On some hyperelliptic Hurwitz–Hodge integrals

Abstract

We address Hodge integrals over the hyperelliptic locus. Recently Afandi computed, via localisation techniques, such one-descendant integrals and showed that they are Stirling numbers. We give another proof of the same statement by a very short argument, exploiting Chern classes of spin structures and relations arising from Topological Recursion in the sense of Eynard and Orantin.

These techniques seem also suitable to deal with three orthogonal generalisations: (1) the extension to the r-hyperelliptic locus; (2) the extension to an arbitrary number of non-Weierstrass pairs of points; (3) the extension to multiple descendants.

1. Introduction

Moduli spaces of curves have been proved in the last decades to be of great interest not only for pure Algebraic Geometry, but a key element at the crossroad of Gromov–Witten theory, Integrable Systems, Random Matrix Models, Topological Recursion, and more.

An important task is the computation of intersection numbers. To fix the ideas, one could think of intersection numbers as integrals packed in polynomials P of the following form:

(1·1) \begin{equation}P(x_1, \dots, x_n) = \int_{\mathcal{M}(g,n)} \frac{\mathcal{C}}{\prod_j (1 - \psi_j x_j)} \cdot [\Delta] \in \mathbb{Q}[x_1, \dots, x_n], \qquad \quad \deg P \leq \dim \mathcal{M}(g,n).\end{equation}

Here $\mathcal{M}(g,n)$ is some moduli space of curves ¹ , either compact or with a virtual fundamental class, $\mathcal{C}$ is a cohomology class of $\mathcal{M}(g,n)$ ² , $[\Delta]$ is a locus of curves of $\mathcal{M}(g,n)$ ³ , $\psi$ -classes are building blocks of $H^*(\mathcal{M}(g,n))$ of complex degree one, and $x_i$ are formal variables.

We study the following intersection numbers, expressable in three different but equivalent ways. We refer to [ 2 , 5 , 17 ] and references therein for an exhaustive state-of-the-art.

1. (1) In terms of the hyperelliptic locus $\overline{\mathcal{H}}_{g, 2g + 2, a} \subseteq \overline{\mathcal{M}}_{g, 2g + 2}$ of algebraic curves admitting a degree 2 map to $\mathbb{P}^1$ , with $2g+2$ marked Weierstrass points and a pairs conjugated by the involution, and $\lambda_i = c_i(\mathbb{E})$ the ith Chern class of the Hodge bundle (see [ 1 ] for more details):

   (1·2) \begin{equation}\int_{\overline{\mathcal{H}}_{g, 2g + 2, 2^{a}}} \lambda_i \operatorname{br}^*\left( \prod_{j=1}^{2g + 2+a} \frac{1}{(1 - \psi_j x_j)} \right) ,\end{equation}
   where $\operatorname{br}$ is the map associating to each hyperelliptic curve its target.

2. (2) In terms of admissible covers and the Hodge class (see [ 15 ] for more details):

   (1·3) \begin{equation}\int_{\overline{\mathcal{M}}_{0, \emptyset - (1^{2g + 2}, 0^{a)}}(\mathbb{B}\mathbb{Z}_r)} \frac{\lambda_i,}{\prod_{j=1}^{2g + 2+a} (1 - x_j\psi_j) } , \qquad \qquad \text{ for } r=2.\end{equation}

3. (3) In terms of the moduli space of stable curves and the $\Omega$ -CohFT (see [ 18 ] and Section 2):

   (1·4) \begin{equation}\int_{\overline{\mathcal{M}}_{0, 2g + 2 + a}} \frac{[\deg_{H^{*}} = i].\Omega(r,0;\; \overbrace{1, \dots, 1}^{2g + 2+a}, \overbrace{0, \dots, 0}^{a})}{\prod_{j=1}^{2g + 2} (1 - \psi_j x_j)}, \qquad \qquad \text{ for } r=2.\end{equation}

The equivalence between the first and the second is well known, the equivalence between the second and the third, for arbitrary r, is achieved in [ 18 ]. This restatement was then employed to address a problem by Goulden–Jackson–Vakil over the existence of an ELSV formula for double Hurwitz numbers with one total ramification [ 8 ]. A really useful Sage package has recently been developed to perform intersection theoretic calculations otherwise by far out of reach with traditional methods [ 7 ].

1·1. Results

Progress on the evaluation of these integrals was recently made in the following formula:

Theorem 1·1 ([ 1 ]). Linear-Hodge one-descendant integrals over the hyperelliptic locus evaluate to:

\begin{align*}& \int_{\overline{\mathcal{H}}_{g, 2g + 2}}\operatorname{br}^*\left( \psi_1^{2g - 1 - i} \right) \lambda_i = \frac{1}{2} \cdot e_i\left(0 + {1}/{2}, 1 + {1}/{2}, \ldots, g-1 + {1}/{2} \right), \\[5pt] & \int_{\overline{\mathcal{H}}_{g, 2g + 2, 2}}\operatorname{br}^*\left( \psi_{2g + 3}^{2g - i} \right) \lambda_i = \frac{1}{2} \cdot e_i(1, 2, \ldots, g).\end{align*}

Here $ e_i(x_1, \ldots, x_n)$ is the $i{th}$ elementary symmetric polynomial. ⁴

The theorem is achieved via virtual localisation techniques. We provide a very short new proof in Section 3, exploiting $\Omega$ -classes and certain vanishings arising from the theory of Topological Recursion in the sense of Eynard and Orantin, introduced in Section 2.

