Linear hyperelliptic Hodge integrals

Abstract

We provide a closed form expression for linear Hodge integrals on the hyperelliptic locus. Specifically, we find a succinct combinatorial formula for all intersection numbers on the hyperelliptic locus with one $\lambda$-class, and powers of a $\psi$-class pulled back along the branch map. This is achieved by using Atiyah–Bott localization on a stack of stable maps into the orbifold $\left[ {\mathbb {P}}^1/{\mathbb {Z}}_2\right]$.

1 Introduction

1.1 Context, history, motivation

The moduli space $\overline{{\mathcal {M}}}_{g, n}$ of stable genus g curves with n marked points has been an object of interest since the pioneering work of Deligne and Mumford [1, 2]. In order to gain a deeper understanding of this moduli space, a promising strategy is to probe its intersection theory

A natural way to construct cycles $[\alpha ] \in A^\bullet (\overline{{\mathcal {M}}}_{g, n})$ is to consider loci of curves with certain geometric properties. The locus of curves that we are primarily interested in is the hyperelliptic locus i.e. the locus of curves that admit a degree 2 map to ${\mathbb {P}}^1$. We denote the hyperelliptic locus by $\overline{{\mathcal {H}}}_{g, 2\,g + 2} \subseteq \overline{{\mathcal {M}}}_{g, 2\,g + 2}$, where the marked points are Weierstrass points. There are two natural maps, the branch map and source map, which we informally and briefly discuss here. Each moduli point in $\overline{{\mathcal {H}}}_{g, 2g + 2}$ consists of the data $[C_g, \varphi : C_g \rightarrow (T, p_1, \ldots , p_{2\,g + 2})]$, where $[C_g] \in \overline{{\mathcal {M}}}_{g, 2\,g + 2}$, and $[T, p_1, \ldots p_{2\,g + 2}] \in \overline{{\mathcal {M}}}_{0, 2\,g + 2}$. The marked points on the latter curve are the branch points of the degree 2 map $\varphi$. Given a point in $\overline{{\mathcal {H}}}_{g. 2g + 2}$, we can either remember the source curve or the target curve of $\varphi$. This is summed up in the following diagram:

There is a slight variant on the above situation that we also consider. The points in $\overline{{\mathcal {H}}}_{g, 2g + 2, 2}$ will correspond to branched coverings as before, but the last two marked points are a conjugate pair of points that are interchanged under the hyperelliptic involution. In this situation, br now maps to $\overline{{\mathcal {M}}}_{0, 2g + 3}$, where the last marked point is the image of the conjugate pair.

Let ${\mathbb {E}}_g$ be the Hodge bundle over $\overline{{\mathcal {M}}}_{g, 2g + 2}$, and let ${\mathbb {L}}_i$ be the $i^{th}$ universal cotangent line bundle over $\overline{{\mathcal {M}}}_{0, 2g + 2}$. We define $\lambda _i:= c_i({\mathbb {E}}_g)$, and $\psi _i:= c_1({\mathbb {L}}_i)$. Our goal is to investigate the following two types of intersection numbers:

$$\begin{aligned}&\int _{\overline{{\mathcal {H}}}_{g, 2g + 2}}br^*\left( \psi _1^{2g - 1 - i} \right) \lambda _i \\&\int _{\overline{{\mathcal {H}}}_{g, 2g + 2, 2}}br^*\left( \psi _{2g + 3}^{2g - i} \right) \lambda _i \end{aligned}$$

These intersection numbers are examples of hyperelliptic Hodge integrals. In this paper, we find a closed form expression for these integrals:

Theorem 1

Let $\displaystyle e_i(x_1, \ldots , x_n):= \sum _{1 \le j_1< \ldots < j_i \le n}x_{j_1}x_{i_2}\ldots x_{j_j}$ be the $i^{th}$ elementary symmetric function on $x_1, \ldots x_n$. Then

$$\begin{aligned}&\int _{\overline{{\mathcal {H}}}_{g, 2g + 2}}br^*\left( \psi _1^{2g - 1 - i} \right) \lambda _i = \left( \frac{1}{2} \right) ^{i + 1}e_i(1, 3, \ldots , 2g - 1) \\&\int _{\overline{{\mathcal {H}}}_{g, 2g + 2, 2}}br^*\left( \psi _{2g + 3}^{2g - i} \right) \lambda _i = \left( \frac{1}{2} \right) ^{i + 1}e_i(2, 4, \ldots , 2g) \end{aligned}$$

There has been much progress made in the computations of hyperelliptic Hodge integrals. The earliest result goes back to Faber and Pandharipande [3], in which they prove that

$$\begin{aligned} \int _{\overline{{\mathcal {H}}}_{g, 2g + 2}}\lambda _{g - 1}\lambda _g = \frac{(2^{2g} - 1)|B_{2g}|}{2g} \end{aligned}$$

where $B_{2g}$ is the $2g^{th}$ Bernoulli number. In [4], Cavalieri generalized Faber and Pandharipande’s result by computing the generating functions of all integrals of the form

$$\begin{aligned} \int _{\overline{{\mathcal {H}}}_{g, 2g + 2}}br^* \left( \psi _1^{i - 1}\right) \lambda _{g - i}\lambda _{g} \end{aligned}$$

In [5], Wise showed that

$$\begin{aligned} \int _{\overline{{\mathcal {H}}}_{g, 2g + 2}}\frac{(1 - \lambda _1 + \cdots + (-1)^g\lambda _g)^2}{1 - br^*\left( \frac{\psi _1}{2}\right) } = \left( \frac{-1}{4} \right) ^g \end{aligned}$$

In [6], Johnson et al found an algorithm to compute linear Hodge integrals over spaces of cyclic covers, in terms of Hurwitz numbers. Furthermore, using the results from [7], one can extract hyperelliptic Hodge integrals as coefficients of a twisted J-function.

Despite the tremendous progress made in the calculation of these intersection numbers, a closed form expression, as stated in the theorem above, does not seem to follow directly from the work of previous authors. As such, the main result of this paper provides a simple, yet highly nontrivial, expression for linear Hodge integrals over the hyperelliptic locus.

1.2 Outline of paper

The main strategy that this paper employs is Atiyah–Bott localization on $\overline{{\mathcal {M}}}_{0, k}(\mathbb {P}^1\times {\mathcal {B}}{\mathbb {Z}}_2, 1)$. We recognize the space $\overline{{\mathcal {H}}}_{g, 2g + 2}$ as the space of genus 0 stable maps into the stacky point ${\mathcal {B}}{\mathbb {Z}}_2$, all of whose marked points have nontrivial isotropy,

$$\begin{aligned} \overline{{\mathcal {H}}}_{g, 2g + 2} \cong \overline{{\mathcal {M}}}_{0, (2g + 2)}({\mathcal {B}}{\mathbb {Z}}_2) \end{aligned}$$

The space $\overline{{\mathcal {H}}}_{g, 2g + 2, 2}$ is isomorphic to the space of genus 0 stable maps into ${\mathcal {B}}{\mathbb {Z}}_2$, but the last marked point has trivial isotropy,

$$\begin{aligned} \overline{{\mathcal {H}}}_{g, 2g + 2, 2} \cong \overline{{\mathcal {M}}}_{0, (2g + 2), 1}({\mathcal {B}}{\mathbb {Z}}_2) \end{aligned}$$

In order to compute linear hyperelliptic Hodge integrals, we compute auxiliary integrals on the space $\overline{{\mathcal {M}}}_{0, k}(\mathbb {P}^1\times {\mathcal {B}}{\mathbb {Z}}_2, 1)$, which vanish for dimension reasons. The integrals that we are interested in appear as ‘vertex terms’ in the localization computation. We find recursions relating these integrals, and check that the purported formula for the integrals satisfy the recursions, thus proving the desired result.

In Chapter 2, we introduce the spaces $\overline{{\mathcal {M}}}_{0, k}(\mathbb {P}^1\times {\mathcal {B}}{\mathbb {Z}}_2, 1)$, and $\overline{{\mathcal {M}}}_{0, k, \ell }({\mathcal {B}}{\mathbb {Z}}_2)$. In Chapter 3, we explain our localization set up, in Chapter 4 we discuss our auxiliary integrals, and in the final Chapter, we put all of the pieces together in order to prove the theorem.

1.3 Future work

To generalize the result in this paper, we can allow arbitrarily many insertions of $\lambda$-classes. The author will investigate this generalization in his PhD thesis.

2 The spaces $\overline{{\mathcal {M}}}_{0, k}(\mathbb {P}^1\times {\mathcal {B}}{\mathbb {Z}}_2, 1)$ and $\overline{{\mathcal {M}}}_{0, k, \ell }({\mathcal {B}}{\mathbb {Z}}_2)$

The intersection numbers that we are interested in come from an auxiliary computation on a larger moduli stack: $\overline{{\mathcal {M}}}_{0, k}({\mathbb {P}}^1 \times {\mathcal {B}}{\mathbb {Z}}_2, 1)$. Intersection numbers on this space are examples of orbifold Gromov-Witten invariants. Foundational material on orbifold Gromov-Witten (GW) theory can be found in [8] and [9]. We will not need the entire machinery of orbifold GW theory. Instead, we highlight enough of the theory in order to begin the relevant computations.

