aainstitutetext: Institute of Physics, University of Amsterdam, Science Park
904, 1090 GL Amsterdam, The Netherlandsbbinstitutetext: Nikhef, Theory Group,
Science Park 105, 1098 XG, Amsterdam, The Netherlands

# Resurgence of large order relations

Coenraad Marinissen a , Alexander van Spaendonck a and Marcel Vonk
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

One of the main applications of resurgence in physics is the decoding of
nonperturbative effects through large order relations. These relations connect
perturbative asymptotic expansions of observables to expansions around other
saddle points. Together, this data is unified in transseries that describe the
nonperturbative structure. It is known that such large order relations
themselves also take the form of transseries. We study these large order
transseries, and show that they in turn are resurgent – that is: their
nonperturbative sectors ‘know about each other’ through Borel residues that
are essentially equal to those of the original transseries. We show that with
an appropriate resummation prescription, large order relations are often
exact: they can be used to exactly compute perturbative coefficients – not
just their large order growth. Finally, we argue that Stokes phenomenon plays
an important role for large order relations, for example if we want to extend
the discrete index of the perturbative coefficients to arbitrary complex
values.

##  1 Introduction

One of the most common approximation methods for physics problems is
perturbation theory. When a problem is too difficult to solve analytically,
one introduces a small parameter $x$ or a large parameter $z$, and writes the
answer to the problem in the form of a power series – in this paper, usually
of the form

$\varphi^{(0)}(z)=\sum_{g=0}^{\infty}\varphi^{(0)}_{g}z^{-g}\,.$ (1)

Unfortunately, even as a method of approximation, this does not always solve
the problem at hand: in many situations, the power series that one finds is
not a convergent Taylor series for an analytic function, but it is asymptotic
in the sense of Poincaré. That is, while every finite sum of its terms may
give a better approximation of some function as $z$ gets larger, the
coefficients $\varphi^{(0)}_{g}$ often grow factorially,
$\varphi^{(0)}_{g}\sim A^{-g}\,g!$, and as a result the infinite sum converges
for no value of $z$.

Nonetheless, a finite value can be assigned to these asymptotic series for
finite $z$ by resumming the series. The simplest method is that of optimal
truncation where, for a given finite value of $z$, one sums all the terms in
the series up to an ‘optimal’ term $\varphi^{(0)}_{G}z^{-G}$ determined by
$G\simeq|Az|$. However, this method is in general only accurate up to
exponentially small contributions of the form $\sim{\rm e}^{-|Az|}$ which are
nonperturbative in the expansion parameter $1/z$. In order to resum asymptotic
series beyond the resolution $\mathcal{O}({\rm e}^{-|Az|})$, we can turn to
the method of Borel summation Borel:1899 . This summation technique plays a
central role in the theory of resurgence introduced by J. Écalle in the 1980s
ecalle1985fonctions , and contrary to optimal truncation provides a way to
resum asymptotic series up to arbitrary precision. In fact, Écalle showed that
Borel summation lies at the basis of a systematic procedure for finding
missing nonperturbative contributions, and that these are intricately
connected to one another in an algebraic sense that he formulated in his alien
calculus.

Over the past decade or so, these techniques from resurgence have become more
popular among physicists and have been applied to various physical models,
including (but definitely not limited to) the Painlevé I equation
Garoufalidis:2010ya ; Aniceto:2011nu ; Baldino:2022aqm ; vanSpaendonck:2022kit
, quantum mechanics AIHPA_1983__39_3_211_0 ; AIHPA:1999:71 ; Zinn-
Justin:2004vcw ; Dunne:2014bca ; vanSpaendonck:2023znn , quantum field
theories Dunne:2012ae ; Gukov:2016njj ; Marino:2019eym ; DiPietro:2021yxb ,
hydrodynamics Heller:2015dha ; Aniceto:2015mto , two-dimensional gravity
Gregori:2021tvs ; Eynard:2023qdr , topological strings
marino2008nonperturbative ; Marino:2008ya ; Couso-Santamaria:2013kmu ;
Gu:2023mgf  and many more. In many of these examples, resurgence analysis
suggests that we should supplement asymptotic series like (1) with additional
nonperturbative contributions, which can be united in what is known as a
transseries (see e.g. edgar2010transseries ), in its simplest form:

$\Phi(\vec{\sigma},z)=\sum_{n=0}^{\infty}\sigma_{n}\,\varphi^{(n)}(z)\,{\rm
e}^{-nAz}\,.$ (2)

The formal power series
$\varphi^{(n)}(z)=\sum_{g=0}^{\infty}\varphi^{(n)}_{g}z^{-g}$ are often called
transseries sectors and they are usually all asymptotically divergent –
including our original series (1), of course. From a physics point of view,
one can often think of the transseries as the semiclassical expansion of a
path integral which yields a sum over saddle point contributions. In this
interpretation, $nA$ is the instanton action of the $n$-th saddle point
solution. In the transseries, each saddle point contribution is weighted by a
parameter $\sigma_{n}$ whose value is determined by boundary conditions
imposed on the path integral.

A well-known result from resurgence is that knowledge of the perturbative
coefficients $\varphi^{(0)}_{g}$ in (1) allows us to decode the coefficients
in other sectors $\varphi^{(n)}$ of the transseries. In practice, this is
achieved through the use of so-called large order relations, which in the
literature often appear in the form

$\frac{\varphi_{g}^{(0)}}{\Gamma(g)}\simeq\sum_{k=1}^{\infty}\frac{S_{1}^{k}}{2\pi{\rm
i}}\sum_{h=0}^{\infty}(kA)^{g-h}\frac{\Gamma(g-h)}{\Gamma(g)}\varphi_{h}^{(k)}\,,$
(3)

and which should be understood to hold in the large $g$ limit. This form of
the large order relation – in which each ‘sector’ $k$ is carries constant of
proporionality $S_{1}^{k}$ – was presented first in Aniceto:2011nu  in the
contexts of ODEs and matrix models, and is a multi-instanton generalization of
the more rigorously derived large order relations found in Costin_1999 ;
garoufalidis2009universality .

The study of the large order behaviour of asymptotic series in physics
predates the conception of resurgence and large order relations. In the
seventies, Bender and Wu studied the large $n$ behaviour of the perturbative
energy levels $E_{n}$ of the quartic anharmonic oscillator Bender:1969si  and
showed that their large order behaviour could be recovered from studying the
one-instanton sector of the ‘negatively coupled’ quartic oscilator
Bender:1973rz . Many other implicit and explicit discussions of large order
behaviour have appeared over the years, e.g. PhysRevD.16.408 ; SHikami_1979 ;
LeGuillou:1990nq . To the best of our knowledge, the first (leading instanton)
large order formulas of the form (3) appeared in Costin_1999  and subsequently
also in marino2008nonperturbative ; Marino:2008ya ;
garoufalidis2009universality  before being generalized to the multi-instanton
version (3) from Aniceto:2011nu .

In the more recent mathematics literature, the focus of resurgence research
has not been on large order behaviour as much as it has been in the physics
literature, but Écalle and Sharma described a method to study large order
relations in zbMATH05976778 , and most of the essectial techniques needed to
discuss large order behaviour can be found in the textbook sauzinbook . In
2020, Sauzin used these techniques to study the large order behavior of the
asymptotic ‘Stirling series’ that occurs when expanding the logarithm of the
gamma function sauzinprivate .

The applications of large order relations go beyond toy models: in a recent
investigation of the resurgent structure of the Adler function of QED and QCD
Laenen:2023hzu , large order relations were studied with the aim of extracting
higher nonperturbative sectors that have an interpretation as renormalon
contributions. For each $k$ in (3) the ratio of gamma functions yields a new
asymptotic series in $1/g$, which in the context of the Adler function (as in
many other examples of asymptotic series) was studied by plotting poles of the
Padé approximant to its Borel transform. This led to the plots shown in figure
1, where the black dots indicate the singularities of the Borel transform of
these series, which in turn should determine nonperturbative contibutions to
the large order relation. It was the interesting singularity structure
observed there – somewhat reminiscent of the ‘peacock patterns’ observed in
Garoufalidis:2020xec  – that led us to the question which underlying structure
gives rise to these patterns.

The observed pattern is not specific to the Adler function: it also appears
for a variety of other models – see e.g. figure 2, which displays a similar
plot for a toy model that we will study extensively later in this paper. In
this case, the Padé poles line up to mimic branch cuts (rather than poles) in
the Borel transform of the large order expansion, but the repeating structure
of singularities is essentially the same. These observations are particularly
interesting in light of the fact, as can be seen from (3), that the $1/g$
expansion of the large order relation (3) itself takes the form of a
transseries Aniceto:2011nu ; Aniceto_2019 . These two ingredients form the
starting point of this work.

In this work we study the resurgent structure of the transseries that large
order relations constitute. We show how this resurgent structure can be
understood using the so-called Stirling transform, which repackages all the
coefficients of the instanton sectors in (2) into new series that appear in
the large order relation (3). In particular, these new series make the
repeated pattern in the imaginary direction in figures 1 and 2(a) manifest,
leading to an underlying geometric structure that we dub the Borel cylinder.
We subsequently derive the Stokes phenomenon stokes_2009a ; stokes_2009b  of
the large order transseries, explain how it relates to the Stokes phenomenon
of the original asymptotic expansion, and show how one can consistenly resum
the large order transseries. In doing so, we test to what extent the
resummation of the large order relation describes the perturbative
coefficients exactly: when we resum (3) for finite $g$, can we calculate the
perturbative coefficients $\varphi^{(0)}_{g}$ to arbitrary precision? We will
find that the answer to this question is often ‘yes’, even in cases where one
ignores potential contributions to the large order relation coming from a
Cauchy integral at infinity. Finally, we will see that the resummation
establishes a natural extension of the large order relation to non-integer and
even complex values of $g$. Throughout the paper, we provide numerous examples
to illustrate various concepts and to numerically verify our statements.

(a)

(b)

Figure 1: Borel transform of the Adler function large order expansion. The
black dots are the singularities of the Borel-Padé transform of the $1/g$
expansion of (3) for $k=1$, as discussed in Laenen:2023hzu . The red
transparant disks are added to aid the eye and are positioned at
$\log(\ell)+\pi{\rm i}n$, with $\ell=1,2,3$ and $n\in\mathbb{Z}$. The two
plots show the contributions to the large order relation coming from the two
leading nonperturbative sectors.

(a)

(b)

Figure 2: Figure 2(a) shows the singularities of the Borel-Padé transform of
(3) for $k=1$ for the quartic free energy. Similar to the plots shown in 1, we
observe singularities at $\log(\ell)+2\pi{\rm i}n$, but now for integers
$\ell\geq 2$ and $n\in\mathbb{Z}$. As the singularities are branch cuts
(mimicked by the accumulation of poles of the Padé approximant), the poles
corresponding to different branch cuts are hard to distinguish. In figure
2(b), we therefore show the Padé singularities after applying a conformal map
to the Borel plane Costin:2020hwg ; Costin:2021bay , which maps the
singularities on the positive real axis to the unit circle. We clearly observe
that the branch cuts are pulled apart and can therefore conclude that the
poles on the real axis in figure 2(a) belong to different branch cuts. We
refer to example 3.2 for more details.

In section 2, we review the tools from resurgence that are necessary for the
study of large order relations. The reader who is already familiar with the
standard ingredients of resurgence – in particular Borel summation,
transseries and the usual derivation of large order formulas – may safely skip
this section, up to perhaps a quick look at the subtleties in the definition
of Borel residues that we point out in section 2.3. In section 3, we study the
large order relations and their transseries structure. In section 3.1 we
introduce the Stirling transform and its Borel transform, which in section 3.2
allow us to establish resurgence relations for the large order transseries. In
section 3.3, we then derive the Stokes automorphism and a consistent way for
resumming these transseries, and finally formulate ‘exact’ large order
relations in section 3.4. Section 4 is devoted to a test of the exact large
order relation for a toy model called the quartic partition function and
section 5 finishes with our conclusions and an outlook. There are three
appendices: in appendix A we provide some more background on the various
possible Borel transforms and Borel summations, in appendix B we discuss the
Stirling transform, and in appendix C we give a derivation of the Stokes
phenomenon for the gamma function that we need in section 4.2.

##  2 Resurgence and asymptotics

In this section, we review material from the theory of Borel summation,
transseries and resurgence that we need in the rest of the paper. In
particular, we collect the ingredients that usually go into the derivation of
a large order formula. We mostly use the notation of the primer Aniceto_2019 ,
to which we also refer for further background information. Another useful
reference in which the material of sections 2.1 – 2.3 is explained in a more
mathematical language is sauzinbook .

In section 2.1, we review the notion of (Gevrey-1) aymptotic series, and how
these divergent expansions can be Borel summed into honest functions. In
section 2.2 we recall how ambiguities in the Borel summation procedure,
originating from Stokes lines, lead to the concepts of Stokes phenomenon and
transseries. We point out an important but often overlooked subtlety in the
description of Stokes phenomenon – more precisely, in the definition of the
so-called Borel residues – in section 2.3. Finally, in section 2.4, we use the
concepts that we have introduced to review the ‘classical’ derivation of the
large order formula for a transseries.

###  2.1 Borel summation of asymptotic expansions

Assume that some physical problem provides us with a power series

$\varphi^{(0)}(z)=\sum_{g=0}^{\infty}\varphi^{(0)}_{g}\,z^{-g},$ (4)

where, as is common in the resurgence literature, we have assumed an expansion
in nonnegative powers around large $z$. This assumption is not very
restrictive: by changing variables and multiplying by an appropriate power of
$z$ we can bring any power series in this form. We will further assume that
this series is asymptotic, of the Gevrey-1 type – that is: we assume that as
$g\to\infty$, the coefficients $\varphi^{(0)}_{g}$ grow factorially as

$\frac{\varphi^{(0)}_{g}A^{g+\beta}}{\Gamma(g+\beta)}\sim c+\mathcal{O}(1/g),$
(5)

for some nonzero complex constants $A,c$ and a real constant $\beta$. In most
of what follows, we will assume for simplicity that $\beta$ is an integer. The
results in this paper are easily generalized to the case of noninteger
$\beta$.

Due to the factorial growth of the coefficients $\varphi_{g}^{(0)}$ in (5),
the series (4) does not converge for any nonzero value of $z^{-1}$. When (4)
arises in a physics setting this is a problem, as one would expect the series
$\varphi^{(0)}(z)$ to not just be purely formal but to describe a function
that has a finite value for some range of $z$-values. The usual way to cure
this problem is to take the Borel transform of the series:

$\mathcal{B}[\varphi^{(0)}](t)\equiv\sum_{g=0}^{\infty}\frac{\varphi^{(0)}_{g}}{g!}\,t^{g}.$
(6)

Dividing the coefficients of our original series by $g!$ greatly improves the
convergence of the series: the new series now has a radius of convergence
equal to $|A|$. In many cases, $\mathcal{B}[\varphi^{(0)}](t)$ can in fact be
analytically continued to the whole complex $t$-plane (or a multiple cover
thereof – something we will come back to in more detail soon) minus a discrete
set of points where the analytic continuation of
$\mathcal{B}[\varphi^{(0)}](t)$ has singularities. We will assume that we are
in such a situation, and will write $\mathcal{B}[\varphi^{(0)}](t)$ for the
function obtained from this analytic continuation as well.

######  Example 2.1.

Let us consider the case where $\varphi^{(0)}_{0}=0$ and
$\varphi^{(0)}_{g}=-A^{-g}(g-1)!$ for $g>0$. This is clearly of the form (5)
with $\beta=0$, $c=-1$, and all $\mathcal{O}(1/g)$ corrections vanishing in
this example. The Borel transform of $\varphi^{(0)}(z)$ is then

$\mathcal{B}[\varphi^{(0)}](t)=-\sum_{g=1}^{\infty}\frac{1}{g}\left(\frac{t}{A}\right)^{g}=\log\left(1-\frac{t}{A}\right),$
(7)

where the series after the first equals sign converges for $|t|<|A|$, and the
second equals sign contains the analytic continuation of this convergent
series into a multi-valued function on the entire $t$-plane except for $t=A$,
where the function has a logarithmic branch point. Equivalently, we can
replace the $t$-plane by its $\mathbb{Z}$-multiple cover branched at $t=A$,
which is the Riemann surface on which the logarithm is single-valued.

When the Borel transform (6) can indeed be analytically continued to most of
the complex plane – more precisely: along paths $\gamma$ that start at the
origin and go off to infinity – then the transform has a well-known ‘inverse’:
one can define a function

$\mathcal{S}_{\gamma}\varphi^{(0)}(z)\equiv
z\int_{\gamma}\mathcal{B}[\varphi^{(0)}](t)\,e^{-tz}\,dt$ (8)

known as a Borel sum (along a path $\gamma$ for which the integral converges)
of the formal power series $\varphi^{(0)}$. Usually, we will take $\gamma$ to
be a straight, semi-infinite ray that has an angle $\theta$ with the positive
real axis; in such a case we denote the Borel sum by
$\mathcal{S}_{\theta}\varphi^{(0)}$. Of course, when
$\mathcal{B}[\varphi^{(0)}]$ has a singularity along the ray in the direction
$\theta$, this Borel sum is not well-defined; in these cases (whenever it is
clear what the singular direction $\theta$ is) we denote the lateral Borel
sums with angles infinitesimally larger or smaller than $\theta$ by
$\mathcal{S}_{+}\varphi^{(0)}$ and $\mathcal{S}_{-}\varphi^{(0)}$
respectively.

