In this section we present asymptotic large time expansions of formal local solutions of the quantum Painlevé equations obtained in the previous section, which describe via Painlevé/gauge correspondence the strong coupling phase of the gauge theory. Our solutions generalize the ones of [5] to general \(\Omega \)-background \(\epsilon \ne 0\) by the formal quantization of the Zak transform.

We first review the structure of the ansatz for the solutions in [5] and illustrate modifications needed to generalize it to the quantum Painlevé case. Then we discuss case by case the concrete forms of the solutions.

Ansatz of [5]

Each solution of [5] is of form (2.29) with \(\epsilon =0\) and \(a=\nu \epsilon _2\). Partition function \(\mathcal \) actually depends on t as fractional power \(\textrm=\kappa \textrm^d\), \(d\in \mathbb \), where \(\textrm\), \(\textrm\) are staying for the variables t, s of [5] and \(\kappa \in \mathbb \) is a convenient normalization factor. The partition function factorizes into three terms with the following dependence on \(\textrm\) and \(\nu \)

1.

A monomial factor of form \(\textrm^ \nu ^2/2}\, e^^2+\upxi _1 \textrm+\updelta \nu \textrm}\), where \(N_\textrm\in \mathbb \).

2.

A \(\textrm\)-independent factor of form \(e^\prod \nolimits _^} \textsf(\nu +\upmu _i)\).

3.

An asymptotic power series in \(\textrm^\) that starts from 1, the k-th coefficient being a polynomial of order 3k in \(\nu \).

The parameters entering the above ansatz are complex numbers or simple polynomial invariants of the masses fixed case by case. We discuss more on these coefficients in the quantum case. The terms in the second and third item respectively are interpreted as the 1-loop and non-perturbative part of the quantum field theory describing the low-energy excitations around the expansion point. In particular number \(N_\textrm\) of the Barnes \(\textsf\)-function factors corresponds to the number of light BPS particles. The first term is needed to provide consistent asymptotic boundary conditions. In the expansion in the electric frame, where a Lagrangian description of the gauge theory is known, it corresponds to the classical term of the partition function. Accordingly, we will call the above factors classical, 1-loop and instanton parts of \(\mathcal \).

These expansions of the Painlevé tau functions are along the canonical rays. In order to make sense to the expansions we are forced to perform them along a specific set of canonical rays rather than within Stokes sectors. Indeed, \(\updelta s\) (see in Item 1 above) must be purely imaginary to prevent an exponential growth of the monomial factor under shift \(\nu \mapsto \nu +n\), \(n\in \mathbb \). As described in Sect. 3, the canonical rays and the corresponding expansions get identified by the action of the rotational symmetry presented in Table 3. This means that we have two different expansions for PV, PIV and PII, actually with same d in \(\textrm=\kappa \textrm^d\). The two expansions are distinguished by their asymptotic behavior, namely, by the monomial factor, as follows. The expansions along red rays of Fig. 3 correspond to \(\upbeta \ne 0\) and \(N_\textrm=1\) while the expansions along the blue rays correspond to \(\upbeta =0\) and \(N_\textrm\) equal to the number of hypermultiplets plus 1. From the gauge theory viewpoint they describe two different phases, one with heavy hypermultiplets and the other with light ones respectively. For PIII\(_\) and PI we have only one expansion, with \(\upbeta \ne 0\) and \(N_\textrm=1\) except PIII\(_1\) where \(N_\textrm=2\).

Ansatz for Quantum Painlevé Equation

The ansatz we use to solve quantum Painlevé equations is given by the operator-like Zak transform (2.29) of

$$\begin \mathcal (a_};\epsilon _1,\epsilon _2|s)=\mathcal _(a_};\epsilon _1,\epsilon _2|s) \mathcal _(a_};\epsilon _1,\epsilon _2) \mathcal _(a_};\epsilon _1,\epsilon _2|s) \end$$

(4.1)

where we denote by \(a_}\) the Cartan variable in the strong coupling expansion/magnetic frame of the gauge theory and \(s=\kappa t^d\) with a convenient \(\kappa \in \mathbb \) is an expansion variable. An important novelty with respect to classical Painlevé equations is that the monodromy parameters \(a_},\eta _}\) obey the non-trivial commutation relations (2.28). We expect the following expressions for the factors of \(\mathcal \) as natural deformations of the above classical ones

1.

