Regular and Floquet bases for gauge and gravity theories: a non perturbative approach

TL;DR

This paper introduces a non-perturbative method using a new 'kink method' to analyze Heun-type equations, connecting gauge and gravity theories through basis changes and phase calculations, with promising applications in supersymmetric theories and gravitational perturbations.

Contribution

It develops a novel 'kink method' for computing phases and quantum momentum in Heun equations, providing an alternative to instanton calculations and insights into non-perturbative effects in gauge and gravity theories.

Findings

Matching of acquired phase with dual gauge period $A_D$.

Compatibility of wave functions with known gravity solutions.

Potential to analyze non-perturbative contributions at all orders.

Abstract

The main topic of the paper is represented by the change of basis, in Heun-type equations, from the one of decaying (at two singular points) solutions to that of Floquet solutions. Crucial in the connection relations is the phase acquired by the Floquet solutions by going from a (ir)regular singularity to another. The new 'kink method' is exploited to compute the quantum momentum of the Floquet solutions as convergent series, explicitly at all orders. Hence, upon integrating it term by term, the acquired phase can be derived explicitly as a similar series. Since Heun equation and its confluences describe ${\cal N}=2$ SYM theories in the NS background as quantisation of Seiberg-Witten differentials and also appear in perturbations of gravity solutions, tests and predictions in both domains can be made. A very encouraging test is the matching of the acquired phase with the dual gauge period, $A_D$, as given by the Nekrasov instanton function. Actually, the procedure can be considered an alternative to instanton computations. On the gravity side, results for the wave functions are at leading order compatible with analogous expressions found by studying the Teukolsky (or others, like Regge-Wheeler) equation. Besides, the whole construction can represent an useful approach to these equations at all orders, shedding also light on non-perturbative contributions, which should reveal very interesting for the consequences in gravity perturbations.

Equations

$$\frac{d^2}{d{y}^2}\psi (y)=\left(2e^{2\theta }\cosh y+P^2\right)\psi (y).$$

$$V_0(y)\simeq \frac{1}{\sqrt{2}}\exp \left(-\frac{\theta }{2}+\frac{y}{4}\right)\exp \left(-2e^{\theta -\frac{y}{2}}\right),$$

$$\frac{d^2\varphi }{d{r}^2}+\left[{\omega }^2\left(1+\frac{L^4}{r^4}\right)-\frac{(l+2{)}^2-\frac{1}{4}}{r^2}\right]\varphi (r)=0$$

$$\Pi (y{)}^2+\frac{d}{dy}\Pi (y)=2e^{2\theta }\cosh y+P^2.$$

$$\Pi (y)\simeq \pm e^{\theta }{e}^{\frac{y}{2}}-\frac{1}{4}\pm \frac{4P^2-1}{32e^{\theta }}{e}^{-\frac{y}{2}}+\frac{4P^2-1}{64e^{2\theta }}{e}^{-y}+O(e^{-3y/2}),$$

$$\Pi (y)\simeq \pm e^{\theta }{e}^{-\frac{y}{2}}+\frac{1}{4}\pm \frac{4P^2-1}{32e^{\theta }}{e}^{\frac{y}{2}}-\frac{4P^2-1}{64e^{2\theta }}{e}^y+O(e^{3y/2}).$$

$${\Pi }_{+}(y)\simeq \pm e^{\theta }{e}^{\pm \frac{y}{2}}\mp \frac{1}{4}\pm \frac{4P^2-1}{32e^{\theta }}{e}^{\mp \frac{y}{2}}\pm \frac{4P^2-1}{64e^{2\theta }}{e}^{\mp y}+O(e^{\mp 3y/2}),$$

$${\psi }_{+}(y)=e^c{e}^{2e^{\theta }{e}^{\frac{y}{2}}+2e^{\theta }{e}^{-\frac{y}{2}}+\frac{y}{4}}\exp \left[{\int }_{-\infty }^{y}d{y}^{\prime }\left({\Pi }_{+}(y^{\prime })-e^{\theta }{e}^{\frac{y^{\prime }}{2}}+e^{\theta }{e}^{-\frac{y^{\prime }}{2}}-\frac{1}{4}\right)\right],$$

$$2\pi ik={\int }_y^{y+2i\pi }d{y}^{\prime }{\Pi }_{+}(y^{\prime }),$$

$${\int }_{-\infty }^{y}d{y}^{\prime }\left({\Pi }_{+}(y^{\prime })-e^{\theta }{e}^{\frac{y^{\prime }}{2}}+e^{\theta }{e}^{-\frac{y^{\prime }}{2}}-\frac{1}{4}\right)={\int }_{-\infty }^{0}d{y}^{\prime }\left({\Pi }_{+}(y^{\prime })-e^{\theta }{e}^{\frac{y^{\prime }}{2}}+e^{\theta }{e}^{-\frac{y^{\prime }}{2}}-\frac{1}{4}\right)+$$

$${\int }_0^{y}d{y}^{\prime }\left({\Pi }_{+}(y^{\prime })-e^{\theta }{e}^{\frac{y^{\prime }}{2}}+e^{\theta }{e}^{-\frac{y^{\prime }}{2}}+\frac{1}{4}\right)-\frac{y}{2},$$

$${\psi }_{+}(y;k)\simeq e^c{e}^{2e^{\theta }{e}^{\frac{y}{2}}-\frac{y}{4}}\exp \left[{\int }_{-\infty }^{0}d{y}^{\prime }\left({\Pi }_{+}(y^{\prime })+e^{\theta }{e}^{-\frac{y^{\prime }}{2}}-\frac{1}{4}\right)+{\int }_0^{+\infty }d{y}^{\prime }\left({\Pi }_{+}(y^{\prime })-e^{\theta }{e}^{\frac{y^{\prime }}{2}}+\frac{1}{4}\right)\right]$$

$$\phi (\theta ,P)={\int }_{-\infty }^{0}d{y}^{\prime }\left({\Pi }_{+}(y^{\prime })+e^{\theta }{e}^{-\frac{y^{\prime }}{2}}-\frac{1}{4}\right)+{\int }_0^{+\infty }d{y}^{\prime }\left({\Pi }_{+}(y^{\prime })-e^{\theta }{e}^{\frac{y^{\prime }}{2}}+\frac{1}{4}\right)\equiv {\phi }_{<}(\theta ,P)+{\phi }_{>}(\theta ,P).$$

$${\phi }_{>}\left(\theta +\frac{i\pi }{2},P\right)={\phi }_{>}(\theta ,P)-i{\int }_0^{\pi }d{y}^{\prime }\left({\Pi }_{+}(i{y}^{\prime })-e^{\theta }{e}^{\frac{i{y}^{\prime }}{2}}+\frac{1}{4}\right).$$

$${\phi }_{<}\left(\theta +\frac{i\pi }{2},P\right)={\phi }_{<}(\theta ,P)-i{\int }_{-\pi }^{0}d{y}^{\prime }\left({\Pi }_{+}(i{y}^{\prime })+e^{\theta }{e}^{-\frac{i{y}^{\prime }}{2}}-\frac{1}{4}\right).$$

$$\phi \left(\theta +\frac{i\pi }{2},P\right)=\phi (\theta ,P)-2\pi ik(\theta ,P).$$

$$k\left(\theta +\frac{i\pi }{2},P\right)={\int }_0^{2\pi }\frac{d{y}^{\prime }}{2\pi }{\Pi }_{+}\left(i{y}^{\prime },\theta +\frac{i\pi }{2}\right)={\int }_0^{2\pi }\frac{d{y}^{\prime }}{2\pi }{\Pi }_{+}(i{y}^{\prime }\pm \pi ,\theta )={\int }_0^{2\pi }\frac{d{y}^{\prime }}{2\pi }{\Pi }_{+}(i{y}^{\prime },\theta )=k(\theta ,P).$$

$$\phi \left(\theta +\frac{i\pi }{2},k\right)=\phi (\theta ,k)-2\pi ik.$$

$${\psi }_{\pm }(y;k)\simeq e^{\mp \frac{\phi }{2}}{e}^{\frac{y}{4}}\exp \left(2e^{\theta }{e}^{-\frac{y}{2}}\right)\left[1+O\left(e^{y/2}\right)\right],$$

$${\psi }_{\pm }(y;k)\simeq e^{\pm \frac{\phi }{2}}{e}^{-\frac{y}{4}}\exp \left(2e^{\theta }{e}^{\frac{y}{2}}\right)\left[1+O\left(e^{-y/2}\right)\right].$$

$$U_0(y;k)=\frac{\sqrt{2}{e}^{\frac{\theta }{2}}}{W[{\psi }_{+},{\psi }_{-}]}\left[e^{-\frac{\phi }{2}}{\psi }_{+}(y;k)-e^{\frac{\phi }{2}}{\psi }_{-}(y;k)\right],V_0(y;k)=\frac{\sqrt{2}{e}^{\frac{\theta }{2}}}{W[{\psi }_{+},{\psi }_{-}]}\left[e^{-\frac{\phi }{2}}{\psi }_{-}(y;k)-e^{\frac{\phi }{2}}{\psi }_{+}(y;k)\right],$$

$$Q(\theta ,k)\equiv W[U_0,V_0]=\frac{\sinh \phi (\theta ,k)}{\sin 2\pi k},$$

$${\Pi }_{+}(y)=\sum _{n=0}^{+\infty }{e}^{2n\theta }{\Pi }_{+}^{(n)}(y),$$

$$2P{\Pi }_{+}^{(1)}(y)+\frac{d}{dy}{\Pi }_{+}^{(1)}(y)=2\cosh y,\sum _{m=0}^n{\Pi }_{+}^{(m)}(y){\Pi }_{+}^{(n-m)}(y)+\frac{d}{dy}{\Pi }_{+}^{(n)}(y)=0\text{for}n\ge 2.$$

$${\Pi }_{+}^{(1)}(y)=\frac{e^y}{1+2P}+\frac{e^{-y}}{2P-1},{\Pi }_{+}^{(2)}(y)=-\frac{e^{2y}}{(1+2P{)}^2(2+2P)}-\frac{1}{P(4P^2-1)}+\frac{e^{-2y}}{(2P-1{)}^2(2-2P)}.$$

$$k(\theta ,P)=\sum _{n=0}^{+\infty }{e}^{4n\theta }{r}_0^{(2n)}(P)=P-\frac{e^{4\theta }}{P(4P^2-1)}+\frac{e^{8\theta }(-60P^4+35P^2-2)}{4(4P^2-1{)}^3(P^2-1)P^3}+O(e^{12\theta }).$$

$${\phi }_{>}(\theta ,P)={\int }_{2\theta }^{+\infty }d{y}^{\prime }\left({\Pi }_{>}(y^{\prime };\theta ,P)-e^{\frac{y^{\prime }}{2}}+\frac{1}{4}\right).$$

$${\Pi }_{>}(y;\theta ,P{)}^2+\frac{d}{dy}{\Pi }_{>}(y;\theta ,P)=e^y+P^2+e^{4\theta }{e}^{-y}.$$

$${\Pi }_{>}(y;\theta ,P)=\sum _{n=0}^{+\infty }{\Pi }_{>}^{(n)}(y;P)e^{4n\theta },$$

$$P^2=\sum _{n=0}^{+\infty }{p}_2^{(n)}(k)e^{4n\theta }=k^2+\frac{2}{4k^2-1}{e}^{4\theta }+\frac{20k^4+7k^2}{2k^2(k^2-1)(4k^2-1{)}^3}{e}^{8\theta }+O\left(e^{12\theta }\right)$$

$${\Pi }_{>}(y;\theta ,k)=\sum _{n=0}^{+\infty }{\tilde{\Pi }}_{>}^{(n)}(y;k)e^{4n\theta },$$

