ( [EMZ03]) for computing saddle connection Siegel-Veech constants (joining two distinct zeros). Consider a generic flat surface with n labeled zeros of orders µ = (m 1 , . . . , m n ). The growth rate of the number of saddle connections of length at most L joining, say, the first two zeros is quadratic and the leading term of the asymptotics (up to a factor of π 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 ( [AEZ16]). 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 hom 1↔2 (µ) of the Siegel-Veech constant counting homologous saddle connections only once. An overview about the variants of Siegel-Veech constants is given in Section 7.
Another application of the volume recursion is a geometric proof of the large genus limit conjecture by Eskin and Zorich ( [EZ15]) for the volumes of the strata and area Siegel-Veech constants. A proof using direct combinatorial arguments was given by Aggarwal ( [Agg18a,Agg18b]). Our proof indeed gives a uniform expression for the second order term as conjectured in [Sau18] (see Section 11).
where the implied constants are independent of µ and g.
Finally we settle another conjecture of Eskin and Zorich on the asymptotic comparison of spin components.
Theorem 1.6 ([EZ15, Conjecture 2]). The volumes of odd and even spin components are comparable for large values of g. More precisely, v(µ) odd v(µ) even = 1 + O(1/g) , where the implied constant is independent of µ and g.
From this theorem one can presumably also deduce that c area (µ) odd c area (µ) even = 1 + O(1/g) , by the strategy of Zorich's appendix to [Agg18b] or by repeating the strategy of the proof for volumes in the context of strict brackets (see Section 10.3).
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 ([Mir07]), by Kontsevich ([Kon92]), and by Okounkov-Pandharipande ( [OP09]) of Witten's conjecture: the generating function of ψ-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 ψ-classes as associated to certain polygon bundles, while we have the interpretation of Abelian differentials as polygons. Okounkov and Pandharipande used Hurwitz spaces of P 1 covers, while we rely on Hurwitz numbers of torus covers. Therefore, we expect that generating functions of Masur-Veech volumes and area Siegel-Veech constants should also satisfy a certain interesting hierarchy as in Witten's conjecture.
In another direction, one can consider saddle connections joining a zero to itself (see [EMZ03,Part 2]) or impose other specific configurations to refine the Siegel-Veech counting (see e.g. the appendix by Zorich to [Agg18b]). 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.
One can also investigate volumes and Siegel-Veech constants for affine invariant manifolds (i.e. SL 2 (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 [DGZZ18] for interesting related results in the case of the principal strata).
We plan to treat these questions in future work.
Organization of the paper. In Section 2 we introduce relevant intersection numbers on Hurwitz spaces of P 1 covers that will appear as coefficients in the volume recursion. In Section 3 we prove that the expression of volumes by intersection numbers satisfies the recursion in (5), thus showing the equivalence of Theorems 1.1 and 1.2. In Section 4 we exhibit another recursion of volumes by using the algebra of shifted symmetric functions and cumulants. In Section 5 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 1.1 and 1.2. In Section 6 we refine the results for the spin and hyperelliptic components of the strata. In Sections 7, 8 and 9 we respectively review the definitions of various Siegel-Veech constants, prove Theorem 1.3 regarding saddle connection Siegel-Veech constants and interpret the result from the perspective of Hurwitz spaces of torus covers. In Section 10 we establish similar intersection and recursion formulas for area Siegel-Veech constants, thus proving Theorem 1.4. Finally in Section 11 we apply our results to evaluate large genus limits of volumes and area Siegel-Veech constants, proving Theorems 1.5 and 1.6. 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.

Hurwitz spaces of P 1 covers
In this section we recall the definition of the moduli space of admissible covers of [HM82] as a compactification of the classical Hurwitz space (see also [HM98]), and prove formulas to compute recursively intersection numbers of ψ-classes on these moduli spaces. These intersection numbers will appear as coefficients and multiplicities in the volume recursion. Along the way we introduce basic notions on stable graphs and level functions.
2.1. Hurwitz spaces and admissible covers. Let d, g, and g ′ be non-negative integers. Let Π = (µ (1) , · · · , µ (n) ) be a ramification profile consisting of n partitions. We define the Hurwitz space H d,g,g ′ (Π) to be the moduli space parametrizing branched covers of smooth connected curves p : X → Y of degree d with profile Π and such that the genera of X and Y are given by g and g ′ respectively. That is, p is ramified over n points and over the i-th branch point the sheets coming together form the partition µ (i) (completed by singletons if |µ (i) | < deg(p)).
The Hurwitz space H d,g,g ′ (Π) has a natural compactification H d,g,g ′ (Π) parametrizing admissible covers. An admissible cover p : X → Y is a finite morphism of connected nodal curves such that i) the smooth locus of X maps to the smooth locus of Y and the nodes of X map to the nodes of Y , ii) at each node of X the two branches have the same ramification order, and iii) the target curve Y marked with the branch points is stable.
The space H d,g,g ′ (Π) is equipped with two forgetful maps obtained by mapping an admissible cover to the stabilization of the source or the target. Here n denotes the number of branch points or equivalently the length of Π and m denotes the number of ramification points or equivalently the number of parts (of length > 1) of all the µ (i) . The Hurwitz number N • d,g,g ′ (Π) is the degree of the map f T , or equivalently the number of connected covers p : X → Y of degree d with profile Π and the location of the branch points fixed in Y . We also denote by N d,g,g ′ (Π) the Hurwitz number of covers without requiring X to be connected. We remark that each cover is counted with weight given by the reciprocal of the order of its automorphism group, as is standard for the Hurwitz counting problem.
By the Riemann-Hurwitz formula, the genus g of the covering surfaces satisfies that The forgetful map f S goes from H P 1 (µ[0], µ[∞]) to M g,n+k , where we assume that the first n marked points are the first n ramification points and the preimages of the last branch point are the k last marked points. Since there are n + 1 branch points in the target surface of genus zero, we conclude that dim For g = 0, define the following intersection numbers on the Hurwitz spaces The definition of ψ-classes will be recalled in Section 3. If n = 2, then is of dimension zero, and hence the intersection number on the right is just the number of points of the Hurwitz space, i.e.
,0 ((m 1 + 1), (m 2 + 1), (p 1 , . . . , p k )) . Again we emphasize that the Hurwitz number on the right-hand side is counted with weight 1/|Aut| for each cover. Correspondingly the intersection numbers are computed on the Hurwitz space treated as a stack. Our goal for the rest of the section is to show the following result.
The definitions of rooted trees and the local contributions h(Γ) are given in Sections 2.3 and 2.4 respectively. The above formula for the Hurwitz number obviously agrees with (6).
There are m 1 ! d m1+1 choices for σ 1 . Fixing σ 1 , to construct σ 2 we first choose • a permutation τ ∈ S k , and then for all ℓ.
This gives k![t m1+1 ] k i=1 (t + · · · + t pi ) choices. Choose c 1 out of the elements in σ 1 , which gives m 1 + 1 choices. Along with the i-indices this determines the elements c 2 , . . . , c k as well as the set of b's as the complement of the union of a's and c's. Finally σ 2 is determined by arranging the b's, which gives (m 2 + 1 − k)! choices. Note that in this process only the cyclic order of (c 1 , . . . , c k ) matters and we cannot actually determine which one is the first c, hence we need to divide the final count by k.
We remark that the above Hurwitz counting problem can also be interpreted by the angular data of the configurations of saddle connections joining two zeros z 1 and z 2 of order m 1 and m 2 respectively in the setting of [EMZ03]. Suppose f : P 1 → P 1 is a branched cover parameterized in the Hurwitz space H d,0,0 ((m 1 + 1), (m 2 + 1), (p 1 , . . . , p k )), where we treat f as a meromorphic function with k poles of order p 1 , . . . , p k . Then the meromorphic differential η = df has two zeros of order m 1 and m 2 as well as k poles of order p 1 + 1, . . . , p k + 1 with no residue. Conversely given such η, integrating η gives rise to a desired branched cover f . Such η can be constructed using flat geometry as in [CC16,Section 2.4]. In particular, it is determined by the angles 2π(a ′ i +1) between the saddle connections (clockwise) at z 1 and the angles 2π (a ′′ i + 1) = m 2 + 1 and a ′ i + a ′′ i + 2 = p i + 1. We see again that the choices involve a partition (a ′ 1 + 1, . . . , a ′ k + 1) of m 1 + 1 such that a ′ i + 1 ≤ p i for all i.
2.3. Level graphs and rooted trees. The boundary of the Deligne-Mumford compactification M g,n is naturally stratified by the topological types of the stable marked surfaces. These boundary strata are in one-to-one correspondence with stable graphs, whose definition we recall below. The boundary strata of Hurwitz spaces and of moduli spaces of Abelian differentials are encoded by adding level structures and twists to stable graphs.
satisfying the following properties: • V is a vertex set with a genus function g; • H is a half-edge set equipped with a vertex assignment a and an involution i (and we let n(v) = |a −1 (v)|); • E, the edge set, is defined as the set of length-2 orbits of i in H (self-edges at vertices are permitted); • (V, E) define a connected graph; • L is the set of fixed points of i, called legs or markings, and is identified with [[1, n]]; • for each vertex v, the stability condition 2g(v) − 2 + n(v) > 0 holds.
We denote by Stab(g, n) the set of stable graphs of genus g and with n legs. A stable graph is said of compact type if h 1 (Γ) = 0, i.e. if the graph has no loops, which is thus a tree.
We will use two extra structures on stable graphs, called level functions and twists. As in [BCG + 18] we define a level graph to be a stable graph Γ together with a level function ℓ : V (Γ) → R ≤0 . An edge with the same starting and ending level is called a horizontal edge. A bi-colored graph is a level graph with two levels (in which case we normalize the level function to take values in {0, −1}) that has no horizontal edges. We denote the set of bi-colored graphs by Bic(g, n).
Definition 2.3. Let Γ be a stable graph in Stab(g, n + k). A twist assignment on Γ of type (µ[0], µ[∞]) is a function p : H(Γ) → Z satisfying the following conditions: • For all 1 ≤ i ≤ n, the twist of the i-th leg is m i + 1 and for all 1 ≤ i ≤ k the twist of the (n + i)-th leg is −p i .
Suppose the graph Γ comes with a level structure ℓ. We say that a twist p is compatible with the level structure if for all edges (h, h ′ ) the condition p(h) > 0 implies that ℓ(a(h)) > ℓ(a(h ′ )), and respectively for the cases < and =. In this case we call the triple (Γ, ℓ, p) a twisted level graph. For the reader familiar with related results of compactifications of strata of Abelian differentials, the above definition characterizes twisted differentials (or canonical divisors) in [BCG + 18] and [FP18] (regarding the m i as the zero orders and p j + 1 as the pole orders of twisted differentials on irreducible components of the corresponding stable curves). In particular, every level graph has only finitely many compatible twists. For a graph of compact type, there exists a unique twist p if the entries of (µ[0], µ[∞]) satisfy the condition that Definition 2.4. Let 1 ≤ i < j ≤ n. A stable rooted tree (or simply a rooted tree) is a twisted level graph (Γ, ℓ, p) of compact type satisfying the following conditions: i) One vertex v j , called the root, carries the i-th and j-th legs and no other of the first n legs, a vertex on the path from v to the root is called an ancestor of v, and a vertex whose ancestors contain v is called a descendant of v; ii) There are no horizontal edges; All vertices of positive genus are leaves and hence on level 0; v) Each vertex of genus zero other than v j carries exactly one of the first n legs.
Since the root is an ancestor of any other vertex, by definition it is the unique vertex lying on the bottom level, hence it has genus zero. Moreover, it is easy to see from the definition that any path towards the root is strictly going down. In particular, any vertex except the root has a unique ancestor. 2.4. The sum over rooted trees. Now we assume that g = 0. Below we define the local contributions from rooted trees in Proposition 2.1 and complete its proof. Consider a graph Γ ∈ RT(µ[0], µ[∞]) 1,2 . Since by assumption every vertex of Γ has genus zero, condition v) implies that Γ has exactly n − 1 vertices and n − 2 edges. Denote by v 2 , . . . , v n the vertices of Γ such that v i carries the i-th leg h i for 3 ≤ i ≤ n and v 2 carries the first two legs. This convention is consistent with our previous notation for the root. We denote by µ[∞] i the list of negative twists at half-edges adjacent to v i . These half-edges are either part of the whole edges joining v i to its descendants (as adjacent vertices to v i on higher level) or part of the k last legs (corresponding to the k marked poles).