This result opens up natural questions about its generalisations in at least three ‘orthogonal’ directions:

1. (Q1) What happens to the integrals over hyperelliptic loci with an arbitrary number of non-Weierstrass pairs of points $\overline{\mathcal{H}}_{g, 2g + 2, 2^a}$ ?

2. (Q2) The hyperelliptic locus ${\overline{\mathcal{H}}_{g, 2g + 2}}$ naturally corresponds to r-admissible covers to the classifying space $B\mathbb{Z}_r$ for $r=2$ . How do these integrals behave for higher r?

3. (Q3) How does the theory with multiple descendants behave?

We address all three questions with the same techniques with which we give a new proof of the theorem above. We provide and prove answers in certain regimes.

The answer to (Q1) is readily obtained.

Proposition 1·2. The integrals do not depend on the additional number a of non-Weierstrass pairs marked:

\begin{align*}& \int_{\overline{\mathcal{H}}_{g, 2g + 2, 2^a}}\operatorname{br}^*\left( \psi_1^{2g - 1 - i + a} \right) \lambda_i = \frac{1}{2} \cdot e_i\left(0 + {1}/{2}, 1 + {1}/{2}, \ldots, g-1 + {1}/{2} \right), \\[5pt] & \int_{\overline{\mathcal{H}}_{g, 2g + 2, 2^{a+1}}}\operatorname{br}^*\left( \psi_{2g + 3}^{2g - i + a} \right) \lambda_i = \frac{1}{2} \cdot e_i(1, 2, \ldots, g).\end{align*}

The answer to Q2 is obtained under certain restrictions on the parametrisations, which we show to be in general sharp.

Theorem 1·3. For arbitrary r and one descendant, if

(1·5) \begin{equation}\max \limits_{i \neq j} (b_i + b_j) \leq r, \end{equation}

then

(1·6) \begin{align}& \int_{{\overline{\mathcal{M}}}_{0,1 + \ell}}\!\!\!\! [\deg_{H^*} = i].\Omega_{0,n}^{[-1]}(r,0;\; -\overline{\mu_1}, b_1, \dots, b_{\ell}) \psi_1^{\ell - 2 -j} = \nonumber\\[5pt] &\frac{1}{r} e_i\left( \frac{\langle \mu \rangle}{r}, \frac{\langle \mu \rangle}{r} + 1, \dots, \frac{\langle \mu \rangle}{r} + [b] - 1 \right).\end{align}

Here $\sum_j b_j = b = [b]r + \langle b \rangle$ , with $0 \leq \langle b \rangle \leq r-1$ by Euclidean division, similarly for $\mu$ . Moreover, $-\overline{\mu_i}$ is the only representative of $-\mu_i$ modulo r between zero and $r-1$ . ⁵

The answer to (Q3) can be given for both generalisations at the same time – that is, for arbitrarily many descendants and for arbitrary r – but imposing $b=\emptyset$ . The formula to compute these integrals is achieved simply by chaining together several existing results, and then simplifying the outcome by the genus-zero restriction. We explain and combine these results in Section 3. Meanwhile we give here a temporary statement.

Proposition 1·4. Let $2g - 2 + n \gt0$ and let r be a positive integer. Fix remainder classes

\begin{align*} -\overline{\mu_1}, \dots, -\overline{\mu_n} \in \{0, \dots, r-1\}. \end{align*}

Then for $\mu_i = \langle \mu_i \rangle + r[\mu_i]$ with $\langle \mu_i \rangle = r - (\!-\!\overline{\mu_i}) $ we have:

(1·7) \begin{equation}\int_{{\overline{\mathcal{M}}}_{0,n}}\frac{\Omega_{0,n}(r,0;\; -\overline{\mu_1}, \dots, -\overline{\mu_n})}{\prod_{i}(1 - \frac{\mu_i}{r}\psi_i)} = \left(\prod_{i=1}^{n} \frac{\left(\frac{\mu_i}{r}\right)^{[\mu_i]}}{[\mu_i]!} \right)^{-1} \!\! \cdot c\sum_{P \in CP}^{\text{finite}} \prod_{l=1}^{\text{finite}} (|I_{l}^{P}||L_{l}^{P}| - |J_{l}^{P}||K_{l}^{P}|).\end{equation}

Here c is an explicit combinatorial prefactor, and $|I_{l}^{P}| \;:\!=\; \sum \mu_{i_l}$ for $i_l$ belonging to the subset $I^{P}_l \subset \{1, \dots, n\}$ .

Moreover, there exists a polynomial $P_{\overline{\mu}}(x_1, \dots, x_n)$ of degree $n-3$ depending on the remainder classes $\overline{\mu_i}$ such that the evaluation $P_{\overline{\mu}}(\mu_1, \dots, \mu_n)$ recovers the expression above.

