aainstitutetext: Department of Physics, Tokyo Institute of Technology,
Tokyo, 152-8551, Japan
bbinstitutetext: Département de Physique Théorique & Section de Mathématiques,
Université de Genève, Genève, CH-1211 Switzerland

# TBA equations and resurgent Quantum Mechanics

Katsushi Ito b Marcos Mariño a Hongfei Shu

###### Abstract

We derive a system of TBA equations governing the exact WKB periods in one-
dimensional Quantum Mechanics with arbitrary polynomial potentials. These
equations provide a generalization of the ODE/IM correspondence, and they can
be regarded as the solution of a Riemann–Hilbert problem in resurgent Quantum
Mechanics formulated by Voros. Our derivation builds upon the solution of
similar Riemann–Hilbert problems in the study of BPS spectra in ${\cal N}=2$
gauge theories and of minimal surfaces in AdS. We also show that our TBA
equations, combined with exact quantization conditions, provide a powerful
method to solve spectral problems in Quantum Mechanics. We illustrate our
general analysis with a detailed study of PT-symmetric cubic oscillators and
quartic oscillators.

##  1 Introduction

In quantum theories one typically obtains a plethora of perturbative series,
associated to different classical configurations. These series are the basic
building block in many computations, but they are almost always divergent. The
theory of resurgence provides a way to organize these series into a coherent
picture of the quantum system. First of all, the divergent series are
converted into meaningful objects by doing Borel resummations in the
perturbation parameter. Second, one discovers that the different series are in
fact intimately related. One way to understand this relation is through the
Stokes phenomenon, which simply states that Borel resummations are
discontinuous along specific rays in the complex plane. It turns out that,
given one specific series, the discontinuities of its Borel transform encode
the information about other series in the problem.

The resurgent program was applied to Quantum Mechanics in one dimension
through an “exact” version of the WKB method bpv ; voros ; voros-quartic ;
dpham ; reshyper ; ddpham . In this context, the different perturbative series
are the WKB series around different classical trajectories, which we will call
WKB or quantum periods. This led to many powerful results, like for example
exact connection formula and exact quantization conditions for spectral
problems. In the case of the pure quartic oscillator, Voros found in voros-
quartic  a complete determination of the structure of the discontinuities for
the different WKB periods involved in the problem. However, this flourishing
of ideas did not provide alternative computational methods in Quantum
Mechanics. One was supposed to calculate the WKB series and their Borel
resummations with traditional tools.

Partially inspired by the results of voros-quartic , a correspondence between
certain quantum mechanical models in one dimension and integrable models (or
ODE/IM correspondence) was proposed by Dorey and Tateo in dt . This
correspondence was originally based on functional relations discovered in the
context of resurgent Quantum Mechanics, which are similar to Baxter-type
equations appearing in integrable systems. The ODE/IM correspondence makes it
possible to write down TBA equations which calculate efficiently and exactly
some of the quantities appearing in the quantum mechanical models, like
spectral determinants and Borel resummations of WKB periods. An important
limitation of the ODE/IM correspondence is that it applies to very special
quantum-mechanical models, namely, monic potentials of the form $V(x)=x^{M}$
(and a limited amount of perturbations thereof).

In this paper we will present TBA equations governing the WKB periods for
general polynomial potentials in one dimension, providing in this way a
generalization of the ODE/IM correspondence. The basic idea was already
pointed out by Voros in his seminal paper voros-quartic , where he called it
the “analytic bootstrap” for the exact WKB method. In this approach, the
fundamental objects are the WKB periods, which are (Borel resummed)
perturbative series in the Planck constant. These periods can be characterized
by two types of data: their classical limit and their discontinuity structure,
which has been known since the work of voros-quartic ; dpham ; ddpham . In the
theory of resurgence, this discontinuity structure is encoded in the action of
the so-called Stokes automorphisms. As Voros explained, we can think about
these data as defining a Riemann–Hilbert problem. Building on recent
developments in seemingly very different contexts ks ; gmn ; gmn2 ; alday-mal
; aldaygm ; y-system ; hito ; tateo-extended , we show that Voros’
Riemann–Hilbert problem has a solution in terms of a TBA-like system, which
determines the exact dependence of the WKB periods on the Planck constant111A
different solution to the problem was proposed in voros-fp  in terms of a
fixed-point equation. It is likely that this solution is closely related to
the TBA system proposed here.. The TBA system of dt  is recovered as a
particular case of our general story, as we fine tune the moduli of the
potential to a point with enhanced symmetry. From the point of view of the
theory of resurgence, the TBA system we obtain here provides an efficient way
to calculate and resum the perturbative expansion of the WKB periods in a
single strike.

The connection between Voros’ Riemann–Hilbert problem for the exact WKB
method, and the Riemann–Hilbert problems appearing in cv ; gmn ; gmn2 ; alday-
mal ; aldaygm ; y-system ; hito , has been already pointed out in the
literature. In particular, Gaiotto argued in oper  that the conformal limit of
the TBA equations for Hitchin systems obtained in gmn ; gmn2  leads to TBA
equations solving Schrödinger problems, and he presented some examples going
beyond the conventional ODE/IM correspondence (see cdz ; dumitrescu ; ito-shu
; bridgeland  for related discussions). In this paper we put the results of
oper  in the context of the general theory of resurgence, and we derive the
TBA equations governing general polynomial potentials in one dimension. In
addition, we provide detailed tests and illustrations of the resulting
formalism: we use the TBA equations to compute exact WKB periods, and we
extract the perturbative expansion from the large $\theta$ expansion of the
TBA system. We also use the TBA system to compute the spectrum of various
interesting quantum-mechanical systems, like PT-symmetric cubic oscillators
and quartic oscillators.

We would like to emphasize that the “resurgent” perspective on the
construction of the TBA system might have many interesting applications in
more general contexts. From the point of view of the theory of resurgence, the
data for the Riemann–Hilbert problem (the classical limit of the quantum
objects and the structure of their Stokes automorphisms) are quite universal,
and characterize many different quantum systems. This opens the possibility to
reformulate and solve quantum theories uniquely in terms of these data,
instead of using path integrals or Schrödinger-type equations222This point of
view has been advocated in particular by M. Kontsevich..

This paper is organized as follows. In section 2 we review some results on the
exact WKB method and the theory of resurgence which will be useful in the
paper. In section 3 we derive the TBA system. We present two different
derivations: the first one gives the TBA system as a solution to a
Riemann–Hilbert problem, while the second one is based on an analysis of the
Schrödinger equation along the lines of dt-stokes ; y-system . We also present
some general properties of the resulting TBA system and we discuss the wall-
crossing structure. In sections 4 and 5 we apply our general formalism to two
important examples in Quantum Mechanics, namely the cubic and the quartic
oscillators. We show in particular how to use our formalism to solve concrete
spectral problems. Finally, in section 6 we state our conclusions and list
some interesting open problems. The Appendix A contains some details on the
perturbative calculation of quantum periods.

##  2 The exact WKB method and resurgent Quantum Mechanics

In this section we give a lightning review of the exact WKB method from the
point of view of the theory of resurgence. This method, in its resurgent
incarnation, was developed in e.g. voros ; voros-quartic ; reshyper ; dpham ;
ddpham ; unfolding , and it has been recently reviewed in in-exactwkb . For
more general reviews of the theory of resurgence, see mmlargen ; abs .

Let us consider the stationary Schrödinger equation for a non-relativistic
particle in a potential $V(q)$ and with energy $E$:

$-\hbar^{2}\psi^{\prime\prime}(q)+(V(q)-E)\psi(q)=0.$ (2.1)

The standard WKB method produces asymptotic expansions in $\hbar$ for the
solutions to this equation. This goes as follows. Let us consider the
following ansatz

$\psi(q)=\exp\left[{{\rm i}\over\hbar}\int^{q}Q(q^{\prime}){\rm
d}q^{\prime}\right].$ (2.2)

Then, when plugged in (2.1), we obtain a Riccati equation for $Q(q)$,

$Q^{2}(q)-{\rm i}\hbar\frac{{\rm d}Q(q)}{{\rm d}q}=p^{2}(q),\quad
p(q)=(E-V(q))^{1/2},$ (2.3)

which can be solved in power series in $\hbar$:

$Q(q)=\sum_{k=0}^{\infty}Q_{k}(q)\hbar^{k}.$ (2.4)

The functions $Q_{k}$ can be calculated recursively. If we split the formal
power series in (2.4) into even and odd powers of $\hbar$,

$Q(q)=P(q)+Q_{\rm odd}(q),$ (2.5)

one finds that $Q_{\rm odd}(q)$ is a total derivative,

$Q_{\rm odd}(q)={{\rm i}\hbar\over 2}{{\rm d}\over{\rm d}q}\log P(q).$ (2.6)

The wavefunction (2.2) can then be written as

$\psi(q)={1\over{\sqrt{P(q)}}}\exp\left[{{\rm
i}\over\hbar}\int^{q}P(q^{\prime}){\rm d}q^{\prime}\right].$ (2.7)

The formal power series $P(q)$ has the structure,

$P(q)=\sum_{n\geq 0}p_{n}(q)\hbar^{2n},$ (2.8)

where $p_{0}(q)=p(q)$ is the classical momentum. Geometrically, we can regard
$P(q){\rm d}q$ as a meromorphic differential on the curve defined by

$y^{2}=2(E-V(q)).$ (2.9)

Note that if $V(q)$ is a polynomial potential, as we will assume in this
paper, then (2.9) is a hyperelliptic curve defining a Riemann surface
$\Sigma_{\rm WKB}$, which we will call WKB curve. This curve depends on a set
of moduli which include the energy $E$ and the parameters of the potential
$V(q)$.

The basic objects in the exact WKB method are the periods of $P(q){\rm d}q$
along the one-cycles $\gamma\in H_{1}(\Sigma_{\rm WKB})$, which we will call
WKB periods or quantum periods. We will denote them as

$\Pi_{\gamma}(\hbar)=\oint_{\gamma}P(q){\rm d}q,\qquad\gamma\in
H_{1}(\Sigma_{\rm WKB}).$ (2.10)

A related quantity is the so-called Voros multiplier or Voros symbol, which is
simply the exponent of the WKB period:

${\cal V}_{\gamma}=\exp\left({{\rm i}\over\hbar}\Pi_{\gamma}\right).$ (2.11)

As we will see, the algebra of Stokes discontinuities has a simpler expression
in terms of Voros multipliers. The WKB periods are formal power series in even
powers of $\hbar$, as $P(q)$,

$\Pi_{\gamma}(\hbar)=\sum_{n\geq
0}\Pi^{(n)}_{\gamma}\hbar^{2n},\qquad\Pi^{(n)}_{\gamma}=\oint_{\gamma}p_{n}(q){\rm
d}q.$ (2.12)

Note that the coefficients $\Pi^{(n)}_{\gamma}$ depend on the moduli of the
WKB curve. The calculation of these coefficients at high order can be quite
involved, even for simple quantum systems. In Appendix A we summarize two
powerful methods to calculate these quantum corrections.

In the exact WKB method, the quantum periods, which start their life as formal
power series in $\hbar$, are promoted to actual functions by the procedure of
Borel resummation. It turns out that, for fixed values of the moduli of the
curve, the series (2.12) has a doubly-factorial growth,

$\Pi^{(n)}_{\gamma}\sim(2n)!.$ (2.13)

The Borel transform of the quantum period is then defined as

$\widehat{\Pi}_{\gamma}(\xi)=\sum_{n\geq
0}{1\over(2n)!}\Pi^{(n)}_{\gamma}\xi^{2n},$ (2.14)

and it is analytic in a neighbourhood of the origin in the $\xi$-plane.
Moreover, it can be analytically continued to a function on the complex plane,
displaying in general various types of singularities (typically poles and
branch cuts.) The Borel resummation of the quantum period is defined by a
Laplace transform,

$s\left(\Pi_{\gamma}\right)(\hbar)={1\over\hbar}\int_{0}^{\infty}{\rm
e}^{-\xi/\hbar}\widehat{\Pi}_{\gamma}(\xi){\rm
d}\xi,\qquad\hbar\in{\mathbb{R}}_{>0}.$ (2.15)

If this integral converges when $\hbar$ is small enough, the quantum period is
said to be Borel summable. We will however need a more general definition of
Borel resummation along a direction in the complex plane, specified by an
angle $\varphi$:

$s_{\varphi}\left(\Pi_{\gamma}\right)(\hbar)={1\over\hbar}\int_{0}^{{\rm
e}^{{\rm i}\varphi}\infty}{\rm e}^{-\xi/\hbar}\widehat{\Pi}_{\gamma}(\xi){\rm
d}\xi.$ (2.16)

Let us note that $\Pi_{\gamma}$ is Borel summable along the direction
$\varphi$ if and only if the series $\Pi_{\gamma}({\rm e}^{{\rm
i}\varphi}\hbar)$ is Borel summable. Therefore, considering general directions
for the Borel resummation is equivalent to considering different rays in the
complex $\hbar$-plane.

One of the basic facts of the theory of Borel resummation is that
singularities of the Borel transform lead to obstructions to Borel
summability. For example, if $\widehat{\Pi}_{\gamma}$ has a singularity on the
positive real axis, then (2.15) is not defined. More precisely, singularities
lead to discontinuities in the Borel resummation: if $\widehat{\Pi}_{\gamma}$
has a singularity along a direction $\varphi$ in the Borel plane of the
variable $\xi$, there will be a jump in the directional Borel resummation
(2.16) as we cross the angle $\varphi$. To measure this discontinuity, it is
useful to introduce the lateral Borel resummations along a direction
$\varphi$:

$s_{\varphi\pm}(\Pi_{\gamma})\left({\rm e}^{{\rm
i}\varphi}\hbar\right)=\lim_{\delta\rightarrow
0}\,s\left(\Pi_{\gamma}\right)\left({\rm e}^{{\rm i}\varphi\pm{\rm
i}\delta}\hbar\right),\qquad\hbar\in{\mathbb{R}}_{>0},$ (2.17)

which are the Borel resummations just after and just before crossing the
discontinuity. Lateral Borel resummations can be equivalently defined in terms
of Borel resummations (2.16) along the directions $\varphi\pm\delta$. The
discontinuity across the direction $\varphi$ is now defined by,

${\rm
disc}_{\varphi}\,\left(\Pi_{\gamma}\right)=s_{\varphi+}(\Pi_{\gamma})-s_{\varphi-}(\Pi_{\gamma}).$
(2.18)

We can think of $s_{\varphi\pm}$ as operators, acting on formal power series.
One then defines the Stokes automorphism in the direction $\varphi$ as

$\mathfrak{S}_{\varphi}=s_{\varphi+}\circ s_{\varphi-}^{-1}.$ (2.19)

In this definition we follow the convention in ddpham  (the opposite
convention, which amounts to exchange $\mathfrak{S}_{\varphi}$ with
$\mathfrak{S}_{\varphi}^{-1}$, is used in e.g. in-exactwkb ). One of the key
points in the resurgent structure of Quantum Mechanics is that the operators
$s_{\varphi\pm}$ and $\mathfrak{S}_{\varphi}$, when acting on a quantum
period, are given by functions involving just the other quantum periods. In
other words, the structure of discontinuities in Quantum Mechanics involves
just the quantum periods on the WKB curve, and no other formal power series
intervene.

Figure 1: The WKB cycles for the Delabaere–Pham formula.