Full text

Keywords: Integrable Field Theories; ODE/IM correspondence; Heun equations; SYM theories; gravity theory

1 Introduction

The Heun equation (HE) and its confluence limits – which can be collectively called Heun-type equations – have ample and relevant applications in mathematics and physics. For instance and importantly, the HE describes the quantisation [1] of the Seiberg-Witten (SW) differential [2] for ${\cal N}=2$ SYM with $N_{f}=4$ flavours in the Nekrasov-Shatashvili (NS) background [3]; besides its different confluences yield the same theories with less flavour multiplets: in specific, the Confluent Heun Equation (CHE) the Doubly Confluent Heun Equation (DCHE) and the Modified Mathieu Equation (MME) play the same rôle for $N_{f}=3,2,0$, respectively. In addition, they may appear in many instances of the dimensional reduction (and linearisation) of the Einstein partial differential equations (PDE), when the metric of some massive object, like a black hole (BH), is subjected to the small perturbation of another field [4]. Famous examples are those (radially symmetric) of scalar, electromagnetic and axial (odd-parity) and polar (even-parity) gravitational perturbations of the Schwarzschild metric which are governed, at least at first order, by the Regge-Wheeler [5] (and Zerilli for polar [6]) or the Teukolsky equations [7], which, in their turn, are separated by the standard angular part (of spherical harmonics) and the radial part represented by a CHE [8, 9]. Without radial symmetry, perturbations of rotating black holes of Kerr metric, for instance, as elaborated by [7], are described, both in their angular and radial part, by CHE as well. But even the HE finds an application in perturbations of asymptotically AdS BHs [10]. Eventually, Heun-type equations govern some of the cosmological (scalar and tensor) perturbations within the $\Lambda CDM$ model ([11] and references therein).

A basis of solutions which is relevant in BH physics is the so-called in/up basis: this is made by a solution satisfying ’incoming’ boundary condition (i.e. which vanishes and enters the black hole) at the horizon and another one satisfying ’outgoing’ boundary condition at infinity (i.e. which very far from the horizon is a wave moving towards infinity). Crucially, as noted in [12], in terms of integrability parameters both these functions are decaying (the incoming at the horizon, the outgoing at infinity) and constitute the basis usually used in the framework of the ODE/IM correspondence: for the specific construction of ODE/IM we discuss in this paper, see [13], for previous foundational works and understanding on different cases, cf [14, 15, 16, 17, 18] and references therein. On the other hand, from the mathematical point of view fundamental and well studied functions are the so-called Floquet solutions with Floquet index $k$. Beside $k$, of these functions we identify as important parameter the parallel transport $\varphi$ from zero to infinity, which together with $k$ constitute the so-called Fenchel-Nielsen coordinates [19]. In this paper we want to highlight the role of the phase $\varphi$ in the connection coefficients between the Floquet basis and the in/up basis and, importantly, in the framework of ${\cal N}=2$ SYM, as the dual quantum period $A_{D}$. In this respect, we find crucial to use the so-called ’kink method’ of integrable QFTs for the systematic computation (as convergent series) of the phase $\varphi$ and then of the connection coefficients between the in/up and the Floquet basis: our results for $A_{D}$ agree with instanton expansions of gauge theory and this is a nontrivial check of the correctness of our method. Importantly, the kink-method is effective also in computing the quantum momentum, the logarithmic derivative, of Floquet solutions of the HE and its confluence limits: thanks to it, the quantum momentum is written as a single converging series in a small parameter (the instanton coupling of the SYM theory). Through the above change of basis, ODE/IM (or in/up) functions can be written as sums of exponentials of converging series. All these results are relevant also for gravity, since, due to the map between the CHE and gravity, the wave functions we construct describe linear perturbations of BHs metric. In this respect our method differs from the approach of, for instance, [8, 9]. In [8] the instanton series for the prepotential of ${\cal N}=2$ SYM is used to compute quasinormal modes for Schwarzschild and Kerr BHs; in [9] the instanton partition function of a quiver gauge theory is the tool to compute solutions of ODEs describing perturbations of Schwarzschild BHs as a double series. In contrast, we approach directly these ODEs by using the kink method and find results both in ${\cal N}=2$ SYM (the quantum periods and then the prepotential) and in gravity (perturbations of BHs as solutions of ODEs, written as single series).

The start is one paradigmatic example, the ODE describing ${\cal N}=2$ SYM with $N_{f}=0$, the MME (Section 2), which is the simplest example of Heun-like equation, with two irregular singularities. In this case we describe in detail the kink method for the computation of the phase $\varphi$ and the connection coefficients and provide for them and for the quantum momentum as well, convergent series expansions. Then, in Section 3 the same method and computations are performed for the DCHE, still with two irregular singularities, relevant for ${\cal N}=2$ SYM with $N_{f}=2$. Finally, Section 4 is dedicated to the HE, with four regular singularities, which is connected to $N_{f}=4$ and, on the gravity perspective, to $AdS$ BHs. Importantly, a confluence limit of the HE, discussed in Subsection 4.2, gives convergent series expansions for the solutions of the CHE, with two regular and one irregular singluarities, which is important for the case $N_{f}=3$ and for studying perturbations of Schwarzschild and Kerr BHs.

From the exposition of this paper it should be clear that the in/up basis is usually used to discuss the ODE/IM correspondence and is also very relevant for applications to BH physics. However, for what concerns applications to gauge theories, the Floquet basis seems more useful. This justifies the emphasis in this paper on the Floquet basis.

2 Case study of the MME ($N_{f}=0$): from the ODE/IM to Floquet basis

The first move is made from the Modified Mathieu Equation (MME), written in the form

[TABLE]

As a particular case, this equation was analysed in [13] as the starting point of the C-ODE/IM correspondence to derive the $Q$-function of the self-dual Liouville model as the Wronskian $W[U_{0},V_{0}]$ between its two solutions $V_{0}(y)$ and $U_{0}(y)=V_{0}(-y)$, forming the ODE/IM basis, with $V_{0}$ uniquely defined by the decaying asymptotic behaviour

[TABLE]

when $\textrm{Re}y\rightarrow-\infty$ with $\left|\textrm{Im}\left(\theta-\frac{y}{2}\right)\right|<\frac{\pi}{2}$. The MME is the quantisation [21] of the SW differential for ${\cal N}=2$ SYM with $N_{f}=0$ in the NS background and, by the map $e^{-\frac{y}{2}}=\frac{r}{L},2e^{\theta}=-i\omega L,P=\frac{l+2}{2}$, becomes the ODE

[TABLE]

which describes scalar field perturbation $\phi(r)=\sqrt{r}\psi(y)$ of the D3 brane [22]. In gravity language $V_{0}\sim e^{i\omega r}$ is the outgoing solution at $r=\infty$, $U_{0}\sim e^{i\omega L^{2}/r}$ is the incoming at the horizon $r=0$111The derivative of the phase of $V_{0}$ with respect to $r$ when $r\rightarrow+\infty$ is $i\omega$; the derivative of the phase of $U_{0}$ with respect to $r$ when $r\rightarrow 0$ is $-i\omega L^{2}/r^{2}$: then, $V_{0}$ describes a wave going towards $r=+\infty$, while $U_{0}$ a wave going towards $r=0$. . Motivated by the relevance of the MME in different areas of physics, we reformulate C-ODE/IM of [13] by using another pair of solutions, the Floquet basis. To identify two linearly independent Floquet solutions of (2.1), it is useful to introduce the logarithmic derivative $\Pi(y)=\frac{d}{dy}\ln\psi(y)$ which satisfies the Riccati equation

[TABLE]

In general when $\textrm{Re}\left(\frac{y}{2}+\theta\right)\rightarrow+\infty$ there are two possible behaviours

[TABLE]

which may each expand, when $\textrm{Re}\left(\frac{y}{2}-\theta\right)\rightarrow-\infty$, in the two alternative ways

[TABLE]

The wave function for generic values of $\theta,P$ cannot decay at both $y=\pm\infty$. The Floquet solutions satisfy $\psi_{\pm}(y+2\pi i)=e^{\pm 2\pi ik}\psi_{\pm}(y)$ and then they have to be different from both $U_{0}(y)$ (since, following notations of [13], $U_{0}(y+2\pi i)=U_{1}(y)$) and $V_{0}(y)$ (since [13] $V_{0}(y+2\pi i)=V_{-1}(y)$). It follows that $\psi_{\pm}(y)=\alpha_{\pm}U_{0}(y)+\beta_{\pm}V_{0}(y)$, with $\alpha_{\pm},\beta_{\pm}$ both different from zero. Then, $\psi_{\pm}(y)$ diverge at both $\pm\infty$. The wave function $\psi_{+}$, corresponds to a solution $\Pi_{+}(y)=\frac{d}{dy}\ln\psi_{+}(y)$ of (2.3) which expands when $\textrm{Re}\left(\frac{y}{2}\pm\theta\right)\rightarrow\pm\infty$ as

[TABLE]

and in addition satisfies the periodicity property $\Pi_{+}(y+2i\pi)=\Pi_{+}(y)$. The corresponding wave function (’ray’ of solutions of equation (2.1), with proportionality constant $e^{c}$) is derived from this ’quantum momentum’ [20] by ’regularising’ the integrand