1·2. Plan of the paper

In the next section we provide the necessary background on $\Omega$ -classes and in Section 3 we prove the four statements above. In Section 4 we provide examples and counterexamples of the statements above.

2. Background

In [ 20 ], Mumford derived a formula for the Chern character of the Hodge bundle on the moduli space of curves $\overline{\mathcal{M}}_{g,n}$ in terms of tautological classes and Bernoulli numbers. Such class appears in the celebrated ELSV formula [ 10 ], named after its four authors Ekedahl, Lando, Shapiro, Vainshtein, that is an equality between simple Hurwitz number and an integral over the moduli space of stable curves.

A generalisation of Mumford’s formula was computed in [ 6 ]. The moduli space $\overline{\mathcal{M}}_{g,n}$ is substituted by the proper moduli stack $\overline{\mathcal{M}}_{g;\;a}^{r,s}$ of rth roots of the line bundle

(2·1) \begin{equation} \omega_{\log}^{\otimes s}\biggl(-\sum_{i=1}^n a_i p_i \biggr),\end{equation}

where $\omega_{\log} = \omega(\sum_i p_i)$ is the log-canonical bundle, r and s are integers with r positive, and $a = (a_1, \ldots, a_n) \in \{ 0,\ldots,r-1 \}^n$ is an integral n-tuple satisfying the modular constraint

(2·2) \begin{equation}a_1 + a_2 + \cdots + a_n \equiv (2g-2+n)s \pmod{r}.\end{equation}

This condition guarantees the existence of a line bundle whose r-th tensor power is isomorphic to $\omega_{\log}^{\otimes s}(\!-\!\sum_i a_i p_i)$ . Let $\pi \colon \overline{\mathcal{C}}_{g;\;a}^{r,s} \to \overline{\mathcal{M}}_{g;\;a}^{r,s}$ be the universal curve, and $\mathcal{L} \to \overline{\mathcal C}_{g;\;a}^{r,s}$ the universal rth root. In complete analogy with the case of moduli spaces of stable curves, one can define $\psi$ -classes and $\kappa$ -classes. There is moreover a natural forgetful morphism

(2·3) \begin{equation}\epsilon \colon\overline{\mathcal{M}}^{r,s}_{g;\;a}\longrightarrow\overline{\mathcal{M}}_{g,n}\end{equation}

which forgets the line bundle, otherwise known as the spin structure. It can be turned into an unramified covering in the orbifold sense of degree $2g - 1$ by slightly modifying the structure of $\overline{\mathcal{M}}_{g,n}$ , introducing an extra ${\mathbb{Z}}_r$ stabilizer for each node of each stable curve (see [ 13 ]).

Let $B_m(x)$ denote the mth Bernoulli polynomial, that is the polynomial defined by the generating series

(2·4) \begin{equation}\frac{te^{tx}}{ e^t - 1} = \sum_{m=0}^{\infty} B_{m}(x)\frac{t^m}{m!}.\end{equation}

The evaluations $B_m(0) = (\!-\!1)^m B_m(1) = B_m$ recover the usual Bernoulli numbers. The following formula provides an explicit formula for the Chern characters of the derived pushforward of the universal rth root $\operatorname{ch}_m(r,s;\;a) = \operatorname{ch}_m(R^{\bullet} \pi_{\ast}{\mathcal L})$ . The formula was obtained by Mumford for $r=1$ and $s=1$ [ 3 ], then generalised by Bini to arbitrary integers s [ 3 ], then generalised by Chiodo to arbitrary positive integers r.

Theorem 2·1 ([ 6 ]). The Chern characters $\operatorname{ch}_m(r,s;\;a)$ of the derived pushforward of the universal rth root have the following explicit expression in terms of $\psi$ -classes, $\kappa$ -classes, and boundary divisors:

(2·5) \begin{equation} \operatorname{ch}_m(r,s;\;a)= \frac{B_{m+1}(\tfrac{s}{r})}{(m+1)!} \kappa_m - \sum_{i=1}^n \frac{B_{m+1}(\tfrac{a_i}{r})}{(m+1)!} \psi_i^m + \frac{r}{2} \sum_{a=0}^{r-1} \frac{B_{m+1}(\tfrac{a}{r})}{(m+1)!} \, j_{a,\ast} \frac{(\psi')^m - (\!-\!\psi'')^m}{\psi' + \psi''}.\end{equation}

Here $j_a$ is the boundary morphism that represents the boundary divisor with multiplicity index a at one of the two branches of the corresponding node, and $\psi',\psi''$ are the $\psi$ -classes at the two branches of the node.

We can then consider the family of Chern classes pushforwarded on the moduli spaces of stable curves

(2·6) \begin{equation} \Omega_{g,n}^{[x]}(r,s;\;\vec{a}) = \epsilon_{\ast} \exp{\Biggl( \sum_{m=1}^\infty (\!-\!1)^m x^{m} (m-1)! \, \operatorname{ch}_m(r,s;\;\vec{a}) \Biggr)} \in H^{\text{even}}(\overline{\mathcal{M}}_{g,n}).\end{equation}

We will omit the variable x when $x = 1$ .

