# Quantum phase transition and Resurgence:
Lessons from 3d $\mathcal{N}=4\>$ SQED

Toshiaki Fujimori1***toshiaki.fujimori018(at)gmail.com, Masazumi
Honda2†††masazumi.honda(at)yukawa.kyoto-u.ac.jp, Syo
Kamata3‡‡‡skamata11phys(at)gmail.com,
Tatsuhiro Misumi1,4§§§tatsuhiromisumi(at)gmail.com, Norisuke
Sakai1¶¶¶norisuke.sakai(at)gmail.com, Takuya
Yoda5∥∥∥t.yoda(at)gauge.scphys.kyoto-u.ac.jp

1 Department of Physics, and Research and Education Center for Natural
Sciences,
Keio University, 4-1-1 Hiyoshi, Yokohama, Kanagawa 223-8521, Japan
2 Center for Gravitational Physics, Yukawa Institute for Theoretical Physics
Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan
3 National Centre for Nuclear Research, 02-093 Warsaw, Poland
4 Department of Mathematical Science, Akita University, Akita 010-8502, Japan
5 Department of Physics, Kyoto University, Kyoto 606-8502, Japan

(March 2021)

###### Abstract

We study a resurgence structure of a quantum field theory with a phase
transition to uncover relations between resurgence and phase transitions. In
particular, we focus on three-dimensional $\mathcal{N}=4$ supersymmetric
quantum electrodynamics (SQED) with multiple hypermultiplets, where a second-
order quantum phase transition has been recently proposed in the large-flavor
limit. We provide interpretations of the phase transition from the viewpoints
of Lefschetz thimbles and resurgence. For this purpose, we study the Lefschetz
thimble structure and properties of the large-flavor expansion for the
partition function obtained by the supersymmetric localization. We show that
the second-order phase transition is understood as a phenomenon where a Stokes
and anti-Stokes phenomenon occurs simultaneously. The order of the phase
transition is determined by how saddles collide at the critical point. In
addition, the phase transition accompanies an infinite number of Stokes
phenomena due to the supersymmetry. These features are appropriately mapped to
the Borel plane structures as the resurgence theory expects. Given the lessons
from the SQED, we provide a more general discussion on the relationship
between the resurgence and phase transitions. In particular, we show how the
information on the phase transition is decoded from the Borel resummation
technique.

YITP-21-13, KUNS-2859

###### Contents

  1. 1 Introduction
  2. 2 Quantum phase transition in the 3d $\mathcal{N}=4$ SQED
  3. 3 Lefschetz thimble structures
    1. 3.1 Real positive $N_{f}$
    2. 3.2 Complex $N_{f}$
    3. 3.3 Phase transition and thimble structures
  4. 4 Borel singularities and resurgence structure
    1. 4.1 Numerical study of Borel singularities
      1. 4.1.1 Padé approximation and its improvement
      2. 4.1.2 Subcritical region
      3. 4.1.3 Supercritical region
    2. 4.2 Analytical study of Borel singularities for large $\lambda$
  5. 5 Lessons from 3d $\mathcal{N}=4\>$ SQED
    1. 5.1 Phase transitions as collisions of saddles
    2. 5.2 Thimbles and Borel singularities around critical points
      1. 5.2.1 Lefschetz thimbles
      2. 5.2.2 Borel singularities
    3. 5.3 Second-order phase transition in the SQED revisited
      1. 5.3.1 Lefschetz thimble analysis
      2. 5.3.2 Borel resummation
  6. 6 Conclusions and discussion
  7. A Details on large flavor expansion
  8. B Lefschetz thimble structures for larger $\arg(N_{f})$
  9. C Comments on the Padé-Uniformized approximation
    1. C.1 Single branch cut
    2. C.2 Multiple branch cuts
    3. C.3 On elimination of multiple branch cuts
  10. D Transseries for a finite $\eta$
    1. D.1 Properties of the shift operator $T$
    2. D.2 Diagonalization of $M(x)$
  11. E Transseries for a finite $\lambda=\eta/N_{f}$
  12. F A possible relation between the Borel singularities and complex SUSY solutions

##  1 Introduction

One of the most important problems in quantum field theory (QFT) is to
determine phase structures in the space of parameters. It is connected to
significant information such as symmetries, energy gap, critical phenomena,
topological order, etc. In particular, second-order phase transitions are
essential as they often describe the starts and goals of renormalization group
flows. It is natural to expect that phase transitions are technically related
to (anti-)Stokes phenomena as they both describe some discontinuous behaviors
of physical quantities in a certain limit of parameters. A well-known example
of this is the connection between first-order phase transitions and anti-
Stokes phenomena as there occur switches of dominant saddle points across
anti-Stokes lines. However, the connections for higher order cases are less
clear, and it seems necessary to study in a systematic framework. One such
approach to describe Stokes phenomena is the resurgence theory [1], which
recently has attracted much attention in the contexts of QFTs111 See e.g.
reviews [2, 3, 4, 5, 6] for details. . This paper aims to study relations
between phase transitions and resurgence in QFTs.

The resurgence theory has a long history in applications to quantum mechanics
and differential equations. It has often been used to cure situations that
perturbative expansions are not convergent. It typically gives relations
between non-perturbative effects and large order behaviors of perturbative
series. There have been applications of resurgence to many physical systems
including quantum mechanics [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36],
hydrodynamics [37, 38, 39, 40, 41, 42, 43, 44], integrable systems [45, 46,
47, 48, 49, 50, 51, 52], non-critical string [53, 54, 55, 56, 57] and string
theory [58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71] as well as
QFTs. Recently there are also various applications to QFTs such as 2d QFTs
[72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88], the 3d
Chern-Simons theory [89, 90, 90, 91, 92, 93, 94, 95, 96, 92] and Skyrme model
[97], 4d non-supersymmetric QFTs [98, 99, 100, 101, 102, 103, 104, 105, 106,
107, 108], and supersymmetric (SUSY) gauge theories in various dimensions
[109, 110, 65, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122].222
There are also studies on relations to renormalization [123, 124, 125, 126,
51, 127], thermalization [128] and large charge expansions [129].  However,
most of the works have focused on systems without phase transitions while we
will mention some works related to phase transitions.

In this paper, we study the resurgence structure of a QFT model with a phase
transition to study relations between resurgence and phase transitions. We
take two approaches to address this problem. The first approach is the
Lefschetz thimble (steepest descents) analysis [9, 10, 130]333 See also [131,
132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142]. . For a Lagrangian
QFT, physical observables admit path integral representations and we can
decompose them in terms of Lefschetz thimbles associated with saddle points in
the field configuration space. In general, the structure of such a thimble
decomposition can change discontinuously as varying parameters continuously in
the theory under consideration. When it happens, asymptotic expansions around
saddle points exhibit discontinuous changes of their forms, called the Stokes
phenomena. Other interesting phenomena related to thimble decomposition are
anti-Stokes phenomena, where dominantly contributing saddles are switched as
varying parameters while thimble structures themselves are unchanged. As
mentioned above, anti-Stokes phenomena typically induce a first-order phase
transition and it is often discussed in theoretical physics simply by
comparing values of actions at saddle points. phase transitions. Here we
mainly focus on relations between second-order phase transitions and Stokes
phenomena. The other approach is to interpret it from the viewpoint of Borel
resummation. The information of such phenomena is expected to be encoded in a
perturbative series. The Borel resummation technique enables us to decode such
information. Therefore, if the resurgence theory works, then we can approach
phase transitions (typically by non-perturbative effects) from perturbative
expansions.

Here we mainly study the three dimensional $\mathcal{N}=4$ supersymmetric
quantum electrodynamics with multiple hypermultiplets (SQED) to obtain lessons
on relations between resurgence and phase transitions. Recently it has been
proposed by Russo and Tierz [143] that there is a quantum second-order phase
transition in the SQED based on saddle point analysis in the large flavor
limit. They argued that the number of dominant saddle points changes across a
particular value of the Fayet-Illiopoulos (FI) parameter and assuming that
they all contribute to the path integral induces the phase transition. We
provide interpretations of the phase transition from the viewpoints of
Lefschetz thimbles and resurgence. We first justify the assumption in [143]
that all the dominant complex saddles contribute to the path integral by the
Lefschetz thimble analysis. Then we interpret the second-order phase
transition as a simultaneous Stokes and anti-Stokes phenomena. Our results
show that the resurgence theory works for describing the second-order phase
transition of the SQED. Given the lessons from the SQED, we finally provide a
more generic discussion on relations between the resurgence and phase
transitions. In particular, we generally show that the orders of phase
transitions are determined by how saddle points collide and scatter as varying
a parameter through a critical point. We also show how the information of the
phase transition is decoded from the Borel resummation technique. We believe
that our results open up potential applications of resurgence to quantum field
theories.

Let us finally comment on previous works closely related to this paper444 See
also [144, 145, 146, 47, 48, 49, 50, 51, 52] for works indirectly related to
this context. . The work [147] studied thimble structures of simple fermionic
systems such as zero-dimensional versions of the Gross-Neveu model and Nambu-
Jona-Lasinio model, and one-dimensional gauge theory coupled to a massive
fermion with a Chern-Simons term. In particular, it was found in the zero-
dimensional Gross-Neveu model that there was a jump in the number of
contributing thimbles at the second-order chiral phase transition point in the
massless case. It was also demonstrated that there was an interesting link
between anti-Stokes lines and Lee-Yang zeros. There are also interesting works
on the two-dimensional pure $U(N)$ Yang-Mills theory on lattice [81, 148,
149], which is technically reduced to a unitary matrix model called the Gross-
Witten-Wadia model [150, 151]555 The Painleve equations were further studied
in [152, 153] . It was found that there occurred a condensation of complex
saddle points at the third order phase transition point in the large-$N$
limit. Historically, physicists have studied simple models to draw lessons and
to uncover general laws. The field of resurgence and its relation to QFT phase
transitions is not an exception either. Now it is a good time to broaden the
reach of resurgence toward more realistic QFTs. The SQED studied in this paper
should be a nice first step along this direction since it is more realistic
and nevertheless its partition function is expressed in a simple manner thanks
to the supersymmetry.

This paper is organized as follows. In Sec. 2, we review the work by Russo and
Tierz [143]. In Sec. 3, we provide interpretations of the phase transition
from the viewpoints of Lefschetz thimble. In Sec. 4, we discuss relations
between the phase transition and resurgence structures. In Sec. 5, given the
lessons from the SQED example, we give a more generic point of view on
relations between the resurgence and phase transitions. Sec. 6 is devoted to
conclusion and discussions. In App. A, we explain details on calculations of
$1/N_{f}$ flavor expansion, where $2N_{f}$ is the number of the
hypermultiplets in the SQED. In App. B, we present thimble structures for
larger ${\rm arg}(N_{f})$ while the main text focuses around ${\rm
arg}(N_{f})=0$. In App. C, we make some comments on the Padé-Uniformized
approximation based on comparisons with the standard Padé approximation in
some simple examples. In App. D and App. E, we study resurgence structures of
the $1/N_{f}$ expansion from the viewpoint of a difference equation for finite
values of the FI parameter $\eta$ and the rescaled parameter
$\lambda=\eta/N_{f}$ respectively. In App. F, we point out a possible relation
between the Borel singularities and complex supersymmetric solutions found in
[115].

##  2 Quantum phase transition in the 3d $\mathcal{N}=4$ SQED

In this section, we review the arguments of [143] to find the quantum phase
transition in the 3d $\mathcal{N}=4$ SQED with a large number of
hypermultiplets. Let us consider a 3d $\mathcal{N}=4$ SUSY $U(1)$ gauge theory
coupled to $2N_{f}$ hypermultiplets with charge 1. We turn on a Fayet-
Illiopoulos (FI) term and a real mass associated with a $U(1)$ subgroup of the
$SU(2N_{f})$ flavor symmetry666 The $U(1)$ subgroup rotates $N_{f}$ hypers
with charge $+1$ and the other $N_{f}$ hypers with charge $-1$. .

Applying the SUSY localization [154], the path integral is dominated by saddle
points and one can exactly compute the $S^{3}$ partition function of this
theory as [155, 156, 157]

$Z=\int_{-\infty}^{\infty}d\sigma\frac{e^{i\eta\sigma}}{\bigl{[}2\cosh{\frac{\sigma+m}{2}}\cdot
2\cosh{\frac{\sigma-m}{2}}\bigr{]}^{N_{f}}},$ (2.1)

where $\eta$ is the FI parameter and $m$ is the real mass. The integral
variable $\sigma$ is the Coulomb branch parameter and the factor in the
denominator is the one-loop determinant of the hypermultiplets777 The integral
(2.1) can be done exactly as $Z$ $=$
$\frac{\sqrt{2\pi}}{2^{N_{f}}}\frac{\Gamma(N_{f}+i\eta)\Gamma(N_{f}-i\eta)}{\Gamma(N_{f})\left(\sinh{m}\right)^{N_{f}-1/2}}P_{-\frac{1}{2}+i\eta}^{\frac{1}{2}-N_{f}}\left(\cosh{m}\right)$
with the associated Legendre polynomial $P_{\ell}^{m}(x)$ [143] but this form
does not seem particularly useful for our purpose. .