Let $\overline{{\mathcal {M}}}_{0, k}({\mathcal {X}}, d)$ be the moduli stack of genus 0 degree d stable maps with k marked points, into the orbifold ${\mathcal {X}}$. The inertia stack of ${\mathcal {X}}$ (see [10], Section 1]), denoted ${\mathcal {I}}{\mathcal {X}}$, is the fiber product

where $\Delta$ is the diagonal map. The product is taken in the 2-category of stacks. The points in ${\mathcal {I}}{\mathcal {X}}$ can be identified with all pairs (x, g), where $x \in {\mathcal {X}}$ and $g \in \text {Aut}_{{\mathcal {X}}}(x)$. In the case that ${\mathcal {X}} = [V/G]$, where V is a smooth projective variety, and G is a finite abelian group, we have

$$\begin{aligned} {\mathcal {I}}{\mathcal {X}} = \coprod _{g \in G}[{\mathcal {X}}^g/G] \end{aligned}$$

In this paper, ${\mathcal {X}} = [{\mathbb {P}}^1/{\mathbb {Z}}_2] ={\mathbb {P}}^1 \times {\mathcal {B}}{\mathbb {Z}}_2$, where ${\mathbb {Z}}_2$ acts trivially on ${\mathbb {P}}^1$. Let us denote the elements in ${\mathbb {Z}}_2$ as ${\mathbb {Z}}_2 = \{0, 1\}$. Then ${\mathcal {I}}{\mathcal {X}}$ is the disjoint union of two copies of ${\mathbb {P}}^1 \times {\mathcal {B}}{\mathbb {Z}}_2$, where each copy is indexed by elements in ${\mathbb {Z}}_2 = \{0, 1\}$. If we define ${\mathcal {I}}{\mathcal {X}}_0$ and ${\mathcal {I}}{\mathcal {X}}_1$ to be the copy indexed by $0 \in {\mathbb {Z}}_2$ and $1 \in {\mathbb {Z}}_2$, respectively, then the inertia stack is

$$\begin{aligned} {\mathcal {I}}{\mathcal {X}} = \left( {\mathbb {P}}^1 \times {\mathcal {B}}{\mathbb {Z}}_2\right) \amalg \left( {\mathbb {P}}^1 \times {\mathcal {B}}{\mathbb {Z}}_2\right) := {\mathcal {I}}{\mathcal {X}}_0 \amalg {\mathcal {I}}{\mathcal {X}}_1 \end{aligned}$$

The main difference between ordinary GW theory and orbifold GW theory is that the source curves are allowed to be orbicurves, that is, the marked points and the nodes are allowed to have non-trivial orbifold structure. Furthermore, the evaluation maps no longer land in the target space, but instead land in the rigidified inertia stack,

$$\begin{aligned} \text {ev}_i : \overline{{\mathcal {M}}}_{0, k}({\mathcal {X}}, d) \rightarrow \overline{{\mathcal {I}}}{\mathcal {X}} \end{aligned}$$

The definition of the rigidified inertia stack is technical, and we refer the reader to [9], Section 3] for details. However, as explained in [9], Section 6], even though there is not a well defined evaluation map from the stack of stable maps to the inertia stack, because there is an isomorphism between the cohomology groups of $\overline{{\mathcal {I}}}{\mathcal {X}}$ and ${\mathcal {I}}{\mathcal {X}}$, there is a well defined map

$$\begin{aligned} \text {ev}_i^*: H^\bullet ({\mathcal {I}}{\mathcal {X}}) \rightarrow H^\bullet (\overline{{\mathcal {M}}}_{0, k}({\mathcal {X}}, d)) \end{aligned}$$

If ${\mathcal {X}} = [{\mathbb {P}}^1/{\mathbb {Z}}_2]$, since ${\mathcal {I}}{\mathcal {X}}$ only consists of two components, the marked points on the source curve are either ’untwisted’ or ’twisted’, i.e. maps to ${\mathcal {I}}{\mathcal {X}}_0$ or maps to ${\mathcal {I}}{\mathcal {X}}_1$. In the former case, the marked point has trivial isotropy, and in the latter, it has non-trivial isotropy. With this, we have the following definitions:

Definition 1

The substack $\overline{{\mathcal {M}}}_{0, k, \ell }({\mathbb {P}}^1\times {\mathbb {Z}}_2, d) \subset \overline{{\mathcal {M}}}_{0, k + l}({\mathbb {P}}^1\times {\mathbb {Z}}_2, d)$ is defined to be the space of degree d maps of genus 0 curves into $[{\mathbb {P}}^1/{\mathbb {Z}}_2]$, in which the first k marked points are twisted, and the last $\ell$ marked points are untwisted. Similarly, the substack $\overline{{\mathcal {M}}}_{0, k, \ell }({\mathcal {B}}{\mathbb {Z}}_2) \subset \overline{{\mathcal {M}}}_{0, k + \ell }({\mathcal {B}}{\mathbb {Z}}_2)$ is the space of degree 0 maps of genus 0 curves into the stack point ${\mathcal {B}}{\mathbb {Z}}_2= [pt./{\mathbb {Z}}_2]$, in which the first k marked points are twisted, and the last $\ell$ marked points are untwisted.

Remark 1

In the case that $\ell = 0$, we suppress $\ell$ from the notation, and simply indicate the number of twisted points. In this paper, the number of twisted points is always greater than zero, so this suppression is unambiguous.

Let $[{\mathcal {C}} \rightarrow {\mathbb {P}}^1 \times {\mathcal {B}}{\mathbb {Z}}_2] \in \overline{{\mathcal {M}}}_{0, k}({\mathbb {P}}^1 \times {\mathcal {B}}{\mathbb {Z}}_2, 1)$. If we compose the map ${\mathcal {C}} \rightarrow {\mathbb {P}}^1 \times {\mathcal {B}}{\mathbb {Z}}_2$ with projection onto the ${\mathcal {B}}{\mathbb {Z}}_2$ factor, we get the map ${\mathcal {C}} \rightarrow {\mathcal {B}}{\mathbb {Z}}_2$, which is equivalent to the data of a principal ${\mathbb {Z}}_2$-bundle over ${\mathcal {C}}$ branched over the k marked points. By Riemann-Hurwitz, the total space of this bundle is a curve C of genus $g = \frac{k - 2}{2}$.

Definition 2

The Hodge bundle ${\mathbb {E}}_g$ over $\overline{{\mathcal {M}}}_{0, k}({\mathbb {P}}^1 \times {\mathcal {B}}{\mathbb {Z}}_2, 1)$ is the vector bundle whose fiber over the point $[{\mathcal {C}} \rightarrow {\mathbb {P}}^1 \times {\mathcal {B}}{\mathbb {Z}}_2]$ is $\Omega ^1(C)$, where C is the ramified cover of ${\mathcal {C}}$ described above. The Chern classes of this vector bundle are denoted $\lambda _i:= c_i({\mathbb {E}}_g)$.

We now identify the moduli stack of stable maps into ${\mathcal {B}}{\mathbb {Z}}_2$ with the moduli stack of hyperelliptic curves. We will be informal in our description. For a rigorous treatment, see for example [11].

If $[{\mathcal {C}} \rightarrow {\mathcal {B}}{\mathbb {Z}}_2] \in \overline{{\mathcal {M}}}_{0, k}({\mathcal {B}}{\mathbb {Z}}_2)$, then again, the map ${\mathcal {C}} \rightarrow {\mathcal {B}}{\mathbb {Z}}_2$ is equivalent to the data of a degree 2 branched covering of ${\mathcal {C}}$, whose branch locus is the k marked points on the source curve. Similarly, if $[{\mathcal {C}} \rightarrow {\mathcal {B}}{\mathbb {Z}}_2] \in \overline{{\mathcal {M}}}_{0, k, 1}({\mathcal {B}}{\mathbb {Z}}_2)$, the map ${\mathcal {C}} \rightarrow {\mathcal {B}}{\mathbb {Z}}_2$ is equivalent to a degree 2 branched covering of ${\mathcal {C}}$, in which the branch locus is the k twisted points on the source curve, but the last marked point, which is untwisted/has trivial isotropy, is not a branch point of the covering. The preimage of the untwisted point is a pair of conjugate points that are interchanged under the hyperelliptic involution. With this geometric description, we see that

$$\begin{aligned}&\overline{{\mathcal {H}}}_{g, 2g + 2} \cong \overline{{\mathcal {M}}}_{0, (2g + 2)}({\mathcal {B}}{\mathbb {Z}}_2) \\&\overline{{\mathcal {H}}}_{g, 2g + 2, 2} \cong \overline{{\mathcal {M}}}_{0, (2g + 2), 1}({\mathcal {B}}{\mathbb {Z}}_2) \end{aligned}$$

Lastly, we also need to know the dimensions of the spaces we’ve discussed (see [11], Section 1.1.3):

$$\begin{aligned}&\text {dim}\left( \overline{{\mathcal {M}}}_{0, k}(\mathbb {P}^1\times {\mathcal {B}}{\mathbb {Z}}_2, 1)\right) = k \\&\text {dim}\left( \overline{{\mathcal {M}}}_{0, k, \ell }({\mathcal {B}}{\mathbb {Z}}_2)\right) = k - 3 + \ell \end{aligned}$$

3 Atiyah–Bott localization

The computational tool we use to prove our main result is Atiyah–Bott localization on $\overline{{\mathcal {M}}}_{0, k}(\mathbb {P}^1\times {\mathcal {B}}{\mathbb {Z}}_2, 1)$. Whenever a space has a torus action, a classical result of Atiyah and Bott says that computing intersection numbers on this space amounts to ’localizing’ to the torus-fixed locus. More precisely:

Theorem 2

([12], Section 3) Let ${\mathcal {X}}$ be a projective variety with a ${\mathbb {C}}^*$-action. Let $\Gamma _1, \ldots , \Gamma _n$ be the irreducible components of the fixed locus of the action. Then for any $\alpha \in H^*({\mathcal {X}}, {\mathbb {Q}})$,

$$\begin{aligned} \int _{\mathcal {X}}\alpha = \sum _{\Gamma _i}\int _{[\Gamma _i]} \frac{\alpha \vert _{\Gamma _i}}{e(N_{\Gamma _i})} \end{aligned}$$

where e(_) denotes the Euler class, and $N_{\Gamma _i}$ is the normal bundle to $\Gamma _i$. $\square$

The ${\mathbb {C}}^*$-action on the coarse space of ${\mathbb {P}}^1 \times {\mathcal {B}}{\mathbb {Z}}_2$ is given by

$$\begin{aligned} \lambda \cdot [x_0 : x_1] = [x_0 : \lambda x_1] \end{aligned}$$

and this action induces a ${\mathbb {C}}^*$-action on $\overline{{\mathcal {M}}}_{0, k}(\mathbb {P}^1\times {\mathcal {B}}{\mathbb {Z}}_2, 1)$ by post composition.