Let us give an important example of such a structure: the Delabaere–Pham
formula (Theorem 2.5.1 of dpham , see also Appendix A of in-exactwkb ). Let us
consider the quantum periods associated to the two WKB cycles shown in Fig. 1:
one of the cycles, $\gamma_{1}$, corresponds to a classically allowed
interval, while the cycle $\gamma_{2}$ corresponds to a classically forbidden
interval (in the language of Stokes graphs in-exactwkb , $\gamma_{2}$
encircles a regular saddle trajectory). The Delabaere–Pham formula states that
the quantum period $\Pi_{\gamma_{1}}$ is not Borel summable along the
direction $\varphi=0$, and therefore there will be a discontinuity as we cross
it. The discontinuity can be expressed in terms of the quantum period
$\Pi_{\gamma_{2}}$ (which is Borel summable), and it is given by

${\rm disc}\left(\Pi_{\gamma_{1}}\right)(\hbar)=-{\rm
i}\hbar\log\left(1+\exp\left(-{{\rm
i}\over\hbar}\Pi_{\gamma_{2}}(\hbar)\right)\right).$ (2.20)

(When no angle is indicated in a discontinuity formula, it is understood that
it refers to $\varphi=0$). In calculating the quantum period associated to the
cycle $\gamma_{2}$ around the forbidden interval, we have to make a choice of
branch cut of the momentum function $p(q)$. The choice is such that ${\rm
i}\Pi_{\gamma_{2}}$ is real and positive, therefore the discontinuity
appearing in the r.h.s. of (2.20) is exponentially small. In other words, it
is a non-perturbative effect in $\hbar$. The Delabaere–Pham formula can be
elegantly expressed in terms of Stokes automorphism and Voros multipliers, and
it reads:

$\mathfrak{S}\,{\cal V}_{\gamma_{1}}=\left(1+{\cal
V}^{-1}_{\gamma_{2}}\right){\cal V}_{\gamma_{1}}.$ (2.21)

Here we have assumed that there is a single cycle $\gamma_{2}$ associated to a
forbidden interval and intersecting the cycle $\gamma_{1}$. When there are
more cycles $\gamma_{a}$, $a=2,\cdots,r$, associated to forbidden intervals
and intersecting $\gamma_{1}$, the above formula generalizes to

$\mathfrak{S}\,{\cal V}_{\gamma_{1}}=\prod_{a=2}^{r}\left(1+{\cal
V}_{\gamma_{a}}^{-1}\right)^{\langle\gamma_{a},\gamma_{1}\rangle}{\cal
V}_{\gamma_{1}},$ (2.22)

where $\langle\gamma_{a},\gamma_{1}\rangle$ is the intersection of the cycles,
which can be appropriately defined after careful choices of orientation (see
dpham  for the details).

The Delabaere–Pham formula (2.21) provides a simple example of a resurgent
structure in Quantum Mechanics: the discontinuity of the Borel resummation of
the quantum period $\Pi_{\gamma_{1}}$ is completely encoded in another quantum
period $\Pi_{\gamma_{2}}$. Note that the discontinuity of $\Pi_{\gamma_{1}}$
is a non-perturbative effect, but at the same time it is determined by its
all-orders perturbative structure through the Borel transform.

One of the goals of the resurgent analysis of Quantum Mechanics is a complete
determination of the discontinuity structure of the quantum periods (or
equivalently, of the Voros multipliers). Such a result was obtained by Voros
in voros-quartic  in the case of the pure quartic oscillator in one dimension,
and further results in one-dimensional Quantum Mechanics were obtained in
dpham ; ddpham  and more recently in in-exactwkb . As we will see, this
discontinuity structure will be a crucial ingredient in our approach.

The other goal of resurgent Quantum Mechanics is to find exact quantization
conditions (EQCs) for spectral problems. The EQC is typically obtained as the
condition that a WKB solution decreasing along one ray in the complex plane
remains decreasing when it is continued along a different ray. To obtain these
conditions, we have to solve the “connection problem” of relating WKB
solutions in different regions. The correct solution of the connection problem
was found by Voros in voros ; voros-quartic , and independently by Silverstone
in silverstone . Using the Voros–Silverstone connection formula, one can
systematically derive EQCs for many problems in one-dimensional Quantum
Mechanics. As emphasized in bpv , they involve in principle all the (Borel
resummed) Voros multipliers associated to the WKB curve, and they typically
have the form of a single functional relation between them:

${\cal Q}\left({\cal V}_{\gamma_{1}},\cdots,{\cal V}_{\gamma_{r}}\right)=0.$
(2.23)

When the Voros multipliers are not Borel summable in the standard sense, one
has to use lateral Borel resummations, and the form of the EQC depends on the
choice of lateral resummation. Different forms of the EQC are then related by
the corresponding Stokes automorphisms. In this paper we will analyze EQCs in
some concrete examples and we will rely on the results obtained in voros ;
voros-quartic ; dpham ; ddpham ; unfolding 333Historically, some of these EQCs
were obtained in the framework of instanton calculus, see e.g. zinn-justin ;
zjj1 ; zjj2 . An alternative approach to derive EQCs is the uniform WKB
method, see e.g. alvarez ; alvarez-casares2 ; alvarez-casares ; dunne-unsal .

Figure 2: The spectrum of a quantum system can be regarded as a discrete
series of energies for a fixed value of $\hbar$, or conversely, as a discrete
series of values of the inverse Planck constants $x=\hbar^{-1}$ for a fixed
value of the energy $E$. In this figure we show the very first energy levels
$E_{n}$, as functions of $x$, and the sequence of $x_{n}$ corresponding to the
energy $E$.

Let us make a final remark on the use of EQCs like (2.23). The Voros
multipliers depend on $\hbar$ and on the moduli of the WKB curve $\Sigma_{\rm
WKB}$, in particular on the energy $E$. Assuming that the spectral problem
leads to a discrete set of energies (or resonances), the EQCs define a
discrete, infinite family of codimension one submanifolds in the “extended”
moduli space parametrized by $\hbar$, $E$ and the additional parameters of the
potential. In particular, if we fix these additional parameters, we obtain a
discrete, infinite family of curves in the $(E,\hbar)$ plane, labelled by the
quantum number $n=0,1,2,\cdots$. Usually, one thinks of these curves as the
discrete values of the energy of the system $E_{n}(\hbar)$, for a fixed value
of $\hbar$. However, one can reverse this relation and think of the spectrum
of a Hamiltonian as a series of discrete values $x_{n}(E)$ of the inverse
Planck constant

$x={1\over\hbar},$ (2.24)

for a fixed value of the energy $E$. By construction, the two spectra
$E_{n}(\hbar)$ and $x_{n}(E)$ are related by

$E_{n}\left({1\over x_{n}(E)}\right)=E.$ (2.25)

This is illustrated in Fig. 2. This alternative view of the spectrum of a
Hamiltonian in terms of the values $x_{n}(E)$ has been advocated by Voros (see
e.g. voros-quartic ) and we will call it the Voros spectrum of the
Hamiltonian. This is in fact the natural form of the spectrum when we think
about the WKB periods as solutions of a Riemann–Hilbert problem. As we will
see in this paper, our TBA equations, combined with EQCs, will give the Voros
spectrum, rather than the traditional spectrum. We will derive the TBA
equations satisfied by the quantum periods in the next section.

##  3 TBA equations and a generalized ODE/IM correspondence

Figure 3: To derive the TBA system obeyed by the WKB periods, we consider a
region in moduli space where the discontinuities of their Borel resummations
are encoded by the Delabarere–Pham formula. In this region all the turning
points are real and different, and we have a sequence of cycles encircling
classically allowed and classically forbidden intervals in an alternate way.

In this section we will derive the TBA equations obeyed by the quantum
periods. We will provide two different derivations. In the first one, we will
consider the Riemann–Hilbert problem for the periods stated by Voros in voros-
quartic , namely: given their classical limit $\Pi_{\gamma}^{(0)}$, and the
discontinuities of their Borel-resummed versions, can we reconstruct the WKB
periods exactly? Then, following gmn , we will write a solution of this
problem, in terms of this TBA system, which satisfies this data. The second
derivation will be more constructive, in the spirit of y-system ; ito-shu .

###  3.1 TBA equations and Riemann-Hilbert problem

Let us consider a polynomial potential of degree $r+1$, $V_{r+1}(q)$. The WKB
curve is a hyperelliptic curve describing a Riemann surface of genus
$g=[r/2]$. Let us consider a region in the moduli space of the curve where all
the turning points $q_{i}$, $i=1,\cdots,r+1$, are real and different. Without
loss of generality, let us order the turning points as

$q_{1}<q_{2}<\cdots<q_{r+1}.$ (3.1)

In addition, let us choose the sign of the term $q^{r+1}$ in the potential in
such a way that the interval

$[q_{i},q_{i+1}],\qquad i=1,\cdots,r,$ (3.2)

corresponds to a classically allowed interval when $i$ is odd, and to a
classically forbidden interval when $i$ is even. We then have a situation like
the one depicted in Fig. 3. The cycle $\gamma_{i}$, $i=1,\cdots,r$, goes
around the interval $[q_{i},q_{i+1}]$. We choose the branch cuts and
orientations of the cycles in such a way that

$m_{2i-1}=\Pi^{(0)}_{\gamma_{2i-1}}=2\int_{q_{2i-1}}^{q_{2i}}p(q){\rm
d}q,\qquad m_{2i}={\rm i}\,\Pi^{(0)}_{\gamma_{2i}}=2{\rm
i}\int_{q_{2i}}^{q_{2i+1}}p(q){\rm d}q,$ (3.3)

are real and positive. Borrowing the language of two-dimensional integrable
theories, we will often refer to these quantities as “masses.” The
Delabaere–Pham formula says that the quantum periods $\Pi_{\gamma_{2i}}$ are
Borel summable along the positive real axis of $\hbar$, while
$\Pi_{\gamma_{2i-1}}$ are not. In addition, we have the following
discontinuity formula,

${\rm disc}\,\Pi_{\gamma_{2i-1}}=-{\rm i}\hbar\log\left(1+\exp\left(-{{\rm
i}\over\hbar}\Pi_{\gamma_{2i-2}}(\hbar)\right)\right)-{\rm
i}\hbar\log\left(1+\exp\left(-{{\rm
i}\over\hbar}\Pi_{\gamma_{2i}}(\hbar)\right)\right),\qquad$ (3.4)

Equivalently, in terms of Stokes automorphisms and Voros multipliers, we have

$\mathfrak{S}\,{\cal V}_{\gamma_{2i-1}}=\left(1+{\cal
V}^{-1}_{\gamma_{2i-2}}\right)\left(1+{\cal V}^{-1}_{\gamma_{2i}}\right){\cal
V}_{\gamma_{2i-1}}.$ (3.5)

A similar discontinuity formula holds along the negative real axis (this
already follows from the fact that the quantum periods are formal power series
in even powers of $\hbar$.) However, there is an additional discontinuity in
the complex plane of $\hbar$. The reason is that, as we rotate
$\hbar\rightarrow\pm{\rm i}\hbar$, $\hbar\in{\mathbb{R}}$, classically allowed
and classically forbidden intervals are exchanged. Along this direction we
have a discontinuity formula like (3.4), (3.5), but now the odd cycles are
exchanged with the even cycles. In order to provide a unified description of
both types of discontinuities, we introduce the functions

$\displaystyle-{\rm i}\epsilon_{2i-1}\left(\theta+{{\rm i}\pi\over 2}\pm{\rm
i}\delta\right)$
$\displaystyle={1\over\hbar}s_{\pm}\left(\Pi_{\gamma_{2i-1}}\right)(\hbar),$
(3.6) $\displaystyle-{\rm i}\epsilon_{2i}\left(\theta\right)$
$\displaystyle={1\over\hbar}s\left(\Pi_{\gamma_{2i}}\right)(\hbar),$

where $0<\delta\ll 1$, and we identify

${\rm e}^{\theta}={1\over\hbar}.$ (3.7)

The Delabaere–Pham discontinuities can now be encoded in a single formula,

${\rm disc}_{\pi/2}\epsilon_{a}(\theta)=L_{a-1}(\theta)+L_{a+1}(\theta),\qquad
a=1,\cdots,r,$ (3.8)

where

$L_{a}(\theta)=\log\left(1+{\rm e}^{-\epsilon_{a}(\theta)}\right),$ (3.9)

and with the understanding that $L_{0}=L_{r+1}=0$. A similar formulae holds
for the discontinuity at $-\pi/2$. We now have to solve the Riemann–Hilbert
problem for the functions $\epsilon_{a}(\theta)$, $a=1,\cdots,r$, which
satisfy the discontinuity formula (3.8) and have the asymptotic behavior

$\epsilon_{a}(\theta)-m_{a}{\rm e}^{\theta}={\cal O}({\rm
e}^{-\theta}),\qquad\theta\rightarrow\infty.$ (3.10)

The solution to this problem is given by the following TBA system

$\epsilon_{a}(\theta)=m_{a}{\rm
e}^{\theta}-\int_{\mathbb{R}}{L_{a-1}(\theta^{\prime})\over\cosh(\theta-\theta^{\prime})}{{\rm
d}\theta^{\prime}\over
2\pi}-\int_{\mathbb{R}}{L_{a+1}(\theta^{\prime})\over\cosh(\theta-\theta^{\prime})}{{\rm
d}\theta^{\prime}\over 2\pi},\qquad a=1,\cdots,r.$ (3.11)

When we expand (3.11) at large $\theta$, we find the all-orders asymptotic
expansion

$\epsilon_{a}(\theta)\sim m_{a}{\rm e}^{\theta}+\sum_{n\geq 1}m_{a}^{(n)}{\rm
e}^{(1-2n)\theta},$ (3.12)

where

$m_{a}^{(n)}={(-1)^{n}\over\pi}\int_{\mathbb{R}}{\rm
e}^{(2n-1)\theta}\left(L_{a-1}(\theta)+L_{a+1}(\theta)\right){\rm d}\theta.$
(3.13)

In view of (3.6), the WKB expansion of the periods can be recovered from the
above expansion as

$m_{2i-1}^{(n)}=(-1)^{n}\Pi^{(n)}_{\gamma_{2i-1}},\qquad m_{2i}^{(n)}={\rm
i}\Pi^{(n)}_{\gamma_{2i}}.$ (3.14)

The TBA system (3.11) is the “conformal limit” of the TBA system considered in
y-system ; hito , in which one replaces $m_{a}\cosh(\theta)$ by $m_{a}{\rm
e}^{\theta}$. This could have been anticipated based on the results of oper .
In the above derivation we have chosen a particular region in moduli space.
From the point of view of resurgence, this region is special in the sense that
the discontinuities of the periods are captured by the Delabaere–Pham formula
in a simple way. In toledo ; toledo-thesis  this region is called the “minimal
chamber” because it leads to a minimal number of $\epsilon$ functions.

###  3.2 Generalized ODE/IM correspondence

Let us now offer a more direct derivation of the TBA system, following the
techniques in y-system . We start from the Schrödinger equation for a
potential of degree $r+1$, which we write as

$\left(-\partial_{z}^{2}+z^{r+1}+\sum_{a=1}^{r}b_{a}z^{r-a}\right){\psi(z,b_{a})}=0.$
(3.15)

Here $z$ is a complex variable, and $b_{a}$, $a=1,\cdots,r$, are complex
coefficients. The case in which $b_{1}=\cdots=b_{r-1}=0$ and $b_{r}\neq 0$ is
the one considered in the usual ODE/IM correspondence, where $b_{r}$ plays the
rôle of spectral parameter. The derivation that follows can be regarded as a
generalization of the techniques of blz-spectral ; dt-stokes  to general
coefficients $b_{a}$.

