these functions, we opt for keeping our expressions uncluttered at the expense of slight abuse of notation. We denote the pullbacks to both $\Omega_{\pm}$ and $\tilde{\Omega}_{\pm}$ by $\hat{\varphi}^{(\pm)} = (\hat{\varphi}_{+}^{(\pm)}, \hat{\varphi}_{-}^{(\pm)})$, and we continue to denote the pullbacks to both $\Omega$ and $\tilde{\Omega}$ simply by $\hat{\varphi}$. In other words, from now on, the symbol “$\hat{\varphi}$” denotes both the holomorphic vector-valued function of $(z, \xi) \in \Xi \subset \mathbb{C}_z \times \mathbb{C}_{\xi}$, the corresponding holomorphic vector-valued function on $\Omega \subset B$ obtained via pullback by $\varrho$, as well as the holomorphic vector-valued function on $\tilde{\Omega} \subset B$ obtained via pullback by $\varrho$, with the intended meaning always clear from the context. Similarly, for the symbols “$\hat{\varphi}^{(\pm)}$”. We summarise these conventions as follows:

$$ \begin{aligned} \hat{\varphi}^{(\pm)} &= (\hat{\varphi}_{+}^{(\pm)}, \hat{\varphi}_{-}^{(\pm)}) := \varrho_{\pm}^{*} \hat{\varphi} & \text{as an element of } \mathcal{O}(\Omega_{\pm}, \mathbb{C}^2) \text{ or } \mathcal{O}(\tilde{\Omega}_{\pm}, \mathbb{C}^2); \\ \hat{\varphi} &= (\hat{\varphi}_{+}, \hat{\varphi}_{-}) := \varrho^{*} \hat{\varphi} & \text{as an element of } \mathcal{O}(\Omega, \mathbb{C}^2) \text{ or } \mathcal{O}(\tilde{\Omega}, \mathbb{C}^2). \end{aligned} $$

More explicitly, for any $\gamma = (\gamma^{+}, \gamma^{-})$ in $B$ or in $\tilde{B}$ with sufficiently small central charge $\xi = Z(\gamma) = \pm Z(\gamma^{\pm})$, we write:

$$ \hat{\varphi}(\gamma) = \hat{\varphi}^{(\pm)}(\gamma^{\pm}) := \hat{\varphi}(\varrho(\gamma)) = \hat{\varphi}(\varrho_{\pm}(\gamma^{\pm})) = \hat{\varphi}(z, \xi), \quad (5.23) $$

where $z = s(\gamma) = s(\gamma^{\pm})$ is such that $(z, \xi) \in \Xi$.

### §2. Lifting the coefficients to the unfolded Borel space.
Correspondingly, we can reinterpret the Initial Value Problem (5.22) over the open subset $\Omega$ or $\tilde{\Omega}$ as follows. We pullback the functions $a = a(z)$, $b = b(z)$, $c = c(z)$ using the corresponding anchor maps. In fact, these functions are globally well-defined meromorphic functions on $\mathbb{C}_z \times \mathbb{C}_{\xi}$. So, abusing notation, we let

$$ a_{\pm} := \varrho_{\pm}^{*} a, \quad b_{\pm} := \varrho_{\pm}^{*} b, \quad c_{\pm} := \varrho_{\pm}^{*} c $$

denote both their pullbacks to globally well-defined meromorphic functions on $\Pi_1(\mathbb{C}_z)$ and their pullbacks to globally well-defined holomorphic functions on $\Pi_1(\mathbb{C}_z^*)$. By a similar abuse of notation, we continue to denote their pullbacks by $\varrho$ to global meromorphic functions on $B$ and global holomorphic functions on $\tilde{B}$ simply by $a, b, c$:

$$ a := \varrho^* a, \quad b := \varrho^* b, \quad c := \varrho^* c. $$

Explicitly, for any $\gamma = (\gamma^{+}, \gamma^{-})$ in $B$ or in $\tilde{B}$ with source $s(\gamma) = s(\gamma^{\pm}) = z$, we have

$$ a(\gamma) = a^{(\pm)}(\gamma^{\pm}) := a(z), \quad b(\gamma) = b^{(\pm)}(\gamma^{\pm}) := b(z), \quad c(\gamma) = c^{(\pm)}(\gamma^{\pm}) := c(z). \quad (5.24) $$

### §3. Lifting the vector fields to the unfolded Borel space.
It remains to interpret the vector fields $\pm V - \partial_{\xi}$ on the left hand side of (5.22) as vector fields on the Borel space $B$.

**Lemma 5.24.** *The pairs*

$$ V_{+} := (V^{s}, V^{s} + 2V^{t}) \quad \text{and} \quad V_{-} := (-V^{s} - 2V^{t}, -V^{s}) $$

define meromorphic vector fields on the algebraic surface $B$. Similarly, they define holomorphic vector fields on the holomorphic manifold $\tilde{B}$. They are the unique such vector fields that satisfy $\varrho_* V_{\pm} = \pm V - \partial_{\xi}$.

**Proof.** By definition of the fibre product, a tangent vector to $B$ at $\gamma = (\gamma^{+}, \gamma^{-})$ is a pair $(v^{+}, v^{-})$ consisting of a tangent vector to $\Pi_1(\mathbb{C}_z)$ at $\gamma^{+}$ and a tangent vector at $\gamma^{-}$ such
that $(\varrho_+)_* v^+ = (\varrho_-)_* v^-$ where $(\varrho_\pm)_*$ is the derivative of $\varrho_\pm$ at $\gamma^\pm$. So, for example, to check that $V_+$ is a well-defined vector field on $B$, we just need to verify the identity $(\varrho_+)_* V^s = (\varrho_-)_*(V^s + 2V^t)$. This is a simple calculation using the properties of source- and target-lifts of $V$ that can be found in §4.2.2: the lefthand side is $V - \partial_\xi$ and the righthand side is $V + \partial_\xi - 2\partial_\xi$. ■

**4. IVP on the unfolded Borel space.** We can now consider the following Initial Value Problem over the unfolded Borel space $B$:

$$ \begin{cases} V_+\varphi_+ = \Phi \\ V_-\varphi_- = \Phi \end{cases} \quad \text{such that} \quad \varphi|_{C_z^*} = c, \qquad (5.25) $$

where $\Phi$ is the following expression in $\varphi_+$, $\varphi_-$ that depends holomorphically on $\gamma \in B$:

$$ \Phi = \Phi(\gamma, \varphi_+, \varphi_-) := a(\gamma)(\varphi_+ - \varphi_-)^{*2}(\gamma) + b(\gamma)(\varphi_+ - \varphi_-)(\gamma) . \qquad (5.26) $$

Recall that the convolution product on $B$ was defined in §5.4. Then we have the following equivalence.

**Lemma 5.25.** *The Initial Value Problems (5.22) and (5.25) are equivalent over $\varrho : \Omega \to \Xi$.*

Now we rewrite (5.25) as an integral equation on the unfolded Borel covering space. Recall that by Lemma 4.21, if the neighbourhood $\Xi \subset C_z^* \times C_\xi$ of $\xi = 0$ is sufficiently small, then the Initial Value Problem (5.22) is equivalent for all $(z, \xi) \in \Xi$ to the following integral equation:

$$ \begin{cases} \varphi_+(z, \xi) = c(z) - \int_0^\xi \Phi(z_{\xi-s}^+, s; \varphi_+(z_{\xi-s}^+, s), \varphi_-(z_{\xi-s}^+, s)) \, ds, \\ \varphi_-(z, \xi) = c(z) - \int_0^\xi \Phi(z_{\xi-s}^-, s; \varphi_+(z_{\xi-s}^-, s), \varphi_-(z_{\xi-s}^-, s)) \, ds, \end{cases} \qquad (5.27) $$

where $z_{\xi-s}^\pm$ is the unique point in $C_z^*$ satisfying the identity

$$ \xi - s = \pm \int_z^{z_{\xi-s}^{\pm}} \lambda . $$

If we interpret $z$ in (5.27) as the source of a pair of curves $\gamma^+, \gamma^- : [0, \xi] \to C_z^*$ with central charge $Z(\gamma^+) = -Z(\gamma^-) = \xi$, then the point $z_{\xi-s}^\pm$ in the integrand can be interpreted as the source of the target-truncation $\bar{\gamma}_s^\pm := \gamma^\pm|_{[\xi-s, \xi]}$ with central charge $\pm Z(\bar{\gamma}_s^\pm) = s$. Consequently, the two equations in (5.27) may be written respectively on $\tilde{\Omega}_+$ and $\tilde{\Omega}_-$ as follows:

$$ \begin{cases} \varphi_+^{(+)}(\gamma^+) = c^{(+)}(\gamma^+) - \int_0^{+Z(\gamma^+)} \Phi(\bar{\gamma}_s^+; \varphi_+^{(+)}(\bar{\gamma}_s^+), \varphi_-^{(+)}(\bar{\gamma}_s^+)) \, ds, \\ \varphi_-^{(-)}(\gamma^-) = c^{(-)}(\gamma^-) - \int_0^{-Z(\gamma^-)} \Phi(\bar{\gamma}_s^-; \varphi_+^{(-)}(\bar{\gamma}_s^-), \varphi_-^{(-)}(\bar{\gamma}_s^-)) \, ds. \end{cases} \qquad (5.28) $$

These equations are valid for all pairs of paths $(\gamma^+, \gamma^-) \in \tilde{\Omega}_+ \times \tilde{\Omega}_-$ with the same source and opposite central charge (i.e., such that $\tilde{\varrho}_+(\gamma^+) = \tilde{\varrho}_-(\gamma^-)$), and any target-truncations $\bar{\gamma}_s^\pm := \gamma^\pm|_{[\xi-s, \xi]}$ with central charge $\pm Z(\bar{\gamma}_s^\pm) = s$. It follows that the pair $\gamma = (\gamma^+, \gamma^-)$
defines a point in the open subset $\tilde{\Omega}$ of the unfolded Borel covering space. Correspondingly, we lift the target-truncations $\bar{\gamma}_s^{\pm}$ to the two pairs $\bar{\gamma}_s^{+} := (\bar{\gamma}_s^{+}, (\bar{\gamma}_s^{+})^{\dagger})$ and $\bar{\gamma}_s^{-} := ((\bar{\gamma}_s^{-})^{\dagger}, \bar{\gamma}_s^{-})$ that define the target-truncations of $\gamma$ as explained in §5.4. Therefore, taking into account our notational conventions (5.23) and (5.24) as well as the fact that $Z(\gamma) = \pm Z(\gamma^{\pm})$ and $Z(\bar{\gamma}_s^{\pm}) = \pm Z(\bar{\gamma}_s^{\pm})$, the system of integral equations (5.28) may be written as follows:

$$ \left\{ \begin{aligned} \varphi_{+}(\gamma) &= c(\gamma) - \int_{0}^{Z(\gamma)} \Phi(\bar{\gamma}_{s}^{+}; \varphi_{+}(\bar{\gamma}_{s}^{+}), \varphi_{-}(\bar{\gamma}_{s}^{+})) \, ds, \\ \varphi_{-}(\gamma) &= c(\gamma) - \int_{0}^{Z(\gamma)} \Phi(\bar{\gamma}_{s}^{-}; \varphi_{+}(\bar{\gamma}_{s}^{-}), \varphi_{-}(\bar{\gamma}_{s}^{-})) \, ds. \end{aligned} \right. \qquad (5.29) $$

Finally, reparameterising the target-truncations $\bar{\gamma}_s^{\pm}$ using their arc-length, we get the following lemma.