The Laplace transformation introduced above is not literally the inverse of
the Borel transform: while $\varphi^{(0)}$ is a formal (and divergent) power
series, $\mathcal{S}_{\gamma}\varphi^{(0)}$ is an honest function – so the two
are clearly not ‘the same object’. However, if one were to plug (6) into (8),
exchange the sum and integral (an operation which is not usually allowed!),
and perform the integrals – say with $\gamma$ the positive real axis – then
one would find back the original asymptotic series. It is in this sense that
the Laplace transform (8) is the ‘inverse’ of the Borel transform (6).

In fact, the Borel transform followed by the Laplace transform solves exactly
the problem we mentioned: it turns an asymptotically divergent series in $z$
into an honest function of $z$, and one can show that for Gevrey-1 series a
particular asymptotic expansion of that function at large $z$ is the original
expansion $\varphi^{(0)}(z)$ that we started with. While this solves our
problem, this is certainly not the end of the story: the construction depends
on the integration path $\gamma$ in the Laplace transform, and therefore the
function one arrives at is generally not unique. As it turns out, this non-
uniqueness is a virtue rather than a problem.

###  2.2 Stokes phenomenon

The existence of many different inverse Borel transforms is the starting point
of the theory of resurgence. Note that if two paths $\gamma$ can be deformed
into one another without crossing any singularities of
$\mathcal{B}[\varphi^{(0)}](t)$, then by Cauchy’s theorem they lead to the
same Borel sum. (This assumes for now the absence of contributions at infinity
– we will discuss the more general situation extensively in sections 3.4 and
4.3.) As a result, the Borel sums $\mathcal{S}_{\theta}\varphi^{(0)}$ do not
change when $\theta$ is varied until $\theta$ crosses a ray on which
$\mathcal{B}[\varphi^{(0)}](t)$ has singularities. These special rays in the
Borel plane are known as Stokes lines 111In the literature, Stokes lines are
often defined to lie in the $z$-plane, rather than the Borel $t$-plane. This
is indeed equivalent when we fix the contour along we resum, as we explain in
appendix A. .

In most of what follows, we will be interested only in what happens at a
single Stokes line. Without loss of generality, by rotating the complex $z$\-
and $t$-variables if necessary, we may assume that this Stokes line is the
positive real axis. The Stokes line can contain more than a single
singularity, but we do make the assumption that any singularities on it are at
$t=\ell A$ for $\ell\in\mathbb{Z}_{>0}$. This is often a reasonable
assumption; for example, there may be multi-instanton configurations in our
physical problem, and those would lead to precisely such a pattern of
singularities.

We now further assume that we are in the setting of simple resurgent
functions, where the Borel transform only has poles and logarithmic branch
cuts as its singularities. We will in particular be interested in the case of
logarithmic singularities, but this is not a very strong restriction, as the
following variation on example 2.1 shows:

######  Example 2.2.

Let us consider the case where $\varphi^{(0)}_{0}=0$ and
$\varphi^{(0)}_{g}=-A^{-g}g!$ for $g>0$. The only difference with example 2.1
is that we replaced $(g-1)!$ by $g!$. The coefficients are again of the form
(5), now with $\beta=1$ and $c=A$. The Borel transform of $\varphi^{(0)}(z)$
is now

$\mathcal{B}[\varphi^{(0)}](t)=-\sum_{g=1}^{\infty}\left(\frac{t}{A}\right)^{g}=-\frac{t}{A-t},$
(9)

where again the second equals sign contains the analytic continuation.

Comparing examples 2.1 and 2.2, we see that the difference between a pole and
a logarithmic branch cut in the Borel plane lies in an asymptotic coefficient
growth proportional to $(g-1)!$ and $g!$ respectively – or equivalently, in a
different value of $\beta$ in (4). One can turn this observation around: if we
replace the standard Borel transform (6) by the ‘$n$-Borel transform’ defined
as

$\mathcal{B}_{n}[\varphi^{(0)}](t)\equiv\sum_{g=0}^{\infty}\frac{\varphi^{(0)}_{g+n}}{g!}\,t^{g},$
(10)

then by taking $n$ sufficiently small (and declaring $\varphi^{(0)}_{h}=0$ for
$h<0$ if needed) one can make sure that at a given point $t=\ell A$ in the
Borel plane only a logarithmic branch cut appears, and no pole. Just like the
ordinary Borel transform, the $n$-Borel transform has a Laplace transform that
acts as its inverse (in the sense described above), and we can work with it in
a completely analogous way to the ordinary transform. We explain this in some
more detail in appendix A. Because of these observations, in most of what
follows we will assume that we are in a situation with only logarithmic branch
cuts in the Borel plane, essentially without loss of generality. The Borel
transform that we use in practice will usually be the Borel transform
$\mathcal{B}_{0}$, and we will leave out the 0-index.

Let us then assume that near some singularity $t=\ell A$, the Borel transform
of $\varphi^{(0)}(z)$ has the form

$\mathcal{B}[\varphi^{(0)}](t)=\frac{{\mathsf{S}}_{0\to\ell}}{2\pi i}\
\mathcal{B}[\varphi^{(\ell)}](t-\ell A)\ \log(t-\ell A)+\text{regular},$ (11)

where ‘regular’ stands for a function that is analytic (and therefore non-
singular) at $t=\ell A$. Here, we have written the prefactor of $\log(t-\ell
A)$ in a particular form suggested by resurgence: after extracting a constant
factor $\frac{\mathsf{S}_{0\to\ell}}{2\pi i}$, we write this prefactor as the
Borel transform of a new asymptotic series $\varphi^{(\ell)}$ whose meaning
and importance will become clear momentarily. Of course, there is some freedom
in scaling $\mathsf{S}_{0\to\ell}$ and $\varphi^{(\ell)}$ in opposite ways;
usually this freedom is fixed by, for example, requiring that the leading term
in $\varphi^{(\ell)}$ is unity.

The constants $\mathsf{S}_{0\to\ell}$ are called Borel residues. An important
observation is that they are formally multivalued, since there are in general
many paths $\gamma$ along which we can analytically continue the Borel
transform from $t=0$ to $t=\ell A$. Therefore, to properly define the
$\mathsf{S}_{0\to\ell}$, we must commit to a convention with regards to these
paths. A standard convention is the one shown in figure 3 on the left side,
where we we analytically continue the Borel transform above the Stokes line to
the singularity of interest. We will come back to this ambiguity in section
2.3.

Figure 3: Conventions for the description (11) of the Borel transforms near
singularities at $t=\ell A$: on the left the standard convention (continuation
above the Stokes line) and on the right an alternative convention
(continuation below the Stokes line).

Whenever we encounter a Stokes line, we will be interested in computing the
discontinuity

$\text{disc}\,\varphi^{(0)}(z)=(\mathcal{S}_{+}-\mathcal{S}_{-})\varphi^{(0)}(z).$
(12)

In computing this discontinuity, it is important that the Laplace integrals
for both $\mathcal{S}_{+}$ and $\mathcal{S}_{-}$ do not cross any branch cuts
associated to the logarithmic contributions to the Borel transform (11). One
may achieve this by aligning all branch cuts with the Stokes line (see the
left hand side of figure 4), but a more conventient choice is to let all
branch cuts go off to infinity in a diagonal direction and deform the
integration contour into a sum of so-called Hankel contours, as depicted in
the middle of figure 4.

Figure 4: Considering the difference of lateral resummations, there are two
natural choices: we either let all branch cuts (and thus the Hankel contours)
run off to infinity below the real line (upper graphs) or we let them run off
to infinity above real line (lower graphs).

By decomposing the integration contour in terms of Hankel contours around
individual singularities in this way, we can use the resurgence relations (11)
to find222Here, we have exchanged the order of summation over $k$ and
integration in $t$. This is an operation that, especially in a resurgence
setting, has to be treated with care, but since the sums over $k$ (as opposed
to those over $g$) generally have a nonzero radius of converge, this exchange
of sums and integrals can often be put on solid footing.

$(\mathcal{S}_{+}-\mathcal{S}_{-})\varphi^{(0)}(z)=-\mathcal{S}_{-}\left(\sum_{k=1}^{\infty}{\mathsf{S}}_{0\to
k}\,\varphi^{(k)}(z)\,{\rm e}^{-kAz}\right).$ (13)

More suggestively, we can write this as

$\mathcal{S}_{+}\varphi^{(0)}(z)=\mathcal{S}_{-}\left(\varphi^{(0)}(z)-\sum_{k=1}^{\infty}{\mathsf{S}}_{0\to
k}\,\varphi^{(k)}(z)\,{\rm e}^{-kAz}\right),$ (14)

which tells us that the Borel summation along one side of the Stokes line is
equal to the resummation along the other side when we modify the asymptotic
expansion in the appropriate way. This is a manifestation of Stokes
phenomenon. Using the Stokes automorphism $\mathfrak{S}$ defined by

$\mathcal{S}_{+}=\mathcal{S}_{-}\circ\mathfrak{S},$ (15)

we can write an equivalent statement in terms of asymptotic expansions which
reads

$\mathfrak{S}\varphi^{(0)}(z)=\varphi^{(0)}(z)-\sum_{k=1}^{\infty}{\mathsf{S}}_{0\to
k}\,\varphi^{(k)}(z)\,{\rm e}^{-kAz}.$ (16)

An important result in the theory of resurgence is that the relation (11)
generalizes to the other formal power series $\varphi^{(k)}$ as well: we have
that

$\mathcal{B}[\varphi^{(k)}](t)=\frac{{\mathsf{S}}_{k\to k+\ell}}{2\pi i}\
\mathcal{B}_{0}[\varphi^{(k+\ell)}](t-\ell A)\,\log(t-\ell A)+\text{regular},$
(17)

for integers $k,\ell$ such that $k\geq 0$ and $k+\ell\geq 0$. We will use the
terminology that the sectors know about each other. (We prefer not to use the
term ‘see each other’, as the paths along which the Borel residues
${\mathsf{S}}_{k\to k+\ell}$ are determined are generally not straight lines –
see figure 3.) As a consequence, the action of the Stokes automorphism (16)
generalizes to

$\mathfrak{S}\varphi^{(k)}(z)=\varphi^{(k)}(z)-\sum_{\ell=1}^{\infty}{\mathsf{S}}_{k\to
k+\ell}\,\varphi^{(k+\ell)}(z)\,{\rm e}^{-\ell Az}\,,$ (18)

for generic sectors $\varphi^{(k)}$. The above expressions (16) and (18) show
that it is useful to unite the formal, asymptotic power series $\varphi^{(n)}$
in a so-called transseries,

$\Phi(\vec{\sigma},z)=\sum_{n=0}^{\infty}\sigma_{n}\,\varphi^{(n)}(z)\,{\rm
e}^{-nAz},$ (19)

which is a double formal expansion, both in the perturbative expansion
parameter $z^{-1}$ and in the nonperturbative transmonomial ${\rm e}^{-Az}$.
In the above expression we have assigned weigths $\sigma_{n}$ – called
transseries parameters – to each transseries sector $\varphi^{(n)}$; these are
all collected in the single vector $\vec{\sigma}$ on the left hand side. The
advantage of introducing these parameters is that we can now also view any
asymptotic expansion before the Stokes jump (18) – that is, simply
$\varphi^{(k)}$ – as a transseries with $\sigma_{n}=\delta_{n,k}$. In fact,
one may be more general and think of the Stokes automorphism as mapping one
transseries expansion – with particular values of $\sigma_{n}$ and summed
above the Stokes line using the $\mathcal{S}_{+}$ procedure – to another one –
with other values of $\sigma_{n}$ and summed below the Stokes line using the
$\mathcal{S}_{-}$ procedure.

Using the transseries notation, we can write the action of the Stokes
automorphism as an action on the transseries parameters:

$\mathfrak{S}\Phi(\vec{\sigma},z)=\Phi(\mathbb{S}\cdot\vec{\sigma},z)\,.$ (20)

Effectively, we are multiplying the (infinite) vector $\vec{\sigma}$ with an
(infinite) lower-triangular Stokes matrix $\mathbb{S}$ whose elements are
given by the Borel residues

$\left(\mathbb{S}\right)^{n}_{\ m}=\begin{cases}-{\mathsf{S}}_{m\to
n}&\text{for }m<n\\\ 1&\text{for }m=n\\\ 0&\text{for }m>n\,,\\\
\end{cases}\qquad=\quad\begin{pmatrix}1&0&0&\ldots\\\ -{\mathsf{S}}_{0\to 1}\
&1&0&\ldots\\\ -{\mathsf{S}}_{0\to 2}\ &-{\mathsf{S}}_{1\to 2}\ &1&\ldots\\\
\ldots&\ldots&\ldots\ &\ldots\end{pmatrix}$ (21)

for integers $m,n\geq 0$. In particular, this means that we have the following
action on a single transseries parameter

$\mathfrak{S}:\sigma_{n}\mapsto\sigma_{n}-\sum_{m=0}^{n-1}\sigma_{m}{\mathsf{S}}_{m\to
n}\,.$ (22)

This generic description expresses the Stokes phenomenon in terms of an
infinite set of Borel residues. However, once we specify the underlying
problem and compute the Borel residues, we often find that they depend on a
finite set of constants called Stokes constants. Moreover, in such cases we
generically find that the number of independent transseries parameters
$\sigma_{n}$ reduces to a finite number as well, as the following example
shows.

######  Example 2.3.

In section 4 we will introduce the quartic free energy – a popular toy model
to study resurgence (see e.g. Aniceto_2019 ). Throughout this section and the
next, we will occasionally refer forward to aspects of this example to clarify
the material. The quartic free energy has a two-parameter transseries solution
that can be written as

$F(x,\sigma,\rho)=\sum_{n=0}^{\infty}\sigma^{n}F^{(n)}(x){\rm
e}^{-nA/x}+\rho\,,$ (23)

where the $F^{(n)}(x)$ are formal power series in $x$. Their Borel transforms
have a Stokes line along the positive real axis and therefore
$F(x,\sigma,\rho)$ undergoes a Stokes transition. Despite the fact that there
are an infinite number of transseries sectors $F^{(n)}$, it turns out that
setting $\sigma_{n}=\sigma^{n}$ is a convenient parametrization under which
the Stokes automorphism along the $\arg(x)=0$ direction acts as

$\mathfrak{S}_{0}F(x,\sigma,\rho)=F(x,\sigma+S_{1},\rho)\,.$ (24)

Thus we can reduce the number of degrees of freedom for this particular
transition to one parameter $\sigma$ and one Stokes constant $S_{1}$. These
relate to our generic formulation (19) and (18) when

${\mathsf{S}}_{k\to k+l}=-\binom{k+l}{k}S_{1}^{l}\,.$ (25)

The second parameter $\rho$ is irrelevant for the above argument; it only
plays a role when we consider another Stokes transition across the negative
real axis.

The structure of this example is a rather generic one for nonlinear problems,
where the transmonomial ${\rm e}^{-nAz}$ is often multiplied by an equal power
$\sigma^{n}$ of a parameter $\sigma$ that can have the interpretation of a
boundary condition or integration constant. The same structure occurs for
example in solutions to ordinary differential equations (such as the Painlevé
equations) or finite difference equations (such as string equations for matrix
models) – see Aniceto:2011nu  for these examples.

In order to deduce the precise map $g(\vec{\sigma})$ that acts on the
transseries parameters, and to prove equations such as (17) that describe how
different nonperturbative sectors know about each other, one must turn to the
alien calculus developed by Écalle ecalle1985fonctions , which introduces a
set of derivative operators $\Delta_{\ell A}$ called alien derivatives. These
operators – see e.g. sauzin2014introduction  for a formal definition – act on
formal power series and produce new formal power series. They are labeled by
‘jumps’ $\ell A$ between the singularities in the Borel plane. A central
result in the theory of resurgence states that one can express the Stokes
automorphism in terms of these operators as follows:

$\mathfrak{S}=\exp\left(\sum_{\ell=1}^{\infty}{\rm e}^{-\ell Az}\Delta_{\ell
A}\right)\,.$ (26)

This equation is the most important piece of machinery under the hood of
resurgence, but for the purposes of this paper, we can work with Borel plane
equations like (17) and will not need the subtleties of alien calculus.

###  2.3 Multivalued Borel residues

As we mentioned in section 2.2, the Borel residues ${\mathsf{S}}_{k\to
k+\ell}$ depend on the path along which we analytically continue the Borel
transforms from one point to another. In computing the discontinuity along a
Stokes line, (13), we assumed that the Hankel contours run off to infinity
below the Stokes line, meaning that we could use the Borel residues
${\mathsf{S}}_{k\to k+\ell}$ as defined by paths above the Stokes line – the
convention in the left of figure 3. Alternatively, we could consider diagonal
Hankel contours above the Stokes line, in which case we would need to use a
different set of residues that we denote by $\tilde{{\mathsf{S}}}_{k\to
k+\ell}$ and which correspond to analytic continuation of the Borel transforms
along paths below the Stokes line, as in figure 3 on the right.