The equation (3.15) is invariant under the Symanzik rotation

$(z,b_{a})\to(\omega z,\omega^{a+1}b_{a}),\qquad\omega={\rm e}^{\frac{2\pi{\rm
i}}{r+3}}.$ (3.16)

Let us define the coefficients $B_{m}$ as

$\left(1+\sum_{a=1}^{r}b_{a}z^{-a-1}\right)^{1/2}:=1+\sum_{m=1}^{\infty}B_{m}z^{-m}.$
(3.17)

By using the WKB approximation, one finds that the fast decaying solution of
(3.15) at infinity around the positive real axis behaves as Sibuya

$y(z,b_{a})\sim\frac{1}{\sqrt{2{\rm
i}}}z^{n_{r}}\exp\Big{(}-\frac{2}{r+3}z^{\frac{r+3}{2}}\Big{)},$ (3.18)

where $n_{r}$ is defined by

$n_{r}=\begin{cases}-\frac{r+1}{4}&\quad r+1:\mbox{odd}\\\
-\frac{r+1}{4}-B_{\frac{r+3}{2}}&\quad r+1:\mbox{even}\end{cases}$ (3.19)

Under the Symanzik rotation we have

$\displaystyle
B_{\frac{r+3}{2}}(\omega^{-k(a+1)}b_{a})=(-1)^{k}B_{\frac{r+3}{2}}(b_{a}).$
(3.20)

Let us now consider the sector in the complex $z$-plane,

${\cal
S}_{k}=\left\\{z\in{\mathbb{C}}:\left|\mbox{arg}(z)-\frac{2k\pi}{r+3}\right|<\frac{\pi}{r+3}\right\\}.$
(3.21)

The solution (3.18) is the subdominant solution at infinity in the sector
${\cal S}_{0}$. By using the Symanzik rotation, we can obtain the subdominant
solution in the sector ${\cal S}_{k}$, as

$y_{k}(z,b_{a})=\omega^{\frac{k}{2}}y(\omega^{-k}z,\omega^{-(a+1)k}b_{a}).$
(3.22)

As is well-known, the Wronskian of two solutions

$\displaystyle W_{k_{1},k_{2}}(b_{a})\equiv
W[y_{k_{1}}(z,b_{a}),y_{k_{2}}(z,b_{a})]=y_{k_{1}}(z,b_{a})\partial_{z}y_{k_{2}}(z,b_{a})-y_{k_{2}}(z,b_{a})\partial_{z}y_{k_{1}},$
(3.23)

is independent of $z$. We introduce the following notation for the shift of
functions:

$\displaystyle f^{[j]}(z,b_{a})$
$\displaystyle:=f(\omega^{-j/2}z,\omega^{-j(a+1)/2}b_{a}).$ (3.24)

It is easy to find that

$\displaystyle W_{k_{1}+1,k_{2}+1}(b_{a})=W_{k_{1},k_{2}}^{[2]}(b_{a}).$
(3.25)

In addition, one can evaluate

$\displaystyle W_{k,k+1}(b_{a})$ $\displaystyle=\begin{cases}1&\quad
r+1:\mbox{odd}\\\ \omega^{(-1)^{k}B_{\frac{r+3}{2}}}&\quad
r+1:\mbox{even}\end{cases}$ (3.26)

Let us now introduce the Y-functions as y-system ; ito-shu

$\displaystyle{\cal Y}_{2j}(b_{a})$
$\displaystyle=\frac{W_{-j,j}(b_{a})W_{-j-1,j+1}(b_{a})}{W_{-j-1,-j}(b_{a})W_{j,j+1}(b_{a})},$
(3.27) $\displaystyle{\cal Y}_{2j+1}(b_{a})$
$\displaystyle=\left[\frac{W_{-j-1,j}(b_{a})W_{-j-2,j+1}(b_{a})}{W_{-j-2,-j-1}(b_{a})W_{j,j+1}(b_{a})}\right]^{[+1]},$

where $j\in{\mathbb{Z}}_{\geq 0}$. By using repeatedly the Plücker type
relation

$\displaystyle W_{k_{1},k_{2}}^{[+2]}W_{k_{1},k_{2}}$
$\displaystyle=W_{k_{1}+1,k_{2}+1}W_{k_{1},k_{2}}=-W_{k_{1}+1,k_{2}}W_{k_{2}+1,k_{1}}-W_{k_{1}+1,k_{1}}W_{k_{2},k_{2}+1},$
(3.28)

we find

$\displaystyle{\cal Y}_{s}^{[+1]}(b_{a}){\cal Y}_{s}^{[-1]}(b_{a})$
$\displaystyle=\Big{(}1+{\cal Y}_{s-1}(b_{a})\Big{)}\big{(}1+{\cal
Y}_{s+1}(b_{a})\Big{)}.$ (3.29)

By definition, ${\cal Y}_{0}=0$. Note that there are $r+3$ sectors on the
$z$-complex plane. Since $y_{j+r+3}(z)$ and $y_{j}({\rm e}^{-2\pi{\rm i}}z)$
are decaying solutions in the same sector, $y_{j+r+3}(z)\propto y_{j}({\rm
e}^{-2\pi{\rm i}}z)$. We then obtain $y_{j+r+3}(z)\propto y_{j}({\rm
e}^{-2\pi{\rm i}}z)=y_{j}(z)$, because the monodromy around the origin is
trivial. We conclude that ${\cal Y}_{r+1}=0$. Therefore, we have $r$
Y-functions, which lead to a Y-system of the $A_{r}$-type. In the special case
where $b_{1}=\cdots=b_{r-1}=0$, (3.29) reproduces the Y-system in the usual
ODE/IM correspondence.

The system (3.29) is a generalization of the Y-system in dt-stokes ; ito-shu .
However, the parameters $b_{a}$ appearing on the l.h.s. are different from the
ones in the r.h.s., and the Y-system is relating the Y-functions at different
points of moduli space. To write down a closed TBA system, one should consider
$b_{a}$ as independent parameters like in the derivation done in masoero .
However, this procedure will be rather complicated in general. To avoid this
problem and find a closed system, we introduce an spectral parameter $\zeta$,
and we rescale the variables in (3.15) as

$\displaystyle q=\zeta^{\frac{2}{r+3}}z,\quad u_{a}$
$\displaystyle=-\zeta^{\frac{2(a+1)}{r+3}}b_{a},\qquad a=1,2,\cdots,r.$ (3.30)

In the new variables, the Schrödinger equation (3.15) can be written as

$\displaystyle\left(-\zeta^{2}\partial_{q}^{2}+q^{r+1}-\sum_{a=1}^{r}u_{a}q^{r-a}\right)\hat{\psi}(q,u_{a},\zeta)=0.$
(3.31)

Therefore, the spectral parameter $\zeta$ plays the rôle of $\hbar$. We can
now write the solution $y(z,b_{a})$ in terms of $q$ and $u_{a}$:

$\hat{y}(q,u_{a},\zeta)=y(z,b_{a})=y(\zeta^{-\frac{2}{r+3}}q,-\zeta^{-\frac{2(a+1)}{r+3}}u_{a}).$
(3.32)

The Symanzik rotation (3.16) can be written as a rotation of $\zeta$:

$\displaystyle(\omega^{-k}z,\omega^{-(a+1)k}b_{a})=\left(({\rm e}^{{\rm i}\pi
k}\zeta)^{-\frac{2}{r+3}}q,-({\rm e}^{{\rm i}\pi
k}\zeta)^{-\frac{2(a+1)}{r+3}}u_{a}\right).$ (3.33)

The solutions $y_{k}(z,b_{a})$ can then be obtained from
$\hat{y}(q,u_{a},\zeta)$ by using

$\hat{y}_{k}(q,u_{a},\zeta)=\omega^{\frac{k}{2}}\hat{y}(q,u_{a},{\rm e}^{{\rm
i}\pi k}\zeta).$ (3.34)

Note that the solution does not go back to itself by rotating ${\rm
e}^{2\pi{\rm i}}\zeta$. Let us now introduce the Wronskian in terms of
$\hat{y}_{j}$:

$\displaystyle\hat{W}_{k_{1},k_{2}}(\zeta,u_{a})$
$\displaystyle=\zeta^{\frac{2}{r+3}}\Big{(}\hat{y}_{k_{1}}(q,u_{a},\zeta)\partial_{q}\hat{y}_{k_{2}}(q,u_{a},\zeta)-\hat{y}_{k_{2}}(q,u_{a},\zeta)\partial_{q}\hat{y}_{k_{1}}(q,u_{a},\zeta)\Big{)}$
(3.35) $\displaystyle=W_{k_{1},k_{2}}(b_{a}).$

We then introduce the Y functions, depending on the spectral parameter
$\zeta$, as

$Y_{2j}(\zeta,u_{a})=\frac{\hat{W}_{-j,j}\hat{W}_{-j-1,j+1}}{\hat{W}_{-j-1,-j}\hat{W}_{j,j+1}}(\zeta,u_{a}),~{}~{}Y_{2j+1}({\rm
e}^{-\frac{\pi{\rm
i}}{2}}\zeta,u_{a})=\frac{\hat{W}_{-j-1,j}\hat{W}_{-j-2,j+1}}{\hat{W}_{-j-2,-j-1}\hat{W}_{j,j+1}}(\zeta,u_{a}).$
(3.36)

They satisfy

$\displaystyle Y_{s}(\zeta{\rm e}^{\frac{\pi{\rm i}}{2}},u_{a})Y_{s}(\zeta{\rm
e}^{-\frac{\pi{\rm i}}{2}},u_{a})$
$\displaystyle=\Big{(}1+Y_{s+1}(\zeta,u_{a})\Big{)}\big{(}1+Y_{s-1}(\zeta,u_{a})\Big{)},$
(3.37)

where $Y_{0}(\zeta,u_{a})=Y_{r+1}(\zeta,u_{a})=0$ (hereafter, we will omit the
argument $u_{a}$ in the Y-functions.) The Y-systems (3.29) and (3.37) are
essentially the same, but with different spectral parameters. Y-system (3.37)
also appears in the study of minimal area surfaces in AdS3 y-system ; hito .

Let us now calculate the small $\zeta$ behavior of the Y-functions (3.36). The
WKB expansion of $\hat{y}_{k}(q,u_{a},\zeta)$ for small $\zeta$ is obtained as

$\hat{y}_{k}(q,u_{a},\zeta)\sim(-1)^{\frac{k}{2}}c(\zeta)\exp\left(-\frac{{\rm
i}{\delta}_{k}}{\zeta}\int^{q}_{q^{(k)}}P(q^{\prime}){\rm
d}q^{\prime}\right).$ (3.38)

In this equation, $c(\zeta)={1\over\sqrt{2{\rm
i}}}\zeta^{\frac{r+1}{2(r+3)}}$, $P(q)$ was defined in (2.5), $q^{(k)}$ is a
point in the sector ${\cal S}_{k}$, and ${\delta}_{k}=\pm(-1)^{k}$ is a sign
factor, where $\pm$ depends on the sheet of the Riemann surface where $q$
lives. Using this WKB form, we can evaluate the Wronskian
$\hat{W}_{k_{1},k_{2}}(\zeta)$, which only depends on the initial points
$q^{(k_{1})}$ and $q^{(k_{2})}$. Substituting this evaluation into the
Y-functions (3.36), we find that, as $\zeta\to 0$, the Y-functions behave as

$\displaystyle\log{Y}_{2k+1}(\zeta,u_{a})$
$\displaystyle\sim-\frac{1}{\zeta}\oint_{\gamma_{r-2k}}p(q){\rm
d}q=-\frac{m_{r-2k}}{\zeta},$ (3.39) $\displaystyle\log{Y}_{2k}(\zeta,u_{a})$
$\displaystyle\sim-\frac{{\rm i}}{\zeta}\oint_{\gamma_{r+1-2k}}p(q){\rm
d}q=-\frac{m_{r+1-2k},}{\zeta},$

which are valid in the region $|\mbox{arg}(\zeta)|<\pi$. Note that, with the
definition (3.36), the $Y_{k}$ function corresponds to the cycle that we
labelled as $r+1-k$. From now on we will therefore set $Y_{k}\rightarrow
Y_{r+1-k}$, which preserves the form of the Y-system (3.37).

It is now a standard exercise to convert the Y-system (3.37) into a set of TBA
equations, as done in y-system . First we introduce an analytic function in
$\left|\arg(\zeta)\right|\leq\frac{\pi}{2}$:

$\ell_{a}(\zeta):=\log Y_{a}(\zeta)+\frac{m_{a}}{\zeta}.$ (3.40)

The Y-system then leads to

$\displaystyle\ell_{a}({\rm e}^{\frac{\pi{\rm i}}{2}}\zeta)+\ell_{a}({\rm
e}^{-\frac{\pi{\rm
i}}{2}}\zeta)=\log\Big{(}(1+Y_{a+1}(\zeta))(1+Y_{a-1}(\zeta))\Big{)}.$ (3.41)

Let us set

$\zeta={\rm e}^{-\theta},\qquad Y_{a}(\zeta)={\rm e}^{-\epsilon_{a}(\theta)}$
(3.42)

and let us introduce the kernel

$K(\theta)=\frac{1}{2\pi\cosh(\theta)}.$ (3.43)

By convoluting (3.41) with this kernel and exploiting the analyticity of
$\ell_{a}(e^{-\theta})$, we obtain again the TBA equations (3.11).

Let us note that the $\epsilon$ functions introduced in (3.42), in terms of
the Y-functions defined in (3.36), are not manifestly equal to the $\epsilon$
functions introduced in (3.6), which are defined as resummed quantum periods.
Of course, it follows from our derivations that they satisfy the same TBA
system with the same asymptotics, therefore they are indeed equal. Their
equality should also follow by a direct comparison of their definitions, along
the lines of allegretti .

###  3.3 Effective central charge and PNP relations

The TBA equations (3.11) have some interesting general properties. First of
all, the functions $\epsilon_{a}(\theta)$ have a constant value as
$\theta\rightarrow-\infty$, which we will denote by

$\epsilon_{a}^{\star}=\lim_{\theta\to-\infty}\epsilon_{a}(\theta).$ (3.44)

As a consequence of (3.41), the corresponding $Y_{a}^{\star}$ values satisfy
the equations

$\left(Y_{a}^{\star}\right)^{2}=(1+Y^{\star}_{a-1})(1+Y^{\star}_{a+1}).$
(3.45)

The solution to these equations is (see e.g. ddt )

$Y_{a}^{\star}={\sin\left(\frac{\pi
a}{r+3}\right)\sin\left(\frac{\pi(a+2)}{r+3}\right)\over\sin^{2}\left(\frac{\pi}{r+3}\right)}.$
(3.46)

It is possible to calculate the “effective central charge” associated to this
TBA system in terms of (3.44), by using the identity zamo-TBA ; kl-mel-1

$c_{{\rm eff}}={6\over\pi^{2}}\sum_{a=1}^{r}m_{a}\int_{\mathbb{R}}{\rm
e}^{\theta}L_{a}(\theta){\rm
d}\theta=r+{3\over\pi^{2}}\sum_{a=1}^{r}\left(\epsilon_{a}^{\star}\log\left(1+{\rm
e}^{\epsilon_{a}^{\star}}\right)+2\,{\rm Li}_{2}(-{\rm
e}^{\epsilon_{a}^{\star}})\right).$ (3.47)

This can be evaluated in closed form and one finds,

$c_{\rm eff}={r(r+1)\over r+3}.$ (3.48)

In particular, this quantity is independent of the moduli. We will now relate
the effective central charge to the so-called PNP relations, which provide a
non-trivial and useful relation between the quantum periods. To simplify our
discussion, we will assume that $r=2g$ is even. In this case, one can
construct the following symplectic basis of cycles,

$A_{j}=\gamma_{2j-1},\qquad B_{j}=\sum_{k=j}^{g}(-1)^{k-j}\gamma_{2k},$ (3.49)

where $\gamma_{a}$, $a=1,\cdots,2g$ are the cycles defined in section 3.1. We
have chosen the cycles such that the intersection number of A and B cycles are
$\langle A_{j},A_{k}\rangle=0=\langle B_{j},B_{k}\rangle$, $\langle
A_{j},B_{k}\rangle=\delta_{jk}$. Let us now introduce periods associated to
this basis:

$\nu_{j}={1\over 2\pi}\oint_{A_{j}}P(q){\rm d}q,\qquad\nu_{D,j}={\rm
i}\oint_{B_{j}}P(q){\rm d}q.$ (3.50)

These are the quantum analogues of the $a$ and the $a_{D}$ variables in
Seiberg–Witten theory. They have a power series expansion of the form,

$\nu_{j}=\sum_{n\geq 0}\nu_{j}^{(n)}\hbar^{2n},\qquad\nu_{D,j}=\sum_{n\geq
0}\nu_{D,j}^{(n)}\hbar^{2n},$ (3.51)

We then define the quantum, or NS free energy, by

${\partial F^{\rm NS}\over\partial\nu_{j}}=\nu_{D,j},\qquad j=1,\cdots,g.$
(3.52)

This is the analogue of the Nekrasov-Shatashvili (NS) free energy ns , and it
can be computed systematically as a formal power series of the form,

$F^{\rm NS}(\nu)=\sum_{n\geq 0}F_{n}^{\rm NS}(\nu)\hbar^{2n}.$ (3.53)

Its classical limit, $F_{0}^{\rm NS}=F_{0}$, is the analogue of the
Seiberg–Witten prepotential sw . We will now introduce an additional quantum
period which can be used to formulate the PNP relations. First of all we
define the so-called “projective coordinates” $X_{j}$, $j=0,1,\cdots,g$, as

$X_{0}={1\over\hbar},\qquad\nu_{j}={X_{j}\over X_{0}},\qquad j=1,\cdots,g.$
(3.54)

The quantum free energy can be promoted to a function of the projective
coordinates as

${\cal F}(X,X_{0})=\sum_{n\geq 0}X_{0}^{2-2n}F_{n}^{\rm NS}(\nu).$ (3.55)

From this we construct the quantum period

${\cal J}=-{1\over X_{0}}{\partial{\cal F}\over\partial X_{0}},$ (3.56)

which is a formal power series of the form

${\cal J}=\sum_{n\geq 0}{\cal J}_{n}\hbar^{2n},$ (3.57)

and we have

${\cal J}_{n}=2(n-1)F_{n}(\nu)+\sum_{j=1}^{g}\nu_{j}{\partial F_{n}^{\rm
NS}\over\partial\nu_{j}},\qquad n\geq 0.$ (3.58)

We note that the classical limit of ${\cal J}$ is given by

${\cal J}_{0}=-2F^{\rm NS}_{0}+\sum_{j=1}^{g}\nu_{j}{\partial
F_{0}\over\partial\nu_{j}}.$ (3.59)

In the case of Seiberg–Witten curves, ${\cal J}_{0}$ is a simple function of
the moduli, in virtue of the so-called Matone relation matone  and its
generalizations eguchi-yang ; sty . Let us now compute the next-to-leading
order. From the definition of the quantum free energy, one has

${\partial F_{1}^{\rm
NS}\over\partial\nu_{j}}=\nu_{D,j}-\sum_{k=1}^{g}\nu_{k}{\partial^{2}F_{0}\over\partial\nu_{j}\partial\nu_{k}}.$
(3.60)

It follows that

${\cal
J}_{1}=\sum_{j=1}^{g}\left(\nu^{(0)}_{j}\nu_{D,j}^{(1)}-\nu^{(1)}_{j}\nu_{D,j}^{(0)}\right).$
(3.61)

Up to a normalization, this is precisely the effective central charge
associated to the TBA system, so one can evaluate

${\cal J}_{1}=-{c_{\rm eff}\over 12}.$ (3.62)

In particular, this quantity is independent of the moduli.