**Lemma 5.26.** *The Initial Value Problem (5.25) is equivalent to the following system of integral equations over the unfolded Borel covering space $\tilde{B}$:*

$$ \left\{ \begin{aligned} \varphi_{+}(\gamma) &= c(\gamma) - \int_{0}^{|\gamma|} \Phi(\bar{\gamma}_{s}^{+}; \varphi_{+}(\bar{\gamma}_{s}^{+}), \varphi_{-}(\bar{\gamma}_{s}^{+})) \, Ds, \\ \varphi_{-}(\gamma) &= c(\gamma) - \int_{0}^{|\gamma|} \Phi(\bar{\gamma}_{s}^{-}; \varphi_{+}(\bar{\gamma}_{s}^{-}), \varphi_{-}(\bar{\gamma}_{s}^{-})) \, Ds, \end{aligned} \right. \qquad (5.30) $$

where $Ds := \pm t^* \lambda_{\gamma_s^{\pm}}$ and $\gamma_s^{\pm}$ are the two arc-length target-truncations of any synchronised parameterisation of $\gamma$.

# §6. Endless Analytic Continuation

In this final section, we construct the global solution $\varphi$ of the Initial Value Problem (5.25), or equivalently the integral equation (5.30), and finish the proof of our main Theorem 2.1 as well as all the remaining results stated in §2. Our construction of $\varphi$, presented in §6.1, uses the Contraction Mapping Principle over the Borel covering space which in particular allows for tight control on the behaviour of $\varphi$ at infinity. This allows us to deduce in §6.2 the resurgence properties of the formal vector-valued solution $\hat{f}$ of the differential system (3.16) and consequently the resurgence properties of the formal solution $\hat{q}$ of the deformed Painlevé I equation. Finally, in §6.3, we flesh out the Borel summability properties of $\hat{f}$ and associated Stokes phenomenon and explain how they imply the analogous properties of $\hat{q}$ that were stated in §2.2.

## §6.1. A Contraction Mapping

In this subsection, we construct the global solution of the integral equation (5.30). The main result, which does the majority of this paper's (mild) analytic heavy-lifting, can be formulated as follows.

**Proposition 6.1.** *There is a unique vector-valued global holomorphic function $\varphi \in \mathcal{O}(\tilde{B}, \mathbb{C}^2)$ on the unfolded Borel covering space $\tilde{B}$ that satisfies the integral equation (5.30) or equivalently the Initial Value Problem (5.25).*

This subsection is devoted to the proof of this Proposition. We employ the familiar strategy of realising the solution $\varphi$ as the fixed point of a contraction mapping $\Upsilon$ on a suitable
function space. This function space is the infinite-dimensional vector space $O(\tilde{B}, \mathbb{C}^2)$ of vector-valued global holomorphic functions over the unfolded Borel covering space $\tilde{B}$. However, the Contraction Mapping Principle for $\Upsilon$ fails if we allow paths on $C_z^*$ to come arbitrarily close to the origin. To address this problem, we will introduce an exhaustion of $C_z^*$ by open subsets of the punctured $z$-plane obtained by excising closed discs around the origin of smaller and smaller radius. This will allow us to introduce a sequence of suitable Banach spaces of functions over the unfolded Borel covering space where $\Upsilon$ can be shown to have a unique fixed point using the Contraction Mapping Principle. We describe this excision process in §6.1.1 and define the relevant Banach spaces in §6.1.2. Then in §6.1.3 we prove Proposition 6.1 through a sequence of lemmas.

**§6.1.1. The excised unfolded Borel covering space.** Let $\bar{D} \subset C_z$ be any compact and simply connected domain around the origin, such as a closed disc centred at the origin of finite nonzero radius, and consider the *excised z-plane*

$$C_z^\circ := C_z \setminus \bar{D}.$$

Then the spaces $C_z^\circ$ and $C_z^*$ are homotopy equivalent, which means any concrete path in $C_z^*$ with source and target outside of $\bar{D}$ is homotopy equivalent to a concrete path entirely contained in $C_z \setminus \bar{D}$. This defines a canonical isomorphism

$$\Pi_1(C_z^\circ) \xrightarrow{\sim} \Pi_1(C_z^*)|_{C_z^\circ}.$$

The upshot is that the constructions of §5 go through without any substantial modifications, resulting in the *excised unfolded Borel covering space*:

$$\tilde{B}^\circ := \Pi_1(C_z^\circ) \times_{\varrho-\varrho_+} \Pi_1(C_z^\circ).$$

Consequently, $\tilde{B}^\circ$ inherits a source map $s: \tilde{B}^\circ \to C_z^\circ$, a central charge $Z: \tilde{B}^\circ \to C_\xi$, and therefore an anchor map $\varrho: \tilde{B}^\circ \to C_z^\circ \times C_\xi$. There is an obvious inclusion $\tilde{B}^\circ \subset \tilde{B}$ which translates into the inclusion $O(\tilde{B}, \mathbb{C}^2) \hookrightarrow O(\tilde{B}^\circ, \mathbb{C}^2)$ of vector spaces.

**§6.1.2. A Banach algebra over the unfolded Borel covering space.** Let $C_z^\circ \subset C_z^*$ be any excised $z$-plane in the above sense, and let $\tilde{B}^\circ$ be the corresponding excised unfolded Borel covering space. Then we can define the following norm on the infinite-dimensional complex vector space of global holomorphic functions over $\tilde{B}^\circ$.

**Definition 6.2.** For any $K > 0$, we define the norm of any $f \in O(\tilde{B}^\circ)$ by

$$\|f\|_K := \sup_{\gamma \in \tilde{B}^\circ} \inf_0^{|\gamma|} e^{-Ks} |f(\gamma_s)Ds|, \qquad (6.1)$$

where the infimum is taken over all source-truncations $\gamma_s$ of $\gamma$. We also extend this norm to the vector spaces $O(\tilde{B}^\circ, \mathbb{C}^n)$ and $O(\tilde{B}^\circ, M_n)$ of vector-valued and matrix-valued functions by replacing the absolute value $|\cdot|$ with any extension to a vector and matrix norm; for definiteness, we take these to be the Euclidean norm and the Frobenius norm, respectively.

**Definition 6.3.** We introduce the vector subspace of $O(\tilde{B}^\circ)$ of functions which are
bounded with respect to this norm:

$$ \mathcal{H}_K(\tilde{\mathbf{B}}^\circ) := \{ f \in \mathcal{O}(\tilde{\mathbf{B}}^\circ) \mid \|f\|_{\mathcal{K}} < \infty \} . \qquad (6.2) $$

We will also consider the closed unit ball in this vector space, which we denote by

$$ \mathcal{H}_K^{\mathcal{H}}(\tilde{\mathbf{B}}^\circ) := \{ f \in \mathcal{O}(\tilde{\mathbf{B}}^\circ) \mid \|f\|_{\mathcal{K}} \le 1 \} . \qquad (6.3) $$

We define the vector subspaces $\mathcal{H}_K(\tilde{\mathbf{B}}^\circ, \mathbb{C}^n)$ and $\mathcal{H}_K(\tilde{\mathbf{B}}^\circ, \mathbb{M}_n)$ as well as the unit balls inside them accordingly.

**Lemma 6.4.** For any $K > 0$ and any $n \ge 1$, the space $\mathcal{H}_K(\tilde{\mathbf{B}}^\circ, \mathbb{C}^n)$ is a Banach space and the space $\mathcal{H}_K(\tilde{\mathbf{B}}^\circ, \mathbb{M}_n)$ is a Banach algebra with respect to the convolution product $*$; i.e., for all $f, g \in \mathcal{H}_K(\tilde{\mathbf{B}}^\circ, \mathbb{M}_n)$, we have the following inequality:

$$ \|f * g\|_{\mathcal{K}} \le \|f\|_{\mathcal{K}} \|g\|_{\mathcal{K}} . \qquad (6.4) $$

In particular, the norm of the constant function is $\|1\|_{\mathcal{K}} = 1/K$.

**Proof.** It is enough to consider the space $\mathcal{H}_K(\tilde{\mathbf{B}}^\circ)$.

**BANACH SPACE STRUCTURE.** First, we show that the norm (6.1) is complete, making $(\mathcal{H}_K(\tilde{\mathbf{B}}^\circ), \| \cdot \|)$ into a Banach space. The argument is very similar to the standard argument for the completeness of the $L^1$-norm.

Let $(f_n)_{n=0}^\infty$ be a Cauchy sequence of functions in $\mathcal{H}_K(\tilde{\mathbf{B}}^\circ)$. Recall that this means that for any $\varepsilon > 0$ there is some $N$ such that $\|f_n - f_m\|_{\mathcal{K}} < \varepsilon$ whenever $n, m > N$. We need to show that it converges in this norm; i.e., there is a function $f \in \mathcal{H}_K(\tilde{\mathbf{B}}^\circ)$ such that $\|f_n - f\|_{\mathcal{K}} \to 0$ as $n \to \infty$.

