aainstitutetext: Département de Physique Théorique et Section de Mathématiques
Université de Genève, Genève, CH-1211 Switzerlandbbinstitutetext: Scuola
Internazionale Superiore di Studi Avanzati (SISSA)
via Bonomea 265, 34136 Trieste, Italy

# Resonances and PT symmetry in quantum curves

Yoan Emery<EMAIL_ADDRESS>a Marcos Mariño<EMAIL_ADDRESS>a,b
Massimiliano Ronzani<EMAIL_ADDRESS>
###### Abstract

In the correspondence between spectral problems and topological strings, it is
natural to consider complex values for the string theory moduli. In the
spectral theory side, this corresponds to non–Hermitian quantum curves with
complex spectra and resonances, and in some cases, to PT-symmetric spectral
problems. The correspondence leads to precise predictions about the spectral
properties of these non–Hermitian operators. In this paper we develop
techniques to compute the complex spectra of these quantum curves, providing
in this way precision tests of these predictions. In addition, we analyze
quantum Seiberg–Witten curves with PT symmetry, which provide interesting and
exactly solvable examples of spontaneous PT-symmetry breaking.

##  1 Introduction

In recent years, various dualities between string/gravity theories and gauge
theories have been proposed, in which the spacetime in the gravity side
emerges from the strong coupling dynamics of the quantum gauge theory. In
these dualities, a precise dictionary relates the geometric moduli of the
spacetime to global parameters of the quantum theory. One interesting and
somewhat puzzling aspect of these dualities is that the entries of the
dictionary have sometimes a very different nature. For example, the gauge
theory side involves global parameters which are discrete and real (like for
example the rank of the gauge group). These parameters correspond, in the
string theory side, to continuous moduli which often can take complex values.
In order to better understand these dualities, it is important to address
these hidden tensions in the dictionary.

Dualities involving topological strings have provided simplified models in
which many of these questions can be addressed in great detail. In ghm ; cgm ,
a precise correspondence was postulated between topological string theory on
toric Calabi–Yau (CY) manifolds, and quantum-mechanical systems on the real
line. This correspondence, sometimes referred to as TS/ST correspondence,
leads to highly non-trivial mathematical predictions, and can be used as an
excellent testing ground to examine the subtleties involved in the dictionary
relating string theory to quantum systems.

In the TS/ST correspondence, some of the Kähler moduli of the CY correspond to
parameters in the Hamiltonian. On the topological string side, Kähler moduli
are naturally complex (in fact, they are mapped to complex structure moduli
through mirror symmetry). However, on the spectral theory side, complex
parameters lead to non–Hermitian operators. In general, the spectral theory of
non–Hermitian operators is much more subtle than for Hermitian operators, and
a precise definition of the spectral problem is needed. Usually, one has to
perform an analytic continuation of the eigenvalue problem as the parameter
under consideration rotates into the complex plane. This often uncovers a rich
structure, as was demonstrated in the pioneering work of Bender and Wu on the
quartic oscillator bw . An important example of such a situation occurs when
there are resonances: in this case, the operator is symmetric but it does not
have conventional bound states, nor a real spectrum. For example, in non-
relativistic quantum mechanics, real potentials which are not bounded from
below, like the cubic oscillator or the inverted quartic oscillator, lead to a
resonant spectrum of complex eigenvalues.

For the operators appearing in the TS/ST correspondence, the non–Hermitian
case is largely unexplored, since one has to deal with difference operators.
We are then in a situation in which one aspect of the theory -its behavior
when the parameters become complex- is easy to understand on one side of the
duality (in this case, on the topological string theory side), but is more
difficult on the other side (in this case, in the spectral theory side). One
can then use the topological string to obtain predictions for the spectral
theory problem in the non–Hermitian case. One such prediction, already noted
in cgum , is that we will have generically an infinite discrete spectrum of
complex eigenvalues, which can be easily computed on the string theory side by
analytic continuation. The analysis of cgum  focused on real values of the
parameters leading to resonances. However, the predictions from topological
string theory were not verified on the spectral theory side111In this paper we
consider the case in which the mass parameters become complex, while $\hbar$
remains real. The case in which $\hbar$ is complex was studied in gm17 ;
kaserg ..

The first goal of this paper is to develop methods to compute this spectrum of
complex eigenvalues, directly in spectral theory. In conventional quantum
mechanics there are two different methods to compute such a spectrum. The
first method is based on complex dilatation techniques (see kp  for an
introductory textbook presentation). The second method is based on the Borel
resummation of perturbative series, and it has been used mostly to compute
resonances of anharmonic oscillators. This second method is not always
guaranteed to work, since there might be additional contributions to the
resonant eigenvalues from non-perturbative sectors alvarez-casares2 . However,
it leads to correct results in many cases (like for example for the resonances
of the cubic oscillator, as proved in calicetiodd ). The results of gu-s
suggest that, in the case of quantum curves, the Borel resummation of the
perturbative series leads to the correct spectrum without further ado. We will
consider both methods in some detail and, as we will see, we will obtain
results in perfect agreement with the predictions of topological string
theory. One unconventional aspect of our analysis is that we consider
perturbative expansions around complex points in moduli space, leading to
perturbative series with complex coefficients. Many of the complex resonances
that we calculate are not obtained by lateral Borel resummation of a non
Borel-summable real series, as it happens in the conventional cubic and
quartic oscillators, but by the conventional Borel resummation of a complex
Borel-summable series.

In general, Hamiltonians with complex parameters lead to complex spectra, but
this is not always the case. A famous example is the pure cubic oscillator
with Hamiltonian

$\mathsf{H}={\mathsf{p}^{2}\over 2}+{\rm i}\mathsf{x}^{3}.$ (1.1)

Due to PT-symmetry (see e.g. bender-review ), the spectrum of this Hamiltonian
is purely real, as originally conjectured by Bessis and Zinn–Justin. In many
examples, the reality of the spectrum can be already seen in perturbation
theory. What is in fact intriguing is the reappearance of complex eigenvalues
in some PT-symmetric Hamiltonians, due to the spontaneous breaking of PT
symmetry. For example, the Hamiltonian

$\mathsf{H}={\mathsf{p}^{2}\over 2}+{\rm i}\mathsf{x}^{3}+{\rm
i}\alpha\mathsf{x}$ (1.2)

has a complex spectrum when $\alpha$ is sufficiently negative. This turns out
to be a genuinely non-perturbative effect in which small exponential terms
become dominant for a certain range of parameters delabaere-trinh ; ben-ber .
The theory of quantum curves leads to Hamiltonians with similar properties. In
this paper we focus on a particular limit of the theory of ghm ; cgm  which
was studied in detail in gm-deformed . The resulting Hamiltonians have the
form

$\mathsf{H}=\cosh(\mathsf{p})+V_{N}(\mathsf{x}),$ (1.3)

where $V_{N}(x)$ is a polynomial of degree $N$. Appropriate choices of
$V_{N}(\mathsf{x})$ lead to PT-symmetric Hamiltonians which display similar
physical phenomena. One can use the conjectural exact quantization conditions
(EQCs) predicted in gm-deformed  to study PT symmetry and its spontaneous
breaking in great detail. This study can be regarded as a precision test of
the predictions of gm-deformed , and as a new family of examples in the world
of PT-symmetric models.

This paper is organized as follows. In section 2 we study non–Hermitian
operators in the theory of quantum mirror curves. We present two basic methods
to compute complex spectra in quantum mechanics, we apply them to the
operators obtained in the theory of quantum curves, and we compare the results
to the predictions of the TS/ST correspondence. In section 3 we consider PT-
symmetric operators appearing in the theory of quantized SW curves. In
particular. we consider a deformed version of the cubic oscillator studied by
Delabaere and Trinh delabaere-trinh . We show that this oscillator displays
the same pattern of PT symmetry breaking, and we show that this is in complete
agreement with the exact quantization conditions obtained in gm-deformed  for
this system. In section 4 we conclude with some open problems for the future.
Appendix A contains some technical details on the WKB analysis of the PT
symmetric, deformed cubic oscillator.

##  2 Resonances and quantum curves

###  2.1 Resonances and non-Hermitian quantum mechanics

In conventional quantum mechanics, we typically require that operators
describing observables are symmetric and self-adjoint. There are however
important situations in which this is not the case. We might have for example
potentials which do not support bound states, but lead to resonant states
(also called Gamow vectors) and to a complex spectrum of resonances (see e.g.
delamg  for a pedagogical overview, and moiseyev  for physical applications).
More generally, one can consider operators which are not even symmetric
(involving for example complex parameters) and define in an appropriate way a
spectral problem leading to complex eigenvalues. In many interesting examples,
resonances are particular cases of this more general spectral problems. In
this paper, we will refer to the complex eigenvalues of a spectral problem as
resonances, whether they come from the resonant spectrum of a symmetric
Hamiltonian, or from the complex spectrum of a non-symmetric Hamiltonian.

The most canonical example of non-Hermitian quantum mechanics is perhaps the
quartic anharmonic oscillator, with Hamiltonian

$\mathsf{H}={\mathsf{p}^{2}\over 2}+{\mathsf{x}^{2}\over 2}+g\mathsf{x}^{4}.$
(2.1)

If $g>0$, this is an essentially self-adjoint operator on
$L^{2}({\mathbb{R}})$ with a trace-class inverse, therefore $\mathsf{H}$ has a
real, discrete spectrum. The behavior at infinity of the eigenfunctions can be
obtained with WKB estimates, and one finds

$\psi(x)\approx\exp\left[\mp{1\over 3}{\sqrt{g\over 2}}x^{3}\right],\qquad
x\rightarrow\pm\infty.$ (2.2)

In fact, the eigenfunction exponentially decrease in a wedge around the real
axis defined by

$\left|{\rm arg}(\pm x)\right|<{\pi\over 6},$ (2.3)

as shown in Fig. 1. Let us now suppose that we start rotating $g$
counterclockwise into the complex plane, with an angle $\varphi_{g}$:

$g=|g|{\rm e}^{{\rm i}\varphi_{g}}.$ (2.4)

Then, the wedges where the WKB wavefunction decreases at infinity rotate
clockwise, and are now defined by

$\left|{\rm arg}(\pm x)+{1\over 6}\varphi_{g}\right|<{\pi\over 6}.$ (2.5)

The center of these rotated regions is at an angle

$\varphi_{x}=-{\varphi_{g}\over 6}.$ (2.6)

For example, if $\varphi_{g}=\pi$, we have to rotate the wedge region an angle
$-\pi/6$, as shown in Fig. 1. We can now consider the following spectral
problem, for arbitrary complex $g$: the eigenfunctions $\psi(x)$ should be
solutions of the Schrödinger equation for the quartic potential, and satisfy

$\lim_{|x|\rightarrow\infty}\psi(x)=0$ (2.7)

inside the wedge (2.5). This defines a discrete spectrum of eigenvalues
$E_{n}(g)$, $n=0,1,2,\cdots$, for any complex $g$ bw ; simon-bw ; bender-
turbiner . Note that, when $g$ is complex, the Hamiltonian (2.1) is not even
symmetric. When $g$ is real but negative, the Hamiltonian (2.1) is symmetric
but it is not essentially self-adjoint (see e.g. hall-book ). One can however
define resonant states or Gamow vectors by imposing so-called Gamow–Siegert
boundary conditions (see delamg  for a pedagogical review). The spectrum of
resonances obtained in this way agrees with the spectrum defined above for
general complex $g$, when we specialize it to $\varphi_{g}=\pi$.

Bender and Wu bw  discovered that the functions $E_{n}(g)$ obtained in this
way display a rich analytic structure: when $|g|$ is large enough, they define
multivalued functions of the coupling with three sheets (in particular,
$E_{n}(g)$ has $6\pi$ periodicity as a function of the argument of $g$). As
$|g|$ decreases, one finds an infinite set of branch points, called Bender–Wu
branch points. Asymptotically, the branch points form a two-dimensional quasi-
lattice in the complex plane. At each of these branch points, two energy
levels coalesce. This implies in particular that one can go from one energy
level to another by a change of sheet in a single multivalued function.

A similar structure is found in the cubic oscillator with Hamiltonian

$\mathsf{H}={\mathsf{p}^{2}\over 2}+{\mathsf{x}^{2}\over 2}-g\mathsf{x}^{3}.$
(2.8)

When $g$ is real, the operator is symmetric but it is not essentially self-
adjoint calicetiodd . An appropriate definition of the spectral problem for
arbitrary complex $g$ leads again to a multivalued structure in the energy
levels, and to Bender–Wu branch points alvarezcubicbw  similar to those
obtained in bw . When $g$ is real, the complex eigenvalues obtained in this
way agree with the resonances defined by Gamow–Siegert boundary conditions.

|
---|---
Figure 1: On the left, we depict the regions in the complex $x$ plane where
the eigenfunctions of the quartic oscillator with $g>0$ decay at infinity. As
$g$ becomes complex, we can define a spectral problem by requiring that the
eigenfunctions decay at infinity in the rotated wedges (2.5) of the complex
plane. The figure on the right shows the relevant wedges for $g<0$.