Throughout, we let t be the equivariant parameter of the ${\mathbb {C}}^*$-equivariant cohomology ring of $\overline{{\mathcal {M}}}_{0, kt}({\mathbb {P}}^1 \times {\mathcal {B}}{\mathbb {Z}}_2, 1)$ i.e. $H_{{\mathbb {C}}^*}^\bullet (pt.) = {\mathbb {C}}[t]$. The variable t has a precise meaning. The classifying space of ${\mathbb {C}}^*$ is ${\mathbb {P}}^\infty$. This space has a tautological line bundle ${\mathcal {O}}_{{\mathbb {P}}^\infty }(-1)$, and t is the first Chern class of the dual to this line bundle.

By Theorem 2, in order to compute intersection numbers on $\overline{{\mathcal {M}}}_{0, k}(\mathbb {P}^1\times {\mathcal {B}}{\mathbb {Z}}_2, 1)$, we need two ingredients: the components of the fixed point locus, and the Euler class to the normal bundle of each component.

The components of the fixed point locus are described as follows. If $[f: {\mathcal {C}} \rightarrow {\mathbb {P}}^1\times {\mathcal {B}}{\mathbb {Z}}_2] \in \overline{{\mathcal {M}}}_{0, k}(\mathbb {P}^1\times {\mathcal {B}}{\mathbb {Z}}_2, 1)$ is a fixed point of the ${\mathbb {C}}^*$-action, the map f must send the marked points on the source curve to the ${\mathbb {C}}^*$-fixed points on ${\mathbb {P}}^1$, namely, $0:= [1: 0]$ and $\infty := [0: 1]$. Since f is a map of degree 1, the only way a marked point can be mapped to either 0 or $\infty$ is if $f^{-1}(0)$ or $f^{-1}(\infty )$ is a marked point, or, the marked points lie on a contracted component over 0 or $\infty$. In the case that a component is contracted to 0 or $\infty$, in order for f to still be a degree 1 map, this contracted component is attached, via a node, to a ${\mathbb {P}}^1$ that maps with degree 1 to the target. Because of this description, we can index/enumerate the components of the ${\mathbb {C}}^*$-fixed locus using simple graphs, which are called localization graphs ([13], Section 4).

Definition 3

A localization graph $\Gamma$ for $\overline{{\mathcal {M}}}_{0, k}({\mathbb {P}}^1 \times {\mathcal {B}}{\mathbb {Z}}_2, 1)$ is a decorated graph with the following properties:

• 1. $\Gamma$ has two vertices, denoted $v_0$ and $v_\infty$. They correspond to contracted components over 0 and $\infty$, respectively.

• 2. $\Gamma$ has one edge connecting $v_0$ and $v_\infty$. This edge corresponds to the component mapping with degree 1 to the target, and we refer to this edge as the central component of $\Gamma$

• 3. The vertices $v_0$ and $v_\infty$ can be incident to half edges.We denote the set of half edges incident to $v_0$ and $v_\infty$ as $e_0$ and $e_\infty$, respectively. We require that $|e_0| + |e_\infty | = k$

• 4. The half edges are labelled, i.e. there is a bijective map $\nu : e_0\cup e_\infty \rightarrow \{1, 2, \ldots , k\}$.

Each localization graph $\Gamma$ represents a fixed locus in $\overline{{\mathcal {M}}}_{0, k}(\mathbb {P}^1\times {\mathcal {B}}{\mathbb {Z}}_2, 1)$. The spaces $\overline{{\mathcal {M}}}_\Gamma$ are isomorphic to components of the fixed loci, up to a difference in gerbe structure due to gluing at the nodes. Consequently, integrals over a component of the fixed locus may be computed as an integral over the corresponding $\overline{{\mathcal {M}}}_\Gamma$, after correcting by a factor, known as the gluing factor [14]. The space $\overline{{\mathcal {M}}}_\Gamma$ is determined by the vertices of the localization graph,

$$\begin{aligned} \overline{{\mathcal {M}}}_{\Gamma } := \overline{{\mathcal {M}}}_{v_0} \times \overline{{\mathcal {M}}}_{v_\infty } \end{aligned}$$

The spaces $\overline{{\mathcal {M}}}_{v_0}$ and $\overline{{\mathcal {M}}}_{v_\infty }$ are described as follows. If $|e_0|$ is odd, then $\overline{{\mathcal {M}}}_{v_0}:= \overline{{\mathcal {M}}}_{0, (|e_0| + 1)}({\mathcal {B}}{\mathbb {Z}}_2)$, and if $|e_0|$ is even, $\overline{{\mathcal {M}}}_{v_0}:= \overline{{\mathcal {M}}}_{0, |e_0|, 1}({\mathcal {B}}{\mathbb {Z}}_2)$. Similarly, if $|e_\infty |$ is odd, $\overline{{\mathcal {M}}}_{v_\infty }:= \overline{{\mathcal {M}}}_{0, (|e_\infty | + 1)}({\mathcal {B}}{\mathbb {Z}}_2)$, and if $|e_\infty |$ is even, $\overline{{\mathcal {M}}}_{v_\infty }:= \overline{{\mathcal {M}}}_{0, |e_\infty |, 1}({\mathcal {B}}{\mathbb {Z}}_2)$.

Example 1

It’s best to understand localization graphs by seeing some examples. Consider the space $\overline{{\mathcal {M}}}_{0, 6}({\mathbb {P}}^1 \times {\mathcal {B}}{\mathbb {Z}}_2, 1)$. Below are two examples of localization graphs for this space:

The graph on the left corresponds to a single component of the ${\mathbb {C}}^*$-fixed locus. Each map in this component has the following properties:

• The map has a rational component mapping with degree 1 to the target, corresponding to the central component of $\Gamma$.

• The rational component mapping with degree 1 is nodal to two rational components, one of which contracts to the point 0, and the other contracts to $\infty$.

• The rational component that contracts to 0 contains the first 3 marked points, and the rational component that contracts to $\infty$ contains the last three marked points

Notice that the nodes are forced to have non-trivial isotropy. If we take a map in this fixed locus, and restrict the map to either of the contracted components, we would get an element in $\overline{{\mathcal {M}}}_{0, 4}({\mathcal {B}}{\mathbb {Z}}_2)$; the first three marked points have non-trivial isotropy, but by Riemann-Hurwitz, the last marked point must also be twisted. Furthermore due to the subtleties that arise from working with stacks, when we integrate against these fixed loci, we must also take into account a gluing factor. The precise formulation/derivation of gluing factors can be found in [14], but in the context of this paper, we can state the gluing factors explicitly in terms of the localization graph: each node contributes a factor of 2 and the central component mapping with degree 1 contributes a factor of $\frac{1}{2}$. By this reasoning, if $\alpha \in A^*\left( \overline{{\mathcal {M}}}_{0, 6}({\mathbb {P}}^1 \times {\mathcal {B}}{\mathbb {Z}}_2, 1\right)$, we have

$$\begin{aligned} \int _{\overline{{\mathcal {M}}}_\Gamma }\frac{\alpha \vert _{\overline{{\mathcal {M}}}_\Gamma }}{e(N_{\overline{{\mathcal {M}}}_\Gamma })} = 2 \cdot \int _{\overline{{\mathcal {M}}}_{0, 4}({\mathcal {B}}{\mathbb {Z}}_2) \times \overline{{\mathcal {M}}}_{0, 4}({\mathcal {B}}{\mathbb {Z}}_2)}\frac{\alpha \vert _{\overline{{\mathcal {M}}}_\Gamma }}{e(N_{\overline{{\mathcal {M}}}_\Gamma })} \end{aligned}$$

Similarly, for the localization graph on the right, we have

$$\begin{aligned} \int _{\overline{{\mathcal {M}}}_\Gamma }\frac{\alpha \vert _{\overline{{\mathcal {M}}}_\Gamma }}{e(N_{\overline{{\mathcal {M}}}_\Gamma })} = 2 \cdot \int _{\overline{{\mathcal {M}}}_{0, 2, 1}({\mathcal {B}}{\mathbb {Z}}_2) \times \overline{{\mathcal {M}}}_{0, 4, 1}({\mathcal {B}}{\mathbb {Z}}_2)} \frac{\alpha \vert _{\overline{{\mathcal {M}}}_\Gamma }}{e(N_{\overline{{\mathcal {M}}}_\Gamma )}} \end{aligned}$$

$\square$

In general, localization graphs will correspond to products of smaller moduli spaces, which is precisely the reason why we obtain recursions of intersection numbers over these spaces.

Now let us describe how one computes $\frac{1}{e(N_{\overline{{\mathcal {M}}}_\Gamma })}$. For a derivation from first principles, see [15]. Let $[f: {\mathcal {C}} \rightarrow {\mathbb {P}}^1 \times {\mathcal {B}}{\mathbb {Z}}_2] \in \overline{{\mathcal {M}}}_{0, k}(\mathbb {P}^1\times {\mathcal {B}}{\mathbb {Z}}_2, 1)$. Then the inverse of the Euler class to the normal bundle to this point is given by

$$\begin{aligned} \frac{e(H^0({\mathcal {C}}, T{\mathcal {C}}))e(H^1({\mathcal {C}}, f^*(T{\mathcal {X}})))}{e(H^0({\mathcal {C}}, f^*(T{\mathcal {X}}))) e(H^1({\mathcal {C}}, T{\mathcal {C}}))} \end{aligned}$$

(1)

where $T{\mathcal {X}}$ and $T{\mathcal {C}}$ are the tangent bundles of ${\mathbb {P}}^1 \times {\mathcal {B}}{\mathbb {Z}}_2$ and ${\mathcal {C}}$, respectively. If $\Gamma$ is a localization graph of $\overline{{\mathcal {M}}}_{0, k}(\mathbb {P}^1\times {\mathcal {B}}{\mathbb {Z}}_2, 1)$, in order to compute $\frac{1}{e(N_{\overline{{\mathcal {M}}}_\Gamma })}$, we take all of the terms appearing in 1 and restrict to $\overline{{\mathcal {M}}}_\Gamma$.