Consider a subsequence $(f_{n_k})_{k=0}^\infty$ of $(f_n)$ such that $\|f_{n_{k+1}} - f_{n_k}\|_{\mathcal{K}} \le 2^{-k}$ for all $k \ge 0$. We claim that the following telescopic series is the desired limit of $(f_n)$:

$$ f := f_{n_0} + \sum_{k=1}^{\infty} (f_{n_{k+1}} - f_{n_k}) . $$

To see this, note that for any positive integer $N$, its $N$-th partial sum is bounded in the $\| \cdot \|_{\mathcal{K}}$-norm by

$$ \|f_{n_0}\|_{\mathcal{K}} + \sum_{k=1}^{N} \|f_{n_{k+1}} - f_{n_k}\|_{\mathcal{K}} \le \|f_{n_0}\|_{\mathcal{K}} + \sum_{k=1}^{N} \frac{1}{2^k} . $$

This means that the series defining $f$ converges in the $\| \cdot \|_{\mathcal{K}}$-norm and that the subsequence $(f_{n_k})$ converges to $f$. Moreover, $f$ is a holomorphic function on $\tilde{\mathbf{B}}^\circ$ because it is the limit of partial sums of a sequence of holomorphic functions on $\tilde{\mathbf{B}}^\circ$. We conclude that $f \in \mathcal{H}_K(\tilde{\mathbf{B}}^\circ)$, and so it remains to show that $f$ is the limit of $(f_n)$. This follows from the fact that $(f_n)$ is a Cauchy sequence. Indeed, for any $\varepsilon > 0$, let $N$ be such that $\|f_n - f_m\|_{\mathcal{K}} < \varepsilon/2$ whenever $n, m > N$. Let $M$ be such that $n_M > N$ and $\|f_{n_M} - f\| < \varepsilon/2$. Then

$$ \|f_n - f\|_{\mathcal{K}} \le \|f_n - f_{n_M}\|_{\mathcal{K}} + \|f_{n_M} - f\|_{\mathcal{K}} < \varepsilon . $$

**BANACH ALGEBRA STRUCTURE.** Next, we show that $(\mathcal{H}_K(\tilde{\mathbf{B}}^\circ), \| \cdot \|)$ is a Banach al-
gebra with respect to the convolution product defined in Definition 5.20. Suppose $f, g \in \mathcal{H}_K(\tilde{\mathbb{B}}^\circ)$; we need to demonstrate the inequality (6.4). For any $\gamma \in \tilde{\mathbb{B}}^\circ$, choose any synchronously parameterised representative and let $L := |\gamma|$. Then we compute as follows:

$$
\begin{align*}
\int_0^L e^{-Ks} |(f * g)(\gamma_s) Ds| &\le \int_0^L e^{-Ks} \int_0^s |f(\gamma_{s,u})| |g(\gamma_{s,s-u})| |DuDs| \\
&= \int_0^L \int_0^s e^{-Ku} |f(\gamma_{s,u})| e^{-K(s-u)} |g(\gamma_{s,s-u})| |DuDs| \\
&\le \int_0^L \int_0^L e^{-Ku} |f(\gamma_u)| e^{-Kr} |g(\gamma_r)| |DuDr| \\
&= \int_0^L e^{-Ku} |f(\gamma_u) Du| \cdot \int_0^L e^{-Kr} |g(\gamma_r) Dr| .
\end{align*}
$$

Here, $\gamma_s$ is the source-truncation of $\gamma$ of length $s$, and $\gamma_{s,u}$ is the source-truncation of $\gamma_s$
of length $u$. Going from the second line to the third, we first noticed that $f(\gamma_{s,u}) = f(\gamma_u)$
and $g(\gamma_{s,s-u}) = g(\gamma_{s-u})$, and then we changed the integration variable $s$ to $r = s-u$
and extended the integration domain in $r$. The last line is the result of applying Fubini's
Theorem. Taking the infimum and then the supremum of both sides of this inequality
yields (6.4). ■

**§6.1.3. A contraction mapping.** We now turn to the proof of Proposition 6.1 by showing that the following integral operator on $O(\tilde{\mathbb{B}}, \mathbb{C}^2)$ admits a unique fixed point.

**Definition 6.5.** We define a mapping

$$
\Upsilon = (\Upsilon_+, \Upsilon_-) : \mathcal{O}(\tilde{\mathbb{B}}, \mathbb{C}^2) \to \mathcal{O}(\tilde{\mathbb{B}}, \mathbb{C}^2), \quad (6.5)
$$

whose components $\Upsilon_{\pm}: \mathcal{O}(\tilde{\mathbb{B}}, \mathbb{C}^2) \to \mathcal{O}(\tilde{\mathbb{B}})$ are given on all $\varphi = (\varphi_+, \varphi_-) \in \mathcal{O}(\tilde{\mathbb{B}}, \mathbb{C}^2)$ and all $\gamma \in \tilde{\mathbb{B}}$ by the following formula:

$$
\Upsilon_{\pm}[\varphi](\gamma) := c(\gamma) - \int_{0}^{|\gamma|} \Phi(\bar{\gamma}_{s}^{\pm}; \varphi_{+}(\bar{\gamma}_{s}^{\pm}), \varphi_{-}(\bar{\gamma}_{s}^{\pm}))Ds, \quad (6.6)
$$

where $Ds = \pm t_{\pm}^* \lambda_{\gamma_s}$ and $\bar{\gamma}_s^{\pm}$ are the two arc-length target-truncations of any concrete representative of $\gamma$. For any excised complex plane $\mathbb{C}_z^\circ \subset \mathbb{C}_z^*$, the mapping $\Upsilon$ readily extends to a mapping $\Upsilon : \mathcal{O}(\tilde{\mathbb{B}}^\circ, \mathbb{C}^2) \to \mathcal{O}(\tilde{\mathbb{B}}^\circ, \mathbb{C}^2)$ by the very same formula.

First, we show that the restriction of $\Upsilon$ to every Banach space $\mathcal{H}_K(\tilde{\mathbb{B}}^\circ, \mathbb{C}^2)$ is well-defined.

**Lemma 6.6.** Let $\mathbb{C}_z^\circ \subset \mathbb{C}_z^*$ be any excised complex z-plane, and let $\tilde{\mathbb{B}}^\circ$ be the corresponding excised unfolded Borel covering space. Then, for any $K > 0$, the integral operator $\Upsilon$ restricts to a mapping

$$
\Upsilon : \mathcal{H}_K(\tilde{\mathbb{B}}^\circ, \mathbb{C}^2) \to \mathcal{H}_K(\tilde{\mathbb{B}}^\circ, \mathbb{C}^2). \qquad (6.7)
$$

**Proof.** Recall that $a(z) = 3z^{-2}$, $b(z) = -3z^{-5}$, and $c(z) = -6z^{-8}$. They are bounded functions on any excised z-plane $\mathbb{C}_z^\circ$, so there is a constant $C > 0$ that gives the following bounds:

$$
|a(z)|, |b(z)|, |c(z)| \le C \quad \forall z \in \mathbb{C}_z^\circ. \tag{6.8}
$$

Since the elements of $\tilde{\mathbb{B}}^\circ$ are pairs of paths entirely contained in $\mathbb{C}_z^\circ$, these bounds auto-
matically translate into the bounds over the space $\tilde{\mathbb{B}}^\circ$:

$$|a(\gamma)|, |b(\gamma)|, |c(\gamma)| \le C \quad \forall \gamma \in \tilde{\mathbb{B}}^\circ. \qquad (6.9)$$

Now, take an arbitrary element $\varphi = (\varphi_+, \varphi_-) \in \mathcal{H}_K(\tilde{\mathbb{B}}^\circ, \mathbb{C}^2)$. We need to show that $\|\Upsilon[\varphi]\|_K < \infty$. To this end, take an arbitrary synchronously parameterised $\gamma \in \tilde{\mathbb{B}}^\circ$, let $L := |\gamma|$, and estimate the individual terms making up the expression

$$\int_0^L e^{-Ks} |\Upsilon_\pm[\varphi](\gamma_s)Ds| .$$

Explicitly using (6.6) and (5.26), this quantity is bounded by

$$\begin{aligned} & \int_0^L e^{-Ks} |c(\gamma_s)Ds| + \int_0^L e^{-Ks} \int_0^s \left| \Phi_\pm \left( \bar{\gamma}_{s,u}^\pm; \varphi_+(\bar{\gamma}_{s,u}^\pm), \varphi_-(\bar{\gamma}_{s,u}^\pm) \right) DuDs \right| \\ & \le \int_0^L e^{-Ks} |c(\gamma_s)Ds| + \int_0^L e^{-Ks} \int_0^s \left| a(\bar{\gamma}_{s,u}^\pm)(\varphi_+ - \varphi_-)^* \bar{\gamma}_{s,u}^\pm DuDs \right| \\ & \quad + \int_0^L e^{-Ks} \int_0^s \left| b(\bar{\gamma}_{s,u}^\pm)(\varphi_+ - \varphi_-)(\bar{\gamma}_{s,u}^\pm) DuDs \right| . \end{aligned}$$

Here, $\gamma_s$ is the source-truncation of $\gamma$ of length $s$, whilst $\bar{\gamma}_{s,u}^\pm$ are the two target-truncations of $\gamma_s$ of length $u$. For the first term, we find:

$$\int_0^L e^{-Ks} |c(\gamma_s)Ds| \le C \int_0^L e^{-Ks} |Ds| .$$

The infimum of this quantity is bounded by $C\|1\|_K = C/K$, which is independent of $\gamma$. Therefore, we conclude that the supremum is finite and

$$\|c\|_K \le C/K. \qquad (6.10)$$

Next, we estimate the terms involving $b\varphi_\pm$ using (6.9) and a similar line of reasoning as the one used in the proof of Lemma 6.4:

$$\begin{aligned} \int_0^L e^{-Ks} \int_0^s \left| b(\bar{\gamma}_{s,u}^\pm) \varphi_\pm(\bar{\gamma}_{s,u}^\pm) DuDs \right| &\le C \int_0^L \int_0^s e^{-K(s-u)} e^{-Ku} \left| \varphi_\pm(\bar{\gamma}_{s,u}^\pm) DuDs \right| \\ &\le C \int_0^L \int_0^L e^{-Kr} e^{-Ku} \left| \varphi_\pm(\bar{\gamma}_u^\pm) DuDr \right| . \end{aligned}$$