[TABLE]

so that, when $y\rightarrow-\infty$, $\psi_{+}(y;k)\simeq e^{c}e^{\frac{y}{4}}e^{2e^{\theta}e^{-\frac{y}{2}}}$. Since $\Pi_{+}(y+2i\pi)=\Pi_{+}(y)$, this is a quasi-periodic solution: $\psi_{+}(y+2\pi i;k)=e^{2\pi ik}\psi_{+}(y;k)$, where the Floquet exponent can be given by the integral

[TABLE]

on a straight line parallel to the imaginary axis, of course independent of $y$. The $y\rightarrow+\infty$ limit is found by manipulating the integral in the exponent in (2.7) as follows

[TABLE]

so that, when $y\rightarrow+\infty$

[TABLE]

It is then natural to define, independently of the normalisation $e^{c}$, the acquired ’scattering phase’ from $y\rightarrow-\infty$ to $y\rightarrow+\infty$:

[TABLE]

This phase satisfies an important functional relation. Indeed, using the symmetry222This symmetry is justified by saying that $\Pi_{+}(y;\theta)$ and $\Pi_{+}\left(y-i\pi;\theta+\frac{i\pi}{2}\right)$ satisfy the same Riccati equation and have the same asymptotic limits (2.6) when $y\rightarrow+\infty$. With a similar reasoning one proves the subsequent identity $\Pi_{+}(y,\theta)=\Pi_{+}\left(y+i\pi;\theta+\frac{i\pi}{2}\right)$. $\Pi_{+}(y;\theta)=\Pi_{+}\left(y-i\pi;\theta+\frac{i\pi}{2}\right)$, for $y>0$, which holds also for the regulator $\left(e^{\theta}e^{\frac{y}{2}}-\frac{1}{4}\right)$ contained in the definition of $\varphi_{>}$, and then simple arguments of complex analysis, one arrives to the relation

[TABLE]

Analogously, due to the symmetry $\Pi_{+}(y,\theta)=\Pi_{+}\left(y+i\pi;\theta+\frac{i\pi}{2}\right)$, for $y<0$, one proves the following property

[TABLE]

The sum of the previous two relations gives the transformation property

[TABLE]

Differently from $\varphi$, the Floquet exponent $k$ does not change under shift $\theta\rightarrow\theta+i\pi/2$:

[TABLE]

Then, if $\theta$ and $k$ are taken as independent variables, the function $\varphi(\theta,k)$ satisfies the functional relation

[TABLE]

It will be convenient for computations and for physical application to have $k$ and not $P$ as independent variable and then this will be our choice from now on. Choosing the constant $c=-\frac{\varphi}{2}$, the function $\psi_{+}$ acquires symmetric asymptotic behaviour at $\pm\infty$. The second Floquet solution, $\psi_{-}(y+2i\pi;k)=e^{-2i\pi k}\psi_{-}(y;k)$, can be defined by the relations $\psi_{\pm}(y;k)=\psi_{\mp}(-y;k)$ 333This solution corresponds to the solution of the Riccati equation (2.3) $\Pi_{-}(y)=-\Pi_{+}(-y)$.. Then, as $\textrm{Re}y\rightarrow-\infty$, with $\left|\textrm{Im}\left(\theta-\frac{y}{2}\right)\right|<\frac{\pi}{2}$, the two Floquet solutions $\psi_{\pm}(y;k)$ behave as

[TABLE]

whilst, when $\textrm{Re}y\rightarrow+\infty$, with $\left|\textrm{Im}\left(\theta+\frac{y}{2}\right)\right|<\frac{\pi}{2}$, they behave as

[TABLE]

As proven in [20], each element of the ODE/IM basis ${V_{0},U_{0}}$ can be expressed as fine-tuned linear combination of elements of the Floquet basis ${\psi_{-},\psi_{+}}$ (with coefficients involving the acquired phase since it appears in the asymptoptics (2.16) and (2.17)), in such a way to cancel the diverging contributions at leading order at $y=-\infty$ and $y=+\infty$ for $V_{0}$ and $U_{0}$, respectively; automatically the whole dominant solutions will cancel out in each combination and the decaying behaviours (at $y=-\infty$ and $y=+\infty$ for $V_{0}$ and $U_{0}$, respectively) will emerge exactly:

[TABLE]

where the Wronskian $W[\psi_{+},\psi_{-}]=-4e^{\theta}\sin 2\pi k$. Immediately, this implies an elegant expression for the connection coefficient

[TABLE]

which is also the $Q$-function of Liouville model at self-dual point [13]. Moreover, (2.15) is equivalent to the ’QQ-system’ $1+Q^{2}\left(\theta+\frac{i\pi}{2},k\right)=Q(\theta,k)Q(\theta+i\pi,k)$ and $Q(\theta)$ can be computed through an integrability or gauge Thermodynamic Bethe Ansatz (TBA) [13].

2.1 Large $\theta\rightarrow-\infty$ converging series

When $\theta$ is negative and very large, the Floquet solutions and then $k$ and $\varphi$ can be computed as converging series. In fact, $\Pi_{+}(y)$ for finite (not too large) $|y|$ allows only for the computation of $k$ (2.8) [23] (not of $\varphi$) as integration on the finite segment on the imaginary axis $[0,2\pi i]$: for these values of $y$ and for $\theta\rightarrow-\infty$, the inequality $|2e^{2\theta}\cosh y|<|P^{2}|$ is valid and then in the Riccati equation (2.3) the potential term is a small perturbation. Therefore, the solution of (2.3) can be written as the converging series

[TABLE]

with $\Pi_{+}^{(n)}(y)$ fixed recursively by (2.3). In fact, $\Pi_{+}^{(0)}(y)=P$ and then the $\Pi_{+}^{(n)}(y)$, $n\geq 1$, solve the ODEs

[TABLE]

These equation can be solved iteratively by Laurent polynomials in $e^{y}$, $\Pi_{+}^{(n)}(y)=\sum\limits_{m=-n}^{n}r_{m}^{(n)}(P)e^{my}$, where the sum runs over $m$ odd (even) if $n$ is odd (even). In fact, the $r_{m}^{(n)}(P)$ can be written in terms of $r_{m^{\prime}}^{(n^{\prime})}(P)$ with $n^{\prime}<n$, $m^{\prime}<m$, so that at the first orders

[TABLE]

Furthermore, the convergence is so good that the series can be integrated term by term in the expression (2.8) (exchange of two limits, integral with series)

[TABLE]

The study of the limit $\theta\rightarrow-\infty$ for $\varphi$ is more challenging, since its definition contains an integration in $y$ on a non compact domain, actually the entire real axis. Therefore, even if $\theta$ is very large and negative, for sufficiently large $y$ the inequality $|2e^{2\theta}\cosh y|<|P^{2}|$ is not guaranteed and the previous expansion of $\Pi_{+}(y)$ cannot be used. To approach the computation of $\varphi$ in the limit $\theta\rightarrow-\infty$ it is useful to get inspiration from the ’kink method’ for solving TBA equations. One starts from the decomposition (2.10) $\varphi=\varphi_{>}+\varphi_{<}$ and rewrites $\varphi_{>}$ in terms of ’right kink’ solution $\Pi_{>}(y)=\Pi_{+}\left(y-2\theta\right)$ as

[TABLE]

The shift in $y$ marks the difference with the previous approach. Now, $\theta$ is present in the integration extreme and, implicitly, in $\Pi_{>}(y)$, which is subjected to a slightly different Riccati equation

[TABLE]

Now, in the region $y>2\theta$ (2.25) can be solved by the convergent series

[TABLE]

since $|e^{4\theta}e^{-y}|\rightarrow 0$ (and even smaller than $|e^{y}|$ and $P^{2}$) and this series can be integrated term by term in (2.24), eventually giving $\varphi_{>}$ as a converging series. In fact, having in mind applications to gauge theories and gravity, one would like to use $\theta$ and $k$ as independent variables. With the aim of rewriting and solving the Riccati equation (2.25) with $k$ as parameter one inverts relation (2.23) to find $P^{2}$ in terms of $k$,

[TABLE]

and insert this into (2.25). Expanding the solution depending on $k$ as a converging series

[TABLE]

the function $\tilde{\Pi}_{>}^{(n)}$ – which acquired a tilde to distinguish its dependence on $k$ – satisfies a first order ODE involving $\tilde{\Pi}_{>}^{(m)}$, with $m<n$:

[TABLE]

Of course, this system can be solved at all orders recursively. In fact $\tilde{\Pi}_{>}^{(0)}(y;k)=\frac{d}{dy}\ln J_{2k}(2ie^{\frac{y}{2}})$, with $J_{2k}$ Bessel function; $\tilde{\Pi}_{>}^{(0)}$ is $2\pi i$ periodic and behaves as $e^{\frac{y}{2}}-\frac{1}{4}+O(e^{-\frac{y}{2}})$ when $y\rightarrow+\infty$ and as $k+\frac{e^{y}}{2k+1}-\frac{e^{2y}}{(2k+2)(2k+1)^{2}}+O(e^{3y})$ when $y\rightarrow-\infty$. The others $2\pi i$ periodic $\tilde{\Pi}_{>}^{(n)}(y;k)$, $n\geq 1$ are found by separation of variables up to an integration whose constant is fixed by integrability at $+\infty$: for instance, for $n=1,2$,

[TABLE]

This peculiar form allows us to argue the general solution (for $n\geq 1$)

[TABLE]

with $P_{m}^{(n)}(e^{-y})$ degree $n$ polynomials which solve simple differential equations derived upon inserting (2.32) into (2.29). Eventually, in this solution (2.28), (2.32) the integral (2.24) can be moved inside the series and yields a convergent series: explicitly,

[TABLE]

where $\tilde{S}_{>}(\theta,k)=\sum\limits_{n=1}^{+\infty}e^{4n\theta}\int_{2\theta}^{+\infty}dy^{\prime}\tilde{\Pi}_{>}^{(n)}(y^{\prime};k)$. Now, from (2.32) one deduces the expansion

[TABLE]

with $a_{m}^{(n)}$ constants444The constants $a_{-n}^{(n)}$ are determined by $a_{0}^{(0)}=k,a_{-1}^{(1)}=\frac{1}{2k-1}$ and by the relation $na_{-n}^{(n)}=\sum\limits_{m=0}^{n}a_{m-n}^{(n-m)}a_{-m}^{(m)}$, valid for $n\geq 2$.. This series can be integrated term by term for $y$ around $-\infty$, but not for $y\rightarrow+\infty$: its importance, however, is that it allows us to compute exactly all the powers of $e^{2\theta}$ coming from the evaluation of the primitive function of $\tilde{\Pi}_{>}^{(n)}(y)$ at $y=2\theta\rightarrow-\infty$. In particular, for $n\geq 1$,

[TABLE]

where $C_{n}$ is the value of the primitive function of $\tilde{\Pi}_{>}^{(n)}(y)$ at $+\infty$. Result (2.35) sheds light on the form of $\tilde{S}_{>}(\theta,k)$, which, as a consequence of it, is expressed through positive (nonzero) powers of $e^{2\theta}$.

