2.1 The Borel transform and resurgent functions

Define the Borel transform as the following linear invertible operator acting on formal power seriesFootnote ¹

(2.1) $$ \begin{align} \mathfrak{B} \colon \mathbb{C}{[\![} \hbar {]\!]} \longrightarrow \mathbb{C}{[\![} s {]\!]} , \qquad \hbar^m \longmapsto \frac{s^m}{m!}\,. \end{align} $$

Throughout the text we denote (when possible) quantities in the $\hbar $ -plane with a tilde and the corresponding quantities in the s-plane (also called the Borel plane) with a hat. For instance, writing the starting series as

(2.2) $$ \begin{align} \widetilde{\varphi}(\hbar) = \sum_{m \ge 0} \varphi_m \hbar^m , \end{align} $$

its Borel transform is given by

(2.3) $$ \begin{align} \widehat{\varphi}(s) = \sum_{m \ge 0} \varphi_m \frac{s^m}{m!} \,. \end{align} $$

Due to the division by the factorial, $\widehat {\varphi }$ is ‘more likely’ to be convergent. This observation leads to the definition of Gevrey-1 series.

The above definition is designed in such a way that the Borel transform maps Gevrey-1 series to holomorphic functions at the origin (and vice versa, every holomorphic function at the origin is the image of a Gevrey-1 series). It is then natural to study the analytic continuation of the Borel transform of Gevrey-1 series. The obstruction to analytic continuation is dictated by singularities in the Borel plane. As we are going to see shortly, singularities in the Borel plane play a crucial role in the theory. Among other things, they control the large-order asymptotics of the coefficients of the original series.

In order to simplify our discussion, we restrict ourselves to a specific class of functions: that of simple resurgent functions.

For simplicity, we restrict ourselves to simple resurgent functions with logarithmic singularities only. Including simple poles in the discussion does not pose any particular difficulties, as shown in [Reference Mitschi and Sauzin63]. For a fixed simple resurgent series $\widetilde {\varphi }$ whose Borel transform has a logarithmic singularity at $A \in \mathbb {C}^{\times }$ , we can write locally at $s = A$

(2.4) $$ \begin{align} \widehat{\varphi}(s) = - \frac{S_A}{2\pi} \widehat{\varphi}_A(s - A) \, \log(s - A) + \text{holomorphic at } A , \end{align} $$

where $\widehat {\varphi }_A(s)$ , called the minor at A, is a holomorphic function at $s = 0$ (so that $\widehat {\varphi }_A(s - A)$ is holomorphic at $s = A$ ). Notice that we might want a specific choice of normalisation for $\widehat {\varphi }_A$ , which motivates the introduction of an additional constant $S_A \in \mathbb {C}$ called the Stokes constant. In what follows, we adopt the physics jargon and refer to the singularities A as instanton actions.

Note that the minor $\widehat {\varphi }_A$ is only well-defined once the Stokes constant $S_A$ is fixed. In our context, which lies within the framework of exponential integrals, it will become clear that the $S_A$ are determined and how, so we consider the notion of minor to be well-defined with respect to that choice.

Since $\widehat {\varphi }_A$ is holomorphic at the origin, we can consider its Taylor expansion:

(2.5) $$ \begin{align} \widehat{\varphi}_A(s) = \sum_{m \ge 0} \varphi_{A,m} \frac{s^m}{m!}. \end{align} $$

We look at $\widehat {\varphi }_A$ as the Borel transform of a formal power series

(2.6) $$ \begin{align} \widetilde{\varphi}_A(\hbar) = \sum_{m \ge 0} \varphi_{A,m} \hbar^m. \end{align} $$

The key point so far is that, given a simple resurgent series $\widetilde {\varphi }$ , we can associate to it the data of its instanton actions, Stokes constants, and minors:

(2.7) $$ \begin{align} \widetilde{\varphi} \longrightarrow \big\{(S_A, \widetilde{\varphi}_A) \big\}_{A \in \mathcal{A}} . \end{align} $$

Here $\mathcal {A}$ denotes the set of singularities of the Borel transform of $\widetilde {\varphi }$ . We call this set of data the Borel plane singularity structure of $\widetilde {\varphi }$ .

2.2 Exponential integrals

A common source of simple resurgent functions, both in mathematics and in physics, is that of exponential integrals. This section essentially relies on results from [Reference Delabaere and Howls26]. We refer the reader to [Reference Aniceto, Başar and Schiappa7, section 3] for an in-depth review of the main results and the available literature on the topic, and to [Reference Kontsevich and Soibelman53] for a short treatment. Let $V \colon \mathbb {C} \to \mathbb {C}$ be a holomorphic function (the potential) and $\mu $ a holomorphic $1$ -form on X. We are interested in computing integrals of the form

(2.8) $$ \begin{align} \varphi(\hbar; \gamma) = \int_{\gamma} e^{-\frac{1}{\hbar} V(t)} \mu(t) \end{align} $$

for $\gamma $ a cycle in X. We assume that V is Morse, that is, has only finitely many critical points $(t_{\alpha })_{\alpha \in I}$ with $V"(t_{\alpha }) \ne 0$ , and we assume that the critical values $V_{\alpha } = V(t_{\alpha })$ are pairwise distinct. The examples we have in mind are the r-Airy integrals, for an integer $r \ge 2$ , for which we have

(2.9) $$ \begin{align} V(t) = \frac{t^{r+1}}{r+1} - t , \qquad\quad \mu(t) = dt . \end{align} $$

We are going to associate to each critical point a formal power series as follows. Fix a critical point $t_{\alpha }$ . From the general theory of Morse functions, one can show that for all values of $\theta = \arg (\hbar )$ such that $\theta \ne \arg (V_{\beta } - V_{\alpha })$ there is a well-defined integral cycle $\gamma _{\alpha ,\theta } \subset \mathbb {C}$ , called the Lefschetz thimble or steepest-descent path, passing through $t_{\alpha }$ along which the integral $\varphi (\hbar ;\gamma _{\alpha ,\theta })$ converges. We define the normalised integrals

(2.10) $$ \begin{align} \varphi^{(\alpha)}(\hbar) := \frac{e^{\frac{V_{\alpha}}{\hbar}}}{\sqrt{2\pi \hbar}} \int_{\gamma_{\alpha,\theta}} e^{-\frac{1}{\hbar} V(t)} \mu(t) . \end{align} $$

The function $\varphi ^{(\alpha )}$ is holomorphic in $\hbar $ in each sector defined by the Stokes rays, that is, the rays defined by $\arg (\hbar ) = \arg (V_{\beta } - V_{\alpha })$ . Again from the general theory of Morse functions and the Laplace method, the above integral has a well-defined divergent asymptotic expansion series:

(2.11) $$ \begin{align} \varphi^{(\alpha)}(\hbar) \sim \sum_{m \ge 0} \varphi_{m}^{(\alpha)} \, \hbar^{m} =: \widetilde{\varphi}^{(\alpha)}(\hbar) , \end{align} $$

where the coefficients $\varphi _{m}^{(\alpha )}$ are independent of $\hbar $ . The first coefficient is given by $\varphi _0^{(\alpha )} = V"(t_{\alpha })^{-1/2}$ . Notice that the asymptotics is the same in every sector. As usual, let $\widehat {\varphi }^{(\alpha )}$ denote the Borel transform of $\widetilde {\varphi }^{(\alpha )}$ . The analytic properties of $\widehat {\varphi }^{(\alpha )}$ are summarised as follows.

Theorem 2.3 (Singularity structure for exponential integrals [Reference Delabaere and Howls26, proposition 4.1])

The asymptotic expansion series $\widetilde {\varphi }^{(\alpha )}$ are simple resurgent. More precisely, the Borel transform $\widehat {\varphi }^{(\alpha )}$ has finitely many singularities at $A_{\alpha ,\beta } = V_{\beta } - V_{\alpha }$ of logarithmic type, with behaviour given by

(2.12) $$ \begin{align} \widehat{\varphi}^{(\alpha)}(s) = \frac{\mathsf{S}_{\alpha,\beta}}{2\pi\mathrm{i}} \, \widehat{\varphi}^{(\beta)}(s - A_{\alpha,\beta}) \, \log(s - A_{\alpha,\beta}) + \mathrm{holomorphic\ at}\ A_{\alpha,\beta} . \end{align} $$

Here $\mathsf {S}_{\alpha ,\beta }$ are the integers defined by the intersection of two Lefschetz thimbles: $\mathsf {S}_{\alpha ,\beta } = \gamma _{\alpha ,\theta _+} \cdot \gamma _{\beta ,\theta _- + \pi }$ for $\theta = \arg (A_{\alpha ,\beta })$ .

In other words, exponential integrals naturally generate simple resurgent series $\widetilde {\varphi }^{(\alpha )}$ labelled by critical points of the potential. Moreover, the Borel plane singularity structure of $\widetilde {\varphi }^{(\alpha )}$ is well understood in terms of:

• • instanton actions – difference of critical values for the locations of singularities: $A_{\alpha ,\beta } = V_{\beta } - V_{\alpha }$ ,

• • Stokes constants – intersection numbers of Lefschetz thimbles: $\mathsf {S}_{\alpha ,\beta }$ ,

• • minors – the asymptotic expansion $\widetilde {\varphi }^{(\beta )}$ for the series attached to the singularity at $A_{\alpha ,\beta }$ .

The hypotheses of theorem 2.3 can be generalised to the higher-dimensional setting, where the domain of V is $\mathbb {C}^n$ (see [Reference Delabaere and Howls26]), and to the punctured setting, where the domain of V is $X = \mathbb {C} \setminus \{ p_1,\dots ,p_M \}$ , V is a rational function with possible poles at the punctures, and $\mu $ is a rational $1$ -form on X (see [Reference Kontsevich and Soibelman53]).

As an example, let us apply the above result to the aforementioned generalised Airy integrals. The critical points are given by $V'(t) = t^r - 1 = 0$ , that is $t_{\alpha } = \zeta ^{\alpha }$ for $\zeta = e^{\frac {2\pi \mathrm {i}}{r}}$ and $\alpha = 1,\dots , r$ . The critical values are $V_{\alpha } = - \frac {r}{r+1} \zeta ^{\alpha }$ . According to the above theorem, the Borel transforms $\widehat {\varphi }^{(\alpha )}$ have logarithmic singularities at $A_{\alpha ,\beta } = \frac {r}{r+1} (\zeta ^{\alpha } - \zeta ^{\beta })$ and Stokes constants

(2.13) $$ \begin{align} \mathsf{S}_{\alpha,\beta} = \begin{cases} +1 & \text{if } \alpha> \beta ,\\ -1 & \text{if } \alpha < \beta . \end{cases} \end{align} $$

See [Reference Bertola, Eynard and Harnad15, section 3.2] for a detailed derivation of the intersection numbers for generalised Airy integrals. It should also be noted that the problem of the Stokes phenomenon for the higher Airy functions was addressed in the literature also beyond the exponential integral context, starting directly from the associated ODE. See for instance [Reference Ohyama65, Reference Suzuki73, Reference Turrittin76].

2.3 Large-order asymptotics from the singularity structure

We are now ready to establish the main theorem of this section, which gives the large-order asymptotics of the coefficients of a simple resurgent function $\widetilde {\varphi }$ in terms of the singularity structure of its Borel transform. The following result justifies the term ‘resurgence’ too. Indeed, we have seen that the singularities of the Borel transform lead to new power series. These new series resurge in the original series through the large-order asymptotics of its coefficients. This is essentially an old theorem of Darboux, which relates the large-order asymptotics of the coefficients of an analytic function at the origin to the behaviour near the closest singularity. It is a well-known fact in the physics community, and perhaps not as widely known in the mathematics community. For the reader’s convenience, we provide the proof here, which essentially relies on the application of Cauchy’s theorem in the Borel plane.

Theorem 2.5 (Borel transform method)

Assume that $\widetilde {\varphi }$ is a simple resurgent series starting at order $\hbar ^{\beta _0}$ , with $\beta _0 \in \mathbb {N}$ called the critical exponent:

(2.14) $$ \begin{align} \widetilde{\varphi}(\hbar) = \sum_{m \ge 0} \varphi_{m} \hbar^{m + \beta_0} . \end{align} $$

Assume that its Borel transform $\widehat {\varphi }$ satisfies the following assumptions.

• • Polynomial growth at infinity. There exists $\nu \in \mathbb {R}$ such that $\widehat {\varphi }(s) = \mathrm {O}(s^{\nu })$ as $s \to \infty $ .

• • Singularity structure. The function $\widehat {\varphi }$ has finitely many logarithmic singularities $A_1, \dots , A_n \in \mathbb {C}^{\times }$ around which it behaves as

  (2.15) $$ \begin{align} \widehat{\varphi}(s) = - \frac{S_i}{2\pi} \, \widehat{\varphi}^{(i)}(s - A_i) \, \log(s - A_i) + \mathrm{holomorphic\ at}\ A_i, \end{align} $$
  for some constants $S_i$ and functions $\widehat {\varphi }^{(i)}$ , the minors, which are holomorphic at the origin.

• • Behaviour of the minors. The functions $\widehat {\varphi }^{(i)}$ have a polynomial growth at infinity and have a finite number of singularities avoiding the ray starting at the origin and passing through $A_i$ . Moreover, their Taylor expansions at the origin have critical exponent $\beta _i \in \mathbb {N}$ , that is, $\widehat {\varphi }^{(i)}(s) = \sum _{k = 0}^{\infty } \varphi ^{(i)}_{k} \frac {s^{k + \beta _i}}{(k + \beta _i)!}$ .

Then the large m asymptotics of $\varphi _m$ is given by

(2.16) $$ \begin{align} \varphi_m = \sum_{i = 1}^n \frac{S_i}{2\pi} \frac{\Gamma(m + \beta_0 - \beta_i)}{A_i^{m + \beta_0 - \beta_i}} \left( \sum_{k = 0}^{K} \frac{A_i^{k}}{(m + \beta_0 - \beta_i - 1)^{\underline{k}}} \, \varphi^{(i)}_{k} + \mathrm{O} \bigg( \frac{1}{m^{K+1}} \bigg) \right) . \end{align} $$

Here $(x)^{\underline {k}} := x (x-1) \cdots (x-k+1)$ denotes the falling factorial.

Before proceeding with a proof of the above result, it is instructive to write down explicitly the first terms of equation (2.16). Without loss of generality, suppose that the singularities are ordered as $|A_1| \le \cdots \le |A_n|$ from the closest to the farthest distance from the origin. Then

(2.17) $$ \begin{align} \begin{aligned} \varphi_m & = \frac{S_1}{2\pi} \frac{\Gamma(m + \beta_0 - \beta_1)}{A_1^{m + \beta_0 - \beta_1}} \left( \varphi^{(1)}_{0} + \frac{A_1}{(m + \beta_0 - \beta_1 - 1)} \, \varphi^{(1)}_{1} + \frac{(A_1)^2}{(m + \beta_0 - \beta_1 - 1)^{\underline{2}}} \, \varphi^{(1)}_{2} + \cdots \right) \\ & + \cdots \\ & \phantom{=} + \frac{S_n}{2\pi} \frac{\Gamma(m + \beta_0 - \beta_n)}{A_n^{m + \beta_0 - \beta_n}} \left( \varphi^{(n)}_{0} + \frac{A_n}{(m + \beta_0 - \beta_n - 1)} \, \varphi^{(n)}_{1} + \frac{(A_n)^2}{(m + \beta_0 - \beta_n - 1)^{\underline{2}}} \, \varphi^{(n)}_{2} + \cdots \right). \end{aligned} \end{align} $$

We can see that the asymptotics presents an overall factorial growth, $\Gamma (m + \beta _0 - \beta _i) = \mathrm {O}(m!)$ , and it is ‘organised’ into different exponential growths corresponding to the rows governed by $1/A_i^{m + \beta _0 - \beta _i} = \mathrm {O}(A_i^{-m})$ with overall constant given by $S_i$ . Moreover, the i-th row contains an asymptotic series in $1/m$ which scales as $A_i$ and whose coefficients are the Taylor coefficients of the corresponding minors.

Proof of theorem 2.5

Since $\widehat {\varphi }$ is holomorphic at the origin, we can extract its expansion coefficients through Cauchy’s formula:

$$ \begin{align*} \frac{\varphi_{m}}{(m + \beta_0)!} = \frac{1}{2\pi\mathrm{i}} \oint_{\mathcal{C}_0} \frac{\widehat{\varphi}(s)}{s^{m + \beta_0 + 1}} \, ds . \end{align*} $$

Here $\mathcal {C}_0$ is a small contour around the origin oriented counter-clockwise. We can deform the contour, avoiding the logarithmic cuts starting at each singularity $A_i$ with partial Hankel contours $\mathcal {H}^{(R)}_i$ starting at $A_i$ in the direction $\theta _i := \arg (A_i)$ , connected by arcs whose union is denoted by $\gamma ^{(R)}$ (see figure 2).

Figure 2

Deformation of the contour $\mathcal {C}_0$ to the union of $\gamma ^{(R)}$ and the partial Hankel contours $\mathcal {H}_i^{(R)}$ avoiding the logarithmic cuts.