A classical term of form (c.f. weak case (2.2))

$$\begin \mathcal _(a_};\epsilon _1,\epsilon _2|s)=s^ a_}^2/2}} e^} s}}, \end$$

(4.2)

We keep same \(N_\textrm\in \mathbb \) because it appears in combination

$$\begin \frac_(a_}+2n\epsilon _1;2\epsilon _1,\epsilon _2\epsilon _1|s) \mathcal _(a_}+2n\epsilon _2;\epsilon _1\epsilon _2,2\epsilon _2|s)}_(a_};2\epsilon _1,\epsilon _2\epsilon _1|s) \mathcal _(a_};\epsilon _1\epsilon _2,2\epsilon _2|s)}=s^. \end$$

(4.3)

Note that we always choose power d in \(s=\kappa t^d\) so that s has mass dimension 1. This implies that \(\beta \) and \(\delta \) are just complex parameters and that \(\xi _1\) and \(\xi _2\) are polynomials of mass dimension 1 and 2 respectively in the masses and in \(\Omega \)-background permutation invariants \(\epsilon =\epsilon _1+\epsilon _2\) and \(\epsilon _1\epsilon _2\).

2.

A 1-loop part of form

$$\begin \mathcal _(a_};\epsilon _1,\epsilon _2)=e^}^2}}\prod _^} \exp \gamma _(a_}+\mu _i-\epsilon /2), \end$$

(4.4)

Function \(\gamma _\) naturally deforms the classical Barnes function according to (B.19) just as in the weak case (2.3). Dimension counting prescribes that \(\chi \) is a complex number and \(\mu _i\) are dimension 1 polynomialsFootnote 4 in the masses and \(\epsilon \).

3.

An instanton part which we write as a power series in \(s^\)

$$\begin \mathcal _(a_};\epsilon _1,\epsilon _2|s)=1+\sum _^ Q_(a_}) (\epsilon _1\epsilon _2s)^+O(s^), \end$$

(4.5)

where \(Q_(a_})\) is a polynomial of order 3k in \(a_}\) and the same mass dimension with coefficients being polynomials in the masses and \(\epsilon ,\, \epsilon _1\epsilon _2\). As in the \(\epsilon =0\) case, this power series turns out to be an asymptotic one.

As it was already mentioned for (Q)PVI at the end of Sect. 2.3, the (quantum) Painlevé equations in the tau form have an additional integration constant replacing the dependence on one of the mass variables. In the classical case this integration constant is fixed by the Hamiltonian form of the equation but in the quantum case we do not know an analog of that. Therefore we fix the \(\epsilon \)-corrections for such dependence indirectly, primarily by comparing the partition functions we obtain with the holomorphic anomaly calculations in Sect. 5.

General Procedure to Determine the Ansatz

Now we describe how we determine the coefficients entering the above ansatz by imposing that it solves the quantum Painlevé equations with a suitable asymptotic behavior. In each case we substitute the above ansatz to the quantum Painlevé equation, i.e. in a system of equations of form (2.32) obtained in Sect. 3. Up to a general factor the relation we obtain is a power series in \(s^\) whose coefficients impose relations on the parameters entering the ansatz, including the coefficients of polynomial \(Q_(a_})\). We will call them \(s^\) relations. Note that a t-independent term \(\mathcal _\) appears in these relations as an analog of (2.20)

$$\begin \mathfrak _n&=\frac_(a_}+2n\epsilon _1;2\epsilon _1,\epsilon _2\epsilon _1)\mathcal _(a_}+2n\epsilon _2;\epsilon _1\epsilon _2,2\epsilon _2)}_(a_}+\mathfrak (2n)\epsilon _1;2\epsilon _1,\epsilon _2\epsilon _1)\mathcal _(a_}+\mathfrak (2n)\epsilon _2;\epsilon _1\epsilon _2,2\epsilon _2)} \end$$

(4.6)

$$\begin&=e^\mathfrak (2n)/4)}\prod _^} \prod \limits _(2n)} \big (a_}+\mu _i+\text (n) (i \epsilon _1+j \epsilon _2)\big ). \end$$

(4.7)

In each case to find \(\mathcal \) we follow the general procedure below

1.

The very first factor we set is the classical one. We find it from the first few \(s^\) relations.