The reasoning goes in a parallel way for what the concerns the ’left kink’ solution $\Pi_{<}(y)=\Pi_{+}\left(y+2\theta\right)$ which works to find

[TABLE]

since $\Pi_{<}(y;\theta,P)$ satisfies this novel Riccati equation

[TABLE]

Inserting (2.27) into (2.37) one has $\theta,k$ as independent parameters and the $2\pi i$-periodic functions entering the series, analogous to (2.28),

[TABLE]

converging in the region $y<-2\theta$, satisfy the system of ODEs

[TABLE]

One finds that $\tilde{\Pi}_{<}^{(0)}(y;k)=\frac{d}{dy}\ln J_{-2k}(2ie^{-\frac{y}{2}})$, whilst the others $\tilde{\Pi}_{<}^{(n)}$ are fixed by integrability at $-\infty$. Since the connection $\tilde{\Pi}_{<}^{(n)}(y;k)=-\tilde{\Pi}_{>}^{(n)}(-y;-k)\Rightarrow\Pi_{<}(y;\theta,k)=-\Pi_{>}(-y;\theta,-k)$ holds, then $\varphi_{<}(\theta,k)=\int_{-\infty}^{-2\theta}dy^{\prime}\left(\Pi_{<}(y^{\prime};\theta,k)+e^{-\frac{y^{\prime}}{2}}-\frac{1}{4}\right)=-\varphi_{>}(\theta,-k)$. Therefore, the use of (2.33) gives

[TABLE]

with $\tilde{S}(\theta,k)=\tilde{S}_{>}(\theta,k)-\tilde{S}_{>}(\theta,-k)$ series of positive powers of $e^{2\theta}$. Expanding the Bessel functions at small argument, one finds

[TABLE]

with the function $S(\theta,k)$ a series of positive powers of $e^{4\theta}$. Indeed, the functional relation (2.13) imposes that $S(\theta,k)$ is invariant under shifts $\theta\rightarrow\theta+i\pi/2$ and this prevents the appearance of terms proportional to $e^{(2n+1)\theta}$, which are present in $\tilde{S}(\theta,k)$. The expansion of $\varphi$ at $\theta\rightarrow-\infty$ in the variables $\theta$ and $k$ is more convenient for physical applications since these are exactly the gauge variables: the instanton coupling $\Lambda=\hbar\sqrt{2}e^{\theta}$ and the gauge period [24] $a=\hbar k$ of ${\cal N}=2$ SYM with $N_{f}=0$ in the NS background. Moreover, it is not difficult to go on in the computation of $\varphi(\theta,k)$ with respect to the first two terms in the r.h.s. of (2.41). From a simple analysis of (2.35) it follows that the $N$-instanton term, i.e. the term in $S(\theta,k)$ proportional to $e^{4\theta N}$, receives contributions from $2N+1$ integrals: from $\int_{2\theta}^{+\infty}dy[\tilde{\Pi}_{>}^{(0)}(y)-e^{\frac{y}{2}}+\frac{1}{4}]$ and from integrals (2.35), with $1\leq n\leq 2N$, minus the same terms with $k\rightarrow-k$. In particular, the first order of the function $S(\theta,k)$, i.e. the ’one-instanton’ contribution to $\varphi(\theta,k)$, is the sum of three terms. The first one comes from $\int_{2\theta}^{+\infty}dy^{\prime}\left[\Pi_{>}^{(0)}(y)-e^{\frac{y}{2}}+\frac{1}{4}\right]-(k\rightarrow-k)$:

[TABLE]

The second one is

[TABLE]

The last one is

[TABLE]

Summing (2.42 -2.44) one finds

[TABLE]

We observe that (2.45) agrees with the instanton expansion of $A_{D}/\hbar$, the dual gauge period of ${\cal N}=2$ SYM with $N_{f}=0$ in the NS background [25, 26].

3 The Doubly Confluent Heun Equation: from ODE/IM to Floquet

The Doubly Confluent Heun Equation (DCHE)

[TABLE]

quantises the SW differential for ${\cal N}=2$ SYM with $N_{f}=2$ flavours in NS background and, by the map given in formula (6.3) of [27] (see also [12]), to which we refer for further details, describes scalar perturbations of the gravitational background given by the intersection of four stacks of D3-branes in type IIB supergravity. Inspired by the case $N_{f}=0$, the starting point is the Riccati equation for $\Pi(y)=\frac{d}{dy}\ln\psi(y)$:

[TABLE]

Floquet solutions are built from a solution $\Pi_{+}$ of (3.2), which is periodic, $\Pi_{+}(y+2\pi i)=\Pi_{+}(y)$, and expands as

[TABLE]

These results imply for $y>0$ the invariance

[TABLE]

whilst, when $y<0$ other symmetry

[TABLE]

In terms of $\Pi_{+}$ the solution of (3.1), diverging both at $\pm\infty$, is constructed as

[TABLE]

where $c(\theta,q_{1},q_{2})$ is a constant. Due to the $2\pi i$-periodicity of $\Pi_{+}(y^{\prime})$, $\psi_{+}$ is quasi-periodic, $\psi_{+}(y+2\pi i)=e^{2\pi ik}\psi_{+}(y)$. Then, it is a Floquet solution, with index $k$ (independent of $y$) defined from the relation

[TABLE]

where integration is on a straight line parallel to the imaginary axis. When $y\rightarrow-\infty$, $\psi_{+}(y;q_{1},q_{2})\simeq e^{c(\theta,q_{1},q_{2})}e^{e^{\theta-y}-\left(q_{2}-\frac{1}{2}\right)y}$. The behaviour of $\psi_{+}$ when $y\rightarrow+\infty$ is obtained after writing $\psi_{+}$ for $y>0$ as

[TABLE]

Then, when $y\rightarrow+\infty$, $\psi_{+}(y;q_{1},q_{2})\simeq e^{c(\theta,q_{1},q_{2})+\varphi(\theta,q_{1},q_{2})}e^{e^{\theta+y}+\left(q_{1}-\frac{1}{2}\right)y}$, where

[TABLE]

can be interpreted as the acquired phase in going from $-\infty$ to $+\infty$. We have defined the important quantities $\varphi,k$. Following the path devised in the case $N_{f}=0$, symmetries (3.5, 3.6) give relations involving them. Relation (3.5) brings

[TABLE]

On the other hand, the use of (3.6) implies

[TABLE]

A functional relation involving both $\varphi$ and $k$ comes then from applying twice (3.11) and (3.12) and after summing the resulting relations. The result is

[TABLE]

which is an exact functional relation we will use in the following.

Now, using the symmetry of the ODE (3.1), $y\rightarrow-y\,,\quad q_{1}\rightarrow q_{2}\,,\quad q_{2}\rightarrow q_{1}\,,\quad P\rightarrow P$, the second Floquet solution, with Floquet exponent $-k(q_{2},q_{1})$, can be defined as $\psi_{-}(y;q_{1},q_{2})=\psi_{+}(-y;q_{2},q_{1})$. Within the choice $c(\theta,q_{1},q_{2})=-\frac{1}{2}\varphi(\theta,q_{1},q_{2})$, when $y\rightarrow-\infty$, $\psi_{+}(y;q_{1},q_{2})\simeq\exp\left[-\frac{1}{2}\varphi(\theta,q_{1},q_{2})+e^{\theta-y}-\left(q_{2}-\frac{1}{2}\right)y\right]$ and $\psi_{-}(y;q_{1},q_{2})\simeq\exp\left[\frac{1}{2}\varphi(\theta,q_{2},q_{1})+e^{\theta-y}-\left(q_{2}-\frac{1}{2}\right)y\right]$ and, as $y\rightarrow+\infty$, $\psi_{+}(y;q_{1},q_{2})\simeq\exp\left[\frac{1}{2}\varphi(\theta,q_{1},q_{2})+e^{\theta+y}+\left(q_{1}-\frac{1}{2}\right)y\right]$ and $\psi_{-}(y;q_{1},q_{2})\simeq\exp\left[e^{\theta+y}+\left(q_{1}-\frac{1}{2}\right)y-\frac{1}{2}\varphi(\theta,q_{2},q_{1})\right]$. The change of basis, from $\psi_{\pm}$ to $\psi_{\pm,0}$ (the ODE/IM basis) with $\psi_{\pm,0}\simeq\exp\left[(-\frac{1}{2}+\tilde{q}_{\pm})(\theta\pm y+\ln 2)-e^{\theta\pm y}\right]$ as $y\rightarrow\pm\infty$ and $\tilde{q}_{+}=q_{1},\tilde{q}_{-}=q_{2}$, $|\tilde{q}_{\pm}|<1/2,$ is [20]

[TABLE]

where, as in the MME case, the above fine-tuned linear combinations of the Floquet solutions are engineered to cancel their leading diverging behaviour, at $+\infty$ for the linear combination giving $\psi_{+,0}$ and at $-\infty$ for the one giving $\psi_{-,0}$. This cancellation extends at all diverging orders, thus leaving subdominant solutions at $+\infty$ and $-\infty$, respectively. Now, differently for the MME case, the Wronskian $W[\psi_{+},\psi_{-}]$ does not have a simple expression in terms of $\theta,k$. However, it is conveniently computed in $y=0$ after using $\psi_{-}(y;q_{1},q_{2})=\psi_{+}(-y;q_{2},q_{1})$. The result is $W[\psi_{+},\psi_{-}]=-\psi_{+}(0;q_{1},q_{2})\psi_{+}^{\prime}(0;q_{2},q_{1})-\psi_{+}(0;q_{2},q_{1})\psi_{+}^{\prime}(0;q_{1},q_{2})$, which by virtue of (3.7) becomes

[TABLE]

The various terms in (3.14) are computable in series of powers of $e^{2\theta}$ by using the ’kink method’. It is particularly important the ’relative’ phase between $\psi_{\pm}$, $\Phi(\theta,q_{1}.q_{2})=\frac{\varphi(\theta,q_{1},q_{2})+\varphi(\theta,q_{2},q_{1})}{2}$, which was claimed [20] to coincide with $A_{D}/\hbar$, with $A_{D}$ the dual gauge period of ${\cal N}=2$ SYM with $N_{f}=2$. Assuming this identification, one can compute $A_{D}$ also by using AGT correspondence [28]. However, this is not enough to capture the relation between in/up and Floquet functions. As (3.15) shows, also $\varphi_{<},\varphi_{>}$ and $\Pi_{+}(0)$ are needed.