To relate the two sets of Borel residues, we observe that both choices should
describe the same discontinuity:

$\displaystyle{\mathrm{disc}}\,\varphi^{(0)}$ $\displaystyle=$
$\displaystyle-\mathcal{S}_{-}\left(\sum_{k=1}^{\infty}{\mathsf{S}}_{0\to
k}\varphi^{(k)}{\rm
e}^{-kAz}\right)=-\mathcal{S}_{-}\left(\mathfrak{S}-1\right)\varphi^{(0)}$
(27) $\displaystyle=$
$\displaystyle-\mathcal{S}_{+}\left(\sum_{k=1}^{\infty}\tilde{\mathsf{S}}_{0\to
k}\varphi^{(k)}{\rm
e}^{-kAz}\right)=-\mathcal{S}_{+}\left(1-\mathfrak{S}^{-1}\right)\varphi^{(0)}\,.$

The top (bottom) line in this equation is obtained by letting the Hankel
contours run to infinity below (above) the real line, as shown by the diagrams
in the top (bottom) of figure 4. This equation (27) implies that two
conventions lead to two distinct transseries expansions that after the
appropriate lateral resummation should reproduce the same discontinuity.
Hence, these respective transseries must consist of different linear
combinations of the formal power series $\varphi^{(k)}$, i.e. they are defined
by two distinct sets of residues $\\{{\mathsf{S}}_{k\to k^{\prime}}\\}$ and
$\\{\tilde{\mathsf{S}}_{k\to k^{\prime}}\\}$. To reproduce the same
discontinuity, the two transseries expressions should be related via

$\mathfrak{S}\,\left(\sum_{k=1}^{\infty}\tilde{\mathsf{S}}_{0\to
k}\varphi^{(k)}{\rm e}^{-kAz}\right)=\sum_{k=1}^{\infty}{\mathsf{S}}_{0\to
k}\varphi^{(k)}{\rm e}^{-kAz}.$ (28)

If we know one set of residues, we can derive the other set straightforwardly
by using this equation and the Stokes automorphism (18). Using the Stokes
matrix $\mathbb{S}$ we can interpret the above equation as saying that

$\mathbb{S}\cdot\begin{pmatrix}0\\\ \tilde{{\mathsf{S}}}_{0\to 1}\\\
\tilde{{\mathsf{S}}}_{0\to 2}\\\ \tilde{{\mathsf{S}}}_{0\to 3}\\\
\dots\end{pmatrix}=\begin{pmatrix}0\\\ {\mathsf{S}}_{0\to 1}\\\
{\mathsf{S}}_{0\to 2}\\\ {\mathsf{S}}_{0\to 3}\\\ \dots\end{pmatrix}$ (29)

Solving this equation for the first three Borel residues
$\tilde{{\mathsf{S}}}_{0\to k}$, we find that

$\displaystyle\tilde{\mathsf{S}}_{0\to 1}$ $\displaystyle=$
$\displaystyle{\mathsf{S}}_{0\to 1}$ $\displaystyle\tilde{\mathsf{S}}_{0\to
2}$ $\displaystyle=$ $\displaystyle{\mathsf{S}}_{0\to 2}+{\mathsf{S}}_{0\to
1}{\mathsf{S}}_{1\to 2}$ $\displaystyle\tilde{\mathsf{S}}_{0\to 3}$
$\displaystyle=$ $\displaystyle{\mathsf{S}}_{0\to 3}+{\mathsf{S}}_{0\to
1}{\mathsf{S}}_{1\to 3}+{\mathsf{S}}_{0\to 2}{\mathsf{S}}_{2\to
3}+{\mathsf{S}}_{0\to 1}{\mathsf{S}}_{1\to 2}{\mathsf{S}}_{2\to 3}\,.$ (30)

The general solution is

$\tilde{\mathsf{S}}_{k\to
k+\ell}=\sum_{r=1}^{\ell}\,\sum_{k=k_{0}<\ldots<k_{r}=k+\ell}\,\prod_{j=0}^{r-1}{\mathsf{S}}_{k_{j}\to
k_{j+1}}.$ (31)

Note that all the ‘one-step’ residues ${\mathsf{S}}_{k\to
k+1}=\tilde{\mathsf{S}}_{k\to k+1}$ are equal. This is as expected, since
their respective paths are in the same homotopy class (see figure 3). Of
course, for the ‘multi-step’ residues, we could have chosen even more
complicated homotopy classes for the paths, going above some singularities and
below others, or even circling the singularities several times – but the two
conventions above are clearly the most natural ones. Regardless of the
conventions that we pick, any complete set of residues contains the same
information about the resurgence relations (11) as any other complete set.

######  Example 2.4.

For the quartic free energy, whose Borel residues ${\mathsf{S}}_{k\to k+l}$
were given in (25), one can also calculate the alternative set of Borel
residues using (31), leading to

$\tilde{\mathsf{S}}_{k\to k+l}=(-1)^{l}\,\binom{k+l}{k}S_{1}^{l}\,.$ (32)

These are again expressed in powers of a single Stokes constant $S_{1}$.

###  2.4 Derivation of large order formulas

Now, let us use the material reviewed in the previous subsections to derive
the standard form in which large order formulas usually appear, expressed in
terms of Borel residues to keep the discussion as generic as possible. We
first note that using Cauchy’s theorem, the series coefficients can be
extracted from the Borel transform in the following way:

$\frac{\varphi_{g}^{(0)}}{g!}=\oint_{0}\frac{\mathop{}\\!\mathrm{d}t}{2\pi{\rm
i}}\frac{\mathcal{B}[\varphi^{(0)}](t)}{t^{g+1}}.$ (33)

We can then deform the contour around $t=0$ to a contour around infinity and
in the process pick up various Hankel contours associated to singularities
that the Borel transform has. Given our definition of the Stokes automorphism,
each of these Hankel contours $H_{kA}^{-}$ grabs the $k$’th singularity
located at $kA$ and ‘leaves’ the Borel plane below the positive real axis – as
shown in figure 4. Using the resurgence relations we can expand the Borel
transform locally near each of these, yielding

$\frac{\varphi_{g}^{(0)}}{g!}=\sum_{k\geq 1}\mathsf{S}_{0\to
k}\int_{H_{kA}^{-}}\frac{\mathop{}\\!\mathrm{d}t}{(2\pi{\rm
i})^{2}}\,\frac{\mathcal{B}[\varphi^{(k)}](t-kA)}{t^{g+1}}\log(t-kA)+\oint_{\infty}\frac{\mathop{}\\!\mathrm{d}t}{2\pi{\rm
i}}\frac{\mathcal{B}[\varphi^{(0)}](t)}{t^{g+1}}\,.$ (34)

The latter term on the right hand side constitutes a contribution coming from
infinity that might be relevant, depending on the details of the Borel
transform $\mathcal{B}[\varphi^{(0)}]$ and the value of $g$. For the rest of
this computation we will assume that its contribution vanishes and therefore
ignore it – but we will come back to this assumption in sections 3.4 and 4.3.
By an appropriate shift, the right hand side can now be rewritten as

$\frac{\varphi_{g}^{(0)}}{g!}=-\sum_{k\geq 1}\mathsf{S}_{0\to
k}\int_{0}^{\infty-{\rm i}\epsilon}\frac{\mathop{}\\!\mathrm{d}t}{2\pi{\rm
i}}\,\frac{\mathcal{B}[\varphi^{(k)}](t)}{(t+kA)^{g+1}}$ (35)

where $\epsilon$ is an infinitesimally small positive number, causing the
contour to run below the singularities. We can then replace the Borel
transform by its Taylor series around $t=0$ and integrate term by term with
respect to $t$, which produces the familiar asymptotic large order relations
that read333To anticipate our discussion in section 4, where we will want to
analytically extend these expressions to complex values of $g$, we already
write $\Gamma(g)$ rather than $(g-1)!$ on the left hand side of this equation.

$\frac{2\pi{\rm i}\,\varphi_{g}^{(0)}A^{g}}{\Gamma(g)}\simeq-\sum_{k\geq
1}\mathsf{S}_{0\to k}k^{-g}\sum_{h\geq
0}\frac{\Gamma(g-h)}{\Gamma(g)}(kA)^{h}\varphi^{(k)}_{h}.$ (36)

Note that this expression only makes sense in the large $g$ limit, given the
fact that the $h\geq g$ terms diverse. This is the form of the large order
formulas that one often encounteres in the literature, and that we wrote in
(3) – there with the Borel residues $\mathsf{S}_{0\to k}$ written out in terms
of a Stokes constant $S_{1}$ as in (32). An important observation (already
underlying e.g. Aniceto:2011nu ) is now that this formula itself has the form
of a transseries: by writing

$k^{-g}=e^{-g\log(k)}$ (37)

we see that it has ‘instanton actions’ $\log(2),\log(3),\log(4)$ etc. The
nature of this transseries is slightly different from the ones we have
discussed so far, though; in particular, the instanton actions are not all
integer multiples of a finite set of ‘basis actions’, and they are (by the
nature of the logarithm function) not single-valued. One may therefore wonder
about the resummation of this transseries, whether it also has a Stokes
automorphism, how its Stokes data are related to those of the original
transseries, and so on. It is to those questions that we turn in the next
section.

##  3 Resurgence of the large order transseries

We finished the previous section with the usual large order relation, which
connects the coefficients in the perturbative sector of a transseries to the
coefficients in its nonperturbative sectors. In particular, we observed that
this large order relation can be written in a form which itself looks like a
transseries, with instanton actions $\log(k)$ for $k\geq 2$. In this section
we study this transseries – which we shall call the large order transseries –
in more detail. In section 3.1 we discuss the so-called Stirling transform,
which relates the large order transseries to the original transseries. We show
in section 3.2 how this Stirling transformation helps in particular to relate
resurgence properties of the original transseries to that of the large order
transseries, and how it explains several properties that this new transseries
has, including the presumed periodicity along the imaginary axis. In section
3.3, we then describe the Stokes phenomenon of the large order transeries and
explain how one can unambiguously resum this transseries. Finally, in section
3.4, we describe how this transseries can be used to derive an exact large
order relation.

###  3.1 The Stirling transform

For each ‘instanton number’ $k$ in the large order transseries (36), the
ratios of gamma functions can be expanded in powers of $1/g$. This yields a
formal power series in $1/g$ for each $k$, whose coefficients encode the
perturbative fluctuations $\varphi^{(k)}_{h}$ around the $k$-th instanton
sector. For $k=1$, the first few terms of this series read

$-\mathsf{S}_{0\to
1}\left(\varphi^{(1)}_{0}+\frac{A\varphi^{(1)}_{1}}{g}+\frac{A\varphi^{(1)}_{1}+A^{2}\varphi^{(1)}_{2}}{g^{2}}+\frac{A\varphi^{(1)}_{1}+3A^{2}\varphi^{(1)}_{2}+A^{3}\varphi^{(1)}_{3}}{g^{3}}+\mathcal{O}(g^{-4})\right)\,.$
(38)

Because the 1-instanton coefficients $\varphi^{(1)}_{h}$ grow factorially as
$h$ becomes large, the above power series will generically be asymptotically
divergent as well. In fact, this is the case for all $k$-sectors in the large
order relation (36).

Working out a few more orders in (38), we find that the integer coefficients
appearing in the series correspond to Stirling numbers of the second kind
NIST:DLMF . The above series therefore constitutes what we will call a
rescaled Stirling transform of the first instanton series. Let us elaborate: a
formal power series $\psi(z)=\sum_{n\geq 0}a_{n}z^{-n}$ defines a sequence of
coefficients $a_{n}$ whose Stirling transform is another sequence of
coefficients $b_{n}$ defined by

$b_{n}=\sum_{k=0}^{n}\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}\,a_{k}\,,$ (39)

where the integers $\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}$ are the Stirling
numbers of the second kind; see appendix B for a definition. We then say that
the formal power series

$\tilde{\psi}(g)=\sum_{n=0}^{\infty}b_{n}g^{-n}$ (40)

is the Stirling transform of the series $\psi(z)$. Furthermore, we can rescale
the argument $z\to z/A$ in $\psi(z)$ to obtain a new series in $z^{-1}$:

$\psi\left(\frac{z}{A}\right)=\sum_{n=0}^{\infty}\left(a_{n}A^{n}\right)z^{-n}\,.$
(41)

Finally, we introduce a formal power series $\hat{\psi}(g)$ which is the
Stirling transform of this rescaled series, and which looks as follows:

$\displaystyle\hat{\psi}(g)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\sum_{k=0}^{n}\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}\left(a_{k}A^{k}\right)g^{-n}$
(42) $\displaystyle=$ $\displaystyle
a_{0}+\frac{Aa_{1}}{g}+\frac{Aa_{1}+A^{2}a_{2}}{g^{2}}+\frac{Aa_{1}+3A^{2}a_{2}+A^{3}a_{3}}{g^{3}}+\mathcal{O}(g^{-4})\,.$

This series is what we call the rescaled Stirling transform of $\psi(z)$ and
it matches exactly the series expansion (38) of the large order relation (36).
If we now identify the coefficients $a_{h}$ with the coefficients
$\varphi^{(1)}_{h}$ of the one-instanton sector, then we conclude that the
asymptotic expansion for $k=1$ in (36) is the rescaled Stirling transform of
the one-instanton series. More generally, the asymptotic expansions in $1/g$
of the large order transseries in its nonperturbative sector multiplying
$k^{-g}=e^{-g\log k}$ can be expressed as a rescaled Stirling transforms of
the $k$-instanton series, where the scaling factor is $kA$ instead of $A$.

Having understood that the asymptotic expansions appearing in the large order
relations are actually rescaled Stirling transforms of the original instanton
series, we can define

$\hat{\varphi}^{(k)}(g)=\sum_{n=0}^{\infty}\sum_{\ell=0}^{n}\genfrac{\\{}{\\}}{0.0pt}{}{n}{\ell}\left(\varphi^{(k)}_{\ell}\left(kA\right)^{\ell}\right)g^{-n}\,,$
(43)

and rewrite the asymptotic large order relations (36) as

$\frac{2\pi{\rm
i}\varphi^{(0)}_{g}A^{g}}{\Gamma(g)}\simeq-\sum_{k=1}^{\infty}\mathsf{S}_{0\to
k}\hat{\varphi}^{(k)}(g)\,{\rm e}^{-g\log(k)}\,.$ (44)

Thus, (44) allows us to order the contributions by size where the one-
instanton coefficients $\varphi_{h}^{(1)}$ determine the leading expansion in
$1/g$, and higher instanton coefficients $\varphi_{h}^{(k)}$ determine
nonperturbative contributions to the large order transseries that are
exponentially suppressed as ${\rm e}^{-g\log(k)}$.

######  Example 3.1.

Let us test our expression (44) by performing a resurgent large order analysis
at leading instanton order $k=1$. (We will perform tests for higher $k$ below
in example 3.4.) To this end (see also Aniceto_2019 ), we study the large
order growth of perturbative coefficients $F^{(0)}_{g}$ of the quartic free
energy, which we already briefly introduced in example 2.3.

The large order relation for the quartic free energy is exactly of the form
(44), with instanton sectors $F^{(k)}$ that carry an instanton action $A=3/2$
and with Borel residues $\mathsf{S}_{0\to k}$ expressed in terms of a Stokes
constant $S_{1}=-2$ by (25). To test (44), we call the sequence on the left
hand side of this equation $r_{g}$ and plot the ratio

$\frac{r_{g}}{S_{1}\hat{\varphi}_{0}^{(1)}}\simeq 1+\mathcal{O}(g^{-1})\,,$
(45)

for $g$ up to 500 in the first plot of figure 5. This indeed convergences
towards 1 as $g$ becomes large. Likewise, in the second plot, we test the
expected subleading $g^{-1}$-behaviour by plotting the ratio

$g\frac{r_{g}-S_{1}\hat{\varphi}_{0}^{(1)}}{S_{1}\hat{\varphi}_{1}^{(1)}}\simeq
1+\mathcal{O}(g^{-1})\,,$ (46)

which again converges towards 1 in the large $g$ limit. We can similarly test
the $g^{-N}$ contributions to the leading sector of (44) by summing the right
hand side of (43) (for $k=1$) up to order $n=N-1$. We then plot the difference
between the (exact) left and (truncated) right hand side of (43) for values of
$g$ up to 500, where we divide by the expected $g^{-N}$ contribution at that
order so that we expact the rescaled difference to become 1 at large $g$. In
figure 5 we see that indeed both sides of (44) match up to order $g^{-12}$
(where the only reason to stop is not to make the figure too large),
confirming the validity of the Stirling numbers appearing in (43).

Figure 5: A test of the rescaled Stirling transform appearing in (44), for the
leading $k=1$ sector and up to order $g^{-12}$.

At this point, we have established how the large order transseries (44) is
constructed in terms of the instanton coefficients $\varphi_{h}^{(k)}$ of the
original transseries. Still, the question remains whether the newly
constructed large order transseries is truly a resurgent transseries, in the
sense that the Borel transforms of its formal power series
$\hat{\varphi}^{(k)}(g)$ know about each other like the original series
$\varphi^{(k)}(g)$ do in (17). In order to answer this question, we want to
derive similar resurgence relations for the Borel transforms of the rescaled
Stirling transforms $\hat{\varphi}^{(k)}$. The first question to address is
therefore what the Borel planes of these series look like.