2.

Then, following the \(s^\) relations step by step, we obtain blowup factors \(\mathfrak _n\) together with the coefficients of polynomials \(Q_(a_})\). It will appear that the coefficients of \(Q_(a_})\) and \(\mathfrak _n\) depend on mass invariants, corresponding to those of the classical Painlevé equations, and on the additional integration constant replacing the highest dimension mass invariant. When available, we can use the factorization (4.7) to fix this freedom completely.

3.

For \(\mathcal _\) we present the first few terms of the power series in \(s^\) of \(\ln \mathcal _\). Dimension counting prescribes that taking the logarithm drastically reduces the power in \(a_}\) of the \(s^\)-coefficients, namely

$$\begin -\epsilon _1\epsilon _2 \ln \mathcal _=\sum _^K P_(a_}) s^+O(s^), \end$$

(4.8)

where \(P_(a_})\) is a polynomial in \(a_}\) of order \(k+2\).

4.

Finally, we compare our results with the undeformed, classical ones by checking their self-consistency and fixing the integration constant freedom up to \(\epsilon \)-corrections. These last are completely fixed in Sect. 5.

The final answer we obtain in the quantum case is the Zak transform in the noncommutative variable \(e^\eta _}}\) with coefficients given by the asymptotic expansion of (4.1).Footnote 5 Note that for asymptotic expansions of individual functions \(\mathcal \) it is not necessary to restrict t along the canonical rays. Though, unlike the classical Painlevé equations, we, strictly speaking, cannot even guarantee the existence of solutions \(\mathcal \) to the considered equations in the form (2.32).

4.1 QPV (linear Exp Singularity)/\(N_\textrm=3\) with Light Hypers

As already mentioned, for Painlevé V we have two different large time expansions. Here we present the quantum deformation of the one on the imaginary canonical rays from [5, Sec. A.4], called there “expansion 2”. For the (quantum) Painlevé V we have \(d=1\) and we set \(s=t\) for the both expansions.

4.1.1 Classical Part

For this expansion we have \(\beta =0\) and \(N_\textrm=4\). The first few leading \(t^\) relations for (3.1) and (3.2) imply that the classical part can be taken as

$$\begin&\mathcal _^}]}(a_},\phi ;\epsilon _1,\epsilon _2|s)=\mathcal _^}]}(\phi ;\epsilon _1,\epsilon _2|s) \mathcal _^}]}(a_};\epsilon _1,\epsilon _2|s) \quad \text\nonumber \\ &\mathcal _^}]}(\phi ;\epsilon _1,\epsilon _2|s)=e^} \, s^-\frac}, \mathcal _^}]}(a_};\epsilon _1,\epsilon _2|t)=e^\sqrt a_} s}} \, s^}^2}}, \end$$

(4.9)

The superscript for \(\mathcal \) here and below will be boldface for the large time expansions and subscript L inside it denotes the Linear behavior in s of the exponent in the classical part. On the contrary, in each special expansion we omit the superscript on the mass invariants for brevity, i.e. here we write \(w_2\) instead of \(w_2^=m_1^2m_2^2m_3^2\). By normalizing the tau function of QPV with \(a_}\)-independent factor \(\mathcal _^]}\)

$$\begin \tau _r(\epsilon _1,\epsilon _2|t)=\frac^]}_(e_;\epsilon _1,\epsilon _2|t)}, \end$$

(4.10)

we obtain from (3.1) to (3.2)

$$\begin & D^1_(\tau _r^,\tau _r^)=0,\quad D^3_(\tau _r^,\tau _r^)=\epsilon D^2_(\tau ^_r,\tau ^_r),\qquad \quad \end$$

(4.11)

$$\begin & D^4_(\tau _r^,\tau _r^)+2\left( \epsilon _1\epsilon _2\frac}\ln t}\right) D^2_(\tau _r^,\tau _r^)\nonumber \\ & \quad - \left( \frac+\frac\epsilon _1\epsilon _2\frac\right) D^2_(\tau _r^,\tau _r^)\nonumber \\ & \quad -\frac\left( \epsilon _1\epsilon _2\frac}\ln t}\right) D^0_\left( \tau ^_r\tau ^_r\right) +\frac t\,\tau ^_r\tau ^_r=0, \end$$

(4.12)

so we see that the quantum Painlevé V actually depends only on two mass parameters, namely \(w_2\) and \(e_3\). The third one appears as an integration constant as for the classical Painlevé V (Appendix D.2).