3.1 The $\theta\rightarrow-\infty$ converging series.

The easiest quantity to study is $\Pi_{+}(y)$ for finite $y$, since, as for the MME, the quantum momentum [20] of the Floquet solution develops as $\Pi_{+}(y)=\sum_{n=0}^{+\infty}e^{n\theta}\Pi_{+}^{(n)}(y)$ and this expansion solves (3.2) iteratively starting from $\Pi_{+}^{(0)}=P$: indeed, $\Pi_{+}^{(n)}(y)$, $n\geq 1$, solve the ODEs

[TABLE]

which again are solved iteratively by Laurent polynomials in $e^{y}$, $\Pi_{+}^{(n)}(y)=\sum\limits_{m=-n}^{n}r_{m}^{(n)}(P)e^{my}$, where the sum runs over $m$ odd (even) if $n$ is odd (even). The first orders are

[TABLE]

The Floquet index comes from (3.8)

[TABLE]

This relation in particular implies that the Floquet exponent of $\psi_{-}(y;q_{1},q_{2})=\psi_{+}(-y;q_{2},q_{1})$ is $-k(q_{1},q_{2})$, i.e. the opposite of the Floquet exponent of $\psi_{+}(y;q_{1},q_{2})$.

Less immediate is the study of $\varphi$, which involves the knowledge of $\Pi_{+}(y)$ for large $y$. As in the case of the MME, to solve the problem of the conflicting $\theta\rightarrow-\infty,y\rightarrow\pm\infty$ limits, one first considers $\varphi_{>}$ and write it in terms of $\Pi_{>}(y)=\Pi_{+}(y-\theta)$ as $\varphi_{>}=\int_{\theta}^{+\infty}dy\left[\Pi_{>}(y)-e^{y}-q_{1}+\frac{1}{2}\right]$. The function $\Pi_{>}(y)$ satisfies the Riccati equation

[TABLE]

which can be solved perturbatively in the domain $y>\theta$. Before doing that, in order to have $k,\theta$ as independent variables, one inverts (3.17) to find $P^{2}$ in terms of $k,q_{1},q_{2}$, $P^{2}=\sum_{n=0}^{+\infty}e^{2n\theta}p^{(2)}_{n}(k,q_{1},q_{2})$ and plugs this expansion into (3.18). The $2\pi i$-periodic functions $\tilde{\Pi}_{>}^{(n)}(y)$ entering the converging series

[TABLE]

satisfy the first order ODEs

[TABLE]

which are solved iteratively. At leading order the solution is (${}_{1}F_{1}$ is the confluent hypergeometric function)

[TABLE]

The higher $\tilde{\Pi}_{>}^{(n)}(y)$, $n\geq 1$ are found by separation of variables, fixing the integration constant by integrability at $+\infty$. The first of these function is

[TABLE]

and, in general,

[TABLE]

with $P_{m}^{(n)}(e^{-y})$ degree $n$ polynomials555To avoid heavy notations, we use the same symbol as for the polynomials appearing (2.32) in solving the MME. The same notations will be used in (4.25). . Eventually, $\varphi_{>}=\int_{\theta}^{+\infty}dy\left[\Pi_{>}(y)-e^{y}-q_{1}+\frac{1}{2}\right]$ equals

[TABLE]

with $\tilde{S}_{>}^{(2)}(\theta,k,q_{1},q_{2})=\sum\limits_{n=1}^{+\infty}e^{2n\theta}\int_{\theta}^{+\infty}dy\tilde{\Pi}_{>}^{(n)}(y)$. Now, (3.23) implies the series expansion

[TABLE]

with $a_{m}^{(n)}$ constants. This series can be integrated term by term around $-\infty$, but not for $y\rightarrow+\infty$. As in the MME case, it gives an estimate of the powers of $e^{\theta}$ coming from the evaluation of the primitive function of $\tilde{\Pi}_{>}^{(n)}(y)$ at $y=\theta\rightarrow-\infty$. In particular, for $n\geq 1$,

[TABLE]

where $C_{n}$ is the value of the primitive function at $+\infty$. Result (3.26) sheds light on the form of $\tilde{S}_{>}^{(2)}(\theta,k)$, which, as a consequence of it, is expressed through positive (nonzero) powers of $e^{\theta}$.

The study of $\varphi_{<}$ proceeds on an analogous path. It is written as $\varphi_{<}(\theta,k,q_{1},q_{2})=\int_{-\infty}^{-\theta}dy\left[\Pi_{<}(y)+e^{-y}+q_{2}-\frac{1}{2}\right]$, where $\Pi_{<}(y)=\Pi_{+}(y+\theta)$ satisfies the Riccati equation

[TABLE]

with $P^{2}$ the function of $k$ coming from the inversion of (3.17). The $2\pi i$-periodic solution of (3.27) is written for $y<-\theta$ as converging series $\Pi_{<}(y)=\sum_{n=0}^{+\infty}\tilde{\Pi}_{<}^{(n)}(y)e^{2n\theta}$, where $\tilde{\Pi}_{<}^{(n)}(y)$ satisfy the ODEs

[TABLE]

At leading order we have

[TABLE]

and the others $\tilde{\Pi}_{<}^{(n)}(y)$, $n\geq 1$, integrable at $-\infty$, come from the iterative solutions of the first orders ODEs (3.28). From the form of (3.29) and of (3.28), it follows that $\tilde{\Pi}_{<}^{(n)}(y,k,q_{1},q_{2})=-\tilde{\Pi}_{>}^{(n)}(-y,-k,q_{2},q_{1})$ and then that $\varphi_{<}(\theta,k,q_{1},q_{2})=-\varphi_{>}(\theta,-k,q_{2},q_{1})$. Then, relation (3.24) implies

[TABLE]

where $S^{(2)}(\theta,k,q_{1},q_{2})=\tilde{S}_{>}^{(2)}(\theta,k,q_{1},q_{2})-\tilde{S}_{>}^{(2)}(\theta,-k,q_{2},q_{1})+\ln\frac{{}_{1}F_{1}\left(\frac{1}{2}-k+q_{2},1-2k,2e^{\theta}\right)}{{}_{1}F_{1}\left(\frac{1}{2}+k+q_{1},1+2k,2e^{\theta}\right)}$. The function $S^{(2)}$, which contains positive powers of $e^{\theta}$, is however bounded from functional relation (3.13) to satisfy the relation $S^{(2)}(\theta+i\pi,k,q_{1},q_{2})=S^{(2)}(\theta,k,q_{1},q_{2})$. This property implies that $S^{(2)}$ must expand only with positive powers of $e^{2\theta}$.

From the analysis of (3.26), it follows again that the $N$-instanton term, i.e. the term in $S^{(2)}$ proportional to $e^{2\theta N}$ comes from $2N+1$ integrals: to be precise, from $\int_{\theta}^{+\infty}dy\left[\tilde{\Pi}_{>}^{(0)}(y)-e^{y}-q_{1}+\frac{1}{2}\right]$ and from integrals (3.26), with $1\leq n\leq 2N$, minus the same terms with $k\rightarrow-k$ and $q_{1}\leftrightarrow q_{2}$. In particular, to derive $S^{(2)}$ up to order $e^{2\theta}$, only $n=0,1,2$ are needed. The terms $n=0$ contributes through the expansion of $\ln\frac{{}_{1}F_{1}\left(\frac{1}{2}-k+q_{2},1-2k,2e^{\theta}\right)}{{}_{1}F_{1}\left(\frac{1}{2}+k+q_{1},1+2k,2e^{\theta}\right)}$ up to order $e^{2\theta}$, while $n=1$ through the integration of the exact solution of $\tilde{\Pi}_{>}^{(1)}$. Finally, the asymptotics $\tilde{\Pi}_{>}^{(2)}\simeq\frac{(2k-1)^{2}-4q_{2}^{2}}{(2k-2)(2k-1)^{2}}e^{-2y}$ when $y\rightarrow-\infty$ gives what comes from $n=2$. Summing everything, one has

[TABLE]

Therefore, the symmetrised phase $\Phi$ reads

[TABLE]

With the map $4e^{\theta}\hbar=\Lambda_{2}$ (instanton coupling), $\hbar q_{j}=m_{j}$, $\hbar k=a$, $\hbar\Phi=A_{D}$ we reproduce the small instanton expansion of the dual gauge period $A_{D}$ for the ${\cal N}=2$ SYM with $N_{f}=2$, with masses $m_{j}$ and gauge period $a$ in the NS background.

4 The HE: from decaying to Floquet solutions

The HE reads

[TABLE]

where the potential

[TABLE]

has four regular singular points at $y=\pm\infty,\pm(\frac{\theta}{2}-\ln 2)$. This equation is the quantisation of SW differential for ${\cal N}=2$ SYM with $N_{f}=4$ in the NS background. On the gravity side, it is worth mentioning that the (angular and radial) Teukolsky equations for Kerr Newman-de Sitter geometries can be transformed [10] into the Heun’s equations. Defining $q_{+}=(q_{1},q_{2})$ and $q_{-}=(q_{3},q_{4})$, the equation (4.1) has the symmetry $y\mapsto-y$ and $q_{+}\leftrightarrow q_{-}$. Choosing $q_{1}>q_{2},q_{1}+q_{2}>1,q_{3}>q_{4},q_{3}+q_{4}>1$, the ODE/IM solutions are uniquely defined by their asymptotics around the singular points

[TABLE]

where $y_{0}=|\frac{\theta}{2}-\ln 2|$. Floquet solutions are different from the ODE/IM ones. Their property is that they are modified only by a phase factor when the variable $e^{y}$ rotates around two regular singularity. For instance the Floquet solution $\psi_{+}$ satisfies

[TABLE]

where $k$ is the Floquet index in the region $[-y_{0},y_{0}]$. It is constructed from the solution $\Pi_{+}$ of the Riccati equation $\Pi^{2}(y)+\frac{d}{dy}\Pi(y)=V(y)$ which satisfies $\Pi_{+}(y+2\pi i)=\Pi_{+}(y)$ and has the asymptotics

[TABLE]

As usual,

[TABLE]

and ($e^{c}$ is a normalisation constant)

[TABLE]

As $\psi_{+}$ is different from the functions of the ODE/IM basis, it diverges at $\pm y_{0}$. In particular