In order to unveil these beautiful structures, it is important to be able to
compute the complex eigenvalues $E_{n}(g)$ from first principles in quantum
mechanics. We will consider two different computational methods. The first one
is the complex dilatation method, which is non-perturbative and is implemented
numerically by combining it with the Rayleigh–Ritz variational method. The
second one is perturbative and is based on the resummation of asymptotic
series.

Let us first describe the complex dilatation method (see e.g. kp  for a
general introduction and bender-resonance  for the original application to the
cubic oscillator). Let $\mathsf{O}(\mathsf{x},\mathsf{p})$ an operator in one-
dimensional quantum mechanics, which is a function of the Heisenberg operators
$\mathsf{x}$, $\mathsf{p}$. One considers the group of complex dilatations

$\mathsf{U}_{{\rm i}\theta}\mathsf{x}\mathsf{U}_{{\rm i}\theta}^{-1}={\rm
e}^{{\rm i}\theta}\mathsf{x},\qquad\mathsf{U}_{{\rm
i}\theta}\mathsf{p}\mathsf{U}^{-1}_{{\rm i}\theta}={\rm e}^{-{\rm
i}\theta}\mathsf{p}.$ (2.9)

In principle $\theta\in{\mathbb{C}}$. Let us now consider the “rotated”
operator,

$\mathsf{O}_{\theta}=\mathsf{U}_{{\rm i}\theta}\mathsf{O}\mathsf{U}^{-1}_{{\rm
i}\theta}=\mathsf{O}\left({\rm e}^{{\rm i}\theta}\mathsf{x},{\rm e}^{-{\rm
i}\theta}\mathsf{p}\right).$ (2.10)

In appropriate circumstances, this operator will have square-integrable
eigenfunctions with complex eigenvalues, which are independent of the precise
value of $\theta$ beyond a given threshold. A numerical approximation to the
resulting eigenvalues can be obtained by using the standard Rayleigh–Ritz
method. To do this, one chooses a basis of $L^{2}({\mathbb{R}})$ depending on
a given parameter $\omega$, which we will denote as
$\\{|\varphi_{n}(\omega)\rangle\\}_{n=0,1,\cdots}$. A convenient choice is the
basis of eigenfunctions of the harmonic oscillator with normalized mass $m=1$
but depending on the frequency $\omega$. Then, one considers the matrix

$\langle\varphi_{n}(\omega)|\mathsf{O}_{\theta}|\varphi_{m}(\omega)\rangle,\qquad
n,m=0,1,2,\cdots$ (2.11)

One can truncate this matrix at a given level $N$, and calculate its
eigenvalues $\lambda_{n}(N,\theta,\omega)$, $n=0,\cdots,N$. For $\theta$
beyond the threshold, one has

$\lim_{N\to\infty}\lambda_{n}(N,\theta,\omega)=\lambda_{n},$ (2.12)

which are the complex eigenvalues one is looking for. In addition, by changing
the value of the angle $\theta$ one can explore the multivalued structure of
the energy levels. In practice, one calculates the truncated versions
$\lambda_{n}(N,\theta,\omega)$, which depend on the size of the matrix, the
threshold angle, and the variational parameter $\omega$. In many cases
precision is dramatically improved, for a given truncation level $N$, by a
judicious choice of $\omega$. Different criteria have been proposed in the
literature to determine the optimal value of $\omega$. For example, one
criterion is to extremize the trace of the truncated matrix, i.e. to choose an
$\omega_{\rm opt}$ such that optimal1 ; optimal2

${\partial\over\partial\omega}\sum_{n=0}^{N}\langle\varphi_{n}(\omega)|\mathsf{O}_{\theta}|\varphi_{n}(\omega)\rangle\biggl{|}_{\omega=\omega_{\rm
opt}}=0.$ (2.13)

This criterion turns out to be very efficient in conventional quantum
mechanics. In the study of resonances for quantum mirror curves, the
stationarity condition is more difficult to implement, since there are
typically many values of $\omega$ satisfying (2.13) and one has to do a
careful study to determine the best choice of variational parameter.

Another method to compute resonances in quantum mechanics is to perform Borel
resummations of perturbative series for energy levels. In many cases, the
energy of the $n$-th level of a quantum-mechanical system has an asymptotic
expansion of the form,

$E_{n}(\lambda)\sim
b(\lambda)+\varphi(\lambda),\qquad\varphi(\lambda)=\sum_{k\geq
0}a_{k}(n)\lambda^{k}.$ (2.14)

Here, $\lambda$ is a small control parameter (it could be a coupling constant
or the Planck constant $\hbar$), $b(\lambda)$ is a $\lambda$-dependent
constant, and the coefficients $a_{k}(n)$ diverge factorially, for large $k$
and fixed $n$,

$|a_{k}(n)|\sim k!,\qquad k\gg 1.$ (2.15)

One can then define the Borel transform of the formal power series above as

$\widehat{\varphi}(\zeta)=\sum_{k\geq 0}{a_{k}(n)\over k!}\zeta^{k}.$ (2.16)

The Borel resummation of the original power series is then given by

$s(\varphi)(\lambda)=\int_{0}^{\infty}{\rm
e}^{-\zeta}\widehat{\varphi}(\lambda\zeta)\,{\rm
d}\zeta=\lambda^{-1}\int_{0}^{\infty}{\rm
e}^{-\zeta/\lambda}\widehat{\varphi}(\zeta)\,{\rm d}\zeta.$ (2.17)

If this integral is well-defined, we say that the original series is Borel
summable. A typical obstruction to the existence of this integral is the
presence of singularities of the Borel transform along the positive real axis.
When this is the case, one can still define a generalized Borel resummation
along the direction $\theta$, as follows

$s_{\theta}(\varphi)(\lambda)=\int_{0}^{{\rm e}^{{\rm i}\theta}\infty}{\rm
e}^{-\zeta}\widehat{\varphi}(\lambda\zeta)\,{\rm d}\zeta.$ (2.18)

In particular, if $\theta=\pm\delta$, with $0<\delta\ll 1$, the integration
contour in (2.18) can be deformed to a contour ${\cal C}_{\pm}$ just above or
below the positive real axis, respectively. The resulting generalized Borel
resummations are called lateral Borel resummations and denoted by

$s_{\pm}(\varphi)(\lambda)=\lambda^{-1}\int_{{\cal C}_{\pm}}{\rm d}\zeta\,{\rm
e}^{-\zeta/\lambda}\widehat{\varphi}(\zeta).$ (2.19)

In favorable cases, (lateral) Borel resummations of the perturbative series
make it possible to recover the exact eigenvalue or resonance
$E_{n}(\lambda)$. A well-known example is the cubic oscillator with
Hamiltonian (2.8). The energy levels have an asymptotic expansion of the form
(2.14), where the control parameter is $\lambda=g^{2}$. When $g$ is real, the
resulting series is not Borel summable, but the lateral Borel resummations are
well-defined and they give precisely the resonances of the cubic oscillator
(see e.g. alvarez-cubic ; calicetiodd ). In some cases, however, one has to
supplement the perturbative series with a full trans-series including
exponentially small corrections. This happens for example in the resonances of
the cubic oscillator when ${\rm Im}(g)<0$ alvarez-casares2 .

In order to build up the perturbative series (2.14), one typically has to
write the Hamiltonian as a perturbation of a harmonic oscillator. In the case
of resonances, this oscillator might have a complex frequency, and even if the
original Hamiltonian has real coefficients, one obtains in this way a
perturbative series with complex coefficients. When this series is Borel
resummable, one can use conventional Borel resummation to obtain a complex,
resonant eigenvalue. An elementary example of such a situation is the
Hamiltonian

$\mathsf{H}={\mathsf{p}^{2}\over 2}-\mathsf{x}-g^{2}{\mathsf{x}^{3}\over
3},\qquad[\mathsf{x},\mathsf{p}]={\rm i}\hbar.$ (2.20)

We can write it as a perturbed harmonic oscillator by introducing the operator

$\mathsf{z}=g^{1/2}\left(\mathsf{x}-{{\rm i}\over g}\right),$ (2.21)

so that

$\mathsf{H}=-{2{\rm i}\over 3g}+\mathsf{H}_{2}-{g^{1/2}\over
3}\mathsf{z}^{3},\qquad[\mathsf{z},\mathsf{p}]={\rm i}g^{1/2}\hbar.$ (2.22)

Here,

$\mathsf{H}_{2}={\mathsf{p}^{2}\over 2}-{\rm i}\mathsf{z}^{2}$ (2.23)

is the Hamiltonian of a harmonic oscillator with frequency

$\omega^{2}_{c}=-2{\rm i}.$ (2.24)

We can now use standard perturbation theory to write down an asymptotic
expansion for the energy levels of this Hamiltonian. For the ground state
energy we find, by using the BenderWu package bwpack ,

$E_{0}(g)\sim-{2{\rm i}\over 3g}+{1-{\rm i}\over 2}+{11g\over 288}-{155\over
27648}(1-{\rm i})g^{2}+\cdots,$ (2.25)

where we set for simplicity $\hbar=g^{-1/2}$. This series is Borel summable
along the real axis and it can be easily resummed by using Borel–Padé
techniques. One finds for example, with high precision,

$E_{0}(1)=0.533167708417867457252...-1.170055989168924629250...{\rm i},$
(2.26)

in agrement with a calculation using complex dilatation techniques. Let us
note that, when we have a resonance, the sign of the imaginary part of the
eigenvalue is not uniquely defined, since it depends on an underlying choice
of branch cut. In the case of the resonances of the cubic oscillator (2.8),
this choice corresponds to the different lateral Borel resummations of a non-
Borel resummable series. In the example above, what determines the sign is the
choice of square root in the frequency of the oscillator (2.24).

As emphasized in gmz , there is no guarantee that the resummation of a Borel-
resummable series reproduces the exact value that one is after. For example,
the perturbative WKB series of the PT-symmetric cubic oscillator (1.2) (which
diverges like $(2k)!$) is Borel resummable, but its resummation misses
important non-perturbative effects delabaere-trinh  (see ims  for a detailed
illustration). However, the perturbative series for the energy levels obtained
in stationary perturbation theory (which typically diverge like $k!$) are
often Borel summable to the right eigenvalues. The same phenomenon has been
observed for quantum mirror curves in the case of conventional bound states
with real energies gu-s 222As shown in serone , even when the perturbative
series is not Borel summable, one can often deform the Hamiltonian so as to
obtain a Borel summable series which leads to the correct spectrum..

###  2.2 Quantum mirror curves

In this section we will focus on the spectral theory of the operators obtained
by quantization of mirror curves. This theory was first considered in km ; hw
, and a complete conjectural description of the exact spectrum of these
operators was put forward in ghm ; cgm . The starting point is the mirror
curve $\Sigma$ of a toric CY manifold $X$, which can be written as a
polynomial in exponentiated variables:

$W_{X}({\rm e}^{x},{\rm e}^{y})=0.$ (2.27)

This equation can be written in many ways, but in order to extract the
relevant operator one should use an appropriate canonical form, as described
in ghm ; cgm . The mirror curve depends on the complex structure moduli of the
mirror CY. It turns out that these moduli are of two types hkp ; hkrs :
$g_{\Sigma}$ “true” moduli, where $g_{\Sigma}$ is the genus of the curve, and
a certain number of mass parameters $\xi_{k}$, $k=1,\cdots,r_{\Sigma}$,
depending on the CY $X$. The quantization of the curve (2.27) involves
promoting $x$, $p$ to Heisenberg operators $\mathsf{x}$, $\mathsf{p}$ on
${\mathbb{R}}$, with the standard commutation relation

$[\mathsf{x},\mathsf{p}]={\rm i}\hbar.$ (2.28)

In this way, one obtains an operator $\mathsf{O}_{X}$ on $L^{2}({\mathbb{R}})$
(when $g_{\Sigma}>1$, one obtains in fact $g_{\Sigma}$ different operators, as
explained in cgm , but for simplicity we will focus for the moment being on
the case $g_{\Sigma}=1$). The mass parameters appear as coefficients in the
operator $\mathsf{O}_{X}$, and its spectral properties depend crucially on
their values, and in particular on their reality and positivity properties.
There is a range of values of the parameters in which $\mathsf{O}_{X}$ is
self-adjoint and its inverse

$\rho_{X}=\mathsf{O}_{X}^{-1}$ (2.29)

is of trace class. Outside this range, the operator obtained in this way is no
longer self-adjoint.