The infimum of this quantity is bounded by $C\|1\|_K\|\varphi_\pm\|_K = (C/K)\|\varphi_\pm\|_K$ which is independent of $\gamma$. Therefore, we conclude that the supremum is finite and

$$\|b\varphi_\pm\|_K \le (C/K)\|\varphi_\pm\|_K. \qquad (6.11)$$

We can similarly estimate the terms involving $a\varphi_+ * \varphi_+$, $a\varphi_+ * \varphi_-$, $a\varphi_- * \varphi_-$. For any one of these terms $a\varphi_\bullet * \varphi_\bullet$, we find:

$$\inf \int_0^L e^{-Ks} \int_0^s \left| a(\bar{\gamma}_{s,u}^\pm)(\varphi_\bullet * \varphi_\bullet)(\bar{\gamma}_{s,u}^\pm) DuDs \right| \le C \|1\|_K \|\varphi_\bullet * \varphi_\bullet\|_K .$$

where the infimum is taken over all synchronised parameterisations of $\gamma$. Therefore, we
conclude that the supremum of the lefthand side is finite and

$$ \|a\varphi_* * \varphi_*\|_K \le (C/K) \|\varphi_*\|_K \|\varphi_*\|_K. \qquad (6.12) $$

We conclude that $\|\Upsilon[\varphi]\|_K < \infty$. ■

Next, we prove the key contractive property of $\Upsilon$ by collecting the estimates obtained in the proof just finished.

**Lemma 6.7.** Let $\mathbb{C}_z^\circ \subset \mathbb{C}_z^*$ be any excised complex z-plane, and let $\tilde{\mathbb{B}}^\circ$ be the corresponding excised unfolded Borel covering space. Then there is a sufficiently large $K > 0$ such that the integral operator $\Upsilon$ restricts to a contraction mapping

$$ \Upsilon : \mathcal{H}_K^{\mathbb{B}^\circ}(\tilde{\mathbb{B}}^\circ, \mathbb{C}^2) \to \mathcal{H}_K^{\mathbb{B}^\circ}(\tilde{\mathbb{B}}^\circ, \mathbb{C}^2). \qquad (6.13) $$

That is, there is a strictly positive real number $\chi < 1$ such that, for all $\varphi, \psi \in \mathcal{H}_K^{\mathbb{B}^\circ}(\tilde{\mathbb{B}}^\circ, \mathbb{C}^2)$,

$$ \|\Upsilon[0]\|_K < 1 - \chi \quad \text{and} \quad \|\Upsilon[\varphi] - \Upsilon[\psi]\|_K \le \chi \|\varphi - \psi\|_K. \qquad (6.14) $$

**Proof.** First, let us show that $\Upsilon$ restricts to a mapping (6.13) for a sufficiently large $K > 0$. Let $C > 0$ be the same constant as in the proof of Lemma 6.6, so that the bounds (6.8) are true. Pick an arbitrary $\gamma \in \tilde{\mathbb{B}}^\circ$. Then, for any $K > 0$ and any $\varphi \in \mathcal{H}_K(\tilde{\mathbb{B}}^\circ, \mathbb{C}^2)$, upon combining the estimates (6.10), (6.11), and (6.12), we find:

$$ \inf \int_0^{|\gamma|} e^{-Ks} |\Upsilon_{\pm}[\varphi](\gamma_s)Ds| \le (C/K) (1+2\|\varphi_+\|_K \|\varphi_-\|_K + \|\varphi_+\|_K^2 + \|\varphi_-\|_K^2 + \|\varphi_+\|_K + \|\varphi_-\|_K), $$

where, as ever, the infimum is taken over all synchronised parameterisations of $\gamma$. Note that the righthand side is independent of $\gamma$, hence the supremum over all $\gamma$ of the lefthand side is also bounded by this quantity. Therefore, if $\varphi \in \mathcal{H}_K^{\mathbb{B}^\circ}(\tilde{\mathbb{B}}^\circ, \mathbb{C}^2)$ so that $\|\varphi_\pm\|_K \le 1$, then

$$ \|\Upsilon_{\pm}[\varphi]\|_K \le 7C/K \quad \text{and so} \quad \|\Upsilon[\varphi]\|_K \le 7C\sqrt{2}/K. $$

We conclude that $\|\Upsilon[\varphi]\|_K \le 1$ provided $K \ge 7C\sqrt{2}$.

Next, we show that $\Upsilon$ is a contraction mapping on the unit ball $\mathcal{H}_K^{\mathbb{B}^\circ}(\tilde{\mathbb{B}}^\circ, \mathbb{C}^2)$ for all sufficiently large $K > 0$. First, let us note that $\Upsilon_{\pm}[0](\gamma) = c(\gamma)$ for all $\gamma \in \tilde{\mathbb{B}}^\circ$. From (6.10), we know that $\|c\|_K \le C/K$ for all $K > 0$, and so

$$ \|\Upsilon[0]\|_K \le C\sqrt{2}/K. $$

Next, for any $K > 0$, any $\varphi, \psi \in \mathcal{H}_K^{\mathbb{B}^\circ}(\tilde{\mathbb{B}}^\circ, \mathbb{C}^2)$, and any synchronously parameterised $\gamma \in \tilde{\mathbb{B}}^\circ$, the expression

$$ \int_0^{|\gamma|} e^{-Ks} |\Upsilon_{\pm}[\varphi](\gamma_s) - \Upsilon_{\pm}[\psi](\gamma_s)| |Ds| \qquad (6.15) $$
is bounded by

$$ \int_0^{|\gamma|} e^{-Ks} \int_0^{|\gamma_s|} |a(\bar{\gamma}_{s,u}^{\pm}) ((\varphi_+ - \varphi_-)^* - (\psi_+ - \psi_-)^*) (\bar{\gamma}_{s,u}^{\pm}) DuDs| \\ + \int_0^{|\gamma|} e^{-Ks} \int_0^{|\gamma_s|} |b(\bar{\gamma}_{s,u}^{\pm}) ((\varphi_+ - \varphi_-) - (\psi_+ - \psi_-)) (\bar{\gamma}_{s,u}^{\pm}) DuDs| . $$

The goal is to estimate this expression in terms of $\|\varphi - \psi\|_K$. The term involving $b$ is easy to deal with because $|b((\varphi_+ - \varphi_-) - (\psi_+ - \psi_-))| \le |b|(|\varphi_+ - \psi_+| + |\varphi_- - \psi_-|)$. Consequently, we can argue just as we did in (6.11) that the infimum of the term involving $b$ is bounded by $2(C/K)\|\varphi - \psi\|_K$. For the term involving $a$, we rewrite $(\varphi_+ - \varphi_-)^* - (\psi_+ - \psi_-)^*$ as follows:

$$ \begin{aligned} & (\varphi_+^{*2} - \psi_+^{*2}) + (\varphi_-^{*2} - \psi_-^{*2}) - 2\varphi_+ * \varphi_- + 2\psi_+ * \psi_- \\ &= (\varphi_+ - \psi_+) * (\varphi_+ + \psi_+) + (\varphi_- - \psi_-) * (\varphi_- + \psi_-) - 2\varphi_+ * (\varphi_- - \psi_-) - 2(\varphi_+ - \psi_+) * \psi_- \\ &= (\varphi_+ - \psi_+) * (\varphi_+ + \psi_+ - 2\psi_-) + (\varphi_- - \psi_-) * (\varphi_- + \psi_- - 2\varphi_+) . \end{aligned} $$

Therefore, using the fact that $\|\varphi_{\pm}\|_K \le \|\varphi\|_K \le 1$, etc., we find that the infimum of the term involving $a$ is bounded by $8(C/K)\|\varphi - \psi\|_K$. Altogether, we find that (6.15) is bounded by $10(C/K)\|\varphi - \psi\|_K$, and so

$$ \|\Upsilon[\varphi] - \Upsilon[\psi]\|_K \le 10\sqrt{2}(C/K)\|\varphi - \psi\|_K . $$

If we put $\chi := 10\sqrt{2}(C/K)$, then in order for (6.14) to be true, we need to choose $K$ such that $10\sqrt{2}(C/K) < 1$ and $C\sqrt{2}/K < 1 - 10\sqrt{2}(C/K)$. Both of these inequalities are satisfied if $K > 11C\sqrt{2}$. For example, we can take $K = 16C$, in which case $\chi = 5\sqrt{2}/8$. ■

Finally, we can be more specific about the choice of constants $C$ and $K$ if we are more specific about the excision $C_z^o = C_z \setminus \bar{D}$.

**Lemma 6.8.** Fix $R > 0$ and let $\bar{D} = \overline{D(0, R)}$ be the closed disc around the origin of radius $R > 0$. Consider the excised complex $z$-plane $C_z^o = C_z \setminus \bar{D}$, and let $\tilde{B}^o$ be the corresponding excised unfolded Borel covering space. Then we can be more specific about the choice of constant $K$ in Lemma 6.7. Namely, we can take $K = 96/R^8$ if $R \le \sqrt[6]{2}$ and $K = 48/R^2$ if $R \ge \sqrt[6]{2}$. In particular, we can take $K$ arbitrarily small provided that $R$ is sufficiently large.