If i = 2, then there is a unique (non-leg) half-edge h i = h i adjacent to v i such that m i := p( h i ) − 1 ≥ 0. Namely, this half-edge is part of the whole edge joining v i to its ancestor (as the adjacent vertex to v i on lower level). With this notation we define the contribution of the rooted tree Γ as Let J ⊂ [[1, n + k]] be a subset such that the cardinalities of J and J c are at least two. Denote by δ J the class of the boundary divisor of M 0,n+k parameterizing curves that consist of a component with the markings in J union a component with the markings in J c . We need the following classical result (see e.g. [ACG11, Lemma 7.4]).
Lemma 2.5. For all 3 ≤ i ≤ n + k, the following relation of divisor classes holds on M 0,n+k : For 3 ≤ i ≤ n, the above lemma implies that Proof. The first part of the claim follows from the same argument as in [Sau17, Proposition 7.1]. Here the vertices of level 0 in the bi-colored graphs correspond to the components of the admissible covers that contain the marked poles. For the other part, suppose that a generic point of a boundary divisor has at least two vertices on level 0 (or on level −1). Then one can scale one of the two functions that induce the covers on the two vertices such that the domain marked curve is fixed while the admissible covers vary. It implies that f S restricted to this boundary divisor has positive dimensional fibers.
Proof of Proposition 2.1, case n ≥ 3. We will prove the result by induction on n. The beginning case n = 2 follows from the definition of h(Γ) in (10) and we have also described it explicitly in Section 2.2. The strategy of the induction for higher n is by successively replacing the ψ i in (9) with the sum over boundary divisors as in the preceding lemma, starting with i = 3. To simplify notation, we write . We also simply write ψ and δ as classes in the Hurwitz space for their pullbacks via f S .
Consider a boundary divisor δ J of M 0,n+k pulled back to H(µ[0], µ[∞]), which is a union of certain boundary divisors of H(µ[0], µ[∞]). We would like to compute the intersection number δ J · n i=4 ψ i on H(µ[0], µ[∞]). By Lemma 2.6 and the projection formula, the only possible non-zero contribution is from the loci in δ J whose bicolored graphs have a unique edge e = (h, h ′ ) connecting two vertices v 0 and v −1 on level 0 and level −1 respectively, such that the last k markings (i.e. the k marked poles) are contained in v 0 . In this case we can assume that p(h) > 0 (and hence p(h ′ ) < 0 as p(h) + p(h ′ ) = 0 by definition). The admissible covers restricted to v 0 and to v −1 belong to Hurwitz spaces of similar type, where at the node (i.e. the edge e) the ramification order of the restricted maps is given by   Therefore, we only need to consider the case when v 0 contains the third marking (hence all markings labeled by J), and consequently v −1 contains the first and second markings (hence all markings labeled by J c ). In this case we obtain that where h P 1 is defined in (9), where in the first factor on the right-hand side the ψ-product skips the third marking and the marking from the half-edge of v 0 , and where in the second factor the ψ-product skips the first and second markings. Now we use the induction hypothesis to decompose the factors h P1 (µ[0] a , µ[∞] a ) for a = 0 and a = −1. It leads to a sum over all possible pairs of rooted trees, where the two rooted trees in each pair generate a new rooted tree. More precisely, one rooted tree in the pair contains the markings of J c ∪ {h ′ } whose root v 2 carries the first and second markings, the other rooted tree contains the markings in J ∪ {h} whose root v 3 carries the third marking and h, and they generate a new rooted tree by gluing the legs h and h ′ as a whole edge and by using v 2 as the new root.
Therefore, if J is a subset of [[3, n]] such that 3 ∈ J, then we obtain that where the sum is over all rooted trees Γ such that the descendants of v 3 are exactly the vertices v j for j ∈ J \ {3}.

Volume recursion via intersection theory
In this section we show that the two main theorems of the introduction, Theorem 1.1 and Theorem 1.2 are equivalent. This section does not yet provide a direct proof of either of them.
We first show that the intersection numbers in Theorem 1.1 are given by a recursion formula of the same shape as in Theorem 1.2. Together with an agreement on the minimal strata this proves the equivalence of the two theorems. Along the way we introduce special classes of stable graphs that are used for recursions throughout the paper.
3.1. Intersection numbers on the projectivized Hodge bundle. Fix g and n such that 2g − 2 + n > 0. We denote by f : X → M g,n the universal curve and by ω X /Mg,n the relative dualizing line bundle. We will use the following cohomology classes: • Let 1 ≤ i ≤ n. We denote by σ i : M g,n → X the section of f corresponding to the i-th marked point and by L i = σ * i ω X /Mg,n the cotangent line at the ith marked point. With this notation, we define ψ i = c 1 (L i ) ∈ H 2 (M g,n , Q).
• For 1 ≤ i ≤ g, we denote by λ i = c i (ΩM g,n ) ∈ H 2 (M g,n , Q) the i-th Chern class of the Hodge bundle. (We use the same notation for a vector bundle and its total space.) • We denote by δ 0 ∈ H 2 (M g,n , Q) the Poincaré-dual class of the divisor parameterizing marked curves with at least one non-separating node. • The projectivized Hodge bundle PΩM g,n comes with the universal line bundle class ξ = c 1 (O(1)) ∈ H 2 (PΩM g,n , Q). Unless otherwise specified, we denote by the same symbol a class in H * (M g,n , Q) and its pull-back via the projection p : PΩM g,n → M g,n . Recall that the splitting principle implies that the structure of the cohomology ring of the projectivized Hodge bundle is given by Let µ = (m 1 , . . . , m n ) be a partition of 2g − 2. We denote by PΩM g,n (µ) the closure of the projectivized stratum PΩM g,n (µ) inside the total space of the projectivized Hodge bundle PΩM g,n . This space is called the (ordered) incidence variety compactification 1 .
In this section we study the intersection numbers for all 1 ≤ i ≤ n. The reader should think of the a i (µ) as certain normalization of volumes. In fact, Theorem 1.1 can be reformulated as vol(ΩM g,n (µ)) = 2(2π) 2g (−1) implying in particular that a i (µ) is independent of i. We prove a collection of properties defining recursively the a i (µ) as the coefficients of some formal series. As the base case for n = 1, i.e. µ = (2g − 2), define the formal series and set For a partition µ, recall that n(µ) denotes the number of its entries and |µ| denotes the sum of the entries.
Theorem 3.1. The generating function A of the intersection numbers a i (2g − 2) is determined by the coefficient extraction identity while the intersection numbers a(µ) = a i (µ) with n(µ) ≥ 2 are given recursively by with the same summation conventions as in Theorem 1.2.
The first identity (15) was proved in [Sau18] and gives By Lagrange inversion, this formula can be written equivalently as and will in fact be proved in this form in Section 4.4. We observe in passing that Q(u) is the asymptotic expansion of ψ(u −1 + 1 2 ) as u → 0, where ψ(x) = Γ ′ (x)/Γ(x) is the digamma function. The proof of the second identity (16) will be completed by the end of Section 3.5.
In the course of proving Theorem 3.1 we will prove the following complementary result, justifying the implicitly used fact that a i (µ) is independent of i. 3.2. Boundary components of moduli spaces of Abelian differentials. In Section 2 we introduced several families of stable graphs to describe the boundary of Hurwitz spaces. Here we show how these graphs encode relevant parts of the boundary of moduli spaces of Abelian differentials.
The recursions in Theorem 1.2 and Theorem 3.1 can be phrased as sums over a small subset of twisted level graphs, with only two levels and more constraints, that we call (rational) backbone graphs, inspired by Figure 2. Recall that a bi-colored graph is a level graph with two levels {0, −1} that has no horizontal edges. Definition 3.3. An almost backbone graph is a bi-colored graph with only one vertex at level −1. For such a graph to be a (rational) backbone graph we require moreover that it is of compact type and that the vertex at level −1 has genus zero.
We denote by BB(g, n) ⊂ ABB(g, n) ⊂ Bic(g, n) the sets of backbone, almost backbone and bi-colored graphs. We denote by BB(g, n) 1,2 ⊂ BB(g, n) the set of backbone graphs such that the first and second legs are adjacent to the vertex of level −1. Moreover, let BB(g, n) ⋆ 1,2 ⊂ BB(g, n) 1,2 be the subset where precisely the first two legs are adjacent to the lower level vertex. Similarly, we define ABB(g, n) 1,2 and ABB(g, n) ⋆ 1,2 and drop (g, n) if there is no source of confusion. Below we fix some notations for these graphs, used throughout in the sequel.
For Γ ∈ BB(g, n) we denote by v −1 the vertex of level −1. backbone graphs will usually have k vertices of level 0. Given a partition µ = (m 1 , . . . , m n ) of 2g − 2, let p be the unique twist of type (µ, ∅) for Γ (see Definition 2.3). We denote by µ[0] −1 the list of m i for all legs i at level −1 and with a slight abuse of notation we denote by p = (p 1 , . . . , p k ) the list of p(h) for half-edges h that are adjacent to the k vertices of level 0. Said differently, the restriction of the twist to level −1 provides v −1 with a twist of type (µ[0] −1 , µ[∞] −1 = p). Finally if v is a vertex of level 0, we denote by µ v the list of p(h) − 1 for all half-edges adjacent to v.
The goal in the remainder of the section is to introduce the classes α Γ,ℓ,p in (18) below that will be used in Proposition 3.12 to compute intersection numbers on PΩM g,n (µ). A stable graph Γ ∈ Stab(g, n) determines the moduli space and comes with a natural morphism ζ Γ : M Γ → M g,n . Let ℓ be a level function on Γ such that (Γ, ℓ) is a bi-colored graph with two levels {0, −1}. We define the following vector bundle This space comes with a natural morphism ζ # Γ,ℓ : ΩM Γ,ℓ → ΩM g,n , defined by the composition ΩM Γ,ℓ → ζ * Γ ΩM g,n → ΩM g,n where the first arrow is the inclusion of a vector sub-bundle and the second is the map on the Hodge bundles induced from ζ Γ by pull-back. The morphism ζ # Γ,ℓ determines a morphism (denoted by the same symbol) on the projectivized Hodge bundles ζ # Γ,ℓ : PΩM Γ,ℓ → PΩM g,n . The image of ζ # Γ,ℓ is the closure of the locus of differentials supported on curves with dual graph Γ such that the differentials vanish identically on components of level −1. In the sequel we will need the following lemma (see [Sau17,Proposition 5.9]). Let be the loci defined by the following three conditions: i) A differential in ΩM 0 has zeros of orders m i at the relevant marked points and of orders p(h) − 1 at the relevant branches of the nodes. ii) For each v of level −1 there exists a non-zero (meromorphic) differential ω v on the component X v corresponding to v that has zeros at the relevant marked points of orders prescribed by µ and poles at the relevant branches of the nodes of orders prescribed by p, i.e. such that the canonical divisor as the corresponding class in H * (PΩM g,n , Q). By [Sau17, Proposition 5.9], we can describe α Γ,ℓ,p with the following two lemmas.
Lemma 3.6. If (Γ, ℓ, p) is a bi-colored graph of compact type, then α Γ,ℓ,p = 0 if and only if there is a unique vertex v −1 of level −1, and in this case α Γ,ℓ,p is divisible by

3.3.
A first reduction of the computation. Recall the (marked and projectivized) Hodge bundle projection p : PΩM g,n → M g,n . As before we usually denote by the same symbol a class in M g,n and its pullback via p. In this section we show that many p-push forwards of intersections of α Γ,ℓ,p with tautological classes vanish or can be computed recursively. The starting point is the following important lemma proved by Mumford in [Mum83, Equation (5.4)].
Lemma 3.7. The Segre class of the Hodge bundle is the Chern class of the dual of the Hodge bundle, i.e.
Lemma 3.8. Let 1 ≤ k ≤ g and let Γ be a stable graph. Then where the sum is over all partitions of k into non-negative integers k v assigned to In As a consequence of the above discussion, we obtain the following result.
Lemma 3.9. Let α = i≥0 ξ i α i be a class in H * (PΩM g,n , Q) where the classes α i are pull-backs from H * (M g,n , Q). Then we have Recall the expressions of the intersection numbers a i (µ) in (12) and in Proposition 3.2. In order to compute a i (µ), by Lemma 3.9 we only need to consider the ξ-degree zero and one parts of the class [PΩM g,n (µ)] in H * (PΩM g,n , Q).