Before we state what each of the terms are in 1, let us explain a product notation we use for elements in $H_{{\mathbb {C}}^*}^*(\overline{{\mathcal {M}}}_{\Gamma })$. Since $\overline{{\mathcal {M}}}_{\Gamma } = \overline{{\mathcal {M}}}_{v_0} \times \overline{{\mathcal {M}}}_{v_{\infty }}$, we have two canonical projection maps $\pi _1: \overline{{\mathcal {M}}}_{\Gamma } \rightarrow \overline{{\mathcal {M}}}_{v_0}$ and $\pi _2: \overline{{\mathcal {M}}}_{\Gamma } \rightarrow \overline{{\mathcal {M}}}_{v_{\infty }}$. Whenever we denote an element $(a) \times (b) \in H_{{\mathbb {C}}^*}^*(\overline{{\mathcal {M}}}_{\Gamma })$, what we mean is $\pi ^*(a) \cdot \pi ^*(b)$.

The formula for $\frac{1}{e(N_{\overline{{\mathcal {M}}}_\Gamma })}$ is summed up in the following Lemma:

Lemma 1

Let $\Gamma$ be a localization graph of $\overline{{\mathcal {M}}}_{0, (2\,g + 2)}([{\mathbb {P}}^1/{\mathbb {Z}}_2], 1)$. We define the following subsets of vertices of $\Gamma$:

• $\text {Val}_0(1):= \{\text {vertices of valence 1 over 0}\}$

• $\text {Val}_\infty (1):= \{\text {vertices of valence 1 over} \ \infty \}$

• $\text {Val}_0(3):= \{\text {vertices of valence 3 over 0}\}$

• $\text {Val}_\infty (3):= \{\text {vertices of valence 3 over} \ \infty \}$

• $\text {Val}_0(\ge 3):= \{\text {vertices of valence at least 3 over 0}\}$

• $\text {Val}_\infty (\ge 3):= \{\text {vertices of valence at least 3 over}\ \infty \}$

With the above definitions, we have

• 1. $e(H^0({\mathcal {C}}, T{\mathcal {C}})) = t^{|\text {Val}_0(1))|}(-t)^{|\text {Val}_\infty (1)|}$

• 2. $e(H^1({\mathcal {C}}, f^*(T[{\mathbb {P}}^1/{\mathbb {Z}}_2]))) = 1$

• 3. $e(H^0({\mathcal {C}}, f^*(T[{\mathbb {P}}^1/{\mathbb {Z}}_2]))) = -t^2$

• 4. $e(H^1({\mathcal {C}}, T{\mathcal {C}})) = \left( (t - \psi _0)^{|\text {Val}_0(\ge 3)|}\right) \times \left( (-t - \psi _{\infty })^{|\text {Val}_\infty (\ge 3)|}\right)$

Incorporating the gluing factor of $\Gamma$, which is $(2)^{\left| \text {Val}_0(3)\right| + \left| \text {Val}_\infty (3) \right| }\left( \frac{1}{2}\right)$, we obtain

$$\begin{aligned} \frac{1}{e(N_{\overline{{\mathcal {M}}}_{\Gamma }})}&= 2^{|\textrm{Val}_0(3)| + |\text {Val}_\infty (3)| - 1}t^{|\textrm{Val}_0(1))|} (-t)^{|\textrm{Val}_\infty (1)|}\\&\quad \left( \frac{-1}{t^2}\right) \left( \frac{1}{t - \psi _0}\right) \times \left( \frac{1}{-t - \psi _{\infty }}\right) \end{aligned}$$

$\square$

Again, we refer the reader to [15] for complete details.

In the context of our computations, the Atiyah–Bott localization theorem can be stated as follows:

Corollary 1

Let $\alpha \in A^\bullet (\overline{{\mathcal {M}}}_{0, k}(\mathbb {P}^1\times {\mathcal {B}}{\mathbb {Z}}_2, 1))$. Then

$$\begin{aligned} \int _{\overline{{\mathcal {M}}}_{0, k}(\mathbb {P}^1\times {\mathcal {B}}{\mathbb {Z}}_2, 1)}\alpha = \sum _{\Gamma }\int _{\overline{{\mathcal {M}}}_\Gamma } \frac{\alpha \vert _{\overline{{\mathcal {M}}}_\Gamma }}{e(N_{\overline{{\mathcal {M}}}_\Gamma })} \end{aligned}$$

where the sum is over all localization graphs $\Gamma$ of $\overline{{\mathcal {M}}}_{0, k}(\mathbb {P}^1\times {\mathcal {B}}{\mathbb {Z}}_2, 1)$. $\square$

4 Auxiliary integrals

In this section, we set up two auxiliary integrals that are zero for dimension reasons. These integrals give us recursions for integrals over the spaces $\overline{{\mathcal {M}}}_{0, k}({\mathcal {B}}{\mathbb {Z}}_2)$ and $\overline{{\mathcal {M}}}_{0, k, 1}({\mathcal {B}}{\mathbb {Z}}_2)$.

4.1 Setup

In order to ease notation, we make the following definition:

Definition 4

For $k = 2g + 2$, we define

$$\begin{aligned}{} & {} D_{i, k}:= \int _{\overline{{\mathcal {M}}}_{0, k}({\mathcal {B}}{\mathbb {Z}}_2)}\psi _1^{k - 3 + i}\lambda _i = \int _{\overline{{\mathcal {H}}}_{g, 2g + 2}}br^*(\psi _1^{2g - 1 - i})\lambda _i \\{} & {} d_{i, k}:= \int _{\overline{{\mathcal {M}}}_{0, k, 1}({\mathcal {B}}{\mathbb {Z}}_2)}\psi _{k + 1}^{k - 2 + i} \lambda _i = \int _{\overline{{\mathcal {H}}}_{g, 2g + 2, 2}}br^* (\psi _{2g + 3}^{2g - i})\lambda _i \end{aligned}$$

From previous results [4, 16], we have the following initial values:

$$\begin{aligned} D_{1, 4} = \frac{1}{4},\qquad D_{0, k} = d_{0, k} = \frac{1}{2}\qquad \text {where} \ k \ge 2 \ \text {is} \ \text {even} \end{aligned}$$

In order to begin the computations, we need the following two standard facts. If $\Gamma$ is a localization graph, then the classes $\text {ev}_i^*(0), \text {ev}_i^*(\infty )$ restrict to $\overline{{\mathcal {M}}}_\Gamma$ as

$$\begin{aligned} \text {ev}_i^*(0)\vert _{\overline{{\mathcal {M}}}_\Gamma } = t, \ \text {ev}_i^* (\infty )\vert _{\overline{{\mathcal {M}}}_\Gamma } = -t \end{aligned}$$

If $\overline{{\mathcal {M}}}_\Gamma = \overline{{\mathcal {M}}}_{0, k_1, l_1}({\mathcal {B}}{\mathbb {Z}}_2) \times \overline{{\mathcal {M}}}_{0, k_2, l_2}({\mathcal {B}}{\mathbb {Z}}_2)$, we have

$$\begin{aligned} \lambda _i\vert _{\overline{{\mathcal {M}}}_\Gamma } = \sum _{i_1 + i_2 = i} \left( \lambda _{i_1}\vert _{\overline{{\mathcal {M}}}_{0, k_1, l_1}({\mathcal {B}}{\mathbb {Z}}_2)} \times \lambda _{i_2}\vert _{\overline{{\mathcal {M}}}_{0, k_2, l_2}({\mathcal {B}}{\mathbb {Z}}_2)} \right) \end{aligned}$$

4.2 First auxiliary integral

Consider the following integral

$$\begin{aligned} I_A^{i, k}:= \displaystyle \int _{\overline{{\mathcal {M}}}_{0, k}(\mathbb {P}^1\times {\mathcal {B}}{\mathbb {Z}}_2, 1)}\text {ev}_1^*(0) \text {ev}_2^*(0)\text {ev}_3^*(\infty )\lambda _i \end{aligned}$$

Since $\text {dim}\left( \overline{{\mathcal {M}}}_{0, k}({\mathbb {P}}^1 \times {\mathcal {B}}{\mathbb {Z}}_2, 1)\right) = k$, as long as $k \ge 6$, this integral vanishes. Indeed, the degree of the classes appearing in the integrand of $I_{A}^{i, k}$ add up to $3 + i$. Since $0 \le i \le g$, where g is the genus of the hyperelliptic cover, and since $k = 2g + 2$, the integral vanishes as long as $3 + i< k \iff 3 + i < 2\,g + 2 \iff g> \frac{3 + i }{2} > 1 \iff g \ge 2 \iff k \ge 6$. We already know the initial conditions

$$\begin{aligned} \int _{\overline{{\mathcal {M}}}_{0, 4}({\mathcal {B}}{\mathbb {Z}}_2)}\lambda _1 = \frac{1}{4}, \int _{\overline{{\mathcal {M}}}_{0, k} ({\mathcal {B}}{\mathbb {Z}}_2)}\psi _i^{ k - 3} = \int _{\overline{{\mathcal {M}}}_{0, k, 1}({\mathcal {B}}{\mathbb {Z}}_2)} \psi _{k + 1}^{k - 2} = \frac{1}{2} \end{aligned}$$

and therefore, we do not need the auxiliary integral for $k = 4$.

Remark 2

The initial conditions above are valid for all $k \ge 2$. In particular, we have the initial condition $\int _{\overline{{\mathcal {M}}}_{0, 2}({\mathcal {B}}{\mathbb {Z}}_2)}1 = \frac{1}{2}$. The moduli space $\overline{{\mathcal {M}}}_{0, 2}({\mathcal {B}}{\mathbb {Z}}_2)$ has dimension $-1$. As such, this integral doesn’t have any geometric meaning. However, our recursions will require this integral as an initial condition, and it turns out that declaring the value of this integral to be $\frac{1}{2}$ is a convention that makes the recursions consistent.