Corollary 2·2. [13] The class $\Omega_{g,n}^{[x]}(r,s;\;\vec{a})$ is equal to

\begin{align*} \hspace{-1pt}\sum_{\Gamma\in {\textsf{G}}_{g,n}} \sum_{w\in {\textsf{W}}_{\Gamma,r,s}}\!\frac{r^{2g-1-h^1(\Gamma)}}{|\operatorname{Aut}(\Gamma)| }\;\xi_{\Gamma *}\!\Bigg[ \!\prod_{v \in V(\Gamma)} \!e^{-\sum\limits_{m = 1} (\!-\!1)^{m-1} x^m \frac{B_{m+1}(s/r)}{m(m+1)}\kappa_m(v)} \! \cdot \prod_{i=1}^n e^{\sum\limits_{m = 1}(\!-\!1)^{m-1} x^m \frac{B_{m+1}(a_i/r)}{m(m+1)} \psi^m_{i}} \cdot \\[5pt] \hspace{+10pt}\prod_{\substack{e\in E(\Gamma) \\ e = (h,h')}}\frac{1-e^{\sum\limits_{m \geq 1} (\!-\!1)^{m-1} x^m \frac{B_{m+1}(w(h)/r)}{m(m+1)} [(\psi_h)^m-(-\psi_{h'})^m]}}{\psi_h + \psi_{h'}} \Bigg]\, .\end{align*}

Here $B_{m+1}(y)$ are Bernoulli polynomials, ${\textsf{G}}_{g,n}$ is the finite set of stable graphs of genus g with n legs, ${\textsf{W}}_{\Gamma,r,s}$ is the finite set of decorating the leg i with $a_i$ and any other half-edge with an integer in $\{0, \dots, r-1\}$ in such a way that decorations of half-edges of the same edge ( $e \in E(\Gamma)$ ) sum up to r and locally on each vertex ( $v \in V(\Gamma)$ ) the sum of all decorations is congruent to $(2g - 2 + n)s$ modulo r.

Remark 2·3. By looking at the formula above is it easy to deduce a few properties of the classes $\Omega$ , see [ 12 ] for a more exhausive list. For instance, $\Omega^{[x]}(r,r;\;\vec{a}) = \Omega^{[x]}(r,0;\;\vec{a})$ , as Bernoulli polynomials satisfy $B_{m+1}(0) = B_{m+1}(1) = B_{m+1}$ . Similarly by $B_{m+1}(1 - X) =$ $(\!-\!1)^{m+1}B_{m+1}(X)$ and Newton identities one can show that $\Omega^{[x]}(r,s;\;a_1, \dots, a_i + r, \dots, a_n) = \Omega^{[x]}(r,s;\;a_1, \dots, a_n) \cdot \left( 1 + \frac{a_i}{r}\psi_i \right)$ .

2·1. Riemann–Roch for $\Omega$ -classes

The Riemann–Roch theorem for an r-th root L of $\omega_{\log}^{\otimes s}(\!-\!\sum_i a_i p_i)$ provides the following relation:

(2·7) \begin{equation} \frac{(2g - 2 + n)s - \sum_i a_i}{r} - g + 1 = h^0(C,L) - h^1(C,L).\end{equation}

In some cases, i.e. for particular parametrisations of $r,s,a_i$ and for topologies (g,n), it can happen that either $h^0$ or $h^1$ vanish, turning $\Omega$ into an actual total Chern class of a vector bundle, so that the Riemann–Roch equation provides the rank of this bundle. If that happens, the Riemann–Roch equation provides a bound for the complex cohomological degree of $\Omega$ :

(2·8) \begin{equation}[\!\deg_{H^*} = k].\Omega_{g,n}(r,s;\;\vec{a}) = 0, \qquad \qquad \text{ for } k \gt \operatorname{rank}(R^{\bullet} \pi_{\ast}{\mathcal L}),\end{equation}

which are usually trivial or not depending on whether the rank $\lt 3g - 3 + n$ . One of these instances of parametrisations is provided by the following result of Jarvis, Kimura, and Vaintrob.

Theorem 2·4 ([ 14 , proposition 4.4]). Let $g=0$ , let $s=0$ , let $n \geq 3$ , consider $a_i$ all strictly positive except at most a single $a_j$ which can be positive, or zero, or equal to $-1$ . Then the rth universal root does not have any global section, that is, we have

\begin{align*} h^0 = 0. \end{align*}

Under the condition of the theorem above, the rank of $R^{\bullet} \pi_{\ast}{\mathcal L}$ equals $h^1$ , and therefore one gets:

(2·9) \begin{equation}[\deg_{H^*} = k].\Omega_{0,n}(r,s;\;\vec{a}) = 0, \qquad \qquad \text{ for } k \gt \frac{\sum_i a_i }{r} - 1.\end{equation}

2·2. String equation for $\Omega$ -classes

It is known [ 12 , 18 ] that if $0 \leq s \leq r$ then

(2·10) \begin{equation}\Omega(r,s;\; a_1, \dots, a_{n}, a_{n+1} = s) = \pi^{*}\Omega(r,s;\; a_1, \dots, a_{n}).\end{equation}

By projection formula, this implies the string equation:

(2·11) \begin{equation}\int_{\overline{\mathcal{M}}_{g, n+1}} \frac{\Omega(r,s;\; a_1, \dots, a_{n}, a_{n+1} = s)}{\prod_{i=1}^n (1 - x_i \psi_i)} = (x_1 + \dots + x_n) \int_{\overline{\mathcal{M}}_{g, n}} \frac{\Omega(r,s;\; a_1, \dots, a_{n})}{\prod_{i=1}^n (1 - x_i \psi_i)}.\end{equation}

By Remark 2·3, $s=r$ and $s=0$ are interchangeable.