Combining Lemmas 3.8 and 3.9 together with Lemmas 3.4 and 3.6 of the previous section, we can already prove the following vanishing result for classes associated with some bi-colored graphs.
Proposition 3.10. If (Γ, ℓ, p) is not a backbone graph, then Proof. For simplicity we write α = α Γ,ℓ,p in the proof. We assume first that Γ is not of compact type, i.e. h 1 (Γ) > 0. Then by Lemma 3.4, the class α is divisible by ξ. Therefore, by Lemma 3.5 we can write Now we assume that Γ is of compact type. By Lemma 3.6, we only need to consider the case when there is a unique vertex v 1 of level −1. Since Γ is not a backbone graph, v 1 has positive genus g 1 . Still by Lemma 3.6 and simplifying the notation ζ Γ * (λ v1,i ) by λ v1,i , the class α is divisible by ξ g1 + ξ g1−1 λ v1,1 + · · · + λ v1,g1 . Consequently we can write where γ 0 and γ 1 are pullbacks from H * (M g,n , Q) and the O(ξ 2 ) term stands for a class divisible by ξ 2 . By Lemma 3.9, we obtain that Using Lemma 3.8, we also obtain that From the projection formula we deduce that ,n(v1) , Q) by Lemma 3.7. Once again the same lemma implies that Putting everything together, we thus conclude that p * (ξ 2g−2 α) = 0.
We define the multiplicity of a twist p to be Proposition 3.11. If (Γ, ℓ, p) is a backbone graph in BB(g, n) 1,2 , then where p v is the entry of p corresponding to the twist on the unique edge of each vertex v of level 0 and µ −1 is the list of entries in µ whose corresponding legs are adjacent to the vertex of level −1.
As a preparation for the proof we relate the space M −1 defined in (17) to the Hurwitz space for backbone graphs. The idea behind this relation was already mentioned in the last paragraph of Section 2.2. If (Γ, ℓ) is a backbone graph, then we claim that H P 1 (µ −1 , p) ∼ = M −1 , where the isomorphism is provided by the source map f S that marks the critical points of the branched covers. To verify the claim, let ω be the meromorphic differential on the unique vertex of Γ of level −1 as in part iii) of the definition for M −1 . Since Γ is of compact type, the global residue condition in [BCG + 18] imposed to ω implies that all residues of ω vanish. Therefore, a point in M −1 can be identified with such a meromorphic differential ω (up to scale) on P 1 without residues, such that ω has zeros of order m i for m i ∈ µ −1 at the corresponding markings and has poles of order p j + 1 for p j ∈ p at the corresponding nodes. In particular, ω is an exact differential and integrating it on P 1 provides a meromorphic function that can be regarded as a branched cover f parameterized in H P 1 (µ −1 , p). Conversely given f in H P 1 (µ −1 , p), we can treat f as a meromorphic function and taking df gives rise to such ω. We thus conclude that where i → v −1 means that the i-th marking belongs to the vertex of level −1. Now we can proceed with the proof of Proposition 3.11.
Therefore we only need to consider the degree zero part of α Γ,ℓ,p , which is given by Multiplying this expression by n i=3 ψ i , we obtain that For the first term on the right-hand side, equality (20) implies that Moreover for all v of level 0, we have where the second identity follows from Lemma 3.9. Since p v is the (positive) twist value assigned to the edge of v, the product of p v over all vertices of level 0 equals m(p) defined in (19). In addition, the sum of g v over all vertices of level 0 equals the total genus g, because Γ is of compact type and v −1 has genus zero. Putting everything together we thus obtain that which is the desired statement.
3.4. The induction formula for cohomology classes. The main tool of the section is the induction formula in [Sau17, Theorem 6 (1)] which we recall now.
Proposition 3.12. For all 1 ≤ i ≤ n, the relation that holds in H * (PΩM g,n , Q), where the sum is over all twisted bi-colored graphs such that the i-th leg is carried by a vertex of level −1.
There are two ways of using equation (22). First one can compute the Poincarédual class of PΩM g,n (µ) in H * (PΩM g,n , Q) in terms of the ψ, λ, ξ classes and boundary classes associated to stable graphs. This strategy is used in [Sau18] to deduce the first formula in Theorem 3.1.
Alternatively, one can compute relations in the Picard group of PΩM g,n (µ) to deduce relations between intersection numbers on PΩM g,n (µ) and intersection numbers on boundary strata associated to twisted graphs. This is the strategy that we will use here. We will use this proposition with i ∈ {1, 2} and multiply the formula by ξ 2g−1 n i=3 ψ i to obtain a 1 (µ) on the left-hand side. Then we will use Propositions 3.10 and 3.11 to compute the right-hand side. A first application of this strategy gives a proof of the complementary proposition.
Proof of Proposition 3.2. We use Proposition 3.12 with i = 1. Multiplying formula (22) by ξ 2g−2 · n i=2 ψ i , we obtain that It suffices to check that each summand in the right-hand side vanishes. Proposition 3.10 implies that if (Γ, ℓ, p) is not a backbone graph, then the corresponding summand vanishes. If (Γ, ℓ, p) is a backbone graph, then we have seen (in the paragraph below Proposition 3.11) that M −1 is birational to a Hurwitz space of admissible covers of dimension n −1 − 2, where n −1 is the number of legs adjacent to the vertex of level −1. Since the ψ-product restricted to level −1 contains n −1 − 1 terms (i.e. it misses ψ 1 only), which is bigger than dim M −1 , it implies that the intersection of α Γ,ℓ,p with n i=2 ψ i vanishes. Now we know that a i (µ) is independent of the choice of 1 ≤ i ≤ n and hence we can drop the subscript i. The second use of the strategy presented above leads to the following induction formula.
Lemma 3.13. The intersection numbers a(µ) satisfy the recursion We remark that this induction formula is not quite the same as the induction formula of Theorem 3.1, e.g. the sums in the two formulas do not run over the same set. Theorem 3.1 will follow further from a combination of Lemma 3.13 and Proposition 2.1 of the previous section.
Proof. We apply the induction formula of Proposition 3.12 with i = 2: We multiply this expression by ξ 2g−1 n i=3 ψ i and apply p * . Since Lemma 3.7 gives p * (ξ 2g [PΩM g,n (µ)]) = 0, the above equality implies that By Proposition 3.10 a term in the sum of the right-hand side vanishes if (Γ, ℓ, p) is not a backbone graph. Suppose (Γ, ℓ, p) is a backbone graph such that the first leg does not belong to the vertex of level −1 (which contains n −1 legs). Then on level −1 the product of ψ-classes contains n −1 −1 terms (i.e. this product misses ψ 2 only), which exceeds the dimension of M −1 (being n −1 −2), hence the corresponding term in the sum also vanishes. Now we only need to consider the case when (Γ, ℓ, p) is a backbone graph in BB(g, n) 1,2 , i.e. the vertex of level −1 carries both the first and second legs. Then the intersection number ξ 2g−1 · n i=3 ψ i · α Γ,ℓ,p is given by Proposition 3.11. We thus conclude that The last equality comes from the fact that the datum of g 1 +· · ·+g k = g, g i ≥ 1 and (m 3 , . . . , m n ) = µ 0 ⊔µ 1 ⊔· · ·⊔µ k determines uniquely a graph (Γ, ℓ, p) in BB(g, n) 1,2 and an automorphism of the backbone graph is determined by a permutation in S k that preserves both the partition of g and the sets µ 1 , . . . , µ k .
3.5. Sums over rooted trees. The purpose of this section is to combine the preceding Lemma 3.13 with Proposition 2.1 that describes the computation of intersection numbers on Hurwitz spaces. We will show that the numbers a(µ) can be expressed as sums over rooted trees in a similar way as we did for intersection numbers on Hurwitz spaces in Section 2.4. Let 2 ≤ i ≤ n and (Γ, ℓ, p) be a rooted tree in RT(g, µ) 1,i (here µ[∞] is empty). Since there is no marked pole, it implies that any vertex of genus zero has at least one edge with a negative twist, hence it is an internal vertex of Γ and lies on a negative level. Denote by µ[∞] 0 the list obtained by taking the (positive) entries p(h) for all half-edges h adjacent to a vertex of level 0. Denote by µ[0] 0 the list of entries of µ from those legs carried by the internal vertices (of genus zero). With this notation we define the rooted tree (Γ 0 , ℓ 0 , p 0 ) in RT(0, µ[0] 0 , µ[∞] 0 ) 1,i obtained by removing the leaves of Γ (i.e. vertices of positive genus and hence on level 0). We also define the multiplicity m 0 (p) of (Γ 0 , ℓ 0 , p 0 ) to be the product of entries of µ[∞] 0 . Now we define the a-contribution of the rooted tree (Γ, ℓ, p) as where h(Γ 0 , ℓ 0 , p 0 ) is the contribution of the rooted tree defined in (10).
Lemma 3.14. The following equality holds: Proof. Removing the leaves of a rooted tree induces a bijection between RT(g, µ) 1,2 and the set which is a partition of RT(g, µ) 1,2 over all possible decorations of the leaves of the rooted trees. Moreover, an automorphism of a rooted tree in RT(g, µ) 1,2 is determined by an automorphism of the backbone graph in BB(g, n) 1,2 , because all internal vertices of the rooted tree (i.e. those of genus zero and hence on negative levels) have marked legs by Definition 2.4. Then we can first use Lemma 3.13 to write a(µ) as a sum over backbone graphs in BB(g, n) 1,2 and then use Proposition 2.1 to express it as the desired sum over the set as claimed in the lemma.
End of the proof of Theorem 3.1. We will prove for n ≥ 2 the equality that This formula together with Lemma 3.14 thus implies Theorem 3.1. Since by definition n i=1 (m i + 1) = 2g − 2 + n, Lemma 3.14 implies that Therefore, the left-hand side of (24) can be rewritten as We claim that there is a bijection Indeed given a rooted tree (Γ, ℓ, p) in RT(g, µ) 1,2 we can construct (Γ ′ , ℓ ′ , p ′ ) ∈ BB(g, n) ⋆ 1,2 by contracting all edges except those adjacent to the root, and the rooted trees ) 1 are the connected components of the graph obtained from (Γ, ℓ, p) by deleting the root. Moreover for a rooted tree (Γ, ℓ, p), by equation (10) we have where as before Γ 0 is obtained from Γ by removing the leaves and the Γ j0 are the connected components after removing the root of Γ 0 . Together with the definition of the a-contribution in (23), it implies that Note also that Combining the above we thus conclude that equality (24) holds.
Proof of the equivalence of Theorems 1.1 and 1.2. We first assume that Theorem 1.2 holds. By Theorem 3.1, the quantities vol(ΩM g,n (m 1 , . . . , m n )) and satisfy the same induction relation that determines both collections of these numbers starting from the case n = 1. The base case (i.e. the minimal strata) that was proved in [Sau18] under a mild assumption of regularity of a natural Hermitian metric on O(−1), and we will give an alternative (unconditional) proof in Section 4.4. Consequently we conclude that Theorem 1.2 implies Theorem 1.1. The converse implication follows similarly.

Volume recursion via q-brackets
In this section we define recursively polynomials in the ring R = Q[h 1 , h 2 , . . .] and show that they compute volumes of the strata after a suitable specialization. The method of proof relies on lifting the E 2 -derivative via the Bloch-Okounkov q-bracket and expressing cumulants in terms of this lift. This recursion looks quite different from the recursion given in Theorem 1.2, since it is only defined on the level of polynomials in the variables h i and requires h i -derivatives.
To define the substitution, we let where the denominator denotes the inverse function of u/P B (u). For the recursion we define for a finite set I = {i 1 , . . . , i n } of positive integers the formal series where we abbreviate H n = H [[1,n]] , H = H 1 and h ℓ1,...,ℓn = [z ℓ1 1 . . . z ℓn n ]H n and where the symmetric bi-differential operator D 2 is defined by Theorem 4.1. The rescaled volume of the stratum with signature µ = (m 1 , . . . , m n ) can be computed as using the recursion (27) and the values of the α ℓ in (26).
4.1. Three sets of generators for the algebra of shifted symmetric functions. We let Λ * be the algebra of shifted symmetric functions (see e.g. [EO01], [Zag16] or [CMZ18]) and recall the standard generators The algebra Λ * is provided with a grading where each p ℓ has weight ℓ + 1. For Hurwitz numbers the geometrically interesting generators are where z ℓ is the size of the conjugacy class of the cycle of length ℓ, completed by singletons. The first few of these functions are The third set of generators, defined implicitly by Eskin and Okounkov, will serve as top term approximations of f ℓ . We define h ℓ ∈ Λ * by Observe that by definition h ℓ has pure weight ℓ + 1. The first few of these functions are Proposition 4.2 ([EO01, Theorem 5.5]). The difference f ℓ − h ℓ /ℓ has weight strictly less than ℓ + 1.