It so happens that the Stirling transforms of functions have a rather
interesting property when it comes to their Borel transforms: consider the
Stirling transform $\tilde{\psi}(g)$ – defined through (39) and (40) from the
arbitrary series $\psi(z)$. Its 0-Borel transform (10) has the following
simple form:

$\mathcal{B}[\,\tilde{\psi}\,](t)=\mathcal{B}[\,\psi\,]({\rm e}^{t}-1)\,.$
(47)

Its generic $n$-Borel transform is straightforwardly obtained through
differentiation, see appendix A. Subsequently, the rescaled Stirling transform
$\hat{\psi}(g)$ given by (42), also has a Borel transform that relates to
$\mathcal{B}[\,\psi\,]$ as follows:

$\mathcal{B}[\,\hat{\psi}\,](t)=\mathcal{B}[\,\psi\,]\left(A({\rm
e}^{t}-1)\right)\,.$ (48)

As a result, we find that the 0-Borel transforms of the asymptotic series
$\hat{\varphi}^{(k)}(g)$ appearing in the large order relations can be
expressed explicitly in terms of the Borel transforms of the original
instanton series $\varphi^{(k)}$, using their proper rescaling:

$\mathcal{B}[\hat{\varphi}^{(k)}](t)=\mathcal{B}[\varphi^{(k)}]\left(kA({\rm
e}^{t}-1)\right)\,.$ (49)

We provide derivations of these statements in appendix B. The above relations
already reveal the underlying reason for the appearance of the towers of
singularities in the Borel plane that we discussed in the introduction: the
Borel transforms (49) of the rescaled Stirling series are functions of ${\rm
e}^{t}$ and are therefore invariant under $t\to t+2\pi{\rm i}$. As a result,
any singularity in the Borel plane of the original instanton series
$\mathcal{B}[\varphi^{(k)}]$ reappears as a tower of evenly spaced
singularities in the Borel plane of $\mathcal{B}[\hat{\varphi}^{(k)}]$. As we
explain in the next subsection, this ‘translational symmetry’ also explains
why it makes sense to write $k^{-g}$ as ${\rm e}^{-g\log(k)}$ in terms of the
‘multivalued instanton action’ $\log(k)$ in the large order transseries
(44).444In a different context, the change of variables $s=kA({\rm e}^{t}-1)$
is the uniformizing map in Costin:2021bay  for a Borel transform with a single
singularity at $s=-kA$.

###  3.2 Resurgent structure

Now that we have seen that the Borel transforms of the rescaled Stirling
transforms of the $\varphi^{(k)}(z)$ can be expressed in terms of the Borel
transforms of these instanton series themselves, we can derive resurgence
relations like (17) for the rescaled Stirling transforms. Using (49) we obtain

$\displaystyle\mathcal{B}[\hat{\varphi}^{(k)}](t)$ $\displaystyle=$
$\displaystyle\mathcal{B}[\varphi^{(k)}]\left(kA({\rm e}^{t}-1)\right)$
$\displaystyle=$ $\displaystyle\frac{{\mathsf{S}}_{k\to k+\ell}}{2\pi{\rm
i}}\mathcal{B}[\varphi^{(k+\ell)}]\left(kA({\rm e}^{t}-1)-\ell
A\right)\log\left(kA({\rm e}^{t}-1)-\ell A\right)+\text{regular}$
$\displaystyle=$ $\displaystyle\frac{{\mathsf{S}}_{k\to k+\ell}}{2\pi{\rm
i}}\mathcal{B}[\hat{\varphi}^{(k+\ell)}]\left(t-\log\left(\frac{k+\ell}{k}\right)\right)\log\left(t-\log\left(\frac{k+\ell}{k}\right)\right)+\text{regular}\,,$

where in going from the first to the second line we expanded $s\equiv kA({\rm
e}^{t}-1)$ around $s=\ell A$ using the ordinary resurgence relations (17). In
the last line we expanded the logarithm around $s=\ell A$ and used (49) to go
back to the Borel transform of $\hat{\varphi}^{(k+\ell)}$. Notice that we are
just interested in the singular terms in these expressions; the higher order
terms in the expansion of the logarithm only yield regular terms, since they
multiply a Borel transform which is analytic at $s=\ell A$.

The resurgence relations for the transseries sectors $\hat{\varphi}^{(k)}$ of
the large order transseries, (3.2), are quite similar to (17) describing the
resurgence of the original transseries sectors $\varphi^{(k)}$. We observe in
particular that the large order transseries sectors labeled by $k$ and
$k+\ell$ are connected by the same Borel residues ${\mathsf{S}}_{k\to k+\ell}$
as in the original transseries. An important difference lies in the fact that
the locations of the Borel singularities are completely different from those
of the original transseries. Within the original transseries, the
$k$-instanton sector knows about the $(k+\ell)$-sector by expanding
$\mathcal{B}[\varphi^{(k)}](t)$ around $t=\ell A$. In the case of our large
order transseries, the $k$-sector knows about the $(k+\ell)$-sector by
expanding $\mathcal{B}[\hat{\varphi}^{(k)}](t)$ around
$t=\log\left(\frac{k+\ell}{k}\right)$. In fact, due to the $t\sim t+2\pi{\rm
i}$ symmetry of (49), we can add any integer multiple of $2\pi{\rm i}$ to the
location of a given singularity of $\mathcal{B}[\hat{\varphi}^{(k)}](t)$ to
find another singularity from which the same $(k+\ell)$-sector resurges with
the same Borel residue ${\mathsf{S}}_{k\to k+\ell}$.

Summarizing this discussion, the Borel transform
$\mathcal{B}[\hat{\varphi}^{(k)}](t)$ will have singularities at

$\mathcal{A}_{\ell,n}^{(k)}=\log\left(\frac{k+\ell}{k}\right)+2\pi{\rm
i}n\qquad\text{for }k+\ell>0\text{ , }\ell\neq 0\text{ and }n\in\mathbb{Z},$
(51)

from which the $(k+\ell)$-sector resurges with residue ${\mathsf{S}}_{k\to
k+\ell}$. When we lift the condition that $k+\ell>0$, which happens for
example in the case with two opposite primitive instanton actions
$A_{1}=-A_{2}$, we can also get singularities at

$\tilde{\mathcal{A}}_{\ell,n}^{(k)}=\log\left(-\frac{k+\ell}{k}\right)+(2n+1)\pi{\rm
i}\qquad\text{for }k+\ell<0\text{ , }\ell\neq 0\text{ and }n\in\mathbb{Z}\,.$
(52)

This pattern of singularities is for example relevant for the plots shown in
figure 1 in the introduction. One can write similar formulas for more general
cases (which we will not consider further in this paper) when there are
several primitive instanton actions $A_{1},A_{2},\ldots$, even when there is
no relation between them, where the addition of $\pi{\rm i}$ in (52) changes.

Using the above notation, the resurgence relations in (3.2) can be written
compactly as

$\mathcal{B}[\hat{\varphi}^{(k)}](t)\bigg{|}_{t=\mathcal{A}^{(k)}_{\ell,n}}=\frac{\mathsf{S}_{k\to
k+\ell}}{2\pi i}\
\mathcal{B}[\hat{\varphi}^{(k+\ell)}]\left(t-\mathcal{A}^{(k)}_{\ell,n}\right)\
\log\left(t-\mathcal{A}^{(k)}_{\ell,n}\right)+\text{regular}\,,$ (53)

which also makes the similarity to (17) more manifest. As for ‘ordinary’
resurgence, these large order resurgence relations tell us that when we
integrate around any singularity $\mathcal{A}^{(k)}_{\ell,n}$ in the Borel
plane, we can extract the formal power series $\hat{\varphi}^{(k+\ell)}$. In
the spirit of section 2, this suggests that we should unite all these series
into what we call the generic large order transseries

$\Phi(\sigma_{k,n},g)=\sum_{n=-\infty}^{\infty}\sum_{k=1}^{\infty}\sigma_{k,n}\
\hat{\varphi}^{(k)}(g)\,{\rm
e}^{-g\mathcal{A}^{(1)}_{k,n}}=\sum_{n=-\infty}^{\infty}\Phi_{n}(\vec{\sigma}_{n},g)\,.$
(54)

The parameters $\sigma_{k,n}$ are again transseries parameters associated to
sectors $(k,n)$. In section 3.3 we will see how the Stokes automorphism acts
on these parameters. In anticipation of what is to come, it is also useful to
write the full transseries $\Phi$ as above, as an infinite sum over
‘horizontal’ subtransseries $\Phi_{n}$. The large order transseries (44) is
then obtained by setting $\sigma_{k,n}=-{\mathsf{S}}_{0\to k}\delta_{n,0}$.
Note that the locations $\mathcal{A}_{\ell,n}^{(k)}$ of the Borel
singularities in the $k$-th Borel plane correspond to the relative distances
between the instanton actions of the sectors $(k+\ell,m+n)$ and $(k,m)$ in the
large order transeries:

$\mathcal{A}^{(k)}_{\ell,n}=\mathcal{A}^{(1)}_{k+\ell,m+n}-\mathcal{A}^{(1)}_{k,m}\,,$
(55)

for arbitrary integers $m$. These distances are visible in the Borel plane of
the leading sector – see also to the singularities shown in figure 1. The fact
that the singularities of the Borel transforms correspond to relative
distances between instanton actions holds in general, but is often concealed
by the fact that in ordinary transseries these actions lie on a semi-lattice
generated by a single primitive instanton action $A$. For the large order
transseries, on the other hand, there are infinitely many primitive instanton
actions $\log(p)$ for $p$ prime.

We conclude that the large order transseries is truly a resurgent transseries,
just like the original transseries. The only differences are that its
singularities no longer depend on the instanton action $A$ of the original
transseries, and that each singularity associated to a sector $k+\ell$ is
copied at locations shifted by integer multiples of $2\pi{\rm i}$ in the Borel
plane of $\mathcal{B}[\hat{\varphi}^{(k)}]$. The domain of the variable $t$ in
terms of which the Borel transforms $\mathcal{B}[\hat{\varphi}^{(k)}]$ are
defined is thus effectively a cylinder that we will henceforth call the Borel
cylinder. This cylinder provides the best way to structure the singularities
found in the figures 1 and 2 that we encountered in section 1; we discuss this
further in example 3.2 below.

######  Example 3.2.

Let us see whether in examples we can reproduce the singularity structure
$\mathcal{A}^{(k)}_{\ell,n}$ and $\tilde{\mathcal{A}}^{(k)}_{\ell,n}$ that is
described in (51) and (52). Since we do not have closed form expressions for
the various Borel transforms, we make use of Padé approximants (see e.g.
bender78:AMM ) to mimic their singularity structure. The singularities
$\mathcal{A}^{(1)}_{\ell,n}$ and $\tilde{\mathcal{A}}^{(1)}_{\ell,n}$, as
approximated by the Padé approximant of $\mathcal{B}[\hat{\varphi}^{(1)}]$,
were already shown in figures 1 and 2 in the introduction, for the Adler
function and the quartic free energy respectively. As further evidence for the
structure, we also consider the second rescaled Stirling transform for the
quartic free energy. In figure 6 we plot the singularities of the Padé
approximant of the Borel transform $\mathcal{B}[\hat{F}^{(2)}]$ of this
series.

Starting with figure 1(a) for the Adler function, we clearly observe
singularities at the anticipated positions $t=\mathcal{A}_{\ell,n}^{(1)}$ with
$\ell=1,2$. For figure 1(b), besides singularities at
$t=\mathcal{A}_{\ell,n}^{(1)}$ with $\ell=2$, we also notice singularities at
positions $t=\tilde{\mathcal{A}}_{\ell,n}^{(1)}$ , with $\ell=1,2$. This is a
consequence of the fact that the Adler function has two opposite primitive
instanton actions $A_{1}=-A_{2}$, which as we have explained leads to
additional contributions in the large order relation (44) that scale with
$(-k)^{-g}=e^{-g\log(k)\pm\pi{\rm i}}$, leading in turn to (52).

For figures 2(a) and 6(a) we also observe singularities at
$t=\mathcal{A}_{\ell,n}^{(1)}$. However, as the branch cuts starting at
$t=\log(k)$, $k\geq 2$ are mimicked by the accumulation of Padé poles along
the real axis, it is hard to distinguish the branch cuts that start at
$t=\log(k)$. We have therefore applied a conformal mapping Costin:2020hwg ;
Costin:2021bay

$t=-\frac{4z\log\left(\frac{k+1}{k}\right)}{(1-z)^{2}}\,,$ (56)

which maps the half-line $[\log\frac{k+1}{k},\infty)$ to the unit circle in
the $z$-plane, where in particular $t=\log\frac{k+1}{k}$ gets mapped to
$z=-1$. The resulting figures for the Padé approximants of these conformally
mapped series are shown in figures 2(b) and 6(b). We observe that the branch
cuts starting on the real positive $t$-axis are now pulled apart in the
$z$-plane and for low values of $k$ become clearly visible.

(a)

(b)

Figure 6: In figure 6(a), we show the singularities of the diagonal order
$800$ Padé approximant of $\mathcal{B}[\hat{F}^{(2)}](t)$. As the branch cuts
are mimicked by an accumulation of poles, the poles corresponding to different
branch cuts on the real line are hard to distinguish. We therefore show in
figure 6(b) the order $300$ diagonal Padé approximant of
$\mathcal{B}[\hat{F}^{(2)}]$ after taking the conformal map of (56). The
branch cuts positioned on the positive real line are now pulled apart as they
are mapped to the unit circle. The singularity at $t=\log(1/2)=-\log(2)$ of
figure 6(a) gets mapped to $z\approx 0.24$. Furthermore, the ‘flower-like’
curves in the $z$-plane correspond to $\mbox{Im}(t)=\pm 2\pi{\rm i}$ and
$\mbox{Im}(t)=\pm 4\pi{\rm i}$. Unfortunately, the branch cut starting at
$z\approx 0.24$ ‘uses’ many poles of the Padé approximant, so we only observe
a limited number of singularities starting on these curves. As a result, the
branch cuts at $\mbox{Im}(t)\pm 2\pi{\rm i}$ and $\mbox{Im}(t)=\pm 4\pi{\rm
i}$ are instead best viewed in the original $t$-plane.

###  3.3 Stokes automorphism

In the previous section, we have verified that the large order transseries
(44) and its generic version (54) truly form resurgent transseries by
establishing the resurgent structure (53) of their nonperturbative sectors.
For practical purposes, we are often interested in resumming the sectors
$\hat{\varphi}^{(k)}$ that appear in these large order transseries. The reason
for this (as explained elaborately in e.g. Aniceto_2019 ), is that we often
want to use the large order behaviour of the perturbative coefficients
$\varphi^{(0)}_{g}$ – which may be easy to obtain in specific examples – to
decode the nonperturbative coefficients $\varphi^{(k)}_{h}$ of the
$k$-instanton sector, which may not be so easy to obtain by other methods, for
example because the nonperturbative effects in the underlying physical problem
are not a priori known. To achieve such a decoding, we need to first ‘remove’
all sectors $\hat{\varphi}^{(k^{\prime})}$ for which $k^{\prime}<k$ from the
large order transseries before we can read off the coefficients in the $k$-th
sector. The only effective way to do this is by resumming these series
$\hat{\varphi}^{(k^{\prime})}$ and substracting them from the large order
transseries. As is so often the case, to achieve this we need to decide along
which contour in the Borel plane – or here: the Borel cylinder – we resum, and
different choices are related by a Stokes phenomenon. In this subsection we
discuss various contours and use the resurgence relations (53) as well as
consistency of resummation to derive the Stokes phenomenon of the generic
large order transseries.

Figure 7: The Hankel contours $H^{-}_{\log(1+\ell)}$ along which we integrate
in equation (59).