**Proof.** From the proof of Lemma 6.7, we know that we can take $K = 16C$ where $C$ is any constant that ensures the bounds (6.8). Recall that $a(z) = 3z^{-2}$, $b(z) = -3z^{-5}$, and $c(z) = -6z^{-8}$, so

$$ |a(z)| \le 3/R^2, \quad |b(z)| \le 3/R^5, \quad |c(z)| \le 6/R^8 \quad \forall z \in C_z^o. \quad (6.16) $$

Thus, we can take $C = 6/R^8$ if $R \le \sqrt[6]{2}$ and $C = 3/R^2$ if $R \ge \sqrt[6]{2}$. ■

**Proof of Proposition 6.1.** Choose any decreasing sequence of strictly positive real numbers $(R_n)$ that converges to 0; i.e., $R_n > R_{n+1}$ and $R_n \to 0$ as $n \to \infty$. For each $n$, consider the excised $z$-plane $C_{z,n}^o := C_z \setminus \bar{D}(0, R_n)$. We obtain a nested increasing sequence of excised $z$-planes which exhausts the punctured $z$-plane $C_z^*$:

$$ C_{z,n}^o \subset C_{z,n+1}^o \subset \cdots \subset C_z^* . $$
In turn, each $\mathbb{C}_{z,n}^\circ$ induces a corresponding excised unfolded Borel covering space that we denote by $\tilde{\mathbb{B}}_n^\circ$, and we also obtain a nested increasing sequence of excised unfolded Borel covering spaces which exhausts the Borel covering space $\tilde{\mathbb{B}}$:

$$ \tilde{\mathbb{B}}_n^\circ \subset \tilde{\mathbb{B}}_{n+1}^\circ \subset \cdots \subset \tilde{\mathbb{B}}. $$

By Lemma 6.7, for each $n$, there is a constant $K_n > 0$ such that the integral operator $\Upsilon$ restricts to a contraction mapping on the unit ball $\mathcal{H}_{K_n}^\Upsilon(\tilde{\mathbb{B}}_n^\circ, \mathbb{C}^2)$. By the Contraction Theorem on a Ball (for example, see [Kre89, Theorem 5.1-4]), $\Upsilon$ has a unique fixed point $\varphi_n \in \mathcal{H}_{K_n}^\Upsilon(\tilde{\mathbb{B}}_n^\circ, \mathbb{C}^2)$. Thus, we have produced a sequence of vector-valued holomorphic functions $\varphi_n \in \mathcal{O}(\tilde{\mathbb{B}}_n^\circ, \mathbb{C}^2)$ each of which is a fixed point of the integral operator $\Upsilon$. We will now argue that this sequence converges to an element $\varphi \in \mathcal{O}(\tilde{\mathbb{B}}, \mathbb{C}^2)$ as $n \to \infty$.

By Lemma 6.8, we can arrange the constants $K_n$ such that $K_n \le K_{n+1}$, yielding inclusions

$$ \mathcal{H}_{K_n}^{\Upsilon}(\tilde{\mathbb{B}}_n^\circ, \mathbb{C}^2) \subset \mathcal{H}_{K_{n+1}}^{\Upsilon}(\tilde{\mathbb{B}}_n^\circ, \mathbb{C}^2). $$

At the same time, each inclusion $\tilde{\mathbb{B}}_n^\circ \hookrightarrow \tilde{\mathbb{B}}_{n+1}^\circ$ induces the reverse inclusion

$$ \mathcal{H}_{K_{n+1}}^{\Upsilon}(\tilde{\mathbb{B}}_{n+1}^\circ, \mathbb{C}^2) \subset \mathcal{H}_{K_{n+1}}^{\Upsilon}(\tilde{\mathbb{B}}_n^\circ, \mathbb{C}^2). $$

This means that for every $n$ we can view both $\varphi_n$ and the restriction of $\varphi_{n+1}$ to the open subset $\tilde{\mathbb{B}}_n^\circ \subset \tilde{\mathbb{B}}_{n+1}^\circ$ as elements of the same function space $\mathcal{H}_{K_{n+1}}^\Upsilon(\tilde{\mathbb{B}}_n^\circ, \mathbb{C}^2)$. It follows that they must be equal because they are fixed points of $\Upsilon$ but, by Lemma 6.7, the integral operator $\Upsilon$ restricts to a contraction mapping on $\mathcal{H}_{K_{n+1}}^\Upsilon(\tilde{\mathbb{B}}_n^\circ, \mathbb{C}^2)$ and so must have a unique fixed point. We deduce that the holomorphic vector-valued function $\varphi_{n+1} \in \mathcal{O}(\tilde{\mathbb{B}}_{n+1}^\circ, \mathbb{C}^2)$ is the analytic continuation to $\tilde{\mathbb{B}}_{n+1}^\circ$ of the holomorphic vector-valued function $\varphi_n \in \mathcal{O}(\tilde{\mathbb{B}}_n^\circ, \mathbb{C}^2)$ from the open subset $\tilde{\mathbb{B}}_n^\circ \subset \tilde{\mathbb{B}}_{n+1}^\circ$. Consequently, the sequence $(\varphi_n)$ converges to a holomorphic function $\varphi \in \mathcal{O}(\tilde{\mathbb{B}}, \mathbb{C}^2)$ that satisfies $\Upsilon[\varphi] = \varphi$.

The uniqueness of $\varphi \in \mathcal{O}(\tilde{\mathbb{B}}, \mathbb{C}^2)$ is the result of holomorphy, uniqueness of fixed points $\Upsilon$, and uniqueness of analytic continuation. Indeed, any other such $\varphi' \in \mathcal{O}(\tilde{\mathbb{B}}, \mathbb{C}^2)$ would necessarily coincide with $\varphi$ on the open subset $\tilde{\mathbb{B}}_1^\circ$ by the uniqueness of fixed points of $\Upsilon$, so its analytic continuation to all of $\tilde{\mathbb{B}}$ is nothing but $\varphi$. ■

As an immediate corollary of the proof of Proposition 6.1 (and in particular of Lemma 6.7), we obtain the following strong bound on the solution $\varphi$.

**Corollary 6.9.** For any excised Borel covering space $\tilde{\mathbb{B}}^\circ$, there is a constant $K > 0$ such that the unique solution $\varphi \in \mathcal{O}(\tilde{\mathbb{B}}, \mathbb{C}^2)$ of the integral equation (5.30) satisfies the bound

$$ \|\varphi|_{\tilde{\mathbb{B}}^\circ}\|_K \le 1. \qquad (6.17) $$

Moreover, $K$ can be taken arbitrarily small provided that the excision $\bar{\mathbb{D}} \subset \mathbb{C}_z$ is sufficiently large. In particular, for any $\gamma \in \tilde{\mathbb{B}}^\circ$,

$$ \inf \int_{0}^{|\gamma|} e^{-Ks} |\varphi(\gamma_s)Ds| \le 1, \qquad (6.18) $$

where $Ds = \pm t_{\pm}^* \lambda$ and the infimum is taken over all arc-length source-truncations $\gamma_s$ of $\gamma$.
## §6.2. Resurgence of the Associated System

Recall that in §3.2 we constructed a local holomorphic solution $\hat{\varphi}$ of the Initial Value Problem (3.25) as the Borel transform of the formal vector-valued solution $\hat{f}$ of the differential system (3.16). By Proposition 3.6, that local solution $\hat{\varphi} = \hat{\varphi}(z, \xi)$ is valid for all nonzero $z$ and all sufficiently small $\xi$; i.e., in an open neighbourhood $\Xi \subset C_z^* \times C_\xi$ of $\xi = 0$. Meanwhile, in §5.5 we explained that this open neighbourhood $\Xi$ is canonically isomorphic to an open neighbourhood $\tilde{\Omega} \subset \tilde{B}$ of the embedding $C_z^* \hookrightarrow \tilde{B}$ using the appropriate anchor map $\varrho$. Furthermore, by Lemma 5.25, the Initial Value Problem (3.25) is equivalent over the isomorphism $\varrho: \tilde{\Omega} \to \Xi$ to the Initial Value Problem (5.25). Therefore, by constructing the global solution $\varphi$ of (5.25) in §6.1, we have constructed the analytic continuation of the local solution $\hat{\varphi}$ to the entire unfolded Borel covering space $\tilde{B}$. We can therefore deduce from Proposition 6.1 the following analogue of our main Theorem 2.1.

**Proposition 6.10.** The formal solution $\hat{f}(z, \hbar)$ of the differential system (3.16) is resurgent.

Much like our main Theorem 2.1, this Proposition packs a lot of information, so we break it down into three parts corresponding to the three defining aspects of resurgence. First, the convergence of the Borel transform of $\hat{f}$ is equivalent to the existence of the local solution $\hat{\varphi}$ which was covered by Proposition 3.6; for completeness, we state this explicitly in §6.2.1, namely, Proposition 6.11. Second, the property of endless analytic continuation of the Borel transform is described in §6.2.2; namely, Proposition 6.12. Third, the property of exponential type of the Borel transform is explained in §6.2.3; namely, Proposition 6.14. Throughout, we indicate how these assertions imply the corresponding results in §2.1 for the deformed Painlevé I equation.

### §6.2.1. Convergence of the Borel transform.
First, we make explicit the convergence of the Borel transform of $\hat{f}$. This follows from Proposition 3.6 and the discussion in §5.5.

**Proposition 6.11 (convergence of the Borel transform).** The Borel transform

$$ \hat{\varphi}(z, \xi) = \sum_{n=0}^{\infty} \varphi_n(z) \xi^n := \mathfrak{B}[\hat{f}](z, \xi) = \sum_{n=0}^{\infty} \frac{1}{n!} f_{n+1}(z) \xi^n \quad (6.19) $$

of the formal solution $\hat{f}(z, \hbar)$ of the differential system (3.16) is a convergent power series in $\xi$, locally uniformly for all nonzero $z$. Moreover, $\hat{\varphi}$ can be canonically regarded via the anchor map $\varrho: \mathfrak{B} \to C_z \times C_\xi$ as a holomorphic vector-valued germ $\hat{\varphi}$ along the embedding $C_z^* \hookrightarrow \mathfrak{B}$. As such, there is an open neighbourhood $\tilde{\Omega} \subset \mathfrak{B}$ of the embedding $C_z^* \hookrightarrow \mathfrak{B}$ such that $\hat{\varphi}$ defines a holomorphic vector-valued function on $\tilde{\Omega}$. Equivalently, $\hat{\varphi}$ is a holomorphic vector-valued germ along the embedding $C_z^* \hookrightarrow \tilde{B}$ via the anchor map $\varrho: \tilde{B} \to C_z^* \times C_\xi$. As such, there is an open neighbourhood $\tilde{\Omega} \subset \tilde{B}$ of the embedding $C_z^* \hookrightarrow \tilde{B}$ such that $\hat{\varphi}$ defines a holomorphic vector-valued function on $\tilde{\Omega}$. In symbols, $\hat{\varphi} \in \mathcal{O}(\tilde{\Omega}, C^2) \cong \mathcal{O}(\tilde{\Omega}, C^2)$.