Here R is the radius of the arcs and is assumed to be greater than $\max _{i=1,\dots ,n}{|A_i|}$ . Taking the limit $R \to +\infty $ gives zero contribution from the contours $\gamma ^{(R)}$ . Indeed, assuming R large enough so that $|\widehat {\varphi }(s)| < C s^{\nu }$ , we find

$$\begin{align*}\left| \frac{1}{2\pi \mathrm{i}} \int_{\gamma_R} \frac{\widehat{\varphi}(s)}{s^{m + \beta_0 + 1}} \, ds \right| \le \frac{C}{R^{m + \beta_0 - \nu}} \end{align*}$$

which tends to zero for $R \to +\infty $ for $m \gg 0$ . Thus we only have to consider the Hankel contours $\mathcal {H}_i$ obtained in the limit $R \to +\infty $ . A simple manipulation of the integral shows that

$$ \begin{align*} \frac{\varphi_{m}}{(m + \beta_0)!} & = \frac{1}{2\pi\mathrm{i}} \sum_{i=1}^n \int_{\mathcal{H}_i} \frac{\widehat{\varphi}(s)}{s^{m + \beta_0 + 1}} \, ds = -\frac{1}{2\pi\mathrm{i}} \sum_{i=1}^n \frac{S_i}{2\pi} \int_{A_i}^{e^{\mathrm{i} \theta_i }\infty} \operatorname{\mathrm{Disc}}\left[ \widehat{\varphi}^{(i)}(s - A_i) \, \log(s - A_i) \right] \frac{ds}{s^{m + \beta_0 + 1}} \\ & = \sum_{i=1}^n \frac{S_i}{2\pi} \int_{A_i}^{e^{\mathrm{i} \theta_i }\infty} \frac{\widehat{\varphi}^{(i)}(s - A_i)}{s^{m + \beta_0 + 1}} \, ds = \sum_{i=1}^n \frac{S_i}{2\pi} e^{-\mathrm{i}\theta_i (m+\beta_0)} \int_{0}^{+\infty} \frac{\widehat{\varphi}^{(i)}(e^{\mathrm{i} \theta_i} s)}{(s + |A_i|)^{m + \beta_0 + 1}} \, ds . \end{align*} $$

In the second equality, we used the behaviour of $\widehat {\varphi }$ at $A_i$ given in equation (2.15). In the third equality, we used the fact that the discontinuity is completely governed by the logarithm and gives a contribution of $-2\pi \mathrm {i}$ . The last equality is simply a change of variable in the integral. The goal now is to approximate $\widehat {\varphi }^{(i)}$ with the first K terms of its Taylor expansion, and estimate the error using Watson’s lemma. Define the tail of $\widehat {\varphi }^{(i)}$ by removing the first K terms of its Taylor expansion around the origin: $\widehat {\varphi }^{(i)}(s; K) := \widehat {\varphi }^{(i)}(s) - \sum _{k=0}^{K} \varphi ^{(i)}_{k} \frac {s^{k + \beta _i}}{(k + \beta _i)!}$ . Then

(2.18) $$ \begin{align} \frac{\varphi_{m}}{(m + \beta_0)!} - \sum_{i=1}^n \frac{S_i}{2\pi} e^{-\mathrm{i}\theta_i (m + \beta_0)} & \sum_{k=0}^K \frac{\varphi^{(i)}_k e^{\mathrm{i} \theta_i (k + \beta_i)}}{(k + \beta_i)!} \int_{0}^{+\infty} \frac{s^{k + \beta_i}}{(s + |A_i|)^{m + \beta_0 + 1}} \, ds \nonumber\\ &\qquad\qquad = \sum_{i=1}^n \frac{S_i}{2\pi} e^{-\mathrm{i}\theta_i (m + \beta_0)} \int_{0}^{+\infty} \frac{\widehat{\varphi}^{(i)}(e^{\mathrm{i} \theta_i} s;K)}{(s + |A_i|)^{m + \beta_0 + 1}} \, ds . \end{align} $$

The left-hand side of the equation above can be explicitly computed using the Euler-type integral $\frac {1}{\Gamma (\mu + 1)} \int _{0}^{+\infty } \frac {s^{\mu }}{(s + A)^{\nu + 1}} ds = A^{\mu -\nu } \, \frac { \Gamma (\nu - \mu ) }{ \Gamma (\nu + 1) }$ valid for $A> 0$ and $\nu> \mu > 0$ . We thus find

$$ \begin{align*} \frac{\varphi_{m}}{(m + \beta_0)!} - \frac{1}{(m + \beta_0)!} \sum_{i=1}^n \frac{S_i}{2\pi} \sum_{k=0}^K \varphi^{(i)}_k \frac{\Gamma(m + \beta_0 - \beta_i - k)}{A_i^{m + \beta_0 - \beta_i - k}} . \end{align*} $$

We are now left with an estimate of the right-hand side of (2.18). In the i-th term of the sum, let us set $s = |A_i|(e^t - 1)$ . We obtain

$$ \begin{align*} \sum_{i=1}^n \frac{S_i}{2\pi} e^{-\mathrm{i}\theta_i (m+\beta_0)} \int_{0}^{+\infty} \frac{\widehat{\varphi}^{(i)}(e^{\mathrm{i} \theta_i} s;K)}{(s + |A_i|)^{m + \beta_0 + 1}} \, ds = \sum_{i=1}^n \frac{S_i}{2\pi} \frac{1}{A_i^{m + \beta_0}} \int_{0}^{+\infty} e^{-t(m + \beta_0)} \widehat{\varphi}^{(i)}( A_i (e^t - 1) ;K) \, dt . \end{align*} $$

We can now apply Watson’s lemma 2.6 to estimate the integral: since $\widehat {\varphi }^{(i)}( A_i(e^t - 1) ;K) = \mathrm {O}(t^{K+\beta _i+1})$ as $t \to 0^+$ (recall that $\widehat {\varphi }^{(i)}(s;K)$ is $\widehat {\varphi }^{(i)}(s)$ pruned of its first K expansion coefficients at the origin) and $\widehat {\varphi }^{(i)}( A_i(e^t - 1) ;K)$ behaves at most exponentially at infinity, we find

$$ \begin{align*} \int_{0}^{+\infty} e^{-t(m + \beta_0)} \widehat{\varphi}^{(i)}( A_i(e^t - 1) ;K) \, dt = \mathrm{O}\bigg( \frac{1}{(m + \beta_0)^{K+\beta_i+2}} \bigg) = \mathrm{O}\bigg( \frac{1}{m^{K+\beta_i+2}} \bigg) . \end{align*} $$

After multiplication by $(m+\beta _0)!$ and some algebraic manipulation, we find the thesis.

The estimate for the error term in the asymptotics was a consequence of Watson’s lemma for Laplace-type integrals, which we report here for the reader’s convenience. See [Reference Olver67] for more details.

Lemma 2.6 (Watson)

Let $f \colon \mathbb {R}_{+} \to \mathbb {C}$ be a continuous function satisfying the following assumptions.

• • Polynomial growth at the origin. There exists an integer K such that $f(t) = \mathrm {O}( t^{d} )$ as $t \to 0^+$ .

• • Exponential growth at infinity. There exists $\nu \in \mathbb {R}$ such that $f(t) = \mathrm {O}( e^{t\nu } )$ as $t \to + \infty $ .

Then, as $x \to +\infty $ :

(2.19) $$ \begin{align} \int_{0}^{+\infty} e^{-tx} \, f(t) \, dt = \mathrm{O}\bigg( \frac{1}{x^{d+1}} \bigg) . \end{align} $$

Remark 2.7 (Gevrey-2 series)

In the next sections, we are going to apply the Borel transform method to formal power series with vanishing even or odd coefficients. In other words, up to relabelling of the coefficients, we have

(2.20) $$ \begin{align} \widetilde{\varphi}(\hbar) = \sum_{g \ge 0} \varphi_{g} \hbar^{2g + \beta_0} . \end{align} $$

Formal power series with $\varphi _{g} = \mathrm {O}((2g)! a^{-g})$ are also known as Gevrey-2 series. In this case, the large g asymptotics, that is equation (2.16), reads

(2.21) $$ \begin{align} \varphi_{g} = \sum_{i = 1}^n \frac{S_i}{2\pi} \frac{\Gamma(2g + \beta_0 - \beta_i)}{A_i^{2g + \beta_0 - \beta_i}} \left( \sum_{k = 0}^{K} \frac{A_i^{k}}{(2g + \beta_0 - \beta_i - 1)^{\underline{k}}} \, \varphi^{(i)}_{k} + \mathrm{O} \bigg( \frac{1}{g^{K+1}} \bigg) \right) . \end{align} $$

We also remark that the above theorem can be generalised to the case of $\widehat {\varphi }$ admitting simple poles. Moreover, one can allow for the critical exponents $\beta _0$ and $\beta _i$ to be any real numbers by suitably adapting the definition of the Borel transform.

2.4 Algebraic properties of resurgent series

So far we have established that the large-order asymptotics of the coefficients of a simple resurgent series is entirely determined by its Borel singularity structure. One advantage of working within the class of simple resurgent functions is that they exhibit favourable properties under addition and multiplication. As will become evident in the subsequent sections, this aspect is of particular significance to us, since our goal is to deduce the singularity structure of specific power series. These series are derived as sums of products of simple resurgent ‘building blocks’, whose singularity structure is completely under control. For all these reasons, let us discuss some algebraic properties of simple resurgent series.

The Borel transform is by definition a linear operation. On the other hand, the Borel transform does not respect the Cauchy product of formal power series, but rather sends it to the convolution product.

Definition 2.8. Given $\widehat {\phi }, \widehat {\psi } \in \mathbb {C}{[\![} s {]\!]} $ , their convolution product is

(2.22) $$ \begin{align} \widehat{\phi} \ast \widehat{\psi} := \mathfrak{B} \big[ \widetilde{\phi} \cdot \widetilde{\psi} \big] , \end{align} $$

with $\widehat {\phi } = \mathfrak {B}[ \widetilde {\phi } ]$ and $\widehat {\psi } = \mathfrak {B}[ \widetilde {\psi } ]$ .

It is easy to check that, if both $\widehat {\phi }$ and $\widehat {\psi }$ converge in a disc of radius R around the origin, then for all s in such a disc we find $(\widehat {\phi } \ast \widehat {\psi })(s) = \frac {d}{ds} \int _{0}^{s} \widehat {\phi }(\sigma ) \widehat {\psi }(s-\sigma ) \, d\sigma $ . In other words, $\widehat {\phi } \ast \widehat {\psi }$ is (the derivative of) the usual convolution product of functions, hence the name. The next two theorems precisely determine how the analytic properties of simple resurgent functions transform under convolution.

Theorem 2.10 (Product of simple resurgent series [Reference Mitschi and Sauzin63, theorem 6.83])

Let $\mathcal {A}$ and $\mathcal {B}$ be nonempty closed discrete subsets of $\mathbb {C}^{\times }$ , and denote $\mathcal {C} := \mathcal {A} \cup \mathcal {B} \cup \left ( \mathcal {A} + \mathcal {B} \right )$ , with $\mathcal {A} + \mathcal {B} := \{ A + B | A \in \mathcal {A}, \, B \in \mathcal {B}\}$ . Let $\widetilde {\phi }$ and $\widetilde {\psi }$ be simple resurgent series with Borel plane singularities in $\mathcal {A}$ and $\mathcal {B}$ respectively, such that

(2.23) $$ \begin{align} \begin{aligned} \widehat{\phi}(s) & = - \frac{S_{A}}{2\pi} \widehat{\phi}_{A}(s - A) \, \log(s - A) + \text{holomorphic at } A && \qquad \forall A \in \mathcal{A} , \\ \widehat{\psi}(s) & = - \frac{S_{B}}{2\pi} \widehat{\psi}_{B}(s - B) \, \log(s - B) + \text{holomorphic at } B &&\qquad \forall B \in \mathcal{B} . \end{aligned} \end{align} $$

If $\mathcal {C}$ is closed and discrete, then $\widetilde {\phi } \cdot \widetilde {\psi }$ is also a simple resurgent series with Borel plane singularities in $\mathcal {C}$ and

(2.24) $$ \begin{align} \begin{aligned} \big( \widehat{\phi} \ast \widehat{\psi} \big)(s) & = - \frac{S_{A}}{2\pi} \big( \widehat{\phi}_{A} \ast \widehat{\psi} \big)(s - A) \, \log(s - A) + \text{holomorphic at } A && \qquad \forall A \in \mathcal{A} , \\ \big( \widehat{\phi} \ast \widehat{\psi} \big)(s) & = - \frac{S_{B}}{2\pi} \big( \widehat{\phi} \ast \widehat{\psi}_{B} \big)(s - B) \, \log(s - B) + \text{holomorphic at } B && \qquad \forall B \in \mathcal{B} . \end{aligned} \end{align} $$

3.1 Airy functions

Consider the ( $\hbar $ -dependent) Airy ODE:

(3.1) $$ \begin{align} \bigg( \bigg( \hbar \frac{d}{dx} \bigg)^2 - x \bigg) \psi(x;\hbar) = 0 . \end{align} $$

A general solution is given by the Airy integral: $\int _{\gamma } e^{-\frac {1}{\hbar }V(t,x)} dt$ , with $V(t,x) := \frac {t^3}{3} - xt$ and $\gamma $ a properly chosen integration contour. In this section, we are interested in a basis of formal (or WKB) solutions, that is, the asymptotic expansion of the (properly normalised) Airy integrals with $\gamma $ being a Lefschetz thimble. In the context of the Lax operator formalism, these are called the wave functions or Baker–Akhiezer functions. Following the prescription outlined in subsection 2.2, such formal solutions and their derivatives are given by

(3.2) $$ \begin{align} \begin{aligned} \psi_{\pm}(x;\hbar) &= \frac{e^{\mp \frac{V(x)}{\hbar}}}{\sqrt{2}} x^{-1/4} \sum_{k = 0}^{\infty} \frac{1}{864^k}\frac{(6k)!}{(2k)!(3k)!} \left(\mp \frac{\hbar}{V(x)} \right)^{k} , \\ \psi_{\pm}'(x;\hbar) &= \pm \frac{e^{\mp \frac{V(x)}{\hbar}}}{\sqrt{2}} x^{1/4} \sum_{k = 0}^{\infty} \frac{1}{864^k}\frac{(6k)!}{(2k)!(3k)!} \frac{1+6k}{1-6k} \left(\mp \frac{\hbar}{V(x)} \right)^{k} , \\ \end{aligned} \end{align} $$

where $\pm V(x) := \mp \frac {2}{3} x^{3/2}$ are the critical values of the potential at the critical point $t = \pm x^{1/2}$ . The reader can recognise $\psi _{-}$ and $\psi _{+}$ as the asymptotic expansions in the appropriate regions of the Airy and Bairy functions respectively. The normalisation is fixed so that the Wronskian is worth $\psi _- \psi _+' - \psi _-' \psi _+ = 1$ .

Consider the Borel transform of the Airy functions. To start with, let us separate the exponential part by setting $\psi _{\pm }(x;\hbar ) =: \exp (\mp \frac {1}{\hbar } V(x)) \, \widetilde {\psi }_{\pm }(x;\hbar )$ , so that $\widetilde {\psi }_{\pm }$ is a formal power series in $\hbar $ . Similarly for the derivatives. Then the Borel transforms of $\widetilde {\psi }_{\pm }$ and $\widetilde {\psi }_{\pm }'$ are expressed in terms of Gauss hypergeometric functions as:

(3.3) $$ \begin{align} \widehat{\psi}_{\pm}(x;s) = \frac{x^{-1/4}}{\sqrt{2}} \, \vphantom{F}_{2}F_{1} \left( \tfrac{5}{6},\tfrac{1}{6}; 1; \pm \frac{s}{A(x)} \right) , \qquad \widehat{\psi}_{\pm}'(x;s) = \pm \frac{x^{1/4}}{\sqrt{2}} \, \vphantom{F}_{2}F_{1} \left( \tfrac{7}{6},-\tfrac{1}{6}; 1; \pm \frac{s}{A(x)} \right) , \end{align} $$

where $\pm A(x) := \pm \frac {4}{3} x^{3/2}$ are the instanton actions.

From the above explicit expression, we deduce that the Borel transforms $\widehat {\psi }_{\pm }$ and $\widehat {\psi }_{\pm }'$ converge in a disc of radius $|A(x)|$ centred at the origin, and they can be extended analytically to functions with logarithmic singularities at $s = \pm A(x)$ . From a direct analysis of the integral representation of the Gauss hypergeometric functions, one can prove the following behaviour at the singularities:

(3.4) $$ \begin{align} \begin{aligned} \widehat{\psi}_{\pm}(x;s) & = - \frac{S}{2\pi} \, \widehat{\psi}_{\mp}\big( x;s \mp A(x) \big) \log\big( s \mp A(x) \big) + \text{holomorphic at} \pm A(x), \\ \widehat{\psi}_{\pm}'(x;s) & = - \frac{S}{2\pi} \, \widehat{\psi}_{\mp}'\big( x;s \mp A(x) \big) \log\big( s \mp A(x) \big) + \text{holomorphic at} \pm A(x) , \end{aligned} \end{align} $$

where $S = 1$ is the Stokes constant. In other words, $\widetilde {\psi }_{\pm }$ and $\widetilde {\psi }_{\pm }'$ are simple resurgent functions with finitely many logarithmic singularities, with Stokes constants and minors that are fully under control. Notice that the above behaviour also follows from theorem 2.3 and the (properly normalised) integral representation of $\psi _{\pm }$ and $\psi _{\pm }'$ , for which the explicit expressions of the Borel transforms are not needed.

In the remaining part of this section, we often keep the generic notation S and $A(x)$ to emphasise how their actual values are not crucial in the following analysis. This fact will allow us to compute the large genus asymptotics of $\Theta $ -class intersection numbers in section 4 by simply changing the values of S and $A(x)$ to the appropriate ones.

3.2 Airy correlators

The Airy ODE can be rewritten as a first-order $2 \times 2$ system in terms of the companion matrix $\mathcal {D}$ :

(3.5) $$ \begin{align} \hbar \frac{d}{dx} \Psi(x;\hbar) = \mathcal{D}(x) \Psi(x;\hbar) , \qquad \mathcal{D}(x) := \begin{pmatrix} 0 & 1 \\ x & 0 \end{pmatrix} , \qquad \Psi(x;\hbar) = \begin{pmatrix} \psi_-(x;\hbar) & \psi_+(x;\hbar) \\ \psi_-'(x;\hbar) & \psi_+'(x;\hbar) \end{pmatrix} . \end{align} $$

We refer to $\Psi $ as the wave matrix. Notice that with the chosen normalisation of the Wronskian, we have $\det {\Psi } = 1$ . We are now interested in computing the correlators associated to the above differential system. To this end, define the matrix

(3.6) $$ \begin{align} M(x;\hbar) := \Psi(x;\hbar) E \Psi^{-1}(x;\hbar) , \qquad\qquad E := \frac{1}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}. \end{align} $$

Notice that E is a generator of the Cartan subalgebra of $\mathfrak {sl}_2(\mathbb {C})$ . More explicitly M is given by

(3.7) $$ \begin{align} M = \begin{pmatrix} \frac{1}{2}(\psi_+' \psi_- + \psi_+ \psi_-') & -\psi_+ \psi_- \\ \psi_+' \psi_-' & - \frac{1}{2}(\psi_+' \psi_- + \psi_+ \psi_-') \end{pmatrix} = \begin{pmatrix} \frac{1}{2}(\widetilde{\psi}_+' \widetilde{\psi}_- + \widetilde{\psi}_+ \widetilde{\psi}_-') & - \widetilde{\psi}_+ \widetilde{\psi}_- \\ \widetilde{\psi}_+' \widetilde{\psi}_-' & - \frac{1}{2}(\widetilde{\psi}_+' \widetilde{\psi}_- + \widetilde{\psi}_+ \widetilde{\psi}_-') \end{pmatrix}. \end{align} $$

In the second equation, we used the simple but crucial fact that the exponential factors $e^{\pm \frac {1}{\hbar }V(x)}$ cancel out in quadratic expressions involving functions labelled by ‘ $+$ ’ and ‘ $-$ ’. In other words, we can substitute the $\psi $ ’s with their respective $\widetilde {\psi }$ ’s. Thus, M is a matrix-valued formal power series in $\hbar $ .

We remark that, from a more geometric point of view, $\Psi $ can be identified with a formal flat section for the $\mathfrak {sl}_2(\mathbb {C})$ -connection $\nabla = \hbar d - \mathcal {D} dx$ on the trivial bundle over the projective line. Besides, M can be seen as a flat section of the adjoint bundle, that is, it satisfies the differential system $\hbar dM - [\mathcal {D},M] dx = 0$ .

We are now ready to define the correlators associated to the Airy differential system. To the best of our knowledge, correlators in the context of ODEs appeared for the first time in [Reference Bergère and Eynard11], inspired by the analogy with random matrix models (see for instance [Reference Dyson35, Reference Mehta58]). Later, they were rediscovered in the context of tau-functions of the KdV hierarchy in [Reference Bertola, Dubrovin and Yang12] and from Fock space formalism (in the form of lemma 3.5) in [Reference Zhou81].

Definition 3.1 (Airy correlators)

Define the n-point Airy correlators $W_n$ asFootnote ²

(3.8) $$ \begin{align} \begin{aligned} & W_1(x_1;\hbar) := - \frac{1}{\hbar} \operatorname{\mathrm{Tr}}{\big( \mathcal{D}(x_1) M(x_1;\hbar) \big)} , \\ & W_n(x_1,\dots,x_n;\hbar) := (-1)^{n-1} \sum_{\sigma \in S_n^{\text{cyc}}} \frac{\operatorname{\mathrm{Tr}}{ \big( \prod_{i=1}^n M(x_{\sigma^i(1)};\hbar) } \big) }{\prod_{i=1}^n ( x_{i} - x_{\sigma(i)} )} \qquad\quad \text{for }n \ge 2 . \end{aligned} \end{align} $$

We refer to the above formula as the determinantal formula. Here $S_n^{\text {cyc}}$ denotes the set of cyclic permutations, that is the subset of $S_n$ consisting of all permutations with a single cycle of length n.