4.1.2 Expansion

We then solve the successive \(s^\) relations, obtaining step by step the leading terms of \(\mathcal _^]}\) together with blowup factors \(\mathfrak _n^]}\). In this way we computed \(\mathfrak _n^]}\) for \(|n|\le \frac\)

$$\begin \mathfrak _0^]}= & \mathfrak _}^]}=1, \qquad \mathfrak _^]}= \prod \limits _ \left( a_}+\frac_1\lambda ' \tilde_2\lambda \lambda ' \tilde_3} \pm \frac\right) , \nonumber \\ \mathfrak _}^]}= & \prod \limits _ \prod \limits _ (i,j)=\\ \scriptscriptstyle (1,3),(3,1) \end}\left( a_}+\frac_1\lambda ' \tilde_2\lambda \lambda ' \tilde_3} \pm \fracj\epsilon _2}\right) \end$$

(4.13)

together with the terms of \(\mathcal _^]}\) up to \(t^\), up to \(t^\) in the logarithmic expansion they are

$$\begin & -\epsilon _1\epsilon _2 \ln \mathcal _^]}(a_},w_2,e_3,\tilde_4;\epsilon _1,\epsilon _2|t)=\nonumber \\ & =\left( 4a_}^3-(w_2\epsilon ^2)a_}+e_3\right) \cdot \frac \nonumber \\ & +\left( 10 a_}^4-(3w_25\epsilon ^2)a_}^2+4e_3 a_} +\frac\epsilon ^2)^2}-\frac_4}\right) \cdot \frac \nonumber \\ & +\left( 44a_}^5-2\frac2\epsilon _1\epsilon _257\epsilon ^2}a_}^3+22e_3a_}^2-\frac a_}-\frac(w_22\epsilon _1\epsilon _213\epsilon ^2)\right) \cdot \frac \nonumber \\ & +\Bigg ( 252a_}^6-2(50w_213\epsilon _1\epsilon _2170\epsilon ^2)a_}^4+146e_3 a_}^3-\fraca_}^2-\frac22\epsilon _1\epsilon _293\epsilon ^2)}a_} \nonumber \\ & +\frac3\epsilon _1\epsilon _27\epsilon ^2}\left( \tilde_4\frac\epsilon ^2)^2}\right) \frac\Bigg )\cdot \frac+\cdots +O\left( \frac\right) , \end$$

(4.14)

where

$$\begin&c_1=44\tilde_4-13w_2^2-2(2\epsilon _1\epsilon _225\epsilon ^2)w_2+4\epsilon _1\epsilon _2\epsilon ^2-37\epsilon ^4, \end$$

(4.15)

$$\begin&c_2=108\tilde_4-39w_2^2-2(16\epsilon _1\epsilon _2115\epsilon ^2)w_2+56\epsilon _1\epsilon _2\epsilon ^2-239\epsilon ^4. \end$$

(4.16)

This expansion has the expected dependence on integration constant \(\tilde_4\), for \(\mathfrak _n^]}\) it is parametrized as

$$\begin w_2=\tilde_1^2+\tilde_2^2+\tilde_3^2, \qquad e_3=\tilde_1\tilde_2\tilde_3, \qquad \tilde_4=\tilde_1^2 \tilde_2^2+\tilde_1^2 \tilde_3^2+\tilde_2^2 \tilde_3^2\, . \end$$

(4.17)

Note that when we substitute the ansatz into the equations we should control the dependence on \(\epsilon _\) of the variable \(\tilde_4\). The substitution gives \(\tilde_4(2\epsilon _1,\epsilon _2\epsilon _1)=\tilde_4(\epsilon _1\epsilon _2,2\epsilon _2)\), and then it is natural to consider that \(\tilde_4\) depends on \(\epsilon \) but not on \(\epsilon _1\epsilon _2\), so the substitution into the equation does not affect \(\tilde_4\). Above \(\mathfrak _n^]}\) are of form (4.7), so our ansatz (4.4) gives

$$\begin \mathcal _^]}(a_},\tilde_;\epsilon _1,\epsilon _2) \prod \limits _ \exp \gamma _ \left( a_}\frac(\lambda \tilde_1\lambda ' \tilde_2\lambda \lambda ' \tilde_3\epsilon )\right) . \end$$