Since the product $\text {ev}_1^*(0) \text {ev}_2^*(0)\text {ev}_3^*(\infty )$ is in the integrand of $I_A^{i, k}$, when we use localization to compute $I_A^{i, k}$, all of the localization graphs that appear in the computation must have the first two marked points over 0, and the third marked point over $\infty$. What remains is a systematic way to enumerate all of the localization graphs that have this property.

Definition 5

Define $A^k_j$ to be the set of all localization of graphs of $\overline{{\mathcal {M}}}_{0, k}({\mathbb {P}}^1 \times {\mathcal {B}}{\mathbb {Z}}_2, 1)$ with the following properties: The first two marked points lie over 0, the third marked point is over $\infty$, there are $k - 3 - j$ marked points over 0, and the remaining j points are over $\infty$ (Fig. 1).

Fig. 1

An element in $A_j^k$

When we localize, by Corollary 1, we have

$$\begin{aligned} I_A^{i, k} = \sum _{j = 0}^{k - 3}\sum _{\Gamma \in A_j^k}(-t^3) \int _{\overline{{\mathcal {M}}}_\Gamma }\frac{\lambda _i\vert _{\overline{{\mathcal {M}}}_\Gamma }}{e\left( N_{\overline{{\mathcal {M}}}_\Gamma }\right) } \end{aligned}$$

We evaluate this integral by computing the contributions coming from each set $A_j^k$. To organize the workflow, we define

$$\begin{aligned} \text {Cont}(A_j^k) := \sum _{\Gamma \in A_j^k}(-t^3) \int _{\overline{{\mathcal {M}}}_\Gamma } \frac{\lambda _i\vert _{\overline{{\mathcal {M}}}_\Gamma }}{e\left( N_{\overline{{\mathcal {M}}}_\Gamma }\right) } \end{aligned}$$

Let us first consider $A_0^k$ and $A_1^k$. The set $A_0^k$ only contains one localization graph $\Gamma$, and the corresponding moduli space is $\overline{{\mathcal {M}}}_\Gamma = \overline{{\mathcal {M}}}_{0, k}({\mathcal {B}}{\mathbb {Z}}_2)$. So the contribution to $I_A^{i, k}$ coming from $A_0^k$ is

$$\begin{aligned} \text {Cont}(A_0^k)&= (-t^3)\int _{\overline{{\mathcal {M}}}_{0, k}({\mathcal {B}}{\mathbb {Z}}_2)} \frac{-\lambda _i}{t^2(t - \psi _k)} \nonumber \\&= \int _{\overline{{\mathcal {M}}}_{0, k}({\mathcal {B}}{\mathbb {Z}}_2)}\frac{\psi _k^{k - 3 - i}}{t^{k - 3 - i}} \lambda _i \nonumber \\&= \frac{1}{t^{k - 3 - i}}D_{i, k} \end{aligned}$$

(2)

Now let us consider $A_1^k$. There are ${k - 3 \atopwithdelims ()1} = k - 3$ localization graphs contained in $A_1^k$, coming from the choice of labeling of the marked point over $\infty$. The moduli space corresponding to each of these localization graphs is $\overline{{\mathcal {M}}}_\Gamma = \overline{{\mathcal {M}}}_{0, (k - 2), 1}({\mathcal {B}}{\mathbb {Z}}_2) \times \overline{{\mathcal {M}}}_{0, 2, 1}({\mathcal {B}}{\mathbb {Z}}_2)$. The second factor is a moduli space of dimension 0, and when we integrate the fundamental class against it, we get $\frac{1}{2}$ ([16], Proposition 3.4). So the contribution of $A_1^k$ to $I_A^{i, k}$ is

$$\begin{aligned} \text {Cont}(A_1^k)&= {k - 3 \atopwithdelims ()1}(-t^3)(2) \int _{\overline{{\mathcal {M}}}_{0, (k - 2), 1}({\mathcal {B}}{\mathbb {Z}}_2)}\nonumber \\&\quad \frac{-\lambda _i}{t^2(t - \psi _{k - 1})} \int _{\overline{{\mathcal {M}}}_{0, 2, 1}({\mathcal {B}}{\mathbb {Z}}_2)}\frac{1}{-t - \psi _3} \nonumber \\&= {k - 3 \atopwithdelims ()1}\left( \frac{-1}{t} \right) \int _{\overline{{\mathcal {M}}}_{0, (k - 2), 1}({\mathcal {B}}{\mathbb {Z}}_2)} \frac{\psi _{k - 1}^{k - 4 - i}}{t^{k - 4 - i}}\lambda _i \nonumber \\&= {k - 3 \atopwithdelims ()1}\frac{-1}{t^{k - 3 - i}}d_{i, k - 2} \end{aligned}$$

(3)

Now we need to consider $A_j^k$ for $j \ge 2$. Notice that the corresponding moduli spaces of each localization graph will vary, depending on whether j is even or odd, so we consider both cases separately. In the case that j is even, each localization graph of $A_j^k$ corresponds to the moduli space

$$\begin{aligned} \overline{{\mathcal {M}}}_\Gamma = \overline{{\mathcal {M}}}_{0, (k - j)}({\mathcal {B}}{\mathbb {Z}}_2) \times \overline{{\mathcal {M}}}_{0, (j + 2)}({\mathcal {B}}{\mathbb {Z}}_2) \end{aligned}$$

There are $k - 3 \atopwithdelims ()j$ localization graphs in $A_j^k$, so the contribution of $A_j^k$ to $I_A^{i, k}$ is

$$\begin{aligned} \text {Cont}(A_j^k)&= {k - 3 \atopwithdelims ()j}(-t^3)(2) \sum _{i_i + i_2 =i}\int _{\overline{{\mathcal {M}}}_{0, (k - j)}({\mathcal {B}}{\mathbb {Z}}_2)}\frac{-\lambda _{i_1}}{t^2(t - \psi _{k - j})}\nonumber \\&\quad \int _{\overline{{\mathcal {M}}}_{0, (j + 2)}({\mathcal {B}}{\mathbb {Z}}_2)} \frac{\lambda _{i_2}}{(-t - \psi _{j + 2})} \nonumber \\&= {k - 3 \atopwithdelims ()j}\left( \frac{-1}{t}\right) (2) \sum _{i_1 + i_2 = i}\int _{\overline{{\mathcal {M}}}_{0, (k - j)}({\mathcal {B}}{\mathbb {Z}}_2)} \frac{\psi _{k - j}^{k - 3 - j - i_1}}{t^{k - 3 - j - i_1}} \lambda _{i_1} \nonumber \\&\quad \times \int _{\overline{{\mathcal {M}}}_{0, (j + 2)}({\mathcal {B}}{\mathbb {Z}}_2)}(-1)^{j - 1 - i_2} \frac{\psi _{j + 2}^{j - 1 - i_2}}{t^{j - 1 - i_2}}\lambda _{i_2}\nonumber \\&= {k - 3 \atopwithdelims ()j}\frac{-1}{t^{k - 3 - i}}(2)\sum _{i_1 + i_2 = i} (-1)^{j - 1 - i_2}D_{i_1, k - j}D_{i_2, j + 2} \nonumber \\&= {k - 3 \atopwithdelims ()j}\frac{2}{t^{k - 3 - i}} \sum _{\ell = 0}^i(-1)^{\ell }D_{i - \ell , k - j}D_{\ell , j + 2} \end{aligned}$$

(4)

In the case that j is odd, the localization graphs contained in $A_j^k$ correspond to

$$\begin{aligned} \overline{{\mathcal {M}}}_\Gamma = \overline{{\mathcal {M}}}_{0, (k - 1 - j), 1}({\mathcal {B}}{\mathbb {Z}}_2) \times \overline{{\mathcal {M}}}_{0, (j + 1), 1}({\mathcal {B}}{\mathbb {Z}}_2) \end{aligned}$$

and the contribution of $A_j^k$ is

$$\begin{aligned} \text {Cont}(A_j^k)&= {k - 3 \atopwithdelims ()j}(-t^3)(2) \sum _{i_i + i_2 = i} \int _{\overline{{\mathcal {M}}}_{0, (k - 1 - j), 1}({\mathcal {B}}{\mathbb {Z}}_2)}\frac{-\lambda _{i_1}}{t^2(t - \psi _{k - j})}\nonumber \\&\quad \int _{\overline{{\mathcal {M}}}_{0, (j + 1), 1}({\mathcal {B}}{\mathbb {Z}}_2)} \frac{\lambda _{i_2}}{(-t - \psi _{j + 2})} \nonumber \\&= {k - 3 \atopwithdelims ()j}\left( \frac{-1}{t}\right) (2) \sum _{i_1 + i_2 = i}\int _{\overline{{\mathcal {M}}}_{0, (k - 1 - j), 1}({\mathcal {B}}{\mathbb {Z}}_2)} \frac{\psi _{k - j}^{k - 3 - j - i_1}}{t^{k - 3 - j - i_1}} \lambda _{i_1} \nonumber \\&\quad \times \int _{\overline{{\mathcal {M}}}_{0, (j + 1), 1}({\mathcal {B}}{\mathbb {Z}}_2)} (-1)^{j - 1 - i_2}\frac{\psi _{j + 2}^{j - 1 - i_2}}{t^{j - 1 - i_2}}\lambda _{i_2} \nonumber \\&= {k - 3 \atopwithdelims ()j}\frac{-1}{t^{k - 3 - i}}(2) \sum _{i_1 + i_2 = i} (-1)^{j - 1 - i_2}d_{i_1, k - 1 - j} d_{i_2, j + 1} \nonumber \\&= {k - 3 \atopwithdelims ()j}\frac{-2}{t^{k - 3 - i}}\sum _{\ell = 0}^i(-1)^\ell d_{i - \ell , k - 1 - j}d_{\ell , j + 1} \end{aligned}$$

(5)