### §6.2.2. Endless Analytic Continuation.
Second, we describe the property of endless analytic continuation of the Borel transform $\hat{\varphi}$ in essence, it says that for every fixed nonzero $z$, the convergent $\xi$-power series $\hat{\varphi}(z, \xi)$ admits maximal analytic continuation to the universal cover of a Riemann surface punctured in ten points which is a fivefold covering of the Borel $\xi$-plane. This Riemann surface is nothing but the Borel surface $B_z = s^{-1}(z)$, the ten points are nothing but the critical elements $\Gamma_z \subset B_z$, and the fivefold cover is nothing but
the central charge $Z_z : \mathbb{B}_z \to \mathbb{C}_\xi$ with branch points at $\xi_\pm = \pm \frac{1}{30}z^5$. Furthermore, as $z$ varies but stays nonzero, these different analytic continuations of $\hat{\varphi}$ patch together into a global holomorphic vector-valued function $\varphi$ on a two-dimensional manifold, which turns out to be the unfolded Borel covering space $\tilde{\mathbb{B}}$. Therefore, the following assertion is an immediate consequence of Proposition 6.1.

**Proposition 6.12 (endless analytic continuation).** The Borel transform $\hat{\varphi}$ admits endless analytic continuation $\varphi$ to the unfolded Borel space $\mathbb{B}$ away from the critical locus $\Gamma \subset \mathbb{B}$. Namely, $\varphi$ defines a global vector-valued holomorphic function on the unfolded Borel covering space $\tilde{\mathbb{B}}$; in symbols, $\varphi \in \mathcal{O}(\tilde{\mathbb{B}}, \mathbb{C}^2)$.

We may sometimes refer to the holomorphic vector-valued function $\varphi \in \mathcal{O}(\tilde{\mathbb{B}}, \mathbb{C}^2)$ as the **global Borel transform** of $\hat{f}$ in order to distinguish it from the holomorphic germ $\hat{\varphi}$. The property of endless analytic continuation of $\hat{\varphi}$ may be stated in the following rather more elementary terms by sacrificing the more refined geometric information.

**Corollary 6.13.** Fix any nonzero $z_0 \in \mathbb{C}_z$ and let $\xi_\pm := \pm \frac{1}{30}z_0^5$. Then the Borel transform $\hat{\varphi}(z_0, \xi) \in \mathbb{C}^2 \otimes \mathbb{C}\{\xi\}$ extends to a holomorphic vector-valued function $\varphi(z_0, \xi)$ on the universal cover of the twice-punctured $\xi$-plane $\mathbb{C}_\xi \setminus \{\xi_+, \xi_-\}$. In symbols, $\hat{\varphi}(z_0, \xi) \in \mathcal{O}(\tilde{\mathbb{C}}_\xi \setminus \{\xi_\pm\}, \mathbb{C}^2)$. Moreover, since the universal cover of a twice-punctured complex plane is (noncanonically) isomorphic to the upper halfplane $\mathbb{H}$, we can (noncanonically) view $\varphi$ as a holomorphic vector-valued function on $\mathbb{H}$; i.e., $\varphi \in \mathcal{O}(\mathbb{H}, \mathbb{C}^2)$.

We can now infer the analogous endless analytic continuation property for the deformed Painlevé I equation and therefore prove Proposition 2.4.

**Proof of Proposition 2.4.** By Lemma 3.5, the Borel transform $\hat{\omega}(t, \xi)$ of the formal power series solution $\hat{q}(t, \hbar)$ is related to the Borel transform $\hat{\varphi}$ by the identity (3.23), which we quote here for convenience:

$$ \hat{\omega}(t, \xi) = f_0^+(z) + f_0^-(z) + \int_0^\xi (\hat{\varphi}_+(z, s) + \hat{\varphi}_-(z, s)) \, ds \quad \text{where } z^4 = -24t. \quad (6.20) $$

So, for any $\gamma \in \tilde{\mathbb{B}}$ and any synchronised parameterisation, we define

$$ \omega(\gamma) := s^* f_0^+(\gamma) + s^* f_0^-(\gamma) + \int_0^{|\gamma|} (\varphi_+(\gamma_s) + \varphi_-(\gamma_s)) \, Ds \quad (6.21) $$

where $Ds = \pm t_\pm^* \lambda_{\gamma_s}$. This expression does not depend on the choice of representative of $\gamma$ because $\varphi_+, \varphi_-$, and $t_\pm^* \lambda$ are holomorphic on $\tilde{\mathbb{B}}$. Therefore, it defines a global holomorphic function $\omega$ on the unfolded Borel covering space $\tilde{\mathbb{B}}$. By Lemma 3.5, we also have under the negation map $z \mapsto -z$ the following symmetries: $\hat{\varphi}_\pm(-z, \xi) = \hat{\varphi}_\mp(z, \xi)$, $f_0^\pm(-z) = f_0^\mp(z)$, and $\hat{\omega}(-z, \xi) = \hat{\omega}(z, \xi)$. But the lift to $\tilde{\mathbb{B}}$ of this negation map is the involution $\sigma$, so $\omega$ is a $\sigma$-invariant function on $\tilde{\mathbb{B}}$; i.e., $\sigma^*\omega = \omega$. Thus, $\omega$ defines a holomorphic function on the quotient $\tilde{\mathbb{B}}/\sigma$, which is the Borel covering space $\tilde{\mathbb{M}}$ by definition. In symbols, $\omega \in \mathcal{O}(\tilde{\mathbb{B}})^\sigma = \mathcal{O}(\tilde{\mathbb{B}}/\sigma) = \mathcal{O}(\tilde{\mathbb{M}})$. Now, all the other assertions of Proposition 2.4 follow from the detailed geometric descriptions in §5. ■
**§6.2.3. Exponential Type.** Finally, we describe the exponential bounds at infinity in $\xi$ on the global Borel transform $\varphi$.

**Proposition 6.14 (exponential type).** The Borel transform $\varphi \in \mathcal{O}(\tilde{\mathcal{B}}, \mathbb{C}^2)$ has exponential type at infinity on $\tilde{\mathcal{B}}$, locally uniformly for all nonzero $z$. That is to say, for every nonzero $z$ and any direction $\alpha$ at infinity in $\tilde{\mathcal{B}}_z$, there are real constants $C, K > 0$ and a sectorial neighbourhood $\Sigma_z \subset \tilde{\mathcal{B}}_z$ whose opening contains $\alpha$ such that

$$|\varphi(\gamma)| \le C e^{K|Z(\gamma)|} \quad \forall \gamma \in \Sigma_{\tau} \subset \tilde{\mathcal{B}}_z. \qquad (6.22)$$

Furthermore, every nonzero $z_0 \in \mathbb{C}_z$ has a neighbourhood $U_0 \subset \mathbb{C}_z$ with the property that $C, K$ can be chosen uniformly for all $z \in U_0$, provided that $\alpha$ and $\Sigma_z$ are chosen continuously for all $z \in U_0$. Moreover, $K$ can be taken arbitrarily small provided that $z_0$ is sufficiently large.

**Proof.** This follows from Corollary 6.9, and more specifically from (6.18), by recognising that the central charge of $\gamma$ is bounded by the arc-length of any synchronised parameterisation of $\gamma$; i.e., $|Z(\gamma)| \le |\gamma|$. ■

The property of exponential type of $\varphi$ may likewise be stated in more elementary terms using the point of view on $\varphi$ as a multivalued function over the twice-punctured $\xi$-plane as follows.

**Corollary 6.15.** Fix any nonzero $z_0 \in \mathbb{C}_z$ and put $\xi_{\pm} := \pm \frac{1}{30} z_0^5$. Then the Borel transform $\varphi(z_0, \xi)$ has exponential type at infinity in all directions of the $\xi$-plane. That is to say, for any choice of branch of the multivalued function $\varphi(z_0, \xi)$ on $\mathbb{C}_{\xi} \setminus \{\pm \xi\}$, there are constants $C, K > 0$ such that, for all $\xi$ sufficiently large,

$$|\varphi(z_0, \xi)| \le C e^{K|\xi|}. \qquad (6.23)$$

Furthermore, every nonzero $z_0 \in \mathbb{C}_z$ has a neighbourhood $U_0 \subset \mathbb{C}_z$ with the property that $C, K$ can be chosen uniformly for all $z \in U_0$, provided that the branch of $\varphi(z, \xi)$ is chosen continuously for all $z \in U_0$. Moreover, $K$ can be taken arbitrarily small provided that $z_0$ is sufficiently large.

We may now complete the proof of the final ingredient in the proof of our main Theorem 2.1 concerning the resurgence of the deformed Painlevé I equation, which is Proposition 2.6 about the exponential type of the Borel transform $\hat{\omega}$ of $\hat{q}$. This relies on the following analogue of Corollary 6.9 which follows immediately from Proposition 2.6 and from the defining formula (6.21) for $\omega$.

**Corollary 6.16.** For any excised Borel covering space $\tilde{\mathcal{B}}^{\circ}$, there is a constant $K > 0$ such that the global Borel transform $\omega \in \mathcal{O}(\tilde{\mathcal{B}})^{\sigma}$ satisfies the bound

$$\|\omega|_{\tilde{\mathcal{B}}^{\circ}}\|_{K} \le 1. \qquad (6.24)$$

Moreover, $K$ can be taken arbitrarily small provided that the excision $\bar{\mathcal{D}} \subset \mathbb{C}_z$ is sufficiently large. In particular, for any $\gamma \in \tilde{\mathcal{B}}^{\circ}$,

$$\inf \int_{0}^{|\gamma|} e^{-Ks} |\omega(\gamma_s)Ds| \le 1, \qquad (6.25)$$
where $D_s = \pm t_{\pm}^* \lambda$ and the infimum is taken over all arc-length source-truncations $\gamma_s$ of $\gamma$.