2·3. Relations for integrals of $\Omega$ -classes arising from Topological recursion

Topological recursion is a universal recursive procedure that produces solutions of enumerative geometric problems (see e.g. [ 11 ] for an introduction). Let us very briefly mention how this can be useful to produce relations between integrals of the $\Omega$ -classes. In [ 4 ], this machinery was employed to generate double Hurwitz numbers. Although they are by definition polynomials in some formal variables $q_1, \dots, q_r$ taking care of ramification conditions, what is generated by topological recursion are formal power series containing poles in exactly one of these variables, namely $q_r$ . Polynomiality implies that the coefficient of $q_r^{-k}$ for $k \gt 0$ , which can be expressed as linear combinations of $\Omega$ -classes integrals, vanishes. For more details on why $\Omega$ -classes integrals appear in double Hurwitz numbers (relation known as ELSV-type formulae) see e.g. [ 18 ]. These vanishing result in the following statement.

Theorem 2·5 ([ 4 ]). Let $2g - 2 + n + \ell \gt 0$ and let r be a positive integer, and $s=0$ .

1. (i) Let $1 \leq \mu_1, \dots, \mu_n \leq r$ , and let $\mu$ be their sum.

2. (ii) Let $1 \leq b_1, \dots, b_{\ell} \leq r-1$ , and let b be their sum.

3. (iii) Impose $b \equiv \mu \; (\!\!\!\! \mod r)$ . Then we can write $b = \mu + r \cdot \delta$ for some integer $\delta$ .

If

(2·12) \begin{equation}\mu \lt b, \text{ or equivalently, } \delta \gt 0 \qquad \qquad \text{ (negativity), }\end{equation}

then the following finite linear combination of $\Omega$ -integrals vanishes:

(2·13) \begin{align} & \sum_{t = 1}^{\ell} \frac{(\!-\!1)^{\ell - t}}{t!} \!\!\!\sum_{\substack{\boldsymbol{\rho} \in (\mathscr{\tilde{P}}_{r - 1})^{k} \\ \sqcup_{\kappa} \rho^{(\kappa)} = \textbf{b}^{\vee}}}\prod_{\kappa = 1}^{t} \frac{\big[\frac{r - |\rho^{(\kappa)}|}{r}\big]_{\ell(\rho^{(\kappa)}) - 1}}{|\operatorname{Aut}{\rho^{(\kappa)}}|} \nonumber \\[5pt] & \int_{\overline{\mathcal{M}}_{g,n + t}} \!\!\!\!\! \frac{\Omega(r,0;\; - \overline{\mu_1},\dots, - \overline{\mu_n},\dots, r - |{\rho}^{(1)}|, \dots, r - |{\rho}^{(t)}|)}{\prod_{i = 1}^n \big(1 - \frac{\mu_i}{r}\psi_i\big)} = 0.\end{align}

Here $\mathscr{\tilde{P}}_{r - 1}$ is the set of partitions of size at most $r-1$ , the Pochhammer symbol $[x]_a \;:\!=\; x(x-1) \cdots (x- a+1)$ stands for the descending factorial, and $\textbf{b}^{\vee} = (r - b_1, \dots, r - b_{\ell})$ .

In particular, if the condition

(2·14) \begin{equation}\max \limits_{i \neq j}(b_i + b_j) \leq r \qquad \qquad \text{ (boundedness) }\end{equation}

is satisfied, then all summands but a single one (the term for $t=\ell$ and for $|{\rho}^{(\kappa)}| = 1$ for $\kappa = 1, \dots, \ell$ ) straighforwardly disappear in the relation above. In this case we obtain: ⁶

(2·16) \begin{equation}\int_{\overline{\mathcal{M}}_{g, n+\ell} } \frac{\Omega(r,0;\; -\overline{\mu_1}, \dots, -\overline{\mu_n}, b_1, \dots, b_{\ell})}{\prod_{i=1}^n (1- \frac{\mu_i}{r}\psi_i)} = 0.\end{equation}

2·3·1. The $r=2$ case

Let us briefly discuss the specialisation of the result above to $r=2$ .

1. (1) All $b_j$ must equal to one, and therefore $b = \ell$ .

2. (2) The boundedness condition is always satisfied.

3. (3) The negativity condition reads $\mu \lt \ell$ .

4. (4) The strong negativity condition reads $\mu + \delta \lt \ell$

5. (5) Strong negativity and negativity are equivalent, as $\delta = (\ell - \mu)/2$ .