It can be shown that $W_n$ is regular along $x_i = x_j$ (see for instance [Reference Bergere, Borot and Eynard10]). Moreover, as M is a formal power series in $\hbar $ , so is $W_{n}$ . The next theorem makes the $\hbar $ -dependence more precise, and most importantly it establishes an enumerative-geometric interpretation of the expansion coefficients: they correspond to $\psi $ -class intersection numbers. For this reason, the $\hbar $ -expansion of $W_{n}$ is also called a genus expansion. A proof of the theorem can be deduced from the original work of Kontsevich [Reference Kontsevich52], together with the fact that correlators of matrix models are computed by determinantal formulae (a fact already hinted at in [Reference Eynard and Orantin42, section 10.5]). As mentioned above, the theorem was later rediscovered in [Reference Bertola, Dubrovin and Yang12] and [Reference Zhou81] from different perspectives.

Theorem 3.2 (Genus expansion of the Airy correlators)

The n-point Airy correlators $W_{n}$ admit the following $\hbar $ -expansion:

(3.9) $$ \begin{align} W_{n}(\boldsymbol{x};\hbar) = \sum_{g = 0}^{\infty} \hbar^{2g-2+n} \, W_{g,n}(\boldsymbol{x}) . \end{align} $$

Moreover, $W_{g,n}$ stores $\psi $ -class intersection numbers: if $2g-2+n>0$ ,

(3.10) $$ \begin{align} W_{g,n}(\boldsymbol{x}) = (-2)^{-(2g-2+n)} \sum_{\substack{ d_{1},\dots,d_n \ge 0 \\ |d| = 3g-3+n }} \langle{ \tau_{d_1} \cdots \tau_{d_n}}\rangle \prod_{i=1}^n \frac{(2d_i+1)!!}{2 \, x_i^{d_i+3/2}} . \end{align} $$

Example 3.3 (1-point correlators)

A direct computation shows that

$$ \begin{align*} \begin{aligned} W_1(x;\hbar) &= \frac{x \, \widetilde{\psi}_- \widetilde{\psi}_+ - \widetilde{\psi}_-' \widetilde{\psi}_+'}{\hbar} = \frac{1}{\hbar} x^{1/2} - \hbar \tfrac{1}{32} x^{-5/2} - \hbar^3 \tfrac{105}{2048} x^{-11/2} - \hbar^5 \tfrac{25025}{65536} x^{-17/2} + \mathrm{O}(\hbar^7) . \end{aligned} \end{align*} $$

In particular, the above expansion agrees with the intersection numbers:

$$ \begin{align*} W_{0,1}(x) &= x^{1/2} , \qquad & W_{1,1}(x) &= - \tfrac{1}{32} x^{-5/2} , \qquad & W_{2,1}(x) &= - \tfrac{105}{2048} x^{-11/2} , \qquad & W_{3,1}(x) &= - \tfrac{25025}{65536} x^{-17/2} , \\ & & \langle{\tau_1}\rangle &= \tfrac{1}{24} , & \langle{\tau_4}\rangle &= \tfrac{1}{1152} , & \langle{\tau_7}\rangle &= \tfrac{1}{82944} . \end{align*} $$

More generally, an explicit computation of the expansion coefficients of $W_1$ from those of the Airy and Bairy functions shows the well-known closed formula $\langle {\tau _{3g-2}}\rangle = \frac {1}{24^g g!}$ for $1$ -point intersection numbers.

Remark 3.4 (Connection with topological recursion and the spectral curve)

The reader familiar with topological recursion can recognise $W_{g,n}$ as essentially the topological recursion correlators computed from the Airy spectral curve $(\mathbb {P}^1, x(z) = z^2, y(z) = z, \omega _{0,2}(z_1,z_2) = \frac {dz_1 dz_2}{(z_1 - z_2)^2})$ . More precisely:

(3.11) $$ \begin{align} \omega_{g,n}(\boldsymbol{z}) = W_{g,n}(\boldsymbol{x}) \, dx_1 \cdots dx_n \big|_{x_i = z_i^2} . \end{align} $$

This can be explained by the fact that both correlators defined through determinantal formulae and topological recursion satisfy the same loop equations [Reference Bergère and Eynard11, Reference Borot, Eynard and Orantin17]. We also emphasise that the wave functions, and consequently the correlators, are defined up to a choice of square root of x. In other words, they are well-defined on the spectral curve $\{x = y^2\}$ . Throughout the rest of the section, we will work with a fixed choice of square root of x, often denoted as $\sqrt {x} = z$ .

The next result gives an alternative expression for the n-point Airy correlators in terms of the Airy kernel, which plays an important role in the Tracy–Widom law and in extreme value statistics [Reference Tracy and Widom74].

Lemma 3.5 (Determinantal formula in kernel form)

For $n \ge 2$ , the n-point Airy correlators are given by the two equivalent expressions

(3.12) $$ \begin{align} W_n(\boldsymbol{x};\hbar) &= (-1)^{n-1} \sum_{\sigma \in S_n^{\text{cyc}}} \prod_{i = 1}^n K_{+,-}(x_i,x_{\sigma(i)};\hbar) \end{align} $$

(3.13) $$ \begin{align} &= (-1)^{n-1} \sum_{\sigma \in S_n^{\text{cyc}}} \prod_{i = 1}^n K_{-,+}(x_i,x_{\sigma(i)};\hbar) \end{align} $$

where $K_{\pm ,\mp }(x,y;\hbar )$ , known as the Airy kernels, are defined as

(3.14) $$ \begin{align} K_{\pm,\mp}(x,y;\hbar) := \frac{\widetilde{\psi}_{\pm}'(x;\hbar) \widetilde{\psi}_{\mp}(y;\hbar) - \widetilde{\psi}_{\pm}(x;\hbar) \widetilde{\psi}_{\mp}'(y;\hbar)}{x-y}. \end{align} $$

Moreover, the kernels are related by the parity relation $K_{-,+}(x,y;\hbar ) = - K_{+,-}(x,y;-\hbar )$ .

Proof. The two expressions for $W_n$ are equivalent, due to the cyclicity of the sum and the symmetry property $K_{+,-}(x,y;\hbar ) = K_{-,+}(y,x;\hbar )$ . Thus, we only need to prove one equality. Let us start from the definition of the n-point correlators, equation (3.8). Since the formula is invariant under shifts of M by matrices proportional to the identity [Reference Eynard and Mitsios40, appendix A], we can substitute M by

$$\begin{align*}M - \tfrac{1}{2} \, \mathrm{Id} = \begin{pmatrix} \widetilde{\psi}_+ \widetilde{\psi}_-' & - \widetilde{\psi}_+ \widetilde{\psi}_- \\ \widetilde{\psi}_+' \widetilde{\psi}_-' & - \widetilde{\psi}_+' \widetilde{\psi}_- \end{pmatrix} = \begin{pmatrix} \widetilde{\psi}_+ \\ \widetilde{\psi}_+' \end{pmatrix} \otimes \begin{pmatrix} \widetilde{\psi}_-' \\ - \widetilde{\psi}_- \end{pmatrix} . \end{align*}$$

Here $u \otimes v$ denotes the outer product of two vectors. Inserting the above expression in the determinantal formula and using the identity $\operatorname {\mathrm {Tr}}( \prod _{k=1}^{n} u_{k} \otimes v_k ) = \prod _{k=1}^{n} \langle {v_k,u_{k+1}}\rangle $ (with the identification $u_{n+1} = u_1$ ) yields the thesis. To conclude, the relations $\psi _{-}(x;\hbar ) = \psi _{+}(x;-\hbar )$ and $\psi _{-}'(x;\hbar ) = - \psi _{+}'(x;-\hbar )$ imply the parity property.

3.3 Singularity structure and large genus asymptotics

The goal of this section is to understand the large genus behaviour of the coefficients in the $\hbar $ -expansion of the correlators via the Borel transform method. Since the singularity structure of the wave functions is well understood (see equation (3.4)), one can deduce a similar statement for the correlator $W_n$ using the algebraic properties of the Borel transform, as explained in subsection 2.4. In the following analysis we assume $n \ge 2$ . The case $n = 1$ can be considered separately and leads to the same result. Moreover, we assume $(x_1,\dots ,x_n)$ to be in generic position, so that the instanton actions lie on distinct rays issuing from the origin, therefore realising the conditions of remark 2.11.

Proposition 3.6 (Singularity structure of the Airy correlators)

The Borel transform $\widehat {W}_{n}(\boldsymbol {x};s)$ of the n-point Airy correlator is simple resurgent. More precisely, on the principal sheet $\widehat {W}_{n}$ has $2n$ logarithmic singularities located at $s = \pm A(x_i)$ and such that

(3.15) $$ \begin{align} \widehat{W}_{n}(\boldsymbol{x};s) = - \frac{S}{2\pi} \, \widehat{W}^{(\pm,i)}_{n}\big( \boldsymbol{x}; s \mp A(x_i) \big) \, \log\big( s \mp A(x_i) \big) + \mathrm{holomorphic\ at}\ \pm A(x_i) , \end{align} $$

where:

• • $A(x) = \frac {4}{3} x^{3/2}$ and $S = 1$ are the instanton action and the Stokes constant associated to the Airy functions,

• • the minor $\widehat {W}^{(\pm ,i)}_{n}$ is the Borel transform of the formal power series

  (3.16) $$ \begin{align} W^{(\pm,i)}_{n}( \boldsymbol{x}; \hbar ) := (-1)^{n-1} \sum_{\sigma \in S_n^{\text{cyc}}} K_{\mp,\mp}(x_i,x_{\sigma(i)};\hbar) \prod_{j \neq i} K_{\pm,\mp}(x_j,x_{\sigma(j)};\hbar) , \end{align} $$
  where the kernels $K_{\pm ,\pm }$ are given by
  (3.17) $$ \begin{align} K_{\pm,\pm}(x,y;\hbar) := \frac{\widetilde{\psi}_{\pm}'(x;\hbar) \widetilde{\psi}_{\pm}(y;\hbar) - \widetilde{\psi}_{\pm}(x;\hbar) \widetilde{\psi}_{\pm}'(y;\hbar)}{x-y} . \end{align} $$

Moreover, the kernels are related by the parity relation $K_{-,-}(x,y;\hbar ) = - K_{+,+}(x,y;-\hbar )$ . Hence, the analogous relation holds for the correlators: $W^{(-,i)}_{n}( \boldsymbol {x}; \hbar ) = (-1)^n \, W^{(+,i)}_{n}( \boldsymbol {x}; -\hbar )$ .

Remark 3.7 (Minors in matrix form and $\mathbb {Z}/2\mathbb {Z}$ -symmetry)

In matrix formulation, the minors are expressed as

(3.18) $$ \begin{align} W^{(\pm,i)}_{n}( \boldsymbol{x}; \hbar ) = (-1)^{n-1} \sum_{\sigma \in S_n^{\text{cyc}}} \frac{\operatorname{\mathrm{Tr}}{ \big( M_{\mp}(x_i;\hbar) \prod_{j=1}^{n-1} M(x_{\sigma^j(i)};\hbar) } \big) }{\prod_{j=1}^n ( x_{j} - x_{\sigma(j)} )} , \end{align} $$

where M and $M_{\pm }$ are given by

(3.19) $$ \begin{align} M = \begin{pmatrix} \frac{1}{2}(\widetilde{\psi}_+' \widetilde{\psi}_- + \widetilde{\psi}_+ \widetilde{\psi}_-') & \widetilde{\psi}_+ \widetilde{\psi}_- \\ \widetilde{\psi}_+' \widetilde{\psi}_-' & - \frac{1}{2}(\widetilde{\psi}_+' \widetilde{\psi}_- + \widetilde{\psi}_+ \widetilde{\psi}_-') \end{pmatrix} , \qquad M_{\pm} = \begin{pmatrix} \widetilde{\psi}^{\prime}_{\pm}\widetilde{\psi}_{\pm} & - ( \widetilde{\psi}_{\pm} )^2 \\ ( \widetilde{\psi}^{\prime}_{\pm} )^2 & - \widetilde{\psi}^{\prime}_{\pm}\widetilde{\psi}_{\pm} \end{pmatrix}. \end{align} $$

From these expressions, it is easy to show that $x_j^{1/2} W^{(\pm ,i)}_{n}$ is an even function of $x_j^{1/2}$ for all $j \ne i$ . This follows from the fact that $x^{1/4} \psi _{\pm }$ is mapped to $x^{1/4} \psi _{\mp }$ under $x^{1/2} \mapsto -x^{1/2}$ , and similarly for the derivatives. Thus $x^{1/2} M$ is even in $x^{1/2}$ . With a similar argument, $x_i^{1/2} W^{(\pm ,i)}_{n}$ is mapped to $x_i^{1/2} W^{(\mp ,i)}_{n}$ under $x_i^{1/2} \mapsto -x_i^{1/2}$ .

Since $W^{(\pm ,i)}_{n}$ are related by the parity relation, we simply consider $W_n^{(+,i)}$ in what follows. To simplify the notation, we drop the ‘ $+$ ’ symbol from the superscript. We can now apply the Borel transform method 2.5 to deduce the large genus asymptotics of the expansion coefficients of $W_n$ from the knowledge of its Borel singularity structure.

Proposition 3.8 (Large genus asymptotics of the Airy correlators)

For every K, the large genus asymptotics of the expansion coefficients of the n-point Airy correlators is given by

(3.20) $$ \begin{align} W_{g,n}(\boldsymbol{x}) = \frac{S}{\pi} \sum_{i=1}^n \frac{\Gamma(2g-2+n)}{A(x_i)^{2g-2+n}} \left( \sum_{k = 0}^K \frac{A(x_i)^{k}}{(2g-3+n)^{\underline{k}}} W^{(i)}_{k,n}(\boldsymbol{x}) + \mathrm{O}\bigg( \frac{1}{g^{K+1}} \bigg) \right) , \end{align} $$

where:

• • $A(x) = \frac {4}{3} x^{3/2}$ and $S = 1$ are the instanton action and the Stokes constant associated to the Airy functions,

• • $W^{(i)}_{k,n}$ is the coefficient of $\hbar ^k$ in the expansion of $W_n^{(i)}$ .

Moreover, the leading term is explicitly given by

(3.21) $$ \begin{align} W_{g,n}(\boldsymbol{x}) = \frac{(-1)^n}{4\pi} \, \frac{\Gamma(2g-2+n)}{\big( \frac{4}{3} \big)^{2g-2+n}} \Bigg( \sum_{\substack{ d_1,\dots,d_n \ge 0 \\ |d| = 3g-3+n}} \prod_{i=1}^n \frac{1}{x_i^{d_i+3/2}} + \mathrm{O}\big( g^{-1} \big) \Bigg) . \end{align} $$

Proof. As a consequence of the Borel transform method theorem 2.5 (see also remark 2.7) and the proposition above, we find

$$\begin{align*}W_{g,n}(\boldsymbol{x}) = \frac{S}{2\pi} \sum_{i=1}^n \frac{\Gamma(2g-2+n)}{A(x_i)^{2g-2+n}} \left( \sum_{k = 0}^K \frac{A(x_i)^k}{(2g-3+n)^{\underline{k}}} \left( W^{(+,i)}_{k,n}(\boldsymbol{x}) + (-1)^{k+n} W^{(-,i)}_{k,n}(\boldsymbol{x}) \right) + \mathrm{O}\bigg( \frac{1}{g^{K+1}} \bigg) \right). \end{align*}$$

The parity relation implies $W^{(-,i)}_{k,n}(\boldsymbol {x}) = (-1)^{n+k} \, W^{(+,i)}_{k,n}(\boldsymbol {x})$ , hence the first part of the proposition.

As for the leading term, it is determined by the coefficient of $\hbar ^0$ in the kernels. From the definition of the kernels and the asymptotic expansion of the Airy functions, we see that the first term in the $\hbar $ -expansion is given by

$$ \begin{align*} K_{\pm,-}(x,y;\hbar) = \pm \frac{1}{2 (x y)^{1/4}} \frac{1}{x^{1/2} \mp y^{1/2}} + \mathrm{O}(\hbar) . \end{align*} $$

Inserting the above expressions in the formula for the $W_{n}^{(i)}$ , we find

$$ \begin{align*} \begin{aligned} W_{0,n}^{(i)}(\boldsymbol{x}) &= \frac{(-1)^n}{2^n \, (x_1 \cdots x_n)^{1/2}} \sum_{\sigma \in S_n^{\text{cyc}}} \frac{1}{x_i^{1/2} + x_{\sigma(i)}^{1/2}} \prod_{j \ne i} \frac{1}{x_j^{1/2} - x_{\sigma(j)}^{1/2}} = - \frac{1}{4} \frac{1}{(x_1 \cdots x_n)^{1/2}} \frac{x_i^{n/2 - 1}}{ \prod_{i \ne j} (x_i - x_j) } . \end{aligned} \end{align*} $$

In the last equation, we applied lemma A.1. As a consequence, recalling that $S = 1$ and $A(x_i) = \frac {4}{3} x^{3/2}$ , we find

$$\begin{align*}W_{g,n}(\boldsymbol{x}) = -\frac{1}{4\pi} \, \frac{\Gamma(2g-2+n)}{\big( \frac{4}{3} \big)^{2g-2+n}} \Bigg( \frac{1}{(x_1 \cdots x_n)^{1/2}} \sum_{i=1}^n \frac{x_i^{-(3g-3+n)-1}}{ \prod_{i \ne j} (x_i - x_j) } + \mathrm{O}\big( g^{-1} \big) \Bigg) . \end{align*}$$

Using the relation $\sum _{i=1}^n \frac {x_{i}^{-D-1}}{\prod _{i \ne j} (x_i - x_j)} = \frac {(-1)^{n-1}}{x_1 \cdots x_n} h_D(\boldsymbol {x}^{-1})$ , with $h_D$ being the complete homogeneous symmetric polynomial of degree D, we find the thesis. This relation is a special case of lemma A.2, but it can be easily proved by taking the generating series on both sides.

As a consequence, we obtain the leading large genus asymptotics of $\psi $ -class intersection numbers.

Corollary 3.9 (Leading large genus asymptotics for $\psi $ -class intersection numbers)

For any given $n \ge 1$ , uniformly in $d_1,\dots ,d_n$ as $g \to \infty $ :

(3.22) $$ \begin{align} \langle{\tau_{d_1} \cdots \tau_{d_n}}\rangle \prod_{i=1}^n (2d_i + 1) = \frac{2^n}{4\pi} \frac{\Gamma(2g-2+n)}{( \frac{2}{3} )^{2g-2+n}} \Big( 1 + \mathrm{O}\big( g^{-1} \big) \Big) . \end{align} $$

The most challenging aspect of proving the corollary lies in the uniformity statement. We postpone the proof to the more general theorem 3.13, which accounts for all subleading corrections. In fact, the next subsection is devoted to the study of the subleading terms $W_{k,n}^{(i)}$ for arbitrary k.

Subleading contributions

From equation (3.21), it is clear that both $W_{g,n}$ and its leading large genus behaviour are polynomials in $x_1^{-1},\dots ,x_n^{-1}$ of degree $3g-3+n$ (up to a prefactor of $(x_1 \cdots x_n)^{-3/2}$ ). It is natural to ask whether the same property holds for the subleading corrections. More precisely, from the analysis of the kernel we see that the leading term $W_{0,n}^{(i)}$ has, a priori, simple poles along $\sqrt {x_i} = - \sqrt {x_j}$ for all $j \ne i$ and along $\sqrt {x_k} = \sqrt {x_l}$ for all $k \ne l$ . However, the only poles appearing are those at $\sqrt {x_i} = \pm \sqrt {x_j}$ . Furthermore, after summing over all contributions, the poles remarkably disappear, yielding a polynomial of the expected form. The next lemmas show that a similar structure holds for all subleading terms.