Now we are interested in the ’t Hooft-like limit:

$N_{f}\rightarrow\infty,\quad\lambda\equiv\frac{\eta}{N_{f}}={\rm fixed}.$
(2.2)

For this purpose, it is convenient to write the partition function as

$\displaystyle
Z=\frac{1}{2^{N_{f}}}\int_{-\infty}^{\infty}\differential{\sigma}\>e^{-S(\sigma)},$
(2.3)

where $S(\sigma)$ is the “action” defined by

$\displaystyle S(\sigma)=N_{f}\Bigl{[}-i\lambda\sigma+\ln(\cosh\sigma+\cosh
m)\Bigr{]}.$ (2.4)

In the large $N_{f}$ limit, the integral is dominated by saddle points
satisfying

$S^{\prime}(\sigma)=N_{f}\left(-i\lambda+\frac{\sinh{\sigma}}{\cosh{\sigma}+\cosh{m}}\right)=0.$
(2.5)

This equation is solved by

$\sigma_{n}^{\pm}=\log{\left(\frac{-\lambda\cosh{m}\pm
i\Delta(\lambda,m)}{i+\lambda}\right)}+2\pi in\quad(n\in\mathbb{Z}),$ (2.6)

where

$\Delta(\lambda,m)=\sqrt{1-\lambda^{2}\sinh^{2}{m}}.$ (2.7)

The action at the $n$-th saddle point is

$\displaystyle S(\sigma_{n}^{\pm})$
$\displaystyle=N_{f}\Biggl{[}-\frac{i\lambda}{2}\ln\left(\frac{-i+\lambda}{\phantom{+}i+\lambda}\frac{-\lambda\cosh
m\pm i\Delta}{-\lambda\cosh m\mp i\Delta}\right)+\ln(\frac{\cosh
m\pm\Delta}{1+\lambda^{2}})+2\pi n\lambda\Biggr{]}.$ (2.8)

Note that the saddles and the action values are complex in general. Also, note
that the action can be written as

$\displaystyle S(\sigma_{n}^{\pm})$ $\displaystyle=S(\sigma_{0}^{\pm})+2\pi
nN_{f}\lambda.$ (2.9)

This implies that the most dominant saddle point for real $\lambda$ is either
$\sigma=\sigma_{0}^{+}$ or $\sigma=\sigma_{0}^{-}$, if it contributes to the
integral.

The authors in [143] have observed that dominant saddles change at
$\lambda=\lambda_{c}$ with

$\lambda_{c}\equiv\frac{1}{\sinh{m}}.$ (2.10)

For the subcritical region $\lambda<\lambda_{c}$, only a single saddle
$\sigma_{0}^{+}$ dominates the integral. For the supercritical region
$\lambda\geq\lambda_{c}$, two saddles $\sigma_{0}^{+}$ and $\sigma_{0}^{-}$
contribute to the integral with equal weights. [143] numerically checked that
these saddle approximations agree with the exact analytic expression of (2.1)
at the large $N_{f}$ limit. The second derivative of the “free energy” jumps
at $\lambda=\lambda_{c}$ as

$\frac{d^{2}F}{d\lambda^{2}}=\begin{cases}\frac{N_{f}}{1+\lambda^{2}}\left(1+\frac{\cosh{m}}{\sqrt{1-\lambda^{2}\sinh^{2}{m}}}\right)&\lambda<\lambda_{c}\cr\frac{N_{f}}{1+\lambda^{2}}&\lambda\geq\lambda_{c}\end{cases}.$
(2.11)

This implies that the system exhibits a second-order phase transition at
$\lambda=\lambda_{c}$.

##  3 Lefschetz thimble structures

In general, saddles with smaller $\real S$ give larger weights. However, such
saddles do not necessarily contribute to the path integral. This is because
its original integration contour may not be deformed to the Lefschetz thimbles
(steepest descent paths) associated with such saddles. Also note that the
“free energy” of a partition function on a general manifold is not necessarily
real since it is not interpreted as a thermodynamic one. This means that
contributing saddles cannot be determined only by requiring the free energy to
be real. For these reasons, we should study Lefschetz thimble structures to
describe quantum phase transitions.

In this section, we interpret the quantum phase transition in terms of
Lefschetz thimbles of the integral (2.1) obtained by SUSY localization. This
provides a more precise justification for the arguments in [143] reviewed in
the previous section. For this purpose, we first extend the Coulomb branch
parameter $\sigma\in{\mathbb{R}}$ to complex values $z\in{\mathbb{C}}$, since
saddle points and the associated Lefschetz thimbles are complex-valued in
general. As we have seen in the last section, we have infinitely many saddle
points $\sigma_{n}^{\pm}$ satisfying the saddle point equation. Contributing
saddles are determined by looking at the Lefschetz thimbles (or the steepest
descents) obtained by deforming the original contour without changing the
value of the integral. This can depend on the original integral contour, the
parameters $(\lambda,m)$ and properties of the (dual) Lefschetz thimbles as
explained below.

The Lefschetz thimbles $\mathcal{J}_{n}^{\pm}$ associated with the saddle
points $\sigma_{n}^{\pm}$ are defined as solutions of the differential
equation called the flow equation,

$\displaystyle\left.\frac{dz}{ds}\right|_{\mathcal{J}_{n}^{\pm}}$
$\displaystyle=\,\overline{\frac{\partial S[z]}{\partial z}}$ (3.1)

with the initial conditions

$\displaystyle\lim_{s\rightarrow-\infty}z(s)=\sigma_{n}^{\pm},$ (3.2)

where $s$ is the flow parameter along the Lefschetz thimbles. Using the flow
equation, we can easily prove the following properties

$\displaystyle\left.\frac{d}{ds}\,{\rm
Re}S[z(s)]\right|_{\mathcal{J}_{n}^{\pm}}\,\geq\,0\quad{\rm
and}\quad\left.\frac{d}{ds}\,{\rm
Im}S[z(s)]\right|_{\mathcal{J}_{n}^{\pm}}\,=\,0,$ (3.3)

which indicate that integrals along Lefschetz thimbles are rapidly convergent
and non-oscillating. We can express the original contour ${\cal
C}_{\mathbb{R}}$ as a linear combination of the Lefschetz thimbles

$\displaystyle{\cal
C}_{\mathbb{R}}\,=\,\sum_{\pm}\sum_{n=-\infty}^{\infty}k_{n}^{\pm}{\cal
J}_{n}^{\pm}.$ (3.4)

If $k_{n}^{\pm}$ is nonzero, it implies that $\sigma_{n}^{\pm}$ contributes to
the integral while we have no contributions from the saddle points with
$k_{n}^{\pm}=0$. It is known that each expansion coefficient $k_{n}^{\pm}$ is
an integer since $k_{n}^{\pm}$ is identified with the intersection number
between the original contour ${\cal C}_{\mathbb{R}}$ and the dual thimble (or
the steepest ascent contour) $\mathcal{K}_{n}^{\pm}$ associated with
$\sigma_{n}^{\pm}$, which is defined by

$\displaystyle\left.\frac{dz}{ds}\right|_{\mathcal{K}_{n}^{\pm}}=\overline{\frac{\partial
S[z]}{\partial z}}\quad\quad{\rm with}\
\lim_{s\rightarrow+\infty}z(s)=\sigma_{n}^{\pm}.$ (3.5)

In general, $k_{n}^{\pm}$ depends on $(g,m)$ but its dependence is not
continuous since $k_{n}^{\pm}$ is an integer. Typically $k_{n}^{\pm}$ is a
constant or a step function, and the latter case leads us to a Stokes
phenomenon.

###  3.1 Real positive $N_{f}$

Figure 1:  Illustrations of the Lefschetz thimble structures of the integral
(2.1) for ${\rm arg}(N_{f})=0$ (In these figures, $m=1$, for which
$\lambda_{c}\simeq 0.85$). The blue circle and orange triangle symbols
indicate the saddles $\sigma_{n}^{+}$ and $\sigma_{n}^{-}$, respectively. The
Lefschetz thimbles associated with them are drawn as the lines with the same
colors. The gray wavy lines and cross symbols indicate the branch cuts and
singularities of the action (2.4), which essentially come from the logarithm.
[Left] At $\lambda=0.4<\lambda_{\text{c}}$ as a representative of the
subcritical region. [Right] At $\lambda=1.2>\lambda_{\text{c}}$ as a
representative of the supercritical region.

First, let us briefly see the Lefschetz thimble structures for real positive
$N_{f}$ i.e. ${\rm arg}(N_{f})=0$. We have numerically solved the flow
equations and drawn the results at some representative values of $(m,\lambda)$
in Fig. 1. We immediately see that the Lefschetz thimbles pass multiple saddle
points both in the subcritical ($\lambda<\lambda_{c}$) and supercritical
($\lambda\geq\lambda_{c}$) regions. Although we have explicitly shown the
results only at the two values of $(m,\lambda)$, we have checked that this
feature remains to hold unless the parameters cross the phase boundary. The
thimble structures imply that the decomposition in terms of the thimbles is
not well-defined at ${\rm arg}(N_{f})=0$ and the Stokes coefficient has a
discrete change. In other words, the present case ${\rm arg}(N_{f})=0$ is on
Stokes lines.

The appearance of the Stokes lines here is natural because we have infinitely
many saddle points with the same imaginary part of the action at ${\rm
arg}(N_{f})=0$ although this is not sufficient but necessary to have the
Stokes lines. This can be explicitly seen as follows. In the subcritical
region $\lambda<\lambda_{c}$, one can easily show that all the saddle points
are purely imaginary and their actions are real:

${\rm Im}(S_{n}^{\pm})=0\quad{\rm for}\ \lambda<\lambda_{c}.$ (3.6)

In the supercritical region $\lambda>\lambda_{c}$, the imaginary parts of the
actions at the saddles are nonzero but they satisfy

${\rm Im}(S_{n}^{\pm})=-{\rm Im}(S_{n}^{\mp})\ \ {\rm and}\ \ {\rm
Im}(S_{n}^{\pm})={\rm Im}(S_{0}^{\pm})\quad{\rm for}\ ^{\forall}n\ {\rm and}\
\lambda>\lambda_{c}.$ (3.7)

The above structure essentially comes from the fact that the action at the
saddles depends on $n$ only via the term “$2\pi n\lambda N_{f}$” as seen from
(2.9). This motivates us to take complex $N_{f}$ to go beyond the Stokes lines
and understand the thimble structures more precisely.

###  3.2 Complex $N_{f}$

Figure 2:  Illustrations of the Lefschetz thimble structure for the
subcritical region $\lambda=0.4<\lambda_{\text{c}}$ with $m=1$. Small phases
$\theta=-0.025$ (left) and $\theta=+0.025$ (right) are given to illustrate
behaviors around $\theta=0$. The solid lines show the Lefschetz thimbles while
the dashed lines are the dual thimbles associated with saddle points. The
opaque saddles and thimbles contribute to the integral, while the translucent
ones do not. In this case, the contributing saddle $\sigma_{0}^{+}$ does not
change for any small $\theta$.

Let us take $N_{f}$ to be complex while keeping $\lambda$ real888 To keep
$\lambda$ real, we should also take ${\rm arg}(\eta)=\theta$ since
$\lambda=\eta/N_{f}$. :

$\displaystyle\theta\equiv{\rm arg}(N_{f}),$ (3.8)

and study the Lefschetz thimble structures. In the main text we study the
thimble structures only around $\theta=0$. See App. B for the non-small
$\theta$ case.

Let us first focus on the subcritical region $\lambda<\lambda_{\text{c}}$
presented in Fig. 2. Regardless of the sign of $\theta$, the dual thimble
$\mathcal{K}_{0}^{+}$ intersects once with the original integration contour
${\cal C}_{\mathbb{R}}$. This means that the original integral contour ${\cal
C}_{\mathbb{R}}$ can be deformed to the thimble $\mathcal{J}_{0}^{+}$. Indeed,
we can apply Cauchy’s integral formula since the integrand decreases at
infinity of the upper-half plane. Thus, we find a unique thimble decomposition

$\displaystyle{\cal
C}_{\mathbb{R}}=k_{0}^{+}\mathcal{J}_{0}^{+}\quad\mbox{with}\quad
k_{0}^{+}=1.$ (3.9)

In other words, there is no Stokes phenomenon in the subcritical region.

Figure 3:  Illustrations of the thimble structure for the supercritical region
$\lambda=1.2>\lambda_{\text{c}}$ with $m=1$. Small phases $\theta=-0.025$
(left-top), $\theta=+0.025$ (right-top), $\theta=-0.015$ (left-bottom), and
$\theta=+0.015$ (right-bottom) are given to illustrate behaviors around
$\theta=0$. Contrary to the subcritical region, the contributing saddles
change discontinuously. As the phase approaches $\theta\rightarrow\pm 0$,
saddle points $\sigma_{n}^{\pm}$ with larger $n$ contribute to the path
integral.

The structures in the supercritical region $\lambda>\lambda_{\text{c}}$ are
shown in Fig. 3. We observe qualitatively different behaviors compared with
the subcritical region999 It is common that the saddle points
$\sigma_{n}^{\pm}$ with $n<0$ never contribute. . Firstly, multiple saddle
points contribute to the integral. In particular, two saddle points
$\sigma_{0}^{\pm}$ always contribute to it as their dual thimbles always
intersect with the original integral contour ${\cal C}_{\mathbb{R}}$. This
justifies the arguments by [143] reviewed in Sec. 2. Note that this fact is a
priori nontrivial since the saddles are complex. Secondly, the intersection
numbers $k^{\pm}_{n\geq 1}$ jump discontinuously as the phase $\theta$ is
changed. The thimble structures depend not only on the sign of $\theta$ but
also on the absolute value $|\theta|$. Specifically, a common feature for each
sign $\pm$ of $\sigma_{n}^{\pm}$ is that the saddle points $\sigma_{n\geq
1}^{\pm}$ do not contribute for ${\rm sign}(\theta)=\mp$. The dependence on
the absolute value $|\theta|$ is more intricate. At $\theta=+0.025$ shown in
Fig. 3 [right top], we see that $\sigma_{1}^{+}$ contributes while
$\sigma_{n}^{+}$ with $n\geq 2$ do not contribute. Similarly, at
$\theta=+0.015$ shown in Fig. 3 [right bottom], we see that $\sigma_{1}^{+}$
and $\sigma_{2}^{+}$ contribute but $\sigma_{n}^{+}$ with $n\geq 3$ do not
contribute. As we further decrease $|\theta|$, we have found the following
structure (although we do not explicitly show the plots). For finite
$\theta>0$ ($\theta<0$), we have contributions from the saddle points
$\sigma_{n}^{+}$ with $1\leq n\leq M_{+}(\theta)$ ($\sigma_{n}^{-}$ with
$1\leq n\leq M_{-}(\theta)$) where $M_{\pm}(\theta)$ is an integer such that

$M_{\pm}(\theta)\rightarrow\infty\quad{\rm as}\ \ \theta\rightarrow\pm 0.$
(3.10)

In summary, the above analysis suggests that the thimble structure in the
supercritical region is

$\displaystyle{\cal
C}_{\mathbb{R}}=k_{0}^{+}\mathcal{J}_{0}^{+}+k_{0}^{-}\mathcal{J}_{0}^{-}+\sum_{\pm,n\geq
1}k_{n}^{\pm}\mathcal{J}_{n}^{\pm},$ (3.11)

where

$\displaystyle k_{0}^{+}=k_{0}^{-}=1,\quad k_{n\geq
1}^{+}=\left\\{\begin{array}[]{ll}0&(\theta=-0)\\\
1&(\theta=+0)\end{array}\right.,\quad
k_{n}^{-}=\left\\{\begin{array}[]{ll}1&(\theta=-0)\\\
0&(\theta=+0)\end{array}\right..$ (3.16)

This indicates an infinite number of Stokes phenomena at $\theta=0$ in the
supercritical region.