In quantum-mechanical models based on a genus one curve, with two independent
quantum periods, one can find a non-trivial relation between them alvarez-
casares ; alvarez ; dunne-unsal ; gorsky ; basar-dunne ; cm-ha ; bdu-quantum .
We will refer to this relation as the PNP relation. It can be formulated in
terms of the formal power series ${\cal J}$ as

${\cal J}=\alpha E+\beta\hbar^{2},$ (3.63)

where $E$ is the energy and $\alpha,\beta$ are constants. The leading order of
this relation is just the analogue of Matone’s relation for the corresponding
elliptic curve, while the next-to-leading order of the equation states that
${\cal J}_{1}$ is a constant. This is precisely the content of (3.62), and one
finds that

$\beta=-{c_{\rm eff}\over 12}.$ (3.64)

Let us note that (3.63) is the “integrated” version of the PNP relation
presented in cm-ha . The version one typically finds in the literature (e.g.
in dunne-unsal ) is obtained from (3.63) by taking an additional derivative
w.r.t. $E$, and it misses the additional information carried by the constant
$\beta$.

###  3.4 Wall-crossing

The derivations we have presented in section 3.1 are valid in the “minimal”
chamber of moduli space, in which the masses are real. However, one can
analytically continue the resulting TBA equations as follows. Let us assume
that $m_{a}$ are now complex numbers, and let us write them as

$m_{a}=|m_{a}|{\rm e}^{{\rm i}\phi_{a}},\qquad a=1,\cdots,r.$ (3.65)

Let us also introduce the shifted functions

$\widetilde{\epsilon}_{a}(\theta)=\epsilon_{a}(\theta-{\rm
i}\phi_{a}),\qquad\widetilde{L}_{a}(\theta)=L_{a}(\theta-{\rm i}\phi_{a}).$
(3.66)

With this definition, $\widetilde{\epsilon}_{a}(\theta)\sim|m_{a}|e^{\theta}$
at large $\theta$ along the positive real axis. Using the shifted functions
(3.66), we obtain the TBA system

$\widetilde{\epsilon}_{a}(\theta)=|m_{a}|e^{\theta}-\int_{{\mathbb{R}}}\frac{\widetilde{L}_{a-1}(\theta^{\prime})}{\cosh(\theta-\theta^{\prime}-{\rm
i}\phi_{a}+{\rm i}\phi_{a-1})}\frac{{\rm
d}\theta^{\prime}}{2\pi}-\int_{{\mathbb{R}}}\frac{\widetilde{L}_{a+1}(\theta^{\prime})}{\cosh(\theta-\theta^{\prime}-{\rm
i}\phi_{a}+{\rm i}\phi_{a+1})}\frac{{\rm d}\theta^{\prime}}{2\pi}.$ (3.67)

We denote the kernels appearing in the integral equations by444Our convention
is different from the one in y-system .

$K_{r,s}(\theta)={1\over 2\pi}{1\over\cosh\left(\theta+{\rm
i}(\phi_{s}-\phi_{r})\right)}.$ (3.68)

The convolution will be denoted by:

$(K\star
f)(\theta)=\int_{\mathbb{R}}K(\theta-\theta^{\prime})f(\theta^{\prime}){\rm
d}\theta^{\prime}.$ (3.69)

Finally, the superscripts $\pm$ denote a shift of the argument by $\pm{\rm
i}\pi/2$:

$f^{\pm}(\theta)=f\left(\theta\pm{\pi{\rm i}\over 2}\right).$ (3.70)

The TBA equations (3.67) can now be written in a more compact form as

$\widetilde{\epsilon}_{a}=|m_{a}|e^{\theta}-K_{a,a-1}\star\widetilde{L}_{a-1}-K_{a,a+1}\star\widetilde{L}_{a+1}.$
(3.71)

Note that the kernel of (3.67) has poles when

$|\phi_{a}-\phi_{a\pm 1}|=\frac{\pi}{2}.$ (3.72)

Therefore, the equations (3.67) are only valid in the region

$|\phi_{a}-\phi_{a\pm 1}|<\frac{\pi}{2}.$ (3.73)

As different $\phi_{a}-\phi_{a\pm 1}$ cross the values
$\pm\frac{\pi}{2},\pm\frac{3\pi}{2},\cdots$, we should modify the TBA
equations to pick the contributions of the poles. This phenomenon is known as
the wall crossing of the TBA equations, and it goes back to the discussion in
e.g. dt-excited . From the point of view of resurgent Quantum Mechanics, what
happens is that, as we move in moduli space, the structure of discontinuities
(or equivalently, of the Stokes automorphisms) becomes more complicated. In
particular, more cycles are involved. This was presciently described in
reshyper  in terms of the appearance (or disappearance) of what they called
geodesic cycles, which are nothing but the BPS trajectories studied in e.g.
selfdual ; shapere-vafa ; gmn2 . In the case of the TBA system considered in
gmn , the wall-crossing phenomenon is indeed due to the change of BPS spectra
in the underlying ${\cal N}=2$ gauge theory, as one moves in moduli space. In
this paper, our study of wall-crossing will rely on the approach put forward
in y-system  and further developed by J. Toledo in toledo ; toledo-thesis ,
based on the analytic continuation of the TBA system.

Let us start by considering the simple case of $r=2$. Let us suppose that we
move in moduli space from a situation in which (3.73) holds, to a situation in
which

$\phi_{2}-\phi_{1}>{\pi\over 2}.$ (3.74)

Wall-crossing occurs when $\phi_{2}-\phi_{1}=\pi/2$. When this happens the
periods satisfy the condition

${\rm Im}\left({\Pi_{\gamma_{1}}\over\Pi_{\gamma_{2}}}\right)=0,$ (3.75)

which defines a codimension one locus in moduli space. In ${\cal N}=2$
theories, this locus is called the curve of marginal stability. The analytic
continuation of the TBA system implies picking the residues of the kernel, and
one easily finds y-system

$\displaystyle\widetilde{\epsilon}_{1}(\theta)$ $\displaystyle=|m_{1}|{\rm
e}^{\theta}-K_{1,2}\star\widetilde{L}_{2}-L_{2}\left(\theta-{\rm
i}\phi_{1}-{{\rm i}\pi\over 2}+{\rm i}\delta\right),$ (3.76)
$\displaystyle\widetilde{\epsilon}_{2}(\theta)$ $\displaystyle=|m_{2}|{\rm
e}^{\theta}-K_{2,1}\star\widetilde{L}_{1}-L_{1}\left(\theta-{\rm
i}\phi_{2}+{{\rm i}\pi\over 2}-{\rm i}\delta\right).$

Figure 4: As we move in moduli space away from the “minimal” chamber, new
cycles intervene in the TBA equations, and correspondingly, in the
discontinuity equations. The figure shows the situation for $r=2$. In the
minimal chamber only the cycles $\gamma_{1}$, $\gamma_{2}$ are involved, but
as they rotate in the complex plane, wall-crossing occurs and one has to
consider as well the cycle $\gamma_{12}=\gamma_{1}+\gamma_{2}$.

Due to the last terms in the r.h.s. of (3.76), the above system is not closed.
We have to add the equations:

$\displaystyle\epsilon_{1}\left(\theta-{\rm i}\phi_{2}+{\pi{\rm i}\over
2}-{\rm i}\delta\right)$ $\displaystyle={\rm i}|m_{1}|{\rm e}^{\theta+{\rm
i}\phi_{1}-{\rm i}\phi_{2}-{\rm
i}\delta}-\int_{\mathbb{R}}{L_{2}\left(\theta^{\prime}-{\rm
i}\phi_{2}\right)\over{\rm i}\,\sinh(\theta-\theta^{\prime}-{\rm
i}\delta)}{{\rm d}\theta^{\prime}\over
2\pi}-\widetilde{L}_{2}\left(\theta\right),$ (3.77)
$\displaystyle\epsilon_{2}\left(\theta-{\rm i}\phi_{1}-{\pi{\rm i}\over
2}+{\rm i}\delta\right)$ $\displaystyle=-{\rm i}|m_{2}|{\rm e}^{\theta-{\rm
i}\phi_{1}+{\rm i}\phi_{2}+{\rm
i}\delta}+\int_{\mathbb{R}}{L_{1}\left(\theta^{\prime}-{\rm
i}\phi_{1}\right)\over{\rm i}\sinh(\theta-\theta^{\prime}+{\rm i}\delta)}{{\rm
d}\theta^{\prime}\over 2\pi}-\widetilde{L}_{1}\left(\theta\right).$

The resulting TBA system has in total four equations. It is possible to
transform this system into another one involving three TBA equations for three
new functions $\epsilon^{\rm n}_{a}(\theta)=-\log Y_{a}^{{\rm n}}(\theta)$,
where $a=1,2,12$. They are associated to the cycles $\gamma_{1}$, $\gamma_{2}$
and $\gamma_{12}=\gamma_{1}+\gamma_{2}$. The corresponding classical periods
are $m_{a}$, $a=1,2$, like before, and

$m_{12}=m_{1}-{\rm i}m_{2}=|m_{12}|{\rm e}^{{\rm i}\phi_{12}}.$ (3.78)

These new functions are defined by,

$\displaystyle Y_{1}^{\rm n}(\theta)$
$\displaystyle=\frac{Y_{1}(\theta)}{1+Y_{2}\left(\theta-\frac{{\rm
i}\pi}{2}\right)},\qquad\quad Y_{2}^{{\rm
n}}(\theta)=\frac{Y_{2}(\theta)}{1+Y_{1}\left(\theta+\frac{{\rm
i}\pi}{2}\right)},$ (3.79) $\displaystyle\quad Y_{12}^{{\rm n}}(\theta)$
$\displaystyle=\frac{Y_{1}(\theta)Y_{2}\left(\theta-\frac{{\rm
i}\pi}{2}\right)}{1+Y_{1}(\theta)+Y_{2}\left(\theta-\frac{{\rm
i}\pi}{2}\right)}.$

In order to write the new TBA system, we will remove the superscript ${\rm n}$
on the $\epsilon$ functions. The system reads,

$\displaystyle\widetilde{\epsilon}_{1}(\theta)$ $\displaystyle=|m_{1}|{\rm
e}^{\theta}-K_{1,2}\star\widetilde{L}_{2}-K_{1,12}^{+}\star\widetilde{L}_{12},$
(3.80) $\displaystyle\widetilde{\epsilon}_{2}(\theta)$
$\displaystyle=|m_{2}|{\rm
e}^{\theta}-K_{2,1}\star\widetilde{L}_{1}-K_{2,12}\star\widetilde{L}_{12},$
$\displaystyle\widetilde{\epsilon}_{12}(\theta)$ $\displaystyle=|m_{12}|{\rm
e}^{\theta}-K_{12,1}^{-}\star\widetilde{L}_{1}-K_{12,2}\star\widetilde{L}_{2}.$

This is the conformal limit of the system obtained in y-system ; hito  in the
study of minimal surfaces in AdS3. In the context of ${\cal N}=2$ gauge
theory, the three TBA functions correspond to the three BPS states appearing
after wall-crossing. A typical evolution in moduli space leading to the new
system (3.80) is shown in Fig. 4.

The pattern observed in this example can be extended to the general case
toledo ; toledo-thesis . In addition to the minimal chamber we described in
section 3.1, there is a “maximal” chamber involving $r(r+1)/2$ Y-functions,
which are associated to cycles pairing all the turning points555It was already
noted in reshyper  that there is such a chamber, involving $r(r+1)/2$ geodesic
cycles. These results were later rediscovered in the study of BPS states of
Argyres–Douglas (AD) theories shapere-vafa ; gmn2 . This is not surprising,
since AD theories are described by the same WKB curve as Quantum Mechanics
with a polynomial potential (see e.g. ito-shu ; gg-qm ; gm-deformed ).. The
TBA equations for this “maximal” chamber, in the case of minimal surfaces in
AdS3, have been obtained in toledo ; toledo-thesis , and they can be
determined by a graphic procedure. The TBA equations for the WKB periods are
simply the conformal limit of the equations in toledo ; toledo-thesis . There
are also intermediate chambers where the number of Y-functions lies in between
the minimal number $r$ and the maximal number $r(r+1)/2$. One can obtain the
TBA equations in each intermediate chamber by carefully picking residues and
redefining the Y-functions, as in the example with $r=2$ that we have
described.

|
---|---
Figure 5: In the case $r=3$, the “maximal” chamber involves six TBA functions.
This is the chamber relevant for a quartic potential in which the energy is
above the potential barrier, as shown on the left. The TBA functions are
associated to the cycles shown on the right.

Let us write down the resulting equations in the next case, $r=3$. The six TBA
functions appearing after wall-crossing in the maximal chamber are associated
to the cycles $\gamma_{a}$, $a=1,2,3$ appearing already in the minimal
chamber, and to the new cycles (see Fig. 5)

$\gamma_{12}=\gamma_{1}+\gamma_{2},\qquad\gamma_{23}=\gamma_{2}+\gamma_{3},\qquad\gamma_{123}=\gamma_{1}+\gamma_{2}+\gamma_{3}.$
(3.81)

The corresponding masses are

$m_{12}=m_{1}-{\rm i}m_{2},\qquad m_{23}=m_{3}-{\rm i}m_{2},\qquad
m_{123}=m_{1}+m_{3}-{\rm i}m_{2},$ (3.82)

and we will denote as usual

$\phi_{a}={\rm arg}(m_{a}),\qquad a=1,2,3,12,23,123.$ (3.83)

The TBA system reads in this case,

$\displaystyle\widetilde{\epsilon}_{1}$ $\displaystyle=|m_{1}|{\rm
e}^{\theta}-K_{1,2}\star\widetilde{L}_{2}-K_{1,12}^{+}\star\widetilde{L}_{12}-K_{1,23}^{+}\star\widetilde{L}_{23}-K_{1,123}^{+}\star\widetilde{L}_{123},$
(3.84) $\displaystyle\widetilde{\epsilon}_{2}$ $\displaystyle=|m_{2}|{\rm
e}^{\theta}-K_{2,1}\star\widetilde{L}_{1}-K_{2,3}\star\widetilde{L}_{3}-K_{2,12}\star\widetilde{L}_{12}-K_{2,23}\star\widetilde{L}_{23}-2K_{2,123}\star\widetilde{L}_{123},$
$\displaystyle\widetilde{\epsilon}_{3}$ $\displaystyle=|m_{3}|{\rm
e}^{\theta}-K_{3,2}\star\widetilde{L}_{2}-K_{3,12}^{+}\star\widetilde{L}_{12}-K_{3,23}^{+}\star\widetilde{L}_{23}-K_{3,123}^{+}\star\widetilde{L}_{123},$
$\displaystyle\widetilde{\epsilon}_{12}$ $\displaystyle=|m_{12}|{\rm
e}^{\theta}-K_{12,1}^{-}\star\widetilde{L}_{1}-K_{12,2}\star\widetilde{L}_{2}-K_{12,3}^{-}\star\widetilde{L}_{3}-K_{12,123}^{-}\star\widetilde{L}_{123},$
$\displaystyle\widetilde{\epsilon}_{23}$ $\displaystyle=|m_{23}|{\rm
e}^{\theta}-K_{23,1}^{-}\star\widetilde{L}_{1}-K_{23,2}\star\widetilde{L}_{12}-K_{23,3}^{-}\star\widetilde{L}_{3}-K_{23,123}^{-}\star\widetilde{L}_{123},$
$\displaystyle\widetilde{\epsilon}_{123}$ $\displaystyle=|m_{123}|{\rm
e}^{\theta}-K_{123,1}^{-}\star\widetilde{L}_{1}-2K_{123,2}\star\widetilde{L}_{2}-K_{123,3}^{-}\star\widetilde{L}_{3}-K_{123,12}^{+}\star\widetilde{L}_{12}-K_{123,23}^{+}\star\widetilde{L}_{23}.$

We will use a simplified version of this TBA system in section 5.