For concreteness, let us consider two operators which will be our main focus
in this section. The first operator is associated to the toric CY known as
local ${\mathbb{F}}_{0}$, and it reads:

$\mathsf{O}_{{\mathbb{F}}_{0}}={\rm e}^{\mathsf{p}}+{\rm e}^{-\mathsf{p}}+{\rm
e}^{\mathsf{x}}+\xi_{{\mathbb{F}}_{0}}{\rm e}^{-\mathsf{x}}.$ (2.30)

This operator involves a mass parameter $\xi_{{\mathbb{F}}_{0}}$. The operator
is self-adjoint and has a trace class inverse as long as
$\xi_{{\mathbb{F}}_{0}}>0$ kama . A closely related example is the operator
associated to the local ${\mathbb{F}}_{2}$ geometry,

$\mathsf{O}_{{\mathbb{F}}_{2}}={\rm e}^{\mathsf{x}}+{\rm e}^{\mathsf{p}}+{\rm
e}^{-2\mathsf{x}-\mathsf{p}}+\xi_{{\mathbb{F}}_{2}}{\rm e}^{-\mathsf{x}}.$
(2.31)

This operator is self-adjoint and it has a trace class inverse as long as
$\xi_{{\mathbb{F}}_{2}}>-2$ kmz . In fact, for this range of parameters, the
operator (2.31) is equivalent to the operator (2.30), in the following precise
sense kama ; gkmr . Let $\lambda_{{\mathbb{F}}_{0}}$ be an eigenvalue of the
operator $\mathsf{O}_{{\mathbb{F}}_{0}}$ with parameter
$\xi_{{\mathbb{F}}_{0}}$. Consider now the operator
$\mathsf{O}_{{\mathbb{F}}_{2}}$ with parameter

$\xi_{{\mathbb{F}}_{2}}=\xi_{{\mathbb{F}}_{0}}^{1/2}+\xi_{{\mathbb{F}}_{0}}^{-1/2}.$
(2.32)

Then, the eigenvalues $\lambda_{{\mathbb{F}}_{2}}$ of
$\mathsf{O}_{{\mathbb{F}}_{2}}$ are given by

$\lambda_{{\mathbb{F}}_{2}}=\xi_{{\mathbb{F}}_{0}}^{-1/4}\lambda_{{\mathbb{F}}_{0}}.$
(2.33)

What happens when the mass parameters are outside the range in which the
operator is self-adjoint and has a trace-class inverse? Let us consider for
example (2.30). When $\xi_{{\mathbb{F}}_{0}}<0$, we are in a situation similar
to the quartic oscillator with negative coupling, in the sense that the
operator is symmetric but does not support bound states. As in conventional
quantum mechanics, it should be possible to perform an analytic continuation
of the eigenvalue problem leading to a discrete spectrum of resonant states,
with complex eigenvalues. This should be also the case for more general,
complex values of $\xi_{{\mathbb{F}}_{0}}$. We expect to have a similar
situation for the operator (2.31) when $\xi_{{\mathbb{F}}_{2}}<-2$. There are
two indications that this is in fact what happens, as discussed in cgum . The
first one is that, in some cases, the spectral traces of the operators (2.30),
(2.31) can be computed in closed form as functions of $\xi_{{\mathbb{F}}_{0}}$
or $\xi_{{\mathbb{F}}_{2}}$. For example, one has, for $\hbar=2\pi$,

${\rm Tr}\,\rho_{{\mathbb{F}}_{0}}={1\over
8\pi}{\log(\xi_{{\mathbb{F}}_{0}})\over\xi_{{\mathbb{F}}_{0}}^{1/2}-1}.$
(2.34)

This function of $\xi_{{\mathbb{F}}_{0}}$ can be analytically continued to the
complex plane of the mass parameter, where it displays a branch cut along the
negative real axis. The second indication is that the TS/ST correspondence
predicts a discrete spectrum of complex resonances for the operators
$\mathsf{O}_{X}$, for any complex value of the mass parameters, as explained
in cgum . This spectrum can be computed from the conjectural form of the
spectral determinants proposed in ghm ; cgm . In the case of genus one curves,
one can also use the EQC proposed in wzh . Unfortunately, the predictions of
the TS/ST correspondence for resonant eigenvalues were not verified against
first principle calculations in spectral theory. One of the goals of the
present paper is to fill this gap by developing computational methods to
compute resonances, directly in the spectral theory.

Let us briefly review what are the predictions for the spectrum from
topological string theory (we refer the reader to the original papers ghm ;
cgm ; wzh  for more details and explanations, and to mmrev  for a review). The
spectrum of the operator $\rho_{X}$ can be found by looking at the zeros of
its spectral determinant,

$\Xi_{X}(\kappa)={\rm det}\left(1+\kappa\rho_{X}\right).$ (2.35)

When $\rho_{X}$ is trace class, this is an entire function of $\kappa$ and it
has an expansion around $\kappa=0$ given by

$\Xi_{X}(\kappa)=1+\sum_{N=0}^{\infty}Z_{X}(N)\kappa^{N},$ (2.36)

where $Z_{X}(N)$ are the fermionic spectral traces of $\rho_{X}$ and can be
computed from the conventional spectral traces ${\rm Tr}\,\rho^{\ell}_{X}$.
According to the TS/ST correspondence, these traces can be obtained as Laplace
transforms,

$Z_{X}(N)={1\over 2\pi{\rm i}}\int_{{\cal C}}{\rm e}^{\mathsf{J}_{X}(\mu)-\mu
N}{\rm d}\mu.$ (2.37)

In this equation, $\mathsf{J}_{X}(\mu)$ is the so-called grand potential of
topological string theory on $X$, which can be explicitly obtained from the
BPS invariants of $X$, as explained in ghm ; cgm ; mmrev .

When the mirror curve has genus one, the condition for the vanishing of the
spectral determinant can be written as an EQC of the form wzh ,

${rCt^{2}\over
2}+D(\boldsymbol{\xi})t+B(\boldsymbol{\xi},\hbar)+\hbar\left(f_{\rm
NS}\left(t,\boldsymbol{\xi},\hbar\right)+f_{\rm NS}\left({2\pi
t\over\hbar},\boldsymbol{\xi}^{2\pi\over\hbar},\frac{4\pi^{2}}{\hbar}\right)\right)=2\pi\hbar\left(n+{1\over
2}\right),\qquad n=0,1,2,\cdots$ (2.38)

In this equation, $\boldsymbol{\xi}=(\xi_{1},\cdots,\xi_{r_{\Sigma}})$ is the
vector of mass parameters,
$\boldsymbol{\xi}^{2\pi\over\hbar}=(\xi_{1}^{2\pi\over\hbar},\cdots,\xi_{r_{\Sigma}}^{2\pi\over\hbar})$,
the function $B(\boldsymbol{\xi},\hbar)$ has the form,

$B(\boldsymbol{\xi},\hbar)=B\left(1+{\hbar^{2}\over
4\pi^{2}}\right)+b(\boldsymbol{\xi}),$ (2.39)

$C$, $r$, and $B$ are constant coefficients depending on the geometry under
consideration, $D(\boldsymbol{\xi})$ and $b(\boldsymbol{\xi})$ are functions
of the mass parameters, and $t$ is related to the true modulus of the curve,

$\kappa={\rm e}^{E},$ (2.40)

through the so-called quantum mirror map acdkv :

$t=t(E,\boldsymbol{\xi},\hbar).$ (2.41)

The function $f_{\rm NS}(t,\boldsymbol{\xi},\hbar)$ can be expressed in terms
of the Nekrasov–Shatashvili (NS) limit of the refined topological string free
energy ns , which we will call NS free energy and will denote by $F_{\rm
NS}(t,\boldsymbol{\xi},\hbar)$. The precise relation is,

$f_{\rm NS}(t,\boldsymbol{\xi},\hbar)=r{\partial F_{\rm NS}^{\rm
inst}\over\partial t}.$ (2.42)

The superscript indicates that we only keep the “instanton” part of the NS
free energy, which can be computed by knowing the BPS invariants of the CY
$X$. We also recall that

$F_{\rm
NS}(t,\boldsymbol{\xi},\hbar)={1\over\hbar}F_{0}(t,\boldsymbol{\xi})+{\cal
O}(\hbar),$ (2.43)

where $F_{0}(t,\boldsymbol{\xi})$ is the genus zero free energy of the toric
CY in the large radius frame,

$F_{0}(t,\boldsymbol{\xi})={C\over 6}t^{3}+D(\boldsymbol{\xi})t+F_{0}^{\rm
inst}(t,\boldsymbol{\xi}).$ (2.44)

The equation (2.38) determines energy levels $E_{n}$, $n=0,1,\cdots$, for the
modulus $E$, while the eigenvalues of $\rho_{X}$ are given by ${\rm
e}^{-E_{n}}$.

In the following, it will be useful to understand the operators obtained by
quantization of mirror curves from the point of view of perturbation theory.
To this end, we expand them around the minimum $(x_{0},p_{0})$ of the
corresponding function. In this way we will make sure that they can be
regarded as perturbed harmonic oscillators. By doing this, we shift

$\mathsf{x}\rightarrow\hbar^{1/2}\mathsf{x}-x_{0},\qquad\mathsf{p}\rightarrow\hbar^{1/2}\mathsf{p}-p_{0},$
(2.45)

so that the operator is written as

$\mathsf{O}=s(\xi)\mathsf{S},$ (2.46)

where $s(\xi)$ is a function of the mass parameter, and the operator
$\mathsf{S}$ can be expanded as gu-s

$\mathsf{S}=f(\xi)+{\hbar\over
2}\left(A(\xi)\mathsf{x}^{2}+B(\xi)\mathsf{p}^{2}+C(\xi)(\mathsf{x}\mathsf{p}+\mathsf{p}\mathsf{x})\right)+{\cal
O}(\hbar^{3/2}),$ (2.47)

where $f(\xi)$ is a $c$-number. One can now perform a linear canonical
transformation to eliminate the cross term
$\mathsf{x}\mathsf{p}+\mathsf{p}\mathsf{x}$, and find in this way a harmonic
oscillator with frequency

$\omega^{2}_{c}=A(\xi)B(\xi)-C^{2}(\xi).$ (2.48)

The extended version of the BenderWu program due to gu-s  makes it possible to
calculate a formal series expansion in powers of $\hbar$ for the eigenvalues
of $\mathsf{O}$:

${1\over s(\xi)}{\rm e}^{E_{n}}\sim f(\xi)+{\hbar\omega_{c}\over
2}\left(n+{1\over 2}\right)+{\cal O}(\hbar^{2}),\qquad n=0,1,\cdots$ (2.49)

In appropriate conditions, the Borel resummation of this series should give
the correct value of the (in general complex) eigenvalue $E_{n}$.

###  2.3 Resonances in local ${\mathbb{F}}_{0}$

The TS/ST correspondence of ghm ; cgm  gives predictions for the spectrum of
non-Hermitian Hamiltonians in which the mass parameters take complex values.
In principle, this spectrum can be found directly in operator theory, by first
defining an appropriate analytic continuation of the spectral problem, and
then computing this spectrum with an appropriate approximation technique. In
this paper we will assume that the analytic continuation can be performed
rigorously, and then we will determine the resulting spectrum with both
complex dilatation techniques and Borel resummation techniques.

Let us start the discussion with the operator (2.30) associated to local
${\mathbb{F}}_{0}$. As explained above, when $\xi_{{\mathbb{F}}_{0}}<0$, and
more generally for complex $\xi_{{\mathbb{F}}_{0}}$, we expect to find a
resonant spectrum. Indeed, if the EQC (2.38) holds also in this regime, it
leads to complex eigenvalues. In this example, the parameters and functions
appearing in the EQC (2.38) are given by,

$r=2,\qquad C=1,\qquad
D(\xi_{{\mathbb{F}}_{0}})=-\log(\xi_{{\mathbb{F}}_{0}}),\qquad
B={2\pi^{2}\over 3},\qquad b(\xi_{{\mathbb{F}}_{0}})=0.$ (2.50)

The quantum mirror map has the structure

$t(E,\xi_{{\mathbb{F}}_{0}},\hbar)=2E-2(1+\xi_{{\mathbb{F}}_{0}}){\rm
e}^{-2E}+{\cal O}\left({\rm e}^{-4E}\right).$ (2.51)

Therefore, the EQC involves both $\xi_{{\mathbb{F}}_{0}}$ and
$\log(\xi_{{\mathbb{F}}_{0}})$, which appears in the function $D(\xi)$ and in
the argument of $f_{\rm NS}$:

$\xi_{{\mathbb{F}}_{0}}^{2\pi\over\hbar}=\exp\left({2\pi\over\hbar}\log(\xi_{{\mathbb{F}}_{0}})\right).$
(2.52)

When $\xi_{{\mathbb{F}}_{0}}<0$, this leads to an imaginary part in $E_{n}$.

Precise predictions about the resonant spectrum can be also found if we take
into account the relationship (2.32), (2.33) between
$\mathsf{O}_{{\mathbb{F}}_{0}}$ and $\mathsf{O}_{{\mathbb{F}}_{2}}$. In
spectral theory, this relation is derived in kmz  when both operators are of
trace class, but since the same relationship can be derived for the mass
parameters of topological string theory gkmr , we expect it to hold as well
for complex values of $\xi_{{\mathbb{F}}_{0}}$, $\xi_{{\mathbb{F}}_{2}}$, once
the analytic continuation is properly implemented. One easy consequence of
this relation is the following: if

$\xi_{{\mathbb{F}}_{0}}={\rm e}^{{\rm i}\phi},\qquad\phi\in{\mathbb{R}},$
(2.53)

we have

$\xi_{{\mathbb{F}}_{2}}=2\cos(\phi),$ (2.54)

so that $-2\leq\xi_{{\mathbb{F}}_{2}}\leq 2$ and the spectrum of
$\mathsf{O}_{{\mathbb{F}}_{2}}$ is real. It follows from (2.33) that

$\xi^{-1/4}_{{\mathbb{F}}_{0}}{\rm e}^{E_{n}(\xi_{{\mathbb{F}}_{0}})}$ (2.55)

should be real. In other words, for complex values of $\xi_{{\mathbb{F}}_{0}}$
in the unit circle of the complex plane, i.e. of the form (2.53), we expect

${\rm Im}\left(E_{n}\right)={\phi\over 4},$ (2.56)

independently of $n$ and of $\hbar$.