We abuse the notation h ℓ for generators of R and for elements in Λ * . This is intentional and should not lead to confusion, since the map h ℓ → h ℓ induces an isomorphism of algebras R ∼ = Λ * , by the preceding proposition.
4.2. The lift of the evaluation map to the Bloch-Okounkov ring. Let f : P → Q be an arbitrary function on the set P of all partitions. Bloch and Okounkov ([BO00]) associated to f the formal power series which we call the q-bracket, and proved that this q-bracket is a quasimodular form of weight k whenever f belongs to the subspace of Λ * of weight k (see [BO00], and [Zag16] or [GM16] for alternative proofs). In [CMZ18, Section 8] we studied in detail an evaluation map Ev (implicitly defined in [EO01]) on the ring of quasimodular forms that measures the growth rate of the coefficients of quasimodular forms. The purpose of this section is to lift this evaluation map to the Bloch-Okounkov ring and to express it in terms of the generators h i introduced in the previous section.
The map Ev is the algebra homomorphism from the ring of quasimodular forms , sending the Eisenstein series E 2 (normalized to have constant coefficient one) to X + 12, E 4 to X 2 , and E 6 to X 3 . In this way, the larger the degree of Ev(f ), the larger the (polynomial) growth of the coefficients of f , see [CMZ18,Proposition 9.4] for the precise statement. It is also convenient to work with the evaluation map 2 We also use the brackets f X := Ev f q (X) and f ℏ := ev f q (ℏ) for f ∈ Λ * as abbreviation. Note that Λ * admits a natural ring homomorphism to Q, the evaluation at the emptyset, explicitly given by the map p ℓ → ℓ!b ℓ+1 .
Proposition 4.3. There is a second order differential operator ∆ : Λ * → Λ * of degree −2 and a derivation ∂ : Λ * → Λ * of degree −1 such that for f ∈ Λ * k homogeneous of weight k. The differential operators are given in terms of the generators p ℓ by Proof. From the definition and [CMZ18, Proposition 9.2] we deduce that the evaluation map can be computed for any F ∈ M k as where d = 12∂/∂E 2 and where a 0 : F → F (∞) is the constant term map from M * to Q. From [CMZ18, Proposition 8.3] we deduce (note that differentiation with respect to Q i in loc. cit. gives the extra p 1 -derivative here) that the differential operators defined above have the property that Since the constant term of the q-bracket of f is in Λ * , the claim follows from these two equations.
To motivate the next section, we recall the notion of cumulants. Let R and R ′ be two commutative Q-algebras with unit and : R → R ′ a linear map sending 1 to 1. (Of course the cases of interest to us will be when R is the Bloch-Okounkov ring Λ * and is the q-, X-, or ℏ-bracket to R ′ = M * , Q[X], or Q[π 2 ][ℏ], respectively.) Then we extend to a multi-linear map R ⊗n → R ′ for every n ≥ 1, called connected brackets, the image of g 1 ⊗ · · · ⊗ g n being denoted by g 1 | · · · |g n , that we define by The most important property of connected brackets is their appearance in the logarithm of the original bracket applied to an exponential: We specialize to the Bloch-Okounkov ring Λ * and we want to compute the leading terms of the connected brackets associated with the · X -or · ℏ -brackets. Recall from [CMZ18, Proposition 11.1]: . . , n) be elements of weight less than or equal to k i and let g ⊤ i ∈ Λ * ki be their top weight components. Let k = k 1 + · · · + k n be the total weight. Then deg( g 1 | · · · |g n X ) ≤ 1 − n + k/2 and The leading terms of the brackets are consequently We call them rational cumulants.
4.3. The cumulant recursion. In this section we prove a formula for computing the connected brackets associated with the · q -or rather the · ℏ -brackets. The core mechanism for their computation is summarized in the following purely algebraic property.
Let R be an N-graded commutative Q-algebra with R 0 = Q, complete with respect to the maximal ideal m = R >0 . The following statement gives a general recursion for expressions that appear in cumulants. We will specialize R to the Bloch-Okounkov ring subsequently.
Key Lemma 4.5. Suppose that D : R → R is a linear map. Then the following statements are equivalent: (1) We have D(x 3 ) − 3xD(x 2 ) + 3x 2 D(x) = 0 for all x ∈ R.
(2) For all x ∈ R and all n ≥ 2 (3) For all x, y, z ∈ R (4) If we denote by D 2 : R 2 → R the symmetric bilinear form for all x 1 , . . . , x n ∈ R.
(5) For any fixed x ∈ R, the bilinear form D 2 (x, y) is a derivation in y.
We now let L(t) = n≥0 L n (X)t n and the claims follow.
4.4. Application to volume computations. We now return to the proof of the main theorem of this section. Recall the main idea from [EO01] that the volume of a stratum is given by the growth rate of the number of connected torus covers and thus to the leading terms of cumulants of the f ℓ . More precisely for 2g − 2 = n i=1 m i , the same argument as in [CMZ18,Proposition 19.1] gives that vol (ΩM g,n (m 1 , . . . , m n )) = (2πi) 2g f m1+1 | · · · |f mn+1 L (2g − 2 + n)! .
Proof of Theorem 4.1, one variable case. First, P B (u) = P (u)| p ℓ →ℓ!b ℓ = P (u)(∅). Next, recall that Lagrange inversion for a power series F ∈ uC[[u]] with non-zero linear term and inverse G(z) states that k[z k ]G n = n[u −n ]F −k for k, n = 0. We apply this to F = u/P B (u) and to k = 2g − 1 and n = −1 to obtain that using Proposition 4.2, (31) and Lagrange inversion.
We pause for a moment to check the initial condition of the theorem in the previous section independently of the Hermitian metric extension problem along the boundary of the strata.
We apply Lagrange inversion to φ(z) = z −2g and F = 1/ A(z) to obtain that using (45) and (15). This implies the claim.
For the general case of the theorem, we apply Section 4.3 to the differential operator Proposition 4.6. The polarization D 2 of D can be computed in terms of the h ℓgenerators by the formula in (28).
Proof. In terms of the p ℓ -generators the polarization is given by The definition (31) of h ℓ in terms of p ℓ implies that We compute that and this implies the claim by the chain rule.
We now define the partition function of h-brackets in the h ℓ -variables. Then the partition function of the rational cumulants for the h ℓ -generators is simply the logarithm of Φ H .
Proof of Theorem 4.1, general case. We first show that the pieces of Φ H sorted by total degree in u can be recursively computed using the D 2 -operator. For this purpose we let h i = ℏ −i h i . From the definition of cumulants, equation (36) and a 0 ( g q ) = g(∅), we obtain that By applying the Key Lemma with X = i≥1 h i u i and undoing the rescaling of the h i using (40) we obtain that with for n > 0. Now define a linear map ℧ n : Q[u] → Q[z] by ℧ n (u ℓ1 · · · u ℓn ) = Symm(z ℓ1 1 · · · z ℓn n ) and zero for monomials of length different from n, where Symm denotes symmetrization with respect to the S n action on the variables z i . In this notation H 1 = ℧ n L n−1 and Consequently, (54) and (27) together with (44) and Proposition 4.2 imply the claim.

Equivalence of volume recursions
In this section we introduce another "averaged volume" recursion that interpolates between the D 2 -recursion introduced in Section 4 and the volume recursion in Theorem 1.2. We will show that the averaged volume recursion and the D 2recursion give the same generating functions, and then Theorem 1.1 will follow from it.
Recall from (27) For any list of positive integers p = (p 1 , . . . , p k ) we define For a finite set I = {i 1 , . . . , i n } of positive integers we define the formal series

.]] inductively by
if n = 1, and otherwise (55) where A For the proof of this theorem we will show that both A I and H I can be written as a sum over a function on certain oriented trees. The two recursions can then be viewed as stemming from cutting the trees at a local maximum (a "top") or a local minimum (a "bottom") respectively. 5.1. Proof of Theorem 1.2 and Theorem 1.1. We assume in this section that Theorem 5.1 holds and finish the proof of Theorems 1.2 and 1.1 under this assumption. We abbreviate λ = (ℓ 1 , . . . , ℓ n ) and recall from Section 4 that we denoted the coefficients of A n for n ≥ 2 by A n = H n = ℓ1,...,ℓn≥1 h λ z ℓ1 i1 · · · z ℓn in .
Proof. We begin by showing the formula for λ with two entries, which in view of (6) is equivalent to show that Since applying (z 1 ∂ ∂z1 + z 2 ∂ ∂z2 ) to the left-hand side above gives A 2 , and since neither side has a constant term, this in turn follows from For λ with more entries, we deduce (for all ℓ r , ℓ s ≥ 1 and all p) from the preceding calculation that Besides, the recursion formula (55) defining A I can be translated for λ with n parts into where λ 1 ⊔ · · · ⊔ λ k partitions λ \ {ℓ r , ℓ s }. Next we remark that the interior sum of (56) is over all backbone graphs with the two markings labelled with r and s at the lower level. In the preceding formula (57) the contribution of vertices with at least one marking is separated from the vertices with no markings. This choice results in a binomial coefficient k+k ′ k and transforms 1 k!k ′ ! into 1 (k+k ′ )! , thus showing that the two recursive formulas (57) and (56) are equivalent.
Proof of Theorem 1.2 and Theorem 1.1. For µ = (m 1 , . . . , m n ) consider the intersection numbers a(µ) that satisfy the recursion in Theorem 3.1, and recall that a(µ) = a i (µ) is independent of the index i by Proposition 3.2. In particular the a(µ) satisfy the recursion (16) for any distinguished pair of indices, and hence satisfy every weighted average of these recursions. We use the weighted average where the recursion with (i, j) distinguished is taken with weight k ∈{i,j} (m k + 1). Conversely, the a(µ) are uniquely determined by this weighted average and the initial values for µ of length one given in (15).

5.2.
Oriented trees. We now start preparing for the proof of Theorem 5.1. An oriented tree is the datum of a graph G = (V, E ⊂ V × V ) whose underlying graph of (V, E) is a tree. In particular it is required to be connected. If (v, v ′ ) ∈ E, we will denote v > v ′ . Moreover, a vertex v ∈ V is called a bottom (respectively a top) if there exists no v ′ ∈ V such that v > v ′ (respectively v < v ′ ). We will denote by B(G) and T (G) the sets of bottoms and tops of G.
For any oriented tree G with n vertices, we define the rational number whose numerator is the number of total orderings on the set of vertices compatible with the orientation of G.
Lemma 5.3. The function f # can be expressed as where in both cases the product is over all connected components G ′ of the oriented graph obtained by deleting the vertex v.
Proof. In order to define a total ordering on V compatible with the orientation of G, we begin by choosing a minimal element v ∈ V . This element is necessarily a bottom. Let us fix such a choice and denote by (G 1 , . . . , G k ) the connected components of G \ {v}. Let n i be the number of vertices of G i for 1 ≤ i ≤ k. A total ordering on V with minimal element v is equivalent to choosing a total ordering on the vertices of G i for all 1 ≤ i ≤ k and a partition of [[1, n − 1]] into k sets of size (n 1 , . . . , n k ). Such an ordering on V is compatible with the orientation of G if and only if each ordering on the vertices of G i is compatible with the orientation of G i for all 1 ≤ i ≤ k. This implies that the number of total orderings on V with minimal element v is equal to Summing over all possible choices of a minimal element, the number of total orderings on vertices of G compatible with the orientation of G is equal to which completes the proof. for all v ∈ V . If I has cardinality greater than one, we denote by OT(I) the set of I-decorated oriented trees. One can easily check that the following two properties hold: • if I has cardinality n ≥ 2, then Γ has n − 1 vertices; • a vertex of a decorated tree is a bottom if and only if it has exactly two markings.
We denote by OT(I) v , OT(I) b and OT(I) t the sets of decorated trees with a choice of an arbitrary vertex, a choice of a bottom and a choice of a top, respectively. If I = {i} has only one element, we define OT({i}) v = {i} as a trivial graph decorated by i.