If (3) or equivalently (4) are satisfied, we have

(2·17) \begin{equation} \int_{\overline{\mathcal{M}}_{g,n + \ell}} \frac{\Omega(2,0;\; -\!\overline{\mu_1}, \dots, -\overline{\mu_n}, 1, 1, \dots, 1)}{\prod_{i=1}^n(1 - \frac{\mu_i}{2}\psi_i)} = 0, \end{equation}

where $-\overline{\mu_i}$ in this case simplifies to the parity of $\mu_i$ (it is one if $\mu_i$ odd, else zero).

3. Proofs

We are now armed to prove the statements presented in the introduction.

3·1. New proof of Theorem 1·1

Proof. Let us first recast Theorem 1·1 as in form of Equation (1.4).

Lemma 3·1. Let x be a formal variable. Theorem 1·1 is equivalent to the following statements.

(3·1) \begin{align}\int_{\overline{\mathcal{M}}_{0,2g + 2}} \frac{\Omega(2,2;\; \overbrace{1, 1, \dots, 1)}^{2g+2}}{(1 - \frac{x}{2}\psi_1)} &= \frac{x^{g-1}}{2^{2g}}\prod_{k=1}^{g}(x - (2k-1)),\end{align}

(3·2) \begin{align}\int_{\overline{\mathcal{M}}_{0,2g + 3}} \frac{\Omega(2,2;\; 0, \overbrace{1,\dots, 1}^{2g+2})}{(1 - \frac{x}{2}\psi_1)} &= \frac{x^{g}}{2^{2g+1}}\prod_{k=1}^{g}(x - 2k). \end{align}

Proof. Simply multiply both sides of both statements of Theorem 1·1 by $(\!-\!1)^i x^i$ and sum over i, then use the general fact for polynomial roots $\prod_{j=1}^d(x - \alpha_j) = \sum_{j=0}^d x^j e_{d-j}(\alpha_i)$ $(\!-\!1)^{d-j}$ , and recast the obtained statements in terms of moduli space of stable curves (from integrals in (1·2) to integrals in (1·4)). This concludes the proof of the Lemma.

Let us now prove Equation (3·1). The LHS in (3·1) is a polynomial $P(x) = c\cdot \prod_{i=1}^{2g-1} (x - \alpha_i)$ , where $\alpha_i$ are the roots. The constant c is immediately computed as $c = [x^{2g-1}].P(x) = 2^{-(2g - 1)} \int_{\overline{\mathcal{M}}_{0,2g + 2}} \psi_1^{2g-1} 2^{-1}= 2^{-2g}$ . It remains to show that zero is a zero of P(x) of order $g-1$ and that $2k-1$ is a simple zero of P(x) for $k = 1, \dots, g$ . The first condition is equivalent to the fact that $\Omega$ has non-trivial cohomological degree at most g. This is indeed true and proved by the Riemann-Roch computation for $\Omega$ in genus zero performed in (2·9): $({2g+2})/{2}-1 =g$ . It only remains to show that $P(2k-1)=0$ for all $k=1, \dots, g$ (in fact, if so, their multiplicity cannot be greater than one by degree constraint). Explicitly, the proof boils down to the following g relations:

(3·3) \begin{equation}\int_{\overline{\mathcal{M}}_{0,2g + 2}} \frac{\Omega(2,2;\; \overbrace{1, 1, \dots, 1)}^{2g+2}}{(1 - \frac{2k-1}{2}\psi_1)} = 0, \qquad \text{ for } k=1, \dots, g.\end{equation}

These relations are immediately implied by Theorem 2·5 specialised as in Equation (2·17), then further specialised to $n=1$ : the vanishing holds for positive odd $\mu_1 \lt 2g + 1$ , or in other words, for $\mu_1 = 2k-1$ for $k=1, \dots, g$ . ⁷ Equation (3·2) is proved similarly. This concludes the proof of Theorem 1·1. ⁸

3·2. Proofs of Proposition 1·2, Theorem 1·3 and Proposition 1·4

Proof of Proposition 1·2. Again, we start by recasting the result.

Lemma 3·2. Let x be a formal variable. Proposition 1.2 is equivalent to the following statements:

(3·5) \begin{align}\int_{\overline{\mathcal{M}}_{0,2g + 2 + a}} \frac{\Omega(2,2;\; \overbrace{1, \dots, 1}^{2g+2}, \overbrace{0, \dots, 0}^{a})}{(1 - \frac{x}{2} \psi_1)} &= \frac{x^{g-1+a}}{4^{g}}\prod_{k=1}^{g}(x - (2k-1)), \end{align}

(3·6) \begin{align} \int_{\overline{\mathcal{M}}_{0,2g + 3 + a}} \frac{\Omega(2,2;\; 0, \overbrace{1,\dots, 1}^{2g+2}, \overbrace{0, \dots, 0}^{a})}{(1 - \frac{x}{2}\psi_1)} &= \frac{x^{g+a}}{2^{2g+1}}\prod_{k=1}^{g}(x - 2k). \end{align}

Both expressions are polynomials in x of degree equal to the dimension of the moduli spaces, that is, of degree $2g-1+a$ and $2g+a$ respectively.