The appearance of complex resonances is also natural when we consider the
operator (2.30) from the point of view of perturbation theory. After expanding
around the minimum

$p_{0}=0,\qquad x_{0}={1\over 2}\log\xi_{{\mathbb{F}}_{0}},$ (2.57)

we find the operator

$\mathsf{O}_{{\mathbb{F}}_{0}}={\rm e}^{\mathsf{p}}+{\rm
e}^{-\mathsf{p}}+\xi_{{\mathbb{F}}_{0}}^{1/2}\left({\rm e}^{\mathsf{x}}+{\rm
e}^{-\mathsf{x}}\right).$ (2.58)

As already pointed out in hw , this is a perturbed harmonic oscillator of
frequency

$\omega_{c}^{2}=4\varpi^{2}=4\xi_{{\mathbb{F}}_{0}}^{1/2},$ (2.59)

where we have introduced the renormalized frequency $\varpi$. When
$\xi_{{\mathbb{F}}_{0}}<0$ this oscillator has a complex frequency, just as in
the example (2.24). By expanding around this oscillator, one can write down
perturbative series for the energy levels. For example, one finds, for the
ground state energy,

${\rm
e}^{E_{0}}=\left(2\varpi^{2}+2\right)+\varpi\hbar+\frac{1}{16}\left(\varpi^{2}+1\right)\hbar^{2}-\frac{3\varpi^{4}-10\varpi^{2}+3}{768\varpi}\hbar^{3}+{\cal
O}(\hbar^{4}).$ (2.60)

The structure (2.56) for mass parameters in the unit circle can be also
deduced from the explicit form of perturbation theory, since in the case
(2.53) the renormalized frequency $\varpi$ introduced in (2.59) is also a pure
phase, $\varpi={\rm e}^{{\rm i}\phi/4}$, and the perturbative series is of the
form $\varpi$, times a formally real series (this can be easily checked in the
first terms of (2.60)). If this real series is Borel summable, as it is the
case, the only imaginary contribution to its Borel–Padé resummation comes from
the overall factor $\varpi$. This leads to the imaginary part (2.56).

Let us now present some detailed calculations of the eigenvalues for different
values of the parameter $\xi_{{\mathbb{F}}_{0}}$ in the resonant region. We
have focused on three values of $\hbar$, namely, $2\pi$, $\pi$ and $2\pi/3$,
where the EQC can be written in a simpler form (the case $\hbar=2\pi$ is the
“self-dual” or “maximally supersymmetric case”, where the theory simplifies
considerably, as already noted in ghm ; the simplification in the cases of
$\hbar=\pi,2\pi/3$ follows from the results of szabolcs ). The first
computational method is complex dilatation, where we use the Rayleigh–Ritz
method with the harmonic oscillator basis and an optimal value of the
frequency satisfying (2.13). The second method involves the Borel–Padé
resummation of the perturbative series, and finally we obtain the prediction
for spectrum based on the EQC and the TS/ST correspondence. We will focus on
the ground state, although we have studied as well excited states.

$\hbar$ | $2\pi$ | $\pi$ | $2\pi/3$
---|---|---|---
c.d. | $2.7606757803$ | $1.97575795104945$ | $1.69010343823507799$
BP | $2.7606757803195866$ | $1.97575795104945438307691$ | $1.690103438235077991455$
EQC | $2.7606757803195866$ | $1.97575795104945438307691$ | $1.690103438235077991$
Table 1: Real part of the first resonance for local ${\mathbb{F}}_{0}$ with
$\log\xi_{{\mathbb{F}}_{0}}=\pi{\rm i}$ and three different values of $\hbar$.
Here, c.d. denotes the complex dilatation method, while BP denotes the
Borel–Padé method.

Let us consider first values with $|\xi_{{\mathbb{F}}_{0}}|=1$, where we
expect the imaginary part of the energy to be given by the simple expression
(2.56). A simple case is $\xi_{{\mathbb{F}}_{0}}=-1$, with the choice of
logarithmic branch given by $\log\xi_{{\mathbb{F}}_{0}}=\pi{\rm i}$. The
imaginary part should be in this case

${\rm Im}(E_{n})={\pi\over 4}.$ (2.61)

We find that the three methods reproduce this imaginary part with arbitrary
precision. The results for the real part of the first resonant state are shown
in Table 1. In the complex dilatation method, we used matrices of rank $200$
for the Rayleigh–Ritz approximation, and in the Borel–Padé resummation we used
$150$ terms generated with the Bender–Wu program. These two methods become
more efficient for smaller values of $\hbar$. In the EQC, we have used $100$
terms in the instanton expansion. As we see, we obtain perfect agreement
between the three methods (in the case of $\hbar=2\pi/3$, the Borel–Padé
method and the EQC can be seen to agree up to $35$ digits). Similar results
can be found when $\xi_{{\mathbb{F}}_{0}}={\rm i}$, and with the choice of
logarithmic branch $\log\xi_{{\mathbb{F}}_{0}}=\pi{\rm i}/2$. The imaginary
part should be in this case

${\rm Im}(E_{n})={\pi\over 8}.$ (2.62)

We show some of the results in Table 2. Again, the imaginary part was
reproduced with arbitrary precision. In the complex dilatation method, we used
matrices of rank $200-400$ for the Rayleigh–Ritz approximation (depending on
the value of $\hbar$), and in the Borel–Padé resummation we used $100$ terms
generated with the Bender–Wu program. In the EQC, we used again $100$ terms in
the instanton expansion.

$\hbar$ | $2\pi$ | $\pi$ | $2\pi/3$
---|---|---|---
c.d. | $2.851995310270838$ | $2.111510733777354897475$ | $1.8523966583371164256660$
BP | $2.8519953102708$ | $2.111510733777354897475$ | $1.852396658337116425665972544$
EQC | $2.851995310270838$ | $2.111510733777354897475$ | $1.852396658337116425665972544$
Table 2: Real part of the first resonance for local ${\mathbb{F}}_{0}$ with
$\log\xi_{{\mathbb{F}}_{0}}=\pi{\rm i}/2$ and three different values of
$\hbar$.

We have performed tests for values of $\xi_{{\mathbb{F}}_{0}}$ which are not
of the form (2.53), and therefore the imaginary part does not have a closed
form. For example, in Table 3 we list the first resonance when
$\xi_{{\mathbb{F}}_{0}}=-4$ and $\hbar=\pi$. All methods agree up to the
numerical precision we have achieved. All these results support the conclusion
that the conjecture of ghm  (or its reformulation in wzh ) is still valid in
the complex realm, when one considers resonances.

$\hbar$ | $\pi$
---|---
c.d. | $2.36465369971376$ | $+$ | $0.94715399106287~{}{\rm i}$
BP | $2.364653699713759905588127$ | $+$ | $0.947153991062872399610128~{}{\rm i}$
EQC | $2.36465369971375990558812739733$ | $+$ | $0.947153991062872399610128166~{}{\rm i}$
Table 3: First resonance for local ${\mathbb{F}}_{0}$ with
$\log\xi_{{\mathbb{F}}_{0}}=2\log(2)+\pi{\rm i}$ and $\hbar=\pi$.

The operator corresponding to local ${\mathbb{F}}_{2}$, given in (2.31), can
be analyzed very similarly to local ${\mathbb{F}}_{0}$. In this case, the
extremum of the underlying function occurs at

$x_{0}=-p_{0}={1\over 2}\log\left(\xi_{{\mathbb{F}}_{2}}+2\right),$ (2.63)

and when we expand around this value we obtain

$\mathsf{O}_{{\mathbb{F}}_{2}}={\sqrt{\xi_{{\mathbb{F}}_{2}}+2}}\,\mathsf{S}_{{\mathbb{F}}_{2}},$
(2.64)

where

$\mathsf{S}_{{\mathbb{F}}_{2}}={\rm
e}^{\mathsf{x}}+{1\over\xi_{{\mathbb{F}}_{2}}+2}{\rm
e}^{\mathsf{p}}+{1\over\xi_{{\mathbb{F}}_{2}}+2}{\rm
e}^{-2\mathsf{x}-\mathsf{p}}+{\xi_{{\mathbb{F}}_{2}}\over\xi_{{\mathbb{F}}_{2}}+2}{\rm
e}^{-\mathsf{x}}.$ (2.65)

This operator is a perturbed harmonic oscillator with frequency

$\omega_{c}^{2}={4\over 2+\xi_{{\mathbb{F}}_{2}}}.$ (2.66)

The lack of trace class property for $\xi_{{\mathbb{F}}_{2}}<-2$ is obvious
from this description: the minimum (2.63) moves to the complex plane, and the
frequency of the oscillator becomes imaginary. We have studied the spectrum of
resonant states for various values of $\xi_{{\mathbb{F}}_{2}}$. Numerically,
this operator leads to less precision than (2.30), but we have obtained again
full agreement between the different methods (complex dilatation, Borel
resummation and exact quantization conditions).

###  2.4 Resonances in perturbed local ${\mathbb{P}}^{2}$

In cgum  the following operator was considered,

$\mathsf{O}_{1,1,\xi}={\rm e}^{\mathsf{x}}+{\rm e}^{\mathsf{p}}+{\rm
e}^{-\mathsf{x}-\mathsf{p}}+\xi{\rm e}^{-2\mathsf{p}}.$ (2.67)

It can be regarded as a perturbation of the operator associated to local
${\mathbb{P}}^{2}$, by the term $\xi{\rm e}^{-2\mathsf{p}}$. It arises by
quantization of the mirror curve of the local $Y^{3,0}$ geometry. This curve
has genus two, so in order to obtain the spectrum of $\mathsf{O}_{1,1,\xi}$
one has to consider the higher genus version of the TS/ST correspondence
presented in cgm  and further studied in cgum . In particular, one has to
calculate the relevant spectral determinant, and the quantization condition
can not be put in the simple form presented in wzh . The operator (2.67) is
trace class for $\xi\geq 0$, and as explained in cgum  it is expected to
display resonances when $\xi<0$.

Some insight on the structure of this operator can be obtained by writing it
as a perturbed harmonic oscillator. In this case, the minimum of the
underlying function occurs at

${\rm e}^{x_{0}}=X_{0}={1\over
2^{1/3}}\left(1+4\xi+{\sqrt{1+8\xi}}\right)^{1/3},\qquad{\rm
e}^{p_{0}}=P_{0}={X_{0}\over 4}{{\sqrt{1+8\xi}}-1\over\xi}.$ (2.68)

If we shift $\mathsf{x}$, $\mathsf{p}$ around this minimum, we find that

$\mathsf{O}_{1,1,\xi}=X_{0}\mathsf{S}_{\xi},$ (2.69)

where

$\displaystyle\mathsf{S}_{\xi}$ $\displaystyle={\rm
e}^{\mathsf{x}}+\frac{\sqrt{8\xi+1}-1}{4\xi}{\rm
e}^{\mathsf{p}}+\frac{8\xi}{\left(\sqrt{8\xi+1}-1\right)\left(4\xi+\sqrt{8\xi+1}+1\right)}{\rm
e}^{-\mathsf{x}-\mathsf{p}}$ (2.70)
$\displaystyle+\frac{2\xi}{\left(4\xi+\sqrt{8\xi+1}+1\right)}{\rm
e}^{-2\mathsf{x}}.$

After expanding around $\mathsf{x}=\mathsf{p}=0$, we find a harmonic
oscillator whose frequency is given by

$\omega_{c}^{2}(\xi)=\frac{24\xi\left(8\xi+\sqrt{8\xi+1}+1\right)}{\left(\sqrt{8\xi+1}-1\right)\left(4\xi+\sqrt{8\xi+1}+1\right)^{2}}.$
(2.71)

When

$-{1\over 8}<\xi<0$ (2.72)

we have a perturbed harmonic oscillator with a positive frequency, and the
perturbative series for the energy levels has real coefficients. Since the
operator does not have bound states but resonances, we expect that the
perturbative series is not Borel summable, as in the conventional cubic
oscillator. For example, when $\xi=-1/9$, one finds for the perturbative
series for the ground state,

${\rm e}^{E_{0}}=\left({2\over 3}\right)^{2/3}\left({15\over 4}+{3\over
4}\hbar-{79\over 48}\hbar^{2}-{12569\over 576}\hbar^{3}-{32066045\over
62208}\hbar^{4}-\cdots\right),$ (2.73)

which has non-alternating signs and is not expected to be Borel summable. A
study of the Borel plane with Padé approximation techniques indicates indeed
the existence of a branch cut singularity along the real axis. We expect that
resonances are obtained in this case by lateral resummation of a non-Borel
summable series. In contrast, when

$\xi<-{1\over 8}$ (2.74)

the perturbative series has complex coefficients, as in the example of local
${\mathbb{F}}_{0}$ with $\xi_{{\mathbb{F}}_{0}}<0$.