Lemma 5.4. If I has cardinality greater than one, then there is a bijection given by cutting at a top vertex, were the union is over all non-empty proper subsets. Similarly, there is a bijection given by cutting at a bottom vertex, were the union is over all partitions of I into k + 1 non-empty sets such that the first distinguished set has precisely two elements and where the element e j = max(I) + j for all 1 ≤ j ≤ k. We denote by ψ t and ψ b the inverses of ϕ t and ϕ b , respectively.
Proof. Given a decorated tree with a chosen top vertex v, we define its image under ϕ t as follows: • If there are two markings on v, then v has no outgoing edges, hence the graph has v as a unique vertex and I has only two elements. It follows that OT(I) v has only one element (and so does the right-hand side of (61)). • If there is only one marking i ∈ I on v and one outgoing edge to a vertex v ′ < v, then I ′ = {i} and the corresponding element in OT(I \ I ′ ) v is the graph obtained by deleting v and choosing v ′ as the distinguished vertex. • If there are two outgoing edges to vertices v ′ < v and v ′′ < v, then v has no I-markings and the graph obtained by deleting v has two connected components. We define I ′ to be the set of markings on the component containing v ′ and define the corresponding elements of OT(I ′ ) v and OT(I \ I ′ ) v to be the connected components containing v ′ and v ′′ as chosen vertices, respectively. The inverse of ϕ t in the first two cases is clear, and in the last case is given by adding a top vertex adjacent to the two chosen vertices.
Given a decorated tree with a chosen bottom vertex v, in the same spirit we define the function ϕ b as follows. Since v is a bottom, it has no outgoing edges, hence it has exactly two I-markings i 1 and i 2 , and the corresponding k graphs on the right-hand side of (62) are the k connected components of the graph obtained by removing v. The inverse of ϕ b is given by gluing these k graphs back to v along the vertices marked by e 1 , . . . , e k . Now we fix a ring R ′ and a function g : OT(I) → R ′ . By slight abuse of notation, we write f # : OT(I) → Q for the composition of f # defined in (60) with the forgetful map of the decorations. As a consequence of the two preceding lemmas we see that the sum S(g) = can be rewritten in two different ways, namely and where n is the cardinality of I (i.e. Γ ∈ OT(I) has n − 1 vertices as remarked before).

Explicit expansions over decorated trees.
In order to finish the proof of Theorem 5.1, we will show that both A I and H I are equal to a generating series S I that is directly defined as a sum over OT(I). Let Γ = (V, E, I → V ) be an oriented tree with decoration by I. A twist assignment on Γ is a function p : E → Z >0 . We work over the ring R[[(z i ) i∈I , (z e ) e∈E ]] of formal series in variables indexed by I ∪ E. Given a twisted decorated oriented tree, we define the contribution of a vertex as End of the proof of Theorem 5.1. We will show that S I = H I and S I = A I for all sets of positive integers I with n = Card(I) > 2. The equalities in the case n = 2 are obvious from the definition. We assume now that n ≥ 3 and that S I ′ = H I ′ = A I ′ for all I ′ such that Card(I ′ ) < n. We first prove that S I = H I . We begin by rewriting the defining equation (27) with two auxiliary "edge" variables z e ′ and z e ′′ for distinct indices e ′ , e ′′ ∈ N \ I as To evaluate the derivative of H I ′ , there are two cases to consider, depending on the cardinality of I ′ . If I ′ = {i}, then ∂H I ′ ∂hp = z p i for all p > 0. Otherwise, we use the induction hypothesis to compute that Now we assume that both I ′ and I \ I ′ have at least two elements. Take two oriented trees ( be the combined graph in OT(I) t as described in Lemma 5.4. The datum of two twist assignments p ′ and p ′′ on Γ ′ and Γ ′′ respectively together with a pair of positive integers (p e ′ , p e ′′ ) is equivalent to the datum of a twist assignment p : V → Z >0 on the graph Γ. Moreover, the contributions of the vertices of Γ with the twist assignment p are given by One checks that this is still true if one of the graphs Γ ′ or Γ ′′ has only one marking. In summary we obtain that where we use (63) to pass from the above second line to the third.
Finally, we prove that S I = A I by a similar argument. It suffices to prove for the case I = [[1, n]]. Using the induction hypothesis that A Ij ∪{n+j} = S Ij ∪{n+j} , we can rewrite the inductive definition (55) of A I as The datum of p together with the twist assignments p (j) for the split graphs Γ j for 1 ≤ j ≤ k is equivalent to a twist assignment for the combined graph Γ = ψ b (Γ 1 , . . . , Γ k ) defined in Lemma 5.4. Moreover, given such a twist assignment the contribution of the vertex carrying i 1 and i 2 is H p i1,i2 . Thus we obtain that where we use (64) to pass from the above second line to the third.

Spin and hyperelliptic components
In this section we prove a refinement of Theorem 1.2 and Theorem 1.1 with spin structures taken into account. We also prove the corresponding refinement for hyperelliptic components in Section 6.5. 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 ( [EOP08]) 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 5 to convert this into the recursion for intersection numbers.
In this section we assume that all entries of µ = (m 1 , . . . , m n ) are even. The spin parity of a flat surface (X, x 1 , . . . , x n , ω) ∈ ΩM g,n (µ) is defined as The parity is constant in a connected family of flat surfaces by [Mum71]. We will denote by ΩM g,n (µ) • with • ∈ {odd, even} the moduli spaces of flat surfaces with a fixed odd or even spin parity. Note that for µ = (2g − 2) with g ≥ 4 and µ = (g −1, g −1) with g ≥ 5 odd, one of the two spin moduli spaces is disconnected, since it contains an extra hyperelliptic component (see [KZ03,Theorem 2]). Moreover, we will denote by ΩM g,n (µ) • their incidence variety compactification and will similarly use this symbol, e.g. in the form v(µ) • , c 1↔2 (µ, C) • , and c area (µ) • for volumes and Siegel-Veech constants.
To state the refined version of the volume recursion we need a generalization of the spin parity. Let (Γ, ℓ, p) be a backbone graph. A spin assignment is a function The parity of the spin assignment is defined as Our goal is the following refinement of Theorems 1.2 and 1.1 under a mild assumption. Recall that the tautological line bundle O(−1) over PΩM g,n (µ) has a natural hermitian metric given by h(X, ω) = i 2 X ω ∧ ω.
This assumption was already present in [Sau18] and will be proved in an appendix using the sequel to [BCG + 18]. Note that we do not need this assumption for Theorem 1.2, as it is stated for the entire stratum whose cohomology class was computed recursively in [Sau17]. However, currently we do not know the cohomology class of each individual spin component.
where the summation conventions for g, µ and p are as in Theorem 1.2 and where the superscript φ(i) indicates the corresponding spin component.
We remark that the same formula holds when replacing "odd" by "even" in the theorem, which follows simply by subtracting the formula in Theorem 6.2 from that in Theorem 1.2.
Theorem 6.3. Let PΩM g,n (µ) • with • ∈ {odd, even} be the connected component(s) of PΩM g,n (µ) with a fixed spin parity. Then the volume can be computed as an intersection number We first show in Section 6.1 that the intersection numbers on the right-hand side of Theorem 6.3 satisfy a recursion as in Theorem 6.2. This is parallel to Section 3. We then complete in Section 6.2 properties of the strict brackets introduced by [EOP08]. The volume recursion in Section 6.3 is parallel to Section 4 and allows efficient computations of volume differences of the spin components. We do not need to prove an analog of Section 5 but can rather apply the results, since the structures of the two recursions are exactly the same as before. Only in Section 6.4 we need Assumption 6.1 to prove the beginning case of Theorem 6.2, i.e. the case of the minimal strata.
6.1. Intersection theory on connected components of the strata. With a view toward Section 6.5 for the hyperelliptic components, we allow here also the profile µ = (g − 1, g − 1) (with g − 1 not necessarily even) and • ∈ {odd, even, hyp}, and study the corresponding union of connected components PΩM g,n (µ) • .
Let (Γ, ℓ, p) be a twisted bi-colored graph and D be an irreducible component of the boundary PΩM p Γ,ℓ . We recall from Section 3 that ζ # Γ,ℓ (D) is a divisor of PΩM g,n (µ) if and only if dim(PΩM p Γ,ℓ ) = dim(PΩM g,n (µ)) − 1. Hence in this case we define α(D) = ζ # Γ,ℓ * (D) ∈ H * (PΩM g,n ), and define α(D) = 0 otherwise. We will denote by PΩM By the same arguments as in Section 3.3 (see Proposition 3.10) one can show that D ·ξ 2g−2 = 0 only if D is an irreducible component of PΩM p Γ,ℓ with (Γ, ℓ, p) a backbone graph. If (Γ, ℓ, p) is a backbone graph, then we let α Proposition 6.5. If n ≥ 2, then the values of a i (µ) • are the same for all 1 ≤ i ≤ n, denoted by a(µ) • , and can be computed as Proof. This follows from the same argument as in the proof of Lemma 3.13.
Proposition 6.6. For µ of length bigger than one and with even entries, we have where the sum is over all choices of backbone graphs with only the first two marked points in the lower level component.
This proposition is a refined combination of Lemma 3.14 and equation (24). 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.
Proof. We apply Proposition 6.4 to PΩM g,n (µ) odd . The proposition then follows from the description of the boundary divisors of connected components of PΩM g,n (µ).
Let (L → X → ∆) be a one-parameter family of theta characteristics, i.e. L is a line bundle such that L| ⊗2 X ≃ ω X for every fiber curve X parametrized by a complex disk (centered at the origin) such that • the restriction of X to ∆ \ 0 is a family of smooth curves; • the central fiber X 0 is 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| X0 equals the sum of the parities over all irreducible components of X 0 (see e.g. [CC16, Proposition 4.1]), which implies that the number of components of X 0 with an odd theta characteristic is odd.
Let (Γ, ℓ, p) ∈ BB(g, n) 1,2 . From the above description we deduce that PΩM p,odd Γ,ℓ can be written as where M −1 and PΩM gv ,nv (p v − 1, µ v ) are defined as in Section 3.2. The arguments in the proof of Lemma 3.13 imply that Then by the same line of arguments as in Section 3.5 (expansions over rooted trees), we get the desired expression.
6.2. Strict brackets and Hurwitz numbers with spin parity. Let f : SP → Q 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 strict bracket defined by The analog of the algebra Λ * is the algebra Λ * = Q[p 1 , p 3 , p 5 , . . .] of supersymmetric functions, where for odd ℓ the functions p ℓ are defined by Note the modification of the constant term and the absence of the shift in comparison to (29). We provide Λ * with the weight grading by declaring p ℓ to have weight ℓ + 1. On the other hand, [EOP08] used characters of the modified Sergeev group C(d) = S(d) ⋉ Cliff(d) to produce elements in Λ * . Here Cliff(d) is generated by involutions ξ 1 , . . . , ξ d and a central involution ε with the relation ξ i ξ j = εξ j ξ i . Irreducible representations of C(d) are V λ indexed by λ ∈ SP. We denote by f µ (λ) the central character of the action of a permutation g µ ∈ S(d) ⊂ C(d) of cycle type µ on V λ by conjugation. The analog of the Burnside formula is [EOP08, Theorem 2] stating that for a fixed profile Π = (µ 1 , . . . , µ n ) where the sum is over all covers p : X → E of a fixed base curve and profile Π.
Theorem 6.7. If we define then the difference f ℓ −ℓh ℓ has weight strictly less than ℓ+1. In particular f ℓ belongs to the subspace Λ * ≤ℓ+1 of weight less than or equal to ℓ + 1. More precisely, This statement was missing in the proof of the following corollary, one of the main theorems of [EOP08].
Corollary 6.8. The strict bracket f ℓ1 f ℓ2 · · · f ℓn str is a quasimodular form of mixed weight less than or equal to n i=1 (ℓ i + 1).
We now prepare for the proof of Theorem 6.7 and prove the corollary along with more precise statements on strict brackets in the next subsection. From [Iva01, Definition 6.3 and Proposition 6.4] we know that the central characters are given by where the objects on the right-hand side are defined as follows. We define for any partition λ the Hall-Littlewood symmetric polynomials These polynomials have cousins where the powers are replaced by falling factorials. That is, writing n ↓k = n(n − 1)(n − 2) · · · (n − k + 1), we define Next, we define X • • (t) to be the base change matrix from the basis of p ρ to the basis P λ (x 1 , . . . , x m ; t), that is, we define them by The existence and the fact that the X • • (t) are polynomials in t is shown in [Mac95, Section III.7]. We abbreviate X ρ λ = X ρ λ (−1) and similarly P λ = P λ ( · ; −1) and P ↓ λ = P ↓ λ ( · ; −1).