### §6.3. Borel Summability and the Stokes Phenomenon

In this final subsection, we describe the Borel summability properties of the formal solution $\hat{f}$ of the differential system (3.16). In addition, the Borel summability properties of the formal solution $\hat{q}$ of the deformed Painlevé I system, described in §2.2, are direct consequences of their analogues for $\hat{f}$, which we make explicit in this subsection. Just like in §2.2, we prioritise clarity and precision over conciseness.

First, we introduce two special directions at every nonzero point in the z-plane in which the Borel resummation of $\hat{f}$ is not well-defined.

**Definition 6.17 (Stokes directions).** For any $z_0 \in \mathbb{C}_z^*$, let $\vartheta_0 := \arg(z_0)$. We define the Stokes directions at $z_0$ to be the directions

$$ \alpha_+ := 5\vartheta_0 \quad \text{and} \quad \alpha_- := 5\vartheta_0 + \pi . $$

See Fig. 3. All other directions are called regular directions, and we put

$$ A_1 := (\alpha_+, \alpha_-) \quad \text{and} \quad A_2 := (\alpha_-, \alpha_+) . $$

#### §6.3.1. Pointwise Borel summability.
First, we spell out the case of Borel summability of $\hat{f}$ for a fixed value of $z$. We start by considering a single regular direction.

**Proposition 6.18 (Pointwise Borel Summability in a Single Direction).** For any nonzero $z_0 \in \mathbb{C}_z$, the formal vector-valued solution $\hat{f}(z_0, \hbar) \in \mathbb{C}^2 \otimes \mathbb{C}[\![\hbar]\!]$ is stably Borel summable in every regular direction $\alpha$ at $z_0$. Thus, the Borel resummation

$$ f_{\alpha}(z_0, \hbar) := s_{\alpha}[\hat{f}](z_0, \hbar) = f_0 + \mathcal{L}_{\alpha}[\hat{\varphi}](z_0, \hbar) \quad (6.26) $$

in the direction $\alpha$ defines a holomorphic vector-valued function on a sectorial neighbourhood $S \subset \mathbb{C}_\hbar$ of the origin with opening $\text{Arc}_\pi(\alpha)$ which is asymptotic to $\hat{f}(z_0, \hbar)$ of uniform factorial type:

$$ f_{\alpha}(z_0, \hbar) \approx \hat{f}(z_0, \hbar) \quad \text{as } \hbar \to 0 \text{ unif. along } \text{Arc}_{\pi}(\alpha) . \quad (6.27) $$

In fact, $f_\alpha(z_0, \hbar)$ is unique with respect to this property. Furthermore, the sectorial neighbourhood $S \subset \mathbb{C}_\hbar$ can be chosen to be the straight sector $S = \text{Sect}_\pi(\alpha; r)$ of some radius $r > 0$ that can be taken arbitrarily large provided that $z_0$ is sufficiently large. In particular, Proposition 2.9 is true.

**Proof.** This is a direct consequence of the exponential type property of $\varphi$, Proposition 6.14. In fact, in this case we may appeal to the more elementary formulation in Corollary 6.15. Indeed, if $\alpha$ is a regular ray at $z_0$, then the infinite straight ray $e^{i\alpha}\mathbb{R}_+ \subset \mathbb{C}_\xi$ does not pass through any of the two Borel singular values $\xi_\pm$. Consequently, the Laplace transform of $\varphi$ in the direction $\alpha$,

$$ \int_{e^{i\alpha}\mathbb{R}_+} e^{-\xi/\hbar}\varphi(z_0, \xi) \, d\xi $$

is well-defined because there is some $K > 0$ such that $\varphi$ satisfies the exponential bound
(6.23) for sufficiently large $\xi$. Furthermore, $\alpha$ necessarily belongs to an open arc $(\alpha - \varepsilon, \alpha + \varepsilon)$ of regular directions, which means the Laplace transform of $\varphi$, and hence the Borel resummation of $\hat{f}$, is well-defined in a sectorial neighbourhood of the origin in the $\hbar$-plane with opening angle $\pi + 2\varepsilon$. Such a sectorial domain necessarily contains a straight sector of opening angle $\pi$. ■

As we vary the ray $\alpha$ through an arc A consisting of regular directions only, the different Borel resummations $f_\alpha(z_0, \hbar)$ can be patched together into a single holomorphic function $f_A(z_0, \hbar)$ on a larger sector in the $\hbar$-plane. This leads to the following description of Borel summability of $\hat{f}$ along an arc of directions.

**Corollary 6.19 (Pointwise Borel Summability in an Arc of Directions).** For any nonzero $z_0 \in \mathbb{C}_z$ and any arc $A \subset \mathbb{S}^1$ of regular directions at $z_0$, the Borel resummations $f_\alpha(z_0, \hbar)$ of $\hat{f}(z_0, \hbar)$ for $\alpha \in A$ assemble into a single holomorphic vector-valued function $f_A(z_0, \hbar)$ defined on a sectorial neighbourhood $S \subset \mathbb{C}_\hbar$ of the origin with opening $\text{Arc}_\pi(A)$ which is asymptotic to $\hat{f}(z_0, \hbar)$ of factorial type:

$$f_A(z_0, \hbar) \approx \hat{f}(z_0, \hbar) \quad \text{as } \hbar \to 0 \text{ along } \text{Arc}_\pi(A). \qquad (6.28)$$

In fact, $f_A(z_0, \hbar)$ is unique with respect to this property. Furthermore, if A is not bounded by a Stokes ray at $z_0$, then the sectorial neighbourhood $S \subset \mathbb{C}_\hbar$ can be chosen to be the straight sector $S = \text{Sect}_\pi(A; r)$ of some radius $r > 0$ that can be taken arbitrarily large provided that $z_0$ is sufficiently large. In particular, Corollary 2.10 is true.

**§6.3.2. Locally uniform Borel summability.** Next, we state the Borel summability property of $\hat{f}$ when $z$ is allowed to vary in a small local neighbourhood of a fixed nonzero point in the $z$-plane. Again, we start by considering summability in a single regular direction.

**Proposition 6.20 (Locally Uniform Borel Summability in a Single Direction).** For any nonzero $z_0 \in \mathbb{C}_z$ and any regular direction $\alpha$ at $z_0$, there is a neighbourhood $U \subset \mathbb{C}_z^*$ around $z_0$ such that $\hat{f}(z, \hbar)$ is stably Borel summable in the direction $\alpha$ uniformly for all $z \in U$. Thus, there is a sectorial neighbourhood $S \subset \mathbb{C}_\hbar$ of the origin with opening $\text{Arc}_\pi(\alpha)$ such that the Borel resummation $f_\alpha(z, \hbar)$ of $\hat{f}(z, \hbar)$ in the direction $\alpha$ defines a holomorphic vector-valued function on the domain $U \times S$ which is uniformly asymptotic to $\hat{f}(z, \hbar)$ of uniform factorial type:

$$f_\alpha(z, \hbar) \approx \hat{f}(z, \hbar) \quad \text{as } \hbar \to 0 \text{ unif. along } \text{Arc}_\pi(\alpha), \qquad (6.29)$$

uniformly for all $z \in U$. In fact, $f_\alpha$ is unique with respect to this property. Furthermore, the sectorial neighbourhood $S \subset \mathbb{C}_\hbar$ can be chosen to be the straight sector $S = \text{Sect}_\pi(\alpha; r)$ of some radius $r > 0$ that can be taken arbitrarily large provided that $z_0$ is sufficiently large and $U$ is sufficiently small. In particular, Proposition 2.11 is true.

As we vary both the ray $\alpha$ in an arc A and $z$ in a neighbourhood $U_0$ of $z_0$, we can patch the Borel resummations provided that A and $U_0$ are sufficiently small in a coherent way as follows.

**Corollary 6.21 (Locally Uniform Borel Summability in an Arc of Directions).** For any nonzero $z_0 \in \mathbb{C}_z$ and any arc $A \subset \mathbb{S}^1$ of regular directions at $z_0$ whose boundary does not contain any Stokes directions at $z_0$, there is a neighbourhood $U \subset \mathbb{C}_z^*$ around $z_0$ such that $\hat{f}(z, \hbar)$ is Borel summable in every direction $\alpha \in A$ uniformly for all $z \in U$. Thus, there is
a sectorial neighbourhood $S \subset \mathbb{C}_h$ of the origin with opening $\text{Arc}_\pi(A)$ such that the Borel resummations $f_\alpha(z, \hbar)$ of $\hat{f}(z, \hbar)$ for $\alpha \in A$ assemble into a single holomorphic vector-valued function $f_A(z, \hbar)$ defined on the domain $U \times S$ which is uniformly asymptotic to $\hat{f}(z, \hbar)$ of factorial type:

$$f_A(z, \hbar) \approx \hat{f}(z, \hbar) \quad \text{as } \hbar \to 0 \text{ along } \text{Arc}_\pi(A), \qquad (6.30)$$

uniformly for all $z \in U$. In fact, $f_A$ is unique with respect to this property. Furthermore, the sectorial neighbourhood $S \subset \mathbb{C}_h$ can be chosen to be the straight sector $S = \text{Sect}_\pi(A; r)$ of some radius $r > 0$ that can be taken arbitrarily large provided that $z_0$ is sufficiently large and $U$ is sufficiently small. In particular, Corollary 2.12 is true.

### §6.3.3. Borel summability in Stokes sectors.
Now we describe the Borel summability properties of $\hat{f}$ in the large. For this purpose, we fix a direction for the resummation and describe the distinguished regions in the $z$-plane where the Borel resummation in this direction is well-defined.

**Definition 6.22 (Stokes lines and sectors).** Fix any phase $\alpha \in \mathbb{S}^1$. A **Stokes line** in the $z$-plane is any infinite straight ray $e^{i\theta}\mathbb{R}_+ \subset \mathbb{C}_z$ with phase $\theta$ satisfying $10\theta + 2\alpha \equiv 0$. The union of all Stokes lines is called the **Stokes graph**. Any connected component $V \subset \mathbb{C}_z$ of the complement of the Stokes graph is called a **Stokes sector**.