In order to test the TS/ST conjecture for the resonant regime of the operator
(2.67), we have calculated the spectral traces from the Airy integral formula
(2.37), used these to calculate an approximation to the spectral determinant,
and finally determined the zeroes of this determinant to locate the
resonances. In our calculations we set $\hbar=2\pi$, we calculated the traces
up to $N=10$ and we used an approximation up to seventh instanton order for
the grand potential $\mathsf{J}_{X}$ in (2.37). The results for the first
resonance, for $\xi=-1/9$ and $\xi=-9/8$, are presented in Table 4, and
compared to results obtained by complex dilatation with a matrix of rank $800$
(only stable digits are shown, as usual). Although the precision is smaller
than the one achieved in the simpler cases of genus one, we still find full
agreement between the predictions of the TS/ST correspondence and a first-
principles computation of the resonant spectrum. The result for $\xi=-9/8$ can
be also reproduced by using the Borel–Padé resummation of the perturbative
series. In the case of $\xi=-1/9$ we have to use lateral resummation and the
standard Borel–Padé method does not converge well, due to the high value of
$\hbar$. It would be interesting to improve on this to obtain these resonances
by resummation techniques.

$\xi$ | $-1/9$ | $-9/8$
---|---|---
c.d. | $17.6475073$ | $+$ | $4.7738402~{}{\rm i}$ | $21.4121476$ | $+$ | $8.3088781~{}{\rm i}$
EQC | $17.647507$ | $+$ | $4.77384~{}{\rm i}$ | $21.41215$ | $+$ | $\,8.30888~{}{\rm i}$
Table 4: Value of ${\rm e}^{E_{0}}$ for the operator (2.67) and $\hbar=2\pi$,
for $\xi=-1/9$ (left) and $\xi=-9/8$ (right).

###  2.5 Multivalued structure

In conventional quantum mechanics with polynomial potentials, like (2.1) and
(2.8), the spectrum has a non-trivial multivalued dependence on the complex
values of the couplings. For example, as we mentioned in section 2, in the
case of the quartic oscillator, the energy levels have $6\pi$ periodicity as
functions of the argument of $g$, when $|g|$ is sufficiently large.

In the case of the operators obtained from mirror curves, the energy levels
depend generically on the logarithm of the mass parameters. Therefore, they
should have an infinitely sheeted structure as we consider complex values of
these parameters, corresponding to the different choices of the branches of
the complex logarithm. In the case of local ${\mathbb{F}}_{0}$, for example,
this is strongly suggested by exact trace formulae like (2.34), by the
structure of the EQC (see e.g. the comment around eq. (2.52)), and also by the
equivalence (2.32) between local ${\mathbb{F}}_{0}$ and ${\mathbb{F}}_{2}$.
Indeed, in order to obtain values of $\xi_{{\mathbb{F}}_{2}}$ in the resonant
region, i.e. $\xi_{{\mathbb{F}}_{2}}<-2$, one should consider values of
$\xi_{{\mathbb{F}}_{0}}$ of the form

$\log\xi_{{\mathbb{F}}_{0}}=2\cosh^{-1}\left({|\xi_{{\mathbb{F}}_{2}}|\over
2}\right)+2k\pi{\rm i},$ (2.75)

where $k$ is an odd integer. This multivalued structure can be also partially
seen in the perturbative calculation of the energy levels. If we consider
again the example of local ${\mathbb{F}}_{0}$, we see that the determination
of the frequency $\omega_{c}$ requires a choice of branch for
$\xi_{{\mathbb{F}}_{0}}^{1/4}$.

In our calculations in the previous sections we have chosen implicitly the
principal branch of the logarithm of the mass parameter, but we can explore
other branches. This is in principle straightforward in the EQC, where one
simply considers different branches of the logarithm. In the complex
dilatation method, reaching other sheets requires rotating $\theta$ or of the
variational parameter $\omega$ in the complex plane. In the Borel resummation
technique, one might have access to other sheets by a choice of different
branches in $\omega_{c}$ (but only to finitely many). We have performed a
preliminary investigation of this multi-sheeted structure. For example, in the
case of local ${\mathbb{F}}_{0}$ with $\hbar=2\pi$, one can consider the value
$\log(\xi_{{\mathbb{F}}_{0}})=3\pi{\rm i}$, instead of $\pi{\rm i}$, as we did
in Table 1. The EQC gives,

$E_{0}=1.510433421361...+{3\pi{\rm i}\over 4}.$ (2.76)

It is possible to access this sheet with the complex dilatation method, by
changing the value of $\omega$ (this is equivalent to choosing a different
value of $\theta$), and one finds:

$E_{0}=1.510433421...+{3\pi{\rm i}\over 4}.$ (2.77)

In the perturbative calculation (2.60), this sheet corresponds to the choice
$\varpi=3\pi{\rm i}/4$, instead of $\varpi=\pi{\rm i}/4$. Although the usual
Borel–Padé method displays in this case an oscillatory behavior, the result is
compatible with the values shown above.

An additional difficulty to explore the multivalued structure is that, in some
cases, the EQC (2.38) does not converge when one tries to access other sheets
for the log of the mass parameter. The EQC is given by a power series
expansion, which in the trace class case converges for values of the modulus
corresponding to the physical spectrum. However, in the complex case, the
convergence depends on the choice of sheet, of level $n$, and of $\hbar$. For
example, for $\xi_{{\mathbb{F}}_{0}}=3\pi{\rm i}$ and $\hbar=\pi$, we have not
been able to obtain the first resonance ($n=0$) with the EQC. This raises
interesting issues that we discuss in the concluding section of the paper.

##  3 PT-symmetric quantum curves

###  3.1 PT symmetry and non-perturbative effects

A particularly interesting subset of non-Hermitian operators are those which
display PT symmetry. A PT transformation changes the Heisenberg operators as

$\mathsf{x}\rightarrow-\mathsf{x},\qquad\mathsf{p}\rightarrow-\mathsf{p},$
(3.1)

and it also changes the imaginary unit ${\rm i}$ into $-{\rm i}$. Operators
which are invariant under this transformation are called PT-symmetric
operators. A simple example is the Hamiltonian (1.1), with a purely cubic,
imaginary potential, but more general examples are possible: the cubic
oscillator (2.8) leads to a PT-symmetric operator when ${\rm Re}(g)=0$.

When an operator is PT-symmetric, its eigenvalues are either real or come in
complex conjugate pairs. In the first case (which is realized in (1.1)) we say
that PT is unbroken, while in the second case we say that PT symmetry is
broken. It turns out that, in many quantum mechanical models, the breaking of
PT symmetry is a non-perturbative effect, due to complex instantons. A
beautiful example which illustrates the phenomenon is the Hamiltonian (1.2),
studied in detail by Delabaere and Trinh in delabaere-trinh  (a similar model
is studied in ben-ber , and related considerations on the importance of non-
perturbative effects in PT symmetry breaking are made in dt-beyond ). When
$\alpha>0$, the eigenvalues of this Hamiltonian are real. However, as $\alpha$
decreases and takes negative values, we find a decreasing sequence of values
$\alpha_{n}<0$, $n=1,2,\cdots$, for which the energy levels $E_{n-1}$ and
$E_{n}$ coalesce and become complex conjugate. Therefore, if

$\alpha_{n+1}<\alpha<\alpha_{n},$ (3.2)

the first $2n$ energy levels come in complex conjugate pairs. This phenomenon
can be understood quantitatively by using EQCs and the complex WKB method
delabaere-trinh  (see also unfolding  for a closely related analysis of
Bender–Wu branch points with exact EQCs). Let us first note that the potential

$V(x)={\rm i}x^{3}+{\rm i}\alpha x,$ (3.3)

has three turning points when $\alpha\geq 0$: one of them, $x_{0}$, is in the
imaginary axis, while the other two, denoted by $x_{\pm}$, are in the fourth
and the third quadrant of the complex plane, respectively. Let $\gamma$ be a
cycle encircling the turning points $x_{0}$ and $x_{+}$. By using the all-
orders WKB method, one obtains a WKB period associated to $\gamma$, which can
be decomposed into real and imaginary parts as

$\Pi_{\gamma}={1\over 2}\Pi_{\rm p}-{{\rm i}\over 2}\Pi_{\rm np},$ (3.4)

where ${\rm p}$, ${\rm np}$ stand for “perturbative” and “non-perturbative.”
The periods $\Pi_{\rm p}$, $\Pi_{\rm np}$ are real formal power series in
$\hbar^{2}$,

$\Pi_{{\rm p},{\rm np}}=\sum_{n\geq 0}\Pi_{{\rm p},{\rm np}}^{(n)}\hbar^{2n},$
(3.5)

which turn out to be Borel summable ddpham ; dpham ; delabaere-trinh . We will
denote by $s(\Pi_{\rm p})$, $s(\Pi_{\rm np})$ their (real) Borel resummations.
The Bohr–Sommerfeld (BS) approximation to the spectrum is given by (see e.g.
bb-pt ; bender-review )

$\Pi_{\rm p}^{(0)}=2\pi\hbar\left(n+{1\over 2}\right),\qquad n=0,1,2,\cdots$
(3.6)

This can be promoted to an all-orders, perturbative WKB quantization condition

$s\left(\Pi_{\rm p}\right)=2\pi\hbar\left(n+{1\over 2}\right),\qquad
n=0,1,2,\cdots$ (3.7)

The perturbative quantization conditions (3.6), (3.7) lead to real spectra, so
they can not explain the breaking of PT symmetry observed in this model.
However, a careful analysis based on the exact WKB method shows that these
conditions miss non-perturbative effects due to complex instantons bpv . The
correct quantization condition is given by

$2\cos\left({1\over 2\hbar}s(\Pi_{\rm p})\right)+{\rm e}^{-{1\over
2\hbar}s(\Pi_{\rm np})}=0.$ (3.8)

The second term in the l.h.s. gives a non-perturbative correction to (3.7).
This correction is crucial to obtain the right results, but it is typically
exponentially small (see ims  for a recent implementation of (3.8) in this
typical regime). However, as shown in delabaere-trinh ; ben-ber , as $\alpha$
becomes more and more negative, the non-perturbative correction becomes of
order one, and the quantization condition (3.8) does not have solutions for
real $E$. Therefore, PT symmetry is broken.

The qualitative structure of PT symmetry breaking is already captured by
considering the leading approximation to the full quantization condition
(3.8), but including the contribution due to $\Pi_{\rm np}$ also at leading
order, as pointed out in ben-ber . This leads to the approximate quantization
condition

$2\cos\left({1\over 2\hbar}\Pi_{\rm p}^{(0)}\right)+{\rm e}^{-{1\over
2\hbar}\Pi^{(0)}_{\rm np}}=0.$ (3.9)

The functions $\Pi_{{\rm p},{\rm np}}^{(0)}$ can be explicitly written down in
terms of elliptic integrals, by using e.g. the results of cm-ha ; ims . The
equation (3.9) describes the curves of possible energy levels in the
$(\alpha,E)$ plane, in very good agreement with numerical results, as shown in
Fig. 2 for $\hbar=1$. It is clear that the breaking of PT symmetry leads to
two different phases in the $(\alpha,E)$ plane: one phase of real values, and
another of complex values. Since the phase transition is triggered by a non-
perturbative effect becoming of order one, the boundary separating the two
phases can be described approximately by

$\Pi^{(0)}_{\rm np}(\alpha,E)=0,$ (3.10)

as we also illustrate in Fig. 2. This is reminiscent of large $N$ phase
transitions triggered by instantons neuberger  (see mmbook  for a textbook
exposition), where the transition point is defined by the vanishing of the
instanton action as a function of the moduli. A careful analysis of (3.9)
leads for example to a semiclassical, asymptotic formula for the values of
$\alpha_{n}$:

$\alpha_{n}=-x_{n}^{4/5},\qquad x_{n}\sim(4n-1){\pi\over\omega_{0}},\qquad
n=1,2,\cdots$ (3.11)

where

$\omega_{0}=\Pi^{(0)}_{\rm p}(1,E_{0}),$ (3.12)

and the energy $E_{0}$ is defined as the zero of (3.10) when $\alpha=1$. The
sequence of values $\alpha_{n}$ is a one-dimensional ray of the two-
dimensional quasi-lattice of Bender–Wu singularities of the cubic oscillator
alvarezcubicbw .

|
---|---
Figure 2: (Left) The curves in the $(\alpha,E)$ plane defined by the
approximate quantization condition (3.9), for $\hbar=1$. The red dots are
numerical values for the spectrum of (1.2) obtained with complex dilatation.
(Right) There are clearly two different phases in the $(\alpha,E)$ plane
separated by the line (3.10), which is shown here in dashed red.