(4.18)

4.1.3 Comparison with the Classical Limit

(\(\epsilon =0\)). Let us compare our result in the \(\epsilon =0\) limit with expansion BLMST [5, (A.49)]. According to (D.19) and (D.20) we check that

$$\begin \tau (a_},\eta _},m_;-2\epsilon _2,2\epsilon _2|t)=e^_}/2}e^_1 \textrm_}/2}\tau _}^}(\nu ,\rho ;\theta _0,\theta _t,\theta _*|\textrm_}). \end$$

(4.19)

The dictionary is given by \(\eta _}=\rho \) and standard scalings (see (D.19) for the all except the first one)

$$\begin a_}=2\epsilon _2\nu , \qquad t=2\epsilon _2 \textrm_}, \qquad m_f=2\epsilon _2\textrm_f|_, \end$$

(4.20)

where masses \(\textrm_\) are given by (D.21). Then, classical part (4.9) becomes

$$\begin \mathcal _^]}(-2\epsilon _2,2\epsilon _2)=e^_1/2)\textrm_}}\, (2\epsilon _2 \textrm_})^. \end$$

(4.21)

1-loop part (4.18), by using (B.19), becomes

$$\begin \mathcal _^]}(-2\epsilon _2,2\epsilon _2)= & \frac\theta _0^2\theta _t^2\frac\frac}}(2\pi )^}\nonumber \\ & \times \prod _ \textsf \left( 1+\nu \pm \tilde_t+\frac_*}\right) \textsf \left( 1+\nu \pm \tilde_0-\frac_*}\right) ,\nonumber \\ \end$$

(4.22)

where the tilded thetas are connected with the tilded masses by same formulas (4.20) and (D.21) as for the non-tilded ones. The Barnes \(\textsf\)-function factor of the above expression coincides with the counterpart from [5, (A.49)] iff in the self-dual case the tilded and the non-tilded masses are equal up to discrete group \(W(A_3)=S_4\), which acts by transpositions of the masses and changing signs for an even number of the masses. This means that \(\tilde_4=w_4|_\), where \(w_4=m_1^2m_2^2m_1^2m_3^2m_2^2m_3^2\). Then, up to the factor in (4.19) and a constant factor in \(\textrm_}\) and \(\nu \), \(\mathcal _^]}\mathcal _^]}\) coincides with the prefactor of the \(\textrm_}^\) series in [5, (A.49)].Footnote 6 Finally, for the first two terms of the instanton expansion we obtain [5, (A.50)]

$$\begin & \mathcal _^]}(-2\epsilon _2,2\epsilon _2)= 1+\left( 4\nu ^3-\text _2 \nu +\text _3\right) \cdot \frac_}} +\left( 8\nu ^6-2(2\text _25)\nu ^4\right. \nonumber \\ & \left. +4\text _3\nu ^3+\frac_2 (\text _26)}\nu ^2 -\text _3(\text _24)\nu +\frac_3^2}\frac_2^2}\frac_4}\right) \cdot \frac_}^2}+O\left( \frac_}^3}\right) , \nonumber \\ \end$$

(4.23)

where \(\textrm_2, \textrm_3, \textrm_4\) are defined in (D.16). In terms of \(\theta \)-variables (D.21) of [5, (A.50)] they read

$$\begin \textrm_2=2\theta _0^2+2\theta _t^2+\theta _*^2, \qquad \textrm_3=\theta _*(\theta _t^2-\theta _0^2), \qquad \textrm_4=(\theta _t^2-\theta _0^2)^2+2\theta _*^2(\theta _0^2\theta _t^2). \end$$

(4.24)

Let us comment more on the connection between the tilded and the non-tilded masses, the latter being the gauge theory ones. The factorization (4.7) prescribes that, without loss of generality, the differences \(\tilde_f-m_f|_\) are proportional to \(\epsilon \) up to a numerical coefficient. The first two relations of (4.17) imply that such coefficients vanish. So, under the assumptions of the ansatz, we have \(\tilde_4=w_4\), as it is further confirmed in Sect. 5.1.