Proof of Theorem 6.7. We need to prove (68). From there one can then derive (67) by expanding the exponential function (just as in [IO02] Proposition 3.5 is derived from Proposition 3.3). We use that for ρ = (ℓ) a cycle, the coefficients X λ ρ in (69) are supported on λ with at most two parts. More precisely, by [Mac95, Example III.7.2] we know that Using and the specialization of this formula for ℓ = 0, our goal is to show that Using that for ℓ odd we see that our goal and the known (71) agree.
6.3. Volume computations via cumulants for strict brackets. We denote by an upper index ∆ the difference of the even and odd spin related quantities, e.g. v(µ) ∆ = v(µ) even − v(µ) odd . Cumulants for strict brackets are defined by the same formula (38) as for q-brackets. We are interested in cumulants for the same reason as we were for the case of the strata as in (44).
Here the subscript L refers to the leading term g 1 | · · · |g n str,L = [ℏ −k−1+n ] g 1 | · · · |g n str,ℏ = lim h→0 ℏ k+1−n ev[ g 1 | · · · |g n str,q ](ℏ) for g i homogeneous of weight k i and k = n i=1 k i . This proposition was certainly the motivation of [EOP08], which stops short of this step. To derive the proposition from (66), we need one more tool, the analog of the degree drop in Proposition 4.4. We use the fact that for strict brackets we have a closed formula (rather than only a recursion as for q-brackets), proved in [EOP08, Section 3.2.2] and in more detail in [BO00, Section 13]. First, and for the more general statement we define the "oddification" of the Eisenstein series to be where D q = q∂/∂q. Then by Proposition 13.3 in loc. cit. the n-point function is given by ℓi≥1, ℓi odd p ℓ1 p ℓ2 · · · p ℓn str z ℓ1 1 · · · z ℓn n ℓ 1 ! · · · ℓ n ! = α∈P(n) A∈α Consequently, the cumulants are simply given by p ℓ1 |p ℓ2 | · · · |p ℓn str = z ℓ1 1 · · · z ℓn n ℓ 1 ! · · · ℓ n ! G odd (z 1 , . . . , z n ) .
Proof of Proposition 6.9. Note that deg Ev p ℓ1 p ℓ2 · · · p ℓn str = 1 2 n i=1 ℓ i , the highest term being contributed by the partition into singletons. From (74) we deduce that deg Ev p ℓ1 |p ℓ2 | · · · |p ℓn str = 1 2 n i=1 ℓ 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 ( [EO01] and [CMZ18, Proposition 19.1]). While (74) 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 Λ * and the Key Lemma 4.5 also works here. The analog of Proposition 4.3 is the following result.
where the evaluation at the empty set is explicitly given by p ℓ → − ζ(−ℓ) 2 . Note that we can regard the differential operator (∆−∂/∂p 1 )/2 appearing in the exponent the same as the operator D defined in (47) when viewing Λ * as a quotient algebra of Λ * with all the even p ℓ set to zero, since the differential operator ∂ sending p ℓ to a multiple of p ℓ−1 is zero on this quotient.
Proof. Using the description (36) of the ℏ-evaluation we need to show that d f str,ℏ = (∆ − ∂/∂p 1 )f str,ℏ . Contrary to the case of q-brackets we will actually show the stronger statement that d f str = (∆ − ∂/∂p 1 )f str . It suffices to check this for all the n-point functions. For n = 1 this can be checked directly from (72). For general n, we write W (z) = s≥1 z 2s−1 /(2s − 1)!. Using (73) and that the commutator [d, D q ] is multiplication by the weight, we compute that where for the factor in the summand with |A 1 | ≥ 2 and where the summation is over all tuples (s a ) a∈A1 . Since 1 2 ∂/∂p 1 ( , the strict bracket of this expression is precisely the second line on the right-hand side of (76). Since its strict bracket matches the first line on the right-hand side of (76), and the part containing the variable for W (z i +z j ) produces of course the special factor (77).
This proposition provides an efficient algorithm to compute the differences of volumes of the spin components. The definitions below are completely analogous to the beginning of Section 4, except that objects with even indices have disappeared and they are written in boldface letters for distinction. For the substitution, we define and α ℓ = u ℓ 1 (u/P Z (u)) −1 .
Proof. Thanks to Proposition 6.10 and the subsequent remark, the proof of Theorem 4.1 can be copied verbatim here. The extra factor 2 −ℓ/2 in the definition of P Z in comparison to the constant term −ζ(−ℓ)/2 of the evaluation of p ℓ compensates for the fact that the strict bracket of the f ℓ gives the counting function in (66) up to a power of two.

Conclusion of the proofs for spin components.
Proof of Theorem 6.3 and Theorem 6.2. Theorem 6.2 is a consequence of Corrolary 6.11. Indeed the arguments of Section 5 adapted to the series H n show that the recursion in Theorem 6.2 is a consequence of the recursion in (78).
6.5. Volume recursion for hyperelliptic components. 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 1.2. It is a consequence of the work of Athreya, Eskin and Zorich ([AEZ16]) on volumes of the strata of quadratic differentials in genus zero.
Recall that only the strata ΩM g (g −1, g −1) and ΩM 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.
Theorem 6.12. For the hyperelliptic components we have provided that Assumption 6.1 holds.
Proposition 6.13. The volumes of the hyperelliptic components ΩM g,2 (g−1, g−1) satisfy the recursion Note in comparison to Theorem 1.2 that only the terms k = 1 and k = 2 appear and that the Hurwitz number h P is identically one here. As a preparation for the proof recall that the canonical double cover construction provides isomorphisms that preserve the Masur-Veech volume and the SL 2 (R)-action. Taking into account the factorials for labeling zeros and poles the main result of [AEZ16] can be translated as where the double factorial notation means (2k)!! = 2 k k! and (2k − 1)!! = (2k)!/2 k k!.
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. ℓ = 0 and ℓ = g), we need to show that For this purpose it suffices to prove the following two identities of generating series and 2 g≥0 16 g g 2 2g g −1 so that we can take the square of the first series and compare the x 2g -terms.
To prove (80) we multiply it by x, differentiate, and are then left with showing that ℓ≥0 2ℓ ℓ x 2ℓ = 1/ √ 1 − 4x 2 , which follows from the binomial theorem. To prove (81) we differentiate and are then left with the identity which is already proved in [Leh85, p. 452, Equation (9)].
The last ingredient is the following straightforward consequence of Proposition 6.5 (analogous to the case of spin components in Proposition 6.6).
7. An overview of Siegel-Veech constants Let (X, ω) be a flat surface, consisting of a Riemann surface X and an Abelian differential ω on X. Siegel-Veech constants measure the asymptotic growth rate of the number of saddle connections (abbreviated s.c.) or cylinders with bounded length (of the waist curve) in (X, ω). There are many variants that we now introduce and compare. 7.1. Saddle connection and area Siegel-Veech constants. For each pair of zeros (z 1 , z 2 ) of ω we let A phy 1↔2 (T ) = |{γ ⊂ X a saddle connection joining z 1 and z 2 , be the counting function. The upper index emphasizes that we count all physically distinct saddle connections. It should be distinguished from the version A hom 1↔2 (T ) = |{γ ⊂ X a homology class of s.c. joining z 1 and z 2 , showed that for almost every flat surface (X, ω) in the sense of the Masur-Veech measure (see [Mas82] and [Vee82]) there is a quadratic asymptotic, i.e. that The constants c phy 1↔2 (X, ω) and c hom 1↔2 (X, ω) are the first type of Siegel-Veech constants we study here, called the saddle connection Siegel-Veech constants. The difference between these two Siegel-Veech constants becomes negligible as the genus of X tends to infinity, which follows from the results of Aggarwal and Zorich (see [Agg18a, Remark 1.1]).
The second type of Siegel-Veech constants counts homotopy classes of closed geodesics, or equivalently flat cylinders. Again, there are two variants, the naive count and the count where each cylinder is weighted by its relative area. As above, the most important counting function with good properties (see e.g. [CMZ18]) and connection to Lyapunov exponents ( [EKZ14]) is the second variant. For the precise definition we consider where w(Z) denotes the width of Z, i.e. the length of its core curve. We then define the cylinder Siegel-Veech constant and the area Siegel-Veech constant by the asymptotic equalities There is a natural action of GL 2 (R) on the moduli space of flat surfaces ΩM g and the orbit closures are nice submanifolds, in fact linear in period coordinates by the fundamental work of Eskin, Mirzakhani and Mohammadi ( [EM18] and [EMM15]). We refer to them as affine invariant manifolds, using typically the letter M. The intersection with the hypersurface of area one flat surfaces (denoted by the same symbol M) comes with a finite SL 2 (R)-invariant ergodic measure ν M with support M. This measure is unique up to scale and for affine invariant manifolds defined over Q there are natural choices of the scaling.
The relevant orbit closures in this paper are the connected components of the strata of Abelian differentials and certain Hurwitz spaces inside the strata. We usually abbreviate by c ⋆ 1↔2 (µ) = c ⋆ 1↔2 (ΩM g,n (µ)) the Siegel-Veech constants for the strata with signature µ.

7.2.
Configurations and the principal boundary. One of the main insights of [EMZ03] is that Siegel-Veech constants can be computed separately according to topological types, called configurations. We formalize their notion of configurations briefly so that it also applies to Hurwitz spaces, and in fact to all SL 2 (R)-orbit closures M provided with the generalization of the Masur-Veech measure ν M . The concept of configurations will be used for showing the equivalence between Theorem 1.3 and Theorem 1.2 in Section 8.
Let (X, z 1 , . . . , z n ) be a pointed topological surface. A configuration C of saddle connections joining z 1 and z 2 for M is a set of simple non-intersecting arcs from z 1 to z 2 up to homotopy preserving the cyclic ordering of the arcs both at z 1 and z 2 . The last condition implies that the tubular neighborhood of the configuration is a well-defined subsurface of X, in fact a ribbon graph R(C) associated with the configuration. The number of arcs in the configuration is called the multiplicity of the configuration.
We say that the saddle connections of length ≤ T joining z 1 and z 2 on a flat surface (X, ω) belong to the configuration C, if the set of these saddle connections is homotopic to C. Each configuration gives rise to a counting function A ⋆ 1↔2 (T, C) for saddle connections belonging to the configuration and to the corresponding Siegel- A full set of saddle connection configurations for an affine invariant manifold M is a finite set of saddle connection configurations C i , with i ∈ I such that the contributions of the configurations C i sum up to the full Siegel-Veech constant, i.e. such that for ⋆ ∈ {phy, hom} respectively. Note that [EMZ03, Section 3.2] in their definition of configurations made a further subdivision of the notion by adding metric data, i.e. specifying angles between saddle connections. In that context, Eskin, Masur and Zorich determined a full set of saddle connection configurations for the strata and used the Siegel-Veech transform to connect the computation to volume computations. The following statement summarizes Proposition 3.3, Corollary 7.2 and Lemma 8.1 of [EMZ03].
Proposition 7.1. For any stratum ΩM g,n (µ) a full set of saddle connection configurations is the set of collections of pairwise homologous simple disjoint arcs joining z 1 and z 2 (up to homotopy).
In this proposition, several configurations are irrelevant, for example those with a connected component of genus zero after removing the saddle connections in the configuration.
The general strategy to compute Siegel-Veech constants is the following relation to volumes, where the submanifold M is in a stratum with labeled zeros.
Proposition 7.2. The saddle connection Siegel-Veech constants of an affine invariant manifold M can be computed as where the sum runs over the full set of saddle connection configurations and where m hom (C) = 1 for all C while m phy (C) is equal to the number of arcs in C.
Proof. This is a direct consequence of the Siegel-Veech transform applied to the characteristic function of a disc of radius ε, see [EMZ03,Lemma 7.3] together with the Eskin-Masur bound on the number of short saddle connections (Theorem on p. 84 of [EMZ03]).
We conclude with remarks on Siegel-Veech constants for general affine invariant manifolds to put the digression on Hurwitz spaces (Section 9) in context. There is another variant, besides (83)  For Hurwitz spaces the two values can be different, but we will see (Proposition 9.2) that their difference becomes negligible as the degree of the covers tends to infinity.