As we saw in (51), the singularities of the large order transseries sectors
$\hat{\varphi}^{(k)}$ accumulate along horizontal lines separated by integer
multiples of $2\pi{\rm i}$. There are two natural resummations
$\mathcal{S}_{0^{+}}$ and $\mathcal{S}_{0^{-}}$, whose contours go above and
below the singularities on the real positive line respectively:

$\mathcal{S}_{0^{\pm}}\hat{\varphi}^{(k)}(g)=g\int_{0}^{\infty\pm{\rm
i}\epsilon}\mathcal{B}[\hat{\varphi}^{(k)}](t)\,{\rm
e}^{-gt}\mathop{}\\!\mathrm{d}t\,.$ (57)

We can then define the discontinuity operator

${\mathrm{disc}}_{0}=\mathcal{S}_{0^{+}}-\mathcal{S}_{0^{-}}\,,$ (58)

in order to study the Stokes phenomenon as discussed in section 2.2. The
discontinuity of the first large order transseries sector
$\hat{\varphi}^{(1)}$ can readily be computed, and yields

$\displaystyle{\mathrm{disc}}_{0}\,\hat{\varphi}^{(1)}$ $\displaystyle=$
$\displaystyle\sum_{\ell=1}^{\infty}\mathsf{S}_{1\to 1+\ell}\
g\int_{H_{\log(1+\ell)}^{-}}\frac{\mathop{}\\!\mathrm{d}t}{2\pi{\rm
i}}\,\mathcal{B}[\hat{\varphi}^{(1+\ell)}](t-\log(1+\ell))\log\left(t-\log(1+\ell)\right){\rm
e}^{-gt}$ (59) $\displaystyle=$
$\displaystyle-\sum_{\ell=1}^{\infty}\mathsf{S}_{1\to 1+\ell}\,{\rm
e}^{-g\log(1+\ell)}\mathcal{S}_{0^{-}}\hat{\varphi}^{(1+\ell)}\,,$

where the Hankel contours $H_{\log(1+\ell)}^{-}$ are shown in figure 7. We can
repeat this computation for generic sectors $\hat{\varphi}^{(k)}$ and find
that

${\mathrm{disc}}_{0}\,\hat{\varphi}^{(k)}=-\sum_{\ell=1}^{\infty}\mathsf{S}_{k\to
k+\ell}\,{\rm
e}^{-g\log\left(\frac{k+\ell}{k}\right)}\mathcal{S}_{0^{-}}\hat{\varphi}^{(k+\ell)}\,.$
(60)

Note that the instanton actions that appear correctly correspond to the
singularities $\mathcal{A}^{(k)}_{l,0}$ in (51). From the discontinuity of the
large order transseries sectors $\hat{\varphi}^{(k)}$, we extract the Stokes
phenomenon

$\mathfrak{S}_{0}\hat{\varphi}^{(k)}=\hat{\varphi}^{(k)}-\sum_{\ell=1}^{\infty}\mathsf{S}_{k\to
k+\ell}\,\hat{\varphi}^{(k+\ell)}\,{\rm
e}^{-g\log\left(\frac{k+\ell}{k}\right)}\,,$ (61)

which is the direct analogue of (18) for the original transseries. The action
of the Stokes automorphism can be formulated within the horizontal
subtransseries $\Phi_{n}(g,\vec{\sigma}_{n})$ of (54) as follows:

$\mathfrak{S}_{0}\Phi_{n}\left(g,\vec{\sigma}_{n}\right)=\Phi_{n}\left(g,\mathbb{S}_{0}\cdot\vec{\sigma}_{n}\right)\,,$
(62)

or at the level of individual transseries parameters as

$\mathfrak{S}_{0}:\sigma_{k,n}\mapsto\sigma_{k,n}-\sum_{j=1}^{k-1}\sigma_{j,n}{\mathsf{S}}_{j\to
k}\,.$ (63)

This mapping should be compared to (22) for the original transseries. The
matrix $\mathbb{S}_{0}$ is essentially the same as $\mathbb{S}$ given in (21),
but with the ‘zeroth’ row and column removed. Since a $\hat{\varphi}^{(0)}$
sector does not exist in the large order transseries, we can assume such a
sector to vanish identically and naturally extend the Stokes matrix
$\mathbb{S}_{0}$ to include also a $j=0$ contribution in (63), in which case
we simply have

$\mathbb{S}_{0}=\mathbb{S}\,.$ (64)

The fact that the Stokes matrix associated to the Stokes phenomenon of the
large order transseries across the positive real axis is equivalent to that of
the ordinary transseries also means that, if we consider our large order
transeries (44) and study its Stokes phenomenon in the anti-clockwise
direction, given by the inverse automorphism $\mathfrak{S}_{0}^{-1}$ or
equivalently by the inverse Stokes matrix $\mathbb{S}^{-1}$, we retrieve the
alternative Borel residues by virtue of (29):

$\mathbb{S}^{-1}\cdot\begin{pmatrix}0\\\ {\mathsf{S}}_{0\to 1}\\\
{\mathsf{S}}_{0\to 2}\\\ {\mathsf{S}}_{0\to 3}\\\
\dots\end{pmatrix}=\begin{pmatrix}0\\\ \tilde{{\mathsf{S}}}_{0\to 1}\\\
\tilde{{\mathsf{S}}}_{0\to 2}\\\ \tilde{{\mathsf{S}}}_{0\to 3}\\\
\dots\end{pmatrix}\,.$ (65)

Said otherwise, we have that

$\mathfrak{S}_{0}\left(\sum_{k=1}^{\infty}\tilde{\mathsf{S}}_{0\to
k}\hat{\varphi}^{(k)}(g){\rm
e}^{-g\log(k)}\right)=\sum_{k=1}^{\infty}\mathsf{S}_{0\to
k}\hat{\varphi}^{(k)}(g){\rm e}^{-g\log(k)}\,.$ (66)

That is, the large order transseries with transseries parameters
$\tilde{\mathsf{S}}_{0\to k}$ resummed above the positive real line (with
$\mathcal{S}_{0^{+}}$) is equivalent to the large order transseries with
parameters ${\mathsf{S}}_{0\to k}$ resummed below the positive real line (with
$\mathcal{S}_{0^{-}}$). This equation is the direct analogue of (28) for the
original transseries.

We can generalize the discontinuity ${\mathrm{disc}}_{0}$, across
singularitities that lie on the positive real line of the Borel plane, to
Stokes transitions with discontinuities that we will label
${\mathrm{disc}}_{n}$ and that arise from singularities
$\mathcal{A}^{(k)}_{l,n}$ on other horizontal lines with fixed $n$ – see
figure 8. For completeness, let us derive the Stokes automorphisms for these
transitions and see how they alter the transeries parameters $\sigma_{k,n}$.

Consider the contours shown in figure 8 on the left, which define Borel
summations

$\mathcal{S}_{n^{\pm}}\hat{\varphi}^{(k)}(g)=g\int_{2\pi{\rm
i}n}^{\infty+2\pi{\rm i}n\pm{\rm
i}\epsilon}\mathcal{B}[\hat{\varphi}^{(k)}](t)\,{\rm
e}^{-gt}\mathop{}\\!\mathrm{d}t\,.$ (67)

By the symmetry of the Borel cylinder, these resummations can be related to
(57) as follows:

$\mathcal{S}_{n^{\pm}}\hat{\varphi}^{(k)}(g)={\rm e}^{-2\pi{\rm
i}ng}\mathcal{S}_{0^{\pm}}\hat{\varphi}^{(k)}(g)\,.$ (68)

For integer $g$, the prefactor on the right hand side equals 1, reflecting the
symmetry of the Borel cylinder under translations by $2\pi{\rm i}$. In what
follows, we will also be interested in cases with non-integer $g$ (where for
real $g$ we can mathematically think of the factors of ${\rm e}^{-2\pi{\rm
i}ng}$ as transition functions on a nontrivial $U(1)$-bundle over the Borel
cylinder), so we will keep these factors explicit in our formulas. We can now
define discontinuities
${\mathrm{disc}}_{n}=\mathcal{S}_{n^{+}}-\mathcal{S}_{n^{-}}$, which can be
expressed as

${\mathrm{disc}}_{n}\,\hat{\varphi}^{(k)}(g)={\rm e}^{-2\pi{\rm
i}ng}\,{\mathrm{disc}}_{0}\,\hat{\varphi}^{(k)}(g)\,.$ (69)

In terms of Stokes automorphisms we then have

$\mathfrak{S}_{n}\hat{\varphi}^{(k)}(g)=\hat{\varphi}^{(k)}(g)-{\rm
e}^{-2\pi{\rm i}ng}\sum_{\ell=1}^{\infty}\mathsf{S}_{k\to
k+\ell}\,\hat{\varphi}^{(k+\ell)}(g)\,{\rm
e}^{-g\log\left(\frac{k+\ell}{k}\right)}\,.$ (70)

We note that the sectors $\varphi^{(k+\ell)}$ and the Stokes residues
$\mathsf{S}_{k\to k+\ell}$ appear, as was to be expected, but with an
instanton action that is shifted by $-2\pi{\rm i}n$.

Next, we can derive the action of this Stokes automorphism on the
subtransseries of (44):

$\mathfrak{S}_{n}\Phi_{m}(\vec{\sigma}_{m},g)=\Phi_{m}(\vec{\sigma}_{m},g)+\Phi_{m+n}\left(\left(\mathbb{S}-\mathrm{I}\right)\cdot\vec{\sigma}_{m},g\right)\,,$
(71)

where $\mathrm{I}$ is an infinite identity matrix. As a consistency check,
note that (70) and (71) reduce to (61) and (62) respectively when we set
$n=0$. For the full transseries, we find that

$\mathfrak{S}_{n}\Phi(\sigma,g)=\sum_{m=-\infty}^{\infty}\Phi_{m}(\vec{\sigma}_{m}+(\mathbb{S}-\mathrm{I})\cdot\vec{\sigma}_{m-n},g)\,,$
(72)

or at the level of individual transseries parameters we find

$\mathfrak{S}_{n}:\sigma_{k,m}\mapsto\sigma_{k,m}+\sum_{j=1}^{k-1}\sigma_{j,m-n}{\mathsf{S}}_{j\to
k}\,.$ (73)

Having derived the actions of the Stokes automorphisms $\mathfrak{S}_{n}$, we
can now consistently resum the (generic) large order transseries along any of
the contours shown in figure 8 on the right, running from $t=0$ to
$t=+\infty$. Changing the contour prescription can then straightforwardly be
compensated for by applying the appropriate automorphism $\mathfrak{S}_{n}$ to
the transseries. For example, as a consequence of (66) we know that the
following two resummations are equal:

$\mathcal{S}_{0^{+}}\left(\sum_{k\geq 1}\tilde{\mathsf{S}}_{0\to
k}\hat{\varphi}^{(k)}{\rm
e}^{-g\log(k)}\right)=\mathcal{S}_{0^{-}}\left(\sum_{k\geq 1}\mathsf{S}_{0\to
k}\hat{\varphi}^{(k)}{\rm e}^{-g\log(k)}\right)\,.$ (74)

We can extend this to any of the contours shown on the right of figure 8 using
our automorphisms $\mathfrak{S}_{n}$.

Recall that the original motivation for resumming large order transseries
sectors was to decode the higher instanton fluctuations $\varphi^{(k)}_{h}$
from the large order behaviour of the perturbative coefficients
$\varphi^{(0)}_{g}$. In general we only know those coefficients for positive
integers $g$. Within that context, the factor ${\rm e}^{-2\pi{\rm i}ng}$
therefore reduces to the identity and all Stokes automorphisms
$\mathfrak{S}_{n}$ are equal to $\mathfrak{S}_{0}$. Thus, for these purposes,
the discussion above simplifies greatly and we can simplify the generic large
order transseries (54) to

$\Phi(\vec{\sigma}_{0},g)=\sum_{k=1}^{\infty}\sigma_{k,0}\
\hat{\varphi}^{(k)}(g){\rm e}^{-g\log(k)}\,.$ (75)

Figure 8: On the left: the contours $[n^{\pm}]$ corresponding to the
resummations $\mathcal{S}_{n^{\pm}}$ defined in (59). Once we have derived the
Stokes phenomena $\mathfrak{S}_{n}$ across the $n$th horizontal ray of
singularities, we are able to resum the large order transseries along any of
the contours shown on the right.

######  Example 3.3.

We can test the relation (69) between different discontinuities numerically
for the $k=1$ sector of the quartic free energy. As already explained in
example 3.2, we do not have a closed form expression for the Borel transform
$\mathcal{B}[\hat{F}^{(1)}](t)$, but we can numerically approximate this
function using Padé approximants. In particular, the singularities of the Padé
approximant encode the singularities of $\mathcal{B}[\hat{F}^{(1)}](t)$. To
test (69), we replaced $\mathcal{B}[\hat{F}^{(1)}](t)$ by its diaganal Padé
approximant of order $N$, $\mathcal{B}\mathcal{P}^{(N)}[\hat{F}^{(1)}](t)$,
and performed the integration required to compute the discontinuity
numerically.

Since this computation does not require the perturbative coefficients, we can
test it for any $g$, not necessarily integer. In particular we have tested the
relation between the discontinuities at $n=0$ and $n=-1$ for diverse values of
$g$. For example, taking $g=5/3$ yields

$\displaystyle{\mathrm{disc}}_{0}\hat{F}^{(1)}(g)$
$\displaystyle\approx-0.20236122\ldots+0.00000000\ldots{\rm i}\,,$
$\displaystyle{\rm e}^{-2\pi{\rm i}g}\,{\mathrm{disc}}_{-1}\hat{F}^{(1)}(g)$
$\displaystyle\approx-0.20236165\ldots+0.00000496\ldots{\rm i}\,,$ (76)

which nicely matches to six decimal places. To obtain (76), we took the
diagonal Padé approximant of order $N=800$. This high order is necessary for
producing enough singularities near $\mathcal{A}^{(1)}_{k,-1}$ to compute
${\mathrm{disc}}_{-1}\hat{F}^{(1)}(g)$ with sufficient accuracy.

In the next subsection we will see that ‘horizontal’ resummations like (74)
naturally appear in the derivation of what we will call exact large order
relations.

###  3.4 Exact large order relations

In the previous subsection we learned that the large order transseries
inherits the resurgence structure – expressed in terms of the Stokes matrix
$\mathbb{S}$ – from the original transseries, despite its distinct singularity
structure. We derived the Stokes automorphism $\mathfrak{S}_{n}$ for several
‘horizontal’ Stokes transitions that allow us to relate the different
resummation prescriptions shown in figure 8 on the right. We arrived at these
results by starting with the conventional large order relation (36) that we
derived at the end of section 2.4. Many properties of the large order
transseries that we have discussed can also be derived in a more direct way,
as we demonstrate in this section.

Let us therefore take a step back and consider the large order relation in its
integral form (35), which we repeat here for convenience and where we have
again discarded the contribution from infinity:

$\frac{\varphi_{g}^{(0)}}{g!}=-\sum_{k\geq 1}\mathsf{S}_{0\to
k}\int_{0}^{\infty-{\rm i}\epsilon}\frac{\mathop{}\\!\mathrm{d}t}{2\pi{\rm
i}}\,\frac{\mathcal{B}[\varphi^{(k)}](t)}{(t+kA)^{g+1}}\,.$ (77)

In section 2.4 we asymptotically expanded the right hand side of this
expression by expanding the Borel transform $\mathcal{B}[\varphi^{(k)}](t)$ as
a power series in $t$ and performing the integral term by term. Let us now
present a different approach: we first introduce the variable $u=\log(t+kA)$
which allows us to rewrite the expression as

$\frac{\varphi_{g}^{(0)}}{g!}=-\sum_{k\geq 1}\mathsf{S}_{0\to
k}\int_{\log(kA)}^{\infty-{\rm
i}\epsilon}\frac{\mathop{}\\!\mathrm{d}u}{2\pi{\rm
i}}\,\mathcal{B}[\varphi^{(k)}]({\rm e}^{u}-kA)\,{\rm e}^{-gu}\,.$ (78)

Subsequently, we shift the integration variable to $v=u-\log(kA)$ and obtain

$\frac{2\pi{\rm i}\,\varphi_{g}^{(0)}A^{g}}{\Gamma(g)}=-\sum_{k\geq
1}\mathsf{S}_{0\to k}{\rm e}^{-g\log(k)}\,\left(g\int_{0}^{\infty-{\rm
i}\epsilon}\mathop{}\\!\mathrm{d}v\,\mathcal{B}[\varphi^{(k)}](kA({\rm
e}^{v}-1))\,{\rm e}^{-gv}\right).$ (79)

On the right hand side we recover the Borel transform (49) of the rescaled
Stirling transforms $\hat{\varphi}^{(k)}$ that we studied before. This
equation expresses the perturbative coefficient $\varphi^{(0)}_{g}$ as the
resummation of the large order transseries (44) along the contour below the
positive real axis. Whenever the sum over $k$ on the right hand side
converges, which is generally the case in examples, (79) is an exact relation
that is also valid for finite $g$. We will therefore call this expression the
exact large order relation from now on.

The brief computation above reproduces much of the hard work of the this
section: it allows us to rederive the resurgence relations (53) and
singularity structure (51) and subsequently the automorphisms $\mathfrak{S}$
that we found in the previous section. Most importantly, it assigns a distinct
contour $[0^{-}]$ to our large order transseries with residues
${\mathsf{S}}_{0\to k}$ along which its resummation computes the correct value
for the perturbative coefficients $\varphi^{(0)}_{g}$.