4.2 QPV (square Exp Singularity)/\(N_\textrm=3\) with Heavy Hypers

Here we present the quantum deformation of the other Painlevé V large time expansion, namely the one on the real canonical rays from [5, Sec. A.4], called there “expansion 1”. Recall that for QPV we set \(s=t\).

4.2.1 Classical Part

For this expansion we have \(\beta \ne 0\) and \(N_\textrm=1\). The first few leading \(t^\) relations for (3.1) and (3.2) imply that the classical part can be taken as

$$\begin&\mathcal _^]}(a_},e_;\epsilon _1,\epsilon _2|t)=\mathcal _^]}(e_;\epsilon _1,\epsilon _2|t) \mathcal _^]}(a_};\epsilon _1,\epsilon _2|t) \quad \text\nonumber \\ &\mathcal _^]}(e_;\epsilon _1,\epsilon _2|t)=e^}e^} \, t^-\frac}, \quad \nonumber \\ &\mathcal _^]}(a_};\epsilon _1,\epsilon _2|t)=e^a_}t}} \, t^}^2}}, \end$$

(4.25)

where subscript S denotes the Square behavior in s of the exponent in the classical part. By normalizing the tau function of QPV with \(a_}\)-independent factor \(\mathcal _^]}\)

$$\begin \tau _r(\epsilon _1,\epsilon _2|t)=\frac^]}_(e_;\epsilon _1,\epsilon _2|t)} \end$$

(4.26)

we obtain from (3.1), (3.2)

$$\begin & D^1_(\tau _r^,\tau _r^)=\frac\tau _r^\tau _r^,\nonumber \\ & D^3_(\tau _r^,\tau _r^)=\frac\epsilon D^2_(\tau ^_r,\tau ^_r)-\frac\left( t^2\frac\right) \tau ^_r\tau ^_r, \nonumber \\ \end$$

(4.27)

$$\begin & D^4_(\tau _r^,\tau _r^)+2\left( \epsilon _1\epsilon _2\frac}\ln t}\right) D^2_(\tau _r^,\tau _r^)\nonumber \\ & \quad + \left( \fracw_22\epsilon _1\epsilon _2\frac\right) D^2_(\tau _r^,\tau _r^)\nonumber \\ & \quad -\frac (t^2\epsilon ^2)\left( \epsilon _1\epsilon _2\frac}\ln t}\right) D^0_\left( \tau ^_r\tau ^_r\right) \nonumber \\ & \quad +\left( \fract^2\fract\frac\left( w_22\epsilon _1\epsilon _2\frac\right) \right) \tau ^_r\tau ^_r=0, \end$$

(4.28)

so again the quantum Painlevé V depends only on two mass parameters \(w_2\) and \(e_3\). And the third mass parameter appears as an integration constant as well.

4.2.2 Expansion

We then solve the successive \(s^\) relations, obtaining step by step the leading terms of \(\mathcal _^]}\) together with blowup factors \(\mathfrak _n^]}\). In this way we computed \(\mathfrak _n^]}\) for \(|n|\le \frac\)

$$\begin & \mathfrak _0^]}=\mathfrak _}^]}=1, \qquad \mathfrak _^]}= \frac}\left( a_}\pm \frac\right) , \nonumber \\ & \mathfrak _}^]}= -\frac\prod \limits _ (i,j)=\\ \scriptscriptstyle (1,3),(3,1) \end}\left( a_}\pm \fracj\epsilon _2}\right) , \nonumber \\ & \mathfrak _^]}=\frac\prod \limits _ (i,j)=\\ \scriptscriptstyle (5,1),(3,3),(1,5) \end}\left( a_}\pm \fracj\epsilon _2}\right) , \nonumber \\ & \mathfrak _}^]}=-\frac\prod \limits _ (i,j)=\\ \scriptscriptstyle (7,1),(3,5),(5,3)(1,7) \end}\left( a_}\pm \fracj\epsilon _2}\right) \end$$