As we mentioned before, the discontinuities of the WKB periods in the maximal
chamber are more complicated than in the minimal chamber (they involve more
WKB periods). However, instead of deriving them from first principles as in
voros-quartic , it is much simpler to deduce them from the TBA equations
obtained after wall-crossing.

The monic potentials appearing in the ODE/IM correspondence correspond to a
point in moduli space with enhanced ${\mathbb{Z}}_{r+1}$ symmetry which lies
in the maximal chamber. The resulting TBA system simplifies, due to this
symmetry, and it coincides with the $A_{r}$ system considered in dt .
Therefore, the TBA system of the standard ODE/IM correspondence is a
particular case of the TBA system presented here, after wall-crossing. We will
illustrate this in detail in the next sections, in the cases $r=2$ and $r=3$
(a closely related observation in the case $r=3$ has been already made in
toledo ).

##  4 Example 1: the cubic potential

Figure 6: The cubic oscillator.

###  4.1 General aspects

Let us now illustrate the use of the TBA equations by applying them to the
cubic oscillator, which is one of the most studied problems in Quantum
Mechanics. The potential is taken to be,

$V(x)={\kappa x^{2}\over 2}-x^{3}.$ (4.1)

There are two WKB periods in this theory, associated to the two cycles shown
in Fig. 6. The turning points will be denoted by $a,b,c$. The “masses”
introduced in (3.3) can be calculated explicitly in terms of elliptic
integrals cm-ha , and one has666We have followed the standard notation for
elliptic integrals, which is different from the one used in e.g. Mathematica.

$\displaystyle m_{1}$ $\displaystyle={4{\sqrt{2}}}{(b-c)^{2}{\sqrt{a-c}}\over
15k^{4}}\left((k^{\prime})^{2}(k^{2}-2)K(k)+2(k^{4}+(k^{\prime})^{2})E(k)\right),$
(4.2) $\displaystyle m_{2}$
$\displaystyle=4\sqrt{2}\frac{(a-b)^{2}(b-c)}{\sqrt{a-c}}\frac{2(1+k^{2}(k^{2}-1))E(k^{\prime})-k^{2}(1+k^{2})K(k^{\prime})}{15k^{2}(k^{2}-1)^{2}},$

where the elliptic modulus $k$ and its complementary $k^{\prime}$ are given by

$k^{2}={b-c\over a-c},\qquad(k^{\prime})^{2}=1-k^{2}={a-b\over a-c}.$ (4.3)

The TBA equations for this theory in the “minimal” chamber are particularly
simple:

$\displaystyle\epsilon_{1}(\theta)$ $\displaystyle=m_{1}{\rm
e}^{\theta}-\int_{\mathbb{R}}{\log\left(1+{\rm
e}^{-\epsilon_{2}(\theta^{\prime})}\right)\over\cosh(\theta-\theta^{\prime})}{{\rm
d}\theta^{\prime}\over 2\pi},$ (4.4) $\displaystyle\epsilon_{2}(\theta)$
$\displaystyle=m_{2}{\rm e}^{\theta}-\int_{\mathbb{R}}{\log\left(1+{\rm
e}^{-\epsilon_{1}(\theta^{\prime})}\right)\over\cosh(\theta-\theta^{\prime})}{{\rm
d}\theta^{\prime}\over 2\pi}.$

As we explained in section 3.4, this system is also valid for complex values
of the masses as long as (3.73) is respected777 The cubic example considered
in oper  corresponds to a value of the masses in which $m_{1}=m_{2}$ and
$\phi_{1,2}=\pi/4$. In this case, the TBA system (4.4) reduces to a single
equation.. The asymptotic values of $\epsilon_{1,2}(\theta)$ as
$\theta\rightarrow-\infty$ are given by

$\epsilon_{\star}=\log\left({{\sqrt{5}}-1\over 2}\right).$ (4.5)

The effective central charge is given by

$c_{\rm eff}={6\over 5}.$ (4.6)

On the other hand, the PNP relation for this theory (3.63) can be written as
cm-ha

${\cal J}=-{2E\over 15}-{\hbar^{2}\over 10},$ (4.7)

so that (3.64) is verified.

As a first application of the TBA system, we can look at the perturbative WKB
expansion of the periods. The higher order corrections can be computed by
various techniques, as we recall in Appendix A. They are highly non-trivial
functions of the moduli of the WKB curve. At the same time, they can be
obtained from the solution of the TBA system by using (3.13). In the original
ODE/IM correspondence of dt ; ddt , which corresponds to a very particular
point in the moduli space of the WKB curve, the integrals appearing in (3.13)
can be computed analytically by relating them to the conserved charges of the
underlying CFT blz1 ; blz2 ; blz-excited . In this more general case, we do
not know how to perform the integrals analytically, but we can solve the TBA
system numerically, compute the coefficients $m_{1,2}^{(n)}$, and verify
(3.14). In Table 1 we compare $m_{1,2}^{(n)}$, $n=1,2,3$, calculated
numerically from the TBA system, to the quantum periods
$\Pi_{\gamma_{1,2}}^{(n)}$, calculated with the techniques in Appendix A, for
fixed values of the moduli $\kappa$ and $E$. As it can be seen in the Table,
the two calculations agree to order $10^{-13}$, and the agreement can be
improved by a more precise numerical integration of the TBA system.

$n$ | $\Pi_{\gamma_{1}}^{(n)}$ | $m_{1}^{(n)}$ | $\Pi_{\gamma_{2}}^{(n)}$ | $m_{2}^{(n)}$
---|---|---|---|---
$1$ | $3.6574758326426$ | $-3.6574758326423$ | $-9.1939620228508$ | $-9.1939620228505$
$2$ | $948.794486716986$ | $948.794486716980$ | $19138.8317303045$ | $19138.8317303047$
$3$ | $1368408.366655312$ | $-1368408.366655315$ | $-228090464.089314$ | $-228090464.089320$
Table 1: The WKB expansion of the quantum periods for the cubic oscillator
with $E=1/200$ and $\kappa=1$. The numerical calculation in the TBA system is
done by Fourier discretization with $2^{12}$ points and a cutoff $(-L,L)$
where $L=35$.

###  4.2 PT-symmetric cubic oscillators

Figure 7: The WKB cycles for the PT-symmetric cubic oscillator with
Hamiltonian (4.8).

In order to illustrate the power of the TBA equations obtained in section 3,
as applied to the cubic oscillator, we will analyze a concrete spectral
problem which cannot be solved in the framework of the conventional ODE/IM
correspondence. Let us consider the following Hamiltonian:

$\mathsf{H}={1\over 2}\mathsf{p}^{2}+{\rm i}\mathsf{q}^{3}-{\rm
i}\lambda\mathsf{q}.$ (4.8)

This Hamiltonian is PT-symmetric (see bender-review  for a review). When
$\lambda\leq 0$, it has a real, positive, discrete spectrum. When $\lambda=0$
we recover the PT-symmetric pure cubic oscillator. When $\lambda$ is positive
and sufficiently large, PT symmetry is broken due to non-perturbative effects,
and one finds an intricate pattern of level crossings studied in delabaere-
trinh .

We will focus here in the case in which $\lambda>0$. By rescaling $\hbar$ and
the energy, it is easy to see that one can set $\lambda=1$. The potential

$V(q)={\rm i}q^{3}-{\rm i}q$ (4.9)

has three turning points: one of them, $q_{0}$, is in the imaginary axis,
while the other two, denoted by $q_{\pm}$, are in the fourth and the third
quadrant of the complex plane, respectively (see Fig. 7). All of them are non-
trivial functions of the energy $E$. There are also three WKB cycles: the
$\gamma_{1}$ cycle goes from $q_{0}$ to $q_{+}$, the $\gamma_{2}$ cycle goes
from $q_{-}$ to $q_{0}$, and the cycle $\gamma_{12}=\gamma_{1}+\gamma_{2}$
goes from $q_{-}$ to $q_{+}$. The corresponding quantum periods are not
independent, since they are related by

$\Pi_{\gamma_{2}}=\Pi_{\gamma_{1}}^{*},\qquad\Pi_{\gamma_{12}}=\Pi_{\gamma_{1}}+\Pi_{\gamma_{2}}.$
(4.10)

In particular, the period $\Pi_{\gamma_{12}}$ is real. We will write

$\Pi_{\gamma_{1}}={1\over 2}\Pi_{\rm p}-{{\rm i}\over 2}\Pi_{\rm np},$ (4.11)

where ${\rm p}$, ${\rm np}$ stand for “perturbative” and “non-perturbative.”
The perturbative period $\Pi_{\rm p}=\Pi_{\gamma_{12}}$ is the relevant one
for the Bohr–Sommerfeld approximation, since the cycle $\gamma_{12}$
corresponds to classical, periodic complex trajectories between the turning
points $q_{-}$ and $q_{+}$ and around them bb-pt ; bender-review . In this
case, both $\Pi_{\rm p}$, $\Pi_{\rm np}$ turn out to be Borel summable ddpham
; dpham ; delabaere-trinh , and the all-orders, perturbative WKB approximation
is given by

$s\left(\Pi_{\rm p}\right)(\hbar)=2\pi\hbar\left(n+{1\over 2}\right),\qquad
n=0,1,2,\cdots$ (4.12)

However, this does not give the correct spectrum of the Hamiltonian (4.8): as
noted already in voros ; bpv , one has to take into account complex
instantons888This is a quantum-mechanical example of how a Borel-resummable
perturbative series fails to reproduce the exact results due to the appearance
of complex instantons, as emphasized in gmz ; cms  in a more general context..
The correct EQC can be easily deduced from the analysis in ddpham ; alvarez-
casares2 ; delabaere-trinh , and it reads

$2\cos\left({1\over 2\hbar}s(\Pi_{\rm p})(\hbar)\right)+{\rm e}^{-{1\over
2\hbar}s(\Pi_{\rm np})(\hbar)}=0.$ (4.13)

The second term in the l.h.s. is an exponentially small correction which is
crucial to obtain the exact results (in fact, this correction can become
exponentially enhanced when $s(\Pi_{\rm np})<0$, leading to PT-symmetry
breaking delabaere-trinh ; ben-ber ). When this correction is neglected, we
recover (4.12).

According to our results, the Borel-resummed quantum periods appearing in the
EQC can be obtained from the TBA equations of Section 3. The PT-symmetric
potential corresponds to a point in moduli space in the “maximal” chamber. We
then consider the three-term TBA equations in (3.80). Let us denote

$\alpha=-{\rm arg}\left(\Pi_{\gamma_{1}}^{(0)}\right).$ (4.14)

This angle is a non-trivial function of the energy $E$. The angles appearing
in the kernels of (3.80) are given by

$\phi_{1}=-\alpha,\qquad\phi_{2}={\pi\over 2}+\alpha,\qquad\phi_{12}=0.$
(4.15)

We also have

$|m_{1}|=|m_{2}|=\left|\Pi_{\gamma_{1}}^{(0)}\right|,\qquad m_{12}=2\,{\rm
Re}\,\left(\Pi_{\gamma_{1}}^{(0)}\right).$ (4.16)

We can now solve the TBA system to obtain the three functions
$\widetilde{\epsilon}_{a}(\theta)$, $a=1,2,12$. There is a conjugation
symmetry

$\widetilde{\epsilon}_{1}(\theta)=\left(\widetilde{\epsilon}_{2}(\theta)\right)^{*},\qquad\theta\in{\mathbb{R}}.$
(4.17)

As in (3.6), the relation between the quantum periods and the TBA functions is

$-{\rm i}\epsilon_{1}\left(\theta+{{\rm i}\pi\over
2}\right)={1\over\hbar}s\left(\Pi_{\gamma_{1}}\right)(\hbar),\qquad-{\rm
i}\epsilon_{2}\left(\theta\right)={1\over\hbar}s\left(\Pi_{\gamma_{2}}\right)(\hbar),$
(4.18)

where $\theta=-\log(\hbar)$. This implies

$\displaystyle{1\over\hbar}s\left(\Pi_{\gamma_{1}}\right)(\hbar)$
$\displaystyle={1\over\hbar}\Pi_{\gamma_{1}}^{(0)}-{\rm
i}\int_{\mathbb{R}}{\widetilde{L}_{2}(\theta^{\prime})\over\cosh\left(\theta-\theta^{\prime}+{\rm
i}\alpha\right)}{{\rm d}\theta^{\prime}\over 2\pi}-{\rm
i}\int_{\mathbb{R}}{\widetilde{L}_{12}(\theta^{\prime})\over\cosh(\theta-\theta^{\prime})}{{\rm
d}\theta^{\prime}\over 2\pi},$ (4.19)
$\displaystyle{1\over\hbar}s\left(\Pi_{\gamma_{2}}\right)(\hbar)$
$\displaystyle={1\over\hbar}\Pi_{\gamma_{2}}^{(0)}+{\rm
i}\int_{\mathbb{R}}{\widetilde{L}_{1}(\theta^{\prime})\over\cosh\left(\theta-\theta^{\prime}-{\rm
i}\alpha\right)}{{\rm d}\theta^{\prime}\over 2\pi}+{\rm
i}\int_{\mathbb{R}}{\widetilde{L}_{12}(\theta^{\prime})\over\cosh(\theta-\theta^{\prime})}{{\rm
d}\theta^{\prime}\over 2\pi}.$

By using these equations and the TBA system (3.80), we can compute
$s\left(\Pi_{\rm p}\right)$, $s\left(\Pi_{\rm np}\right)$ numerically, see
Fig. 8.

Figure 8: Numerical calculation of the functions $s\left(\Pi_{\rm
p}\right)(\hbar)/\hbar$ (top) and $s\left(\Pi_{\rm np}\right)(\hbar)/\hbar$
(bottom), as a function of $\theta=-\log(\hbar)$, by using the TBA equations.
Here we set $E=1$.

One can now determine the Voros spectrum $x_{n}(E)$ for a fixed energy, by
combining the EQC (4.13) with the TBA calculation of the resummed WKB periods.
The accuracy of this method can be tested by calculating numerically the
energies $E_{n}(x_{n}^{-1})$ corresponding to those values of the Planck
constant. This is done by combining the standard Rayleigh–Ritz method with
complex dilatation techniques bender-resonance . We can then compare the
numerical values obtained in this way to the expected relation (2.25). The
results are shown in Table 2, for $E=1$. The numerical approximation to the
TBA system involves a discretization with $2^{18}$ points and a cutoff at
$L=25$. As we can see, the numerical calculation of the spectrum by using the
TBA system produces values which are correct with a precision of $10^{-10}$
(for the ground state) and of $10^{-12}$ for all the states with $n\geq 2$. As
in any semiclassical scheme, the approximation becomes better as the quantum
number $n$ increases.