###  3.3 Phase transition and thimble structures

We clarify a relation between the phase transition and the above Lefschetz
thimble analysis. Whether the saddles contribute to the integral was a priori
nontrivial since all of them are complex for $\lambda>0$. Our Lefschetz
thimble analysis showed that only a single saddle $\sigma_{0}^{+}$ contributes
to the integral in the subcritical region $\lambda<\lambda_{\text{c}}$, while
multiple saddles $\sigma_{n}^{\pm}$ with $n\geq 0$ contribute in the
supercritical region $\lambda\geq\lambda_{\text{c}}$. Among them, only two
saddles $\sigma_{0}^{\pm}$ survive the large-flavor limit. Thus, the dominant
saddles jump at the critical point $\lambda=\lambda_{\text{c}}$ from
$\sigma_{0}^{+}$ to $\sigma_{0}^{\pm}$, which causes the phase transition.
These provide a more precise interpretation of the phase transition in terms
of the Lefschetz thimble analysis. All of these behaviors come from
(anti-)Stokes phenomena and therefore these motivate us to study, in more
detail, a relation between the phase transition and (anti-)Stokes phenomena.
This will be summarized in Sec. 5.

Also, note that the periodicity of the action (2.4) along the imaginary axis
causes an infinite number of Stokes phenomena at $\arg(N_{f})=0$. Such
property is typical in sphere partition functions of $\mathcal{N}\geq 4$
supersymmetric gauge theories with FI terms and without diagonal Chern-Simons
terms. These infinite number of Stokes phenomena are inevitably related to the
phase transition as we will see more details in Sec. 5.

Finally, we shortly provide some preparations for the next section. We have
collected the thimble structures for larger values of $\arg(N_{f})$ as
summarized in App. B. For non-small $\arg(N_{f})$, we have encountered a
subtlety essentially coming from the logarithmic branch cuts in the action
(2.4): when a thimble crosses the branch cuts once, the action changes its
value by $\pm 2\pi iN_{f}$ where the sign depends on a direction of the
crossing. Note that this modifies the condition for having Stokes phenomena as
the imaginary part of the action is changed. App. B demonstrates that the
Stokes phenomena due to this effect indeed happen in our problem. For
instance, in the subcritical region $(\lambda,m)=(0.4,1)$, we have numerically
found that the Lefschetz thimble associated with the saddle $\sigma_{0}^{+}$
crosses one of the branch cuts once and then passes the neighboring saddle
$\sigma_{1}^{+}$ around $\arg(N_{f})\simeq-1.190$. The appearance of this
Stokes phenomenon cannot be understood without taking the effect of the branch
cuts into account as follows. The existence of the branch cut implies that the
condition for having Stokes phenomena between the saddles $\sigma_{n}^{+}$ and
$\sigma_{0}^{+}$ is modified as

$\displaystyle\imaginary\left[S(\sigma_{n}^{+})-S(\sigma_{0}^{+})+2\pi
iN_{f}\mathbb{Z}\right]=0,$ (3.17)

which is solved by

$\displaystyle\tan{\theta}=-\frac{1}{n\lambda}\mathbb{Z}.$ (3.18)

This condition specifically for the above case corresponds to
$\theta=\left.{\rm Arctan}(-1/\lambda)\right|_{\lambda=0.4}\simeq-1.190$,
which agrees with the result in App. B. Similarly in the supercritical region
$(\lambda,m)=(1.2,1)$, we have found the Stokes phenomenon between
$\sigma_{1}^{-}$ and $\sigma_{0}^{+}$ coming from the effect of the branch
cuts for $\theta\simeq=-0.039$.

From the viewpoint of the resurgence theory, the information of Stokes
phenomena is encoded in the $1/N_{f}$ expansion of the partition function
(2.1). Stokes phenomena observed in the figures in App. B are associated with
the Borel singularities of the $1/N_{f}$ expansion. Relations between Stokes
phenomena and Borel singularities in one-dimensional integrals are shown in
[158, 159, 160] although some of the assumptions there are violated in the
SQED due to the logarithmic branch cuts in the action. We will discuss
corresponding Borel singularities in the next section.

##  4 Borel singularities and resurgence structure

In this section, we consider the $1/N_{f}$ expansion of the partition function
(2.1) and study its resurgence structure from the viewpoint of the Borel
resummation method. We numerically compute the $1/N_{f}$ expansion up to 50th
order and then study the structures of the Borel singularities. We confirm
that the locations of the Borel singularities are consistent with the
Lefschetz thimble structure. The resurgence structure of trans-series with
respect to $\eta$ or $\lambda$, instead of $1/N_{f}$, is discussed in App. D
and App. E.

###  4.1 Numerical study of Borel singularities

Let us focus on the $1/N_{f}$ expansion around the saddle point
$\sigma=\sigma_{0}^{+}$. It can be computed in the standard way and the
expansion takes the form (see App. A for details)

$\displaystyle\int_{\mathcal{J}_{0}^{+}}\differential{\sigma}\>e^{-N_{f}\tilde{S}(\sigma)}$
$\displaystyle=e^{-N_{f}\tilde{S}(\sigma_{0}^{+})}\sqrt{\frac{2\pi}{N_{f}\tilde{S}^{\prime\prime}(\sigma_{0}^{+})}}\sum_{\ell=0}^{\infty}\frac{a_{\ell}}{N_{f}^{\ell}},$
(4.1)

where

$S(\sigma)=N_{f}\tilde{S}(\sigma).$ (4.2)

The coefficients are given by

$\displaystyle
a_{\ell}=\sum_{n=0}^{2\ell}\frac{(-1)^{n}\Gamma\left(\frac{1}{2}+\ell+n\right)\tilde{c}_{2\ell-n}(n)}{\Gamma(1/2)\Gamma(n+1)}$
(4.3)

where

$\displaystyle\tilde{c}_{k}(n)=\left.\left(\sum_{k^{\prime}=0}^{\infty}c_{k^{\prime}}\epsilon^{k^{\prime}}\right)^{n}\right|_{\epsilon^{k}},\quad
c_{k}=\frac{2^{\frac{k+3}{2}}}{(k+3)!}\frac{\tilde{S}^{(k+3)}(\sigma_{0}^{+})}{\bigl{(}\tilde{S}^{\prime\prime}(\sigma_{0}^{+})\bigr{)}^{\frac{k+3}{2}}}.$
(4.4)

This implies that the coefficient $a_{\ell}$ grows factorially and the formal
$1/N_{f}$ expansion is not convergent. Therefore we apply the Borel
resummation technique to control the divergence. Let us write the perturbation
series as

$\displaystyle
F\left(\frac{1}{N_{f}}\right)=\sum_{\ell=0}^{\infty}\frac{a_{l}}{N_{f}^{\ell}},$
(4.5)

and define its Borel transformation by

$\displaystyle\mathcal{B}F(t)=\sum_{l=0}^{\infty}\frac{a_{\ell}}{\Gamma(\ell+1)}t^{\ell}=\sum_{\ell=0}^{\infty}b_{\ell}t^{\ell}.$
(4.6)

Then, the Borel resummation of the function $F(1/N_{f})$ is given by

$\displaystyle
F_{C}\left(\frac{1}{N_{f}}\right)=N_{f}\int_{C}\differential{t}\>e^{-N_{f}t}\widetilde{\mathcal{B}F}(t),$
(4.7)

where $\widetilde{\mathcal{B}F}(t)$ is a simple analytic continuation of the
series (4.6) and the integration contour $C$ is chosen so that
$\arg(N_{f}t)=0$. The Borel resummation discontinuously changes if the
integration contour crosses a singularity of $\mathcal{B}F(t)$. From the
viewpoint of resurgence, the Borel singularities must correspond to Stokes
phenomena summarized in App. B.

####  4.1.1 Padé approximation and its improvement

As seen from (4.6), the Borel transformation is defined in terms of an
infinite number of the perturbative coefficients. In practice, we often
encounter the situation where we know only the finite number of the
coefficients and have to estimate (the analytic continuation of) the Borel
transformation from the limited perturbative data in some way. One of the
standard ways to do this is the so-called Borel-Padé approximation, where we
replace the Borel transformation $\widetilde{\mathcal{B}F}(t)$ in (4.7) by its
Padé approximation. The Padé approximation with degrees $(m,n)$ is defined by
a rational function

$\displaystyle\mathcal{P}_{m,n}(t)=\frac{P_{m}(t)}{Q_{n}(t)},$ (4.8)

where $P_{m}(t)$ and $Q_{n}(t)$ are polynomials of degrees $m$ and $n$,
respectively. The explicit forms of the polynomials are determined such that
the small-$t$ expansion of $\mathcal{P}_{m,n}(t)$ agrees with the Borel
transformation up to a desired order $L$:

$\displaystyle\sum_{\ell=0}^{L}b_{\ell}t^{\ell}=\mathcal{P}_{m,n}(t)+\mathcal{O}(t^{L+1})\quad{\rm
with}\ m+n=L.$ (4.9)

While there are various possible choices of $(m,n)$ given $L$, it is
empirically known that the Padé approximation often has better accuracy when
$m$ and $n$ are close. Therefore we here take $m=n=L/2$ with even $L$ and do
not pursue $(m,n)$-dependence. This particular case is called the diagonal
Padé approximation. In practice, we will present results for $L=50$ as a
representative.

In general, the Padé approximation is good at approximating meromorphic
functions since it has only pole-type singularities. For cases with branch
cuts, the Padé approximation typically becomes worse as it is impossible to
express branch cuts in terms of a rational function in the exact sense. It is
known that when the Padé approximation works for the cases with branch cuts,
there appear sets of dense poles around locations of the branch cuts101010 In
other words, many poles in the Padé are consumed to resemble the jump around
the branch cuts and we typically need larger $(m,n)$. . It is also known that
the Padé approximation generically gives better descriptions for singularities
closer to the origin $t=0$. In particular, the location of the closest
singularity is expected to be predicted when $(m,n)$ is larger because this
information is closely related to the radius of the convergence of the
small-$t$ expansion. In other words, it is typically hard to detect Borel
singularities far away from the origin when $(m,n)$ is not-so-large.

There are various ways to improve the Padé approximation (4.8). Here we use
one of the improvements called the Padé-Uniformized approximation [161], which
can be used when we know information on the location of a branch cut in the
Borel transformation $\widetilde{\mathcal{B}F}(t)$. This is constructed as
follows111111 Another way to treat branch cuts is to use functions with branch
cuts for approximation (e.g. fractional power of some simple functions). See
[162, 163, 164, 165, 166, 167] for such approaches. . Suppose that the
function $\widetilde{\mathcal{B}F}(t)$ has a branch cut ending at $t=s$. We
send the Borel $t$-plane to a $u$-plane by the uniformization map

$\displaystyle t\mapsto u(t)=-\ln\left(1-\frac{t}{s}\right),$ (4.10)

which can be inverted as

$\displaystyle u\mapsto t(u)=s(1-e^{-u}).$ (4.11)

Note that the singularity at $t=s$ in the $t$-plane is mapped to infinity in
the $u$-plane. Then we construct the standard Padé approximation in the
$u$-plane, meaning that we construct a rational approximation
$\mathcal{P}_{m,n}(u)$ such that

$\displaystyle\mathcal{P}_{m,n}(u)=\sum_{\ell=0}^{L}b_{\ell}^{\prime}u^{\ell}+\mathcal{O}(u^{L+1})\quad{\rm
with}\ m+n=L,$ (4.12)

where the coefficient $b_{\ell}^{\prime}$ is defined to satisfy

$\displaystyle\sum_{\ell=0}^{L}b_{\ell}^{\prime}u^{\ell}=\sum_{\ell=0}^{L}b_{\ell}\left(t(u)\right)^{\ell}+\mathcal{O}(t^{L+1}).$
(4.13)

Finally, we come back to the $t$-plane and approximate the Borel
transformation as

$\displaystyle\widetilde{\mathcal{B}F}(t)\simeq\mathcal{P}_{m,n}(u(t)).$
(4.14)

The uniformization map sends the branch cut singularity to infinity, where the
standard Padé approximation does not see. Instead, it sends a region away from
the branch cut singularity to a region around the origin. Thus, we can avoid
pole resources of the Padé approximation being wasted on the branch cut. Also,
thanks to the logarithm of the uniformization map, the order of Borel
singularities affect only the scale of $u$. For example, $1/(1-t)^{1/n}$ is
mapped just to $e^{u/n}$ and logarithmic singularities are mapped to regular
points. Another good example is $\ln(1-t)$, which is simply mapped to $-u$.
For the above reasons, the Padé-Uniformized approximation is safer than the
standard Padé approximation in application to our problem.

####  4.1.2 Subcritical region

Figure 4:  Illustrations of the Borel plane structure for the subcritical
region $\lambda=0.4<\lambda_{\text{c}}$ obtained by the standard Padé
approximation (left panel) and the Padé-Uniformized approximation (right
panel). The red cross symbols indicate the location of poles found by the
approximations. The circle/triangle symbols on the real axis indicate the
values of the saddle point actions relative to the action of $\sigma_{0}^{+}$
which are specifically $N_{f}t=S(\sigma_{n}^{\pm})-S(\sigma_{0}^{+})$. The
same symbols beyond the real axis are their counterparts on different Riemann
sheets obtained by shifting the ones on the real axis by $2\pi i\mathbb{Z}$.
The colored symbols among them denote the saddles at which Stokes phenomena
are expected to occur from the Lefschetz thimble analysis.