In the first part of [EMZ03] on recursive computations of Siegel-Veech constants, Eskin, Masur and Zorich called the locus of degenerate surfaces that contribute to the Siegel-Veech counting the principal boundary. At that time the notion of principal boundary was used only as a partial topological compactification. Presently, we dispose of a complete and geometric compactification for the strata ([BCG + 18]) and for Hurwitz spaces (by admissible covers), and we can then identify the principal boundary as part of the compactification (see [CC16] for the case of the strata and Section 9 for the case of Hurwitz spaces). The reader should keep in mind that the locus "principal boundary" depends on the type of saddle connections under consideration.
Finally we remark that there is a zoo of possibilities of associating weights with saddle connections and cylinders and to define Siegel-Veech constants accordingly. This started with [Vor97], and see also [BG15] for computations and conversions of Siegel-Veech constants.

Saddle connection Siegel-Veech constants
In this section we deduce from the volume recursion and its refinement for spin and hyperelliptic components a proof of Theorem 1.3. Almost all we need here has been proven already in [EMZ03]. We start with two more auxiliary statements.
Proposition 8.1. The full set of saddle connection configurations for the strata given in Proposition 7.1 is in bijection with (possibly unstable) backbone graphs and a cyclic ordering of its vertices at level zero. The subset of relevant configurations is in bijection with stable backbone graphs.
Although not needed in the sequel, we relate for the convenience of the reader our notion of twists and the angle information that [EMZ03] be the arcs of a configuration realized by (X, ω) labeled cyclically and let a ′ i and a ′′ i be the angles between γ i and γ i+1 at z 1 and z 2 respectively. If an edge e is separated by the loop formed by γ i and γ i+1 , then the twist is The above proposition can be seen as follows. Given a collection of k homologous short saddle connections there is a sphere (with z 1 as its south pole and z 2 as its north pole, see [EMZ03, Figure 5]) supporting the k saddle connections. This sphere is the source of the backbone of the graph. The components at level zero are bounded by the arcs γ i and γ i+1 . The converse is obvious, given that all the edges of a (stable) backbone graph are separating by definition. Finally formula (90) is just a restatement of the Gauss-Bonnet theorem.
Recall that a backbone graph (being of compact type) is compatible with a unique twist p(·). If the vertices at level zero are labeled as 1, . . . , k as usual and if (h j , i(h j )) is the edge connecting the j-th vertex to level −1, we write p j = |p(h j )| as we did in Section 3.
Proof. This is mainly contained in [EMZ03, Corollary 7.2, Formulas 8.1 and 8.2], stating that the volume of the locus with an ε-short configuration is πε 2 times the volume of the corresponding boundary. We now explain the combinatorial factors that appear. First, the factor of two and the factorials result from the passage of the boundary volume element in the ambient stratum to the product of the volume elements of the components at the boundary, as explained in detail in [EMZ03,Section 6]. The 1/k! stems from labelling the level zero vertices. Second, we need to count the ways to obtain a surface in ΩM ε g,n (µ, C) by gluing a collection of surfaces (X i , ω i ) in ΩM gi,ni+1 (µ i , p i − 1).
Suppose we are given a branched cover b : P 1 → P 1 that has ramification profile Π = ((m 1 + 1), (m 2 + 1), (p 1 , . . . , p k )) over the points 0, 1 and ∞. We provide the domain with the differential ω −1 = b * dz. Since this differential has no residues we can glue t · ω −1 with the surfaces (X i , ω i ) by cutting the pole of order p i + 1 and gluing it to an annular neighborhood of the zero of order p i − 1 of ω i . For t ≤ ε this provides a surface in ΩM ε g,n (µ, C), see e.g. [BCG + 18, Section 4] for details of the construction. The plumbing construction also depends on the choice of a p i -th root of unity at each pole (from the choice of a horizontal slit at a zero of order p i − 1). In total there are h P 1 ((m 1 , m 2 ), p) · k i=1 p i possibilities involved in the construction, thus justifying the remaining combinatorial factors in the formula.
We claim that this construction provides a collection of maps to ΩM ε g,n (µ, C) that are almost everywhere injections if none of the surfaces (X i , ω i ) has a period of length smaller than ε. In fact, if two such plumbed surfaces are isomorphic, this isomorphism restricts to an automorphism of (P 1 , ω −1 ) (see [EMZ03,Lemma 8.1] for more details) and this happens only on a measure zero set. The locus where one of the (X i , ω i ) has a short period is subsumed in the o(ε 2 ) ([EMZ03, Lemma 7.1]). Conversely, for each surface (X, ω) in ΩM ε g,n (µ, C) we can cut a ribbon graph around the configuration C. The restriction of ω has no periods since the boundary curves are homologous by definition of C. It can thus be integrated and completed to a map b : P 1 → P 1 with ramification profile as above.
Proof of Theorem 1.3. We first focus on the case that ΩM g,n (µ) is connected. A decomposition g = k i=1 g i and (m 3 , . . . , m n ) = µ 1 ⊔ · · · ⊔ µ k (as in equation (5) we proved) determines uniquely a configuration and the converse is true up to the labeling of the k vertices at level zero by Proposition 8.1. The configuration is relevant if and only if the volumes on the right-hand side of (5) are non-zero. Since the saddle connection Siegel-Veech constant is the sum of ratios of the boundary volumes over the total volume (by Proposition (7.2)), comparing the formula in Proposition 8.2 with equation (5) thus implies Theorem 1.3. More precisely, note that the rescaled volume v(µ) defined in (4) involves a product of all (m i + 1), while on the right-hand side (m 1 + 1)(m 2 + 1) is missing and this factor gives the right hand side of the desired formula in Theorem 1.3. The factor of π in Proposition 8.2 cancels with the one in Proposition 8.1.
For disconnected strata with components parameterized by S ⊆ {odd, even, hyp} the same proof gives the averaged version that

The viewpoint of Hurwitz spaces
This section is a digression on how to interpret the volume recursion and the saddle connection Siegel-Veech constant from the viewpoint of Hurwitz spaces. The results in this section are not needed for proving any of the theorems stated in the introduction. We will rather explain and motivate • why the homologous count of saddle connections is more natural than the physical count from the viewpoint of intersection theory, • how to heuristically deduce the value of the saddle connection Siegel-Veech constant in Theorem 1.3 from an equidistribution of cycles in Hurwitz tuples, and • why backbone graphs correspond to configurations.
Theorem 9.1. There exists a constant M (µ) such that the Hurwitz numbers N • d (Π) for connected torus covers of profile Π can be approximated as where Π \ {µ (1) , µ (2) } = Π 1 ⊔ · · · ⊔ Π k is the decomposition of the profile according to the leg assignment in Γ and where p = (p 1 , . . . p k ) is the unique twist compatible with Γ.
At the end of the section we will show by combining Theorem 1.2 together with Theorem 9.1 that M (µ) = (m 1 + 1)(m 2 + 1). Indeed an independent proof of this equality would provide an alternative proof of Theorem 1.2 (and hence Theorem 1.1) that would bypass the complicated combinatorics in Sections 4 and 5.
The strategy to prove Theorem 9.1 consists of comparing the Hurwitz number N • d (Π), that is the fiber cardinality of the forgetful map f T : H d (Π) → M 1,n to the target curve with the fiber cardinality of the extension of f T to the space of admissible covers H d (Π) = H d,g,1 (Π) over degenerate targets of the following type. 9.1. Admissible torus covers. Let E 0,{1,2} be the stable curve of genus one consisting of a P 1 -component carrying precisely the first two marked points and of an elliptic curve E carrying the remaining marked points, joint at a node q E . If p : X → E 0,{1,2} is an admissible cover, we denote by X 0 and X −1 the (possibly reducible) curves mapping to E and to P 1 respectively, both deprived of their unramified P 1 -components. See Figure 3 for examples of such admissible covers.
The admissible covers of E 0,{1,2} come in two types. One possibility is that the first two branch points are in the same (hence the unique) component of X −1 . The stable dual graph of the cover is thus a graph Γ ∈ ABB ⋆ 1,2 and we denote by N • d (Π, E 0,{1,2} , Γ) the number of such covers. The second possibility is that each of the two branch points is on its own component X is a cyclic cover of degree (m i +1). By contracting the components over P 1 we see that such covers are (up to the automorphism group of size |Aut(X −1 /P 1 )| = (m 1 + 1)(m 2 + 1)) in bijective correspondence with covers of E with the profile Π (12) = ((µ (1) , µ (2) ), µ (3) , . . . , µ (n) ), where the first two ramification points are piled over the same branch point.
where Π i and p are associated with Γ as in Theorem 9.1.
Proof. Recall from [EO01] or the proof of [CMZ18,Proposition 9.4] that if Π is the profile for a cover π with π * ω ∈ ΩM g,n (µ), then there exist C 1 , C 2 = 0 such that Suppose that Γ has k components on level 0, each of genus g i and with n i marked points or nodes. Let g 0 be the genus of the component on level −1, and let b = h 1 (Γ). Then The cover of the P 1 -component has finitely many choices independent of d. Over the elliptic component of E 0,{1,2} , the number of choices of covers has asymptotic growth given by B d1+···+d k =d d 2gi−2+ni i for some constant B independent of d. This quantity is a polynomial of degree We thus conclude that the total number of admissible covers N 0 d (Π, E 0,{1,2} , Γ) has asymptotic growth given by a polynomial of degree r(µ) − 2 − b − 2g 0 ≤ r(µ) − 2, with equality attained if and only if b = g 0 = 0, i.e. if and only if Γ ∈ BB ⋆ 1,2 . To justify equation (92) we refer to the computation of the Hurwitz numbers in Proposition 2.1 and divide by k! to account for our auxiliary labeling of the k components of X 0 .
Proposition 9.2 reveals the geometric reason for homologous count of saddle connections behind the recursions in Sections 3 to 5. The factor h P 1 ((m 1 , m 2 ), p) (possibly with 1/k! if all branches are labeled) appears in the direct count of admissible covers and in the count of configurations. There is no extra factor k in (92), which corresponds to our setting of the coefficient m hom (C) = 1 (instead of k) in (89) for homologous count of saddle connections (instead of physical count).
Proof of Theorem 9.1. We first show that To see this, note that the ramification order of f T over E 0,{1,2} at the branch through a cover π : X → E is equal to the product of ramification orders at the nodes of X. This results in the factors p i inside the product of the right-hand side and cancels the factor 1/|Aut(X −1 /P 1 )| when counting E 0,{1,2} -covers instead of counting N • d (Π (12) ). On the other hand, since the volume of the stratum can be approximated by counting covers of profile Π (12) and since the generating function of counting these covers is a quasimodular form, arguments as in [CMZ18,Proposition 9.4] imply the existence of a constant M (µ) such that The combination of equations (93) and (94) thus implies the desired formula (91). Figure 3. Configurations for Hurwitz spaces in ΩM 2 (1, 1) We now address the equidistribution heuristics for saddle connection Siegel-Veech constants. Recall that N • (Π) is the number (weighted by |Aut(p)|) of transitive Hurwitz tuples (α, β, (γ i ) n i=1 ) ∈ S n+2 d with [α, β] = n i=1 γ i and γ i of type µ (i) . Proposition 9.3. If the pairs (γ 1 , γ 2 ) appearing in the Hurwitz tuples of profile Π equidistribute among pairs of (m i + 1)-cycles in S 2 d as d → ∞, then M (µ) = (m 1 + 1)(m 2 + 1).
Proof. If the non-trivial cycles in γ 1 and γ 2 have no letter in common, then taking (α, β, γ 1 • γ 2 , γ 3 , . . . , γ n ) is a Hurwitz tuple of profile Π (12) . Assuming equidistribution and comparing to the total number of Hurwitz tuples, the number of Hurwitz tuples with γ 1 and γ 2 having two letters in common is negligible, while the ratio of those having one letter in common is (m 1 + 1)(m 2 + 1)/d + o(1/d).
Example 9.4. For the reader's convenience we illustrate the contributions to the right-hand side of (93) for the stratum ΩM 2 (1, 1) explicitly in Figure 3. The picture on the left gives stable graphs in BB ⋆ 1,2 , while the pictures in the middle and on the right give graphs in ABB ⋆ 1,2 \ BB ⋆ 1,2 . The preimages of E in the middle and on the right are unramified and thus again are elliptic curves, while on the left the preimage of E is a curve of genus two.