$n$ | $x_{n}$ | $E_{n}(x_{n}^{-1})$
---|---|---
$0$ | $0.\,560\,405\,699\,850$ | $0.\,999\,999\,999\,855$
$1$ | $1.\,496\,238\,960\,118$ | $0.\,999\,999\,999\,980$
$2$ | $2.\,509\,053\,231\,083$ | $0.\,999\,999\,999\,993$
$3$ | $3.\,507\,747\,393\,173$ | $0.\,999\,999\,999\,997$
Table 2: The first column shows the spectrum of values of $x_{n}$ for the
Hamiltonian (4.8), with $E=\lambda=1$. It is obtained by combining the EQC
(4.13) with a numerical solution of the TBA system, involving a discretization
with $2^{16}$ points and a cutoff at $L=25$. The second column shows the
values of the energy levels $E_{n}(x_{n}^{-1})$, computed by complex
dilatation techniques.

It is interesting to recover the results of the standard ODE/IM correspondence
as a particular case of our more general results. Let us suppose that
$\lambda=0$ in (4.8), so that we obtain the pure PT-symmetric cubic oscillator

$\mathsf{H}={1\over 2}\mathsf{p}^{2}+{\rm i}\mathsf{q}^{3}.$ (4.20)

In this case we can set $E=1$ without loss of generality. The turning points
are now located at the unit circle

$q_{0}={\rm i},\qquad q_{+}={\rm e}^{-\pi{\rm i}/6},\qquad q_{-}=-{\rm
e}^{\pi{\rm i}/6},$ (4.21)

and the classical periods can be calculated in closed form:

$\Pi_{\gamma_{1}}^{(0)}={\sqrt{6\pi}\Gamma(1/3)\over 3\Gamma(11/6)}{\rm
e}^{-{\rm
i}\pi/3},\qquad\Pi_{\gamma_{2}}^{(0)}=\left(\Pi_{\gamma_{1}}^{(0)}\right)^{*}.$
(4.22)

Higher order corrections can be computed by using the techniques of bpv . We
find, for example,

$\displaystyle\Pi_{\gamma_{1}}^{(1)}$
$\displaystyle={\sqrt{6\pi}\Gamma\left(\frac{5}{3}\right)\over
8\Gamma\left(\frac{1}{6}\right)}{\rm e}^{-2\pi{\rm i}/3},$ (4.23)
$\displaystyle\Pi_{\gamma_{1}}^{(2)}$ $\displaystyle=0,$
$\displaystyle\Pi_{\gamma_{1}}^{(3)}$
$\displaystyle={89\sqrt{6\pi}\Gamma\left(\frac{1}{3}\right)\over
120960\Gamma\left(-\frac{19}{6}\right)}{\rm e}^{-\pi{\rm i}/3}.$

The exact WKB theory of the Hamiltonian (4.20) has a ${\mathbb{Z}}_{3}$
symmetry which manifests itself in the TBA system. In particular, since

$|m_{1}|=|m_{2}|=|m_{12}|,$ (4.24)

the $\widetilde{\epsilon}$ functions in (3.80) are now all equal (and in
particular they are real):

$\widetilde{\epsilon}_{a}(\theta)=\epsilon(\theta),\qquad a=1,2,3.$ (4.25)

The TBA system reduces to one TBA equation for the function $\epsilon(\theta)$
with periodicity $\epsilon(\theta)=\epsilon(\theta+\frac{5\pi i}{3})$. This
equation can be easily written down, by using that the angle in (4.14) is
$\alpha=\pi/3$, and one finds

$\epsilon(\theta)={\sqrt{6\pi}\Gamma(1/3)\over 3\Gamma(11/6)}{\rm
e}^{\theta}+\int_{\mathbb{R}}\Phi(\theta-\theta^{\prime})L(\theta^{\prime}){\rm
d}\theta^{\prime},$ (4.26)

where $L(\theta)=\log(1+{\rm e}^{-\epsilon(\theta)})$, as usual, and

$\Phi(\theta)={\sqrt{3}\over\pi}{\sinh(2\theta)\over\sinh(3\theta)}.$ (4.27)

The TBA equation (4.26) appears in the analysis of the thermodynamics of the
Lee–Yang model and it has been extensively studied in e.g. zamo-TBA ; blz-
excited . It is the TBA equation that one would obtain for the $A_{2}$ system
appearing in dt . The relation between the quantum periods and the TBA
functions is still given by (4.18), and (4.19) further simplifies due to the
${\mathbb{Z}}_{3}$ symmetry. We conclude that, when the potential is monic,
the TBA system in the maximal chamber for $r=2$ (3.80) reduces to the TBA
equation of the conventional ODE/IM correspondence, as we mentioned in section
3.4. We have a similar result for the standard monic, cubic potential
$V(q)=q^{3}$: the relation (4.25) still holds, where $\epsilon(\theta)$
satisfies (4.26), but the angles entering into the TBA system are in this case
$\phi_{1}=-\phi_{12}=-\pi/6$, $\phi_{2}=\pi$.

$n$ | $E_{n}^{\rm num}$ | $E_{n}^{\rm TBA}$ | $E_{n}^{\rm p}$ | $E_{n}^{\rm WKB}$
---|---|---|---|---
$0$ | $1.\,156\,267\,071\,988$ | $1.\,156\,267\,071\,988$ | $1.\,134\,513\,239\,424$ | $1.\,094\,269\,500\,533$
$1$ | $4.\,109\,228\,752\,810$ | $4.\,109\,228\,752\,806$ | $4.\,109\,367\,351\,095$ | $4.\,089\,496\,119\,273$
$2$ | $7.\,562\,273\,854\,979$ | $7.\,562\,273\,854\,971$ | $7.\,562\,273\,170\,784$ | $7.\,548\,980\,437\,586$
Table 3: The energy levels for the PT cubic oscillator for $\hbar={\sqrt{2}}$.
The first column shows a numerical calculation with complex dilatation
techniques. The second column is a calculation by using the TBA system with
$L=25$ and $n=2^{16}$ steps in the discretization. The third column is
obtained by using the all-orders WKB perturbative quantization condition
(4.12) and the TBA calculation of $\Pi_{\rm p}(\hbar)$. The final column is
the result of the Bohr–Sommerfeld approximation.

The EQC for (4.20) is again given by (4.13). Note that, in this case, the
energy $E$ and the Planck constant always appear in the combination

${E^{5/6}\over\hbar}$ (4.28)

and the Voros spectrum can be easily translated into the usual spectrum. In
Table 3 we compare different approaches to the calculation of the energy
levels of this oscillator. We have chosen $\hbar={\sqrt{2}}$ to facilitate the
comparison with results in the literature bender-review ; bb-pt 999Note
however that the some of the numerical results for the excited energy levels
of (4.20) presented in Table 1 of bender-review  are slightly off.. The first
column is a numerical calculation of the energy levels, by combining standard
Rayleigh–Ritz techniques with complex dilatation. The second column uses the
EQC (4.13) and the TBA calculation of the quantum periods. The third column is
obtained by using the all-orders WKB perturbative quantization condition
(4.12), and the TBA calculation of $s\left(\Pi_{\rm p}\right)$. The final
column gives the result of the Bohr–Sommerfeld quantization condition, which
approximates the l.h.s. of (4.12) by the classical period.

|
---|---
Figure 9: The singularity structure of the Borel transforms of $\Pi_{\rm p}$
(left) and $\Pi_{\rm np}$ (right) for the PT-symmetric pure cubic oscillator
(4.20) can be approximated numerically by using Borel–Padé techniques, and it
reproduces the discontinuities of the TBA equations. The diagonal rays are at
the angles $\pm\pi/6$.

As we have already noted, although the structure of discontinuities of the WKB
periods in the maximal chamber is more complicated than the one in the minimal
chamber, it can be determined directly from the TBA equations (3.80). We would
also like to emphasize that many ingredients of this structure can be
determined empirically by looking at the singularities of the Borel transform,
in agreement with the theory of resurgence. The PT-symmetric pure cubic
oscillator (4.20) provides a beautiful illustration of this fact. The
discontinuities for $\Pi_{\rm p}/\hbar$ and $\Pi_{\rm np}/\hbar$ can be easily
obtained from (3.80). In the case of the perturbative period, they are located
at the rays with angles $\pm\pi/6$. For the non-perturbative period, there are
discontinuities at the rays with angles $\pm\pi/6$ and $\pm\pi/2$. A numerical
calculation of the Borel transforms of $\Pi_{\rm p}$ and $\Pi_{\rm np}$ is
shown in Fig. 9, by using standard Borel–Padé techniques applied to $125$
terms of their formal power series. The location of the singularities
reproduces precisely the location of the discontinuities obtained from the TBA
system.

##  5 Example 2: the quartic potential

Figure 10: The symmetric double-well potential.

###  5.1 General aspects

As another example of our general results, we consider Quantum Mechanics with
a quartic potential. For simplicity, we will mostly focus on the case of a
parity symmetric potential obeying $V(x)=V(-x)$. A convenient parametrization
of such a potential is

$V(x)={1\over 32}-{\kappa x^{2}\over 4}+{x^{4}\over 2}.$ (5.1)

When $\kappa=1$, we have the double-well potential

$V(x)={1\over 32}\left(4x^{2}-1\right)^{2},$ (5.2)

with the structure shown in Fig. 10. In this case, the turning points are real
when the energy satisfies

$0<E<{1\over 32}.$ (5.3)

When $\kappa<0$, we have a perturbed harmonic oscillator.

Let us now consider the TBA equations governing the WKB periods of this
potential. Like in the cubic case, we first consider the minimal chamber. Due
to the parity symmetry, $\gamma_{1}=\gamma_{3}$, so the corresponding TBA
functions appearing in (3.11) are equal,

$\epsilon_{1}=\epsilon_{3},$ (5.4)

and the general equations (3.11) for $r=3$ simplify to

$\displaystyle\epsilon_{1}(\theta)$ $\displaystyle=m_{1}{\rm
e}^{\theta}-\int_{\mathbb{R}}{\log\left(1+{\rm
e}^{-\epsilon_{2}(\theta^{\prime})}\right)\over\cosh(\theta-\theta^{\prime})}{{\rm
d}\theta^{\prime}\over 2\pi},$ (5.5) $\displaystyle\epsilon_{2}(\theta)$
$\displaystyle=m_{2}{\rm e}^{\theta}-2\int_{\mathbb{R}}{\log\left(1+{\rm
e}^{-\epsilon_{1}(\theta^{\prime})}\right)\over\cosh(\theta-\theta^{\prime})}{{\rm
d}\theta^{\prime}\over 2\pi}.$

The classical periods can be computed explicitly in terms of elliptic
integrals (see e.g. cm-ha ), and one finds,

$\displaystyle m_{1}$ $\displaystyle={2a^{3}\over 3}\left[(1+k^{\prime
2})E(k)-2k^{\prime 2}K(k)\right],$ (5.6) $\displaystyle m_{2}$
$\displaystyle={4a^{3}\over 3}\left[(1+k^{\prime
2})E(k^{\prime})-k^{2}K(k^{\prime})\right],$

where the elliptic moduli are

$k^{2}=1-{b^{2}\over a^{2}},\qquad\left(k^{\prime}\right)^{2}=1-k^{2}.$ (5.7)

By solving the TBA system numerically, we can compute quantum corrections to
the periods and compare them to the analytic results obtained in Appendix A.
In particular, we can check again the identification (3.14) with high
precision and to high orders.

The limiting behavior of $\epsilon_{1}$, $\epsilon_{2}$ is easily found from
(3.46) to be

$\epsilon_{1}^{\star}=-\log(2),\qquad\epsilon_{2}^{\star}=-\log(3),$ (5.8)

and the effective central charge is

$c_{\rm eff}=2.$ (5.9)

Although our general discussion of the PNP relation in section 3.3 was carried
out for simplicity for odd potentials, it can be extended straightforwardly to
the double-well potential. By using the period and dual period defined in e.g.
cm-ha , one finds that the PNP relation for this potential reads

${\cal J}=-{E\over 3}-{\hbar^{2}\over 6},$ (5.10)

so that $\alpha=-1/6$, and (3.64) is again satisfied.

Figure 11: The functions ${\cal J}_{\epsilon}(\hbar)$ for $\epsilon=-1$
(bottom) and $\epsilon=1$ (top) as a function of $x=-\log\hbar$. The
horizontal lines are $\pi$ and $3\pi$, and intersect ${\cal
J}_{\epsilon}(\hbar)$ in four points which give the first four values of the
spectrum of $\hbar$.

###  5.2 Quartic oscillators

Let us now illustrate how to use the TBA system to compute the spectrum of
various quartic oscillators.

Let us first consider the double-well potential. When the energy of the system
is below the top of the barrier, we are in the minimal chamber and we can use
the TBA equations (5.5) to compute the Borel resummation of the WKB periods.
In this case, as it follows from the Delabaere–Pham formula, the quantum
period $\Pi_{\gamma_{1}}$ is not Borel summable for real values of $\hbar$,
and one has to specify an appropriate resummation. As it should be clear from
the discussion in section 3.1, the lack of Borel summability of
$\Pi_{\gamma_{1}}$ is manifest in the TBA system: after shifting
$\theta\rightarrow\theta+\pi{\rm i}/2$ in $\epsilon_{1}(\theta)$, as
prescribed by (3.6), we hit a pole in the kernel and the integral becomes
singular. The two deformations of the integration contour that avoid this pole
correspond to the two lateral Borel resummations, as indicated in (3.6) by the
explicit introduction of a small $\delta$. In many cases, however, the most
appropriate resummation is the so-called “median” resummation dpham ; ddpham
(see mmnp ; as13  for other applications of the median resummation). We will
denote this resummation by $s_{\rm med}\left(\Pi_{\gamma_{1}}\right)$, which
in this particular example is just the average of the two lateral resummations
above and below the Stokes line. In the TBA system, the median resummation is
easily implemented: it corresponds to the principal value of the singular
integral.

To compute the resummed quantum periods, we first solve the TBA system (5.5)
to obtain the functions $\epsilon_{1,2}(\theta)$ along the real line. We then
have,

$\displaystyle{1\over\hbar}s_{\rm med}\left(\Pi_{\gamma_{1}}\right)(\hbar)$
$\displaystyle=m_{1}{\rm e}^{\theta}+{\rm
P}\int_{\mathbb{R}}{L_{2}(\theta^{\prime})\over\sinh(\theta-\theta^{\prime})}{{\rm
d}\theta^{\prime}\over 2\pi},$ (5.11) $\displaystyle{{\rm
i}\over\hbar}s\left(\Pi_{\gamma_{2}}\right)(\hbar)$
$\displaystyle=\epsilon_{2}(\theta),$

where as before we have to identify $\theta=-\log(\hbar)$.

The EQC for this problem was famously conjectured by Zinn–Justin in zinn-
justin  (it follows however by a simple analytic continuation from previous
results by Voros voros  for the pure quartic oscillator). In the language of
resurgence, Zinn-Justin’s EQC has been derived and reformulated in ddpham ,
and it can be written in the following way:

$\cos\left({1\over\hbar}s_{\rm
med}\left(\Pi_{\gamma_{1}}(\hbar)\right)\right)+{1\over{\sqrt{1+{\rm e}^{-{\rm
i}s\left(\Pi_{\gamma_{2}}(\hbar)\right)/\hbar}}}}=0.$ (5.12)

This can be solved more explicitly as follows,

${\cal J}_{\epsilon}(\hbar)=2\pi\left(k+{1\over 2}\right),\qquad
k\in{\mathbb{Z}}_{\geq 0},$ (5.13)

where

${\cal J}_{\epsilon}(\hbar)={1\over\hbar}s_{\rm
med}\left(\Pi_{\gamma_{1}}\right)(\hbar)+\epsilon\tan^{-1}\left({\rm
e}^{-{{\rm i}\over 2\hbar}s\left(\Pi_{\gamma_{2}}\right)(\hbar)}\right).$
(5.14)

Each integer value of $k$ leads to two states of the spectrum, depending on
the value of the parity $\epsilon=\pm 1$. In this way, the $n$-th state of the
spectrum, where $n=0,1,2,\cdots$, corresponds to

$n=2k-{\epsilon-1\over 2}.$ (5.15)

The median resummation $s_{\rm med}\left(\Pi_{\gamma_{1}}\right)$ can be
regarded as the all-orders, perturbative WKB contribution, while the Borel
resummed period ${\rm i}s\left(\Pi_{\gamma_{2}}\right)$ gives a non-
perturbative correction due to tunneling effects.