Let us focus on the subcritical region $\lambda<\lambda_{\text{c}}$. In the
left panel of Fig. 4, we present the locations of poles of the standard Padé
approximation for the Borel transformation $\widetilde{\mathcal{B}F}(t)$,
which is expected to approximate the Borel singularities. The red crosses
indicate the poles of the Padé approximant while the other symbols denote the
values of the saddle point actions subtracted by the one of $\sigma_{0}^{+}$
and their counterparts on different Riemann sheets121212 Recall the arguments
at the end of Sec. 3.3. The thimble structures for non-small $\arg(N_{f})$
studied in App. B imply that we have to take care of situations that thimbles
cross the branch cuts and then the action is shifted by $2\pi
iN_{f}\mathbb{Z}$. . The colored symbols among them denote the saddles at
which Stokes phenomena are expected to occur from the Lefschetz thimble
analysis in the last section. In other words, a symbol associated with a
saddle $\sigma_{n}^{\pm}$ is colored if there exists $\arg(N_{f})$ such
that131313 For example, as discussed in the end of Sec. 3.3 and demonstrated
in App. B, this condition for $(\lambda,m)=(0.4,1)$ is satisfied when
$\arg(N_{f})=-1.190$. Therefore the circles on a ray
$\arg(t)=-\arg(N_{f})=1.190$ are colored.

$\displaystyle\imaginary[S(\sigma_{n}^{\pm})-S(\sigma_{0}^{+})+2\pi
iN_{f}\mathbb{Z}]=0.$ (4.15)

From the viewpoint of resurgence, we expect that the Borel singularities are
located at these color symbols.

In the left panel of Fig. 4, we see that a bunch of poles appear around the
point corresponding to the saddle $\sigma_{0}^{-}$ and are stretched along the
negative real axis. According to general expectation on the Padé
approximation, this signals that the Borel transformation has a branch cut
type singularity ending on the point corresponding to $\sigma_{0}^{-}$ along
the negative real axis. This is consistent with the expectation that we have
Stokes phenomena with $\sigma_{0}^{-}$. We also see a good agreement between
the location of the poles and the action values at
$N_{f}t=S(\sigma_{1}^{+})\pm 2\pi i-S(\sigma_{0}^{+})$ as expected from the
thimble analysis. However, it seems that we do not have a similar agreement
for the other saddles in particular when we go away from the origin. One
reason is that the Padé approximation becomes worse outside the convergence
radius. Another reason is that there is a branch cut on the Borel $t$-plane.
The Padé approximant has limited pole resources to resemble the genuine Borel
plane structure. In our case, there are only $L/2=25$ poles. If there is a
branch cut, a lot of poles are consumed to resemble it. Indeed, a lot of poles
are accumulated on the negative real axis. Thus, it seems more appropriate to
use the Padé-Uniformized approximation using the input that we have a branch
cut ending on the point $N_{f}t=S(\sigma_{0}^{-})\pm 2\pi
i-S(\sigma_{0}^{+})$.

In the right panel of Fig. 4, we show the result of the Padé-Uniformized
approximation where we have eliminated the expected branch cut by the
uniformization map (4.10). We first see that there is no longer a bunch of
dense poles, which appeared in the standard Padé approximant. This confirms
that the uniformization map has successfully removed the branch cut. Because
of this, we expect that the Padé-Uniformized approximation has a better
description of other singularities. Indeed the result shows better agreements
between the locations of poles and the expected Borel singularities around the
origin. In particular, note that there is no Borel singularity on the positive
real axis. This is consistent with the Lefschetz thimble structure around
$\arg(N_{f})=0$ as shown in Fig. 1. However, we see some poles around the real
axis which do not coincide with action values. It seems that they are
artifacts by the Padé approximation. For details, see App. C.

####  4.1.3 Supercritical region

Figure 5:  Illustrations of the Borel plane structure for the supercritical
region $\lambda=1.2>\lambda_{\text{c}}$ obtained by the Padé approximation
(left panels) and by the Padé-Uniformized approximation (right panels). The
lower panels are the zoomed versions of the upper ones.

The results for the supercritical region $\lambda\geq\lambda_{\text{c}}$ are
shown in Fig. 5. As in the subcritical case, the left panels are the result of
the standard Padé approximation while the right panels denote the one of the
Padé-Uniformized approximation. The upper and lower panels are essentially the
same but we plot them in different scales for convenience. Note that the
actions of the saddles $\sigma_{n}^{+}$ and $\sigma_{n}^{-}$ are different by
purely imaginary values even on the same Riemann surface.

Let us first focus on the result of the standard Padé approximation shown in
the left panels of Fig. 5. We easily see that there are a bunch of poles
around the point corresponding to the saddle $\sigma_{0}^{-}$ again but now
they are stretched along the upper imaginary axis in contrast to the
subcritical case. This again implies that the Borel transformation has a
branch cut ending on the point corresponding to $\sigma_{0}^{-}$ along the
upper imaginary axis. While this agrees with the expectation from the
resurgence, we do not see good agreements beyond that saddle. Therefore we
again improve the Padé approximation assuming the information on the branch
cut as in the subcritical region.

The right panels of Fig. 5 show the result of the Padé-Uniformized
approximation. The expected branch cut has been eliminated by the
uniformization map (4.10) and one can check that a bunch of dense poles are
indeed absent in this case. We now see better agreements: there are poles
around expected locations of the Borel singularities. This is consistent with
the Lefschetz thimble structure around $\arg(N_{f})=0$ as shown Fig. 2.
However, we still have missing singularities away from the origin. For
details, see App. C.

###  4.2 Analytical study of Borel singularities for large $\lambda$

In Sec. 4.1, we have numerically found the Borel singularities at $2\pi
i\mathbb{Z}$ in the supercritical region as demonstrated in the right panel of
Fig. 5. We interpret that this class of singularities corresponds to the
saddle $\sigma_{0}^{+}$ on different Riemann sheets. Here we provide an
analytical justification for that: We analytically prove that the Borel
transformation of $1/N_{f}$ expansion around the saddle point
$\sigma_{n}^{\pm}$ has singularities at $2\pi i\mathbb{Z}$ in the large
$\lambda$ limit.

Let us consider the large $\lambda$ limit $\lambda\gg\lambda_{\text{c}}$ and
$\lambda\gg 1$. The saddle point $\sigma_{n}^{\pm}$ in this limit is expanded
as

$\sigma_{\pm}^{n}=\mp m+(2n+1)\pi
i+\frac{i}{\lambda}+\mathcal{O}\left(\frac{1}{\lambda^{2}}\right).$ (4.16)

The action values at these saddle points are

$S(\sigma_{\pm}^{n})=N_{f}\Bigl{[}\pm i\lambda
m+(2n+1)\pi\lambda-\log{\lambda}+1+\log{(i\sinh{m})}\Bigr{]}+\mathcal{O}\left(\frac{1}{\lambda}\right).$
(4.17)

We are interested in the perturbative coefficients in the leading order of the
large-$\lambda$ limit. Let us expand the action around the saddle points:

$S(\sigma)=S(\sigma_{\pm}^{n})+\sum_{n=2}^{\infty}\frac{1}{n!}S^{(n)}(\sigma_{\pm}^{n})\delta\sigma^{n},$
(4.18)

where we regard $\delta\sigma=\mathcal{O}(N_{f}^{-1/2})$ and the first few
derivatives of the action are

$\displaystyle S(\sigma)$ $\displaystyle=$ $\displaystyle
N_{f}\Bigl{[}-i\lambda\sigma+\log{\left(\cosh{\sigma}+\cosh{m}\right)}\Bigr{]},$
$\displaystyle S^{(1)}(\sigma)$ $\displaystyle=$ $\displaystyle
N_{f}\Biggl{[}-i\lambda+\frac{\sinh{\sigma}}{\left(\cosh{\sigma}+\cosh{m}\right)}\Biggr{]},$
$\displaystyle S^{(2)}(\sigma)$ $\displaystyle=$ $\displaystyle
N_{f}\Biggl{[}\frac{\cosh{\sigma}}{\left(\cosh{\sigma}+\cosh{m}\right)}-\frac{\sinh^{2}{\sigma}}{\left(\cosh{\sigma}+\cosh{m}\right)^{2}}\Biggr{]}.$
(4.19)

Noting

$\left.\frac{1}{\cosh{\sigma}+\cosh{m}}\right|_{\sigma=\sigma_{\pm}^{n}}\simeq\mp
i\frac{\lambda}{\sinh{m}},$ (4.20)

we can approximate $S^{(n)}(\sigma_{\pm}^{n})$ as

$S^{(n)}(\sigma_{\pm}^{n})\simeq\left.-N_{f}(-1)^{n}(n-1)!\left(\frac{\sinh{\sigma}}{\cosh{\sigma}+\cosh{m}}\right)^{n}\right|_{\sigma=\sigma_{\pm}^{n}}=-N_{f}(n-1)!(i\lambda)^{n}.$
(4.21)

Therefore the action becomes

$\displaystyle S[\sigma]$ $\displaystyle\simeq$ $\displaystyle
S[\sigma_{\pm}^{n}]-N_{f}\sum_{n=2}^{\infty}\frac{1}{n}(i\lambda\delta\sigma)^{n}$
(4.22) $\displaystyle=$ $\displaystyle
S[\sigma_{\pm}^{n}]+N_{f}\left(i\lambda\delta\sigma+\log{(1-i\lambda\delta\sigma)}\right).$

Then the perturbative series in the large $\lambda$ limit is generated by

$F(N_{f};\lambda)=\int_{-\infty}^{\infty}\differential{\delta\sigma}\>e^{-N_{f}\left(i\lambda\delta\sigma+\log{(1-i\lambda\delta\sigma)}\right)}.$
(4.23)

One can rewrite this integral in the form of a Laplace transformation as in
the Borel resummation formula if we make a change of variable as

$t=i\lambda\delta\sigma+\log{(1-i\lambda\delta\sigma)}.$ (4.24)

Noting that this equation is rewritten as

$-e^{t-1}=(i\lambda\delta\sigma-1)e^{i\lambda\delta\sigma-1},$ (4.25)

we can write the solution as

$\delta\sigma=\frac{1}{i\lambda}\left(1+W(-e^{t-1})\right),$ (4.26)

where $W(z)$ is the Lambert $W$ function defined as a solution of the
following equation

$z=ue^{u}\quad\Leftrightarrow\quad u=W(z).$ (4.27)

Then, using

$\frac{dW(z)}{dz}=\frac{W(z)}{z(1+W(z))},$ (4.28)

we find

$F(N_{f};\lambda)=\frac{1}{i\lambda}\int dt\
e^{-N_{f}t}\frac{W(-e^{t-1})}{1+W(-e^{t-1})}$ (4.29)

It seems natural to identify

$\widetilde{\mathcal{B}F}(t)=\frac{1}{i\lambda}\frac{W(-e^{t-1})}{1+W(-e^{t-1})}.$
(4.30)

Some important features of the Lambert $W$ function are

  * •

$W(z)$ has a branch cut along $(-\infty,-e^{-1})$.

  * •

$W(-e^{-1})=-1$.

  * •

Small $z$ expansion of $W(z)$ has a radius of convergence $e^{-1}$.

Thus the Borel transformation in the large $\lambda$ limit has the branch cut
singularities at

$t=2\pi i\mathbb{Z}.$ (4.31)

One might wonder why we now do not have singularities beyond the imaginary
axis which appeared in the numerical study represented in Fig. 5. This is
because of the large $\lambda$ limit: the singularities beyond the the
imaginary axis go to infinity as $\lambda\rightarrow\infty$. It is most
transparent in the formula (4.17) for the asymptotic behaviors for the action.

We can see that the above Borel singularities come from the branch cuts in the
$\sigma$-plane as follows. The variable $t$ of the Borel plane is related to
the $\sigma$-plane by the map (4.26). Therefore the origin $t=0$ is associated
with a saddle $\delta\sigma=0$ while the infinity is associated with the
branch cut singularity $\delta\sigma=1/i\lambda$. An interval $[0,2\pi i]$ is
associated with a closed loop which starts from the saddle $\delta\sigma=0$
and runs around the branch cut singularity $\delta\sigma=1/i\lambda$ back to
the saddle. Since there is a logarithmic branch cut on the
$\delta\sigma$-plane, we reach the next Riemann sheet once we move along the
closed loop. Thus the Borel singularities at $t=2\pi i\mathbb{Z}$ are
associated with the saddles on the different Riemann sheets. Such a relation
should hold even when $\lambda$ is not large as long as we are in the
supercritical region.

The above structures technically come from the fact that the action (2.4) has
the periodic structure and the logarithm branch cuts. Physically this type of
factor is originated from one-loop contributions of hypermultiplets in the
localization formula of $S^{3}$ partition functions [155, 156, 157]. This
indicates that the above structures hold not only for the SQED but also for
more general supersymmetric gauge theories.

##  5 Lessons from 3d $\mathcal{N}=4\>$ SQED

In this section, given the lessons from the SQED obtained in the previous
sections, we provide a more generic discussion on relations between the
resurgence and phase transitions. In particular, we discuss how the order of
phase transitions are described from the viewpoint of (anti-)Stokes phenomena.

###  5.1 Phase transitions as collisions of saddles

Let us consider a generic theory whose partition function is described by a
one-dimensional integral of the form

$\displaystyle
e^{-NF(\lambda)}=\int\differential{\sigma}\>e^{-N\tilde{S}(\lambda;\sigma)}$
(5.1)

where $\tilde{S}(\lambda;\sigma)$ is the “action” and $(N,\lambda)$ are some
parameters specifying the theory. Suppose that the theory undergoes a phase
transition at $\lambda=\lambda_{\text{c}}$ in the limit $N\rightarrow\infty$,
accompanying a collision and a scattering of $n$ saddles at
$\sigma=\sigma_{\text{c}}$. We do not consider phase transitions simply by
anti-Stokes phenomena which have been often discussed in the context of the
Lefschetz thimble analysis. Here we show that the order of phase transition is
determined by the scattering angle of saddles. More specifically, we prove the
following statement: if the $n$-saddles collide and scatter with a scattering
angle $\beta\pi$ as we vary the parameter $\lambda$ through the critical point
$\lambda=\lambda_{c}$ (as illustrated in Fig. 6), then we have the phase
transition of the order $\lceil(n+1)\beta\rceil$, where $\lceil x\rceil$ is
the smallest integer larger than or equal to $x$.