The saddle connection Siegel-Veech counting in this case was carried out in [EMS03] in a similar way as summarized in Theorem 9.1, despite that only primitive torus covers were considered. 9.2. The principal boundary of Hurwitz spaces. We focus on saddle connections joining the first two marked zeros and determine a full set of configurations and the corresponding principal boundary of the Hurwitz spaces. We say that Γ ∈ ABB ⋆ 1,2 is realizable in H d (Π) if there is an admissible cover p : X → E 0,{1,2} whose stable graph is Γ and such that the vertices with ℓ(v) = 0 correspond bijectively to the components of X 0 . Recall also the definition of ribbon graphs associated to configurations in Section 7.2.
Proposition 9.5. Associating with a configuration C the boundary curves of the ribbon graph R(C) induces a map ϕ : C → Γ(C) from a full set of saddle connection configurations onto the subset of ABB ⋆ 1,2 that is realizable in H d (Π). The image of ϕ is independent of d for d large enough. The fibers of ϕ are finite with cardinality bounded independently of d.
Moreover if a graph Γ ∈ BB ⋆ 1,2 is realizable, then the configurations in ϕ −1 (Γ) are in bijection with cyclic orderings of the components at level 0.
Proof. To define ϕ, we pinch the boundary curves of R(C) to obtain a pointed nodal curve. The configuration C of saddle connections remains in one component of the curve that contains z 1 and z 2 . We provide the dual graph of the curve with the level structure such that the component containing z 1 and z 2 is the unique one at level −1 and all the other components are on level 0. This way we thus obtain a graph ϕ(C) ∈ ABB ⋆ 1,2 . We leave the straightforward verification of the other statements to the reader.
An application of the Riemann-Hurwitz formula shows that any configuration in ϕ −1 (Γ) has multiplicity |E(Γ)| + 2g(X −1 ). In the special case Γ ∈ BB ⋆ 1,2 (i.e. if g(X −1 ) = 0 and Γ is of compact type), the configuration consists of k = |E(Γ)| pairwise homologous arcs. However, the cover on the right-hand side of Figure 3 shows that graphs in Γ ∈ ABB ⋆ 1,2 \ BB ⋆ 1,2 also contribute. It is not hard to give an example that the fiber cardinality of ϕ over a target graph with g(X −1 ) > 0 can indeed be larger than one, and we leave it to the reader since it is irrelevant to our applications.
Finally we address that Theorem 9.1 and an a priori knowledge that M (µ) = (m 1 + 1)(m 2 + 1) would give an alternative proof of Theorem 1.2. In terms of our volume normalization, [CMZ18,Proposition 9.4] says that as D → ∞, where r = 2g + n − 1. To sum the right-hand side of (91) we let (a+b−1)! used recursively implies the following result.
Lemma 9.6. Suppose that S D (Π i ) = v i D ri + O(D ri−1 log D) as D → ∞ and that there exists a constant C depending on Π i only such that N • d (Π i ) < Cd ri−1 for i = 1, . . . , k. Then Alternative proof of Theorem 9.1 (assuming M (µ) = (m 1 + 1)(m 2 + 1)). With the abbreviation r i = 2g i + n(µ i ) we obtain from (95) that Since r 1 + · · · + r k = 2g + n − 2, Lemma 9.6 implies that Summing over all backbone graphs and taking the limit after dividing by D 2g+n−1 thus implies the desired formula (5).
Conversely, the above argument shows that the mere knowledge of Theorem 9.1 gives the recursion in Theorem 1.2 with M (µ) on the left-hand side that replaces (m 1 + 1)(m 2 + 1), and hence the two theorems taken together thus determine the value M (µ) = (m 1 + 1)(m 2 + 1) as claimed in the beginning of the section.

Area Siegel-Veech constants
The goal of this section is to show that area Siegel-Veech constants are ratios of intersection numbers, i.e. to prove Theorem 1.4. For this purpose we introduce and then Theorem 1.4 can be reformulated as The proof proceeds similarly to the proof of Theorem 1.1 by showing a recursive formula for both the intersection numbers and the area Siegel-Veech constants. The difference in the formulas is that one vertex at level zero of the backbone graphs is distinguished by carrying the Siegel-Veech weight. We remark that in this section area Siegel-Veech constants for disconnected strata are volume-weighted averages of the constants for the individual components. The intersection number recursion leads to the remarkable formula where c area (C) is defined to be the sum of the area Siegel-Veech constants of the splitting pieces induced by the saddle connection configuration. The other recursion leads to a very efficient way to compute area Siegel-Veech constants, given in Theorem 10.6.
10.1. A recursion for the d i (µ) via intersection theory. We have seen that the values of a i (µ) do not depend on the index i. Similarly for d i (µ) it suffices to focus on the case i = 1. To state the recursion, we introduce the generating series whose coefficients are determined using the following proposition.
Proposition 10.1. The generating function of the intersection numbers d 1 (2g − 2) is determined by the coefficient extraction identity while the intersection numbers d(µ) = d i (µ) with n(µ) ≥ 2 are given recursively by for n = n(µ) ≥ 2, with the usual summation conventions as in Theorem 1.2 and Theorem 3.1.
The first identity (101) was proved in [Sau18]. The proof of the second identity (102) will be completed by the end of this subsection. This identity together with the conversions in Section 8 implies (100) immediately. We start the proof with the following analog of Proposition 3.11.
Proposition 10.2. If (Γ, ℓ, p) is a backbone graph in BB(g, n) 1,2 , then with the conventions for p v as in Proposition 3.11.
Proof. We have the equality that . Combining with the fact that δ 0 λ g = 0, it implies that Therefore, we obtain that λ g−1 · δ 0 · α 0 Γ,ℓ,p = v∈V (Γ),ℓ(V )=0 Using the last formula in Lemma 3.9, the rest of the proof then follows from the same argument as in Proposition 3.11.
We also need the following analog of Lemma 3.13.
Proof. We use the formula in Proposition 3.12 for i = 2, multiply by ξ 2g−2 δ 0 n i=3 ψ i and apply p * . The left-hand side then evaluates (by Lemma 3.9 and the fact that δ 0 λ g = 0) to the left-hand side of (103). The right-hand side evaluates (by Proposition 3.10) to the weighted sum over all (Γ, ℓ, p) ∈ BB(g, n) 1,2 of the expression in Proposition 10.2. To prove the lemma we interpret as usual a backbone graph as a decomposition of g and the marked points. The factor (k − 1)! (instead of k! in Lemma 3.13) comes from the fact that one of the top level vertices of the backbone graph is distinguished.
End of the proof of Proposition 10.1. Now the proof of the proposition can be completed similarly to the end of the proof of Theorem 3.1 at the end of Section 3.
As a consequence, d i (µ) does not depend on i and we simply write d(µ) from now on.
To evaluate the numerator of this fraction, recall from [CMZ18, Section 16] the definition of the modified q-bracket where ∂ 2 is the differential operator This bracket is useful, since its effect can be computed by differential operators acting (contrary to T −1 ) within the Bloch-Okounkov algebra. In fact, [CMZ18,Theorem 16.1] states that where ρ i,j are differential operators of degree j that shift the weight by −i − 2j, whose definition we recall in (109) below. Motivated by the action of these operators we define and we let Φ H (u) = exp( ℓ≥1 h ℓ u ℓ ) such that Φ H (u) q = Φ H (u) q .
Proof. First note that Ev(G (j−1) 2 )(X) = −1 24 (j!X + (j − 1)!) by the defining formulas in [CMZ18,Section 9]. This is the reason for the factor j! in (107). From the non-vanishing of the area Siegel-Veech constant, we know that the leading degree contribution is as in (40). Lower weight terms before passing to the cumulant quotient will contribute to lower order in the growth polynomial. Since ∂ 2 is of degree −2, its contribution in (106) is negligible and we can work with the starbrackets. For the same reason, the terms with i > 0 in the definition f ⋆ q are dominated by the corresponding term with i = 0 and can be neglected.
Our goal is to compute the h-evalutaion of C • −1 (u) and its leading term using Proposition 4.3.
Proof. We will check the relation on the n-point function for every n. Since we will recall formulas from [CMZ18] we use the rescalings Q k = p k−1 /(k−1)! of the generators of Λ * , where Q 0 = 1 and Q 1 = 0. The following identities even hold on the polynomial ring R = Q[Q 0 , Q 1 , Q 2 , . . .] mapping to Λ * . We set W (z) = k≥0 Q k z k−1 .
We recall from [CMZ18, Theorem 14.2] the action of the operators ρ 0,j , namely where z J = j∈J z j and N = {1, . . . , n}. On the other hand, in terms of the Q i , the operator D defined in Section 4 is just D = 1 2 (∆−∂ 2 ), where ∂ is the differential operator sending Q i to Q i−1 . From [CMZ18, Proposition 10.5] we know that e D W z 1 ) · · · W z n = e −z 2 Using these identities we can evaluate both sides of (108) to be of the form where R({z a }) are polynomials that are visibly different on the two sides, but in fact agree by using the identity To verify this expression, let e i = (−1) i [x n−i ] a∈A (x − z a ) be the elementary symmetric functions in the z a . Then the contribution with |J| = j to the righthand side is e n−j 1 (e 1 e j − (j + 1)e j+1 ). This means that the right-hand side is a telescoping sum where only the first term remains after summation.
The preceding Lemma 10.5, Proposition 4.3 for the computation of the h-brackets, Lemma 10.4 and (105) now imply immediately our goal: Proof of Theorem 1.4. We start with the case of a single zero. Comparing (110) and (101) we need to show that D(u) = 2∆(u)/u, i.e. in view of (15) we need to show that This equality can be implied by showing that which in turn follows since the derivative A(u) 2g−1 + uA ′ (u) (2g − 1)uA ′ (u) has no (−1)-term. Finally, to deal with the case of multiple zeros, we recall from (59) that a i (µ) = [z m1+1 1 · · · z mn+1 n ]/(2g − 2 + n) n j=1 (m j + 1) H n and hence we need to show that 1 . . . z mn+1 n ]∂ 2 (H n ) (2g − 2 + n) n i=1 (m i + 1) h ℓ →α ℓ after knowing that this is true for the case of n(µ) = 1. This follows immediately from differentiating (56), since after substituting h ℓ → α ℓ this is exactly the sum of the recursions (102) (known to hold for the d(µ)), averaging over all pairs (m r , m s ) of the entries of µ, as in (58).
Given Theorem 1.4 for the area Siegel-Veech computation of the strata on one hand, and the refined Theorem 6.3 for the volume computation of the spin components on the other hand, it is natural to suspect that area Siegel-Veech constants for the spin components can also be computed as ratios of intersection numbers c area (µ) • = −1 4π 2 PΩMg,n(µ) • β i · δ 0 PΩMg,n(µ) • β i · ξ for all 1 ≤ i ≤ n, where • ∈ {even, odd}. Using Assumption 6.1 to deal with the case of the minimal strata, the validity of (111) is equivalent to the validity of c area (µ) odd = −1 16π 2 [z m1+1 1 · · · z mn+1 n ] (2πi) 2g (∂ 2 (H n ) − ∂ ∆ 2 (H n )) (2g − 2 + n)!v(µ) odd h ℓ →α ℓ h ℓ →α ℓ where H n and H n are recursively defined as in Sections 4 and 6, and where is the analog of the differential operator ∂ 2 on the algebra of super-symmetric functions. There is a clear strategy towards this goal: • Show that there are operators like the T p for p ≥ −1 odd as in [CMZ18, Section 12], whose strict brackets compute Siegel-Veech weighted and spinweighted Hurwitz numbers. • Show that the action of T p inside strict brackets can be encoded by differential operators like the ρ ij in [CMZ18, Section 14]. • Show that these operators satisfy a commutation relation as in Lemma 10.5, with ∂ replaced by ∂ ∆ 2 .
Therefore, using the fact that c area (µ) = d(µ)/v(µ) we thus deduce the second part of Theorem 1.5.
11.3. Spin asymptotics. The goal here is to prove that the volumes of the odd and even spin components are asymptotically equal. This is the content of Theorem 1.6 in the introduction that refines the conjecture of Eskin and Zorich ([EZ15, Conjecture 2]). Recall that we defined in Section 6.3 P Z (u) = exp j≥1 1 2 j/2 ζ(−j) 2 u j+1 .
For the reader's convenience we give a table for low genus values: Proof of Proposition 11.1. The first statement is a reformulation of a special case of Corollary 6.11.
The power series P Z (u) is a very rapidly divergent series, just as P B (u) is, since the coefficients ℓ!b ℓ and ζ(−ℓ)/2 = ℓ!b ℓ · √ 2(2 ℓ − 2 −ℓ ) differ by a factor that grows only geometrically. The asymptotic statement thus follows from the method of very rapidly divergent series.