Using the TBA equations, we have computed the functions ${\cal
J}_{\pm}(\hbar)$ appearing in the l.h.s. of the exact quantization condition,
as well as the Voros spectrum $x_{n}(E)$. In Fig. 11 we plot the functions
${\cal J}_{\epsilon}$ for $E=1/64$, as a function of $\theta$, as well as the
horizontal lines at $\pi$ and $3\pi$, which intersect ${\cal J}_{\pm}(\hbar)$
in four points which give the first four values of the spectrum of $\hbar$.
For the numerical solution of the TBA system we have used a Fourier
discretization in an interval $(-L,L)$ with $L=35$ and $N=2^{18}$ points. The
principal value is computed as the limit

${\rm P}\int_{\mathbb{R}}{f(\theta^{\prime})\over\sinh(\theta-\theta)}{\rm
d}\theta^{\prime}=\lim_{\delta\to
0}\int_{{\mathbb{R}}}{\sinh(\theta-\theta^{\prime})\cos(\delta)f(\theta^{\prime})\over\sinh^{2}(\theta-\theta^{\prime})\cos^{2}(\delta)+\cosh^{2}(\theta-\theta^{\prime})\sin^{2}(\delta)}{\rm
d}\theta^{\prime},$ (5.16)

which is obtained by considering the real part of

${1\over\sinh(\theta-\theta^{\prime}\pm{\rm i}\delta)}.$ (5.17)

Numerical approximations to the principal value are then obtained by picking a
small value of $\delta$. In our numerical computation, we have used
$\delta=10^{-15}$. The values of $x_{n}$ obtained in this way can be tested
numerically by verifying the identity (2.25), where the energy levels are
those of the Hamiltonian

$\mathsf{H}(\hbar)={\mathsf{p}^{2}\over 2}-{\mathsf{q}^{2}\over
4}+{\mathsf{q}^{4}\over 2}+{1\over 32}.$ (5.18)

An example of such a computation, for $E=1/64$, is shown in Table 4. Although
the precision obtained for the quartic potential is lower than for the cubic
potential discussed in the previous section, the discrepancies are at worst of
$10^{-8}$ (for the ground state).

Level | $x_{n}$ | $E_{n}(x_{n}^{-1})$
---|---|---
$0$ | $27.\,792\,941\,509$ | $0.\,015\,625\,014$
$1$ | $30.\,383\,260\,063$ | $0.\,015\,625\,011$
$2$ | $90.\,374\,172\,968$ | $0.\,015\,625\,000$
$3$ | $90.\,400\,201\,849$ | $0.\,015\,625\,000$
Table 4: The first column shows the spectrum of values of $x_{n}$, as computed
from the exact quantization condition (5.14) and the TBA equations. We used a
discretization with $2^{18}$ points and $L=35$. The second column shows the
values of $E_{n}(\hbar_{n})$ for the Hamiltonian (5.18).

Figure 12: The curve of marginal stability for the quartic oscillator with
$\kappa=1$, in the complex energy plane. In the real axis, it stretches from
$E=0$ to the top of the barrier at $E=1/32$.

In the previous example we considered a quartic potential in the minimal
chamber: the double-well potential with an energy below the barrier. However,
as soon as the energy increases above the barrier, we move into the maximal
chamber of moduli space, as shown in Fig. 5. More generally, one can easily
determine the counterpart of the curve of marginal stability for the double-
well potential in the complex energy plane, defined by the condition (3.75).
It is shown in Fig. 12. The minimal chamber, where (5.5) holds, corresponds to
the region inside the curve.

Let us then focus on the maximal chamber. The corresponding TBA system was
written down in (3.84) and it involves generically six functions. However, due
to the parity symmetry of the symmetric quartic potential, we have

$m_{3}=m_{1},\qquad m_{12}=m_{23},$ (5.19)

and we can set

$\widetilde{\epsilon}_{3}=\widetilde{\epsilon}_{1},\qquad\widetilde{\epsilon}_{23}=\widetilde{\epsilon}_{12}.$
(5.20)

The resulting TBA system involves only four functions,
$\widetilde{\epsilon}_{a}$ with $a=1,2,12,123$, and it reads

$\displaystyle\widetilde{\epsilon}_{1}$ $\displaystyle=|m_{1}|{\rm
e}^{\theta}-K_{1,2}\star\widetilde{L}_{2}-2K_{1,12}^{+}\star\widetilde{L}_{12}-K_{1,123}^{+}\star\widetilde{L}_{123},$
(5.21) $\displaystyle\widetilde{\epsilon}_{2}$ $\displaystyle=|m_{2}|{\rm
e}^{\theta}-2K_{2,1}\star\widetilde{L}_{1}-2K_{2,12}\star\widetilde{L}_{12}-2K_{2,123}\star\widetilde{L}_{123},$
$\displaystyle\widetilde{\epsilon}_{12}$ $\displaystyle=|m_{12}|{\rm
e}^{\theta}-2K^{-}_{12,1}\star\widetilde{L}_{1}-K_{12,2}\star\widetilde{L}_{2}-K^{-}_{12,123}\star\widetilde{L}_{123},$
$\displaystyle\widetilde{\epsilon}_{123}$ $\displaystyle=|m_{123}|{\rm
e}^{\theta}-2K^{-}_{123,1}\star\widetilde{L}_{1}-2K_{123,2}\star\widetilde{L}_{2}-2K^{+}_{123,12}\star\widetilde{L}_{12}.$

Let us now apply this TBA system to the double-well potential (with
$\kappa=1$) in which the energy is above the energy barrier. In this problem
there are two turning points on the real axis, connected by the cycle
$\gamma_{123}$, and two complex turning points along the imaginary axis,
connected by the cycle $\gamma_{2}$. The angles appearing in the TBA system
are

$\phi_{1}=-\phi_{12}=-\alpha,\qquad\phi_{2}=\pi,\qquad\phi_{123}=0,$ (5.22)

where $\alpha$ depends on the energy $E$. The EQC for this potential requires
two ingredients. First of all, it involves the WKB period associated to the
cycle $\gamma_{123}$, which corresponds to the classical motion between the
two turning points on the real axis. This is the cycle that one would consider
in an all-orders, perturbative WKB calculation. It is related to the TBA
function by,

${1\over\hbar}s_{\pm}\left(\Pi_{\gamma_{123}}\right)(\hbar)=-{\rm
i}\epsilon_{123}\left(\theta+{\pi{\rm i}\over 2}\pm{\rm i}\delta\right),$
(5.23)

under the usual identification $\theta=-\log(\hbar)$. Note that this WKB cycle
is not Borel summable, and as we discussed before, the two lateral
resummations are related to the two infinitesimal deformations of the
integration contour in the TBA system. Its median resummation is given, in
terms of TBA functions, by

$\displaystyle{1\over\hbar}s_{\rm med}\left(\Pi_{\gamma_{123}}\right)(\hbar)$
$\displaystyle=m_{123}{\rm e}^{\theta}+2{\rm
i}\int_{\mathbb{R}}\left({\widetilde{L}_{1}(\theta^{\prime})\over\cosh(\theta-\theta^{\prime}-{\rm
i}\alpha)}-{\widetilde{L}_{12}(\theta^{\prime})\over\cosh(\theta-\theta^{\prime}+{\rm
i}\alpha)}\right){{\rm d}\theta^{\prime}\over 2\pi}$ (5.24)
$\displaystyle-2\,{\rm
P}\int_{\mathbb{R}}{\widetilde{L}_{2}(\theta^{\prime})\over\sinh(\theta-\theta^{\prime})}{{\rm
d}\theta^{\prime}\over 2\pi}.$

Although in this problem there are no real instantons, there are complex
instantons associated to the cycle $\gamma_{2}$ between the two complex
turning points, which make a crucial contribution to the EQC, as noted in bpv
in a related context. The EQC reads

${1\over\hbar}s_{\rm
med}\left(\Pi_{\gamma_{123}}\right)(\hbar)-2(-1)^{n}\tan^{-1}\left({\rm
e}^{-{{\rm i}\over
2\hbar}s\left(\Pi_{\gamma_{2}}\right)(\hbar)}\right)=2\pi\left(n+{1\over
2}\right),\qquad n\in{\mathbb{Z}}_{\geq 0}.$ (5.25)

We have implemented this equation together with the TBA system in order to
calculate the Voros spectrum of $x_{n}$. The results for $E=1$ are shown in
Table 5. We have employed a discretization of the TBA equations with $2^{19}$
points. This gives an error of at most $10^{-6}$ (for the ground state) which
decreases rapidly to at most $10^{-12}$ for the states with $n\geq 3$.

Level | $x_{n}$ | $E_{n}(x_{n}^{-1})$
---|---|---
$0$ | $0.\,564\,245\,815$ | $1.\,000\,001\,185$
$1$ | $1.\,424\,136\,265\,$ | $1.\,000\,000\,021$
$2$ | $2.\,368\,078\,806$ | $1.\,000\,000\,000$
$3$ | $3.\,307\,532\,741$ | $1.\,000\,000\,000$
Table 5: The spectrum of values of $x_{n}$, as computed from the exact
quantization condition (5.25), and the values of $E_{n}(\hbar_{n})$ for the
Hamiltonian (5.18) with $E=1$.

Exactly the same TBA system and the same quantization condition can be used in
the case of the quartic oscillator with a non-negative harmonic term, e.g.
with a potential of the form

$V(x)=-{\kappa x^{2}\over 4}+{x^{4}\over 2},\qquad\kappa\leq 0.$ (5.26)

The case with $\kappa=0$ is the pure quartic oscillator studied in the
pioneering works bpv ; voros ; voros-quartic  and revisited in the context of
the ODE/IM correspondence dt ; ddt . As we mentioned in section 3.4, in this
case the TBA system (5.21) acquires an additional symmetry and it reduces to
the TBA system of dt  (this has been already pointed out in toledo ). Indeed,
in the pure quartic case one has

$|m_{1}|=|m_{12}|={4{\sqrt{2}}\over 3}K({\rm
i}),\qquad|m_{2}|=|m_{123}|={8\over 3}K({\rm i}),$ (5.27)

where $K$ is the elliptic integral of the first kind. The angles appearing in
the kernels of the TBA system (5.21) are as in (5.22), with $\alpha=\pi/4$.
The solutions of the TBA system satisfy

$\widetilde{\epsilon}_{1}=\widetilde{\epsilon}_{12},\qquad\widetilde{\epsilon}_{2}=\widetilde{\epsilon}_{123}$
(5.28)

with periodicity
$\widetilde{\epsilon}_{a}(\theta)=\widetilde{\epsilon}_{a}(\theta+\frac{3\pi
i}{2})$. The four equations (5.21) reduce to two different equations for, say,
$\widetilde{\epsilon}_{a}$, $a=1,2$, which can be easily obtained from (5.21),

$\displaystyle\widetilde{\epsilon}_{1}(\theta)$ $\displaystyle=|m_{1}|{\rm
e}^{\theta}+{1\over\pi}\int_{\mathbb{R}}{\widetilde{L}_{1}(\theta^{\prime})\over\cosh(\theta-\theta^{\prime})}{\rm
d}\theta^{\prime}+{{\sqrt{2}}\over\pi}\int_{\mathbb{R}}{\cosh(\theta-\theta^{\prime})\over\cosh(2(\theta-\theta^{\prime}))}\widetilde{L}_{2}(\theta^{\prime}){\rm
d}\theta^{\prime},$ (5.29) $\displaystyle\widetilde{\epsilon}_{2}(\theta)$
$\displaystyle=|m_{2}|{\rm
e}^{\theta}+{2{\sqrt{2}}\over\pi}\int_{\mathbb{R}}{\cosh(\theta-\theta^{\prime})\over\cosh(2(\theta-\theta^{\prime}))}\widetilde{L}_{1}(\theta^{\prime}){\rm
d}\theta^{\prime}+{1\over\pi}\int_{\mathbb{R}}{\widetilde{L}_{2}(\theta^{\prime})\over\cosh(\theta-\theta^{\prime})}{\rm
d}\theta^{\prime}.$

This is precisely the TBA system proposed in dt  for the pure quartic
oscillator. This example, together with the cubic example discussed in the
previous section, make it clear that the TBA system introduced and studied in
this paper provides a generalization of the ODE/IM correspondence for
arbitrary polynomial potentials.

##  6 Conclusions and outlook

In this paper we have provided a solution to Voros’ analytic bootstrap voros-
quartic  for arbitrary polynomial potentials in one-dimensional Quantum
Mechanics. The solution takes the form of a TBA system for the (resummed)
quantum periods which generalizes the ODE/IM correspondence dt ; ddt . Our
approach builds upon recent progress in the solution of similar
Riemann–Hilbert problems in e.g. gmn ; gmn2 ; y-system ; oper . We have also
illustrated our general considerations with concrete examples in Quantum
Mechanics. In particular, we have seen how our results can be combined with
exact quantization conditions to solve spectral problems in an efficient way.

There are clearly many questions open by our research. Let us list some of
them.

From the TBA equations, one could identify the integrable model corresponding
to the Schrödinger equation with general polynomial potential. It is important
to see how the integral model “roams” under the wall crossing of TBA
equations. It is also interesting to study the Schrödinger equation with the
potential $q^{6}-\alpha q^{2}$ to recover the hidden $U_{q}(\widehat{gl}(2|1)$
structure suzuki ; ddt1 . It was shown in blz-spectral  that the original
ODE/IM correspondence of dt  can be extended to incorporate terms in the
potential of the form $1/q^{2}$. It would be interesting to see whether our
formalism can also accommodate such terms. It was also pointed out in blz-
spectral ; dt-beyond  that the WKB expansion of monic potentials corresponds,
under the ODE/IM correspondence, to an expansion of the TBA system in terms of
local charges of the corresponding CFT. It would be very interesting to see
whether this expansion can be generalized to general polynomial potentials, in
which the perturbative series involves complicated functions on moduli space.

In section 3.3 we found a link between the effective central charge of the TBA
system and the first quantum correction to the quantum period ${\cal J}$. This
suggests that this period is the appropriate object to formulate and extend
the PNP relation. In particular, it would be interesting to use these insights
to find a higher genus version of the PNP relation. We also found that the
effective central charge computed in section 3.3 is equivalent to the
effective central charge of the 2d chiral algebra ${\cal A}_{-(2\ell+1)}$ in
CSVY . It would be interesting to study what this equivalence means in the
context of the 4d/2d correspondence BLLPRR .

Perhaps the most interesting avenue open by our results is the possibility to
apply the same logic to more general problems in quantum theory. The data
required by the Riemann–Hilbert problem (namely, classical limits and
discontinuity structures) are provided by the resurgent structure of many
quantum theories. It is then possible that one can use the logic employed here
to obtain TBA equations governing the coupling constant dependence for other,
potentially more complicated theories, including quantum field theories and
string theories.

A good starting point for these more ambitious generalizations would be the
deformed quantum mechanics studied in gm-deformed . The WKB periods in this
theory are the quantum periods of Seiberg–Witten theory mirmor , and the
solution of the corresponding Riemann–Hilbert problem would provide a new
perspective on the NS limit of ${\cal N}=2$ supersymmetric gauge theories. One
could also make a step further and consider the quantum periods of local
Calabi–Yau threefolds acdkv . Finally, one could try to apply the same
techniques to field theory models where the resurgent structure is relatively
well understood, like (complex) Chern–Simons theory gmp ; ghatsuda , or to
field-theoretical models where semiclassical methods are reliable (see e.g.
du-review  and references therein.)

###### Acknowledgements.

We would like to thank Jorgen Andersen, Carl Bender, Sergio Cecotti, Santiago
Codesido, Bertrand Eynard, Davide Gaiotto, Jie Gu, Lotte Hollands, Maxim
Kontsevich, Greg Moore, Andy Neitzke, Ricardo Schiappa, Amit Sever, Junji
Suzuki and Roberto Tateo for discussions and correspondence. We are
particularly grateful to Jonathan Toledo for making available to us his
unpublished note on wall-crossing. M.M. and H.S. would like to thank the
conference String-Math 2018 for hospitality. M.M. would like to thank the BIRS
for hospitality during the workshop Geometry and physics of quantum curves,
and the Erwin Schrödinger Institute for hospitality during the workshop
Geometric correspondences of gauge theories. H.S. would like to thank the
University of Geneva for hospitality. The work of K.I. is supported in part by
Grant-in-Aid for Scientific Research 15K05043, 18K03643 and 16F16735 from
Japan Society for the Promotion of Science (JSPS). The work of M.M. was
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 H.S. is supported in part by JSPS Research
Fellowship 17J07135 for Young Scientists, from Japan Society for the Promotion
of Science (JSPS).