Now, recall that $I_A^{i, k} = 0$ for $k \ge 6$ and $i \ge 0$. After combining all of the contributions from Eqs. 2, 3, 4, and 5, and multiplying through by $t^{k - 3 - i}$, we get

$$\begin{aligned} 0&= D_{i, k} - {k - 3 \atopwithdelims ()1}d_{i, k - 2} + 2 \sum _{\begin{array}{c} j \ \text {even} \\ j \ge 2 \end{array}} {k - 3 \atopwithdelims ()j}\sum _{\ell = 0}^i (-1)^{\ell }D_{i - \ell , k - j}D_{\ell , j + 2} \\&\quad - 2\sum _{\begin{array}{c} j \ \text {odd} \\ j \ge 3 \end{array}} {k - 3 \atopwithdelims ()j} \sum _{\ell = 0}^i(-1)^{\ell } d_{i - \ell , k - 1 - j}d_{\ell , j + 1} \end{aligned}$$

Notice that

$$\begin{aligned}&-{k - 3 \atopwithdelims ()1}d_{i, k - 2} - 2\sum _{\begin{array}{c} j \ \text {odd} \\ j \ge 3 \end{array}} {k - 3 \atopwithdelims ()j} \sum _{\ell = 0}^i(-1)^{\ell }d_{i - \ell , k - 1 - j}d_{\ell , j + 1} \\&\quad = -2\sum _{\begin{array}{c} j \ \text {odd} \\ j \ge 1 \end{array}} {k - 3 \atopwithdelims ()j} \sum _{\ell = 0}^i(-1)^{\ell }d_{i - \ell , k - 1 - j}d_{\ell , j + 1} \end{aligned}$$

Indeed, when $j = 1$, the only term that is nonzero in the sum $\sum _{\ell = 0}^i(-1)^{\ell }d_{i - \ell , k - 1 - j}d_{\ell , j + 1}$ is the term corresponding to $\ell = 0$, in which case one obtains

$$\begin{aligned} -2{k - 3 \atopwithdelims ()1}d_{i, k - 2}d_{0, 2} = -2{k - 3 \atopwithdelims ()1}d_{i, k - 2} \left( \frac{1}{2}\right) = -{k - 3 \atopwithdelims ()1}d_{i, k - 2} \end{aligned}$$

as desired.

We therefore have the following recursion for $D_{i, k}$:

$$\begin{aligned} D_{i, k}&= 2\sum _{\begin{array}{c} j \ge 1 \\ j \ \text {odd} \end{array}} {k - 3 \atopwithdelims ()j}\left( \sum _{\ell = 0}^i (-1)^\ell d_{i - \ell , k - 1 - j}d_{\ell , j + 1} \right) \nonumber \\&\quad - 2\sum _{\begin{array}{c} j \ge 2 \\ j \ \text {even} \end{array}}{k - 3 \atopwithdelims ()j}\left( \sum _{\ell = 0}^i (-1)^\ell D_{i - \ell , k - j}D_{\ell , j + 2} \right) \end{aligned}$$

(6)

4.3 Second auxiliary integral

Now consider the integral

$$\begin{aligned} I_B^{i, k}:= \displaystyle \int _{\overline{{\mathcal {M}}}_{0, k}(\mathbb {P}^1\times {\mathcal {B}}{\mathbb {Z}}_2, 1)}\text {ev}_1^*(0) \text {ev}_2^*(0) \lambda _i \end{aligned}$$

Again, $I_B^{i, k} = 0$ for dimension reasons, so evaluating it will result in a recursion as before. As in the previous auxiliary integral, we enumerate the relevant localization graphs.

Definition 6

$B_j^k$ is the set of all localization graphs of $\overline{{\mathcal {M}}}_{0, k}(\mathbb {P}^1\times {\mathcal {B}}{\mathbb {Z}}_2, 1)$ with the following properties: the first two marked points lie over 0, j marked points lie over $\infty$, and the remaining $k - 2 - j$ marked points lie over 0 (Fig. 2).

Fig. 2

An element in $B_j^k$

By Corollary 1, we see that

$$\begin{aligned} I_B^{i, k} = \sum _{j = 0}^{k - 2}\sum _{\Gamma \in B_j^k}(t^2) \int _{\overline{{\mathcal {M}}}_\Gamma }\frac{\lambda _i\vert _{\overline{{\mathcal {M}}}_\Gamma }}{e(N_{\overline{{\mathcal {M}}}_\Gamma })} \end{aligned}$$

As in the case of the first auxiliary integral, we define

$$\begin{aligned} \text {Cont}(B_j^k) := \sum _{\Gamma \in B_j^k}(t^2) \int _{\overline{{\mathcal {M}}}_\Gamma }\frac{\lambda _i\vert _{\overline{{\mathcal {M}}}_\Gamma }}{e(N_{\overline{{\mathcal {M}}}_\Gamma })} \end{aligned}$$

We begin by considering $B_0^k, B_1^k$, and $B_2^k$. There is one localization graph in $B_0^k$, whose corresponding moduli space is $\overline{{\mathcal {M}}}_\Gamma = \overline{{\mathcal {M}}}_{k, 1}({\mathcal {B}}{\mathbb {Z}}_2)$, so the contribution of $B_0^k$ to $I_B^{i, k}$ is

$$\begin{aligned} \text {Cont}(B_0^k)&= (t^2)\int _{\overline{{\mathcal {M}}}_{0, k, 1}({\mathcal {B}}{\mathbb {Z}}_2)} \frac{-t}{-t^2(t - \psi _{k + 1})}\lambda _i \nonumber \\&= \int _{\overline{{\mathcal {M}}}_{0, k, 1}({\mathcal {B}}{\mathbb {Z}}_2)}\frac{\psi _{k + 1}^{k - 2 - i}}{t^{k - 2 - i}}\lambda _i \nonumber \\&= \frac{1}{t^{k - 2 - i}}d_{i, k} \end{aligned}$$

(7)

Each localization graph of $B_1^k$ has corresponding moduli space $\overline{{\mathcal {M}}}_\Gamma = \overline{{\mathcal {M}}}_{0, k}({\mathcal {B}}{\mathbb {Z}}_2)$, and we get

$$\begin{aligned} \text {Cont}(B_1^k)&= {k - 2 \atopwithdelims ()1}(t^2) \int _{\overline{{\mathcal {M}}}_{0, t} ({\mathcal {B}}{\mathbb {Z}}_2)}\frac{-1}{t^2(t - \psi _{k})}\lambda _i \nonumber \\&= {k - 2 \atopwithdelims ()1}\left( \frac{-1}{t}\right) \int _{\overline{{\mathcal {M}}}_{0, k} ({\mathcal {B}}{\mathbb {Z}}_2)}\frac{\psi _k^{k - 3 - i}}{t^{k - 3 - i}}\lambda _i \nonumber \\&= {k - 2 \atopwithdelims ()1}\left( \frac{-1}{t^{k - 2 - i}} \right) D_{i, k} \end{aligned}$$

(8)

For $B_2^k$, each localization graph has corresponding moduli space $\overline{{\mathcal {M}}}_\Gamma = \overline{{\mathcal {M}}}_{0, (k - 2), 1}({\mathcal {B}}{\mathbb {Z}}_2) \times \overline{{\mathcal {M}}}_{0, 2, 1}({\mathcal {B}}{\mathbb {Z}}_2)$, and we get

$$\begin{aligned} \text {Cont}(B_2^k)&= {k - 2 \atopwithdelims ()2}(t^2) \int _{\overline{{\mathcal {M}}}_{0, (k - 2), 1}({\mathcal {B}}{\mathbb {Z}}_2)}\frac{-\lambda _i}{t^2 (t - \psi _{k - 1})}\int _{\overline{{\mathcal {M}}}_{0, 2, 1}({\mathcal {B}}{\mathbb {Z}}_2)} \frac{1}{-t - \psi _3} \nonumber \\&= {k - 2 \atopwithdelims ()2}\frac{1}{t^2}\int _{\overline{{\mathcal {M}}}_{0, (k - 2), 1}({\mathcal {B}}{\mathbb {Z}}_2)} \frac{\psi _{k - 1}^{k - 4 - i}}{t^{k - 4 - i}} \nonumber \\&= {k - 2 \atopwithdelims ()2}\frac{1}{t^{k - 2 - i}}d_{i, k - 2} \end{aligned}$$

(9)

Now lets consider $B_j^k$ for $j \ge 3$. As in the previous auxiliary integral, we consider the two subcases where j is either even or odd. If j is odd, each localization has corresponding moduli space $\overline{{\mathcal {M}}}_\Gamma = \overline{{\mathcal {M}}}_{0, (k - j + 1)}({\mathcal {B}}{\mathbb {Z}}_2) \times \overline{{\mathcal {M}}}_{0, (j + 1)}({\mathcal {B}}{\mathbb {Z}}_2)$, and we get

$$\begin{aligned} \text {Cont}(B_j^k)&= {k - 2 \atopwithdelims ()j}(t^2)(2) \sum _{i_1 + i_2 = i}\int _{\overline{{\mathcal {M}}}_{0, (k - j + 1)}({\mathcal {B}}{\mathbb {Z}}_2)}\frac{-\lambda _{i_1}}{t^2(t - \psi _{k - j + 1})}\nonumber \\&\quad \int _{\overline{{\mathcal {M}}}_{0, (j + 1)}({\mathcal {B}}{\mathbb {Z}}_2)} \frac{\lambda _{i_2}}{-t - \psi _{j + 1}} \nonumber \\&= {k - 2 \atopwithdelims ()j}\frac{1}{t^2}(2)\sum _{i_1 + i_2 = i}\int _{\overline{{\mathcal {M}}}_{0, (k - j + 1)}({\mathcal {B}}{\mathbb {Z}}_2)}\frac{\psi _{k - j + 1}^{k - j - 2 - i_1}}{t^{k - j - 2 - i_1}}\lambda _{i_1} \nonumber \\&\quad \times \int _{\overline{{\mathcal {M}}}_{0, (j + 1)}({\mathcal {B}}{\mathbb {Z}}_2)}(-1)^{j - 2 - i_2} \frac{\psi _{j + 1}^{j - 2 - i_2}}{t^{j - 2 - i_2}}\lambda _{i_2} \nonumber \\&= {k - 2 \atopwithdelims ()j}\frac{-1}{t^{k - 2 - i}}(2)\sum _{i_1 + i_2 = i} (-1)^{i_2}D_{i_1, k - j + 1}D_{i_2, j + 1} \nonumber \\&= {k - 2 \atopwithdelims ()j}\frac{-2}{t^{k - 2 - i}} \sum _{\ell = 0}^i(-1)^\ell D_{i - \ell , k - j + 1}D_{\ell , j + 1} \end{aligned}$$