###  3.2 PT symmetry in deformed quantum mechanics

In gm-deformed , a deformed version of one-dimensional quantum mechanics was
introduced. The Hamiltonian of this theory is given by

$\mathsf{H}_{N}=\Lambda^{N}\left({\rm e}^{\mathsf{p}}+{\rm
e}^{-\mathsf{p}}\right)+V_{N}(\mathsf{x}),$ (3.13)

where

$V_{N}(\mathsf{x})=\sum_{k=0}^{N-1}(-1)^{k}\mathsf{x}^{N-k}h_{k}$ (3.14)

is a degree $N$ potential, and we take $h_{0}=1$, $h_{1}=0$ without loss of
generality. We will refer to $\Lambda$, $h_{k}$, $k=2,\cdots,N-1$ as the
moduli of the potential. As explained in detail in gm-deformed , the
Hamiltonian (3.13) can be obtained by quantizing the Seiberg–Witten (SW) curve
of $SU(N)$, ${\cal N}=2$ Yang–Mills theory sw ; af ; klty , and in this case
the parameters $\Lambda$, $h_{k}$ correspond to the dynamically generated
scale and the Coulomb branch moduli, respectively. One interesting aspect of
the Hamiltonian (3.13) is that its spectrum of resonances or bound states is
encoded in an EQC which can be written in closed form for any $N$. As in the
conventional exact WKB method, this condition can be written in terms of
resummed WKB periods of the underlying SW curve:

$\Lambda^{N}\left({\rm e}^{p}+{\rm
e}^{-p}\right)+\sum_{k=0}^{N}(-1)^{k}x^{N-k}h_{k}=0,$ (3.15)

where $(-1)^{N-1}h_{N}$ can be regarded as the eigenvalue of the Hamiltonian
(3.13). As shown in mirmor , these WKB periods are nothing but the quantum
periods appearing in the NS limit ns  of ${\cal N}=2$ super Yang–Mills theory.
They are denoted by

$\Pi_{A_{i}}=2\pi a_{i},\qquad\Pi_{B_{i}}={\partial F_{\rm NS}\over\partial
a_{i}},\qquad i=1,\cdots,N-1,$ (3.16)

and they have a formal power series expansion in $\hbar^{2}$. The zero-th
order terms of this expansion are given by the usual SW periods. In this
equation, $F_{\rm NS}$ is the free energy of SW theory in the NS limit of the
so-called Omega-background. It has the structure,

$F_{\rm NS}=F_{\rm NS}^{\rm pert}+F_{\rm NS}^{\rm inst},$ (3.17)

where $F_{\rm NS}^{\rm pert}$ is the so-called perturbative piece, and $F_{\rm
NS}^{\rm inst}$ is the instanton piece, which can be computed by using
instanton calculus in the gauge theory n . This leads to an expansion in
powers of $\Lambda^{2N}$, of the form

$F^{\rm inst}_{\rm NS}=\sum_{\ell\geq 1}F^{(\ell)}_{\rm
NS}(\hbar,a_{i})\Lambda^{2N\ell}.$ (3.18)

In addition to this series, one can also obtain gauge theory expansions for
the moduli $h_{k}$ of the SW curve in terms of the A-periods $a_{i}$,

$h_{k}=h_{k}^{\rm pert}+\sum_{\ell\geq
1}h_{k}^{(\ell)}(\hbar,a_{i})\Lambda^{2N\ell},$ (3.19)

where $h_{k}^{\rm pert}$ is a classical or perturbative piece which can be
determined by elementary algebraic considerations. These expansions for the
moduli are sometimes called “quantum mirror maps” and they provide a
dictionary between the moduli of the curve and the quantum A-periods $a_{i}$.
More details, explicit expressions and examples for the series (3.18) and
(3.19) can be found in gm-deformed . Let us note that, in contrast to the
standard WKB expansions in $\hbar^{2}$, the instanton expansions (3.18) and
(3.19) are expected to be convergent in the so-called large radius region or
semiclassical region of the SW moduli space, where the moduli $a_{i}$ are
large in absolute value. They provide explicit resummations of the quantum
periods and make unnecessary the use of Borel techniques, as emphasized in gm-
deformed .

It is clear that, by an appropriate choice of the moduli $\Lambda$, $h_{k}$,
$k=2,\cdots,N-1$, we can easily engineer Hamiltonians of the form (3.13) which
are PT symmetric. In fact, any PT symmetric Hamiltonian of the form

$\mathsf{H}={\mathsf{p}^{2}\over 2}+V(\mathsf{x}),$ (3.20)

where $V(x)$ is a polynomial potential, leads to a deformed Hamiltonian of the
form (3.13). For concreteness, we will focus on the deformed version of the
Hamiltonian (1.2), which is obtained by taking $N=3$ and $\Lambda={\rm i}$.
One has

$\mathsf{H}={\rm i}\mathsf{H}_{3}=2\cosh(\mathsf{p})+{\rm
i}\mathsf{x}^{3}+{\rm i}\alpha\mathsf{x},$ (3.21)

where $\alpha=h_{2}$, and we want to solve the spectral problem

$\mathsf{H}|\phi\rangle=E|\phi\rangle,\qquad E={\rm i}h_{3}.$ (3.22)

The spectrum is expected to have a structure qualitatively similar to the one
obtained in delabaere-trinh  for (1.2), and indeed our analysis will confirm
this. We will rely on the EQC conjectured in gm-deformed . The comparison of
our explicit numerical results with the predictions of this EQC will provide
additional support for the conjecture of gm-deformed , and therefore for the
TS/ST correspondence of ghm ; cgm .

Before presenting the conjecture of gm-deformed , let us make a cautionary
remark concerning its general applicability in the study of PT-symmetric
Hamiltonians. In defining the spectral problem for these Hamiltonians, one has
to make a careful choice of boundary conditions for the eigenfunctions. In the
case of the PT symmetric cubic oscillator, the boundary condition is square
integrability along the real axis. This agrees with the boundary condition
that one obtains by analytic continuation in the coupling constant
alvarezcubicbw . However, there are cases in which the boundary conditions
appropriate for PT symmetry are different from the boundary conditions
obtained by analytic continuation. For example, the quartic oscillator with
potential $-x^{4}$, which is PT symmetric, leads to a real spectrum when
appropriate boundary conditions are used. If one does instead an analytic
continuation in the coupling constant, the appropriate boundary condition is
exponential decay in the wedge (2.5), and one obtains a spectrum of complex
resonances bender-review . The EQCs obtained in gm-deformed  describe
resonances obtained by analytic continuation in the parameters. Therefore,
they cannot be used to describe e.g. the real spectrum of an inverted, PT-
symmetric quartic oscillator.

Let us now present the EQC for the Hamiltonian (3.21). By setting
$\Lambda={\rm i}$ in the EQC for $N=3$ in gm-deformed , one obtains

$1+\frac{1-{\rm e}^{-2\pi a_{1}/\hbar}}{1-{\rm
e}^{-2\pi(a_{1}+a_{2})/\hbar}}{\rm e}^{-\pi a_{2}/\hbar+{\rm
i}\phi_{2}}+\frac{1-{\rm e}^{-2\pi a_{2}/\hbar}}{1-{\rm
e}^{-2\pi(a_{1}+a_{2})/\hbar}}{\rm e}^{-\pi a_{1}/\hbar+{\rm i}\phi_{1}}=0,$
(3.23)

where

$\phi_{1}=\frac{1}{\hbar}\left(\frac{\partial F_{\rm NS}}{\partial
a_{2}}-2\frac{\partial F_{\rm NS}}{\partial
a_{1}}\right),\qquad\phi_{2}=\frac{1}{\hbar}\left(2\frac{\partial F_{\rm
NS}}{\partial a_{2}}-\frac{\partial F_{\rm NS}}{\partial a_{1}}\right).$
(3.24)

When $N=3$, the perturbative piece of the NS free energy is determined by

$\frac{\partial F^{\rm pert}_{\rm NS}}{\partial
a_{j}}=2\gamma(a_{j},\hbar)+2\gamma(a_{1}+a_{2},\hbar),\qquad j=1,2.$ (3.25)

where the function $\gamma(a,\hbar)$ was introduced in hm  and it is given
by333The definition of $\gamma$ differs from the one in gm-deformed  by the
factor $a\log\Lambda$ which in our case ($\Lambda={\rm i}$) is already
included in (3.23).

$\gamma(a,\hbar)=a\log\hbar-\frac{\pi\hbar}{4}-\frac{i\hbar}{2}\log\frac{\Gamma(1+{\rm
i}a/\hbar)}{\Gamma(1-{\rm i}a/\hbar)}.$ (3.26)

The quantum mirror maps (3.19) relating $\alpha,E$ to the quantum periods
$a_{1},a_{2}$ are given, at the very first orders in $\Lambda^{6}$, by

$\displaystyle\begin{aligned}
\alpha=&-\frac{1}{3}(a_{1}^{2}+a_{1}a_{2}+a_{2}^{2})\\\
&+\frac{2(a_{1}^{2}+a_{2}^{2}+a_{1}a_{2}+3\hbar^{2})}{(\hbar^{2}+a^{2}_{1})(\hbar^{2}+a^{2}_{2})\left(\hbar^{2}+(a_{1}+a_{2})^{2}\right)}(-\Lambda^{6})+{\cal
O}(\Lambda^{12}),\end{aligned}$ (3.27) $\displaystyle\begin{aligned}
E=&-\frac{{\rm i}}{27}(a_{1}-a_{2})(a_{1}+2a_{2})(2a_{1}+a_{2})\\\
&-\frac{{\rm
i}(4a_{1}^{3}+6a_{2}a_{1}^{2}-6a_{2}^{2}a_{1}-4a_{2}^{3})}{3(\hbar^{2}+a^{2}_{1})(\hbar^{2}+a^{2}_{2})\left(\hbar^{2}+(a_{1}+a_{2})^{2}\right)}(-\Lambda^{6})+{\cal
O}(\Lambda^{12}).\end{aligned}$ (3.28)

Although we have written down explicitly the powers of $\Lambda$ to make clear
the instanton order, we have to set $\Lambda={\rm i}$ in actual computations.

It will be useful to write down the EQC (3.23) in a form similar to (3.8):

$F(U,Y)=1+2{\rm e}^{U}\cos Y=0,$ (3.29)

where the functions $U$, $Y$ are given by

$\displaystyle U$ $\displaystyle=\frac{1}{2}\log\left(\frac{(1-{\rm e}^{-2\pi
a_{1}/\hbar})(1-{\rm e}^{-2\pi a_{2}/\hbar})}{(1-{\rm
e}^{-2\pi(a_{1}+a_{2})/\hbar})^{2}}\right)-\frac{\pi}{2\hbar}(a_{1}+a_{2})+\frac{3{\rm
i}}{2\hbar}\left(\frac{\partial F_{\rm NS}}{\partial a_{2}}-\frac{\partial
F_{\rm NS}}{\partial a_{1}}\right),$ (3.30) $\displaystyle Y$
$\displaystyle=\frac{1}{2{\rm i}}\log\left(\frac{1-{\rm e}^{-2\pi
a_{1}/\hbar}}{1-{\rm e}^{-2\pi a_{2}/\hbar}}\right)+\frac{\pi}{2{\rm
i}\hbar}(a_{1}-a_{2})+\frac{1}{2\hbar}\left(\frac{\partial F_{\rm
NS}}{\partial a_{2}}+\frac{\partial F_{\rm NS}}{\partial a_{1}}\right).$
(3.31)

The EQC (3.29) is a prediction of the TS/ST correspondence for the spectral
problem defined by the Hamiltonian (3.21). A first question to ask is how this
EQC compares to standard semiclassical results, like for example the BS
quantization condition. If we write the SW curve (3.15) in the form

$2\cosh(p)+V(x)=E,$ (3.32)

we notice that the turning points are the solutions of the equation

$E-V(x)=2.$ (3.33)

In the case

$V(x)={\rm i}x^{3}+{\rm i}\alpha x,$ (3.34)

and for $\alpha$ smaller than a certain $\alpha_{c}>0$, there are three
turning points $x_{0}$, $x_{\pm}$ in the imaginary axis and the fourth and
third quadrant, respectively, as in the undeformed cased considered in (3.3).
As in bb-pt ; bender-review , the relevant period corresponds to a cycle going
from $x_{-}$ to $x_{+}$, given by

$\Pi_{\rm BS}=2\int_{x_{-}}^{x_{+}}p(x){\rm d}x,$ (3.35)

and the BS quantization condition is

$\Pi_{\rm BS}(E)=\pi\hbar\left(n+{1\over 2}\right).$ (3.36)

In contrast to the situation in conventional quantum mechanics, this can not
be expressed in terms of elliptic integrals, since the underlying SW curve has
genus two. One possibility to calculate the period above is to write down an
appropriate Picard–Fuchs operator which annihilates $\Pi_{\rm BS}$, and then
use it to obtain a power series expansion for large energies. The details are
presented in Appendix A, and one obtains at the end of the day an expression
of the form

$\displaystyle\Pi_{\rm BS}(E)$
$\displaystyle=\frac{1}{2}\epsilon\left(6\sqrt{3}\log(\epsilon)+\pi+3\sqrt{3}(\log(3)-2)\right)+\frac{\alpha\left(-6\sqrt{3}\log(\epsilon)+\pi-3\sqrt{3}\log(3)\right)}{6\epsilon}$
(3.37)
$\displaystyle+\frac{-6\sqrt{3}\left(\alpha^{3}+18\right)\log(\epsilon)-\pi\left(\alpha^{3}+18\right)-3\sqrt{3}\left(\alpha^{3}(\log(3)-1)-6+18\log(3)\right)}{162\epsilon^{5}}$
$\displaystyle+{\cal O}(\epsilon^{-7}),$

where $\epsilon=E^{1/3}$. One can verify that (3.36) provides an excellent
approximation to the energy levels of the PT-symmetric Hamiltonian (3.21), in
particular for highly excited states, as expected from the BS approximation.