Proof. The proof is the same as in Lemma 3·1. This concludes the proof of the Lemma.

Now it suffices to apply String Equation (2·11) to each of the last a marked points:

(3·7) \begin{equation}\int_{\overline{\mathcal{M}}_{0,2g + 2 + a}} \frac{\Omega(2,2;\; \overbrace{1, \dots, 1}^{2g+2}, \overbrace{0, \dots, 0}^{a})}{(1 - \frac{x}{2} \psi_1)} = x^a \cdot \int_{\overline{\mathcal{M}}_{0,2g + 2}} \frac{\Omega(2,2;\; \overbrace{1, \dots, 1}^{2g+2})}{(1 - \frac{x}{2} \psi_1)}.\end{equation}

This concludes the proof of the Proposition.

Proof of Theorem 1·3. Consider the polynomial

(3·8) \begin{equation}P (x) \;:\!=\; \int_{{\overline{\mathcal{M}}}_{0,1 + \ell}}\frac{\Omega_{0,n}(r,0;\; -\overline{\mu_1}, b_1, \dots, b_{\ell})}{(1 - \frac{x}{r}\psi_1)} = c\cdot \prod_{i} (x - \alpha_i).\end{equation}

It is a polynomial of degree $\deg(P) = \dim_{\mathbb{C}}({\overline{\mathcal{M}}}_{0,1 + \ell}) = \ell - 2$ , and its leading coefficient can be easily computed as

\begin{align*} c = \int_{{\overline{\mathcal{M}}}_{0,1 + \ell}} \Omega_{0,n}(r,0;\; -\!\overline{\mu_1}, b_1, \dots, b_{\ell})\frac{\psi_1^{\ell-2}}{r^{\ell-2}} = \frac{1}{r^{\ell-2}\cdot r} \int_{{\overline{\mathcal{M}}}_{0,1 + \ell}} \psi_1^{\ell-2} = \frac{1}{r^{\ell - 1}}. \end{align*}

Since $\mu_1 = \mu \equiv b$ modulo r, we must have that $\langle \mu \rangle = \langle b \rangle$ , and therefore $\bar{-\mu_1} = r - \langle \mu \rangle$ . The lowest degree in x of P(x) can be computed as

\begin{align*}\deg(P) - \operatorname{maxdeg}_{H^{*}}\Omega_{0,n}(r,0;\; -\!\overline{\mu_1}, b_1, \dots, b_{\ell}) & = \ell - 2 - \left( \frac{ r - \langle \mu \rangle + [b]r + \langle b \rangle }{r} - 1\right) \\[5pt] & = \ell - 2 - [b]. \end{align*}

So far we achieved to show that P has the form

\begin{align*} P(x) = \frac{x^{\ell - 2 - [b]}}{r^{\ell - 1}}\prod_{i=1}^{[b]}(x - \alpha_i).\end{align*}

for some suitable roots $\alpha_i$ . The question is whether the TR vanishing (2·16) can guarantee at least (and therefore all) [b] roots. ⁹ Assuming boundedness, we only have to worry about negativity $\mu \lt b$ holding true, which means that

\begin{align*} \langle \mu \rangle + rk \lt \langle b \rangle + r[b] \end{align*}

for a few possible non-negative integers k. As $\langle \mu \rangle = \langle b \rangle$ , negativity holds true for the values $k=0, \dots, [b]-1$ , which amounts to [b] different values of $\mu$ providing [b] simple roots $P(\langle \mu \rangle + rk) =0$ as required. This concludes the proof of the Theorem.

Proof of Proposition 1·4. As anticipated in the introduction, the proposition is simply obtained by a chain of several known results, simplified in genus zero.

1. (1) By [ 18 , section 5] we have:

   \begin{align*} \int_{{\overline{\mathcal{M}}}_{g,n}}\frac{\Omega_{g,n}(r,0;\; -\overline{\mu_1}, \dots, -\overline{\mu_n})}{\prod_{i}(1 - \frac{\mu_i}{r}\psi_i)} = \int_{{\overline{\mathcal{M}}}_{g,\emptyset - \mu}(\mathcal{B}\mathbb{Z}_r)}\frac{\sum_{i=0} (\!-\!1)^i \lambda_i^U}{\prod_{i}(1 - \frac{\mu_i}{r}\bar{\psi}_i)} \end{align*}
   for U the representation of the cyclic group $\mathbb{Z}_r$ sending 1 to $e^{2\pi i/r}$ .