The right hand side of the exact large order relation constitutes a function
of $g$ that we can expand in a large $g$ limit. Equation (79) tells us that
one possible transseries expansion is the large order transseries (44), which
after resummation along the $[0^{-}]$ contour gives back the same function. If
we repeat the computation above, but with Hankel contours running above the
real positive $t$ axis in (77) – see again the bottom graphs in figure 4 –
then we obtain a similar expression:

$\frac{2\pi{\rm
i}\,\varphi_{g}^{(0)}A^{g}}{\Gamma(g)}=-\sum_{k=1}^{\infty}\tilde{\mathsf{S}}_{0\to
k}{\rm e}^{-g\log(k)}\,\left(g\int_{0}^{\infty+{\rm
i}\epsilon}\mathop{}\\!\mathrm{d}v\,\mathcal{B}[\varphi^{(k)}](kA({\rm
e}^{v}-1))\,{\rm e}^{-gv}\right)\,.$ (80)

This is simply the $[0^{+}]$-resummation of the large order transseries with
the alternative Borel residues $\tilde{\mathsf{S}}_{0\to k}$. The resulting
function is of course equal to (79):

$\displaystyle\frac{2\pi{\rm i}\,\varphi_{g}^{(0)}A^{g}}{\Gamma(g)}$
$\displaystyle=$
$\displaystyle\mathcal{S}_{0^{+}}\left(-\sum_{k=1}^{\infty}\tilde{\mathsf{S}}_{0\to
k}{\rm e}^{-g\log(k)}\hat{\varphi}^{(k)}(g)\right)$ (81) $\displaystyle=$
$\displaystyle\mathcal{S}_{0^{-}}\left(-\sum_{k=1}^{\infty}\mathsf{S}_{0\to
k}{\rm e}^{-g\log(k)}\hat{\varphi}^{(k)}(g)\right)\,.$

The equality of both resummations is consistent with the Stokes automorphism
(66) that we derived in the previous subsection, and can be seen as an
alternative derivation of that equation. More generally, by considering (78)
and substituting $v=u-\log(kA)+2\pi{\rm i}n$ where $n$ is any integer, we can
see that

$\displaystyle\frac{2\pi{\rm i}\varphi^{(0)}A^{g}}{\Gamma(g)}$
$\displaystyle=$ $\displaystyle{\rm e}^{-2\pi{\rm
i}ng}\,\mathcal{S}_{n^{+}}\left(-\sum_{k=1}^{\infty}\tilde{\mathsf{S}}_{0\to
k}{\rm e}^{-g\log(k)}\hat{\varphi}^{(k)}(g)\right)$ (82) $\displaystyle=$
$\displaystyle{\rm e}^{-2\pi{\rm
i}ng}\,\mathcal{S}_{n^{-}}\left(-\sum_{k=1}^{\infty}\mathsf{S}_{0\to k}{\rm
e}^{-g\log(k)}\hat{\varphi}^{(k)}(g)\right)\,,$

which is consistent with (68). In example 3.4, we check the exact relation
(79), and in particular (81), for the quartic free energy.

######  Example 3.4.

We consider again the quartic free energy. Recall example 3.1, where we tested
the large order transserie (44) for $k=1$ and in the large $g$-limit by
applying ratio tests on

$r_{g}\equiv\frac{2\pi{\rm i}A^{g}}{\Gamma(g)}F_{g}^{(0)}\,.$ (83)

We can now also perform tests for $k>1$ by resumming the asymptotic large
order relation, i.e. we can test the exact large order relation (79). Similar
to example 3.3, we do not have an exact Borel transform
$\mathcal{B}[\hat{F}^{(k)}](t)$ at our disposal, so we replace this function
by its (diagonal) Padé approximant in the integration and define

$r_{g}^{(k)}\equiv S_{1}^{k}{\rm e}^{-g\log(k)}\int_{0}^{\infty-{\rm
i}\epsilon}dv\,\mathcal{B}\mathcal{P}^{[N]}[\hat{F}^{(k)}](v)e^{-gv}\,.$ (84)

A numerical evaluation of the integral yields, for example for $g=20$:

$\displaystyle r_{20}$
$\displaystyle\approx\phantom{-0.0000000000000000000+}1.386394325206770660\,{\rm
i}\,,$ $\displaystyle r_{20}-r_{20}^{(1)}$
$\displaystyle\approx\phantom{-}0.000000879112757638+0.000000000111827370\,{\rm
i}\,,$ $\displaystyle r_{20}-(r_{20}^{(1)}+r_{20}^{(2)})$
$\displaystyle\approx\phantom{-}0.000000000000643417-0.000000000223651298\,{\rm
i}\,,$ $\displaystyle r_{20}-(r_{20}^{(1)}+r_{20}^{(2)}+r_{20}^{(3)})$
$\displaystyle\approx-0.000000000000643249-0.000000000000010324\,{\rm i}\,.$
(85)

To obtain these numbers, we used the $N=100$ diagonal Padé approximant. We
observe that $r_{20}^{(1)}$ already matches the exact value $r_{20}$ to $7$
decimal places and adding further $r_{20}^{(k)}$ improves the approximation
even further.

In Aniceto:2011nu , a nice way to visualize approximations with an increasing
number of nonperturbative sectors was introduced. In figure 9(a), we compare
$r_{g}$ with the resummed large order relation up to twelve nonperturbative
sectors in this way. The vertical axis shows the precision, defined by

$\log_{10}\left|\frac{r_{g}}{r_{g}-\sum_{k=1}^{n}r_{g}^{(k)}}\right|\,,$ (86)

for $g$ ranging from 1 to 30 and $n$ ranging from 1 to 12. In figure 9(b), we
also test (80) by changing the integration path in (84) to the $[0^{+}]$
contour and using the Borel residues (32) – i.e. we replace $S_{1}^{k}$ by
$-(-S_{1})^{k}$. It is important to realize that we must do both replacements:
the large order relation only gives exact results when we use the Borel
residues that correspond to the chosen integration contour. We observe that
once we do this, both resummations of the large order transseries match the
exact values $r_{g}$ with an extraordinarily small error.

(a)

(b)

Figure 9: In 9(a) we show the precision – defined by (86) – of the resummed
large order transseries along the $[0^{-}]$ contour. We plotted the values for
$g$ ranging from 1 to 30 and $n$ ranging from 1 to 12. In 9(a) we used an
integration contour below the real line as in (84). In 9(b) we show a similar
plot where we used an integration path along the $[0^{+}]$ contour instead.

In section 2.4 we discussed the derivation of the classical ‘asymptotic’ large
order relation which expresses the large $g$ behaviour of the perturbative
coefficients $\varphi_{g}^{(0)}$ in terms of the large order transseries shown
in (44). In this section we learned that we can also express those same
perturbative coefficients as the resummation of the large order transseries,
without invoking any large $g$ limit. It is therefore natural to wonder if the
equation (79) is an exact large order relation, in the sense that its right
hand side computes the perturbative coefficients $\varphi^{(0)}_{g}$ also at
finite $g$ to arbitrary precision. Figures 9(a) and 9(b) support this idea, as
both resummations describe the perturbative coefficients with extraordinary
precision. Let us discuss this possibility of exactness somewhat further.

One possible objection that one might raise against this assertion is that a
finite number of perturbative coefficients $\varphi^{(0)}_{g}$ can be changed
‘by hand’ by adding a polynomial to $\varphi^{(0)}(z)$, without altering the
instanton coefficients that determine the resummation on the right hand side
of (79). Therefore, how can that equation be used to exactly compute all
$\varphi^{(0)}_{g}$? As we will explain now, this apparent contradiction is
resolved by the contribution at infinity to the Laplace transform that
appeared in (34) but that we have ignored in what followed.

Imagine that indeed we wish to alter a select number of perturbative
coefficients by adding a polynomial to the original series $\varphi^{(0)}(z)$
as follows:

$\tilde{\varphi}^{(0)}(z)=\varphi^{(0)}(z)+\sum_{g=0}^{G}a_{g}z^{-g}\,,$ (87)

thereby altering the values of $\varphi_{g}^{(0)}$ with $g\leq G$ to
$\tilde{\varphi}_{g}^{(0)}=\varphi_{g}^{(0)}+a_{g}$. Clearly, the large order
behaviour of the perturbative coefficients $\tilde{\varphi}_{g}^{(0)}$ is
unaffected by such a modification and hence the instanton coefficients
$\tilde{\varphi}^{(k)}_{h}=\varphi^{(k)}_{h}$ also remain unchanged. However,
the exact large order relation for $\tilde{\varphi}^{(0)}_{g}$ at finite
$g\leq G$ must differ from the original by $a_{g}$. To see how this happens,
we first note that the Borel transform of our new series
$\tilde{\varphi}^{(0)}$ reads

$\mathcal{B}[\tilde{\varphi}^{(0)}](t)=\mathcal{B}[\varphi^{(0)}](t)+\sum_{h=0}^{G}\frac{a_{h}}{h!}t^{h}\,.$
(88)

The first term on the right hand side, when inserted into (34), yields the
familiar exact large order relation of $\varphi^{(0)}_{g}$ when we follow the
contour deformation as explained in section 2.4. The second term on the right
hand side, however, leads to poles at $t=0$ and $t=\infty$. If we deform the
contour away from zero to $t=\infty$ we obtain

$\frac{\tilde{\varphi}_{g}^{(0)}}{g!}=\left(\text{exact large order relation
of }\varphi^{(0)}_{g}\right)+\sum_{h=0}^{G}\frac{a_{h}}{h!}\frac{1}{2\pi{\rm
i}}\oint_{t=\infty}t^{h-g-1}\,\mathop{}\\!\mathrm{d}t\,.$ (89)

An elementary residue computation shows that the latter term is simply
$a_{g}/g!$, as required. We can even extend the argument to the case that
$G=\infty$, as long as the power series that is then added in (87) has a
finite radius of convergence: the large order behaviour is unaltered and
therefore the instanton coefficients of the large order relation will remain
unchanged. The Borel transform of such a series with a nonzero radius of
convergence will moreover be analytic in the whole complex plane and hence, as
with the polynomial, a residue computation at infinity will again simply add
the coefficient $a_{g}/g!$ to the exact large order relation. In the next
section, we will discuss a contribution at infinity of this kind in some more
detail for the quartic partition function.

A final interesting aspect of the exact large order relation is the fact that,
since the right hand side is now a function of $g$, we can carry out the
resummation even when $g$ is not a positive integer. In fact, we can resum the
large order transseries for arbitrary complex values of $g$. If we regard the
perturbative coefficients as a prescription to assign values
$\varphi^{(0)}_{g}$ to positive integers $g$, one can wonder whether the
resummation of the large order transseries for arbitrary $g$ provides a
natural extension of this map $\varphi^{(0)}_{g}:\mathbb{N}\to\mathbb{C}$ to
the complex domain. We will discuss this question for a specific example in
the next section.

##  4 A worked out example

In the previous section we derived the exact large order relation (79) for the
perturbative coefficients of an asymptotic transseries. Although our
derivation of this relation is only valid for integer values of $g$ – for
example because in the Cauchy integral representation (33) one can only
extract residues when $g$ is integer – nothing prevents us from studying the
resulting equation for generic real or even complex values of $g$. In this
section, we want to find out to what extent the large order relation is truly
exact. We do so by studying the example of the quartic partition function –
closely related to the quartic free energy that featured in several previous
examples – for which the perturbative coefficients $\varphi_{g}^{(0)}$ have a
closed form expression which admits a natural extension to complex $g$.

In the introduction we stated that one can often think of a transseries as the
semiclassical expansion of a path integral around all of its saddle points. To
exemplify this, in section 4.1 we discuss a well-known toy model called the
quartic partition function, which is a zero-dimensional version of the
textbook example of $\phi^{4}$ theory, as well as its logarithm – the quartic
free energy. In section 4.2 we verify that the exact large order relation for
the quartic paritition function indeed computes the perturbative coefficients,
as well as their natural extension to complex values of the index $g$, with
arbitrary numerical precision. In the extension to the complex domain, we will
see that also for the large order transseries, Stokes phenomenon plays a
crucial role. We discuss the interpretation and significance of these results
in section 4.3.

###  4.1 Quartic integral and free energy

So far in this work, we have used the quartic free energy in various examples
to illustrate some of the concepts and perform numerical checks on the
formulas we derived. For the purpose of the storyline, we kept the examples
small and restricted ourself to the necessary details, postponing a more
detailed introduction of the model to the present section. Of course, the
quartic model is a favorite toy model in many resurgence studies, so the
interested reader can find many further applications of this model elsewhere,
notably in Aniceto_2019 .

Let us start by considering the quartic partition function

$Z(\hbar)=\frac{1}{2\pi}\int_{\Gamma}dw\,e^{-\frac{1}{\hbar}V(w)}\,,$ (90)

where $V(w)$ is a quartic potential defined by

$\qquad V(w)=\frac{1}{2}w^{2}-\frac{\lambda}{24}w^{4}\,,$ (91)

and $\Gamma$ is an infinite contour in the complex $w$-plane chosen such that
the integral converges. The parameters $\hbar$ and $\lambda$ can be arbitrary
nonzero complex numbers. One can think of (90) as a zero-dimensional path
integral, hence the name ‘partition function’ for $Z(\hbar)$.

By defining $x\equiv\lambda\hbar$ and changing the variable
$w\to\sqrt{\hbar}\,w$, the partition function (90) can be written as555For
convenience, we change the overall normalisation by a factor
$\sqrt{\hbar}/(2\pi)$; see Aniceto_2019  for more details.

$Z(x)=\frac{1}{\sqrt{2\pi}}\int_{\tilde{\Gamma}}dw\,e^{-\frac{1}{2}w^{2}+\frac{x}{24}w^{4}}\,,$
(92)

where $\tilde{\Gamma}$ is the curve obtained from $\Gamma$ by the above
rescaling. The partition function can be shown to satisfy the second-order
linear differential equation

$16x^{2}Z^{\prime\prime}(x)+(32x-24)Z^{\prime}(x)+3Z(x)=0\,,$ (93)

which is essentially a Ward-Takahashi identity for the zero-dimensional path
integral. In the next subsection, we will consider the transseries solution to
this differential equation and test the exact large order relation (79) for
this transseries.

Before we do so, let us introduce the quartic free energy $F=\log Z$, which is
the model we discussed in the examples 2.3 and 3.1 – 3.4. Using (93), it is
straightforward to show that $F$ satisfies the following nonlinear second
order differential equation

$16x^{2}F^{\prime\prime}(x)+16x^{2}\left(F^{\prime}(x)\right)^{2}+(32x-24)F^{\prime}(x)=0\,.$
(94)

Being a second order ODE, the solution to (94) has two integration constants
and is therefore described by a two-parameter transseries which turns out to
have the form

$F(x,\sigma,\rho)=\sum_{n=0}^{\infty}\sigma^{n}\,e^{-\frac{nA}{x}}\sum_{g=0}^{\infty}F^{(n)}_{g}x^{g}+\rho\,,$
(95)

with instanton action $A=3/2$ and $\sigma$ and $\rho$ the transseries
parameters. This expression was used in example 2.3. As (94) is a non-linear
ODE, one obtains infinitely many instanton sectors $F^{(n)}$ with evenly
spaced actions $nA$, leading to the pattern of Borel plane singularities of
figure 3. By plugging the transseries (95) into (94), one can obtain recursive
relations to determine the instanton coefficients $F_{g}^{(n)}$ that can be
found in appendix B of Aniceto_2019 .666In Aniceto_2019 , the instanton
$F_{g}^{(n)}$ are labeled as $F_{g}^{(n,0)}$. Furthermore, because the
integration constant $\rho$ is simply an additive constant, the two-parameter
transseries in (95) is dubbed a ‘1.5-parameter’ transseries there.

###  4.2 Test of the exact large order relations

We want to test the exact large order relation (79) for the quartic partition
function. We therefore need the solution to the second-order linear ODE (93),
which takes the form

$\mathcal{Z}(\sigma_{0},\sigma_{1},x)=\sigma_{0}Z^{(0)}(x)+\sigma_{1}\,{\rm
e}^{-\frac{A}{x}}Z^{(1)}(x)\,,$ (96)

with the same instanton action $A=3/2$ as for the quartic free energy, but now
only two asymptotic series:

$\displaystyle Z^{(0)}(x)=$
$\displaystyle\sum_{n=0}^{\infty}Z_{g}^{(0)}x^{n}\,,\qquad\qquad
Z^{(0)}_{g}=\left(\frac{2}{3}\right)^{g}\frac{(4g)!}{2^{6g}\,(2g)!\,g!}\,,$
$\displaystyle Z^{(1)}(x)=$
$\displaystyle\sum_{n=0}^{\infty}Z_{g}^{(1)}x^{n}\,,\qquad\qquad
Z^{(1)}_{g}=-\frac{{\rm
i}}{\sqrt{2}}\left(-\frac{2}{3}\right)^{g}\frac{(4g)!}{2^{6g}\,(2g)!\,g!}\,.$
(97)

The corresponding $0$-Borel transforms of the two asymptotic sectors can be
expressed in closed form in terms of hypergeometric functions and are given by

$\displaystyle\mathcal{B}[Z^{(0)}](t)=$ $\displaystyle\
{}_{2}F_{1}\left(\frac{1}{4},\frac{3}{4},1;\frac{2t}{3}\right)\,,$
$\displaystyle\mathcal{B}[Z^{(1)}](t)=$ $\displaystyle-\frac{{\rm
i}}{\sqrt{2}}{}_{2}F_{1}\left(\frac{1}{4},\frac{3}{4},1;-\frac{2t}{3}\right)\,.$
(98)

With the expressions for the partition function at our disposal, we can now
plug it into the exact large order relation (79). The exact large order
relation for the coefficients $Z_{g}^{(0)}$ consists of only a single term on
the right hand side:

$Z^{(0)}_{g}=-\frac{S_{1}}{2\pi{\rm i}}\ \Gamma(g+1)A^{-g}\ \int_{0}^{{\rm
e}^{-{\rm
i}\theta}\infty}\mathop{}\\!\mathrm{d}t\,\mathcal{B}[Z^{(1)}]\left(\frac{3}{2}({\rm
e}^{t}-1)\right){\rm e}^{-gt}\,,$ (99)

where $\theta=\arg(g)$ and the Stokes constant reads $S_{1}=-2$. Let us
explain the integration path in (99). The extension for our exact large order
relation to generic real and positive $g$ is straightforward, but in the case
of complex $g$ we need to be careful: when $\mbox{Re}(g)$ is negative, any
Laplace integral going from $0$ to $+\infty$ along the real $t$-axis will no
longer converge. Hence we need to adjust the angle along which we integrate as
we move $g$ through the complex plane. The correct way to do this is to set
the angle of integration to $-\theta$. This will moreover ensure that the
Laplace integral converges as rapidly as possible, which is also desirable
from a numerical computational point of view.