Lemma 3.10 (Pole structure of the minors)

The minor $W_n^{(i)}$ is regular along $\sqrt {x_k} = \sqrt {x_l}$ for all $k \ne l$ with $k,l \ne i$ . Moreover, it has simple poles at $\sqrt {x_i} = \pm \sqrt {x_j}$ for all $j \neq i$ , with residue

(3.23) $$ \begin{align} \operatorname{\mathrm{Res}}_{\sqrt{x_j} \to \pm \sqrt{x_i}} W_n^{(i)}(x_1,\dots,x_n;\hbar) = - \frac{1}{2 \sqrt{x_i}} \, W_{n-1}^{(i)}(x_1,\dots,\widehat{\,x_j\,},\dots,x_{n};\hbar) . \end{align} $$

On the right-hand side, the superscript $(i)$ means that the ‘special variable’ is $x_i$ .

Proof. Throughout the proof, we denote $z_m = \sqrt {x_m}$ . Consider the poles at $z_k = z_l$ for all $k \ne l$ with $k,l \ne i$ . The only terms in the definition of $W_n^{(i)}$ contributing to the residue are those permutations $\sigma \in S_n^{\text {cyc}}$ such that $\sigma (k) = l$ or $\sigma (l) = k$ . If both k and l are different from i, we find

$$ \begin{align*} \operatorname{\mathrm{Res}}_{z_k \to z_l} W_n^{(i)}(\boldsymbol{x};\hbar) = (-1)^{n-1} & \operatorname{\mathrm{Res}}_{z_k \to z_l} \Bigg( \sum_{\substack{\sigma \in S_n^{\text{cyc}} \\ \sigma(k) = l}} K_{-,-}(x_i,x_{\sigma(i)};\hbar) K_{+,-}(x_k,x_l;\hbar) \prod_{j \ne i,k} K_{+,-}(x_j,x_{\sigma(j)};\hbar) \\ &\quad + \sum_{\substack{\sigma \in S_n^{\text{cyc}} \\ \sigma(l) = k}} K_{-,-}(x_i,x_{\sigma(i)};\hbar) K_{+,-}(x_l,x_k;\hbar) \prod_{j \ne i,l} K_{+,-}(x_j,x_{\sigma(j)};\hbar) \Bigg). \end{align*} $$

The pole at $z_k \to z_l$ is given by $K_{+,-}(x_k,x_l;\hbar )$ in the first sum and by $K_{+,-}(x_l,x_k;\hbar )$ in the second one. Moreover, the residues are

$$\begin{align*}\operatorname{\mathrm{Res}}_{z_k \to z_l} K_{+,-}(x_k,x_l;\hbar) = \frac{1}{2z_l} , \qquad\qquad \operatorname{\mathrm{Res}}_{z_k \to z_l} K_{+,-}(x_l,x_k;\hbar) = - \frac{1}{2z_l} , \end{align*}$$

since they coincide (up to a sign) with the Wronskian of the differential system. After taking the residue, the two terms cancel out. This proves the first part of the statement.

For the second part, assume without loss of generality that $j = n$ (and $i < n$ ). Consider the pole at $z_n = z_i$ . With the same computation as before, we find

$$ \begin{align*} \operatorname{\mathrm{Res}}_{z_n \to z_i} W_n^{(i)}(\boldsymbol{x};\hbar) = (-1)^{n-1} \operatorname{\mathrm{Res}}_{z_n \to z_i} \Bigg( \sum_{\substack{\sigma \in S_n^{\text{cyc}} \\ \sigma(n) = i}} K_{-,-}(x_i, & x_{\sigma(i)};\hbar) K_{+,-}(x_n,x_i;\hbar) \prod_{j \ne i,n} K_{+,-}(x_j,x_{\sigma(j)};\hbar) \\ & + \sum_{\substack{\sigma \in S_n^{\text{cyc}} \\ \sigma(i) = n}} K_{-,-}(x_i,x_n;\hbar) \prod_{j \ne i} K_{+,-}(x_j,x_{\sigma(j)};\hbar) \Bigg). \end{align*} $$

Now the residues from the two sums give different contributions:

$$\begin{align*}\operatorname{\mathrm{Res}}_{z_n \to z_i} K_{+,-}(x_n,x_i;\hbar) = \frac{1}{2z_i} , \qquad\qquad \operatorname{\mathrm{Res}}_{z_n \to z_i} K_{-,-}(x_i,x_n;\hbar) = 0 . \end{align*}$$

The first residue is again the Wronskian, while the second one is simply a cancellation of the numerator of $K_{-,-}$ in the limit. Thus, the second sum gives no contribution, while the first sum can be rewritten as a sum over $S_{n-1}^{\text {cyc}}$ recovering $- W_{n-1}^{(i)}$ .

Finally, consider the pole at $z_n = - z_i$ . With the same computation as before, but with the residue contributions

$$\begin{align*}\operatorname{\mathrm{Res}}_{z_n \to -z_i} K_{+,-}(x_n,x_i;\hbar) = 0 , \qquad\qquad \operatorname{\mathrm{Res}}_{z_n \to -z_i} K_{-,-}(x_i,x_n;\hbar) = \frac{\mathrm{i}}{2z_i} . \end{align*}$$

and the property $K_{+,-}(x_n,x_{\sigma (n)};\hbar ) \big |_{z_n \mapsto -z_i} = - \mathrm {i} \, K_{-,-}(x_i,x_{\sigma (n)};\hbar )$ , we conclude the proof.

We now turn our attention to the k-th subleading term in the asymptotic expansion of the n-point Airy correlators. For simplicity, denote

(3.24) $$ \begin{align} U_{k,g,n}(\boldsymbol{x}) := (-1)^n \, 2^{k+2} \, (x_1 \cdots x_n)^{3/2} \sum_{i=1}^n x_i^{-\frac{3}{2}(2g-2+n-k)} \, W_{k,n}^{(i)}(\boldsymbol{x}) . \end{align} $$

With this notation, (3.20) reads

(3.25) $$ \begin{align} W_{g,n}(\boldsymbol{x}) = \frac{(-1)^n}{4\pi} \frac{\Gamma(2g-2+n)}{(\frac{4}{3})^{2g-2+n}} \left( \sum_{k = 0}^K \frac{(\frac{2}{3})^{k}}{(2g-3+n)^{\underline{k}}} \frac{U_{k,g,n}(\boldsymbol{x})}{(x_1 \cdots x_n)^{3/2}} + \mathrm{O}\bigg( \frac{1}{g^{K+1}} \bigg) \right) . \end{align} $$

Equation (3.21) is equivalent to $U_{0,g,n}(\boldsymbol {x}) = h_{3g-3+n}(\boldsymbol {x}^{-1})$ , that is, the leading term is the complete homogeneous symmetric polynomial of degree $3g-3+n$ in the variables $x_i^{-1}$ . The next lemma extends the result to subleading corrections, showing that the genus dependence is completely captured by a complete homogeneous polynomial of degree $3g-3+n$ minus a small defect that depends on k. We refer to appendix B and [Reference Macdonald56] for more details and notations on symmetric functions.

Lemma 3.11 (Polynomial properties of subleading corrections)

For every $k \ge 0$ , $n \ge 1$ , and g large enough, $U_{k,g,n}$ is a homogeneous symmetric polynomial in $x_i^{-1}$ of degree $3g-3+n$ of the form

(3.26) $$ \begin{align} U_{k,g,n}(\boldsymbol{x}) = \sum_{d=0}^{\min\{ \left\lfloor {\frac{3k+n-1}{2}} \right\rfloor ,\, 3k-1 \}} P_{k,n}^{(d)}(\boldsymbol{x}^{-1/2}) \, h_{3g-3+n-d}(\boldsymbol{x}^{-1}), \end{align} $$

where $P_{k,n}^{(d)}$ is a homogeneous symmetric polynomial of degree $2d$ , both even and of degree $\le 3k$ in each individual variable. The sequence of polynomials $(P_{k,n})_{n \ge 1}$ , withFootnote ³

(3.27) $$ \begin{align} P_{k,n} := \sum_{d=0}^{\min\{ \left\lfloor {\frac{3k+n-1}{2}} \right\rfloor ,\, 3k-1 \}} P_{k,n}^{(d)} , \end{align} $$

satisfies the following two specialisation properties:

(3.28) $$ \begin{align} P_{k,n+1}(u_1,\dots, u_n, u_{n+1}) \big|_{u_{n+1} = \pm 1} = P_{k,n}(u_1,\dots,u_n) . \end{align} $$

In turn, the degree condition and the specialisation properties uniquely determine $P_{k,n}$ for all n by an explicit algorithm from the first terms of the sequence, namely $P_{k,1},\dots , P_{k,3k-1}$ . Moreover, the coefficient of $m_{2\nu }$ in the expansion of $P_{k,n}$ in the monomial basis is a polynomial in n of degree $\le 3k-1 - |\nu |$ .

As a consequence of the above lemma, we get a polynomiality statement for the coefficients of $U_{k,g,n}$ in terms of n and the multiplicities of the exponents.

Corollary 3.12 (Polynomial properties of subleading corrections: the coefficients)

For every $k \ge 0$ , $n \ge 1$ , g large enough, and $d_1, \dots , d_n \ge 0$ satisfying $d_1 + \dots + d_n = 3g-3+n$ , the coefficients of $U_{k,g,n}$ are of the form

(3.29) $$ \begin{align} \big[ x_1^{-d_1} \cdots x_n^{-d_n} \big] \, U_{k,g,n}(\boldsymbol{x}) = \alpha_k\big( n, p_0, \dots, p_{ \left\lfloor {\frac{3k}{2}} \right\rfloor -1} \big) , \end{align} $$

where $\alpha _k$ is a polynomial in n and $p_m := \#\{ d_i = m\}$ for $m = 1, \dots , \left \lfloor {\frac {3k}{2}} \right \rfloor -1$ that can be effectively computed. Moreover, the degree is $\le 3k-1$ , under the assignment $\deg {n} = 1$ and $\deg {p_m} = m+1$ .

The first few polynomials $P_{k,n}$ and coefficients $\alpha _k$ are given in table 2. The intuitive explanation of the above statement can be given as follows. Proposition 3.8 shows that the k-th subleading correction in the large genus asymptotics of the correlators is proportional to

(3.30) $$ \begin{align} \sum_{i=1}^n x_i^{-\frac{3}{2}(2g-2+n-k)} \, W_{k,n}^{(i)} . \end{align} $$

In the large genus limit, most of the degree is then coming from the factors $x_i^{-\frac {3}{2}(2g-2+n)}$ , which in turn give rise to complete homogeneous polynomials of large degree times some polynomials of small degree. Thus, the extraction of coefficients depends only on the number of variables with a small degree, as all monomials appear in the complete homogeneous polynomial with coefficient $1$ . We also emphasise that lemma 3.11 gives a concrete algorithm to compute the subleading corrections. Indeed, the proof shows that, for fixed k, the computation of $P_{k,n}$ for $n \le 3k-1$ gives a closed, explicit expression for $P_{k,n}$ for arbitrary n. In turn, this gives a closed, explicit expression for the coefficients $\alpha _k$ , see figure 3.

Table 2

The polynomials $P_{k,n}(u_1,\dots ,u_n)$ and the coefficients $\alpha _k$ for $k = 0,1,2$ . Here $e_{k}$ denotes the elementary symmetric polynomial in the variables $u_1^2,\ldots,u_n^2$ .

Table 2 Long description

The table consists of three columns and three data rows.

* Row 1: k equals 0. P sub 0 comma n equals 1. alpha sub 0 (n, p) equals 1.

* Row 2: k equals 1.

- P sub 1 comma n equals minus numerator 17 minus 15 n plus 3 n-squared all over 12 minus numerator 3 minus n all over 2 e sub 1 minus 1 over 2 e sub 2.

- alpha sub 1 (n, p) equals minus numerator 17 minus 15 n plus 3 n-squared all over 12 minus numerator (3 minus n)(n minus p sub 0) all over 2 minus numerator (n minus p sub 0) to the underline 2 power all over 4.

* Row 3: k equals 2.

- P sub 2 comma n equals numerator 1225 minus 1632 n plus 741 n-squared minus 138 n-cubed plus 9 n to the 4 power all over 288 plus numerator 105 minus 98 n plus 30 n-squared minus 3 n-cubed all over 24 e sub 1 plus numerator 3(10 minus 7 n plus n-squared) all over 8 e sub 1-squared plus 5 over 8 e sub 1-cubed minus numerator 121 minus 75 n plus 9 n-squared all over 24 e sub 2 minus numerator 3(2 n minus 3) all over 8 e sub 1 e sub 2 plus numerator 12 n minus 43 all over 8 e sub 3 plus 3 over 4 e sub 1 e sub 3 minus 9 over 4 e sub 4.

- alpha sub 2 (n, p) equals numerator 1225 minus 1632 n plus 741 n-squared minus 138 n-cubed plus 9 n to the 4 power all over 288 plus numerator (105 minus 98 n plus 30 n-squared minus 3 n-cubed)(n minus p sub 0) all over 24 plus numerator 3(10 minus 7 n plus n-squared)(n minus p sub 0 minus p sub 1) all over 8 plus numerator (59 minus 51 n plus 9 n-squared)(n minus p sub 0) to the underline 2 power all over 48 plus numerator 5(n minus p sub 0 minus p sub 1 minus p sub 2) all over 8 plus numerator 3(4 minus n)(n minus p sub 0 minus 1)(n minus p sub 0 minus p sub 1) all over 4 plus numerator (7 minus 3 n)(n minus p sub 0) to the underline 3 power all over 24 plus numerator 3(n minus p sub 0 minus 1) to the underline 2 power (n minus p sub 0 minus p sub 1) all over 8 plus numerator 3(n minus p sub 0) to the underline 4 power all over 96.

Figure 3

The flowchart representing the algorithm for the computation of the subleading contributions.

Figure 3 Long description

The flowchart consists of four rectangular process blocks connected by downward-pointing arrows.

1. The top block contains the instruction: Compute P sub k,1 comma ... comma P sub k,3k-1 via determinantal formulae.

2. An arrow points down to the second block: Compute (P sub k,n) sub n greater than or equal to 1 in the elementary basis.

3. Between the second and third blocks, a horizontal arrow from the right introduces the input: Kostka numbers.

4. The third block contains: Compute (P sub k,n) sub n greater than or equal to 1 in the monomial basis.

5. Between the third and fourth blocks, a horizontal arrow from the right introduces the input: M sub n,mu,nu via theorem B.1.

6. The final bottom block contains the instruction: Compute alpha sub k.

Proof of lemma 3.11

We first claim that the functions $W_{k,n}^{(i)} = [\hbar ^k]W_{n}^{(i)}$ are of the form

$$ \begin{align*} W_{k,n}^{(i)}(\boldsymbol{x}) = - \frac{2^{-k-2}}{(x_1 \cdots x_n)^{1/2}} \frac{x_i^{n/2 - 1 - 3k/2}}{\prod_{j \ne i} (x_i - x_j)} \, P_{k,n-1} \bigg( \Big( \frac{x_i}{x_1} \Big)^{1/2}, \dots, \widehat{ \Big( \frac{x_i}{x_i} \Big)^{1/2} }, \dots, \Big( \frac{x_i}{x_n} \Big)^{1/2} \bigg) , \end{align*} $$

where $P_{k,n-1}$ is a symmetric polynomial in $n-1$ variables of total degree $\le \min \{ 3k + n - 2, 2(3k - 1)\}$ , both even and of degree $\le 3k$ in each individual variable, and satisfying the two specialisation properties $P_{k,n+1}( \,\cdot , \pm 1) = P_{k,n}(\,\cdot \,)$ . Indeed, from the homogeneity property and the parity relation satisfied by the Airy functions, that is, $\psi _{\pm }(x;\hbar ) = x^{-1/4} \, \psi _{\pm }(1;\frac {\hbar }{2A(x)})$ , $\psi _{+}(x;\hbar ) = \psi _{-}(x;-\hbar )$ , and similarly for the derivatives, we deduce that

$$\begin{align*}K_{\pm,-}(x,y;\hbar) = \pm \frac{1}{(x y)^{1/4}} \frac{1}{x^{1/2} \mp y^{1/2}} \sum_{m \ge 0} A_m\big( \mp x^{-1/2},y^{-1/2} \big) \hbar^m , \end{align*}$$

where $A_m$ is a homogeneous symmetric polynomial of degree $3m$ . From lemma 3.10 and a degree consideration, we conclude that

$$\begin{align*}(x_1 \cdots x_n)^{1/2} \, x_i^{3k/2 - n/2 + 1} \Bigg( \prod_{j \ne i} (x_i - x_j) \Bigg) W_{k,n}^{(i)}(\boldsymbol{x}) \end{align*}$$

is a symmetric polynomial in $t_j = \sqrt {x_i}/\sqrt {x_j}$ for $j \neq i$ of degree $\le 3k + n - 2$ . In the $t_j$ variables the above polynomial is given by

$$\begin{align*}(-1)^n \prod_{j \neq i} (1 - t_j^2) \sum_{m \neq i} \sum_{\substack{k_1 + \dots + k_{n} = k \\ \sigma \in S_n^{\text{cyc}}, \, \sigma(m) = i}} \frac{A_{k_i}(1, t_{\sigma(i)})}{1 + t_{\sigma(i)}} \frac{A_{k_m}(-t_{m},1)}{t_{m} - 1} \prod_{\substack{j \ne i, \, m}} \frac{A_{k_j}(-t_j, t_{\sigma(j)})}{t_j - t_{\sigma(j)}} . \end{align*}$$

The polynomiality and the specialisation properties, which are not obvious from the above expression, are guaranteed by lemma 3.10. It is clear that the degree in each individual variable is $\le 3k$ . Moreover, from remark 3.7 we deduce that the above expression is even in $t_j$ . Thus, after normalising by $-2^{-k-2}$ , we have that

$$\begin{align*}W_{k,n}^{(i)}(\boldsymbol{x}) = - \frac{2^{-k-2}}{(x_1 \cdots x_n)^{1/2}} \frac{x_i^{n/2 - 1 - 3k/2}}{\prod_{j \ne i} (x_i - x_j)} \, P_{k,n-1} \bigg( \Big( \frac{x_i}{x_1} \Big)^{1/2}, \dots, \widehat{ \Big( \frac{x_i}{x_i} \Big)^{1/2} }, \dots, \Big( \frac{x_i}{x_n} \Big)^{1/2} \bigg) , \end{align*}$$

where $P_{k,n-1}$ is a symmetric polynomial of total degree $\le 3k + n - 2$ , both even and of degree $\le 3k$ in each individual variable. Our next goal is to prove that there exists a value $N_k$ such that

$$ \begin{align*} \deg(P_{k,n+1}) = \deg(P_{k,n}) \qquad\quad \text{ for all } n \geq N_k . \end{align*} $$

To this end, consider for $P_{k,n}(u_1, \dots , u_n)$ the two changes of variables (which naturally follow from the specialisation properties)

$$ \begin{align*} \hat{u}_j := u_j - 1 , \quad \hat{e}_s := e_s(\hat{u}_1, \dots, \hat{u}_n) , \qquad\qquad \check{u}_j := u_j + 1 , \quad \check{e}_s := e_s(\check{u}_1, \dots, \check{u}_n) . \end{align*} $$