(10)

If j is even, each localization graph has corresponding moduli space $\overline{{\mathcal {M}}}_\Gamma = \overline{{\mathcal {M}}}_{0, (k - j), 1}({\mathcal {B}}{\mathbb {Z}}_2) \times \overline{{\mathcal {M}}}_{0, j, 1}({\mathcal {B}}{\mathbb {Z}}_2)$, and we get

$$\begin{aligned} \text {Cont}(B_j^k)&= {k - 2 \atopwithdelims ()j}(t^2)(2) \sum _{i_1 + i_2 = i}\int _{\overline{{\mathcal {M}}}_{0, (k - j), 1}({\mathcal {B}}{\mathbb {Z}}_2)}\nonumber \\&\quad \frac{-\lambda _{i_1}}{t^2(t - \psi _{k - j + 1})}\int _{\overline{{\mathcal {M}}}_{0, j, 1}({\mathcal {B}}{\mathbb {Z}}_2)}\frac{\lambda _{i_2}}{-t - \psi _{j + 1}} \nonumber \\&= {k - 2 \atopwithdelims ()j}\frac{1}{t^2}(2)\sum _{i_1 + i_2 = i}\int _{\overline{{\mathcal {M}}}_{0, (k - j), 1}({\mathcal {B}}{\mathbb {Z}}_2)}\frac{\psi _{k - j + 1}^{k - j - 2 - i_1}}{t^{k - j - 2 - i_1}}\lambda _{i_1} \nonumber \\&\quad \times \int _{\overline{{\mathcal {M}}}_{0, j, 1}({\mathcal {B}}{\mathbb {Z}}_2)}(-1)^{j - 2 - i_2}\frac{\psi _{j + 1}^{j - 2 - i_2}}{t^{j - 2 - i_2}} \lambda _{i_2}\nonumber \\&= {k - 2 \atopwithdelims ()j}\frac{1}{t^{k - 2 - i}}(2)\sum _{i_1 + i_2 = i}(-1)^{i_2}d_{i_1, k - j}d_{i_2, j} \nonumber \\&= {k - 2 \atopwithdelims ()j}\frac{2}{t^{k - 2 - i}}\sum _{\ell = 0}^i(-1)^\ell d_{i - \ell , k - j}d_{\ell , j} \end{aligned}$$

(11)

Since $I_B^{i, k} = 0$, for $k \ge 4$ and $i \ge 0$, when we combine all of the contributions in Eqs. 7, 8, 9, 10, and 11, and multiply through by $t^{k - 2 - i}$, we get

$$\begin{aligned} 0&= d_{i, k} - {k - 2 \atopwithdelims ()1} D_{i, k} + {k - 2 \atopwithdelims ()2}d_{i, k - 2}\nonumber \\&\quad - 2\sum _{\begin{array}{c} j \ \text {odd} \\ j \ge 3 \end{array}} {k - 2 \atopwithdelims ()j}\sum _{\ell = 0}^i(-1)^\ell D_{i - \ell , k - j + 1} D_{\ell , j + 1} \\&\quad + 2 \sum _{\begin{array}{c} j \ \text {even} \\ j \ge 4 \end{array}} {k - 2 \atopwithdelims ()j}\sum _{\ell = 0}^i(-1)^\ell d_{i - \ell , k - j}d_{\ell , j} \end{aligned}$$

Now, we have

$$\begin{aligned}&- {k - 2 \atopwithdelims ()1}D_{i, k} - 2\sum _{\begin{array}{c} j \ \text {odd} \\ j \ge 3 \end{array}} {k - 2 \atopwithdelims ()j}\sum _{\ell = 0}^i(-1)^\ell D_{i - \ell , k - j + 1}D_{\ell , j + 1} \\&\quad = - 2\sum _{\begin{array}{c} j \ \text {odd} \\ j \ge 1 \end{array}} {k - 2 \atopwithdelims ()j}\sum _{\ell = 0}^i(-1)^\ell D_{i - \ell , k - j + 1}D_{\ell , j + 1} \end{aligned}$$

Indeed, when $j = 1$, the only nonzero term in the sum $\sum _{\ell = 0}^iD_{i - \ell , k - j + 1}D_{\ell , j + 1}$ is achieved when $\ell = 0$ (see Remark 2), in which case one obtains

$$\begin{aligned} -2{k - 2 \atopwithdelims ()1}D_{i, k}D_{0, 2} = -2{k - 2 \atopwithdelims ()1}D_{i, k} \left( \frac{1}{2}\right) = -{k - 2 \atopwithdelims ()1}D_{i, k} \end{aligned}$$

as desired. A similar argument also shows that

$$\begin{aligned}&{k - 2 \atopwithdelims ()2}d_{i, k - 2} + 2 \sum _{\begin{array}{c} j \ \text {even} \\ j \ge 4 \end{array}} {k - 2 \atopwithdelims ()j}\sum _{\ell = 0}^i(-1)^\ell d_{i - \ell , k - j}d_{\ell , j} \\&\quad = 2 \sum _{\begin{array}{c} j \ \text {even} \\ j \ge 2 \end{array}} {k - 2 \atopwithdelims ()j}\sum _{\ell = 0}^i(-1)^{\ell }d_{i - \ell , k - j}d_{\ell , j} \end{aligned}$$

Therefore, we have the following recursion for $d_{i, k}$:

$$\begin{aligned} d_{i, k}&= 2\sum _{\begin{array}{c} j \ge 1 \\ j \ \text {odd} \end{array}} {k - 2 \atopwithdelims ()j}\left( \sum _{\ell = 0}^i(-1)^\ell D_{i - \ell , k - j + 1} D_{\ell , j + 1} \right) \nonumber \\&\quad - 2\sum _{\begin{array}{c} j \ge 2 \\ j \ \text {even} \end{array}}{k - 2 \atopwithdelims ()j} \left( \sum _{\ell = 0}^i(-1)^\ell d_{i - \ell , k - j}d_{\ell , j}\right) \end{aligned}$$

(12)

5 Proof of main Theorem

With the two recursions obtained in the previous chapter, we are ready to prove our main theorem. The elementary symmetric functions play a big role in our discussion.

Definition 7

Let $x_1, \ldots , x_n$ be indeterminates. The i th elementary symmetric function on $x_1, \ldots , x_n$, denoted $e_i(x_1, \ldots , x_n)$, is defined as

$$\begin{aligned} e_i(x_1, \ldots , x_n) = \sum _{1 \le j_1< j_2< \ldots < j_i \le n} x_{j_1}x_{j_2}\ldots x_{j_i} \end{aligned}$$

First, we need some preliminary results.

Lemma 2

For $p \le n$, we have the following identity:

$$\begin{aligned} \sum _{k = 0}^{2n - 1}(-1)^k {2n - 1 \atopwithdelims ()k}k^p = \sum _{k = 0}^{2n}(-1)^k{2n \atopwithdelims ()k}k^p = 0 \end{aligned}$$

Proof

The proof of this identity can be found in [17] on pg. 30. $\square$

Furthermore, we have a simple corollary to the above lemma

Corollary 2

If $(m_1, \ldots , m_n) \in {\mathbb {R}}^n$, we have

$$\begin{aligned} \sum _{k = 0}^{2n - 1}(-1)^k{2n - 1 \atopwithdelims ()k}\prod _{i = 1}^n (m_i - k) = \sum _{k = 0}^{2n}(-1)^k{2n \atopwithdelims ()k}\prod _{i = 1}^n (m_i - k) = 0 \end{aligned}$$

Proof

We have

$$\begin{aligned}&\sum _{k = 0}^{2n - 1}(-1)^k{2n - 1 \atopwithdelims ()k} \prod _{i = 1}^n(m_i - k) \\&\quad = \sum _{k = 1}^{2n - 1}(-1)^k{2n - 1 \atopwithdelims ()k} \left( \sum _{i = 0}^ne_i(m_1, \ldots , m_n)k^{n - i} \right) \\&\quad = \sum _{i = 0}^ne_i(m_1, \ldots , m_n) \left( \sum _{k = 0}^{2n - 1}(-1)^k{2n - 1 \atopwithdelims ()k}k^{n - i} \right) \\&\quad = 0 \end{aligned}$$

A similar argument is used to show the vanishing of the second expression. $\square$

We are now ready to state and prove our main theorem.