Before moving onto the proof, let us recall some basics on phase transition.
We have an $p$-th order phase transition at $\lambda=\lambda_{c}$ when the
$p$-th derivative of the “free energy” $F(\lambda)$ becomes singular at
$\lambda=\lambda_{c}$ given its non-singular lower derivatives, that is

$\displaystyle\absolutevalue{F^{(p)}(\lambda_{\text{c}}\pm
0)}=\infty\quad\text{or}\quad F^{(p)}(\lambda_{\text{c}}-0)\neq
F^{(p)}(\lambda_{\text{c}}+0),$ (5.2)

with

$\displaystyle\absolutevalue{F^{(k<p)}(\lambda_{\text{c}}\pm
0)}<\infty\quad\text{and}\quad
F^{(k<p)}(\lambda_{\text{c}}-0)=F^{(k<p)}(\lambda_{\text{c}}+0).$ (5.3)

The order $p$ is related to a behavior of the free energy around the critical
point as follows. Suppose that the free energy is expanded around the critical
point as

$\displaystyle
F(\lambda=\lambda_{\text{c}}+\delta\lambda)\simeq\left\\{\begin{array}[]{ll}C+A(\delta\lambda)^{\gamma}&{\rm
for}\ \delta\lambda<0\\\ C+B(\delta\lambda)^{\gamma}&{\rm for}\
\delta\lambda>0\end{array}\right.,$ (5.6)

where $A,B,C$ are complex constants and $\gamma>0$. Note that the free energy
is not necessarily real since it is not necessarily interpreted as the
thermodynamic one for QFT on a generic manifold141414 It is interpreted as the
thermodynamic one when QFT is put on a manifold including $S^{1}$. . If
$\gamma\notin\mathbb{Z}$, the phase transition is of the order
$p=\lceil\gamma\rceil$. Similarly, if $\gamma\in\mathbb{Z}$ and $A\neq B$,
then the order of the phase transition is $p=\lceil\gamma\rceil$ while there
is no phase transition for $A=B$. In what follows, we consider only the case
with $A\neq B$ and the exponent $\gamma$ independent of the sign of
$\delta\lambda$. This is a consequence of a saddle collision as we will see
soon.

Figure 6:  An illustration of collision and scattering of saddles for $n=2$.
As the parameter $\lambda$ is varied through the critical point
$\lambda=\lambda_{c}$, two saddles $\sigma_{0},\sigma_{1}$ collide at
$\sigma=\sigma_{c}$ and scattered with an angle $\beta\pi$.

Now we provide the proof. As mentioned above, we are interested in the
situation that the $n$-saddles collide and scatter at
$\sigma=\sigma_{\text{c}}$ in varying the parameter $\lambda$ thorough the
critical point $\lambda_{\text{c}}$. This means that the saddle point equation
at $\lambda=\lambda_{\text{c}}$ has the root with the degeneracy $n$.
Therefore the action around $\sigma=\sigma_{\text{c}}$ is expanded as151515
Note that the configuration $\sigma=\sigma_{\text{c}}$ is not a saddle point
for $\lambda\neq\lambda_{\text{c}}$ generically.

$\displaystyle\tilde{S}(\lambda;\sigma=\sigma_{\text{c}})=a_{0}(\lambda)+a_{1}(\lambda)\delta\sigma+\cdots+a_{n+1}(\lambda)\delta\sigma^{n+1}+\cdots.$
(5.7)

The coefficient $a_{i}(\lambda)$ is constrained by the condition that the
saddles collide at $\lambda=\lambda_{\text{c}}$ as

$\displaystyle a_{i}(\lambda)=c_{i}\delta\lambda^{\alpha_{i}}+\cdots.$ (5.8)

Without loss of generality, one can shift the action by an appropriate
constant to make $a_{0}(\lambda)$ independent of $\lambda$:

$\displaystyle a_{0}(\lambda)=c_{0}.$ (5.9)

Furthermore, using the condition that the saddle point equation has the root
with the degeneracy $n$, we find

$\displaystyle a_{i}(\lambda_{\text{c}})=0\quad(i=1,\dots,n),$ (5.10)
$\displaystyle a_{n+1}(\lambda_{\text{c}})\neq 0.$ (5.11)

Combining the above conditions for the coefficient leads us to

$\displaystyle c_{i}=0,\quad\text{or}\quad c_{i}\neq
0,\>\alpha_{i}>0\quad(i=1,\dots,n),$ (5.12) $\displaystyle c_{n+1}\neq
0,\>\alpha_{n+1}=0.$ (5.13)

Solving the saddle point equation

$\displaystyle
0=a_{1}(\lambda)+\cdots+(n+1)a_{n+1}(\lambda)\delta\sigma^{n}+\cdots$ (5.14)

around the collision point $\delta\sigma=0$, the saddle point around the
critical point is simply written as

$\displaystyle\delta\sigma_{m}\simeq
s_{m}\delta\lambda^{\beta}\quad(m=0,\dots,n-1),$ (5.15)

where $s_{m}$ is a some constant and

$\displaystyle\beta=\min\left(\frac{\alpha_{1}}{n},\frac{\alpha_{2}}{n-1},\dots,\frac{\alpha_{n}}{1}\right).$
(5.16)

Around the critical point $\delta\lambda=0$, each saddle acquires a phase
$(-1)^{\beta}$. This implies that the $n$ saddles collide and scatter with an
angle $\beta\pi$. Then the action at the saddle $\sigma_{m}$ takes the value

$\displaystyle\tilde{S}_{m}\simeq c_{0}+T_{m}(\delta\lambda)^{(n+1)\beta},$
(5.17)

with a constant $T_{m}$.

At the phase transition point, there is a jump of contributions saddle points
in various ways. For example, in the case where the contributing saddles jump
as $\sigma_{0}\rightarrow\sigma_{1}$, the free energy changes as

$\displaystyle
F\simeq\left\\{\begin{array}[]{ll}c_{0}+T_{0}(\delta\lambda)^{(n+1)\beta}&{\rm
for}\ \delta\lambda<0\\\ c_{0}+T_{1}(\delta\lambda)^{(n+1)\beta}&{\rm for}\
\delta\lambda>0\end{array}\right..$ (5.20)

In the case where contributing saddles jump as
$\sigma_{0}\rightarrow\sigma_{0},\dots,\sigma_{n-1}$, the free energy changes
as

$\displaystyle
F\simeq\left\\{\begin{array}[]{ll}c_{0}+T_{0}(\delta\lambda)^{(n+1)\beta}&{\rm
for}\ \delta\lambda<0\\\
c_{0}+(T_{0}+\cdots+T_{n-1})(\delta\lambda)^{(n+1)\beta}&{\rm for}\
\delta\lambda>0\end{array}\right..$ (5.23)

In any case, the phase transition is of the order $\lceil(n+1)\beta\rceil$ and
this completes the proof. Our argument also shows a connection between the
order of the phase transition and the anti-Stokes line. The formula (5.17) for
the action shows that the anti-Stokes line is given by
$\real\left[(\delta\lambda)^{(n+1)\beta}\right]=0$. Thus one can also read off
the order of the phase transition by looking at the anti-Stokes line.

###  5.2 Thimbles and Borel singularities around critical points

In this subsection, we demonstrate the discussion in the last subsection using
the integral representation of the Airy function whose “action” is given by

$\displaystyle\tilde{S}(\lambda;\sigma)=\frac{i\sigma^{3}}{3}-i\lambda\sigma.$
(5.24)

We refer to this example as the Airy-type model. As we will see soon, this
example corresponds to $n=2,\>\alpha=1/2$, and has common features with the
SQED (2.1) in the context of the argument in this section.

####  5.2.1 Lefschetz thimbles

The Airy-type model corresponds to

$\displaystyle c_{0}=0,\quad c_{1}=-i,\quad\alpha_{1}=1,\quad c_{2}=0,\quad
c_{3}=\frac{i}{3},\quad\alpha_{3}=0.$ (5.25)

The saddle points in this example are simply given by

$\displaystyle\sigma_{\pm}=\pm\lambda^{1/2},$ (5.26)

which indicates that the two saddles collide at $\sigma=0$ for $\lambda=0$.
Therefore, in the notation of the last subsection, we have

$\displaystyle\sigma_{\text{c}}=0,\quad\lambda_{\text{c}}=0,\quad\delta\sigma_{\pm}=\pm\delta\lambda^{1/2},$
(5.27)

and

$\displaystyle n=2,\quad\alpha=\frac{1}{2}.$ (5.28)

The action at the saddle $\sigma_{\pm}$ is given by

$\displaystyle\tilde{S}_{\pm}$
$\displaystyle=\mp\frac{2i}{3}(\delta\lambda)^{3/2}.$ (5.29)

This leads us to the standard Airy-type Stokes graph

$\displaystyle\text{Stokes line:}\
\imaginary[i(\delta\lambda)^{3/2}]=0,\qquad\text{Anti-Stokes line:}\
\real[i(\delta\lambda)^{3/2}]=0.$ (5.30)

We can find the (dual) thimbles by solving

$\displaystyle\mathcal{J}_{\pm}:\;\derivative{\sigma}{s}=\left.\overline{\derivative{S(\sigma)}{\sigma}}\right|_{\sigma\simeq\sigma_{\pm}},\quad\mathcal{K}_{\pm}:\;\derivative{\sigma}{s}=-\left.\overline{\derivative{S(\sigma)}{\sigma}}\right|_{\sigma\simeq\sigma_{\pm}}.$
(5.31)

Figure 7:  Illustrations of the thimble structures of the Airy-type model for
$\absolutevalue{\delta\lambda}=1$. The blue circle and orange triangle symbols
indicate the saddle $\sigma_{0}$ and $\sigma_{1}$, respectively. The Lefschetz
thimbles associated with them are drawn as the lines with the same colors. The
dual thimbles are drawn as the dashed lines. The opaque saddles and thimbles
contribute to the integral, while the translucent ones do not. The phase of
$\delta\lambda$ is $\arg(\delta\lambda)=-\pi+0.01$ [top left], $-2\pi/3$ [top
right], $-\pi/3+0.01$ [bottom left], and $0$ [bottom right].

In Fig. 7, we show how the thimbles change as increasing the phase
$\arg(\delta\lambda)$. As $\arg(\delta\lambda)$ is increased from a negative
value, we encounter a Stokes phenomenon at $\arg(\delta\lambda)=-\pi$, an
anti-Stokes phenomenon at $\arg(\delta\lambda)=-2\pi/3$, a Stokes phenomenon
at $\arg(\delta\lambda)=-\pi/3$ and an anti-Stokes phenomenon at
$\arg(\delta\lambda)=0$. In particular, we observe the jump of the
contributing saddles at $\arg(\delta\lambda)=-\pi/3$ (as well as
$\arg(\delta\lambda)=+\pi/3$): we have a contribution only from
$\sigma=\sigma_{+}$ for $\arg(\delta\lambda)<-\pi/3$ while we have
contributions from the two saddles $\sigma=\sigma_{+}$ and $\sigma=\sigma_{-}$
at $\arg(\delta\lambda)=-\pi/3+0$. This is a manifestation of the Stokes
phenomenon. The free energy also jumps as

$\displaystyle
F\simeq\left\\{\begin{array}[]{lll}\tilde{S}_{+}&=\frac{2}{3}(-\delta\lambda)^{3/2}&{\rm
for}\ \delta\lambda<0\\\ \tilde{S}_{+}+\tilde{S}_{-}&=0&{\rm for}\
\delta\lambda>0\end{array}\right.,$ (5.34)

which implies the second-order phase transition.

Next, let us increase $\delta\lambda$ from $-1$ to $1$, keeping
$\imaginary\delta\lambda=0$. As $\delta\lambda$ goes from $-1$ to $0$, the two
saddles (in the left top panel of Fig. 7) approach the origin along the
imaginary axis. At $\delta\lambda=0$, they collide and change their
directions. As $\delta\lambda$ goes from $0$ to $+1$, the two saddles (in the
bottom right panel of Fig. 7) depart the origin along the real axis. In other
words, the two saddles collide with an angle $\pi/2$ at the phase transition.
Also, we remark that, during the phase transition, we cross the anti-Stokes
line $\arg(\delta\lambda)=-2\pi/3$ and the Stokes line
$\arg(\delta\lambda)=-\pi/3$. Thus, the second-order phase transition is
understood as a phenomenon in which an anti-Stokes and a Stokes phenomenon
occur simultaneously. To summarize, the second-order phase transition in the
Airy-type model is interpreted as follows

  1. i.

Contributing saddles jump as $\sigma_{+}\rightarrow\sigma_{+},\sigma_{-}$.

  2. ii.

The two saddles collide and scatter with a scattering angle $\pi/2$.

  3. iii.

A Stokes phenomenon and an anti-Stokes phenomenon occur simultaneously.