We also refer to $\hat {P}_{k,n}$ (resp. $\check {P}_{k,n}$ ) as $P_{k,n}$ expanded in the basis of elementary symmetric polynomials $\hat {e}_s$ (resp. $\check {e}_s$ ). From the specialisation properties (3.28) and the fact that setting any of the $\hat {u}_j$ (resp. $\check {u}_j$ ) to zero is the same as setting $\hat {e}_{n+1}$ (resp. $\check {e}_{n+1}$ ) to zero, we obtain the two decompositions

(3.31) $$ \begin{align} \hat{P}_{k,n+1} = \hat{P}_{k,n} + \hat{e}_{n+1}\hat{Q}_{k,n} , \qquad \qquad \check{P}_{k,n+1} = \check{P}_{k,n} + \check{e}_{n+1}\check{Q}_{k,n} , \end{align} $$

for some $\hat {Q}_{k,n}$ and $\check {Q}_{k,n}$ , polynomials in $\hat {e}_s$ and $\check {e}_s$ respectively. With the natural degree assignment $\deg {\hat {e}_s} = \deg {\check {e}_s} = s$ , we find $\deg (\hat {Q}_{k,n}) = \deg (\check {Q}_{k,n}) \leq 3k - 1$ . Define the linear unitriangular ring isomorphism $\mathsf {T}_n$ performing the change of basis from $\hat {e}$ to $\check {e}$ :

$$ \begin{align*} \mathsf{T}_n \colon \mathbb{Q}[\hat{e}] \longrightarrow \mathbb{Q}[\check{e}] , \qquad \hat{e}_s \longmapsto \sum_{m=0}^s \binom{n-s+m}{m} (-2)^m \check{e}_{s-m} . \end{align*} $$

The factor $-2$ arises from the difference $\hat {u}_j - \check {u}_j$ . Applying $\mathsf {T}_{n+1}$ to the first decomposition of equation (3.31), we obtain

$$ \begin{align*} \check{P}_{k,n+1} - \mathsf{T}_{n+1}(\hat{P}_{k,n}) = \mathsf{T}_{n+1}(\hat{e}_{n+1} \hat{Q}_{k,n}). \end{align*} $$

Here we used the fact that $\mathsf {T}_{n+1}(\hat {P}_{k,n+1}) = \check {P}_{k,n+1}$ . However, $\mathsf {T}_{n+1}(\hat {P}_{k,n})$ does not necessarily coincide with $\check {P}_{k,n}$ , as the change of basis depends on the number of variables. Adding and subtracting $\check {P}_{k,n}$ in the above equation and applying the second decomposition of equation (3.31) yields

$$ \begin{align*} \check{P}_{k,n} - \mathsf{T}_{n+1}(\hat{P}_{k,n}) - \mathsf{T}_{n+1}(\hat{e}_{n+1}) \mathsf{T}_{n+1}(\hat{Q}_{k,n}) = \check{e}_{n+1}\check{Q}_{k,n} . \end{align*} $$

Notice that, as the right-hand side is divisible by $\check {e}_{n+1}$ , so is the left-hand side. The only factors that (might) contain $\check {e}_{n+1}$ are $\mathsf {T}_{n+1}(\hat {e}_{n+1})$ and $\mathsf {T}_{n+1}(\hat {Q}_{k,n})$ . However, for $n \geq 3k - 1$ there are no such terms in $\mathsf {T}_{n+1}(\hat {Q}_{k,n})$ for degree reasons. Therefore, after removing the leading term from $\mathsf {T}_{n+1}(\hat {e}_{n+1})$ , we find

$$ \begin{align*} \check{P}_{k,n} - \mathsf{T}_{n+1}(\hat{P}_{k,n}) = \left(\mathsf{T}_{n+1}(\hat{e}_{n+1}) - \check{e}_{n+1}\right) \mathsf{T}_{n+1}(\hat{Q}_{k,n}) . \end{align*} $$

By the unitriangularity of the operator $\mathsf {T}_n$ , the degree of $\check {P}_{k,n} - \mathsf {T}_{n+1}(\hat {P}_{k,n})$ is lower by one compared to the degree of $\check {P}_{k,n}$ , as the leading terms cancel each other out. Explicitly, we obtain that $\deg (\check {P}_{k,n} - \mathsf {T}_{n+1}(\hat {P}_{k,n})) \leq 3k + n - 2,$ so that $\deg (\hat {Q}_{k,n}) \leq 3k - 2$ , which in turn forces

$$ \begin{align*} \deg(P_{k,n+1}) = \deg(P_{k,n}) \qquad \text{for } n \geq 3k-1 . \end{align*} $$

In other words, we proved that for $n \ge 3k-1$ the degree of $P_{k,n}$ is independent of n, which in turn implies that $\deg {( P_{k,n} )} \le \min \{ 3k + n - 1, 2(3k - 1) \}$ . Being even in all variables, we find that $\deg {( P_{k,n} )} \le 2 \min \{ \left \lfloor {\frac {3k + n - 1}{2}} \right \rfloor , 3k - 1 \}$ Footnote ⁴ .

From the above proof it follows that, for any fixed k, the first few values of n determine the whole sequence $(P_{k,n})_{n \ge 1}$ . Moreover, the coefficients of $P_{k,n}$ in the monomial basis are polynomials of n. Indeed, as the degree of $P_{k,n}$ is constant in n for n large enough, we deduce from equation (3.31) that eventually $\hat {P}_{k,n+1} = \hat {P}_{k,n}$ . The change of basis from $\hat {e}_s = e_s(u_1-1,\dots ,u_n-1)$ to $e_s = e_s(u_1,\dots ,u_n)$ , which is analogous to the transformation $\mathsf {T}_n$ , depends polynomially on n. Furthermore, the change of basis from elementary to complete homogeneous polynomials is independent of n (it involves Kostka numbers, see [Reference Macdonald56]). Hence, the coefficients of $P_{k,n}$ in the monomial basis are polynomials of n. As for the number of polynomials $P_{k,n}$ that determine the whole sequence, one can show that $P_{k,1}, \dots , P_{k,3k - 1}$ are sufficient by degree considerations and the parity property. Moreover, as polynomials of n, it can be shown that the degree of $m_{2\nu }$ is bounded by $3k-1-|\nu |$ by the explicit change of basis from $\hat {e}$ to e.

We now proceed with the proof of equation (3.26) for $U_{k,g,n}$ . From the definition of $U_{k,g,n}$ and the definition of $P_{k,n-1}$ , we find

$$ \begin{align*} U_{k,g,n}(\boldsymbol{x}) = (-1)^{n-1} \, x_1 \cdots x_n \sum_{i=1}^n \frac{ x_i^{-(3g-3+n)-1} }{ \prod_{j \ne i} (x_i - x_j) } \, P_{k,n-1} \bigg( \Big( \frac{x_i}{x_1} \Big)^{1/2}, \dots, \widehat{ \Big( \frac{x_i}{x_i} \Big)^{1/2} }, \dots, \Big( \frac{x_i}{x_n} \Big)^{1/2} \bigg) . \end{align*} $$

We claim that, for g large enough,

$$ \begin{align*} \sum_{i=1}^n \frac{ x_i^{-(3g-3+n)-1} }{ \prod_{j \ne i} (x_i - x_j) } \, P_{k,n-1} \bigg( \Big( \frac{x_i}{x_1} \Big)^{1/2}, & \dots, \widehat{ \Big( \frac{x_i}{x_i} \Big)^{1/2} }, \dots, \Big( \frac{x_i}{x_n} \Big)^{1/2} \bigg) \\ & = \frac{(-1)^{n-1}}{x_1 \cdots x_n} \sum_{d=0}^{\min\{ \left\lfloor {\frac{3k+n-1}{2}} \right\rfloor ,\, 3k-1 \}} P_{k,n}^{(d)} ( \boldsymbol{x}^{-1/2} ) \, h_{3g-3+n-d} ( \boldsymbol{x}^{-1} ), \end{align*} $$

where $P_{k,n}^{(d)}$ is the homogeneous component of degree $2d$ in $P_{k,n}$ (notice the shift in n). This would complete the proof of the lemma. In order to prove the claimed formula, consider the function

$$\begin{align*}f(q) := \frac{P_{k,n}(\frac{q}{\sqrt{x_1}}, \dots, \frac{q}{\sqrt{x_n}})}{\prod_{i=1}^n (q^2 - x_i)} . \end{align*}$$

We omit its dependence on $x_1^{1/2},\dots ,x_n^{1/2}$ for simplicity. As a function of q, it has simple poles at $q = \pm \sqrt {x_i}$ with residues

$$ \begin{align*} \operatorname{\mathrm{Res}}_{q = \pm\sqrt{x_i}} f(q) = \frac{P_{k,n}\big( \pm \frac{\sqrt{x_i}}{\sqrt{x_1}}, \dots, \pm \frac{\sqrt{x_i}}{\sqrt{x_i}}, \dots, \pm \frac{\sqrt{x_i}}{\sqrt{x_n}} \big)}{\pm 2\sqrt{x_i} \prod_{j \ne i} (x_i - x_j)} = \pm \frac{P_{k,n-1}\big( \frac{\sqrt{x_i}}{\sqrt{x_1}}, \dots, \widehat{ \frac{\sqrt{x_i}}{\sqrt{x_i}} }, \dots, \frac{\sqrt{x_i}}{\sqrt{x_n}} \big)}{2\sqrt{x_i} \prod_{j \ne i} (x_i - x_j)} . \end{align*} $$

Here we used the fact that $P_{k,n}$ is even and it satisfies $P_{k,n}( \,\cdot , \pm 1) = P_{k,n-1}(\,\cdot \,)$ . As a consequence, we obtain

$$\begin{align*}\begin{aligned} \big[ q^{6g-6+2n} \big] f(q) & = \big[ q^{6g-6+2n} \big] \sum_{i=1}^n \left( \frac{P_{k,n-1}\big( \frac{\sqrt{x_i}}{\sqrt{x_1}}, \dots, \widehat{ \frac{\sqrt{x_i}}{\sqrt{x_i}} }, \dots, \frac{\sqrt{x_i}}{\sqrt{x_n}} \big)}{(q - \sqrt{x_i}) \, 2\sqrt{x_i} \prod_{j \ne i} (x_i - x_j)} - \frac{P_{k,n-1}\big( \frac{\sqrt{x_i}}{\sqrt{x_1}}, \dots, \widehat{ \frac{\sqrt{x_i}}{\sqrt{x_i}} }, \dots, \frac{\sqrt{x_i}}{\sqrt{x_n}} \big)}{(q + \sqrt{x_i}) \, 2\sqrt{x_i} \prod_{j \ne i} (x_i - x_j)} \right) \\ & = \big[ q^{6g-6+2n} \big] \sum_{i=1}^n \frac{1}{q^2 - x_i} \frac{P_{k,n-1}\big( \frac{\sqrt{x_i}}{\sqrt{x_1}}, \dots, \widehat{ \frac{\sqrt{x_i}}{\sqrt{x_i}} }, \dots, \frac{\sqrt{x_i}}{\sqrt{x_n}} \big)}{\prod_{j \ne i} (x_i - x_j)} \\ & = - \sum_{i=1}^n \frac{x_i^{-(3g-3+n)-1}}{\prod_{j \ne i} (x_i - x_j)} \, P_{k,n-1} \bigg( \Big( \frac{x_i}{x_1} \Big)^{1/2}, \dots, \widehat{ \Big( \frac{x_i}{x_i} \Big)^{1/2} }, \dots, \Big( \frac{x_i}{x_n} \Big)^{1/2} \bigg) . \end{aligned} \end{align*}$$

On the other hand, extracting the coefficient of $q^{6g-6+2n}$ from the definition of f gives

$$\begin{align*}\big[ q^{6g-6+2n} \big] f(q) = \frac{(-1)^{n}}{x_1 \cdots x_n} \sum_{d=0}^{\min\{ \left\lfloor {\frac{3k+n-1}{2}} \right\rfloor ,\, 3k-1 \}} P_{k,n}^{(d)} ( \boldsymbol{x}^{-1/2} ) \, h_{3g-3+n-d} ( \boldsymbol{x}^{-1} ) . \end{align*}$$

This completes the proof.

Proof of corollary 3.12

From lemma 3.11, we see that $P_{k,n}$ in the monomial basis reads

$$\begin{align*}P_{k,n}(u_1,\dots,u_n) = \sum_{\substack{ |\nu| \le \min\{ \left\lfloor {\frac{3k+n-1}{2}} \right\rfloor ,\, 3k-1 \} \\ \nu_1 \le \left\lfloor {\frac{3k}{2}} \right\rfloor ,\, \ell(\nu) \le n }} C_{k,n,\nu} m_{\mu}(\boldsymbol{u}^2) . \end{align*}$$

The condition $|\nu | \le \min \{ \left \lfloor {\frac {3k+n-1}{2}} \right \rfloor ,\, 3k-1 \}$ follows from the bound on the total degree, while $\nu _1 \le \left \lfloor {\frac {3k}{2}} \right \rfloor $ follows from the bound on the degrees in each individual variable. Notice that we also employed the parity condition, together with the property $m_{2\nu }(\boldsymbol {u}) = m_{\nu }(\boldsymbol {u}^2)$ , so that only monomial symmetric polynomials in the squared variables appear. Moreover, we know that $C_{k,n,\nu }$ is an (explicit) polynomial of n of degree $\le 3k-1-|\nu |$ . Inserting the above expansion in the formula for $U_{k,g,n}$ , we obtain:

$$\begin{align*}U_{k,g,n}(\boldsymbol{x}) = \sum_{\substack{ |\nu| \le \min\{ \left\lfloor {\frac{3k+n-1}{2}} \right\rfloor ,\, 3k-1 \} \\ \nu_1 \le \left\lfloor {\frac{3k}{2}} \right\rfloor ,\, \ell(\nu) \le n }} C_{k,n,\nu} \, m_{\nu}(\boldsymbol{x}^{-1}) \, h_{3g-3+n-|\nu|}(\boldsymbol{x}^{-1}) . \end{align*}$$

Since we are interested in extracting coefficients, it is natural to decompose $U_{k,g,n}$ in the basis of monomial symmetric polynomials. For products of the form $m_{\nu } h_{D - |\nu |}$ , this is discussed in appendix B:

$$\begin{align*}m_{\nu} \, h_{D - |\nu|} = \sum_{|\mu| = D} M_{n,\mu,\nu} \, m_{\mu} , \end{align*}$$

where $M_{n,\mu ,\nu }$ are (explicit) polynomials of n and $p_m = \#\{ \mu _i = m\}$ for $m = 0,\dots , \nu _1 - 1$ . For a fixed tuple $d_1, \dots , d_n \ge 0$ satisfying $|d| = 3g-3+n$ , consider the associated partition $\mu $ (since $U_{k,g,n}$ is symmetric, the order does not matter). Then

$$\begin{align*}\alpha_k\big( n, p_0, \dots, p_{ \left\lfloor {\frac{3k}{2}} \right\rfloor -1} \big) = \big[ x_1^{-d_1} \cdots x_n^{-d_n} \big] \, U_{k,g,n}(\boldsymbol{x}) = \sum_{\substack{ |\nu| \le \min\{ \left\lfloor {\frac{3k+n-1}{2}} \right\rfloor ,\, 3k-1 \} \\ \nu_1 \le \left\lfloor {\frac{3k}{2}} \right\rfloor ,\, \ell(\nu) \le n} } C_{k,n,\nu} \, M_{n,\mu,\nu} \end{align*}$$

is a polynomial of n and $p_m = \#\{ d_i = m\}$ for $m = 1, \dots , \left \lfloor {\frac {3k}{2}} \right \rfloor -1$ . This concludes the proof.

The three main ingredients that entered into the proof of the above results are:

1. 1. the Wronskian equation: $\psi ^{\prime }_+ \psi _- - \psi _+ \psi _-' = 1$ ;

2. 2. the homogeneity property and the parity relation satisfied by the formal Airy functions: $\psi _{\pm }(x;\hbar ) = x^{-1/4} \, \psi _{\pm }(1;\frac {\hbar }{2A(x)})$ , $\psi _{+}(x;\hbar ) = \psi _{-}(x;-\hbar )$ and similarly for the derivatives;

3. 3. the exponent $3/2$ in the instanton action $A(x) = \frac {4}{3} x^{3/2}$ , giving rise to the value $ \left \lfloor {3k/2} \right \rfloor $ .

This observation will allow us to obtain a similar result for $\Theta $ -class and r-spin intersection numbers for which, mutatis mutandis, the same properties hold.

We can now extract coefficients from both sides of equation (3.25) to get the large genus asymptotics of Witten–Kontsevich intersection numbers. The leading term, already computed in corollary 3.9, recovers Aggarwal’s result. The polynomiality structure of the subleading corrections proved in the previous proposition confirms a conjecture of Guo–Yang [Reference Guo and Yang49, conjecture 1].

Theorem 3.13 (Large genus asymptotics for $\psi $ -class intersection numbers)

For any given $n \ge 1$ and $K \ge 0$ , uniformly in $d_1,\dots ,d_n$ as $g \to \infty $ :

(3.32) $$ \begin{align} \langle{\tau_{d_1} \cdots \tau_{d_n}}\rangle & \prod_{i=1}^n (2d_i + 1)!! \nonumber\\ & = \frac{2^n}{4\pi} \frac{\Gamma(2g-2+n)}{( \frac{2}{3} )^{2g-2+n}} \left( \sum_{k = 0}^K \frac{( \frac{2}{3} )^{k}}{(2g-3+n)^{\underline{k}}} \, \alpha_k\big( n, p_0, \dots, p_{ \left\lfloor {\frac{3k}{2}} \right\rfloor -1} \big) + \mathrm{O}\bigg( \frac{1}{g^{K+1}} \bigg) \right) . \end{align} $$

Each term $\alpha _k$ in the asymptotic expansion is a polynomial in n and $p_m = \#\{ d_i = m\}$ for $m = 1, \dots , \left \lfloor {\frac {3k}{2}} \right \rfloor -1$ of degree $\le 3k-1$ , with the assignment $\deg {n} = 1$ and $\deg {p_m} = m+1$ . Moreover, the term $\alpha _k$ can be effectively computed from equation (3.29).

Proof. Our starting point is the application of the Borel transform method, theorem 2.5, to the n-point function. From the proof of theorem 2.5, after the application of Cauchy’s theorem and a deformation of the contour around the origin into several Hankel contours along the logarithmic branch-cuts, we find

(3.33) $$ \begin{align} & \frac{W_{g,n}(\boldsymbol{x})}{\Gamma(2g-2+n)} - \sum_{i=1}^n \frac{S}{\pi} \frac{1}{A(x_i)^{2g-2+n}} \sum_{k=0}^K \frac{A(x_i)^{2g-2+n-k}}{(2g-3+n)^{\underline{k}}} \, W_{k,n}^{(i)}(\boldsymbol{x}) \nonumber\\ &\qquad\quad = (2g - 2 + n) \sum_{i=1}^n \frac{S}{\pi} \, \frac{1}{A(x_i)^{2g-2+n}} \int_{0}^{+\infty} e^{-t(2g-2+n)} \, \widehat{W}_{n}^{(i)} \big(\boldsymbol{x};A(x_i)(e^t-1);K \big) \, dt . \end{align} $$

Here we recall that $S=1$ and $A(x) = \frac {4}{3} x^{3/2}$ are the Stokes constant and the instanton action of the Airy functions, and $\widehat {W}_{n}^{(i)}(\boldsymbol {x};s;K)$ denotes the correlator $\widehat {W}_{n}^{(i)}(\boldsymbol {x};s)$ minus its Taylor expansion at $s = 0$ up to order K. We also notice that the application of the above theorem requires all singularities to be distinct. Since the singularities are located at $\pm A(x_i)$ , we require $x_i \neq \pm x_j$ for all $i \ne j$ . From equation (3.33), it is clear that the dominant contribution to the asymptotics comes from the closest singularity to the origin (see appendix C for a visualisation of the phenomenon). Without loss of generality, we can suppose that $R_1 < \cdots < R_n$ , so that $i = 1$ is the dominant contribution.