##  Appendix A Perturbative calculation of the quantum periods

In the exact WKB method, we are often interested in computing the perturbative
expansion of the WKB periods (2.12) to high order. Different methods have been
developed in the literature in order to achieve this. In this Appendix we will
present two different methods. The first one uses the holomorphic anomaly
equations of (refined) topological string theory bcov ; hk ; kw , and is based
on the results of cm-ha . The second one is based on the construction of
differential operators.

###  A.1 Holomorphic anomaly equation

Let us assume that the WKB curve is elliptic, so that there are only two
independent cycles in the problem which we will call the $A$-cycle and the
$B$-cycle (curves of higher genus can be also analyzed with this method fkn ).
They define respectively the quantum periods $\nu$ and $\nu_{D}$, as in
(3.50). As shown in cm-ha , the holomorphic anomaly equations of topological
string theory give the expansion of the quantum free energy

$F^{\rm NS}(\nu)=\sum_{n\geq 0}\hbar^{2n}F_{n}^{\rm NS}(\nu)$ (A.1)

as a function of $\nu$. However, in this formalism the quantum period $\nu$ is
parametrized by a value of the modulus, which we will denote by $\xi_{0}$
which is related to $\nu$ by the equation defining the classical period. We
then have

$\nu=\Pi_{A}^{(0)}(\xi_{0}).$ (A.2)

At the same time, the actual value of the modulus, $\xi$, has a non-trivial
expansion in $\hbar$ when expressed in terms of $\xi_{0}$, and one finds:

$\xi(\nu,\hbar)=\xi_{0}(\nu)+\sum_{n\geq 1}\xi_{n}(\nu)\hbar^{2n}.$ (A.3)

The PNP relations make it possible to obtain this power series from the power
series of $F^{\rm NS}$. Then, we have the identity:

$\Pi_{A}(\xi(\nu,\hbar))=\Pi_{A}^{(0)}(\xi_{0}).$ (A.4)

By expanding in $\hbar$, we find equations for the $\Pi_{A}^{(n)}$. For
example, at leading order in $\hbar^{2}$, (A.4) gives

${\partial\Pi_{A}^{(0)}\over\partial\xi}\xi_{1}+\Pi_{A}^{(1)}=0,$ (A.5)

and we conclude that

$\Pi_{A}^{(1)}=-\xi_{1}{\partial\Pi_{A}^{(0)}\over\partial\xi}.$ (A.6)

Similarly, for the B-periods we have by definition

${\rm i}\Pi_{B}(\xi(\nu,\hbar))={\partial F^{\rm NS}\over\partial\nu}.$ (A.7)

From this identity we deduce for example

${\rm i}\Pi^{(1)}_{B}=-{\rm
i}\xi_{1}{\partial\Pi_{B}^{(0)}\over\partial\xi}+{\partial F_{1}^{\rm
NS}\over\partial\nu}.$ (A.8)

The results of cm-ha  give the values of $\xi_{n}$, $F_{n}^{\rm NS}$ in terms
of (quasi)modular forms of the $\tau$ parameter of the WKB curve. The modular
forms are the Eisenstein series $E_{2}(\tau)$ and

$\displaystyle K_{2}\left(q\right)$
$\displaystyle:=\vartheta_{3}^{4}\left(\tau\right)+\vartheta_{4}^{4}\left(\tau\right),$
(A.9) $\displaystyle K_{4}\left(q\right)$
$\displaystyle:=\vartheta_{2}^{8}\left(\tau\right),$

Using the above results, we can compute the quantum periods $\Pi_{A,B}^{(n)}$
systematically in terms of modular forms. In order to do this, it is useful to
express all the ingredients of the derivation in terms of modular forms as
well, which is easily done by using the results of cm-ha .

Proceeding in this way, we obtain for example, for the cubic oscillator (4.1)
with $\kappa=1$:

$\displaystyle\Pi_{A}^{(0)}$
$\displaystyle={\pi\over\sqrt{2}}{2E_{2}(K_{2}^{2}+3K_{4})+K_{2}(9K_{4}-K_{2}^{2})\over
45(K_{2}^{2}+3K_{4})^{5/4}},$ (A.10) $\displaystyle\Pi_{A}^{(1)}$
$\displaystyle=\frac{\pi\left(K_{2}^{2}+3K_{4}\right)^{5/4}\left(\left(K_{2}^{2}+3K_{4}\right)^{2}-2E_{2}\left(K_{2}^{3}-9K_{2}K_{4}\right)\right)}{18\sqrt{2}K_{4}\left(K_{2}^{2}-K_{4}\right)^{2}},$

For the double-well (5.1) with $\kappa=1$, we find

$\displaystyle\Pi_{A}^{(0)}$
$\displaystyle={\sqrt{2}}\pi{2E_{2}K_{2}+3K_{4}-K_{2}^{2}\over
72K_{2}^{3/2}},$ (A.11) $\displaystyle\Pi_{A}^{(1)}$
$\displaystyle=\sqrt{2}\pi\frac{K_{2}^{3/2}\left(-2E_{2}K_{2}^{2}+3E_{2}K_{4}+K_{2}^{3}\right)}{9K_{4}\left(K_{2}^{2}-K_{4}\right)}.$

The $B$-periods can be easily obtained from the $A$-periods by the following
rule,

$\Pi_{B}^{(n)}=\mu\,\tau\Pi_{A}^{(n)}\bigl{|}_{E_{2}\rightarrow E_{2}^{D}}.$
(A.12)

where $\mu=1,2$ for the cubic potential and the double-well, respectively, and
coms

$E^{D}_{2}=E_{2}+{6\over\pi{\rm i}\tau}.$ (A.13)

It is easy to see that general values of $\kappa$ can be incorporated by
including a factor $\kappa^{-5n+5/2}$ in the case of the cubic well, and a
factor $\kappa^{-3n+3/2}$ in the case of the double-well, and changing the
value of $\tau$ accordingly.

###  A.2 Differential operators

We present another approach to the calculation of higher order corrections to
the quantum periods for the cubic and quartic potentials huang ; ito-okubo .
In this approach it is convenient to set $V(q)-E$ in (2.1) as

$V(q)-E=q^{r+1}-u_{1}q^{r-1}-\dots-u_{r},$ (A.14)

which is a $(r+1)$-th order polynomial in $q$. We consider the cycle on the
WKB curve $\tilde{y}^{2}=V(q)-E$, which is slightly different from the
convention used in the main text101010The quantum periods will differ by some
overall numerical factors.. $\Pi^{(0)}_{\gamma}$ is the period integral of the
curve $\tilde{y}^{2}=V(q)-E$

$\Pi^{(0)}_{\gamma}={\rm i}\oint_{\gamma}\tilde{y}{\rm d}q.$ (A.15)

Since

$\partial_{u_{i}}\tilde{y}\,{\rm d}q=-{q^{r-i}\over 2\tilde{y}}{\rm d}q,\qquad
i=1,\cdots,r,$ (A.16)

provide a basis of meromorphic differentials on the WKB curve, one can expand
the $p_{n}(q){\rm d}q$ in (2.12) in terms of this basis:

$p_{n}(q){\rm d}q={\rm
i}\sum_{j=1}^{r}b_{j}^{(n)}\partial_{u_{j}}\tilde{y}{\rm d}q+\partial_{q}(*),$
(A.17)

where the second term denotes a total derivative. It follows that the quantum
corrections to the periods are given by

$\Pi^{(n)}_{\gamma}=\sum_{i=1}^{r}b_{i}^{(n)}\partial_{u_{i}}\Pi^{(0)}_{\gamma}.$
(A.18)

We can find the coefficients $b_{i}^{(n)}$ by solving (A.17). Here we present
the results for the cubic ($r=2$) and the quartic ($r=3$) cases up to some
higher order.

Let us first consider the cubic curve given by

$\tilde{y}^{2}=q^{3}-u_{1}q-u_{2}$ (A.19)

whose discriminant is given by $\Delta=4u_{1}^{3}-27u_{2}^{2}$. By solving
(A.17) we find that the first four corrections are given by

$\Pi^{(n)}=b^{(n)}_{1}\partial_{u_{1}}\Pi^{(0)}+b^{(n)}_{2}\partial_{u_{2}}\Pi^{(0)}$
(A.20)

with

$\displaystyle b^{(1)}_{1}$ $\displaystyle={9u_{2}\over 4\Delta},\quad
b^{(1)}_{2}={u_{1}^{2}\over 2\Delta},$ (A.21) $\displaystyle b^{(2)}_{1}$
$\displaystyle={21u_{1}^{2}(308u_{1}^{3}+5697u_{2}^{2})\over
160\Delta^{3}}\quad b^{(2)}_{2}={63u_{1}u_{2}(788u_{1}^{3}+2467u_{2}^{2})\over
320\Delta^{3}},$ $\displaystyle b^{(3)}_{1}$
$\displaystyle={81u_{1}u_{2}(24549712u_{1}^{6}+344562120u_{1}^{3}u_{2}^{2}+304562349u_{2}^{4})\over
17290\Delta^{5}},$ $\displaystyle b^{(3)}_{2}$ $\displaystyle={1\over
17290\Delta^{5}}(69623840u_{1}^{9}+5529102768u_{1}^{6}u_{2}^{2}+26089138530u_{1}^{3}u_{2}^{4}+6057679446u_{2}^{6}),$
$\displaystyle b^{(4)}_{1}$ $\displaystyle={1\over
573440\Delta^{7}}81u_{1}^{2}u_{2}(277770953152u_{1}^{9}+11292809333904u_{1}^{6}u_{2}^{2}+63249377035572u_{1}^{3}u_{2}^{4}$
$\displaystyle+47953365866955u_{2}^{6}),$ $\displaystyle b^{(4)}_{2}$
$\displaystyle={1\over
573440\Delta^{7}}6(421453455808u_{1}^{12}+62854121561616u_{1}^{9}u_{2}^{2}+782827801494228u_{1}^{6}u_{2}^{4}$
$\displaystyle+1375370060382675u_{1}^{3}u_{2}^{6}+139863680471790u_{2}^{8}).$

In particular, at $u_{1}=0$, up to the 18-th order, they are found to be

$\displaystyle b_{1}^{(0)}$ $\displaystyle=0,\quad b^{(0)}_{2}={6u_{2}\over
5},$ (A.22) $\displaystyle b^{(1)}_{1}$ $\displaystyle=-{1\over 12u_{2}},\quad
b_{2}^{(1)}=0$ $\displaystyle b_{1}^{(3)}$ $\displaystyle=0,\quad
b^{(3)}_{2}=-{21983\over 933120u_{2}^{4}}$ $\displaystyle b^{(4)}_{1}$
$\displaystyle=-{26317819\over 188116992u_{2}^{6}},\quad b_{2}^{(4)}=0$
$\displaystyle b_{1}^{(6)}$ $\displaystyle=0,\quad
b^{(6)}_{2}=-{14175476893921\over 2298682146816u_{2}^{9}}$ $\displaystyle
b^{(7)}_{1}$ $\displaystyle=-{61331166685678399\over
391193907167232u_{2}^{11}},\quad b_{2}^{(7)}=0$ $\displaystyle b_{1}^{(9)}$
$\displaystyle=0,\quad b^{(9)}_{2}=-{2807892430989520754556859\over
47734804593462214656u_{2}^{14}}$

and other $b_{i}^{(n)}$ for $n=2,5,8$ are zero. These coefficients are
confirmed by the CFT approach to the T-function blz1 .

Next we consider the quartic case with the curve

$\tilde{y}^{2}=q^{4}-u_{1}q^{2}-u_{2}q-u_{3}$ (A.23)

whose discriminant is given by

$\Delta=4u_{1}^{3}u_{2}^{2}-27u_{2}^{4}-16u_{1}^{4}u_{3}+144u_{1}u_{2}^{2}u_{3}-128u_{1}^{2}u_{3}^{2}-256u_{3}^{3}.$
(A.24)

Solving (A.17), we find that the coefficients of the first two corrections are
given by

$\displaystyle b_{1}^{(1)}$
$\displaystyle={4(u_{1}^{4}-9u_{1}u_{2}^{2}+16u_{1}^{2}u_{3}+48u_{3}^{2})\over
3\Delta}$ (A.25) $\displaystyle b_{2}^{(1)}$ $\displaystyle=0$ $\displaystyle
b_{3}^{(1)}$
$\displaystyle={3u_{1}^{2}u_{2}^{2}-16u_{1}^{3}u_{3}+36u_{2}^{2}u_{3}-64u_{1}u_{3}^{2}\over
3\Delta}$ $\displaystyle b_{1}^{(2)}$ $\displaystyle={1\over
45\Delta^{3}}2\Bigl{(}1792u_{1}^{13}-39768u_{1}^{10}u_{2}^{2}+316302u_{1}^{7}u_{2}^{4}-1022301u_{1}^{4}u_{2}^{6}+1137240u_{1}u_{2}^{8}+40800u_{1}^{11}u_{3}$
(A.26)
$\displaystyle-598896u_{1}^{8}u_{2}^{2}u_{3}+2354760u_{1}^{5}u_{2}^{4}u_{3}-2599128u_{1}^{2}u_{2}^{6}u_{3}+390528u_{1}^{9}u_{3}^{2}-3069696u_{1}^{6}u_{2}^{2}u_{3}^{2}$
$\displaystyle+10014624u_{1}^{3}u_{2}^{4}u_{3}^{2}-15688080u_{2}^{6}u_{3}^{2}+1555456u_{1}^{7}u_{3}^{3}-12363264u_{1}^{4}u_{2}^{2}u_{3}^{3}+29310336u_{1}u_{2}^{4}u_{3}^{3}$
$\displaystyle+405504u_{1}^{5}u_{3}^{4}-9934848u_{1}^{2}u_{2}^{2}u_{3}^{4}-12656640u_{1}^{3}u_{3}^{5}+74207232u_{2}^{2}u_{3}^{5}-23101440u_{1}u_{3}^{6}\Bigr{)},$
$\displaystyle b_{2}^{(2)}$ $\displaystyle=0,$ $\displaystyle b_{3}^{(2)}$
$\displaystyle={1\over
45\Delta^{3}}\Bigl{(}-1344u_{1}^{11}u_{2}^{2}+26676u_{1}^{8}u_{2}^{4}-185139u_{1}^{5}u_{2}^{6}+355752u_{1}^{2}u_{2}^{8}+1792u_{1}^{12}u_{3}-61296u_{1}^{9}u_{2}^{2}u_{3}$
$\displaystyle+611736u_{1}^{6}u_{2}^{4}u_{3}-500904u_{1}^{3}u_{2}^{6}u_{3}-2449440u_{2}^{8}u_{3}+38784u_{1}^{10}u_{3}^{2}-958464u_{1}^{7}u_{2}^{2}u_{3}^{2}-48096u_{1}^{4}u_{2}^{4}u_{3}^{2}$
$\displaystyle+6792336u_{1}u_{2}^{6}u_{3}^{2}+708096u_{1}^{8}u_{3}^{3}-1660416u_{1}^{5}u_{2}^{2}u_{3}^{3}-432000u_{1}^{2}u_{2}^{4}u_{3}^{3}+5582848u_{1}^{6}u_{3}^{4}$
$\displaystyle-25288704u_{1}^{3}u_{2}^{2}u_{3}^{4}+58102272u_{2}^{4}u_{3}^{4}+16957440u_{1}^{4}u_{3}^{5}-121614336u_{1}u_{2}^{2}u_{3}^{5}+10321920u_{1}^{2}u_{3}^{6}$
$\displaystyle-21626880u_{3}^{7}\Bigr{)}.$

In this Appendix we have presented two different approaches to compute the
corrections of WKB periods. The quantum corrections computed in these two
approaches match very well, up to the overall numerical factors due to the
different normalizations.

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