####  5.2.2 Borel singularities

The “partition function” of the Airy-type model is defined as

$\displaystyle
Z(\lambda)=\int\differential{\sigma}\>e^{-N\tilde{S}(\lambda;\sigma)}.$ (5.35)

Let us consider the $1/N$ expansion around the “trivial” saddle
$\sigma_{+}=\lambda^{1/2}$. Using the formula in App. A, the perturbative
series is formally given by

$\displaystyle\int_{\mathcal{J}_{+}}\differential{\sigma}\>e^{-N\tilde{S}(\lambda;\sigma)}=\sqrt{\frac{\pi}{i\lambda^{1/2}N}}e^{\frac{2i\lambda^{3/2}}{3}N}F\left(\frac{1}{N}\right),$
(5.36)

where

$\displaystyle
F\left(\frac{1}{N}\right)=\sum_{l=0}^{\infty}\frac{a_{l}}{N^{l}},\quad
a_{l}=\frac{\Gamma(3l+1/2)}{3^{2l}(-i\lambda^{3/2})^{l}\Gamma(1/2)\Gamma(2l+1)}.$
(5.37)

Note that the coefficient grows factorially. The analytic continuation of its
Borel transformation is

$\displaystyle\widetilde{\mathcal{B}F}(t)={}_{2}F_{1}\left(\frac{1}{6},\frac{5}{6},1;\frac{3t}{4(-i\lambda^{3/2})}\right).$
(5.38)

This function has a Borel singularity (branch cut singularity) at

$\displaystyle t=\frac{4(-i\lambda^{3/2})}{3}.$ (5.39)

This Borel singularity corresponds to the “non-trivial saddle”
$\sigma_{-}=-\lambda^{1/2}$, and it collides with the origin corresponding to
the “trivial saddle” at the critical point $\lambda=0$. The scattering angle
is $-3\pi/2\sim\pi/2$. After the collision ($\lambda>0$), the Borel
singularity is on the imaginary axis. This means that there occurs an anti-
Stokes phenomenon: $\real\tilde{S}_{+}=\real\tilde{S}_{-}=0$. Thus, the
collision of saddles is appropriately encoded in the perturbative series as
expected by the resurgence theory.

###  5.3 Second-order phase transition in the SQED revisited

In this section, we revisit the second-order phase transition in the SQED
based on the previous subsections to clarify more the relationship between the
phase transition and resurgence.

####  5.3.1 Lefschetz thimble analysis

From the Lefschetz thimble analysis in Sec. 3, we have seen that the SQED
around the second-order phase transition point has the following properties

  1. i.

Contributing saddle points jump as
$\sigma_{0}^{+}\rightarrow\sigma_{0}^{+},\sigma_{0}^{-}$.

  2. ii.

The two saddles collide and scatter with a scattering angle $\pi/2$.

  3. iii.

An infinite number of Stokes phenomena associated with saddles
$\sigma_{n>0}^{\pm}$ occur.

The first two points are common with the Airy-type model in the last
subsection. This is because the “action” of the SQED (2.4) has a similar
expansion to one of the Airy-type models around the critical point. Thus, the
second-order phase transition in the SQED is interpreted in a similar way as
the Airy-type model. The third point is particular for the SQED. The
difference essentially comes from the fact that the SQED has infinite number
of saddles periodically distributed along the imaginary axis. Once thimbles
run along the imaginary axis after a phase transition, they inevitably path
through the periodic saddles. Such behavior of thimbles causes an infinite
number of Stokes phenomena. Technically the appearance of the periodic saddles
is due to the ${\rm cosh}$ factors originated from the one-loop determinant of
the hypermultiplets in the SUSY localization of the $S^{3}$ partition
function. Therefore we expect that the above features appear also in other
SUSY gauge theories on $S^{3}$.

####  5.3.2 Borel resummation

In the language of the Borel resummation, the second-order phase transition
has the following features

  1. I.

In the supercritical region, the two Borel singularities line up along the
imaginary axis on the Borel plane.

  2. II.

The two Borel singularities collide and scatter with a scattering angle
$\pi/2$ as we cross the critical point.

  3. III.

The $1/N_{f}$-expansion becomes Borel non-summable along the positive real
axis in the supercritical region.

(I), (II) and (III) here correspond to (i), (ii) and (iii) of the Lefschetz
thimble analysis, respectively, as expected from the resurgence theory. The
first point means that the saddle points associated with the Borel
singularities along the imaginary axis have the same real part of the actions
and therefore contribute to the integral with the equal weights in the
supercritical region. The second point is a counterpart of the collision of
the two saddles from the viewpoint of the Borel resummation. The relation
between Borel singularities and saddle points implies that the collision of
the two saddles in the $\sigma$-plane leads to one of the two Borel
singularities in the $t$-plane. Thus we can also decode the order of the phase
transition purely from how the Borel singularities collide. The third point
means that the thimbles cross the multiple saddle points for $\arg(N_{f})=0$
as shown in Fig. 1. In the case of the SQED, the Borel non-summability detects
the infinite number of periodic saddles which come from the contribution from
the hypermultiplets.

Finally, let us see the Stokes graph. The second-order phase transition is
interpreted in terms of the Stokes graph as follows.

  * •

The anti-Stokes line is given by $\real[(\delta\lambda)^{3/2}]=0$.

  * •

A Stokes phenomenon and an anti-Stokes phenomenon associated with saddles
$\sigma_{0}^{\pm}$ occur simultaneously.

These points are analogous to the Airy-type model discussed in Sec. 5.2. The
only difference is that the infinite number of Stokes phenomena associated
with $\sigma_{n>0}^{\pm}$ occur simultaneously.

##  6 Conclusions and discussion

We have studied the resurgence structure of a quantum field theory with a
phase transition to uncover relations between resurgence and phase
transitions. In particular we have focused on the three dimensional
$\mathcal{N}=4$ SQED, which undergoes the second-order quantum phase
transition in the large-flavor limit [143]. We have approached the problem
from the viewpoints of the Lefschetz thimbles and Borel resummation. In the
Lefschetz thimble approach, we have specifically studied the thimble
structures of the integral representation of the partition function obtained
by the supersymmetric localization [155, 156, 157]. We have first justified
the assumption in [143] that all the dominant complex saddles contribute to
the integral by the Lefschetz thimble analysis. Then we have found that there
are a collision of the two saddles and a jump of the contributing saddle
points as we cross the critical value of the parameter $\lambda=\eta/N_{f}$.
While this is the Stokes phenomenon, we have seen that an anti-Stokes
phenomenon also occurs at the same time. Thus we interpret the second-order
phase transition as the simultaneous Stokes and anti-Stokes phenomena. Our
result also shows that the phase transition accompanies an infinite number of
Stokes phenomena associated with the other saddles. This behavior technically
comes from the fact that the action (2.4) has the periodic structure which is
physically originated from one-loop contributions of hypermultiplets in the
localization formula of $S^{3}$ partition functions. This indicates that the
above structures hold not only for the SQED but also for more general
supersymmetric gauge theories.

In the Borel resummation approach, we have seen that the thimble structures
are appropriately mapped to the Borel plane structures of the large-flavor
expansion as expected from the resurgence theory. We have found the Borel
singularities, two of which correspond to the two saddles. The two Borel
singularities line up vertically along the imaginary axis after the phase
transition. It is a sign that the two saddles contribute to the integral with
equal weights. At the phase transition, the two Borel singularities collide as
the two saddles. The scattering angle of the Borel singularities at the
collision is related to the order of the phase transition. We have also seen
that the large-flavor expansion becomes Borel non-summable along the positive
real axis in the supercritical region, due to an infinite number of the Borel
singularities. This reflects the infinite number of Stokes phenomena.

Given the lessons from the SQED, we have provided more generic discussion on
relations between the resurgence and phase transitions. We have considered the
one-dimensional integral of the form (5.1) and shown that if the $n$-saddles
collide and scatter with a scattering angle $\beta\pi$ as we vary the
parameter $\lambda$ through the critical point $\lambda=\lambda_{c}$, then we
have the phase transition of the order $\lceil(n+1)\beta\rceil$. Our argument
has also shown that we have anti-Stokes phenomena at the critical point where
the anti-Stokes line is given by
$\real\left[(\delta\lambda)^{(n+1)\beta}\right]=0$. This implies that one can
read off the order of the phase transition also by looking at the anti-Stokes
line. We have also argued that the above behaviors are naturally translated
into the language of the Borel plane. This means that the order of phase
transitions can be determined also by tracking how Borel singularities move as
varying the parameter. This implies that we can read off information on phase
structures purely in terms of perturbative expansions. The above results apply
to more general theories as long as they reduce to the form (5.1).

Finally, we have revisited the second-order quantum phase transition in the
SQED from the above viewpoints. In the case of the SQED, the two saddles
$\sigma_{0}^{+}$ and $\sigma_{0}^{-}$ collide and scatter with $\pi/2$ as we
cross the critical point $\lambda=\lambda_{c}$. Therefore we have
$(n,\beta)=(2,1/2)$ in the formula $\lceil(n+1)\beta\rceil$ for the order of
the phase transition and this agrees with the fact that the second-order phase
transition occurs. From the viewpoint of Stokes graphs, the second-order phase
transition is essentially described as the standard Airy-type graph. This
clarifies that the second-order phase transition is understood as a phenomenon
where the Stokes and anti-Stokes phenomena occur at the same time. This is a
clear contrast to the common understanding that a first-order phase transition
is associated with an anti-Stokes phenomenon. All of the above results support
that resurgence works for describing the second-order phase transition in the
SQED.

We have obtained a good news which may be useful to develop studies of
resurgence on the technical side. Originally, the correspondence between
saddles and Borel singularities in one dimensional integrals was shown in
[158, 159, 160]. It is not guaranteed that we can naively apply the
correspondence to the SQED because some of their assumptions are violated due
to the logarithmic branch cuts in the action. Nevertheless our results suggest
that the correspondence still holds even in the SQED. This seems to imply that
one can extend the correspondence beyond the class of integrals studied in
[158, 159, 160]. It would be interesting to pursue this direction.

We believe that our results give a good step to understand connections between
phase transitions and resurgence. Yet there are still various questions and
tasks which should be addressed as next steps. First, it is important to
understand the physical meaning of the second order phase transition in the
SQED. For instance, we have not understood yet whether there is a change of
symmetries around the critical point, whether the critical point describes
some conformal field theory and so on. Second, we have not identified
interpretations of the saddle points in the SQED in the language of the
original path integral. It seems that they are closely related to the complex
supersymmetric solutions found in [115] as discussed in App. F. We need
further studies to clarify the relations more precisely. Third, it would be
interesting to study relations between Lee-Yang zeros and the Stokes graph.
While the authors in [147] found that Lee-Yang zeros are on anti-Stokes curve
in the zero dimensional Gross-Neveu-like model, the SQED studied in this paper
does not seem to have such a property. This may suggest that the situation in
the SQED is different from the zero dimensional Gross-Neveu-like model.
Fourth, it would be illuminating to study resurgence structures with respect
to other parameters in the SQED such as the FI parameter $\eta$. There may be
interesting relations to the resurgence structure of the large flavor
expansion as in the two-dimensional pure $U(N)$ Yang-Mills theory on lattice,
where there were found interesting connections among expansions by $1/N$,
Yang-Mills coupling and ’t Hooft coupling [81, 148, 149]. Finally it is
technically important to improve the Padé-Uniformized approximation for cases
with multiple branch cuts in Borel planes. It seems that an improvement of the
approximation is hindered by the non-trivial topology of Riemann sheets due to
the branch cuts. Such a problem often arises in the context of resurgence and
therefore further studies are desired.

While this paper has focused on connections between resurgence and phase
transitions, more generally, it would be very interesting to explore relations
between resurgence and phases themselves rather than their transitions. It is
known that information on phases in quantum field theories are partially
captured by ’t Hooft anomalies, including phases beyond the Ginzburg-Landau or
Nambu paradigm. While ’t Hooft anomalies are typically easy to calculate and
give quite robust information on phases, it relies on existence of
symmetries161616 This is not necessarily true for “anomalies in the space of
coupling constants” which was recently proposed [168, 169]. . In contrast,
analysis of resurgence does not require symmetries and gives detailed
information while it is technically much more complicated. Therefore they play
complementary roles. In this paper we have discussed that some features of
phase transitions are captured by qualitative behaviors of the objects
appearing in the analysis of resurgence. It would be great if one can find
similar connections for ’t Hooft anomalies171717 See [36] for a very recent
work on quantum mechanics in a similar spirit. . It might open a door to a
shining world of non-perturbative physics.

### Acknowledgment

The authors would like to thank Okuto Morikawa, Naohisa Sueishi, Hiromasa
Takaura and Yuya Tanizaki for valuable discussions. Preliminary results of
this work have been presented in the KEK workshop “Thermal Quantum Field
Theory and its Application” (Aug. 2020), the YITP workshop YITP-W-20-08
“Progress in Particle Physics 2020” (Sep. 2020), the JPS meetings (Sep. 2020
and Mar. 2021), the YITP-RIKEN iTHEMS workshop YITP-T-20-03 “Potential Toolkit
to Attack Nonperturbative Aspects of QFT -Resurgence and related topics-”
(Sep. 2020), the 15th Kavli Asian Winter School on Strings, Particles and
Cosmology (Jan. 2021) and the Osaka City University Workshop “Randomness,
Integrability and Representation Theory in Quantum Field Theory 2021” (Mar.
2021). Discussions during the workshops were helpful to complete this work. M.
H. is partially supported by MEXT Q-LEAP. This work is supported in part by
Grant-in-Aid for Scientific Research (KAKENHI) (B) Grant Number 18H01217 (T.
F., T. M. and N. S.). S. K is supported by the Polish National Science Centre
grant 2018/29/B/ST2/02457.