On the other hand, after multiplication by $(x_1 \cdots x_n)^{3/2}$ , the left-hand side is a homogeneous symmetric polynomial in $\boldsymbol {x}^{-1}$ of degree $3g-3+n$ (thanks to theorem 3.2 and lemma 3.11). Our goal is to extract the coefficient of a fixed monomial $x_1^{-d_1} \cdots x_{n}^{-d_n}$ and show that the resulting value has the appropriate asymptotics as $g \to \infty $ , uniform in $d_1,\dots ,d_n$ . As the dominant contribution comes from $i = 1$ , it is natural to normalise the left-hand side of equation (3.33) as:

$$ \begin{align*} \Bigg( \prod_{i=1}^{n} 2x_i^{3/2} \Bigg) (-2A(x_1))^{2g-2+n} \, x_1^{n/2} & \Bigg[ \frac{W_{g,n}(\boldsymbol{x})}{\Gamma(2g-2+n)} \\ &\qquad\qquad - \sum_{i=1}^n \frac{S}{\pi} \frac{1}{A(x_i)^{2g-2+n}} \sum_{k=0}^K \frac{A(x_i)^{2g-2+n-k}}{(2g-3+n)^{\underline{k}}} \, W_{k,n}^{(i)}(\boldsymbol{x}) \Bigg]. \end{align*} $$

Indeed, a simple degree counting shows that the left-hand side is a homogeneous polynomial of degree $3g-3+n$ in $x_1,x_2^{-1},\dots ,x_n^{-1}$ . Extracting the coefficient of $x_1^{d_1} x_2^{-d_2} \cdots x_n^{-d_n}$ yields

$$ \begin{align*} \mathcal{C}_{g,d,K} := \frac{( \frac{2}{3} )^{2g-2+n}}{\Gamma(2g-2+n)} \langle{\tau_{d_1} \cdots \tau_{d_n}}\rangle \prod_{i=1}^n (2d_i + 1)!! - \frac{2^n}{4\pi} \sum_{k = 0}^K \frac{( \frac{2}{3} )^{k}}{(2g-3+n)^{\underline{k}}} \, \alpha_k\big( n, p_0, \dots, p_{ \left\lfloor {\frac{3k}{2}} \right\rfloor -1} \big) . \end{align*} $$

The thesis is equivalent to showing that $\mathcal {C}_{g,d,K} = \mathrm {O}( g^{-K-1} )$ , uniformly in d. In other words, there must exist $g_0 \ge 0$ and $C> 0$ , depending on n and K but not on d, such that for $g \ge g_0$ we have $|\mathcal {C}_{g,d,K}| \le C \, g^{-K-1}$ . In order to prove such a claim, let us consider the appropriately normalised right-hand side of equation (3.33). The coefficient extraction can be achieved by applying the operator

$$ \begin{align*} \mathcal{O}_{d} \colon Z \longmapsto \left( \prod_{i=1}^n \oint_{|x_i| = R_i} \frac{dx_i}{2\pi\mathrm{i}} \, x_i^{\widetilde{d}_i-1} \right) Z(\boldsymbol{x}) . \end{align*} $$

Here Z is any continuous function on $\{ \boldsymbol {x} \in \mathbb {C}^n | |x_i| = R_i\}$ , $\widetilde {d}_1 = -d_1$ , and $\widetilde {d}_j = d_j$ for all $j = 2,\dots ,n$ . Thus, looking at the right-hand side of equation (3.33), we find

$$ \begin{align*} \mathcal{C}_{g,d,K} = (2g-2+n) \, \mathcal{O}_{d} \bigg[ \sum_{i=1}^n \left( \frac{x_1}{x_i} \right)^{3g-3+n} \int_{0}^{+\infty} e^{-t(2g-2+n)} \, \widehat{Z}_{n}^{(i)}(\boldsymbol{x};t;K) \, dt \bigg] , \end{align*} $$

where we have set

$$ \begin{align*} \widehat{Z}_{n}^{(i)}(\boldsymbol{x};t;K) := (-1)^n \frac{S}{\pi} \, \Bigg( \prod_{i=1}^{n} 2x_i^{3/2} \Bigg) \, \widehat{W}_{n}^{(i)} \big(\boldsymbol{x};A(x_i)(e^t-1);K \big) . \end{align*} $$

Notice that summing over n we find a polynomial in $x_1,x_2^{-1},\dots ,x_n^{-1}$ , though the j-th summand has simple poles at $x_i = x_j$ . However, since we are integrating along circles of different radii $R_1 < \cdots < R_n$ , there is no issue with the application of $\mathcal {O}_{d}$ . Furthermore, it should be noted that $\widehat {Z}_{n}^{(i)}$ satisfies the following properties as a function of t for $|x_i| = R_i$ fixed:

• • Polynomial growth at the origin. $\widehat {Z}_{n}^{(i)}(\boldsymbol {x};t;K) = \mathrm {O}(t^{K+1})$ as $t \to 0^+$ , since the left-hand side is obtained from the truncation of the Taylor series of $\widehat {W}_{n}^{(i)}$ at the origin.

• • Exponential growth at infinity. $\widehat {Z}_{n}^{(i)}(\boldsymbol {x};t;K) = \mathrm {O}(e^{t\nu })$ as $t \to +\infty $ for some exponent $\nu \in \mathbb {R}_+$ , since $\widehat {W}_{n}^{(i)}(\boldsymbol {x};A(x_i)s)$ is a sum of convolutions of wave functions with polynomial growth at $s = \infty $ . The latter property can be easily checked from the integral representation of the Airy functions.

From the homogeneity properties of the Airy function, it can be shown that both estimates are uniform in $R_i$ , as long as all $R_i$ are fixed in a compact set, say $R_i \in [1,2]$ . In order to prove the claim, let us analyse separately the term corresponding to $i = 1$ (the leading term), and the one corresponding to $i> 1$ (the exponentially subleading terms).

Leading term. Consider

$$ \begin{align*} \mathcal{C}_{g,d,K}^{\text{lead}} := (2g-2+n) \, \mathcal{O}_{d} \bigg[ \int_{0}^{+\infty} e^{-t(2g-2+n)} \, \widehat{Z}_{n}^{(1)}(\boldsymbol{x};t;K) \, dt \bigg] . \end{align*} $$

As the integrand is analytic in the integration domain, we can exchange the integral in t with the one in $\boldsymbol {x}$ to get

$$ \begin{align*} | \mathcal{C}_{g,d,K}^{\text{lead}} | \le (2g-2+n) \int_{0}^{+\infty} e^{-t(2g-2+n)} \left| \mathcal{O}_{d} \Big[ \widehat{Z}_{n}^{(1)}(\boldsymbol{x};t;K) \Big] \right| dt . \end{align*} $$

Notice that the integrand $| \mathcal {O}_{d} [ \widehat {Z}_{n}^{(1)}(\boldsymbol {x};t;K) ] |$ can be bounded by

$$\begin{align*}\left| \mathcal{O}_{d} \Big[ \widehat{Z}_{n}^{(1)}(\boldsymbol{x};t;K) \Big] \right| \le \frac{R_{2}^{d_2} \cdots R_{n}^{d_n}}{R_1^{d_1}} \, f_{n,K}(t) , \end{align*}$$

where $f_{n,K}$ satisfies the hypothesis of Watson’s lemma 2.6: it behaves like $\mathrm {O}(t^{K+1})$ as $t \to 0^+$ and it has exponential growth as $t \to + \infty $ . Both estimates depend on n and K, but not on $R_i$ and $d_i$ . Applying Watson’s lemma 2.6, we find

$$\begin{align*}\mathcal{C}_{g,d,K}^{\text{lead}} = (2g - 2 + n) \frac{R_{2}^{d_2} \cdots R_{n}^{d_n}}{R_1^{d_1}} \, \mathrm{O}\bigg( \frac{1}{(2g - 2 + n)^{K+2}} \bigg) = \frac{R_{2}^{d_2} \cdots R_{n}^{d_n}}{R_1^{d_1}} \, \mathrm{O}\bigg( \frac{1}{g^{K+1}} \bigg) . \end{align*}$$

Again, $\mathrm {O}( g^{-K-1} )$ depends on n and K but not on d.

Exponentially subleading terms. With the same argument as before, we deduce that

$$ \begin{align*} \begin{aligned} \mathcal{C}_{g,d,K}^{\text{sub}} & := (2g-2+n) \, \mathcal{O}_{d} \Bigg[ \sum_{i=2}^n \left( \frac{x_1}{x_i} \right)^{3g-3+n} \int_{0}^{+\infty} e^{-t(2g-2+n)} \, \widehat{Z}_{n}^{(i)}(\boldsymbol{x};t;K) \, dt \Bigg] \\ & = \frac{R_{2}^{d_2} \cdots R_{n}^{d_n}}{R_1^{d_1}} \, \sum_{i=2}^n \left( \frac{R_1}{R_i} \right)^{3g-3+n} \mathrm{O}\bigg( \frac{1}{g^{K+1}} \bigg) . \end{aligned} \end{align*} $$

Notice that $(R_1/R_i)^{3g-3+n}$ are exponentially suppressed terms.

Conclusion. To conclude, let us choose the radii $R_i = 1 + \frac {i-1}{n-1} \frac {1}{g}$ . In the limit $g \to + \infty $ all radii coincide. It is then natural to expect that all terms would give a contribution behaving as $\mathrm {O}(g^{-K-1})$ . Indeed, since $|d| = 3g-3+n$ , we find

$$\begin{align*}\begin{aligned} \mathcal{C}_{g,d,K}^{\text{lead}} & = \left(1 + \frac{1}{g} \right)^{3g-3+n} \mathrm{O}\bigg( \frac{1}{g^{K+1}} \bigg) = \mathrm{O}\bigg( \frac{1}{g^{K+1}} \bigg) , \\ \mathcal{C}_{g,d,K}^{\text{sub}} & = \left(1 + \frac{n-1}{g} \right)^{3g-3+n} \, (n-1) \left( \frac{(n-1)g}{1 + (n-1)g} \right)^{3g-3+n} \mathrm{O}\bigg( \frac{1}{g^{K+1}} \bigg) = \mathrm{O}\bigg( \frac{1}{g^{K+1}} \bigg) . \end{aligned} \end{align*}$$

Here we used the fact that $(1 + \frac {1}{g} )^{3g-3+n} = (1 + \frac {n-1}{g} )^{3g-3+n} = (n-1) ( \frac {(n-1)g}{1 + (n-1)g} )^{3g-3+n} = \mathrm {O}(1)$ . All estimates depend on n and K, but not on d. Altogether, we find the thesis: $\mathcal {C}_{g,d,K} = \mathrm {O}(g^{-K-1})$ uniformly in d.

3.4 Trans-series structure

For the reader familiar with resurgence and trans-series analysis, it can be shown that the correlators have a full trans-series expansion that can be constructed as follows. Consider the matrix

(3.34) $$ \begin{align} \mathsf{M}(x;\hbar) = \Psi(x;\hbar) \,\mathsf{E}\, \Psi^{-1}(x;\hbar) , \qquad\qquad \mathsf{E} = \begin{pmatrix} \frac{\sigma}{2} & -\sigma_- \\ \sigma_+ & -\frac{\sigma}{2} \end{pmatrix} , \end{align} $$

which depends on trans-series parameters $\sigma , \sigma _+, \sigma _-$ (but we omit its dependence for the sake of notational simplicity). Notice that $\mathsf {E}$ is a generic element of $\mathfrak {sl}_2(\mathbb {C})$ , rather than a Cartan element as in equation (3.6). We decompose $\mathsf {M}$ as

(3.35) $$ \begin{align} \mathsf{M} = \sigma \, M + \sigma_{+} \, e^{\frac{A(x)}{\hbar}} M_+ + \sigma_{-} \, e^{-\frac{A(x)}{\hbar}} M_- , \end{align} $$

where M and $M_{\pm }$ are the formal power series in $\hbar $ previously encountered. We can then define the n-point correlators with the exact same determinantal formula as in equation (3.8):

(3.36) $$ \begin{align} \mathsf{W}_1(x_1;\hbar) := - \frac{1}{\hbar} \operatorname{\mathrm{Tr}}{\big( \mathcal{D}(x_1) \mathsf{M}(x_1;\hbar) \big)} , \qquad \mathsf{W}_n(\boldsymbol{x};\hbar) := (-1)^{n-1} \sum_{\sigma \in S_n^{\text{cyc}}} \frac{\operatorname{\mathrm{Tr}}{ \big( \prod_{i=1}^n \mathsf{M}(x_{\sigma^i(1)};\hbar) } \big) }{\prod_{i=1}^n ( x_{i} - x_{\sigma(i)} )} . \end{align} $$

In the above definition we have chosen different constants $\sigma _{i}$ , $\sigma _{i,+}$ , $\sigma _{i,-}$ for different $i=1,\dots ,n$ . Thus, we have the trans-series expansion

(3.37) $$ \begin{align} \mathsf{W}_n(\boldsymbol{x};\hbar) = \sum_{I_{+} \sqcup I_{-} \subseteq [n]} \prod_{i \not \in I_{+} \sqcup I_{-} } \sigma_{i} \prod_{i \in I_{+} } \sigma_{i,+} \prod_{i \in I_{-} } \sigma_{i,-} \, e^{\frac{1}{\hbar} \big( \sum_{i \in I_{+}} A(x_i) - \sum_{i \in I_{-}} A(x_i) \big)} \, \mathsf{W}^{(I_{+},I_{-})}_n(\boldsymbol{x};\hbar) . \end{align} $$

Notice that, by definition, the perturbative sector $\mathsf {W}^{(\varnothing ,\varnothing )}_{n}$ coincides with the $W_n$ defined in equation (3.8) and stores $\psi $ -class intersection numbers. Moreover, the $(1,0)$ - and $(0,1)$ -instanton sectors coincide with the correlators defined in equation (3.16): $\mathsf {W}^{(\{i\},\varnothing )}_n = W^{(+,i)}_n$ and $\mathsf {W}^{(\varnothing ,\{i\})}_n = W^{(-,i)}_n$ . In particular, they govern the large genus asymptotics of the intersection numbers, but they do not have (to the best of our knowledge) an enumerative-geometric interpretation. It would be interesting to explore whether such a geometric interpretation actually exists for all multiinstanton sectors.

Finally, we remark that the trans-series expansion is in accordance with the fact that $\mathsf {W}_n$ satisfies a linear ODE of order $3^n$ (see [Reference Eynard, Mitsios and Oukassi41] for a proof). For instance, the $1$ -point correlator satisfies

(3.38) $$ \begin{align} \mathsf{W}^{"'}_1 - 4x \, \mathsf{W}^{\prime}_1 + 2 \hbar \, \mathsf{W}_1 = 0 , \end{align} $$

where $f'$ stands for $\hbar \frac {d}{dx}f$ . In particular, a perturbative ansatz of the form $W_1 = \sum _{g \ge 0} w_g \frac {\hbar ^{2g-1}}{x^{3g-1/2}}$ would give rise to a recursive relation for the coefficients $w_g$ that can be explicitly solved, and is equivalent to $\langle {\tau _{3g-2}}\rangle _{g} = \frac {1}{24^g g!}$ after setting $w_0 = 1$ .

5.1 Higher Airy functions

Consider the ( $\hbar $ -dependent) r-Airy ODE:

(5.5) $$ \begin{align} \left( \left( \hbar \frac{d}{dx} \right)^r - x \right) \psi(x;\hbar) = 0 . \end{align} $$

The general solution is given by the r-Airy integral: $\int _{\gamma } e^{-\frac {1}{\hbar }V(t,x)} dt$ with $V(t,x) := \frac {t^{r+1}}{r+1} - xt$ and $\gamma $ a properly chosen integration contour. Again, we are interested in asymptotic solutions constructed through Lefschetz thimbles: $\psi _{\alpha }(x;\hbar )$ for $\alpha = 1, \dots , r$ . Their expression, together with that of their derivatives, is given by

(5.6) $$ \begin{align} \psi_{\alpha}^{(m)}(x;\hbar) := (-1)^{\frac{r-\alpha+2}{2}} \zeta^{\alpha(m + \frac{1}{2})} \frac{e^{-\frac{V_{\alpha}(x)}{\hbar}}}{\sqrt{r}} x^{-\frac{r-1-2m}{2r}} \sum_{k = 0}^{\infty} a_k^{(m)} \left( - \frac{\hbar}{V_{\alpha}(x)} \right)^{k} , \end{align} $$

where $V_{\alpha }(x) := - \zeta ^{\alpha } \frac {r}{r+1} x^{(r+1)/r}$ and $\zeta := e^{\frac {2\pi \mathrm {i}}{r}}$ . The coefficients $a_{k}^{(m)}$ are independent of $\alpha $ and are computed recursively from the differential equation (see for instance [Reference Charbonnier, Chidambaram, Garcia-Failde and Giacchetto21]) as

(5.7) $$ \begin{align} \begin{cases} a_{k}^{(m)} = a_{k}^{(m-1)} - \left( k - \frac{1}{2} - \frac{m}{r+1} \right) a_{k-1}^{(m-1)} & \text{if } m = 1,\dots, r \\ a_{k}^{(0)} = a_{k}^{(r)} \end{cases} \end{align} $$

with $a_{0}^{(m)} = 1$ . With the above normalisation, we can see that the Wronskian is constantly $1$ . Indeed, the ODE implies that the Wronskian is constant, and we can compute its value at $x \to \infty $ as

(5.8) $$ \begin{align} \det\left( (-1)^{\frac{r-\alpha+2}{2}} \zeta^{\alpha(m + \frac{1}{2})} \frac{ e^{-\frac{V_{\alpha}(x)}{\hbar}} }{ \sqrt{r} } x^{-\frac{r-1-2m}{2r}} \right)_{\substack{ m = 0,\dots,r-1 \\ \alpha = 1,\dots,r}} = \frac{(-1)^{\frac{(r-1)(r-2)}{4}}}{r^{r/2}} \det\big( \zeta^{\alpha m} \big)_{\substack{ m = 0,\dots,r-1 \\ \alpha = 1,\dots,r}} . \end{align} $$

The Vandermonde determinant on roots of unity was computed by Schur [Reference Schur71] and it reads $\det ( \zeta ^{\alpha m} ) = r^{r/2} (-1)^{\frac {(r-1)(3r+2)}{4}}$ , so that the Wronskian simplifies to $1$ .