Let us now compare the BS approximation with the EQC (3.29). First of all, let
us note that the functions $U$ and $Y$ have an asymptotic expansion in powers
of $\hbar^{2}$ of the form

$Y\sim\sum_{n\geq 0}Y_{n}\hbar^{2n-1},\qquad U\sim\sum_{n\geq
0}U_{n}\hbar^{2n-1}.$ (3.38)

Note that the logarithmic terms in (3.30), (3.31) do not contribute to this
asymptotic expansion, as they are exponentially small in $\hbar$. The
asymptotic series (3.38) can be obtained by using the gauge theory instanton
expansion for $F^{\rm inst}_{\rm NS}$ and expanding each order in
$\Lambda^{6}$ in powers of $\hbar^{2}$. This leads to explicit expressions for
$Y_{n}$, $U_{n}$ as expansions in $\Lambda^{6}$, which are suitable for the
large radius region in which $|a_{1}|,\,|a_{2}|\gg 1$. Let us give an explicit
example of such an expansion. To do this, we parametrize $a_{1,2}$ as

$a_{1}=\rho\,{\rm e}^{{\rm i}\phi},\qquad a_{2}=\rho\,{\rm e}^{-{\rm i}\phi}.$
(3.39)

When $\rho,\phi\in{\mathbb{R}}$, $a_{1}$ and $a_{2}$ are complex conjugate

$a_{1}=\overline{a_{2}},$ (3.40)

and this leads to real values of $\alpha$ and $E$, so this describes the phase
of unbroken PT symmetry. In addition, $U$ and $Y$ are also real. The gauge
theory instanton expansion of the function $Y_{0}$ is, in the polar variables
above,

$\displaystyle Y_{0}$
$\displaystyle=\rho\big{(}(\pi-2\phi)\sin\phi+(-6+6\log\rho+4\log(2\cos\phi))\cos\phi\big{)}$
(3.41)
$\displaystyle-\frac{(7\cos(2\phi)+2\cos(4\phi)+3)\sec^{3}(\phi)}{4\rho^{5}}(-\Lambda^{6})+{\cal
O}\left(\Lambda^{12}\right),$

while for the function $U_{0}$ one finds,

$\displaystyle U_{0}$
$\displaystyle=\rho\big{(}(6\phi-\pi)\cos\phi+(-6+6\log\rho)\sin\phi\big{)}+\frac{3(4\cos(2\phi)+3)\tan(\phi)\sec(\phi)}{2\rho^{5}}(-\Lambda^{6})$
(3.42) $\displaystyle+{\cal O}\left(\Lambda^{12}\right).$

Note that the power of $\Lambda$ is correlated with the inverse power of
$\rho$, so the expansion in powers of $\Lambda$, typical of the gauge theory
instanton expansion, is also an expansion around the large radius region
$\rho\rightarrow\infty$, as expected.

The BS quantization condition should be recovered from (3.29) by setting ${\rm
e}^{-U}\sim 0$ and keeping the first term in the asymptotic expansion of $Y$,
i.e.

$\cos(Y_{0}/\hbar)=0\Rightarrow Y_{0}=\pi\hbar\left(n+{1\over 2}\right),$
(3.43)

so we should have

$Y_{0}=\Pi_{\rm BS}.$ (3.44)

This can be verified by using the quantum mirror map and expanding $\Pi_{\rm
BS}$ in power series around $\rho=\infty$ and $\phi=\pi/3$ (which corresponds
to $\alpha=0$). This can be easily compared to the series (3.41), and one
finds that (3.44) holds to very high order.

In view of the structure of the EQC (3.29), which is formally identical to the
one in the undeformed case (3.8), the deformed Hamiltonian (3.21) might also
undergo PT symmetry breaking. As in the undeformed case, this can happen if
the non-perturbative correction ${\rm e}^{-U}$ becomes of order one.
Semiclassically, this occurs when

$U_{0}(\alpha,E)=0.$ (3.45)

The solutions to this equation, if they exist, define a curve in the
$(\alpha,E)$ plane separating the two phases. Note that $U_{0}$ is completely
determined by the explicit expression (3.30), which is in turn a prediction of
the TS/ST correspondence.

Figure 3: The black lines show the values of $(\alpha,E)$ in the spectrum of
(3.21) at $\hbar=1$. The dashed red line is the solution of the equation
(3.45), which defines the boundary between the phases of broken and unbroken
PT symmetry.

It is easy to verify numerically that indeed, PT symmetry breaking takes place
in this model, when $\alpha<0$. As in the undeformed case, there is a sequence
of values

$\alpha_{k},\qquad k=1,2,\cdots,$ (3.46)

for which the energy levels become degenerate $E_{k-1}=E_{k}$, and take
complex values afterwards. The curve determined by (3.45) provides a
qualitatively accurate boundary between the resulting two phases, as shown in
Fig. 3. The EQC (3.29) turns out to capture as well the precise pattern of
symmetry breaking. To see this, we consider the leading order approximation to
(3.29), but taking into account the contribution of $U$, which is the
responsible for symmetry breaking, also at leading order, as we did in (3.9).
We obtain in this way the approximate quantization condition,

$2\cos(Y_{0}/\hbar)+{\rm e}^{-U_{0}/\hbar}=0,$ (3.47)

similar to the condition (3.8) in the undeformed case. The equation (3.47)
defines a discrete set of curves in the $(\alpha,E)$ plane, which we show in
Fig. 4 for $\hbar=1/5$. The functions $Y_{0}$, $U_{0}$ are defined by the
expansions (3.41), (3.42). In drawing the gray lines, we have just used the
very first term in this expansion (which takes into account only the
perturbative contribution to $F_{\rm NS}$), while in drawing the black lines
we have included the first gauge theory instanton correction. The red dots are
numerical values for the spectrum of the operator (3.21), and as we can see
they are located with good precision on the curves defined by (3.47).

Figure 4: The black lines show the curves satisfying the approximate EQC
(3.47), in which $U_{0}$, $Y_{0}$ are given by the gauge theory instanton
expansion up to order $\Lambda^{6}$. In the gray lines, only the perturbative
part (i.e. the very first terms in the expansions (3.41), (3.42)) have been
used. The red dots represent numerical values of the spectrum of the operator
(3.21), obtained with complex dilatation techniques. Here $\hbar=1/5$.

Another consequence of the EQC is an asymptotic expression for the branch
points of the spectrum. i.e. for the values of $\alpha$ (3.46) at which the
energy levels coalesce. These are the points satisfying (3.29), and in
addition the singularity condition

${\partial F\over\partial E}=0.$ (3.48)

Since $\alpha_{k}$ with $k\gg 1$ occurs at higher energies, in order to obtain
the leading asymptotic value of $\alpha_{k}$ it suffices to use the leading
terms in (3.41), (3.42). A careful analysis shows that the branching points
obey two different conditions. The first one, not surprisingly, is (3.45),
which is solved by

$\rho(\phi)\approx\exp\left({\pi\over 6\phi}\right)$ (3.49)

The second condition is

$\sin(Y)\approx-1,$ (3.50)

which is solved by

$Y\approx 2k\pi-{\pi\over 2},\qquad k=1,2,\cdots.$ (3.51)

This result is illustrated in Fig. 5, where we show the curves (3.49), (3.51)
in the $(\alpha,E)$ plane. Their intersection agrees with good precision with
the branching points of the spectrum. The values of $\rho$ satisfying the two
conditions (3.49), (3.51) can be written down explicitly as

$\rho_{k}=\frac{X_{k}}{W\left({\rm e}^{\frac{c}{6}}X_{k}\right)},$ (3.52)

where $W(z)$ is the Lambert $W$ function defined as the solution of the
equation $W(z)e^{W(z)}=z$, and

$X_{k}=\frac{\hbar\pi(2k-1/2)}{6},\qquad c=4\log 2-6.$ (3.53)

Figure 5: The black dashed line represents the solution of the spectrum, for
$\hbar=1$. The red dots are the branch points. The green and orange lines are
the curves (3.49) and (3.51).

##  4 Conclusions and outlook

The correspondence between quantum spectral problems and (topological) strings
put forward in ghm ; cgm ; wzh  has been mostly studied in the case in which
reality and positivity conditions lead to trace class, self-adjoint operators.
However, it is natural to extend this correspondence to general complex values
of the mass parameters. On the topological string side, the mass parameters
are identified with complex moduli of the CY target, and we can perform a
straightforward analytic continuation. In the spectral theory side, this
continuation is more delicate since one has to consider a problem in
non–Hermitian quantum mechanics. In this paper we have started to explore the
TS/ST correspondence in the complex realm of the mass parameters. In the
spectral theory side, we have developed techniques to compute accurately the
complex spectra, by using complex dilatation techniques and Borel–Padé
resummation of perturbative series. We have found, in various models, that
these techniques provide results in precise agreement with the predictions of
the TS/ST correspondence.

In addition, we have explored PT-symmetric Hamiltonians arising in the
quantization of SW curves. These Hamiltonians provide an interesting
generalizations of the more conventional PT-symmetric quantum-mechanical
models, and they can be analyzed in detail by using the exact quantization
conditions conjectured in gm-deformed , which also followed from the TS/ST
correspondence. In this paper we have analyzed in detail the deformed model of
gm-deformed  with a PT-symmetric cubic potential. We have shown that the
quantization conditions conjectured in gm-deformed  describe very precisely
the properties of the spectrum. In addition, we have shown that the deformed
model displays as well the spontaneous breaking of PT-symmetry observed in
delabaere-trinh ; ben-ber .

There are many interesting problems open by this investigation. The most
important one, in our view, is a deeper understanding of the multivalued
structure of the spectrum of quantum mirror curves. As we have pointed out,
the exact solution conjectured in ghm ; cgm ; wzh  indicates that there should
be an infinitely-sheeted structure due to the logarithmic dependence on the
mass parameters. Although we have been able to access some of these sheets in
spectral theory, a full understanding of this structure is still lacking. In
addition, the exact quantization conditions in ghm ; cgm ; wzh  involve series
expansions which do not always converge when one considers arbitrary sheets of
the logarithm. This raises the possibility that these conditions have to be
reformulated in order to describe the full analytic continuation of the
spectral problem.

In our analysis of PT-symmetric models, we have focused for simplicity on
quantum SW curves, but it would be nice to find interesting examples of PT-
symmetric quantum mirror curves444Spectral problems with PT symmetry have been
recently considered in ar-pt , in the closely related context of the Fermi gas
approach to ABJM theories and generalizations thereof.. Conversely, in the
case of quantum SW curves, we have not studied in detail the general
multivalued structure of the spectral problem. Since these models are
deformations of conventional quantum mechanical potentials, we should have
analogues of the Bender–Wu branch points for the quartic bw  and the cubic
case alvarezcubicbw . It would be very interesting to explore the analytic
structure of the energy levels with the exact quantization conditions of gm-
deformed , similar to what was done in unfolding ; delabaere-trinh . This
might lead to a beautiful interplay between the complex geometry of the
spectral problem, and the complex geometry of the quantum moduli space of
Seiberg–Witten theory.

## Acknowledgements

We are grateful to C. Bender and J. Gu for useful discussions. We are
particularly grateful to S. Zakany for providing us his powerful Mathematica
packages to implement exact quantization conditions. Many of the computations
reported in this paper were performed at University of Geneva on the Baobab
cluster. The work of Y.E. and M.M. is supported in part by the Fonds National
Suisse, subsidies 200020-175539 and 200020-141329, and by the Swiss-NSF grant
NCCR 51NF40-141869 “The Mathematics of Physics” (SwissMAP). The work of M.R.
is supported by the funds awarded by the Friuli Venezia Giulia autonomous
Region Operational Program of the European Social Fund 2014/2020, project
“HEaD - HIGHER EDUCATION AND DEVELOPMENT SISSA OPERAZIONE 3”, CUP
G32F16000080009, by INFN via the Iniziativa Specifica GAST and by National
Group of Mathematical Physics (GNFM-INdAM). M.R. would like to thank the GGI
for hospitality during the workshop “Supersymmetric Quantum Field Theories in
the Non-perturbative Regime” and CERN for the hospitality during his stay in
Geneva.