##  Appendix A Details on large flavor expansion

In this appendix, we compute the coefficients of the $1/N_{f}$ expansion of
the partition function around a general saddle point $\sigma_{\ast}$. First,
to make $N_{f}$-dependence transparent, we introduce

$S(\sigma)=N_{f}\tilde{S}(\sigma).$ (A.1)

Then we expand $\tilde{S}(\sigma)$ as

$\displaystyle\tilde{S}(\sigma)=\tilde{S}(\sigma_{\ast})+\frac{\tilde{S}^{\prime\prime}(\sigma_{\ast})}{2!}(\sigma-\sigma_{\ast})^{2}+\sum_{k=0}^{\infty}\frac{\tilde{S}^{(k+3)}(\sigma_{\ast})}{(k+3)!}(\sigma-\sigma_{\ast})^{k+3},$
(A.2)

and regard the last term as a perturbation. Next we rewrite the contribution
from the saddle $\sigma_{\ast}$ to the integral (2.1) as

$\displaystyle\int_{\mathcal{J}_{\ast}}\differential{\sigma}\>e^{-S(\sigma)}$
$\displaystyle=e^{-N_{f}\tilde{S}(\sigma_{\ast})}\int_{\mathcal{J}_{\ast}}\differential{\sigma}\>\sum_{n=0}^{\infty}\frac{1}{n!}\left(-\sum_{k=0}^{\infty}\frac{N_{f}\tilde{S}^{(k+3)}(\sigma_{\ast})}{(k+3)!}(\sigma-\sigma_{\ast})^{k+3}\right)^{n}e^{-\frac{N_{f}\tilde{S}^{\prime\prime}(\sigma_{\ast})}{2!}(\sigma-\sigma_{\ast})^{2}}$
$\displaystyle=e^{-N_{f}\tilde{S}(\sigma_{\ast})}\sqrt{\frac{2}{N_{f}\tilde{S}^{\prime\prime}(\sigma_{\ast})}}\int_{-\infty}^{\infty}\differential{\xi}\>\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\left(\sum_{k=0}^{\infty}c_{k}\frac{\xi^{k+3}}{N_{f}^{\frac{k+1}{2}}}\right)^{n}e^{-\xi^{2}},$
(A.3)

where $\mathcal{J}_{\ast}$ is the Lefschetz thimble associated with
$\sigma_{\ast}$ and

$\displaystyle
c_{k}=\frac{2^{\frac{k+3}{2}}}{(k+3)!}\frac{\tilde{S}^{(k+3)}(\sigma_{\ast})}{\bigl{(}\tilde{S}^{\prime\prime}(\sigma_{\ast})\bigr{)}^{\frac{k+3}{2}}}.$
(A.4)

To proceed, we also introduce

$\displaystyle\left(\sum_{k^{\prime}=0}^{\infty}c_{k^{\prime}}\epsilon^{k^{\prime}}\right)^{n}\equiv\sum_{k=0}^{\infty}\tilde{c}_{k}(n)\epsilon^{k}.$
(A.5)

Then, exchanging the integration and summation, the formal $1/N_{f}$ expansion
is computed as

$\displaystyle\ \ \ \
e^{-N_{f}\tilde{S}(\sigma_{\ast})}\sqrt{\frac{2}{N_{f}S^{\prime\prime}(\sigma_{\ast})}}\sum_{n,k=0}^{\infty}\frac{(-1)^{n}\tilde{c}_{k}(n)}{n!}\frac{1}{N_{f}^{\frac{n+k}{2}}}\int_{-\infty}^{\infty}\differential{\xi}\>\xi^{3n+k}e^{-\xi^{2}}$
$\displaystyle=e^{-N_{f}\tilde{S}(\sigma_{\ast})}\sqrt{\frac{2\pi}{N_{f}\tilde{S}^{\prime\prime}(\sigma_{\ast})}}\sum_{n,k=0,\>3n+k=\text{even}}^{\infty}\frac{(-1)^{n}\Gamma\left(\frac{1}{2}+\frac{3n+k}{2}\right)\tilde{c}_{k}(n)}{\Gamma(1/2)\Gamma(n+1)}\frac{1}{N_{f}^{\frac{n+k}{2}}}$
$\displaystyle=e^{-N_{f}\tilde{S}(\sigma_{\ast})}\sqrt{\frac{2\pi}{N_{f}\tilde{S}^{\prime\prime}(\sigma_{\ast})}}\sum_{\ell=0}^{\infty}\frac{a_{\ell}}{N_{f}^{\ell}},$
(A.6)

where

$\displaystyle
a_{\ell}=\sum_{n=0}^{2\ell}\frac{(-1)^{n}\Gamma\left(\frac{1}{2}+\ell+n\right)\tilde{c}_{2\ell-n}(n)}{\Gamma(1/2)\Gamma(n+1)}.$
(A.7)

##  Appendix B Lefschetz thimble structures for larger $\arg(N_{f})$

In this appendix, we study the thimble structures for larger values of
$\theta=\arg(N_{f})$ than the one in Sec. 3.2 to understand the Stokes
phenomena more precisely. It appears that the larger $\arg(N_{f})$ region is
not directly related to the phase transition itself since originally the
parameters were real. However, this is essential to understand the Borel plane
structures as discussed in Sec. 4.

As mentioned in the last of Sec. 3.3, for non-small $\arg(N_{f})$, we have to
take the effects of the branch cuts into account. Namely, when thimbles cross
the branch cuts, the action is shifted by $2\pi iN_{f}\mathbb{Z}$, and this
effect modifies the condition for having Stokes phenomena. We will see soon
that the Stokes phenomena due to this effect indeed occurs in this problem.

In Fig. 8, we summarize the Lefschetz thimble structures for
$(\lambda,m)=(0.4,1)$ with various $-\pi<\arg(N_{f})<0$ as a representative of
the subcritical region $\lambda<\lambda_{\text{c}}$. Reflecting these figures
along the vertical axis corresponds to flipping the sign of $\arg(N_{f})$.
Thus, these figures practically cover the full region
$-\pi\leq\arg(N_{f})<+\pi$. In the figures on the left side, Stokes phenomena
occur. For example, at $\arg(N_{f})=-1.190$, the Lefschetz thimble associated
with a saddle $\sigma_{0}^{+}$ (blue line) passes also through other saddles
$\sigma_{n>0}^{+}$. This is nothing but a Stokes phenomenon. One can easily
check that the codition for having Stokes phenomena:

$\displaystyle\imaginary[S(\sigma_{n>0}^{+})-S(\sigma_{0}^{+})+2\pi
iN_{f}\mathbb{Z}]=0,$ (B.1)

is satisfied by $\arg(N_{f})=-1.190$.

In Fig. 9, we summarize the thimble structures for the supercritical region
$\lambda>\lambda_{\text{c}}$ with various $-\pi<\arg(N_{f})<+\pi$
(specifically $(\lambda,m)=(1.2,1)$ in the figures). In the figures on the
left side, Stokes phenomena occur. For example, at $\arg(N_{f})=-0.039$, the
Lefschetz thimble associated with a saddle $\sigma_{0}^{+}$ (blue line) passes
also through another saddle $\sigma_{1}^{-}$. This means that

$\displaystyle\imaginary[S(\sigma_{1}^{-})-S(\sigma_{0}^{+})+2\pi
iN_{f}\mathbb{Z}]=0$ (B.2)

holds at $\arg(N_{f})=-0.039$. This is also a Stokes phenomenon. These
structures are consistent with the Borel plane structures discussed in Sec. 4.

Figure 8:  Illustrations of the Lefschetz thimble structures for the
subcritical region $\lambda<\lambda_{\text{c}}$ (in these figures, $m=1$, for
which $\lambda=0.4$). Larger phases are given $-\pi\leq\theta\leq 0$ so that
we can observe Stokes phenomena (left ones) and thimble structures between
them (right ones). Reflecting figures along the imaginary axis corresponds to
flipping the sign of $\theta$.

Figure 9:  Illustrations of the Lefschetz thimble structures for the
supercritical region $\lambda>\lambda_{\text{c}}$ (in these figures, $m=1$,
for which $\lambda=0.4$). Larger phases are given $-\pi\leq\theta\leq 0$ so
that we can observe Stokes phenomena (left ones) and thimble structures
between them (right ones).

##  Appendix C Comments on the Padé-Uniformized approximation

Figure 10:  Illustrations of the Borel plane structure for the supercritical
region. In this figure, we set $m=10$ and set $\lambda=1.0\times
10^{-3}\geq\lambda_{\text{c}}=9.0\times 10^{-5}$. The left panel is obtained
by the Padé approximation, and the right panel is by the Padé-Uniformized
approximation.

In the main text, we have seen that the Padé approximation becomes worse due
to branch cuts. The branch cut singularities were associated with a saddle
$\sigma_{0}^{-}$ both in the subcritical and supercritical regions. To make
matters worse, there are other signs of branch cuts on the Borel $t$-plane.
Fig. 10 is an example. We set $m=10$ (for which $\lambda_{\text{c}}=9.0\times
10^{-5}$) so that singularities gather around the origin and the Padé
approximation works better. We can see signs of branch cuts associated not
only with a saddle $\sigma_{0}^{-}$ but also with four other saddles
$\sigma_{\pm 1}^{\pm}$. We claim that even the Padé-Uniformized approximation
is obstructed by these branch cuts. In this appendix, we discuss this point by
studying some simple examples of applications of the Padé-Uniformized
approximation.

###  C.1 Single branch cut

Figure 11:  Comparison of the Padé and the Padé-Uniformized approximations
(25,25) in the case of a single branch cut. [Left] The Padé approximation.
[Right] The Padé-Uniformized approximation, where the branch cut is removed by
the uniformization map.

Let us start with the simplest case. Consider, for example, a function which
has a single branch cut

$\displaystyle\mathcal{B}F(t)=\frac{1}{(1-t)^{1/5}},$ (C.1)

and a uniformization map

$\displaystyle u=\psi(t)=-\ln(1-t).$ (C.2)

In Fig. 11, we compare the standard Padé approximation with the Padé-
Uniformized approximation for this example. We see that the Padé-Uniformized
approximation works well in this case.

###  C.2 Multiple branch cuts

Figure 12:  Comparison of the Padé and the Padé-Uniformized approximation
(25,25) in the case with the two branch cuts. One of the branch cuts is
removed by the uniformization map.

Next let us consider the following example with two branch cuts

$\displaystyle\mathcal{B}F(t)=\frac{1}{(1-t)^{1/5}(1+t)^{1/5}}.$ (C.3)

The result is shown in Fig. 12. We see that the approximation becomes worse.
In the Padé-Uniformized approximation (right panel), we can see that there is
a pair of singularities above and below the negative real axis. This is an
artifact due to the uniformization map as explained below. The $n$-th Riemann
sheet of the Borel $t$-plane is sent to a region

$\displaystyle-n\pi<\imaginary u<+n\pi$ (C.4)

by the uniformization map. The branch cut which starts from the singularity
$t=-1$ is sent to

$\displaystyle\real u<-\ln 2,\quad\imaginary u=2\pi n.$ (C.5)

Then, the Padé-Uniformized approximation on the $t$-plane (or the standard
Padé approximation on the $u$-plane) resembles singularities at

$\displaystyle u=-\ln 2+2\pi in.$ (C.6)

Ideally, all of which are sent back to the same point $t=-1$ by the inverse
map

$\displaystyle t=\phi(u)=1-e^{-u}.$ (C.7)

However, since the approximation becomes worse away from the origin $u=0$,
found singularities (particularly with $\absolutevalue{n}>0$) are not sent
back exactly to $t=-1$. As a result, the Padé-Uniformized approximation
returns multiple singularities around $t=-1$. The pair of singularities in the
right panel of Fig. 12 corresponds to $n=\pm 1$. Other pairs of singularities
which correspond to larger $\absolutevalue{n}$ are missing simply because they
are too far away from the origin $u=0$.

Figure 13:  Comparison of the Padé and the Padé-Uniformized approximation
(25,25) in the case of two branch cuts and two poles. One of the branch cuts
is removed by the uniformization map.

Such artifacts cause trouble since they are indistinguishable from other
genuine singularities. Let us consider, for example, the following function

$\displaystyle\mathcal{B}F(t)=\frac{1}{(1-t)^{1/5}(1+t)^{1/5}}\frac{1}{(i-t)(i+t)}.$
(C.8)

The result of the Padé(-Uniformized) approximation is shown in Fig. 13. The
two poles $t=\pm i$ are sent to

$\displaystyle u=-\ln(1\mp i)$ (C.9)

while the branch cut singularities $t=\pm 1$ are sent to

$\displaystyle u=-\ln 2+2\pi in.$ (C.10)

These two singularities $u=-\ln 2+2\pi in$ are further away from the poles
$u=-\ln(1\mp i)$. Then, the approximation for the two singularities is
disturbed by the poles. As a result, the Padé-Uniformized approximation
returns the poles which originate in $t=\pm i$, but with much worse artifacts
which originate in $t=\pm 1$. Indeed, in the right panel of Fig. 13, the pair
of artifacts is indistinguishable from the genuine poles. We claim that some
of singularities found in Sec. 4.1 are these types of artifacts.

###  C.3 On elimination of multiple branch cuts

Figure 14:  Comparison of the Padé and the Padé-Uniformized approximation
(25,25) in the case of two branch cuts. The two singularities $t=\pm 1$ are
mapped to $u=\pm\infty$. The multiple branch cuts are eliminated by the map.

One possible way to avoid such multiple branch cuts is to consider a map

$\displaystyle u=\psi(t)=-\ln(1-t)+\ln(1+t),$ (C.11)

which sends the singularities at $t=\pm 1$ to $u=\pm\infty$. However, this map
causes other branch cuts on the $u$-plane. As a result, the Padé-Uniformized
approximation returns a lot of artifacts along the imaginary axis as shown in
Fig. 14. It seems that the problem resides in the non-trivial topology of the
Borel $t$-plane due to multiple branch cuts.

For the above reasons, the multiple branch cuts worsen the Padé-Uniformized
approximation. Some artifacts are indistinguishable from other genuine
singularities. Also, multiple branch cuts are not eliminated simultaneously at
least by naive maps. Further improvements of the Padé-Uniformized
approximation are left for future works.

##  Appendix D Transseries for a finite $\eta$

In this appendix we derive the transseries for the finite $\eta$ from the
viewpoint of difference equation. See [170, 2] for technical details. We
consider the formal transseries satisfying the difference equation
($x=N_{f}$).