(a)

(b)

(c)

(d)

Figure 10: Results of the numerical integration of the exact large order
relation for the quartic partition function for four different values of
$\arg(g)$ and with $|g|$ (horizontal axis) ranging from 0 to 2. In the upper
graphs in each panel, we show in red (solid line) the real part of the exact
value (100) of $Z^{(0)}_{g}$. The round blue markers display the numerical
results for the exact large order relation (99), whereas the green triangle
markers show the results for (101), with Stokes phenomenon taken into account.
The Stokes phenomenon, which occurs at $\arg(g)=\pi/2$, explains why the blue
markers match the red line in figures 10(a) and 10(b), and the green markers
do so in 10(c) and 10(d). In the lower graphs in each panel, we show the ratio
of the exact and computed values. We achieve a good accuracy for $|g|>0.5$,
whereas for smaller values of $|g|$, the numerical accuracy decreases due to
the exponential factor in the Laplace integral which becomes large in this
regime. In figure 10(d), we shifted $|g|$ by $1/1000$ in the numerical
integrations to avoid computing ratios of infinities. The lower graph shows
that even close to the singularities, the exact large order relation works to
great accuracy.

If we regard the perturbative coefficients $Z^{(0)}_{g}$ in (97) as a function
whose domain consists of integers $g\in\mathbb{N}$, then this function admits
a natural extension to the domain of complex numbers:

$Z^{(0)}_{g}=\left(\frac{2}{3}\right)^{g}\frac{\Gamma(4g+1)}{2^{6g}\,\Gamma(2g+1)\Gamma(g+1)}\,,$
(100)

i.e. we replace the factorials by gamma functions.777Strictly speaking we also
need to define $c^{g}$ for constant $c$ as $e^{g\log c}$ and pick a branch for
the logarithm, but we can ignore this subtlety in what follows and simply
assume that $c^{g}$ is always extended in the ‘natural’ way from integer $g$.
The first question that we want to address is whether the exact large order
relation can reproduce the exact value of $Z^{(0)}_{g}$, defined in this way,
for real positive $g$. The answer to this question is shown in the upper graph
of figure 10(a) for the real part of $Z^{(0)}_{g}$. (One obtains similar plots
for its imaginary part.) The solid red line plots the exact value (100), and
the round blue markers are computed using the right hand side of (99). The
triangular green markers in this plot will be explained shortly. The lower
graph depicts the ratio between the exact and computed values. Clearly, the
two expressions match nicely also for non-integer $g$. Notice that there are
no singularities on the integration path – i.e. on the real positive line – of
(99), so we do not need to deform the contour here.

We can also consider negative real values of $g$. For this, we need to rotate
the phase of $g$ from $0$ to $\pi$ and rotate the direction of the integration
contour in (99) to $-\infty$ in the opposite direction, as we explained
before. In figure 10(d) we show the result. At first sight our conjecture of
exactness of (99) seems to fail, as the blue markers clearly fail to
accurately describe the analytic expression for $Z^{(0)}_{g}$. However, there
is a simple reason for this: the Borel transform
$\mathcal{B}[Z^{(1)}]\left(\frac{3}{2}({\rm e}^{t}-1)\right)$ has numerous
singlarities along the imaginary axis, and these induce a Stokes phenomenon
when the integration contour crosses this axis. Hence, when we rotate the
contour of integration to the negative axis in the Borel plane, we pick up
nonperturbative corrections that we need to include. Taking these
nonperturbative contributions into account we find for $\mbox{Re}(g)$ negative
that

$Z^{(0)}_{g}=-\frac{S_{1}}{2\pi{\rm i}}\ \Gamma(g+1)A^{-g}\left(1-\frac{2{\rm
e}^{2\pi{\rm i}g}}{1+{\rm e}^{4\pi{\rm i}g}}\right)\ \int_{0}^{{\rm e}^{-{\rm
i}\theta}\infty}\mathcal{B}Z^{(1)}\left(A({\rm e}^{t}-1)\right){\rm
e}^{-gt}\mathop{}\\!\mathrm{d}t\,,$ (101)

The nonperturbative corrections are therefore described by including the
additional factor

$-\frac{2{\rm e}^{2\pi{\rm i}g}}{1+{\rm e}^{4\pi{\rm i}g}}\,,$ (102)

as we derive in appendix C. Using the expression (101) we obtain the green
(triangular) markers in the different panels of figure 10. Indeed, these
markers describe the exact values of the perturbative coefficients for
$\mbox{Re}(g)<0$, as one can see in figure 10(d).

Now, we can also test the exact large order relation for complex values of
$g$. The most interesting values are those for which the integration contour
in the Borel plane is near the Stokes line. To clearly see the Stokes
phenomenon, we have therefore tested the exact large order relation for
$\arg(g)=0.45\pi$ (figure 10(b)) and $\arg(g)=0.55\pi$ (figure 10(c)). Before
the Stokes phenomenon takes place – which happens at $\theta=0.5\pi$ – we
observe that the blue markers obtained using (99) correctly describe
$Z^{(0)}_{g}$. Conversely, the green markers (101) describe $Z^{(0)}_{g}$
correctly after the Stokes phenomenon has occurred. Note that in particular
the difference between the blue and green markers in figures 10(b) and 10(c)
becomes more significant when $g$ is very small, since – as is always the case
for a Stokes transition – the two differ by a nonperturbative, exponentially
small contribution in the large $g$ expansion.

###  4.3 Discussion

The analysis in this section has shown that for the quartic partition
function, the exact large order relation (101) describes the natural extension
(100) of the perturbative coefficients to the complex $g$-plane to high
numerical precision. To make this work, we needed to account for the Stokes
phenomenom, showing that this phenomenon plays an equally important role for
large order transseries as it does for ‘ordinary’ ones.

It is important to realize, though, that as the perturbative coefficients
$\varphi_{g}^{(0)}$ are only defined for positive integers $g$, the extension
to complex $g$ is in general not unique. The extension of the factorials to
gamma functions was the natural choice – in a sense, we make the same choice
as one does when using the Laplace integral (which defines the gamma function)
in (8) to reintroduce the factorials after having performed a Borel transform.
Less natural options are certainly available: one could for example extend the
factorials to Hadamard’s gamma function, but with this replacement (99) and
(100) would no longer lead to the same result.

One may now wonder whether the exact large order relation for complex $g$ only
worked for this specific example, or if it works in general. Let us first
emphasize that for more complicated (non-linear) examples one cannot even give
a ‘natural’ extension of $\varphi_{g}^{(0)}$ to complex values of $g$.
Furthermore, deriving the integral expression of the exact large order
relation by other means that provide a natural smooth $g$-dependence – as we
were able to do in appendix C for (99) – is not possible in general. In fact,
we can turn things around and say that the exact large order relation itself
defines the most natural extension to the complex $g$-plane of
$\varphi_{g}^{(0)}$. Of course, as the example of the partition function
shows, if we want to define an extension to complex $g$ in this way, Stokes
pheneomenon must be taken into account carefully. A final subtelty is that we
have ignored a potential contribution at infinity in the Cauchy integral that
is used to extract $\varphi^{(0)}_{g}$ from the Borel transform, also when we
took $g$ non-integer. Let us now discuss this subtlety for the quartic
parition function.

To this end, let us ask under which conditions a contribution at infinity of
the form

$\oint_{\infty}\frac{\mathop{}\\!\mathrm{d}t}{2\pi{\rm
i}}\frac{\mathcal{B}[\varphi^{(0)}](t)}{t^{g+1}}\,,$ (103)

is nonzero. In the explicit example of the quartic partition function, we
already found a match between the resummed large order transseries without
such a contribution and the exact value of the perturbative coefficients for
values of $g$ throughout the complex plane. This seems to suggest that in this
example the contribution at infinity vanishes for any $g$, but is this really
the case?

Because we have an explicit expression for the Borel transform of the quartic
partition function, we can expand it around $t=\infty$ and obtain

$\mathcal{B}[Z^{(0)}](t)\simeq
t^{-\frac{1}{4}}\,\sum_{n=0}^{\infty}a_{n}\left(\frac{1}{\sqrt{t}}\right)^{n}\,,$
(104)

where the coefficients $a_{n}$ depend on a branch choice at infinity of the
hypergeometric function. Since the Borel transform goes to zero as $t$
approaches infinity, the integral (103) vanishes for $\mbox{Re}(g)>0$:
consider a circular path $C_{R}$ of radius $R$, then the integral scales as

$\int_{C_{R}}\frac{\mathop{}\\!\mathrm{d}t}{2\pi{\rm
i}}\frac{\mathcal{B}[Z^{(0)}](t)}{t^{g+1}}\simeq\mathcal{O}\left(R^{-g-1/4}\right)\,,$
(105)

which for $\mbox{Re}(g)>0$ vanishes as $R\to\infty$. When we complexify $g$
and rotate into the half-plane where $\mbox{Re}(g)<0$, the situation is
complicated. Formally, the contribution at infinity should diverge and
therefore the exact large order relation should break down. However, this is
not what we found in this section: remarkably, the exact large order relation
seems to hold, even when $\mbox{Re}(g)<0$, if we account for the Stokes
phenomenon of $\mathcal{B}[Z^{(1)}]\left(\frac{3}{2}({\rm e}^{t}-1)\right)$.
It is noteworthy however, that the integrand of (103), when we plug in the
expansion (104), has a simple pole – with residue $a_{n}$ – at infinity
whenever $g=-\frac{1}{4}-\frac{n}{2}$ for a nonnegative integer $n$. These are
exactly the values of $g$ for which both sides of the large order relation
(101) diverge and where a finite contribution $a_{n}$ would be neglectable
anyway – see also figure 10(d). Formally, this argument is invalidated by the
fact that the residue is defined as the coefficient of the simple pole in a
Laurent series expansion in $t$, which the above series is not - it is an
expansion in fractional powers of $t$. Yet, the fact that the series has
simple poles for the same values of $g$ for which the coefficient
$Z^{(0)}_{g}$ diverges, does not seem like a coincidence.

It would be interesting to better understand the nature of the contribution at
infinity and how in the end it does or does not contribute to the exact large
order relation for complex values of $g$, as well as to what extent this holds
true for other transseries. In this regard, note that this example is much
simpler than generic large order transseries, since there is only a single
instanton sector and hence no Stokes transitions $\mathfrak{S}_{n}$ across
horizontal lines on the Borel cylinder to account for. In fact, ‘unwrapping’
the Borel cylinder for a fully nonlinear problem, one would find an infinite
number of diagonal Stokes lines in the large order Borel plane, each
containing a single primitive instanton action plus all its positive integer
multiples. Combining all of the corresponding Stokes automorphisms into a
single one that rotates from the positive to the negative real $t$-axis seems
to be a very difficult task.

Finally, let us comment on the nature of the Stokes phenomenon, computed in
(127), of the Borel transform
$\mathcal{B}[\hat{Z}^{(1)}](t)\equiv\mathcal{B}[Z^{(1)}]\left(\frac{3}{2}({\rm
e}^{t}-1)\right)$ in the example of the quartic partition function. Usually,
when an asymptotic series undergoes a Stokes phenomenon, it ‘jumps’ by an
exponentialy small factor ${\rm e}^{-Az}$ multiplied by a diferent asymptotic
series. On the contrary, here we observe that the series $\hat{Z}^{(1)}(x)$
jumps by an exponentially small factor multiplied by itself. This phenomenon
is known as self-resurgence and appears also in the context of WKB solutions
(see e.g. section 5 of inbook ). Using asymptotic large order relations, we
have checked explicitly that the higher order terms of the formal power series
$\hat{Z}^{(1)}(x)$ probe the lower order terms of that exact same series. From
the dependence on $e^{t}-1$, one sees that the singularities at $t=2\pi{\rm
i}n$ for nonzero integers $n$ in the Borel cylinder of
$\mathcal{B}[\hat{Z}^{(1)}]$ correspond to the origin in the original Borel
plane of $\mathcal{B}[Z^{(1)}]$. This shows that there is actually a nonzero
Borel residue at the origin on the other sheets of the multisheeted (original)
Borel plane – i.e. if one takes a path around some of the other singularities.
This is a particular example of the multivaluedness of the Borel residues that
we discussed in section 2.3. This particular type of Stokes phenomenon for
large order transseries appears in this example because both $Z^{(0)}$ and
$Z^{(1)}$ know about each other – that is: ${\mathsf{S}}_{0\to 1}$ and
${\mathsf{S}}_{1\to 0}$ are both nonzero. In the context of nonlinear ODEs,
one always has that the residues ${\mathsf{S}}_{k\to 0}$ vanish, and hence the
same effect does not occur in the Borel cylinders of the first sector
$\mathcal{B}[\hat{\varphi}^{(1)}]$ – see e.g. figures 1(a), 1(b) and 2(a).
Even in those examples, though, it does occur for the higher sectors
$\mathcal{B}[\hat{\varphi}^{(k>1)}]$ – see e.g. figure 6(a), where we do find
singularities at $t=2\pi{\rm i}n$ for integer $n\neq 0$.

##  5 Conclusion

Large order relations are a powerful tool for decoding fluctuations in higher
nonperturbative (e.g. instanton) sectors. They have been used extensively in a
wide variety of physical models. In this work, originally motivated by the
desire to understand the repeating singularity structure found in figure 1, we
have taken a closer look at the generic underlying transseries structure of
these large order relations. We have argued that for rather generic
transseries – only assuming the resurgent structure (17) – we can fully
explain this observed singularity structure, as described by (51). We have
seen that the large order transseries is constructed out of rescaled Stirling
transforms $\hat{\varphi}^{(k)}$ of the original instanton sectors, and that
these formal power series have a Borel cylinder rather than a Borel plane:
$t\sim t+2\pi{\rm i}$.

We derived the Stokes automorphism $\mathfrak{S}_{0}$ of the large order
transseries across the positive real axis of this Borel cylinder, which turns
out to be qualitatively the same as that of the original transseries,
expressed in terms of the Stokes matrix $\mathbb{S}$ in (21). After
establishing further automorphisms $\mathfrak{S}_{n}$ across copies of the
positive real axis, we obtained a clear and unambiguous resummation procedure
for the large order transseries which, after establishing the exact large
order relations (82), is expected to compute the perturbative coefficients
$\varphi^{(0)}$ exactly in many cases. We have checked the ‘exactness’ of this
statement for the quartic partition function, where we found that the exact
large order relation holds for complex values of $g$, and that Stokes
phenomenon for the large order transseries plays an important role to make
this happen.

From a practical point of view, what we have gained is a clear understanding
of the precise contour along which one is supposed to resum the large order
transseries sectors with a given set of Borel residues in order to succesfully
probe higher order instanton coefficients $\varphi^{(k>1)}_{h}$. Having
established a clear understanding of its Stokes phenomenon, we can now also
resum the large order transseries along other contours using the automorphisms
$\mathfrak{S}_{n}$. We would like to stress that our analysis makes no
assumptions about the Stokes data of the underlying transseries: we formulated
our results in terms of generic Borel residues rather than specific Stokes
constants.

From a more conceptual point of view, it is interesting to note that the exact
large order relation does not only compute the perturbative coefficients
$\varphi^{(0)}_{g}$, but also allows us to extend them to complex values of
$g$. It is striking that the exact large order relation for the quartic
partition function, which we tested thoroughly in section 4, seems to work for
all complex values of $g$ – if we take the Stokes phenomenon explained in
appendix C into account – and does not seem to be spoiled by the contribution
at infinity in the Cauchy integral. Of course, this is only a single example,
but it would be interesting to see more examples where this contribution might
or might not be relevant.

A first open question that would be interesting to address is therefore
whether the example of the quartic partition function is special or whether
the exactness that we found holds for large classes of examples, either
because contributions from infinity are absent or because we can pinpoint such
contributions and correctly take them into account. This question is
intimately related to the question whether our techniques (and in particular
the Stirling transform) can lead to numerical gain: while from the
computations that we have performed the use of the Stirling transform does not
seem to lead to much faster numerics for large values of $g$, it may lead to
improvements for small values of $g$ – but to compute at such values, it is
crucial that many contributions to the exact large order relation – including
potential contributions from infinity – are properly taken into account.