Following the prescription of subsection 2.2, we find that, after separating the exponential part as $\psi _{\alpha }^{(m)} =: e^{-V_{\alpha }/\hbar } \, \widetilde {\psi }_{\alpha }^{(m)}$ , the Borel transform of $\widetilde {\psi }_{\alpha }^{(m)}$ is simple resurgent with logarithmic singularities at $s = A_{\alpha ,\beta }(x)$ for $\beta \neq \alpha $ , where

(5.9) $$ \begin{align} A_{\alpha,\beta}(x) := V_{\beta}(x) - V_{\alpha}(x) = \frac{r}{r+1} \big( \zeta^{\alpha} - \zeta^{\beta} \big) x^{(r+1)/r} . \end{align} $$

The behaviour at the singularities is given by

(5.10) $$ \begin{align} \begin{aligned} \widehat{\psi}_{\alpha}(x;s) & = - \frac{S_{\alpha,\beta}}{2\pi} \, \widehat{\psi}_{\beta}\big( x;s - A_{\alpha,\beta}(x) \big) \log\big( s - A_{\alpha,\beta}(x) \big) + \text{holomorphic at } A_{\alpha,\beta}(x) , \end{aligned} \end{align} $$

where the Stokes constants $S_{\alpha ,\beta }$ are given in terms of constants $\mathsf {S}_{\alpha ,\beta }$ computed as Lefschetz thimble intersection numbers (see equation (2.13)) and are equal to

(5.11) $$ \begin{align} S_{\alpha,\beta} = (-1)^{\frac{\beta-\alpha+1}{2}} \, \mathsf{S}_{\alpha,\beta} = \begin{cases} +(-1)^{\frac{\beta-\alpha+1}{2}} & \text{if } \alpha> \beta ,\\ -(-1)^{\frac{\beta-\alpha+1}{2}} & \text{if } \alpha < \beta . \end{cases} \end{align} $$

The above ODE can be rewritten as a first-order $r \times r$ system:

(5.12) $$ \begin{align} \hbar \frac{d}{dx} \Psi(x;\hbar) = \mathcal{D}(x) \Psi(x;\hbar) , \qquad\qquad \mathcal{D}(x;\hbar) = \begin{pmatrix} 0 & 1 & & \\ & \ddots & \ddots & \\ & & \ddots & 1 \\ x & & & 0 \end{pmatrix} . \end{align} $$

A simple computation shows that the matrix $\Phi := \Psi ^{-t}$ , called the dual wave matrix, satisfies the differential system $-\hbar \frac {d}{dx}\Phi = \mathcal {D}^{t} \Phi $ . In other words, $\Phi $ is a (permutation of the) companion matrix associated to the higher Airy ODE, with $\hbar \mapsto - \hbar $ . In order to fix the normalisation, it is sufficient to compute $\Phi $ at $x \to \infty $ . We find that

(5.13) $$ \begin{align} \Phi(x;\hbar) = \left( (-1)^{r-\alpha} \phi_{\alpha}^{(r-m-1)} \right)_{\substack{ m = 0,\dots,r-1 \\ \alpha = 1,\dots,r}} , \qquad \phi_{\alpha}^{(k)}(x;\hbar) := \psi_{\alpha}^{(k)}(x;-\hbar) . \end{align} $$

Since the $\phi $ ’s are obtained from the $\psi $ ’s simply by substituting $\hbar $ with $-\hbar $ , they enjoy similar properties. In particular, we can deduce that, after separating the exponential part as $\phi _{\alpha }^{(m)} =: e^{+V_{\alpha }/\hbar } \, \widetilde {\phi }_{\alpha }^{(m)}$ , the Borel transform of $\widetilde {\psi }_{\alpha }^{(m)}$ is simple resurgent with logarithmic singularities at $s = - A_{\alpha ,\beta }$ for $\beta \neq \alpha $ . Its behaviour is given by

(5.14) $$ \begin{align} \begin{aligned} \widehat{\phi}_{\alpha}(x;s) & = - \frac{S_{\alpha,\beta}}{2\pi} \, \widehat{\phi}_{\beta}\big( x;s + A_{\alpha,\beta}(x) \big) \log\big( s + A_{\alpha,\beta}(x) \big) + \text{holomorphic at } -A_{\alpha,\beta}(x) , \end{aligned} \end{align} $$

where the Stokes constants are the same as before.

5.2 Higher Airy correlators

The Lie algebra $\mathfrak {sl}_r(\mathbb {C})$ admits the root space decomposition into traceless diagonal matrices, upper-diagonal matrices, and lower-diagonal matrices: $\mathfrak {sl}_r(\mathbb {C}) = \mathfrak {h} \oplus \mathfrak {n}_+ \oplus \mathfrak {n}_-$ . A basis of $\mathfrak {h}$ is given in terms of elementary matrices $e_{i,j} = (\delta _{i,k} \delta _{j,l})$ by

(5.15) $$ \begin{align} E_a := e_{a,a} - e_{a+1,a+1} \in \mathfrak{h} , \qquad\qquad a = 1,\dots,r-1 . \end{align} $$

Define the matrix

(5.16) $$ \begin{align} M(x;\hbar) := \Psi(x;\hbar) E \Psi^{-1}(x;\hbar) , \qquad\qquad E := \frac{1}{r} \sum_{a=1}^{r-1} a E_a , \end{align} $$

which is an $\mathfrak {h}$ -valued formal power series in $\hbar $ . Indeed, as in the Airy case, M contains only quadratic expressions in the functions $\psi $ ’s and $\phi $ ’s with opposite exponential parts. The result is then a formal power series in $\hbar $ . As before, one can consider the n-point r-Airy correlators defined through determinantal formulae:

(5.17) $$ \begin{align} \begin{aligned} & W_1(x_1;\hbar) := - \frac{1}{\hbar} \operatorname{\mathrm{Tr}}{\big( \mathcal{D}(x_1) M(x_1;\hbar) \big)} , \\ & W_n(x_1,\dots,x_n;\hbar) := (-1)^{n-1} \sum_{\sigma \in S_n^{\text{cyc}}} \frac{\operatorname{\mathrm{Tr}}{ \big( \prod_{i=1}^n M(x_{\sigma^i(1)};\hbar) } \big) }{\prod_{i=1}^n ( x_{i} - x_{\sigma(i)} )} \qquad\quad \text{for }n \ge 2 . \end{aligned} \end{align} $$

Again, the genus expansion of the correlators stores interesting enumerative invariants, the r-spin intersection numbers. The determinantal formula for such intersection numbers was proved by Bertola–Dubrovin–Yang [Reference Bertola, Dubrovin and Yang13, Reference Bertola, Dubrovin and Yang14] as a consequence of the Faber–Shadrin–Zvonkine result [Reference Faber, Shadrin and Zvonkine43]. Adjusted to our chosen normalisation, and with the insertion of $\hbar $ by homogeneity, the result of Bertola–Dubrovin–Yang reads as follows.

Theorem 5.1 (Genus expansion of the higher Airy correlators [Reference Bertola, Dubrovin and Yang13, Reference Bertola, Dubrovin and Yang14])

The higher Airy correlators $W_{n}$ admit the following $\hbar $ -expansion:

(5.18) $$ \begin{align} W_{n}(\boldsymbol{x};\hbar) = \sum_{g = 0}^{\infty} \hbar^{2g-2+n} \, W_{g,n}(\boldsymbol{x}) . \end{align} $$

Moreover, $W_{g,n}$ stores r-spin intersection numbers: if $2g-2+n>0$ ,

(5.19) $$ \begin{align} W_{g,n}(\boldsymbol{x}) = \!\!\! \sum_{\substack{ d_{1},\dots,d_n \ge 0 \\ a_1,\dots,a_n \in \{1,\dots,r-1\} \\ r|d|+|a| = (r+1)(2g-2+n) }} \!\!\! (-r)^{g-1-|d|} \langle{\tau_{d_1,a_1} \cdots \tau_{d_n,a_n}}\rangle^{r\text{-spin}} \prod_{i=1}^n \frac{(rd_i+a_i)!_{(r)}}{r \, x_i^{d_i+\frac{a_i}{r}+1}} . \end{align} $$

Here $m!_{(r)}$ denotes the r-factorial, see equation (1.11). The degree condition is equivalent to $D_{g,n}^r(a) + |d| = 3g-3+n$ , where $D_{g,n}^r(a)$ is the complex cohomological degree of the Witten class.

Remark 5.2 (Connection with topological recursion)

For the reader familiar with topological recursion, $W_{g,n}$ coincides with the topological recursion differentials computed from the r-Airy spectral curve $( \mathbb {P}^1, \, x(z) = z^r, \, y(z) = z, \, B(z_1,z_2) = \frac {dz_1 dz_2}{(z_1 - z_2)^2} )$ – see [Reference Bouchard and Eynard18]:

(5.20) $$ \begin{align} \omega_{g,n}(\boldsymbol{z}) = W_{g,n}(\boldsymbol{x}) \, dx_1 \cdots dx_n \big|_{x_i = z_i^r} . \end{align} $$

Throughout this section, we will work with a fixed choice of r-th root of x.

The n-point correlators can be expressed through determinantal formulae in the kernel form. We omit the proof, which is completely analogous to that of lemma 3.5.

Lemma 5.3. For $n \ge 2$ , the n-point correlators are given by the two equivalent expressions

(5.21) $$ \begin{align} W_n(\boldsymbol{x};\hbar) = (-1)^{n-1} \sum_{\sigma \in S_n^{\text{cyc}}} \prod_{i=1}^n K_{+,-}(x_i,x_{\sigma(i)};\hbar) = (-1)^{n-1} \sum_{\sigma \in S_n^{\text{cyc}}} \prod_{i=1}^n K_{-,+}(x_i,x_{\sigma(i)};\hbar) , \end{align} $$

where the kernels $K_{\pm }$ are defined by

(5.22) $$ \begin{align} \begin{aligned} K_{+,-}(x,y;\hbar) & := \frac{ \sum_{m=0}^{r-1} \widetilde{\psi}_{r}^{(m)}(x;\hbar) \, \widetilde{\phi}_{r}^{(r-1-m)}(y;\hbar) }{x-y} , \\ K_{-,+}(x,y;\hbar) & := - \frac{ \sum_{m=0}^{r-1} \widetilde{\phi}_{r}^{(m)}(x;\hbar) \, \widetilde{\psi}_{r}^{(r-1-m)}(y;\hbar) }{x-y} . \end{aligned} \end{align} $$

Moreover, the kernels are related by the parity relation $K_{-,+}(x,y;\hbar ) = - K_{+,-}(x,y;-\hbar )$ .

5.3 Singularity structure and large genus asymptotics

From the usual properties of the Borel transform, we immediately find the singularity structure of $W_n$ in the Borel plane from the knowledge of the singularity structure of the functions $\widehat {\psi }$ and $\widehat {\phi }$ . Again, we assume $(x_1,\dots ,x_n)$ in a generic position.

Proposition 5.4 (Singularity structure of the higher Airy correlators)

The Borel transform $\widehat {W}_n(\boldsymbol {x};s)$ of the n-point r-Airy correlator is simple resurgent. More precisely, on the principal sheet, $\widehat {W}_n$ has $2(r-1)n$ logarithmic singularities located at $s = \pm A_{r,\alpha }(x_i)$ (with $\alpha = 1,\dots ,r-1$ and $i = 1,\dots ,n$ ) and such that

(5.23) $$ \begin{align} \widehat{W}_{n}(\boldsymbol{x};s) = - \frac{S_{r,\alpha}}{2\pi} \, \widehat{W}_{n}^{(\pm\alpha,i)}\big( \boldsymbol{x}; s \mp A_{r,\alpha}(x_i) \big) \log\big( s \mp A_{r,\alpha}(x_i) \big) + \text{holomorphic at } \pm A_{r,\alpha}(x_i) , \end{align} $$

where:

• • $A_{r,\alpha }(x) = \frac {r}{r+1} (1 - \zeta ^{\alpha }) x^{(r+1)/r}$ and $S_{r,\alpha } = (-1)^{\frac {\alpha -r+1}{2}}$ are the instanton actions and the Stokes constants associated to the r-Airy functions;

• • the minor $\widehat {W}^{(\pm \alpha ,i)}_{n}$ is the Borel transform of the formal power series

  (5.24) $$ \begin{align} W_{n}^{(\pm \alpha,i)}( \boldsymbol{x}; \hbar ) := (-1)^{n-1} \sum_{\sigma \in S_n^{\text{cyc}}} K_{\pm\alpha,\mp}(x_i,x_{\sigma(i)};\hbar) \prod_{j \ne i} K_{\pm,\mp}(x_j,x_{\sigma(j)};\hbar) , \end{align} $$
  where the kernels $K_{\pm \alpha ,\mp }$ are given by
  (5.25) $$ \begin{align} \begin{aligned} K_{+\alpha,-}(x,y;\hbar) & := \frac{ \sum_{m=0}^{r-1} \widetilde{\psi}_{\alpha}^{(m)}(x;\hbar) \, \widetilde{\phi}_{r}^{(r-1-m)}(y;\hbar) }{x-y} , \\ K_{-\alpha,+}(x,y;\hbar) & := - \frac{ \sum_{m=0}^{r-1} \widetilde{\phi}_{\alpha}^{(m)}(x;\hbar) \, \widetilde{\psi}_{r}^{(r-1-m)}(y;\hbar) }{x-y} . \end{aligned} \end{align} $$

Moreover, the kernels are related by the parity relation $K_{-\alpha ,-}(x,y;\hbar ) = - K_{+\alpha ,+}(x,y;-\hbar )$ . Hence, the analogous relation holds for the correlators: $W^{(-\alpha ,i)}_{n}( \boldsymbol {x}; \hbar ) = (-1)^n \, W^{(+\alpha ,i)}_{n}( \boldsymbol {x}; -\hbar )$ .

Again, since $W^{(\pm \alpha , i)}_{n}$ are related by the parity relation, we can simply use $W_n^{(+\alpha ,i)}$ and drop the ‘ $+$ ’ symbol from the superscript. From the analysis of the singularity structure of $W_n$ in the Borel plane, we find the large genus asymptotics of its coefficients through the Borel transform method.

Proposition 5.5 (Large genus asymptotics of the r-Airy correlators)

The large genus asymptotics of the expansion coefficients of the n-point r-Airy correlators is given by

(5.26) $$ \begin{align} W_{g,n}(\boldsymbol{x}) = \sum_{\alpha=1}^{r-1} \frac{S_{r,\alpha}}{\pi} \sum_{i=1}^n \frac{\Gamma(2g-2+n)}{A_{r,\alpha}(x_i)^{2g-2+n}} \left( \sum_{k = 0}^K \frac{A_{r,\alpha}(x_i)^{k}}{(2g-3+n)^{\underline{k}}} W^{(\alpha,i)}_{k,n}(\boldsymbol{x}) +\mathrm{O}\bigg( \frac{1}{g^{K+1}} \bigg) \right) . \end{align} $$

Moreover, the leading term is explicitly given by

(5.27) $$ \begin{align} W_{g,n}(\boldsymbol{x}) & = \frac{(-1)^{g-1+n}}{\pi} \frac{ \Gamma(2g-2+n) }{ \left( \frac{r}{r+1} \right)^{2g-2+n} } \frac{2^n}{r^n x_1 \cdots x_n} \nonumber\\ &\qquad \times \Bigg[ \sum_{\alpha = 1}^{ \left\lfloor {\frac{r-1}{2}} \right\rfloor } \frac{(-1)^{\alpha n}}{\left( 2\sin(\frac{\alpha}{r} \pi) \right)^{2g-1+n}} \Big( h_{(r+1)(2g-2+n)}^{(r,\alpha)}(\boldsymbol{x}^{-1/r}) + \mathrm{O}\big( g^{-1} \big) \Big) \nonumber\\ &\qquad\qquad\qquad + \frac{\delta_{r}^{\text{even}}}{2} \, \frac{(-1)^{\frac{r}{2}n}}{2^{2g-1+n}} \Big( h_{(r+1)(2g-2+n)}^{(r,\frac{r}{2})}(\boldsymbol{x}^{-1/r}) + \mathrm{O}\big( g^{-1} \big) \Big) \Bigg] . \end{align} $$

Here $\delta _{r}^{\text {even}}$ gives one if r is even and zero otherwise, and $h_{D}^{(r,\alpha )}(\boldsymbol {u})$ denotes the polynomial

(5.28) $$ \begin{align} h_{D}^{(r,\alpha)}(\boldsymbol{u}) := \sum_{ \substack{k_1,\dots,k_n \ge 0 \\ |k| = D} } \prod_{i=1}^n \sin\Big( \frac{\alpha k_i}{r}\pi \Big) u_i^{k_i} = \sum_{ \substack{d_1,\dots,d_n \ge 0 \\ a_1,\dots,a_n \in \{1,\dots,r-1\} \\ r|d|+|a| = D} } (-1)^{\alpha|d|} \prod_{i=1}^n \sin\left( \frac{\alpha a_i}{r}\pi \right) u_i^{r d_i+a_i} . \end{align} $$

Proof. Equation (5.26) follows from theorem 2.5 and the parity relation. For the leading term, notice that $\widetilde {\psi }_r^{(m)}(x;\hbar ) = \frac {1}{\sqrt {r}} x^{-\frac {r-1-2m}{2r}} + \mathrm {O}(\hbar )$ and $\widetilde {\phi }_r^{(r-1-m)}(y;\hbar ) = \frac {1}{\sqrt {r}} y^{\frac {r-1-2m}{2r}} + \mathrm {O}(\hbar )$ . Thus,

$$\begin{align*}K_{+,-}(x,y;\hbar) = \frac{1}{r} \frac{1}{x-y} \left( \frac{x}{y} \right)^{\frac{1-r}{2m}} \sum_{m=0}^{r-1} \left( \frac{x}{y} \right)^{\frac{m}{r}} + \mathrm{O}(\hbar) = \frac{1}{r} \frac{(xy)^{\frac{1-r}{2r}}}{x^{1/r} - y^{1/r}} + \mathrm{O}(\hbar) . \end{align*}$$

A similar computation shows that $K_{+\alpha ,-}(x,y;\hbar ) = - \frac {(-1)^{\frac {r-\alpha }{2}}}{r} \frac {(xy)^{\frac {1-r}{2r}}}{\zeta ^{\alpha /2} x^{1/r} - \zeta ^{-\alpha /2} y^{1/r}} + \mathrm {O}(\hbar )$ . Inserting the above expressions in the definition of $W^{(\alpha ,i)}_{n}$ , we find

$$\begin{align*}\begin{aligned} W^{(\alpha,i)}_{0,n}(\boldsymbol{x}) &= (-1)^{n-1} \frac{- (-1)^{\frac{r-\alpha}{2}}}{r^n (x_1 \cdots x_n)^{\frac{r-1}{r}}} \sum_{\sigma \in S_n^{\text{cyc}}} \frac{1}{\zeta^{\alpha/2} x_i^{1/r} - \zeta^{-\alpha/2} x_{\sigma(i)}^{1/r}} \prod_{j \ne i} \frac{1}{ x_j^{1/r} - x_{\sigma(j)}^{1/r}} \\ &= (-1)^{n-1} \frac{(-1)^{\frac{r-\alpha}{2}}}{r^n (x_1 \cdots x_n)^{\frac{r-1}{r}}} \frac{ \zeta^{\frac{\alpha}{2}} (1-\zeta^{\alpha})^{n-2} \, x_i^{\frac{n-2}{r}} }{ \prod_{j \ne i} (x_i^{1/r} - x_j^{1/r})(\zeta^{\alpha} x_i^{1/r} - x_j^{1/r}) } . \end{aligned} \end{align*}$$

In the second line, we applied lemma A.1 to simplify the sum over permutations. Recalling the values of $S_{r,\alpha } = (-1)^{\frac {\alpha -r+1}{2}}$ and $A_{r,\alpha }(x) = \frac {r}{r+1} (1 - \zeta ^{\alpha }) x^{(r+1)/r}$ , we find

$$ \begin{align*} W_{g,n}(\boldsymbol{x}) & = \frac{(-1)^{n}}{\pi\mathrm{i} \, r^n (x_1 \cdots x_n)^{\frac{r-1}{r}}} \frac{\Gamma(2g-2+n)}{\left( \frac{r}{r+1} \right)^{2g-2+n}} \sum_{\substack{\alpha = 1,\dots,r-1 \\ i = 1,\dots,n}} \left( \frac{ \zeta^{\frac{\alpha}{2}} (1-\zeta^{\alpha})^{-2g} \, x_i^{-\frac{(r+1)(2g-2+n)+n-2}{r}} }{ \prod_{j \ne i} (x_i^{1/r} - x_j^{1/r})(\zeta^{\alpha} x_i^{1/r} - x_j^{1/r}) } + \mathrm{O}\big( g^{-1} \big) \right) \\ & = \frac{(-1)^n}{\pi \mathrm{i} \, r^n (x_1 \cdots x_n)^{\frac{r-1}{r}}} \frac{\Gamma(2g-2+n)}{\left( \frac{r}{r+1} \right)^{2g-2+n}} \sum_{i=1}^n \Bigg( \sum_{\alpha = 1}^{ \left\lfloor {\frac{r-1}{2}} \right\rfloor } \Bigg( \frac{ \zeta^{\frac{\alpha}{2}} (1-\zeta^{\alpha})^{-2g} }{ \prod_{j \ne i} (x_i^{1/r} - x_j^{1/r})(\zeta^{\alpha} x_i^{1/r} - x_j^{1/r}) } \\ &\qquad\qquad\qquad\qquad - \frac{ \zeta^{-\frac{\alpha}{2}} (1-\zeta^{-\alpha})^{-2g} }{ \prod_{j \ne i} (x_i^{1/r} - x_j^{1/r})(\zeta^{-\alpha} x_i^{1/r} - x_j^{1/r}) } \Bigg) x_i^{-\frac{(r+1)(2g-2+n)+n-2}{r}} \\ &\qquad\qquad\qquad\qquad + \delta_{r}^{\text{even}} \frac{ \mathrm{i} \, 2^{-2g} }{ \prod_{j \ne i} (x_i^{1/r} - x_j^{1/r})(- x_i^{1/r} - x_j^{1/r}) } x_i^{-\frac{(r+1)(2g-2+n)+n-2}{r}} + \mathrm{O}\big( g^{-1} \big) \Bigg) . \end{align*} $$

In the second line, we have rearranged the sum as

$$\begin{align*}\sum_{\alpha=1}^{r-1} w_{\alpha} = \sum_{\alpha=1}^{ \left\lfloor {\frac{r-1}{2}} \right\rfloor } ( w_{\alpha} + w_{r-\alpha}) + \delta_{r}^{\text{even}} \, w_{\frac{r}{2}} . \end{align*}$$

The reason behind it is that the singularities corresponding to $(1-\zeta ^{\alpha })$ and $(1-\zeta ^{r-\alpha })$ are complex conjugate, hence at the same distance from the origin (cf. figure 1). It is natural to combine them, since they contribute to the same leading asymptotic. We can now perform the sum over i employing lemma A.2 to get the thesis.