Notice that the equation $10\theta + 2\alpha \equiv 0$ has exactly ten distinct solutions in $\mathbb{S}^1$ distributed evenly around the circle. So for the differential system (3.16), there are a total of ten Stokes lines and therefore ten Stokes sectors each of which is an infinite sector in the $z$-plane with opening angle $\pi/5$.

**Proposition 6.23 (Borel Summability in Stokes Sectors).** Fix any phase $\alpha \in \mathbb{S}^1$ and select a Stokes sector $V \subset \mathbb{C}_z$. Then $\hat{f}(z, \hbar)$ is stably Borel summable in the direction $\alpha$, locally uniformly for all $z \in V$. Thus, the Borel resummation $\hat{f}_\alpha(z, \hbar) = s_\alpha[\hat{f}](z, \hbar)$ defines a holomorphic vector-valued function on a domain $V \subset V \times \mathbb{C}_h$ with the following property: every $z_0 \in V$ has a neighbourhood $V_0 \subset V$ such that there is a sectorial neighbourhood $S_0 \subset \mathbb{C}_h$ of the origin with opening $\text{Arc}_\pi(\alpha)$ satisfying $V_0 \times S_0 \subset V$. Furthermore, $f_\alpha$ is locally uniformly asymptotic to $\hat{f}$ of uniform factorial type:

$$f_\alpha(z, \hbar) \approx \hat{f}(z, \hbar) \quad \text{as } \hbar \to 0 \text{ unif. along } \text{Arc}_\pi(\alpha), \qquad (6.31)$$

locally uniformly for all $z \in V$. In fact, $f_\alpha$ is the unique holomorphic vector-valued function on $V$ with this property. Moreover, $V$ is a maximal domain of Borel summability of $\hat{f}$ in the direction of $\alpha$; i.e., there does not exist another domain $V'$ properly containing $V$ such that $\hat{f}(z, \hbar)$ is stably Borel summable in the direction $\alpha$, locally uniformly for all $z \in V$. In particular, Proposition 2.14 is true.

### §6.3.4. The Stokes phenomenon.
Finally, we describe the failure of Borel summability in a Stokes direction and the associated Stokes phenomenon.

**Corollary 6.24 (Lateral Borel Summability).** Fix any nonzero $z_0 \in \mathbb{C}_z$ and suppose $\alpha$ is a Stokes ray at $z_0$. Then $\hat{f}(z_0, \hbar)$ is laterally Borel summable in the direction $\alpha$. Thus, the left
and right lateral Borel resummations

$$f_{\alpha}^{L/R}(z_0, \hbar) := s_{\alpha}^{L/R}[\hat{f}](z_0, \hbar) = f_0 + \mathcal{L}_{\alpha}^{L/R}[\varphi](z_0, \hbar) \quad (6.32)$$

in the direction $\alpha$ define two holomorphic vector-valued functions on a sectorial neighbourhood $S \subset \mathbb{C}_h$ of the origin with opening $\text{Arc}_\pi(\alpha)$, each of which is asymptotic to $\hat{f}(z_0, \hbar)$ of factorial type:

$$f_{\alpha}^{L/R}(z_0, \hbar) \approx \hat{f}(z_0, \hbar) \quad \text{as } \hbar \to 0 \text{ along } \text{Arc}_{\pi}(\alpha). \quad (6.33)$$

**Definition 6.25 (Stokes jump).** For any nonzero $z_0 \in \mathbb{C}_z$, suppose $\alpha$ is a Stokes ray at $z_0$, and let $f_{\alpha}^{L/R}(z_0, \hbar) \in \mathcal{O}(S, \mathbb{C}^2)$ be the two lateral Borel resummations of $\hat{f}(z_0, \hbar)$ in the direction $\alpha$ from Corollary 6.24. The **Stokes jump** across $\alpha$ from right to left is the difference between the left and right lateral Borel resummations of $\hat{f}$ in the direction $\alpha$:

$$\Delta_{\alpha} \hat{f}(z_0, \hbar) := f_{\alpha}^{L}(z_0, \hbar) - f_{\alpha}^{R}(z_0, \hbar). \quad (6.34)$$

The Stokes jump is clearly a holomorphic vector-valued function on $S$; i.e., $\Delta_{\alpha} \hat{f}(z_0, \hbar) \in \mathcal{O}(S, \mathbb{C}^2)$. Furthermore, since both $f_{\alpha}^{L}$ and $f_{\alpha}^{R}$ admit the same asymptotic expansion as $\hbar \to 0$ in $S$, the Stokes jump is asymptotic to 0 of factorial type:

$$\Delta_{\alpha} \hat{f}(z_0, \hbar) \approx 0 \quad \text{as } \hbar \to 0 \text{ along } \text{Arc}_{\pi}(\alpha). \quad (6.35)$$

Again, let us point out the absence of the qualifier “uniformly” in (6.35), for otherwise it would force the Stokes jump to be identically zero.

**Definition 6.26 (variation).** For any nonzero $z_0 \in \mathbb{C}_z$, suppose $\alpha$ is a Stokes ray at $z_0$, and let $\xi_0 \in \mathbb{C}_{\xi}$ be the Borel singular value corresponding to the visible singularity in the direction $\alpha$ at $z_0$. The **variation** of the multivalued vector function $\varphi(z_0, \xi)$ at $\xi_0$ is the difference between its values on two consecutive sheets:

$$\Delta_{\xi_0} \varphi(z_0, \xi) := \varphi(z_0, \xi_0 + \xi^L) - \varphi(z_0, \xi_0 + \xi^R). \quad (6.36)$$

where $\xi^L$ and $\xi^R$ are two preimages of $\xi$ on the universal cover of the punctured neighbourhood of $\xi_0$ related to each other by the anti-clockwise primitive generator of the deck transformations; i.e., $\xi^R = e^{2\pi i \xi^L}$.

The variation $\Delta_{\xi_0} \varphi(z_0, \xi)$ may be regarded as a sectorial germ at the origin in the Borel $\xi$-plane. But since both $\varphi(z_0, \xi_0 + \xi^L)$ and $\varphi(z_0, \xi_0 + \xi^R)$ admit analytic continuation in the direction $\alpha$ of exponential type at infinity, we obtain the following characterisation of the Stokes jump in terms of the Laplace transform of the Borel transform's variation at the visible singularity.

**Proposition 6.27 (Stokes Phenomenon).** Fix any nonzero $z_0 \in \mathbb{C}_z$, suppose $\alpha$ is a Stokes direction at $z_0$, and let $f_{\alpha}^{L/R}(z_0, \hbar) \in \mathcal{O}(S)$ be the two lateral Borel resummations of $\hat{f}(z_0, \hbar)$ in the direction $\alpha$. Let $\xi_0 \in \mathbb{C}_{\xi}$ be the Borel singular value corresponding to the visible singularity in the direction $\alpha$ at $z_0$. Then for all $\hbar \in S$, the Stokes jump across $\alpha$ is

$$\Delta_{\alpha} \hat{f}(z_0, \hbar) = e^{-\xi_0 / \hbar} \mathcal{L}_{\alpha} [\Delta_{\xi_0} \varphi](z_0, \hbar), \quad (6.37)$$
where $\Delta_{\xi_0} \varphi(z_0, \xi)$ is the variation of $\varphi$ at $\xi_0$ given by (6.36).

**Proof.** This is a matter of a simple calculation. Expanding (6.34) according to its definition (6.32), we have:

$$ \Delta_{\alpha} \hat{f}(z_0, \hbar) = f_{\alpha}^{L}(z_0, \hbar) - f_{\alpha}^{R}(z_0, \hbar) = \mathcal{L}_{\alpha}^{L}[\varphi](z_0, \hbar) - \mathcal{L}_{\alpha}^{R}[\varphi](z_0, \hbar) = \int_{\mathcal{C}} e^{-\xi/\hbar} \varphi(z_0, \xi) \, d\xi $$

where the integration contour $\mathcal{C}$ is homotopic to the keyhole contour around $\xi_0$ obtained by concatenating the inverse of the contour $e^{i\alpha}\mathbb{R}_+$ with $e^{i\alpha}\mathbb{R}_+^L$. The two arms of this keyhole contour lift via the central charge $Z_{z_0} : \mathbb{B}_{z_0} \to \mathbb{C}_{\xi}$ to infinite geodesics on two consecutive sheets of the Borel surface emanating from the visible Borel singularity above $\xi_0$. Therefore, the above contour integral may be written like so:

$$ \int_{\xi_0}^{e^{i\alpha}\cdot\infty} e^{-\xi/\hbar} \left( \varphi(z_0, \xi^L) - \varphi(z_0, \xi^R) \right) d\xi , $$

where $\varphi(z_0, \xi^L)$ and $\varphi(z_0, \xi^R)$ denote the values at $(z_0, \xi)$ of the two relevant branches of the multivalued function $\varphi(z_0, \xi)$. Shifting the integration variable $\xi \to \xi_0 + \xi$, the factor $e^{-\xi_0/\hbar}$ appears and the integral becomes the Laplace transform of the variation (6.36). So we get (6.37). ■

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MOHAMAD ALAMEDDINE

Université Jean Monnet, Institut Camille Jordan<fnref n="1" />

OLIVIER MARCHAL

Université Jean Monnet, Institut Camille Jordan

Institut Universitaire de France

NIKITA NIKOLAEV

School of Mathematics, University of Birmingham<fnref n="2" />

NICOLAS ORANTIN

Section de Mathématiques, Université de Genève<fnref n="3" />

<footnote n="1">¹Institut Camille Jordan, CNRS UMR 5208, Les Forges 2, 20 Rue du Dr Annino, 42000 Saint-Étienne, France.</footnote>

<footnote n="2">²Watson Building, Edgbaston, Birmingham, B15 2TT, United Kingdom.</footnote>

<footnote n="3">³24 rue du Général Dufour, 1211 Genève 4, Suisse.</footnote>