$\displaystyle Z(x+1)=f(x)Z(x)+g(x)Z(x-1),$ (D.1)

where

$\displaystyle f(x)=\frac{(2x-1)\cosh(m)}{2x\sinh^{2}(m)},\qquad
g(x)=-\frac{(x-1)^{2}+\eta^{2}}{4x(x-1)\sinh^{2}(m)}.$ (D.2)

We introduce $P(x)=Z(x-1)$, and eq.(D.1) can be written as a vectorial
expression, given by

$\displaystyle{\bf Z}(x+1)=M(x){\bf Z}(x),$ (D.3)

where ${\bf Z}(x)=(Z(x),P(x))^{\top}$, and the $2$-by-$2$ matrix $M(x)$ is
defined as

$\displaystyle M(x)=\begin{pmatrix}f(x)&g(x)\\\
1&0\end{pmatrix}\sim\Lambda\left({\mathbb{I}}+x^{-1}A\right)+O(x^{-2}),$ (D.4)
$\displaystyle\Lambda=\begin{pmatrix}\frac{\cosh(m)}{\sinh^{2}(m)}&-\frac{1}{4\sinh^{2}(m)}\\\
1&0\end{pmatrix},\qquad A=\begin{pmatrix}0&0\\\ 2\cosh(m)&-1\end{pmatrix}.$
(D.5)

The transseries structure generally depends on the form of a difference
equation and is uniquely determined from the change of asymptotic series by
acting the shift operator. See Appendix D.1 in detail. In order to obtain the
transseries based on the difference equation, one needs to diagonalize
$\Lambda$ and $A$. By using an invertible matrix $U$ to diagonalize $\Lambda$,
one finds that

$\displaystyle\hat{\Lambda}=U\Lambda U^{-1}\,=:\,{\rm
diag}(\lambda_{-},\lambda_{+}),$ (D.9)
$\displaystyle\lambda_{\pm}=\frac{\cosh(m)\pm
1}{2\sinh^{2}(m)}=\frac{1}{2(\cosh(m)\mp 1)},$
$\displaystyle\tilde{A}:=UAU^{-1}=\begin{pmatrix}-\frac{1}{2}&-\frac{1}{2}+\frac{\cosh(m)}{\cosh(m)-1}\\\
-\frac{1}{2}+\frac{\cosh(m)}{\cosh(m)+1}&-\frac{1}{2}\end{pmatrix},$
$\displaystyle
U=\begin{pmatrix}-\sinh^{2}(m)&\frac{1}{2}+\frac{\cosh(m)}{2}\\\
\sinh^{2}(m)&\frac{1}{2}-\frac{\cosh(m)}{2}\end{pmatrix},$
$\displaystyle\Rightarrow\quad$ $\displaystyle\tilde{\bf
Z}(x+1)=\tilde{M}(x)\tilde{\bf Z}(x),$ (D.10)

where $\tilde{M}(x):=UM(x)U^{-1}$ and $\tilde{\bf Z}(x):=U{\bf Z}(x)$. Next,
we consider the diagonalization for $\tilde{A}$ by employing the technique in
Appendix D.2. In order to perform it, we act $W(x)$ from the left as

$\displaystyle\check{\bf Z}(x+1)$ $\displaystyle=$
$\displaystyle\check{M}(x)\check{\bf Z}(x),\qquad\check{\bf
Z}(x):=W(x)\tilde{\bf Z}(x),$ (D.11)

where

$\displaystyle\check{M}(x)=W(x+1)\tilde{M}(x)W^{-1}(x),\qquad
W(x):={\mathbb{I}}+x^{-1}V,$ (D.12)

and

$\displaystyle{\rm diag}_{M}[\tilde{A}]=:{\rm diag}(a_{-},a_{+}),\qquad
a_{\pm}=-\frac{1}{2}.$ (D.13)

Hence,

$\displaystyle
Z(x)=\sum_{s=\\{+,-\\}}\sum_{n=0}^{\infty}\sigma_{s}c_{s,n}e^{\log\lambda_{s}\cdot
x}x^{a_{s}-n},$ (D.14)

where the transseries in terms of $x$ is completely determined but it is not
for other parameters such as $m$ and $\eta$. Hence,
$\sigma_{s}=\sigma(m,\eta),c_{s,n}=c_{s,n}(m,\eta)$ in general. Since
$c_{\pm,0}$ is relevant only to the normalization, one can take $c_{\pm,0}=1$
without loss of generality. $\sigma_{\pm}$ can be determined from the
partition function, and they are given by

$\sigma_{+}(m,\eta)=0,\qquad\sigma_{-}(m,\eta)=\sqrt{2\pi(1+\cosh m)}.$ (D.15)

Notice that $c_{\pm,n>0}$ are recursively determined from the difference
equation.

###  D.1 Properties of the shift operator $T$

We define the shift operator $T$ as $T[f(x)]=f(x+1)$. It satisfies the
following homomorphic properties for summation and multiplication:

$\displaystyle T[f(x)+g(x)]=T[f(x)]+T[g(x)],$ (D.16) $\displaystyle
T[f(x)\cdot g(x)]=T[f(x)]\cdot T[g(x)].$ (D.17)

The action of $T$ to transmonomials gives

$\displaystyle T[x^{-1}]=\frac{1}{x+1}=\sum^{\infty}_{n=0}(-1)^{n}x^{-1-n},$
(D.18) $\displaystyle
T[x^{a}]=(x+1)^{a}=x^{a}(1+x^{-1})^{a}=x^{a}\sum^{\infty}_{n=0}\frac{a\cdot(a-1)\cdots(a-n+1)}{n!}\,x^{-n},$
(D.19) $\displaystyle T[e^{-\mu x}]=e^{-\mu(x+1)}=e^{-\mu}\cdot e^{-\mu x},$
(D.20) $\displaystyle
T[\log(x)]=\log(x+1)=\log(x)+\log(1+x^{-1})=\log(x)+\sum_{n=1}^{\infty}\frac{(-1)^{n+1}x^{-n}}{n}.$
(D.21)

Notice that when $g(x+1)=g(x)$, the action of $T$ to $g(x)$ gives the identity
map. As one can see easily, for example, the below type of transseress is
closed under action of the shift operator $T$:

$\displaystyle\sum_{n,k=0}^{\infty}e^{-n\log\mu}c_{n,k}x^{a-k}\
\xrightarrow[]{\ \ T\ \ }\
\sum_{n,k=0}^{\infty}e^{-n\log\mu}c^{\prime}_{n,k}x^{a-k}.$ (D.22)

The type of transseries and the propagation of integration constants are
determined by the form of difference equation such as (non)linearity,
(non)autonomous, and so on.

###  D.2 Diagonalization of $M(x)$

Suppose $f(x)$ is a transseries determined by a difference equation having a
form $f(x+1)=M(x)f(x)$, where $M(x)$ is a function satisfying
$M(x)\sim\Lambda(1+x^{-1}A)+O(x^{-2})$ with a constant $\Lambda$ and $A$.
Assume that $\Lambda$ is positive. If the transseries includes $e^{-\mu
x}x^{a}$, since the action of $T$ gives

$\displaystyle T[e^{-\mu x}x^{a}]=e^{-\mu}\cdot e^{-\mu
x}x^{a}\left(1+ax^{-1}+O(x^{-2})\right),$ (D.23)

hence, $\mu$ and $a$ are determined from $\Lambda$ and $A$, respectively, as

$\displaystyle\mu=-\log(\Lambda),\qquad a=A.$ (D.24)

We would extend $f(x)$ to multi-variables, ${\bf
f}(x)=(f_{1}(x),\cdots,f_{N}(x))^{\top}$,

$\displaystyle{\bf f}(x+1)=M(x){\bf f}(x),$ (D.25)

where $M(x)$ is an $N$-by-$N$ matrix given by

$\displaystyle M(x)=\Lambda\left({\mathbb{I}}+x^{-1}A\right)+O(x^{-2}),$
(D.26)

with constants matrix $\Lambda$ and $A$. Assume that $\Lambda$ is
diagonalizable by an invertible matrix $U$ and that the all of eigenvalues are
positive. By acting $U$ from the left in the both sides, one can obtain that

$\displaystyle\tilde{\bf f}(x+1)=\tilde{M}(x)\tilde{\bf f}(x),$ (D.27)

where

$\displaystyle\tilde{M}(x)=\hat{\Lambda}\left({\mathbb{I}}+x^{-1}\tilde{A}\right)+O(x^{-2}),$
(D.28) $\displaystyle\hat{\Lambda}=U\Lambda U^{-1}=:{\rm
diag}(\lambda_{1},\cdots,\lambda_{N}),$ (D.29)
$\displaystyle\tilde{A}=UAU^{-1}.$ (D.30)

In order to diagonalize $\hat{\Lambda}\tilde{A}$, we redefine $\tilde{\bf
f}(x)$ and the difference equation as

$\displaystyle\check{\bf f}(x+1)=\check{M}(x)\check{\bf f}(x),$ (D.31)
$\displaystyle\check{\bf f}(x)=W(x)\tilde{\bf f}(x),$ (D.32) $\displaystyle
W(x)={\mathbb{I}}+x^{-1}V,$ (D.33)

with an $N$-by-$N$ constant matrix $V$ and $\check{M}(x)$ is given by

$\displaystyle\check{M}(x)$ $\displaystyle=$ $\displaystyle
W(x+1)\tilde{M}(x)W^{-1}(x)$ (D.34) $\displaystyle=$
$\displaystyle\left({\mathbb{I}}+(x+1)^{-1}V\right)\tilde{M}(x)\left({\mathbb{I}}+x^{-1}V\right)^{-1}$
$\displaystyle\sim$
$\displaystyle\hat{\Lambda}+x^{-1}\left([V,\hat{\Lambda}]+\hat{\Lambda}\tilde{A}\right)+O(x^{-2}),$

where $[A,B]:=AB-BA$. Notice that all the diagonal parts of $V$ can be taken
as zero and one can determine other $N(N-1)$-components of $V$ such that

$\displaystyle[V,\hat{\Lambda}]+\hat{\Lambda}\tilde{A}={\rm
diag}_{M}[\hat{\Lambda}\tilde{A}]=\hat{\Lambda}{\rm diag}_{M}[\tilde{A}],$
(D.35) $\displaystyle{\rm diag}_{M}[\tilde{A}]=:{\rm
diag}(a_{1},\cdots,a_{N}),$ (D.36)

where ${\rm diag}_{M}[\tilde{A}]$ is a diagonal matrix having diagonal parts
of $\tilde{A}$.

##  Appendix E Transseries for a finite $\lambda=\eta/N_{f}$

Let us rewrite the recursion relation (D.1) in terms of a fixed
$\lambda=\eta/N_{f}$

$\displaystyle Z(x+1)=f(x)Z(x)+g(x)Z(x-1),$ (E.1)

where

$\displaystyle f(x)=\frac{(2x-1)\cosh(m)}{2x\sinh^{2}(m)},\qquad
g(x)=-\frac{(x-1)^{2}+x^{2}\lambda^{2}}{4x(x-1)\sinh^{2}(m)}.$ (E.2)

If we ignore the extra dependence of $Z(x,\eta)$ on $x$ through
$\eta=\lambda/x$, we find that the vectorial recursion relation for ${\bf
Z}(x)=(Z(x),P(x))^{\top}$ is modified to

$\displaystyle{\bf Z}(x+1)=M(x){\bf Z}(x),$ (E.3) $\displaystyle
M(x)=\begin{pmatrix}f(x)&g(x)\\\
1&0\end{pmatrix}\sim\Gamma\left({\mathbb{I}}+x^{-1}{\cal A}\right)+O(x^{-2}),$
(E.4)
$\displaystyle\Gamma=\begin{pmatrix}\frac{\cosh(m)}{\sinh^{2}(m)}&-\frac{\lambda^{2}+1}{4\sinh^{2}(m)}\\\
1&0\end{pmatrix},\qquad{\cal A}=\begin{pmatrix}0&0\\\
\frac{2\cosh(m)}{\lambda^{2}+1}&\frac{\lambda^{2}-1}{\lambda^{2}+1}\end{pmatrix}.$
(E.5)

By diagonalizing the matrix $\Gamma$, we find two exponents
$\gamma_{1},\gamma_{2}$

$\displaystyle\tilde{M}(x)=\hat{\Gamma}\left({\mathbb{I}}+x^{-1}\tilde{\cal
A}\right)+O(x^{-2}),$ (E.6) $\displaystyle\hat{\Gamma}=U\Gamma U^{-1}=:{\rm
diag}(\gamma_{+},\gamma_{-}),$ (E.7) $\displaystyle\tilde{\cal A}=U{\cal
A}U^{-1}.$ (E.8) $\displaystyle\gamma_{\pm}=\frac{\cosh
m\pm\sqrt{1-\lambda^{2}\sinh^{2}m}}{2\sinh^{2}m}.$ (E.9)

These two exponents are degenerate when

$\displaystyle\lambda=\frac{1}{\sinh m},$ (E.10)

which agrees with the critical point $\lambda_{c}$ in (2.10) for the phase
transition.

##  Appendix F A possible relation between the Borel singularities and complex
SUSY solutions

In this appendix, we point out that a path integral interpretation of the
Borel singularities appearing the main text may be complex supersymmetric
solutions (CSS) found in [115]. It was proposed that Borel transformation of
large Chern-Simons level expansion includes the following factor

$\mathcal{B}Z(t)\supset\prod_{\rm CSS}\frac{1}{(t-S_{c})^{n_{B}-n_{F}}},$
(F.1)

where $n_{B}(n_{F})$ is the number of bosonic (fermionic) solutions. In the
SQED studied in this paper, there are two types of CSS with $n_{B}-n_{F}\neq
0$, more precisely $n_{B}-n_{F}=N_{f}$. Actions of the solutions are

$S_{\rm CSS}=N_{f}(2\pi n\lambda\mp i\lambda m),$ (F.2)

while the action at the saddle points $\sigma_{n}^{\pm}$ in the localization
formula is

$\displaystyle S_{n}^{\pm}$ $\displaystyle=$ $\displaystyle N_{f}\Biggl{[}2\pi
n\lambda-\frac{i\lambda}{2}\log{\frac{(-\lambda\cosh{m}\mp
i\Delta)(-i+\lambda)}{(-\lambda\cosh{m}\pm
i\Delta)(i+\lambda)}}+\log{\frac{\cosh{m}\pm\Delta}{1+\lambda^{2}}}\Biggr{]}.$
(F.3)

Comparing this with (F.2), we find that these actions agree when $\lambda$ is
large $\lambda\gg 1$. Thus it seems plausible that the Borel singularities
correspond to the CSS in the original path integral at least for large
$\lambda$. It would be interesting to extend the analysis in this appendix to
finite $\lambda$.

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