As a consequence, we obtain the following asymptotic formula for r-spin intersection numbers, expressing all $\mathrm {O}(1)$ terms for each exponential contribution.

Corollary 5.6 (Leading large genus asymptotics for r-spin intersection numbers)

For any given $n \ge 1$ and $a_1,\dots ,a_n \in \{1,\dots ,r-1\}$ , uniformly in $d_1,\dots ,d_n$ as $g \to \infty $ :

(5.29) $$ \begin{align} \langle\tau_{d_1,a_1} & \cdots \tau_{d_n,a_n}\rangle^{r\text{-spin}} \prod_{i=1}^n (rd_i + a_i)!_{(r)} \nonumber\\ & = \frac{2^n}{2\pi} \, \frac{\Gamma(2g-2+n)}{r^{g-1-|d|}} \Bigg[ \sum_{\alpha = 1}^{ \left\lfloor {\frac{r-1}{2}} \right\rfloor } \frac{ (-1)^{(\alpha-1)(|d|+n)} }{ \left( \frac{2r}{r+1} \sin(\frac{\alpha}{r}\pi) \right)^{2g-2+n} } \Bigg( \frac{\prod_{i=1}^n \sin(\frac{\alpha a_i}{r}\pi)}{\sin(\frac{\alpha}{r}\pi)} + \mathrm{O}\big( g^{-1} \big) \Bigg) \nonumber\\ &\qquad\qquad\qquad + \frac{\delta_{r}^{\text{even}}}{2} \, \frac{ (-1)^{(\frac{r}{2} - 1)(|d|+n)} }{ \left( \frac{2r}{r+1} \right)^{2g-2+n} } \Bigg( \prod_{i=1}^n \sin\left( \tfrac{a_i}{2}\pi \right) + \mathrm{O}\big( g^{-1} \big) \Bigg) \Bigg] . \end{align} $$

Remark 5.7. At a practical level, only the term corresponding to $\alpha = 1$ contributes in the asymptotic formula (for $r> 2$ ), since all other values of $\alpha $ give exponentially suppressed terms. In this case, the formula simplifies to

(5.30) $$ \begin{align} \langle{\tau_{d_1,a_1} \cdots \tau_{d_n,a_n}}\rangle^{r\text{-spin}}& \prod_{i=1}^n (rd_i + a_i)!_{(r)} \nonumber\\ & = \frac{2^n}{2\pi \sin(\frac{\pi}{r})} \, \frac{ \Gamma(2g-2+n) }{ r^{g-1-|d|} \, \left( \frac{2r}{r+1} \sin(\frac{\pi}{r}) \right)^{2g-2+n} } \Bigg( \prod_{i=1}^n \sin\left( \tfrac{a_i}{r}\pi \right) + \mathrm{O}\big( g^{-1} \big) \Bigg) . \end{align} $$

Specialised to $n = 1$ , the above asymptotic formula retrieves the one proved by Dubrovin–Yang–Zagier in [Reference Dubrovin, Yang and Zagier34, theorem 5 (vii)]. We also emphasise that, despite being exponentially suppressed, the terms corresponding to $\alpha> 1$ can be detected numerically as explained in appendix C.

Subleading contributions

Similarly to the $r = 2$ case, the minors exhibit a distinct pole structure and polynomiality structure. We provide a brief outline of the proof, as it resembles the $r = 2$ case with a few minor variations that we outline below.

Proposition 5.8. For any $k \ge 0$ , $n \ge 2$ , and $\alpha = 1,\dots , r-1 $ :

(5.31) $$ \begin{align} W_{k,n}^{(\alpha,i)}(\boldsymbol{x}) = \frac{(-1)^{n-1+\frac{r-\alpha}{2}}}{r^n (x_1 \cdots x_n)^{\frac{r-1}{r}}} & \frac{ \zeta^{\alpha/2} (1-\zeta^{\alpha})^{n-2} \, x_i^{\frac{n-2-(r+1)k}{r}} }{ \prod_{j \ne i} (x_i^{1/r} - x_j^{1/r})(\zeta^{\alpha}x_i^{1/r} - x_j^{1/r}) } \nonumber\\ &\qquad\qquad\qquad \times R_{k,n-1}^{(\alpha)}\left( \Big( \frac{x_i}{x_1} \Big)^{1/r} , \dots, \widehat{ \Big( \frac{x_i}{x_i} \Big)^{1/r} } , \dots, \Big( \frac{x_i}{x_n} \Big)^{1/r} \right), \end{align} $$

where $R_{k,n-1}^{(\alpha )}$ is a symmetric polynomial of degree $\le \min \{ (r+1)k + n - 2, 2((r+1)k - 1)\}$ and degree $\le (r+1)k$ in each individual variable satisfying the $\mathbb {Z}/r\mathbb {Z}$ -symmetry

(5.32) $$ \begin{align} (-\zeta^{\alpha})^k \cdot R_{k,n}^{(\alpha)}(\zeta^{-\alpha}u_1, \dots, \zeta^{-\alpha}u_n) = R_{k,n}^{(r-\alpha)}(u_1, \dots, u_n) . \end{align} $$

The sequence of polynomials $(R_{k,n}^{(\alpha )})_{n \ge 1}$ satisfies the following specialisation properties:

(5.33) $$ \begin{align} \begin{aligned} & R_{k,n+1}^{(\alpha)}(u_1,\dots, u_n, u_{n+1}) \big|_{u_{n+1} = 1} = R_{k,n}^{(\alpha)}(u_1,\dots,u_n) , \\ & R_{k,n+1}^{(\alpha)}(u_1,\dots, u_n, u_{n+1}) \big|_{u_{n+1} = \zeta^{-\alpha}} = R_{k,n}^{(\alpha)}(u_1,\dots,u_n) . \end{aligned} \end{align} $$

In turn, the degree condition and the specialisation properties uniquely determine $R_{k,n}^{(\alpha )}$ for all n by an explicit algorithm from the first terms of the sequenceFootnote ⁷ . Moreover, the coefficients of $R_{k,n}^{(\alpha )}$ in the monomial basis are polynomials of n.

Sketch of the Proof

The Wronskian condition appearing in the residue computation of the kernels (cf. lemma 3.10) is no longer the key fact. This is because, for $\Psi \in \mathrm {SL}(2,\mathbb {C})$ , matrix elements of $\Psi ^{-1}$ are linear in the matrix elements of $\Psi $ . However, this linearity does not hold for elements of $\mathrm {SL}(r,\mathbb {C})$ with $r> 2$ . Nonetheless, we can employ the relation $\Phi ^t \Psi = \mathrm {Id}$ , which indeed yields quadratic relations that generalise the Wronskian condition. This, together with homogeneity arguments, yields equation (5.31) and the specialisation properties. From the specialisation properties, one can prove that the degree of $R_{k,n}^{(\alpha )}$ is eventually independent of n, and more precisely

$$\begin{align*}\deg{( R_{k,n}^{(\alpha)} )} \le \min\{ (r+1)k + n - 1, 2(r+1)k - 2 \} . \end{align*}$$

Thus, from the degree bound and the specialisation properties, we get an algorithm for computing $(R_{k,n}^{(\alpha )})_{n \ge 1}$ from the first few values of n, together with the polynomiality statement for the expansion coefficients in the monomial basis. As for the $\mathbb {Z}/r\mathbb {Z}$ -symmetry, which substitutes evenness for the corresponding polynomials in the $r=2$ case, it follows from analogous relations between the wave functions.

As a consequence of the above result, one can prove that the subleading corrections to $W_{g,n}$ exhibit the following structure. Firstly, let us group together subleading corrections in accordance with the absolute value of the instanton actions. In other words, let us group together terms corresponding to $\alpha $ and $r-\alpha $ :

(5.34) $$ \begin{align} W_{g,n}(\boldsymbol{x}) = \frac{(-1)^n}{\pi} \Gamma(2g-&2+n) \Bigg[ \sum_{\alpha=1}^{ \left\lfloor {\frac{r-1}{2}} \right\rfloor } \Bigg( \sum_{k=0}^K \frac{1}{(2g-3+n)^{\underline{k}}} \frac{V_{k,g,n}^{(\alpha)}(\boldsymbol{x})}{x_1 \cdots x_n} + \mathrm{O}\bigg( \frac{1}{|A_{r,\alpha}(x_i)|^{2g} g^{K+1}} \bigg) \Bigg) \nonumber\\ & + \frac{\delta_{r}^{\text{even}}}{2} \Bigg( \sum_{k=0}^K \frac{1}{(2g-3+n)^{\underline{k}}} \frac{V_{k,g,n}^{(\frac{r}{2})}(\boldsymbol{x})}{x_1 \cdots x_n} + \mathrm{O}\bigg( \frac{1}{|A_{r,\frac{r}{2}}(x_i)|^{2g} g^{K+1}} \bigg) \Bigg) \Bigg], \end{align} $$

where we have set

(5.35) $$ \begin{align} V_{k,g,n}^{(\alpha)}(\boldsymbol{x}) := (-1)^n \, x_1 \cdots x_n \sum_{i=1}^n \left( S_{r,\alpha} \frac{W_{k,n}^{(\alpha,i)}(\boldsymbol{x})}{A_{r,\alpha}(x_i)^{2g-2+n-k}} + S_{r,r-\alpha} \frac{W_{k,n}^{(r-\alpha,i)}(\boldsymbol{x})}{A_{r,r-\alpha}(x_i)^{2g-2+n-k}} \right) . \end{align} $$

With the residue strategy of lemmata 3.11 and A.2, but taking $f_{\alpha }(q) = \frac {1}{\pi \mathrm {i}} \frac {\zeta ^{\alpha /2} (1-\zeta ^{\alpha })^{k+1-2g}}{\prod _{i=1}^n (q-z_i)(\zeta ^{\alpha }q-z_i)} R_{k,n}^{(\alpha )}(q \boldsymbol {z}^{-1})$ , we find

(5.36) $$ \begin{align} V_{k,g,n}^{(\alpha)}(\boldsymbol{x}) = (-1)^{g-1+\alpha n} & \frac{2^n}{2 \sin(\frac{\alpha}{r}\pi)} \frac{1}{r^n \left( \frac{2r}{r+1} \sin(\frac{\alpha}{r}\pi) \right)^{2g-2+n-k}} \nonumber\\ & \sum_{d= 0}^{\min{\{ (r+1)k + n - 1, 2(r+1)k - 2 \}} } \bar{R}_{k,n}^{(\alpha,d)}(\boldsymbol{x}^{-1/r}) \, h^{(r,\alpha)}_{(r+1)(2g-2+n)-d}(\boldsymbol{x}^{-1/r}), \end{align} $$

where $h^{(r,\alpha )}_{D}$ is the polynomial defined in equation (5.28), $\bar {R}_{k,n}^{(\alpha ,d)}$ is the homogeneous component of $R_{k,n}^{(\alpha )}$ of degree d, normalised according to the following formula:

(5.37) $$ \begin{align} R_{k,n}^{(\alpha)}(q\boldsymbol{u}) = (-1)^{\frac{k}{2}} \sum_{d \ge 0} q^d \, \zeta^{\frac{\alpha (d-k)}{2}} \bar{R}_{k,n}^{(\alpha,d)}(\boldsymbol{u}) . \end{align} $$

Except for having to group together instanton actions according to their distance from the origin, we highlight two more differences between the $r = 2$ case and the general r case. Firstly, the $\mathbb {Z}/r\mathbb {Z}$ -symmetry, which substitutes the evenness property, is a much weaker constraint. Thus, some simplifications no longer occur. Secondly, the complete homogeneous polynomial is essentially substituted by the polynomials $h^{(r,\alpha )}_D$ . While in the complete homogeneous polynomial $h_D$ all monomials of degree D appear with coefficient $1$ , in $h^{(r,\alpha )}_D$ all monomials appear with coefficient $\sin (\frac {\alpha k}{r}\pi )$ , where k is the exponent of the corresponding variable. The latter depends on the (parity of the) quotient and on the remainder of the Euclidean division of k by r: writing $k = r[k] + \langle {k}\rangle $ for $[k] \ge 0$ and $\langle {k}\rangle \in \{ 0,\dots ,r-1 \}$ ,

(5.38) $$ \begin{align} \sin\left( \frac{\alpha k}{r}\pi \right) = (-1)^{\alpha [k]} \sin\left( \frac{\alpha \! \langle{k}\rangle}{r}\pi \right) . \end{align} $$

This last difference complicates the extraction of coefficients from products of the form $m_{\nu } h^{(r,\alpha )}_{D}$ . Nonetheless, the strategy employed in appendix B can still be adapted to obtain the following formula. For all partitions $\mu = (\mu _1,\dots ,\mu _n)$ and $\nu = (\nu _1,\dots ,\nu _n)$ (zero parts are allowed):

(5.39) $$ \begin{align} \big[ u_1^{\mu_1} \cdots u_n^{\mu_n} \big] m_{\nu} h^{(r,\alpha)}_{|\mu| - |\nu|} = (-1)^{\alpha ([\mu] - [\nu])} \sum_{ A \in \{-(r-1),\dots,r-1\}^n } M_{n,\mu,\nu}^{(r,\alpha)}(A) \, \prod_{i=1}^n \sin\left( \frac{\alpha A_i}{r}\pi \right) . \end{align} $$

Here $[\mu ] = \sum _{i=1}^n [\mu _i]$ is the sum of the quotients of the Euclidean division of the parts $\mu _i$ by r, and the quantity $M_{n,\mu ,\nu }^{(r,\alpha )}(A)$ is the explicit polynomial in the multiplicities $p_{m}(\mu )$ given by

(5.40) $$ \begin{align} M_{n,\mu,\nu}^{(r,\alpha)} := \frac{1}{z_{\nu}} \prod_{k=0}^{\nu_1} \prod_{s = -(r-1)}^{r-1} \left( n - \sum_{i=0}^{k-1} p_i(\mu) - \sum_{\substack{ i=k, \dots, \mu_1 \\ i \not\equiv s-k \pmod{r}}} p_{i}(\mu) - \sum_{j = k+1}^{\nu_1} p_{j,s-k-j}(\nu,A) \right)^{\underline{p_{k,s}(\nu,A)}} , \end{align} $$

and $p_{k,s}(\nu ,A) := \#\{ i | \nu _i = k \,\wedge \, A_i = s \}$ is the joint multiplicity of $(k,s)$ in $(\nu ,A)$ .

As a consequence, we obtain the following asymptotic formula for the r-spin intersection numbers. We omit the proof, which is parallel to that of theorem 3.13.

Theorem 5.9 (Large genus asymptotics for r-spin intersection numbers)

For any given $n \ge 1$ and $a_1,\dots ,a_n \in \{1,\dots ,r-1\}$ , uniformly in $d_1,\dots ,d_n$ as $g \to \infty $ :

(5.41) $$ \begin{align} & \langle{\tau_{d_1,a_1} \cdots \tau_{d_n,a_n}}\rangle^{r\text{-spin}} \prod_{i=1}^n (rd_i + a_i)!_{(r)} \nonumber\\ &\quad = \frac{2^n}{2\pi} \, \frac{\Gamma(2g-2+n)}{r^{g-1-|d|}} \Bigg[ \sum_{\alpha = 1}^{ \left\lfloor {\frac{r-1}{2}} \right\rfloor } \frac{ 1 }{ \left( \frac{2r}{r+1} \sin(\frac{\alpha}{r}\pi) \right)^{2g-2+n} } \Bigg( \sum_{k=0}^K \frac{ ( \frac{2r}{r+1} \sin(\frac{\alpha}{r}\pi) )^{k} }{(2g-3+n)^{\underline{k}}} \, \gamma^{(r,\alpha)}_{k} + \mathrm{O}\bigg( \frac{1}{g^{K+1}} \bigg) \Bigg) \nonumber\\ &\qquad\qquad\qquad + \frac{\delta_{r}^{\text{even}}}{2} \, \frac{ 1 }{ \left( \frac{2r}{r+1} \right)^{2g-2+n} } \Bigg( \sum_{k=0}^K \frac{ ( \frac{2r}{r+1} )^{k} }{(2g-3+n)^{\underline{k}}} \, \gamma^{(r,\frac{r}{2})}_{k} + \mathrm{O}\bigg( \frac{1}{g^{K+1}} \bigg) \Bigg) \Bigg] . \end{align} $$

Each $\gamma ^{(r,\alpha )}_{k}$ is a function of n and the multiplicities $p_{m} = \#\{i | rd_i + a_i = m\}$ . Moreover, it depends polynomially on the multiplicities, up to an overall factor of $(-1)^{\alpha |d|}$ .

Remark 5.10. Contrary to the case $r = 2$ , it is not clear whether the subleading corrections $\gamma ^{(r,\alpha )}_{k}$ have a polynomiality behaviour in n. Moreover, from equation (5.40), there seems to be no bound on the number of multiplicities appearing in $\gamma ^{(r,\alpha )}_{k}$ (unless some vanishing condition would hold). Nonetheless, our approach provides an algorithm to compute the subleading corrections for any value of k.