[TABLE]

where $\varphi$, which has the meaning of acquired phase when going from $-y_{0}$ to $+y_{0}$, reads

[TABLE]

The first line identifies $\varphi_{<}$, the second $\varphi_{>}$. From the symmetry $\Pi_{+}(\theta+2\pi i,y-i\pi)=\Pi_{+}(\theta,y)$, valid for $y>0$, we find the property

[TABLE]

From the symmetry $\Pi_{+}(\theta+2\pi i,y+i\pi)=\Pi_{+}(\theta,y)$, valid for $y<0$, we find the property

[TABLE]

Summing

[TABLE]

The other Floquet solution is defined by $\psi_{-}(y,q_{+},q_{-})=\psi_{+}(-y,q_{-},q_{+})$ and then satisfies $\psi_{-}(y+2\pi i,q_{+},q_{-})=e^{-2\pi ik(q_{-},q_{+})}\psi_{-}(y,q_{+},q_{-})$ for $|\text{Re}y|<y_{0}$. Choosing $c=-\frac{1}{2}\varphi$, the connection between the Floquet and the ODE/IM basis, done in the same spirit of cancelling diverging orders as (2.18) for the MME and (3.14) for the DCHE, is

[TABLE]

where, similarly to the DCHE case, the Wronskian between Floquet solutions has the the form

[TABLE]

As before, in (4.14) the relative factor between $\psi_{\pm}$ depends on the symmetrised phase $\Phi(q_{+},q_{-})=\frac{1}{2}[\varphi(q_{+},q_{-})+\varphi(q_{-},q_{+})]$, whilst the change of basis needs also $\varphi_{>},\varphi_{<}$ and $\Pi_{+}(0)$. However, everything is computable in converging series when $\theta\rightarrow-\infty$, as results of next subsection will show.

4.1 Converging expansion around $\theta\rightarrow-\infty$

One has $y_{0}=\ln 2-\frac{\theta}{2}$ and for finite $y$, $|y|<y_{0}$, $\Pi_{+}(y)\simeq P+[(\frac{2P-1}{8}+\frac{q_{1}q_{2}}{4P+2})e^{y}+(\frac{2P-1}{8}+\frac{q_{3}q_{4}}{4P+2})e^{-y}]e^{\theta/2}+\left[Ce^{2y}+De^{-2y}+\frac{1}{2P}\left(\frac{4P^{2}+1}{32}-\frac{q_{1}q_{2}q_{3}q_{4}}{2(4P^{2}-1)}\right)\right]e^{\theta}+O(e^{3\theta/2})$. The form of $C,D$ is not relevant for our aim, which is the use of (4.7) to find the Floquet index in the region $[-y_{0},y_{0}]$, $k(q_{+},q_{-})$, as a function of $P$:

[TABLE]

This in particular implies that the Floquet exponent of $\psi_{-}(y,q_{+},q_{-})=\psi_{+}(-y,q_{-},q_{+})$ is $-k(q_{+},q_{-})$, i.e. the opposite of the Floquet exponent of $\psi_{+}$. The inversion of (4.16) gives $P^{2}=k^{2}+\left(\frac{q_{1}q_{2}q_{3}q_{4}}{2(4k^{2}-1)}-\frac{4k^{2}+1}{32}\right)e^{\theta}+O(e^{2\theta})$ and, using this expression, $V(y)$ is made dependent parametrically on $k$. The quantity $\varphi_{>}(q_{+},q_{-})$ is written in terms of the right kink quantum momentum

[TABLE]

as

[TABLE]

and $\Pi_{>}$ satisfies the modified Riccati equation

[TABLE]

We are interested (see (4.18)) to the region $\theta/2<y<\ln 2$, where we can expand in converging series $V_{>}(y)=\sum\limits_{n=0}^{\infty}V_{>}^{(n)}(y)e^{n\theta}$ and

[TABLE]

The function $\tilde{\Pi}_{>}^{(0)}(y)$ is a solution of

[TABLE]

and then is expressed in terms of an hypergeometric function:

[TABLE]

The first correction $\tilde{\Pi}_{>}^{(1)}(y)$ satisfies

[TABLE]

and has the form

[TABLE]

For $n\geq 2$ the equation is $2\tilde{\Pi}_{>}^{(0)}(y)\tilde{\Pi}_{>}^{(n)}(y)+\frac{d}{dy}\tilde{\Pi}_{>}^{(n)}(y)+\sum\limits_{m=1}^{n-1}\tilde{\Pi}_{>}^{(m)}(y)\tilde{\Pi}_{>}^{(n-m)}(y)=V^{(n)}_{>}(y)$ and the claim is that its solution reads

[TABLE]

with $P_{m}^{(n)}(e^{-y})$ degree $n$ polynomials satisfying a simple system of ODEs. Then, the quantity $\varphi_{>}$ (4.18) can be written as

[TABLE]

with $\tilde{S}^{(4)}_{>}(\theta,k)=\sum_{n=1}^{+\infty}e^{n\theta}\int_{\frac{\theta}{2}}^{\ln 2}dy\tilde{\Pi}_{>}^{(n)}(y)$. In general, for $n\geq 1$ the expansion $\tilde{\Pi}_{>}^{(n)}(y)=\sum\limits_{m=-n}^{+\infty}a_{m}^{(n)}e^{my}$, with $a_{m}^{(n)}$ constants, holds. This series can be integrated term by term around $y=-\infty$, but not for $y\rightarrow+\infty$ and then gives the value of the primitive function at $\frac{\theta}{2}\rightarrow-\infty$:

[TABLE]

where $C_{n}$ is the value of the primitive function at $+\infty$. This result implies that $\tilde{S}^{(4)}_{>}(\theta,k)$ expands in positive powers of $e^{\frac{\theta}{2}}$. On the other hand, $\varphi_{<}(q_{+},q_{-})$ is written in terms of the left kink quantum momentum

[TABLE]

as

[TABLE]

The function $\Pi_{<}(y)$ satisfies the other modified Riccati equation

[TABLE]

Similarly to the other cases, it is solved by the simple symmetry

[TABLE]

which implies that $\varphi_{<}(k,q_{+},q_{-})=-\varphi_{>}(-k,q_{-},q_{+})$. The sum $\varphi_{>}+\varphi_{<}$ gives the acquired phase

[TABLE]

where, using that $\tilde{S}^{(4)}_{<}(\theta,k,q_{+},q_{-})=-\tilde{S}^{(4)}_{>}(\theta,-k,q_{-},q_{+})$,

[TABLE]

Now, the functional relation (4.13) implies that $S^{(4)}(\theta,k)+\frac{q_{3}+q_{4}-q_{1}-q_{2}}{2}\ln\left(1-\frac{1}{2}e^{\frac{\theta}{2}}\right)$ is $2\pi i$-periodic and then must contain only integer powers of $e^{\theta}$. Then, the symmetrised phase $\Phi(q_{+},q_{-})=\frac{1}{2}[\varphi(q_{+},q_{-})+\varphi(q_{-},q_{+})]$ reads

[TABLE]

with $\frac{1}{2}[S^{(4)}(\theta,k,q_{+},q_{-})+S^{(4)}(\theta,k,q_{-},q_{+})]$ containing only integer positive powers of $e^{\theta}$. For the quantity $\Phi$, the term $e^{N\theta}$ ($N\geq 1$), the so-called $N$-instanton contribution, comes from the last term in (4.33) (symmetrised upon the exchange $q_{+}\leftrightarrow q_{-}$) and from a finite number ($2N$) of terms, coming from $\sum\limits_{n=1}^{2N}e^{n\theta}\int_{\frac{\theta}{2}}^{\ln 2}dy\tilde{\Pi}_{>}^{(n)}(y)$, contained in $\tilde{S}^{(4)}_{>}(\theta,k)$. In particular, the computation of the symmetrised phase $\Phi(q_{+},q_{-})=\frac{1}{2}[\varphi(q_{+},q_{-})+\varphi(q_{-},q_{+})]$ up to one instanton needs the knowledge of $\tilde{\Pi}_{>}^{(1)}$ and of the limit for $y\rightarrow-\infty$ of $\tilde{\Pi}_{>}^{(2)}(y):a_{>}e^{-2y}$, with

[TABLE]

The result is

[TABLE]

With the mapping $\frac{e^{\theta}\hbar}{4}=\Lambda_{4}$ (instanton coupling), $\hbar q_{j}=m_{j}$, $\hbar k=a$, expression (4.36) reproduces the small instanton expansion of the dual gauge period $A_{D}/\hbar$ for ${\cal N}=2$ SYM with $N_{f}=4$, with masses $m_{j}$ and gauge period $a$, in the NS background. Then, our claim (to be rigorously proved in an incoming publication) is $\Phi=A_{D}/\hbar$.

4.2 The limit to the Confluent Heun Equation

The ’confluence’ limit from the HE (4.1, 4.2) to the CHE is made through the following steps. First, one defines the new independent variable $y^{\prime}=y+\frac{\theta}{2}$, in which the two finite (for $\theta$ finite) singularity points occur at $y^{\prime}=\theta-\ln 2$ and $y^{\prime}=\ln 2$, respectively. The second step is to perform on the (Heun) equation in the variable $y^{\prime}$ the scaling limit $\theta\rightarrow-\infty,|q_{4}|\rightarrow+\infty$ with fixed $q_{4}e^{\theta}=4e^{\theta_{3}}$, which will be called for simplicity $\lim\limits_{H->CHE}$. This limit sends the regular singularity at $y^{\prime}=\theta-\ln 2$ onto the other (regular) singularity at $y^{\prime}=-\infty$, thus creating an irregular singularity at $y^{\prime}=-\infty$, whilst the other two regular singularities, at $y^{\prime}=\ln 2,+\infty$, stay untouched. Following this prescription, $\psi^{(3)}(y^{\prime})=\lim\limits_{H->CHE}\psi\left(y=y^{\prime}-\frac{\theta}{2}\right)$ satisfies the equation

[TABLE]

where

[TABLE]

which is a CHE. Interestingly, the CHE has relevance in gravity: using the maps

[TABLE]

it maps into the Teukolsky equation for Schwarzschild background (in its homogeneous version, with spin $s=-2$) [7]:

[TABLE]

The confluence limit on the Floquet index in the region $[-y_{0},y_{0}]$ of the HE, $k$ (4.16), gives the corresponding interval $[-\infty,\ln 2]$ Floquet index of the CHE, $k_{3}=\lim\limits_{H->CHE}k=P-\frac{q_{1}q_{2}q_{3}}{P(4P^{2}-1)}e^{\theta_{3}}+O(e^{2\theta_{3}})$. Analogously, the confluence limit on (4.36) gives the acquired phase on the interval $[-\infty,\ln 2]$

[TABLE]

With the identification $e^{\theta_{3}}\hbar=\Lambda_{3}$ (instanton coupling), $\hbar q_{j}=m_{j}$, $\hbar k_{3}=a$, the quantity $\Phi_{3}$ (4.41) reproduces the small instanton expansion of the dual gauge period $A_{D}/\hbar$ for ${\cal N}=2$ SYM with $N_{f}=3$, with masses $m_{j}$ and gauge period $a$, in the NS background.