##  Appendix A Picard–Fuchs equation for the deformed cubic oscillator

An efficient way to compute the BS period (3.35) is to find out the
Picard–Fuchs (PF) equation that it satisfies, and then solve this equation in
a power series in appropriate variables. As a starting point, one can derive
the following PF equation when $\alpha=0$, corresponding to the pure cubic
potential:

$\Pi^{\prime}(E)+\frac{3}{4}\left(3\left(E^{2}-4\right)\Pi^{(3)}(E)+7E~{}\Pi^{\prime\prime}(E)\right)=0.$
(A.1)

To simplify notation, we have denoted $\Pi=\Pi_{\rm BS}$. This ODE can be
solved in closed form as

$\Pi^{\prime}(E)=c_{1}\frac{P\left(-\frac{1}{2},\frac{1}{6},\frac{E}{2}\right)}{\sqrt[12]{E^{2}-4}}+c_{2}\frac{Q\left(-\frac{1}{2},\frac{1}{6},\frac{E}{2}\right)}{\sqrt[12]{E^{2}-4}},$
(A.2)

where $P(n,m,x)$ are Legendre polynomials, and $Q(n,m,x)$ are Legendre
functions of the second kind. The PF equation for arbitrary $\alpha$ is more
involved:

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
192\left(8\,\alpha^{3}+\,E^{2}-216\right)\Pi^{\prime}(E)-\left(-3483\,E^{3}-18064\,\alpha^{3}\,E+487728\,E\right)\Pi^{\prime\prime}(E)$
(A.3)
$\displaystyle-\frac{1}{3}\left(3328\,\alpha^{6}+6912\,\alpha^{3}-18063\,E^{4}-71088\,\alpha^{3}\,E^{2}+1942704\,E^{2}-2612736\right)\Pi^{(3)}(E)$
$\displaystyle-\frac{2}{3}\left(-3645\,E^{5}-11232\,\alpha^{3}\,E^{3}+326592\,E^{3}+1744\,\alpha^{6}\,E-864\,\alpha^{3}\,E-1248048\,E\right)\Pi^{(4)}(E)$
$\displaystyle-\frac{1}{9}(-128\,\alpha^{9}+10368\,\alpha^{6}-279936\,\alpha^{3}-2187\,E^{6}-5184\,\alpha^{3}\,E^{4}+174960\,E^{4}+1680\,\alpha^{6}\,E^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}+2592\,\alpha^{3}\,E^{2}-1294704\,E^{2}+2519424)\Pi^{(5)}(E)$

To solve this ODE, we use the following ansatz

$\Pi^{\prime}(E)=\log(E)\sum_{n=0}^{\infty}a_{n}E^{-\frac{2n}{3}-\frac{2}{3}}+\sum_{n=0}^{\infty}b_{n}E^{-\frac{2n}{3}-\frac{2}{3}}$
(A.4)

By plugging (A.4) into (A.3), we can solve for $a_{n}$ and $b_{n}$. Doing so,
we are left with only four undetermined constants, say $a_{n}$ and $b_{n}$
with $n\in\\{0,1\\}$. These can be fixed by various boundary conditions for
the problem, and they turn out to be given by

$\displaystyle
a_{0}=-\frac{1}{\sqrt{3}},~{}~{}~{}~{}b_{0}=\frac{1}{6}\left(-\pi-3\sqrt{3}\log(3)\right),$
(A.5) $\displaystyle
a_{1}=-\frac{\alpha}{3\sqrt{3}},~{}~{}~{}~{}b_{1}=\frac{\alpha}{18}\left(\pi-3\sqrt{3}(\log(3)-2)\right).$

By using the above results, one can derive the expansion (3.37). Let us note
that (3.35) is a period of $SU(3)$ SW theory, so it should be possible to
write it as a linear combination of the basis of periods obtained in klt  in
terms of Appell functions.

## References

  * (1) A. Grassi, Y. Hatsuda and M. Mariño, _Topological strings from quantum mechanics_ , _Annales Henri Poincaré_ 17 (2016) 3177–3235, [1410.3382].
  * (2) S. Codesido, A. Grassi and M. Mariño, _Spectral theory and mirror curves of higher genus_ , _Annales Henri Poincaré_ 18 (2017) 559–622, [1507.02096].
  * (3) C. M. Bender and T. T. Wu, _Anharmonic oscillator_ , _Phys. Rev._ 184 (1969) 1231–1260.
  * (4) S. Codesido, J. Gu and M. Mariño, _Operators and higher genus mirror curves_ , _JHEP_ 02 (2017) 092, [1609.00708].
  * (5) A. Grassi and M. Mariño, _The complex side of the TS/ST correspondence_ , _J. Phys._ A52 (2019) 055402, [1708.08642].
  * (6) R. M. Kashaev and S. M. Sergeev, _Spectral equations for the modular oscillator_ , _Rev. Math. Phys._ 30 (2018) 1840009, [1703.06016].
  * (7) K. Konishi and G. Paffuti, _Quantum mechanics: a new introduction_.  Oxford University Press, 2009.
  * (8) G. Álvarez and C. Casares, _Exponentially small corrections in the asymptotic expansion of the eigenvalues of the cubic anharmonic oscillator_ , _J. Phys._ A33 (2000) 5171.
  * (9) E. Caliceti, S. Graffi and M. Maioli, _Perturbation theory of odd anharmonic oscillators_ , _Commun. Math. Phys._ 75 (1980) 51–66.
  * (10) J. Gu and T. Sulejmanpasic, _High order perturbation theory for difference equations and Borel summability of quantum mirror curves_ , _JHEP_ 12 (2017) 014, [1709.00854].
  * (11) C. M. Bender, _Making sense of non-Hermitian Hamiltonians_ , _Rept. Prog. Phys._ 70 (2007) 947, [hep-th/0703096].
  * (12) E. Delabaere and D. T. Trinh, _Spectral analysis of the complex cubic oscillator_ , _J. Phys._ A33 (2000) 8771.
  * (13) C. M. Bender, M. Berry, P. N. Meisinger, V. M. Savage and M. Simsek, _Complex WKB analysis of energy-level degeneracies of non-Hermitian Hamiltonians_ , _J. Phys._ A34 (2001) L31.
  * (14) A. Grassi and M. Mariño, _A Solvable Deformation Of Quantum Mechanics_ , 1806.01407.
  * (15) R. De La Madrid and M. Gadella, _A pedestrian introduction to Gamow vectors_ , _American Journal of Physics_ 70 (2002) 626–638.
  * (16) N. Moiseyev, _Non-Hermitian quantum mechanics_.  Cambridge University Press, 2011.
  * (17) B. Simon and A. Dicke, _Coupling constant analyticity for the anharmonic oscillator_ , _Annals Phys._ 58 (1970) 76–136.
  * (18) C. M. Bender and A. Turbiner, _Analytic continuation of eigenvalue problems_ , _Phys. Lett._ A173 (1993) 442–446.
  * (19) B. C. Hall, _Quantum theory for mathematicians_.  Springer–Verlag, 2013.
  * (20) G. Álvarez, _Bender-Wu branch points in the cubic oscillator_ , _J. Phys._ A28 (1995) 4589.
  * (21) R. Yaris, J. Bendler, R. A. Lovett, C. M. Bender and P. A. Fedders, _Resonance calculations for arbitrary potentials_ , _Phys. Rev. A_ 18 (1978) 1816.
  * (22) P. Kościk and A. Okopińska, _The optimized Rayleigh–Ritz scheme for determining the quantum-mechanical spectrum_ , _J. Phys._ A40 (2007) 10851.
  * (23) A. Kuroś, P. Kościk and A. Okopińska, _Determination of resonances by the optimized spectral approach_ , _J. Phys._ A46 (2013) 085303.
  * (24) G. Álvarez, _Coupling-constant behavior of the resonances of the cubic anharmonic oscillator_ , _Physical Review_ A37 (1988) 4079.
  * (25) T. Sulejmanpasic and M. Ünsal, _Aspects of Perturbation theory in Quantum Mechanics: The BenderWu Mathematica package_ , _Comput. Phys. Commun._ 228 (2018) 273–289, [1608.08256].
  * (26) A. Grassi, M. Mariño and S. Zakany, _Resumming the string perturbation series_ , _JHEP_ 1505 (2015) 038, [1405.4214].
  * (27) K. Ito, M. Mariño and H. Shu, _TBA equations and resurgent Quantum Mechanics_ , _JHEP_ 01 (2019) 228, [1811.04812].
  * (28) M. Serone, G. Spada and G. Villadoro, _The Power of Perturbation Theory_ , _JHEP_ 05 (2017) 056, [1702.04148].
  * (29) J. Kallen and M. Mariño, _Instanton effects and quantum spectral curves_ , _Annales Henri Poincaré_ 17 (2016) 1037–1074, [1308.6485].
  * (30) M.-x. Huang and X.-f. Wang, _Topological strings and quantum spectral problems_ , _JHEP_ 1409 (2014) 150, [1406.6178].
  * (31) M.-X. Huang, A. Klemm and M. Poretschkin, _Refined stable pair invariants for E-, M- and $[p,q]$-strings_, _JHEP_ 1311 (2013) 112, [1308.0619].
  * (32) M.-x. Huang, A. Klemm, J. Reuter and M. Schiereck, _Quantum geometry of del Pezzo surfaces in the Nekrasov–Shatashvili limit_ , _JHEP_ 1502 (2015) 031, [1401.4723].
  * (33) R. Kashaev and M. Mariño, _Operators from mirror curves and the quantum dilogarithm_ , _Commun. Math. Phys._ 346 (2016) 967, [1501.01014].
  * (34) R. Kashaev, M. Mariño and S. Zakany, _Matrix models from operators and topological strings, 2_ , _Annales Henri Poincaré_ 17 (2016) 2741–2781, [1505.02243].
  * (35) J. Gu, A. Klemm, M. Mariño and J. Reuter, _Exact solutions to quantum spectral curves by topological string theory_ , _JHEP_ 10 (2015) 025, [1506.09176].
  * (36) X. Wang, G. Zhang and M.-x. Huang, _New exact quantization condition for toric Calabi–Yau geometries_ , _Phys. Rev. Lett._ 115 (2015) 121601, [1505.05360].
  * (37) M. Mariño, _Spectral theory and mirror symmetry_ , _Proc. Symp. Pure Math._ 98 (2018) 259, [1506.07757].
  * (38) M. Aganagic, M. C. Cheng, R. Dijkgraaf, D. Krefl and C. Vafa, _Quantum geometry of refined topological strings_ , _JHEP_ 1211 (2012) 019, [1105.0630].
  * (39) N. A. Nekrasov and S. L. Shatashvili, _Quantization of integrable systems and four dimensional gauge theories_ , in _16th International Congress on Mathematical Physics, Prague, August 2009, 265-289, World Scientific 2010_ , 2009.  0908.4052.
  * (40) S. Zakany, _Quantized mirror curves and resummed WKB_ , 1711.01099.
  * (41) P. Dorey, A. Millican-Slater and R. Tateo, _Beyond the WKB approximation in PT-symmetric quantum mechanics_ , _J. Phys._ A38 (2005) 1305–1332, [hep-th/0410013].
  * (42) E. Delabaere and F. Pham, _Unfolding the quartic oscillator_ , _Ann. Physics_ 261 (1997) 180–218.
  * (43) E. Delabaere, H. Dillinger and F. Pham, _Exact semiclassical expansions for one-dimensional quantum oscillators_ , _J. Math. Phys._ 38 (1997) 6126–6184.
  * (44) E. Delabaere and F. Pham, _Resurgent methods in semi-classical asymptotics_ , _Annales de l’IHP_ 71 (1999) 1–94.
  * (45) C. M. Bender and S. Boettcher, _Real spectra in non-Hermitian Hamiltonians having PT symmetry_ , _Phys. Rev. Lett._ 80 (1998) 5243–5246, [physics/9712001].
  * (46) R. Balian, G. Parisi and A. Voros, _Quartic oscillator_ , in _Feynman Path Integrals_ , vol. 106, pp. 337–360.  Springer–Verlag, 1979.
  * (47) S. Codesido and M. Mariño, _Holomorphic Anomaly and Quantum Mechanics_ , _J. Phys._ A51 (2018) 055402, [1612.07687].
  * (48) H. Neuberger, _Nonperturbative Contributions in Models With a Nonanalytic Behavior at Infinite $N$_, _Nucl. Phys._ B179 (1981) 253–282.
  * (49) M. Mariño, _Instantons and large $N$. An introduction to non-perturbative methods in quantum field theory_.  Cambridge University Press, 2015.
  * (50) N. Seiberg and E. Witten, _Electric-magnetic duality, monopole condensation, and confinement in $\mathcal{N}=2$ supersymmetric Yang–Mills theory_, _Nucl. Phys._ B426 (1994) 19–52, [hep-th/9407087].
  * (51) P. C. Argyres and A. E. Faraggi, _The vacuum structure and spectrum of N=2 supersymmetric SU(n) gauge theory_ , _Phys. Rev. Lett._ 74 (1995) 3931–3934, [hep-th/9411057].
  * (52) A. Klemm, W. Lerche, S. Yankielowicz and S. Theisen, _Simple singularities and $N=2$ supersymmetric Yang-Mills theory_, _Phys. Lett._ B344 (1995) 169–175, [hep-th/9411048].
  * (53) A. Mironov and A. Morozov, _Nekrasov functions and exact Bohr–Sommerfeld integrals_ , _JHEP_ 1004 (2010) 040, [0910.5670].
  * (54) N. A. Nekrasov, _Seiberg–Witten prepotential from instanton counting_ , _Adv. Theor. Math. Phys._ 7 (2004) 831–864, [hep-th/0206161].
  * (55) Y. Hatsuda and M. Mariño, _Exact quantization conditions for the relativistic Toda lattice_ , _JHEP_ 05 (2016) 133, [1511.02860].
  * (56) L. Anderson and M. M. Roberts, _Mass deformed ABJM and $\mathcal{PT}$ symmetry_, 1807.10307.
  * (57) A. Klemm, W. Lerche and S. Theisen, _Nonperturbative effective actions of N=2 supersymmetric gauge theories_ , _Int. J. Mod. Phys._ A11 (1996) 1929–1974, [hep-th/9505150].