The results of this paper can most likely be generalized to broader settings,
and it would be interesting to explore some of those. Here, one can think of
transseries with multiple Stokes lines, transseries with logarithmic terms
(such as resonant transseries), transseries that have additional parameters
and display the higher order Stokes phenomenon of Howls2004 , etc. While we do
not expect major qualitative differences in such cases, it would still be good
to extend the large order toolbox that we have developed to these settings.

The geometry of the Borel cylinder also begs to be explored further. While the
$t\sim t+2\pi{\rm i}$ symmetry in the Borel plane of the large order
transseries is exact when we explore positive integer $g$, in more general
cases our expressions involved factors of $e^{-2\pi{\rm i}g}$ for each shift
in the imaginary direction. This seems to hint at a structure where the
quantities that we compute, rather than functions on the Borel cylinder, are
sections of a bundle over this cylinder. Exploring this possibility may lead
to an even better, more geometric understanding of large order relations.
Another geometric question arises from the fact that the locations of the
singularities on the Borel cylinder are the logarithms of the locations of the
singularities of the original transseries. This may suggest that we should not
only think of sectors $\hat{\varphi}^{(k)}$ associated to singularities at
$\log(k)$, but that one can also think of a sector $\hat{\varphi}^{(0)}$ that
encodes the original perturbative series and that has been shifted to the end
of the Borel cylinder at $-\infty$.

###### Acknowledgements.

We thank David Sauzin for useful correspondence and for sharing an unpublished
worked out example with us. The research of AvS was supported by the grant
OCENW.KLEIN.128, ‘A new approach to nonperturbative physics’, from the Dutch
Research Council (NWO).

##  Appendix A $n$-Borel transforms

Assume that we are given a sequence of complex numbers
$\varphi=\\{\varphi_{g}\,|\,g\in\mathbb{Z}_{\geq 0}\\}$. We will think of
these coefficients as coefficients of some power series, e.g.

$\qquad\varphi(z)=\sum_{g=0}^{\infty}\varphi_{g}z^{-g},$ (106)

but for most purposes in this appendix the exact origin of the sequence
$\varphi$ is irrelevant.

We define the $0$-Borel transform of the sequence $\varphi$ as

$\mathcal{B}_{0}[\varphi](t)\equiv\sum_{g=0}^{\infty}\frac{\varphi_{g}}{g!}\,t^{g}.$
(107)

The $0$-Borel transform is a formal power series in the variable $t$, but as
usual, when this power series converges in some domain we use the same
notation for the function to which it converges, as well as for its analytic
continuation.

The $1$-Borel transform of $\varphi$ is then defined to be the $t$-derivative
of the $0$-Borel transform:

$\mathcal{B}_{1}[\varphi](t)\equiv\frac{d}{dt}\mathcal{B}_{0}[\varphi](t)=\sum_{g=0}^{\infty}\frac{\varphi_{g+1}}{g!}\,t^{g}.$
(108)

More generally, we define the $n$-Borel transform of $\varphi$ to be

$\mathcal{B}_{n}[\varphi](t)\equiv\sum_{g=0}^{\infty}\frac{\varphi_{g+n}}{g!}\,t^{g},$
(109)

which can be viewed as the $n$th derivative of the $0$-Borel transform of
$\varphi$. In fact, for general real (or even complex) $\nu$, one may define

$\mathcal{B}_{\nu}[\varphi](t)\equiv\sum_{g=0}^{\infty}\frac{\varphi_{g}}{\Gamma(g-\nu+1)}t^{g-\nu},$
(110)

which for $\nu=n$ integer agrees with (109). It is then generally true that

$\frac{d}{dt}\mathcal{B}_{\nu}[\varphi](t)=\mathcal{B}_{\nu+1}[\varphi](t).$
(111)

Once a $\nu$-Borel transform is analytically continued beyond the disk of
convergence, it can be mapped back to the Borel sum via

$\mathcal{S}_{\gamma}\varphi(z)=z^{1-\nu}\int_{\gamma}\mathcal{B}_{\nu}[\varphi](t){\rm
e}^{-zt}\mathop{}\\!\mathrm{d}t\,,$ (112)

where to ensure convergence, the contour $\gamma$ is a straight path starting
at the origin and approaching infinity in the half plane where
$\mbox{Re}(zt)>0$. Note that the left hand side of this equation is
independent of $\nu$. This resummation constructs a function of $z$ which we
can asymptotically expand again to retrieve the formal power series
$\varphi(z)$. We could also rewrite the Borel sum (112) as

$\mathcal{S}_{\gamma}\varphi(z)=z^{-\nu}\int_{0}^{\infty}\mathcal{B}_{\nu}[\varphi]\left(\frac{s}{z}\right){\rm
e}^{-s}\mathop{}\\!\mathrm{d}s$ (113)

and fix the integration contour to run from zero to infinity along the
positive real axis. Under such conventions, convergence is ensured and the
locations of the Borel singularities become dependent on $z$. We then find
that Borel singularities will lie on the positive real $s$-axis for certain
values of $z$ that paramatrize a set of lines in the complex $z$-plane that
are often also called Stokes lines. In this work however, we use (112) and
employ a definition in which Stokes lines are rays in the Borel $t$-plane on
which singularities lie.

When we start from a sequence that grows factorially as

$\frac{\varphi_{g}}{A^{-(g+\beta)}\Gamma(g+\beta)}=\mbox{const}+\mathcal{O}\left(\frac{1}{g}\right)\qquad\mbox{as~{}}g\to\infty$
(114)

for some nonzero complex constant, and some real choice of $\beta$, then
$\mathcal{B}_{1-\beta}[\varphi](t)$ has a simple pole at $t=A$. Using (111),
we see that similarly, for integer $k\geq 1$,
$\mathcal{B}_{k-\beta}[\varphi](t)$ has a pole of order $k$ at $t=A$, and
$\mathcal{B}_{-\beta}[\varphi](t)$ has a logarithmic branch point. We use
these facts in this paper to assume, without too much loss of generality, that
our functions in the Borel plane have only logarithmic branch points as their
singularities – for any given Gevrey-1 asymptotic power series and any given
singular point $t=A$, we can always choose an appropriate $n$-Borel transform
such that this is the case.

Of course, the above discussion should not be taken to imply that all sectors
in a given transseries have logarithmic branch points at all singularities in
the Borel plane: the Painlevé I equation is an example where the value of
$\beta$ in (114) differs from sector to sector and has no upper bound – see
Aniceto:2011nu  – so that there will always be sectors where poles in the
Borel plane do appear. Extending the results of this paper so that they also
apply to these cases is completely straightforward: we insist on logarithmic
branch points only to keep our formulas concise. Similarly, it is
straightforward to derive versions of most formulas appearing in this paper
for cases where $\beta$ is non-integer.

##  Appendix B Properties of Stirling transforms

Stirling numbers of the second kind can be defined as NIST:DLMF

$\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}=\frac{1}{k!}\sum_{j=0}^{k}(-1)^{j}\binom{k}{j}\left(k-j\right)^{n}\,.$
(115)

They allow us to express the $0$-Borel transform of the Stirling transform
$\tilde{\psi}(g)$ defined in (40) in terms of the $0$-Borel transform of
$\psi(z)$:

$\mathcal{B}_{0}[\,\tilde{\psi}\,](t)=\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\sum_{j=0}^{k}\frac{(-1)^{j}(k-j)^{n}a_{k}}{k!n!}\binom{k}{j}t^{n}=\mathcal{B}_{0}[\,\psi\,]({\rm
e}^{t}-1)\,.$ (116)

The $n$-Borel transform is then obtained by simply differentiating both sides
$n$ times with respect to $t$. One finds for example

$\mathcal{B}_{1}[\,\tilde{\psi}\,](t)={\rm
e}^{t}\mathcal{B}_{1}[\,\psi\,]({\rm e}^{t}-1)\,,$ (117)

which can also be obtained by explicitly computating
$\mathcal{B}_{1}[\tilde{\psi}]$ using (109) and (115), just as we showed for
the $0$-Borel transform in (116). The $0$-Borel transform of the rescaled
Stirling transform (42) is essentially obtained by repeating the computation
(116), only with $a_{k}$ replaced by $a_{k}A^{k}$, leading to

$\mathcal{B}_{0}[\,\hat{\psi}\,](t)=\mathcal{B}_{0}[\,\psi\,]\left(A({\rm
e}^{t}-1)\right)\,,$ (118)

which again can be differentiated with respect to $t$ to obtain the
corresponding $n$-Borel transforms.

##  Appendix C Stokes phenomenon for gamma functions

In the example of the quartic partition function, the Borel transform
$\mathcal{B}[Z^{(1)}]\left(\frac{3}{2}({\rm e}^{t}-1)\right)$ induces a Stokes
phenomenon as we cross the imaginary $t$-axis. In order to derive the exact
form of this Stokes phenomenon, we start from the exact large order relation
as it holds for positive $g$:

$\int_{0}^{\infty}\mathcal{B}[Z^{(1)}]\left(\frac{3}{2}({\rm
e}^{t}-1)\right){\rm e}^{-gt}\mathop{}\\!\mathrm{d}t=-\frac{2\pi{\rm
i}}{S_{1}}\,\frac{\Gamma(4g+1)}{2^{6g}\,\Gamma(2g+1)\Gamma(g+1)^{2}}\,.$ (119)

Our goal is then to derive the Stokes phenomenon that the gamma functions on
the right hand side of this expression undergo and combine them into a Stokes
phenomenon for the whole expression. To this end, we use Binet’s Log Gamma
formula,

$\log\Gamma(z)=\underbrace{\left(z-\frac{1}{2}\right)\log(z)-z+\frac{1}{2}\log(2\pi)}_{C(z)}+\int_{0}^{\infty}\left(\frac{1}{2}-\frac{1}{t}+\frac{1}{{\rm
e}^{t}-1}\right)\frac{{\rm e}^{-zt}}{t}\mathop{}\\!\mathrm{d}t\,,$ (120)

for $\mbox{Re}(z)>0$. This expression consists of singular and constant terms
(as $1/z\to 0$) that we collect in the function $C(z)$, and $1/z$ corrections
which are resummed using Borel-Laplace resummation. Let us first shift the
argument by one:

$\log\Gamma(z+1)=C(z+1)+\int_{0}^{\infty}\left(\left(\frac{1}{2}-\frac{1}{t}+\frac{1}{{\rm
e}^{t}-1}\right)\frac{{\rm e}^{-t}}{t}\right){\rm
e}^{-zt}\mathop{}\\!\mathrm{d}t\,.$ (121)

In the latter term we interpret the integrand as a 1-Borel transform of some
formal power series $F(z)$:

$\mathcal{B}_{1}[F](t)=\left(\frac{1}{2}-\frac{1}{t}+\frac{1}{{\rm
e}^{t}-1}\right)\frac{{\rm e}^{-t}}{t}\,.$ (122)

By shifting and rescaling the argument we straightforwardly find that

$\displaystyle\log\Gamma(z+1)$ $\displaystyle=$ $\displaystyle
C(z+1)+\int_{0}^{\infty}\mathcal{B}_{1}[F]\left(t\right){\rm
e}^{-zt}\mathop{}\\!\mathrm{d}t\,,$ $\displaystyle\log\Gamma(2z+1)$
$\displaystyle=$ $\displaystyle
C(2z+1)+\int_{0}^{\infty}\frac{1}{2}\mathcal{B}_{1}[F]\left(\frac{t}{2}\right){\rm
e}^{-zt}\mathop{}\\!\mathrm{d}t\,,$ $\displaystyle\log\Gamma(4z+1)$
$\displaystyle=$ $\displaystyle
C(4z+1)+\int_{0}^{\infty}\frac{1}{4}\mathcal{B}_{1}[F]\left(\frac{t}{4}\right){\rm
e}^{-zt}\mathop{}\\!\mathrm{d}t\,.$ (123)

The large $z$ expansion of the function $C(z)$ has a finite radius of
convergence and will therefore not contribute to any Stokes phenomenon that
the large $z$ expansion of $\log\Gamma(cz+1)$ undergoes. Let

$\mathcal{B}_{1}[\mathcal{F}_{n}](t)=\frac{1}{n}\mathcal{B}_{1}[F]\left(\frac{t}{n}\right)\,,$
(124)

then each $\mathcal{B}_{1}[\mathcal{F}_{n}](t)$ has simple poles located at
$t=2\pi{\rm i}nm$, with residue $1/(2\pi{\rm i}m)$, for all nonzero integers
$m$. When we therefore move the contour of integration across the imaginary
$t$-axis – as depicted in figure 11 for $\mathcal{F}_{1}$ – we pick up poles
which lead to the discontinuity

$\text{disc}_{-\frac{\pi}{2}}\mathcal{F}_{n}=-\log\left(1-{\rm e}^{2\pi{\rm
i}ng}\right)\,.$ (125)

Note that the contour of integration runs along an angle $-\theta$ in the
lower half plane in order to keep ${\rm e}^{-gt}$ real and exponentially
suppressed. The Stokes automorphism on the product of gamma functions then
acts as

$\mathfrak{S}_{-\frac{\pi}{2}}{\rm
e}^{\mathcal{F}_{4}-\mathcal{F}_{2}-2\mathcal{F}_{1}}=\left(1-\frac{2{\rm
e}^{2\pi{\rm i}g}}{1+{\rm e}^{4\pi{\rm i}g}}\right){\rm
e}^{\mathcal{F}_{4}-\mathcal{F}_{2}-2\mathcal{F}_{1}}\,.$ (126)

This Stokes jump is independent of the Borel transform that we chose. Taking
the above jump into account in (119) and now using the $0$-Borel transform
that we use throughout the paper, we find that after the Stokes transition,
i.e. for $g$ in the left half of the complex plane, it becomes

$\displaystyle\int_{0}^{-{\rm i}\infty+\epsilon}\mathcal{B}_{0}[Z^{(1)}]$
$\displaystyle\left(\frac{3}{2}({\rm e}^{t}-1)\right){\rm
e}^{-gt}\mathop{}\\!\mathrm{d}t$ $\displaystyle=\left(1-\frac{2{\rm
e}^{2\pi{\rm i}g}}{1+{\rm e}^{4\pi{\rm i}g}}\right)\int_{0}^{-{\rm
i}\infty-\epsilon}\mathcal{B}_{0}[Z^{(1)}]\left(\frac{3}{2}({\rm
e}^{t}-1)\right){\rm e}^{-gt}\mathop{}\\!\mathrm{d}t\,.$ (127)

This implements the Stokes phenomenon that we use in section 4 to analytically
continue the exact large order relation (99) to (101) when
$\frac{\pi}{2}<\theta<\frac{3\pi}{2}$.

Figure 11: The Stokes phenomenon of $\mathcal{F}_{1}$

## References

  * (1) E. Borel, _Mémoire sur les séries divergentes_ , _Ann. Sc. ENS_ 16 (1899) 9.
  * (2) J. Écalle, _Les fonctions resurgentes_ , Publications mathématiques d’Orsay, Université de Paris-Sud. Departement de Mathématique (1985).
  * (3) S. Garoufalidis, A. Its, A. Kapaev and M. Marino, _Asymptotics of the instantons of Painlevé I_ , _Int. Math. Res. Not._ 2012 (2012) 561 [1002.3634].
  * (4) I. Aniceto, R. Schiappa and M. Vonk, _The resurgence of instantons in string theory_ , _Commun. Num. Theor. Phys._ 6 (2012) 339 [1106.5922].
  * (5) S. Baldino, R. Schiappa, M. Schwick and R. Vega, _Resurgent Stokes data for Painlevé equations and two-dimensional quantum (super) gravity_ , _Commun. Num. Theor. Phys._ 17 (2023) 385 [2203.13726].
  * (6) A. van Spaendonck and M. Vonk, _Painlevé I and exact WKB: Stokes phenomenon for two-parameter transseries_ , _J. Phys. A_ 55 (2022) 454003 [2204.09062].
  * (7) A. Voros, _The return of the quartic oscillator. The complex WKB method_ , _Annales de l’I.H.P. Physique théorique_ 39 (1983) 211.
  * (8) E. Delabaere and F. Pham, _Resurgent methods in semi-classical asymptotics_ , _Annales de l’I.H.P. Physique théorique_ 71 (1999) 1.
  * (9) J. Zinn-Justin and U.D. Jentschura, _Multi-instantons and exact results I: Conjectures, WKB expansions, and instanton interactions_ , _Annals Phys._ 313 (2004) 197 [quant-ph/0501136].
  * (10) G.V. Dunne and M. Unsal, _Uniform WKB, Multi-instantons, and Resurgent Trans-Series_ , _Phys. Rev. D_ 89 (2014) 105009 [1401.5202].
  * (11) A. van Spaendonck and M. Vonk, _Exact instanton transseries for quantum mechanics_ , 2309.05700.
  * (12) G.V. Dunne and M. Unsal, _Resurgence and Trans-series in Quantum Field Theory: The CP(N-1) Model_ , _JHEP_ 11 (2012) 170 [1210.2423].
  * (13) S. Gukov, M. Marino and P. Putrov, _Resurgence in complex Chern-Simons theory_ , 1605.07615.
  * (14) M. Mariño and T. Reis, _Renormalons in integrable field theories_ , _JHEP_ 04 (2020) 160 [1909.12134].