Theorem 3

The following equalities hold for $D_{i, k}$ and $d_{i, k}$:

$$\begin{aligned}&D_{i, k} = \left( \frac{1}{2}\right) ^{i + 1} e_i(1, 3, \ldots , k - 3) \end{aligned}$$

(13)

$$\begin{aligned}&d_{i, k} = \left( \frac{1}{2} \right) ^{i + 1} e_i(2, 4, \ldots , k -2) \end{aligned}$$

(14)

Proof

In order to prove the theorem, we simply need to check that the purported expressions of $D_{i, k}$ and $d_{i, k}$ in Eqs. 13 and 14 satisfy the recursions obtained in the previous chapter, along with the initial conditions. The initial conditions follow easily. Indeed,

$$\begin{aligned}&D_{0, k} = \frac{1}{2} = \frac{1}{2}e_0(1, 3, \ldots , k - 3) \\&D_{1, 4} = \frac{1}{4} = \left( \frac{1}{2}\right) ^{1 + 1}e_1(1) \\&d_{0, k} = \frac{1}{2} \end{aligned}$$

When we plug in 13 and 14 into the recursion obtained in Eq. 6, we get

$$\begin{aligned}&\left( \frac{1}{2}\right) ^{i + 1}e_i(1, 3, \ldots , k - 3) \nonumber \\&\quad = 2 \sum _{\begin{array}{c} j \ge 1 \\ j \ \text {odd} \end{array}}{k - 3 \atopwithdelims ()j}\nonumber \\&\qquad \left( \sum _{\ell = 0}^i(-1)^\ell \left( \frac{1}{2}\right) ^{i + 2}e_{i - \ell }(2, 4, \ldots , k - 3 - j) e_{\ell }(2, 4, \ldots , j - 1) \right) \nonumber \\&\qquad - 2 \sum _{\begin{array}{c} j \ge 2 \\ j \ \text {even} \end{array}} {k - 3 \atopwithdelims ()j} \nonumber \\&\qquad \left( \sum _{\ell = 0}^i (-1)^\ell \left( \frac{1}{2}\right) ^{i + 2}e_{i - \ell }(1, 3, \ldots , k - 3 - j) e_{\ell }(1, 3, \ldots , j - 1) \right) \end{aligned}$$

(15)

Lemma 3

Eq. 15 is equivalent to the following equality of polynomials:

$$\begin{aligned}&\prod _{n = 1}^\frac{k - 2}{2}(1 + (2n - 1)t) \nonumber \\&\quad = \sum _{\begin{array}{c} j \ge 1 \\ j \ \text {odd} \end{array}}{k - 3 \atopwithdelims ()j} \prod _{n = 1}^\frac{k - 3 - j}{2}(1 + 2nt) \prod _{n = 1}^\frac{j - 1}{2}(1 - 2nt) \nonumber \\&\qquad - \sum _{\begin{array}{c} j \ge 2 \\ j \ \text {even} \end{array}}{k - 3 \atopwithdelims ()j}\prod _{n = 1}^\frac{k - 2 - j}{2} (1 + (2n - 1)t)\prod _{n = 1}^\frac{j}{2}(1 - (2n - 1)t) \end{aligned}$$

(16)

where we make the convention that $\prod _{n = 1}^0(1 - 2nt):= 1$

Before we can proceed with the proof of Lemma 3, we need a few standard combinatorial facts (see [18, 19]). The elementary symmetric functions have a very nice description via their generating functions,

$$\begin{aligned} e_i(x_1, x_2, \ldots , x_n) = [t^i] \cdot \prod _{j = 1}^n(1 + x_jt) \end{aligned}$$

where by $[t^i] \cdot p(t)$, we mean the degree i coefficient of p(t). Furthermore, recall that, if $\displaystyle f(t) = \sum _{i \ge 0} a_it^i$ and $\displaystyle g(t) = \sum _{i \ge 0}b_it^i$ are generating functions for sequences $a_i$ and $b_i$, then $f(t)g(-t)$ is the generating function of the sequence $c_i$, where

$$\begin{aligned} c_i = \sum _{j = 0}^i(-1)^ja_{i - j}b_j \end{aligned}$$

Proof of Lemma 3

Using the above facts, we see that the left hand side of Eq. 15 is

$$\begin{aligned} \left( \frac{1}{2}\right) ^{i + 1} \ [t^i] \cdot \prod _{n = 1}^\frac{k - 2}{2}(1 + (2n - 1)t) \end{aligned}$$

The first sum on the right hand side of Eq. 15 is

$$\begin{aligned} 2 \left( \frac{1}{2}\right) \left( \frac{1}{2}\right) ^{i + 1} \sum _{\begin{array}{c} j \ge 1 \\ j \ \text {odd} \end{array}} {k - 3 \atopwithdelims ()j} [t^i] \cdot \prod _{n = 1}^{\frac{k - 3 - j}{2}} (1 + 2nt)\prod _{n = 1}^\frac{j - 1}{2}(1 - 2nt) \end{aligned}$$

The second sum on the right hand side of Eq. 15 is

$$\begin{aligned} 2\left( \frac{1}{2}\right) \left( \frac{1}{2}\right) ^{i + 1} \sum _{\begin{array}{c} j \ge 2 \\ j \ \text {even} \end{array}}{k - 3 \atopwithdelims ()j} [t^i] \cdot \prod _{n = 1}^\frac{k - 2 - j}{2} (1 + (2n - 1)t)\prod _{n = 1}^\frac{j}{2}(1 - (2n - 1)t) \end{aligned}$$

Cancelling the terms $\left( \frac{1}{2}\right) ^{i + 1}$ on both sides, Lemma 3 follows. $\square$

Making the variable substitution $g:= \frac{k - 2}{2}$ in Equation (16), we get

$$\begin{aligned} \prod _{n = 1}^g(1 + (2n - 1)t)&= \sum _{\begin{array}{c} j \ge 1 \\ j \ \text {odd} \end{array}}{2g - 1 \atopwithdelims ()j}\prod _{n = 1}^\frac{2g - 1 - j}{2} (1 + 2nt)\prod _{n = 1}^\frac{j - 1}{2}(1 - 2nt) \nonumber \\&- \sum _{\begin{array}{c} j \ge 2 \\ j \ \text {even} \end{array}}{2g - 1 \atopwithdelims ()j} \prod _{n = 1}^\frac{2g - j}{2}(1 + (2n - 1)t)\prod _{n = 1}^\frac{j}{2} (1 - (2n - 1)t) \end{aligned}$$

(17)

Lemma 4

Eq. 17 is equivalent to the vanishing of the polynomial

$$\begin{aligned} P(t) := \sum _{j = 0}^{2g - 1}(-1)^j{2g - 1 \atopwithdelims ()j} \prod _{n = 1}^{g}(1 + (2g - 1 - j - 2(n - 1))t) \end{aligned}$$

Proof of Lemma 4

The first claim is that

$$\begin{aligned} j \ \text {odd} \implies \prod _{n = 1}^\frac{2g - 1 - j}{2}(1 + 2nt) \prod _{n = 1}^\frac{j - 1}{2}(1 - 2nt) = \prod _{n = 1}^{g} (1 + (2g - 1 - j - 2(n - 1))t) \end{aligned}$$

Indeed, if j is odd, we have

$$\begin{aligned}&\prod _{n = 1}^\frac{2g - 1 - j}{2}(1 + 2nt) \prod _{n = 1}^\frac{j - 1}{2}(1 - 2nt) = \prod _{\begin{array}{c} -(j - 1) \le x \le 2g - 1 - j \\ x \ \text {even} \end{array}}(1 + xt) \\&\quad = (1 + (2g - 1 - j)t)(1 + (2g - 1 - j - 2)t) \ldots \\&\qquad (1 + (2g - 1 - j - 2(g - 1))t) \\&\quad = \prod _{n = 1}^{g}(1 + (2g - 1 - j - 2(n - 1))t) \end{aligned}$$

A similar argument shows that

$$\begin{aligned} j \ \text {even}&\implies \prod _{n = 1}^\frac{2g - j}{2} (1 + (2n - 1)t)\prod _{n = 1}^\frac{j}{2}(1 - (2n - 1)t) \\&= \prod _{n = 1}^{g}(1 + (2g - 1 - j - 2(n - 1))t) \end{aligned}$$

Therefore, the right hand side of Eq. 17 becomes

$$\begin{aligned} -\sum _{j = 1}^{2g - 1}(-1)^j{2g - 1 \atopwithdelims ()j}\prod _{n = 1}^{g} (1 + (2g - 1 - j - 2(n - 1))t) \end{aligned}$$

Now, if we take the product $\prod _{n = 1}^{g}(1 + (2\,g - 1 - j - 2(n - 1))t)$ and set j equal to zero, we obtain

$$\begin{aligned} \prod _{n = 1}^g(1 + (2g - 1 - 2(n - 1))t)&= \prod _{n = 1}^g(1 + (2g + 1 - 2n)t) \\&= (1 + (2g - 1)t)(1 + (2g - 3)t)\ldots (1 + t) \\&= \prod _{\begin{array}{c} 1 \le x \le 2g - 1 \\ x \ \text {odd} \end{array}} (1 + xt) \\&= \prod _{n = 1}^g(1 + (2n - 1)t) \end{aligned}$$

Putting all of these calculations together gives the desired result. $\square$

It remains to show that the polynomial P(t) in Lemma 4 actually vanishes. Consider the following variable transformation for P(t)

$$\begin{aligned} \widehat{P}(t) := t^gP\left( \frac{1}{t}\right) = \sum _{j = 0}^{2g - 1}(-1)^j{2g - 1 \atopwithdelims ()j} \prod _{n = 1}^g((t + 2g - 1- 2(n - 1)) - j) \end{aligned}$$

For $1 \le n \le g$, define $m_n(t):= t + 2g - 1 - 2(n - 1)$, so that $\widehat{P}(t)$ becomes

$$\begin{aligned} \widehat{P}(t) = \sum _{j = 1}^{2g - 1} (-1)^j {2g - 1 \atopwithdelims ()j} \prod _{n = 1}^g(m_n(t) - j) \end{aligned}$$

By direct application of Corollary 2,

$$\begin{aligned} \widehat{P}(1) = \widehat{P}(2) = \ldots = \widehat{P}(g + 1) = 0 \end{aligned}$$

Therefore, $\widehat{P}(t)$ has $g + 1$ distinct roots. But since the degree of $\widehat{P}(t)$ is g, it follows that $\widehat{P}(t) = 0$, and therefore, $P(t) = 0$. This shows that

$$\begin{aligned} D_{i, k} = \left( \frac{1}{2} \right) ^{i + 1}e_i(1, 3, \ldots , k - 3) \end{aligned}$$

In order to prove that $d_{i, k} = \left( \frac{1}{2} \right) ^{i + 1} e_i(2, 4, \ldots , k - 2)$, we plug in this expression into the recursion in 12, and run through the same arguments as before. After going through the analogous computations as in the case for $D_{i, k}$, the desired result follows after showing the vanishing of the polynomial

$$\begin{aligned} \sum _{j = 0}^{2g}(-1)^j{2g \atopwithdelims ()j}\prod _{n = 1}^g (1 + (2g + 1 - j - 2(n - 1))t) \end{aligned}$$

$\square$