Coming to wave functions, Floquet solutions of the CHE can be derived from Floquet solutions of the HE, constructed in last subsection: thus, the main idea is again to expand the quantum momentum or eikonal rather the wave function. Before going on, it is convenient to interpret in a more general way the two regions of $y$, as argument of the Floquet $\psi_{+}(y)$, where the kink method is applied and where the wave function is expressed as converging series. In fact, the limitation on their separation ${\cal{A}}$, which before was chosen zero, in general is to be between the two singularities at $y=\pm y_{0}$, i.e. $|{\cal{A}}|<|\ln 2-\theta/2|$. The right region is ${\cal{A}}<y<\ln 2-\theta/2$; then, the argument of $\Pi_{>}(y)=d/dy\ln\psi_{>}(y)$ in equation (4.19), with $\psi_{>}(y)=\psi_{+}(y-\theta/2)$, has to range in the interval ${\cal{A}}+\theta/2<y<\ln 2$. The left region of the argument of $\psi_{+}(y)$ is $\theta/2-\ln 2<y<{\cal{A}}$; then, the argument of $\Pi_{<}(y)=d/dy\ln\psi_{<}(y)$ in equation (4.30), with $\psi_{<}(y)=\psi_{+}(y+\theta/2)$, has to range in the interval $-\ln 2<y<{\cal{A}}-\theta/2$.

Now, a possibility for studying the CHE limit is ${\cal{A}}=a-\frac{\theta}{2}$, with $a<\ln 2$, which gives the validity intervals $-\ln 2<y<a-\theta$ of $\psi_{<}(y)$ and $a<y<\ln 2$ of $\psi_{>}(y)$. In the right region of $\psi_{+}(y)$, ${\cal{A}}=a-\frac{\theta}{2}<y<\ln 2-\theta/2$, which means $a<y^{\prime}<\ln 2$, these equalities hold $\psi_{+}^{(3)}(y^{\prime})=\lim\limits_{H->CHE}\psi_{+}(y^{\prime}-\frac{\theta}{2})=\lim\limits_{H->CHE}\psi_{>}(y^{\prime})$ and imply that the scaling limit to $N_{f}=3$ can be performed directly on relations (4.20, 4.22, 4.24, 4.25): this corresponds to expanding the potential (4.38) in (4.37) at large negative $\theta_{3}\rightarrow-\infty$ (without any kink procedure) and solving at any order in positive integer powers $e^{n\theta_{3}}$. Explicitly, the quantum momentum $\Pi_{+}^{(3)}(y^{\prime})=\frac{d}{dy^{\prime}}\ln\psi_{+}^{(3)}(y^{\prime})$ enjoys the expansion for $\theta_{3}\rightarrow-\infty$ in the right interval $y^{\prime}\in[a,\ln 2)$

[TABLE]

which becomes at first orders

[TABLE]

with $\tilde{\Pi}_{>}^{(0)}(y^{\prime})$ denoting (with little abuse) the CHE limit of the HE $\tilde{\Pi}_{>}^{(0)}(y^{\prime})$ given by (4.22):

[TABLE]

Thanks to (4.25) we can claim that in general

[TABLE]

with $P_{m}^{(n,r)}(e^{-y^{\prime}})$ degree $n$ polynomials satisfying a simple system of ODEs, which are the CHE limit of the polynomials appearing in (4.25).

On the contrary, in the left region of $\psi_{+}(y)$, $\frac{\theta}{2}-\ln 2<y<{\cal{A}}=a-\theta/2$, which means $\theta-\ln 2<y^{\prime}<a$, the CHE Floquet is given by a kink method as $\psi_{+}^{(3)}(y^{\prime})=\lim\limits_{H->CHE}\psi_{+}\left(y^{\prime}-\frac{\theta}{2}\right)=\lim\limits_{H->CHE}\psi_{<}(y^{\prime}-\theta)$. Nevertheless, this still implies that the corresponding quantum momentum enjoys a simple expansion in the left interval $y^{\prime}\in]-\infty,a]$ as $\theta_{3}\rightarrow-\infty$

[TABLE]

Signal of the kink method on (4.37), (4.38) is that in the CHE limit the ${}_{2}F_{1}$ functions (present in $\tilde{\Pi}_{<}^{(n)}(y)$) loose one parameter, as $q_{4}e^{\theta}\rightarrow 4e^{\theta_{3}}$ and become ${}_{1}F_{1}$ functions (this is ostensive in the series representation or in the differential equation). At the two first orders, the series (4.46) contains the functions

[TABLE]

[TABLE]

Thanks to (4.25) and the symmetry (4.31) we can claim that

[TABLE]

with $P_{m}^{(n,l)}(e^{y^{\prime}-\theta_{3}})$ degree $n$ polynomials satisfying a simple system of ODEs.

Besides, the logarithmic derivative of the other Floquet solution, $\Pi_{-}^{(3)}(y^{\prime})=\frac{d}{dy^{\prime}}\ln\psi_{-}^{(3)}(y^{\prime})$ is obtained from $\Pi_{+}^{(3)}(y^{\prime})$ by replacing $k_{3}\rightarrow-k_{3}$; hence the same holds for the expressions at all orders (4.42, 4.45) and (4.46, 4.49)

[TABLE]

In fact, if ${\cal{A}}<y<\ln 2-\theta/2$, we have the chain of equalities

[TABLE]

in which $a<y^{\prime}<\ln 2$. If $\theta/2-\ln 2<y<{\cal{A}}$,

[TABLE]

where $\theta-\ln 2<y^{\prime}<a$.

Moreover, it is interesting to consider the CHE limit of (4.14) in the form

[TABLE]

[TABLE]

Equation (4.51) becomes proportional to $\psi_{0,0}(y^{\prime})$ [29], identified by the condition that, as $y^{\prime}\rightarrow\ln 2$, $\psi_{0,0}(y^{\prime})\simeq\frac{1}{\sqrt{2}}(e^{y^{\prime}}-2)^{\frac{1+q_{1}+q_{2}}{2}}$ and proportional to the wave function used in gravity set-up $\psi_{in}(y^{\prime})$: $\psi_{in}(y^{\prime})=\hat{C}\psi_{0,0}(y^{\prime})$. Eventually, we can write (4.51) in the CHE limit as

[TABLE]

with $\tilde{C}$ given in terms of the proportionality factor $\hat{C}$ and the Wronskian $W[\psi_{-}^{(3)},\psi_{+}^{(3)}]$. In fact, this Wronskian emerges in the exact expression of $\psi_{0,0}$ as the decaying linear combination of the Floquet solutions, in the same way as for (4.51) (and previous cases (3.14, 2.18)). Then, thanks to the expansion of $\Phi_{3}$ (4.41) around $\theta_{3}\rightarrow-\infty$ and those of the two quantum momenta (4.46, LABEL:paiexp-), given at all orders for all $y^{\prime}<\ln 2$ ($r>2M$), we can write both terms in (4.53) as exponentials of a single series of integer powers of $e^{\theta_{3}}$. Now, there are two possible regimes of expansions around $\theta_{3}\rightarrow-\infty$: one is (4.46) $\forall y^{\prime}<\ln 2$ (more effective for $y^{\prime}\rightarrow-\infty$, i.e. large distance $r\rightarrow\infty$ from the horizon), while the other (4.42) for finite $y^{\prime}\in[a,\ln 2)$ (more effective around the horizon $y^{\prime}=\ln 2$, i.e. $r=2M$). In the first, $\psi_{\pm}^{(3)}(y^{\prime})$ are dominated at $e^{\theta_{3}}\ll 1$ by the leading orders (4.47),

[TABLE]

while in the second the two expansions start, at leading order (4.43,4.44), with

[TABLE]

In (4.54, 4.55) $\tilde{A}_{\pm}$ and $\tilde{B}_{\pm}$ are quantities independent of $y^{\prime}$. Actually, to make explicit this calculations, we are now going to compute the proportionality factor $\hat{C}$ in (4.53). To normalise $\psi_{in}(y^{\prime})$ one constructs the gravity function which has the physical meaning of a wave incoming in the horizon

[TABLE]

The, requires that its behaviour at $r\rightarrow+\infty$ is exactly

[TABLE]

with some constant $B$ which is fixed. In the style of the ODE/IM, as detailed in [29], the basis (at spatial infinity) is given by the subdominant solutions of the CHE when $y^{\prime}\rightarrow-\infty$:

[TABLE]

and the particular dominant, obtained by acting on $\psi_{-,0}$ with the discrete symmetry [29] $\Omega_{-}:\theta_{3}\rightarrow\theta_{3}+i\pi$, $q_{3}\rightarrow-q_{3}$: $\psi_{-,1}=\Omega_{-}\psi{-,0}$,

[TABLE]

Now, it is easy to express $\psi_{0,0}$ in terms of the basis $\psi_{-,0},\psi_{-,1}$

[TABLE]

with the Wronskian (connection coefficient) $Q_{1}=W[\psi_{0,0},\psi_{-,0}]$. Eventually, upon comparing the $y^{\prime}\rightarrow-\infty$ limits of $\psi_{0,0}(y^{\prime})$, coming from (4.60) and (4.58, 4.59) with the same limit, descending from (4.56, 4.57), for $\psi_{in}(y^{\prime})$, one eventually finds the exact ratio

[TABLE]

from which one gets the behaviour of the incoming $R_{in}(r)$ at the horizon

[TABLE]

and as a by-product the exact value for the coefficient $B$:

[TABLE]

For large negative $\theta_{3}\rightarrow-\infty$, the coefficient in (4.53), $e^{-\Phi_{3}(\theta_{3},k_{3})}\sim e^{k_{3}\theta_{3}}\cdot\exp(\textrm{series of }e^{n\theta_{3}})$, see (4.41), is a subdominant non-perturbative contribution and can be approximated to zero. Yet, if $y^{\prime}\rightarrow\ln 2$, i.e. in physical terms in the vicinity of the horizon, both Floquet functions are needed in (4.53) to reproduce the decaying behaviour of $\psi_{in}$ at $y^{\prime}=\ln 2$, because the diverging contributions of the Floquet solutions have to cancel each other (at leading order and then automatically at all orders), as in the very same construction of all the other formulæ, (2.18, 3.14, 4.14), connecting Floquet to decaying solutions.