2. (2) By [ 15 , theorem 1] we have:

   \begin{align*} h_{g,\mu}^{(r), \circ}= r^{2g - 2 + n + \frac{|\mu|}{r}}\left(\prod_{i=1}^{n} \frac{\left(\frac{\mu_i}{r}\right)^{[\mu_i]}}{[\mu_i]!} \right) \cdot \int_{{\overline{\mathcal{M}}}_{g,\emptyset - \mu}(\mathcal{B}\mathbb{Z}_r)}\frac{\sum_{i=0} (\!-\!1)^i \lambda_i^U}{\prod_{i}(1 - \frac{\mu_i}{r}\bar{\psi}_i)}, \end{align*}
   where $h_{g,\mu}^{(r), \circ}$ are Hurwitz numbers enumerating connected genus g degree $d = |\mu|$ ramified covers of the Riemann sphere with $b = b(g) = 2g - 2 + {|\mu|}/{r} + n$ simple ramifications, except the ramification with profile $(r,r,\dots, r)$ above zero and the ramification with profile $(\mu_1, \dots, \mu_n)$ above infinity, with $n = \ell(\mu)$ . Moreover, the integral is a polynomial of degree $3g - 3 + n$ in the $\mu_i$ depending on the remainder classes $\langle \mu_i \rangle$ modulo r, whereas the exponential prefactor is manifestly not polynomial in the parts $\mu_i$ . This property is known as quasi-polynomiality, and has been shown independently in [ 9 ] in the framework of Topological Recursion.

3. (3) By Okounkov [ 12 ] and Okounkov and Pandharipande [ 21 ], we have that Hurwitz numbers can be efficiently written as vacuum expectation of operators in the Fock space, which in this case form a handy algebra closed under commutation relations:

   (3·9) \begin{equation}h_{g, \mu}^{(r), \circ} = \frac{[z_1^2 \cdots z_{b(g)}^2]}{\prod \mu_i \cdot r^{d/r}}\Bigg{\langle} \mathcal{E}_{\mu_1}(0) \dots \mathcal{E}_{\mu_n}(0) \mathcal{E}_{0}(z_1) \dots \mathcal{E}_{0}(z_{b(g)}) \mathcal{E}_{-r}(0)^{d/r} \Bigg{\rangle}^{\circ},\end{equation}
   where $[x^a]f(x)$ selects the coefficient of $x^a$ in the formal power series f(x), $b(g) = 2g - 2 + n + d/r$ is the Riemann–Hurwitz count of simple ramifications, and the following relations hold:
   \begin{align*}[\mathcal{E}_{a}(z), \mathcal{E}_b(w)] = 2\sinh \left(\frac{aw - bz}{2}\right) \mathcal{E}_{a+b}(z + w), \end{align*}
   and
   (3·10) \begin{equation} \bigg{\langle} \mathcal{E}_0(z) \bigg{\rangle} = \frac{1}{2\sinh(z/2)}, \qquad \qquad \qquad \mathcal{E}_{k} \bigg{\rangle} = 0 = \bigg{\langle} \mathcal{E}_{-k} , \qquad \text{ for } k \gt0.\end{equation}

4. (4) By [ 16 ] we have an algorithm that computes the vacuum expectation explicitly, iteratively commuting the operators $\mathcal{E}$ with negative indices from left to right until they hit the vacuum $\big \rangle$ and vanish. Along the way, they generate a large amount of summands from the commutation relation (intuitively speaking, the number of summands “doubles” at every commutation, although many terms end up vanishing at some further iteration of the algorithm). The algorithm defines a finite sum over P running over the set of Commutation Patterns CP (see [ 16 ]), obtaining

   (3·11) \begin{equation}h_{g, \mu}^{(r)} = \frac{[u^{2g - 2 + n + d/r}]}{\prod \mu_i \cdot r^{d/r}}\frac{1}{2\sinh(u\cdot d/2)} \sum_{P \in CP} \prod_{l=1}^{n-1+d/r} 2\sinh \left((u/2)(|I_{l}^P||L_{l}^P| - |J_{l}^P||K_{l}^P|)\right)\end{equation}

   Here each $I^P_t$ , $J^P_t$ , $K^P_t$ , $I^P_t$ is a sum of a certain subset of the $\mu_i$ .

5. (5) Restricting to genus zero forces to collect the minimal power of u, that is, to substitute all $\sinh (X)$ simply with their arguments X. We obtain:

   (3·12) \begin{equation}h_{0, \mu}^{(r)} = \frac{1}{\prod \mu_i \cdot r^{d/r} \cdot d} \sum_{P \in CP} \prod_{l=1}^{n-1+d/r} (|I_{l}^P||L_{l}^P| - |J_{l}^P||K_{l}^P|).\end{equation}

Putting everything together, one obtains

\begin{align*} & \int_{{\overline{\mathcal{M}}}_{0,n}} \!\!\!\! \frac{\Omega(r,0;\; -\overline{\mu_1}, \dots, -\overline{\mu_n})}{\prod_{i}(1 - \frac{\mu_i}{r}\psi_i)} = \frac{\left(r^{-1}\right)^{2g - 2 + n + 2\frac{|\mu|}{r}}}{\prod \mu_i \cdot d} \\[5pt] & \left(\prod_{i=1}^{n} \frac{\left(\frac{\mu_i}{r}\right)^{[\mu_i]}}{[\mu_i]!} \right)^{-1} \!\!\! \sum_{P \in CP} \prod_{l=1}^{n-1+d/r} (|I_{l}^P||L_{l}^P| - |J_{l}^P||K_{l}^P|).\end{align*}

This concludes the proof fo the Proposition.