A physical consequence of this is that only far from the horizon, at $y^{\prime}\ll\ln 2$, or at least not too close to it, we may neglect the second term in the r.h.s. of (4.53) and approximate

[TABLE]

up to non-perturbative contributions; this is the so-called Floquet approximation. Now, if $y^{\prime}\rightarrow-\infty$, i.e. at large distances $r\rightarrow+\infty$ from the horizon, the $\theta\rightarrow-\infty$ expansion (4.46) is in general the more effective and moreover in the Floquet region it is dominated as in (4.64) by the lower subscript $-$ result of (4.54):

[TABLE]

Otherwise, – it is important to reiterate –, for generic values of $y^{\prime}$ closer to the horizon, both Floquet functions $\psi_{\pm}^{(3)}(y^{\prime})$ would contribute. In gravity variables this becomes:

[TABLE]

Now, the normalisation constant $\tilde{A}^{\prime}=\tilde{C}\tilde{A}_{-}/\sqrt{2M}$ is such that the asymptotic behaviour (4.57) at $r\rightarrow+\infty$ is satisfied. By expanding ${}_{1}F_{1}$ at large argument, one finds the two exponentials in (4.57) and matches the coefficient of the second with $e^{-i\omega r^{\ast}}$ to determine $\tilde{A}^{\prime}$ and obtain

[TABLE]

With the map $k_{3}=\frac{1}{2}+\hat{l}$, it gives a partial resummation of the quiver gauge partition function (33) of [9]; moreover it is the leading term of a single series expansion obtainable by exploiting the expressions (4.53). As emphasised above, it misses non peturbative contribution $\sim e^{-\Phi_{3}}$ present in the original expressions (4.53), which instead we are able to compute. Actually, we have an extra prefactor $1/2$, which seems to be missing in [9], and a little improvement, as we found a resummation to all instantons of the first two arguments of the ${}_{1}F_{1}$ functions: indeed, $k_{3}=P+O(e^{\theta_{3}})=l+\frac{1}{2}+O(e^{\theta_{3}})$. Finally, the match with $e^{i\omega r^{\ast}}$ gives the approximate (if $M\omega\ll 1$) coefficient $B$ appearing in (4.57):

[TABLE]

One can also verify that in the limit $M\omega\ll 1$ the exact expression (4.63) expands at leading order as (4.68) by using the methods of [29] (then (4.57) is reproduced).

Although (4.42 - 4.45) have been devised as exact converging expansions as $\theta_{3}\rightarrow-\infty$ around the horizon, $a<y^{\prime}<\ln 2$, $a$ is generic and then they can reach the Floquet regime rather far from the horizon. Since $y^{\prime}$ keeps finite, whilst $\theta_{3}$ is large negative, then $e^{\theta_{3}}\ll e^{2y^{\prime}}$, that in gravity variables (4.39) becomes $\omega r\ll\frac{M}{r}$. Parametrising $k_{3}=\hat{l}+\frac{1}{2}$, the Floquet function $\psi_{-}^{(3)}(r)$, normalised such that $\psi_{-}^{(3)}(r)\simeq\left(\frac{2M}{r}-1\right)^{-\frac{1}{2}+2iM\omega}$ as $r\rightarrow 2M$, is, according to (4.44),

[TABLE]

Implementing the Floquet approximation $\psi_{in}(r)\simeq\tilde{C}\psi_{-}^{(3)}(r)$ (for $r$ sufficiently far from $2M$) in gravity variables, we obtain

[TABLE]

This expression is proportional to (33) of [9], apart from the usual improvement given by the resummation at all instantons in the arguments of the ${}_{2}F_{1}$ and the $\Gamma$ functions and from the fact that we find a different exponent for the binomial $\left(1-\frac{2M}{r}\right)$. We stress again that the validity of (4.70) cannot be extended at the horizon, $r=2M$; the behaviour of $R_{in}$ at the horizon is given by (4.62).

Analogously, the CHE limit of (4.52) is proportional to $\psi_{-,0}(y^{\prime})$ and thanks to this it is also proportional to the wave function used in gravity set-up $\psi_{up}(y^{\prime})$: then we write $\psi_{up}(y^{\prime})=\hat{D}\psi_{-,0}(y^{\prime})$ and

[TABLE]

Again, we find the exact proportionality constant $\hat{D}$. The function $\psi_{up}(y^{\prime})$ is defined by the fact that as $r\rightarrow+\infty$,

[TABLE]

i.e. its gravity equivalent $R_{up}$ satisfies outgoing boundary conditions at infinity. Then, using (4.58), it is simple to find that

[TABLE]

where $R_{-,0}(r)=\frac{r\sqrt{r-2M}}{\sqrt{2M}}\psi_{-,0}(y^{\prime})$. As in the case of $\psi_{in}(y^{\prime})$, thanks to the expansion (4.46), we can write if $y^{\prime}<a$ ($r>2M$) both Floquet functions in (4.71) as exponentials of series of $e^{\theta_{3}}$ and this gives a complete understanding of the wave functions at all orders in $e^{\theta_{3}}$. Moreover, if we are not close to $r=+\infty$, in the limit $\theta_{3}\rightarrow-\infty$, the term $e^{-\Phi_{3}(\theta_{3},k_{3})}\sim e^{k_{3}\theta_{3}}$ is suppressed and then we have the approximate proportionality $\psi_{up}(y^{\prime})\simeq\tilde{D}\psi_{+}^{(3)}(y^{\prime})$.

We have here built up the foundations for more detailed analysis and calculations on this interesting case: we will leave this aim for the near future.

5 Conclusions and outlook

We have studied the change from the ODE/IM (in/up in gravity) to the Floquet basis for the MME, DCHE, CHE and HE. The expressions for the four cases (2.18) for the MME, (3.14) for the DCHE, (4.53, 4.71) for the CHE, (4.14) for the Heun, depend on the acquired phase $\varphi$, the two contributions to it, $\varphi_{<}$ (from the region $y<0$), $\varphi_{>}$ (from the region $y>0$) and the quantum momentum at zero, $\Pi_{+}(y=0)$. Whilst this last quantity can be easily computed as $\theta\rightarrow-\infty$, the computation of $\varphi$ and $\varphi_{<}$, $\varphi_{>}$ would naively diverge as suffering from the conflicting $\theta\rightarrow-\infty$ and $y\rightarrow\pm\infty$ limits. A new method has solved the issue by introducing two kink Riccati equations, (2.25, 2.37) for the MME, (3.18, 3.27) for the DCHE, (4.21, 4.30) for the HE and their confluent limits for the CHE, one for region $y>0$ and the other for $y<0$. The solutions of these equations have been written as converging series and found explicitly by solving recursive first order ODEs. Eventually, this provides converging series for $\varphi_{>}$, $\varphi_{<}$, $\varphi$, with explicit formulæ for their lowest orders and, importantly, also for the quantum momentum corresponding to Floquet eigenfunctions: (2.28, 2.32) for the MME, (3.19, 3.23) for the DCHE, (4.42, 4.45) and (4.46, 4.49) for the CHE, (4.20, 4.25) for the HE. Through the change of basis to ODE/IM functions, also these functions can be given expressions as sums of exponentials of converging series.

The quantity $\varphi$ for the MME (2.45) and the symmetrised $\Phi$ for the DCHE (3.32), the CHE (4.41) and the HE (4.34) is conjectured to coincide with $A_{D}/\hbar$, where $A_{D}$ is dual gauge period of ${\cal N}=2$ SYM with $N_{f}=0,2,3,4$, respectively, in the NS background. The rigorous proof of this identification is left for an incoming publication, where we will also prove the existence of a function ${\cal F}(\theta,k)$ (such that $\varphi=\frac{\partial{\cal F}}{\partial k},P^{2}=-\frac{1}{2}\frac{\partial{\cal F}}{\partial\theta}$ for $N_{f}=0$ and similar formulæ for other flavours), which is then identified with the prepotential of gauge theory666The relation $P^{2}=-\frac{1}{2}\frac{\partial{\cal F}}{\partial\theta}$ is the Matone relation [30, 31]..

An important remark is that (2.30, 2.1) for $\Pi_{>}(y)$ in the case of the MME imply for the wave function the form

[TABLE]

with $\frac{d\ln\psi_{>}^{(0)}(y)}{dy}=\Pi^{(0)}_{>}(y)$ and ${\cal P}^{(n)},{\cal Q}^{(n)}$ polynomials in $e^{-y}$ of degree $n$. Similar formulæ hold for the DCHE, the CHE and the HE. Expressions of this kind for the Floquet solutions of the CHE appear in Section IV of [32] and also in [33], concerning solutions of the BPZ equation and in these papers are obtained by solution of the Schroedinger-like equations for the wave function. We believe that solving Riccati equations, insted of Schroedinger-like equations, is a more effective way to understand the final structure (5.1) of the wave functions, as started in [20] for analysing the acquired phase in the context of Painlevé and gauge theories.

Finally, we observe that, if in the system of Riccati ODEs the function $\tilde{\Pi}_{<}^{(0)}(y)$ is chosen as a function decaying at $y\rightarrow-\infty$, for instance in the MME case $\tilde{\Pi}_{<}^{(0)}(y;k)=\frac{d}{dy}\ln K_{-2k}(2ie^{-\frac{y}{2}})$, one can construct directly a series for the subdominant solution in the limit $y\rightarrow-\infty$. Analogously, choosing $\tilde{\Pi}_{>}^{(0)}(y;k)$ as a solution decaying at $y\rightarrow+\infty$, for instance $\frac{d}{dy}\ln K_{2k}(2ie^{\frac{y}{2}})$ in the MME case, one constructs the series for the solution subdominant at $y\rightarrow+\infty$.

Acknowledgements Special thanks go to D. Gregori, for the effort done in [29]. Discussions with G. Bonelli, P. Di Vecchia, K. Gupta, C. Heissenberg, P. Longhi, A. Nervo, F. Olivi, R. Russo, A. Tanzini, K. Zarembo are kindly acknowledged. This research was supported in part by the grants GAST (INFN), the Cost action CaLISTA CA21109, MSCA CaLIGOLA 2021-SE-01-101086123, the EC Network Gatis, the MIUR-PRIN contract 2017CC72MK_003 and the National Science Foundation under Grant No. NSF PHY-1748958.