(4.29)

together with the terms of \(\mathcal _^]}\) up to \(t^\), up to \(t^\) in the logarithmic expansion with \(\alpha _}=\textrma_}\) they are

$$\begin & -\epsilon _1\epsilon _2 \ln \mathcal _^]}(a_},w_2,e_3,\check_4;\epsilon _1,\epsilon _2|t) =\Bigg ( \frac}^3}+\left( 4w_2\frac\epsilon _1\epsilon _2\frac\epsilon ^2\right) \alpha _}+8e_3\Bigg )\cdot \frac\nonumber \\ & +\Bigg ( -\frac\alpha _}^4-\left( 12w_2\frac\epsilon _1\epsilon _2\frac\epsilon ^2\right) \alpha _}^2-64e_3\alpha _}-8\check_42w_2\epsilon _1\epsilon _2 \nonumber \\ & -\frac(\epsilon _1\epsilon _2)^2\frac\epsilon _1\epsilon _2\epsilon ^2\Bigg )\cdot \frac +\Bigg ( \frac\alpha _}^5+\left( \fracw_2\frac\epsilon _1\epsilon _2\frac\epsilon ^2\right) \alpha _}^3\nonumber \\ & +448e_3\alpha _}^2 +\frac\alpha _}+\frac(8w_210\epsilon _1\epsilon _217\epsilon ^2)\Bigg )\cdot \frac \nonumber \\ & \qquad +\Bigg ( -\frac\alpha _}^6-\left( 230w_2\frac\epsilon _1\epsilon _2\frac\epsilon ^2\right) \alpha _}^4-3008e_3\alpha _}^3-c_2 \alpha _}^2 \nonumber \\ & \qquad -16e_3(96w_2146\epsilon _1\epsilon _2165\epsilon ^2)\alpha _}+c_3 \Bigg )\cdot \frac+\cdots +O\left( \frac\right) , \end$$

(4.30)

where

$$\begin c_1&=448\check_4+16w_2^2+(166\epsilon _1\epsilon _2\epsilon ^2)w_2+\frac(\epsilon _1\epsilon _2)^2-\frac\epsilon _1\epsilon _2\epsilon ^2-\frac\epsilon ^4, \end$$

(4.31)

$$\begin c_2&=1728\check_4+96w_2^2+7(106\epsilon _1\epsilon _2\epsilon ^2)w_2+\frac(\epsilon _1\epsilon _2)^2-\frac\epsilon _1\epsilon _2\epsilon ^2-\frac\epsilon ^4, \end$$

(4.32)

$$\begin c_3&=-16(4w_210\epsilon _1\epsilon _219\epsilon ^2)\check_4-320e_3^2-16\epsilon _1\epsilon _2w_2^2\nonumber \\&\quad -\left( 44(\epsilon _1\epsilon _2)^2\frac\epsilon _1\epsilon _2\epsilon ^2\frac\epsilon ^4\right) w_2 \nonumber \\&\quad -10(\epsilon _1\epsilon _2)^3+\frac(\epsilon _1\epsilon _2)^2\epsilon ^2-\frac\epsilon _1\epsilon _2\epsilon ^4-\frac\epsilon ^6. \end$$

(4.33)

This expansion has the expected dependence on integration constant \(\check_4\), and \(\check_4(2\epsilon _1,\epsilon _2\epsilon _1)=\check_4(\epsilon _1\epsilon _2,2\epsilon _2)\) so we consider that \(\check_4\) depends on \(\epsilon \) but not on \(\epsilon _1\epsilon _2\) as in the linear exp case. Above \(\mathfrak _n^]}\) are of form (4.7), so our ansatz (4.4) gives

$$\begin \mathcal _^]}(a_};\epsilon _1,\epsilon _2)=(2\textrm)^}^2}} \exp \gamma _ (a_}-\epsilon /2). \end$$

(4.34)

4.2.3 Comparison with the Classical Limit

(\(\epsilon =0\)). Let us compare our result in the \(\epsilon =0\) limit with expansion [5, (A.45)]. Namely, we check (4.19) under \(\eta _}=\rho \) and scaling (4.20) with (D.21). Then, classical part (4.25) becomes

$$\begin \mathcal _^]}(-2\epsilon _2,2\epsilon _2)=e^_}^2/32}e^\nu /2+\theta _t+\theta _*/2)\textrm_}}\, (2\epsilon _2 \textrm_})^. \end$$

(4.35)

1-loop part (4.34), by using (B.19), becomes

$$\begin \mathcal _^]}(-2\epsilon _2,2\epsilon _2)=(2\textrm)^}\frac-\frac}}(2\pi )^}\textsf(1+\nu ). \end$$

(4.36)

Then, up to the factor in (4.19) and a constant factor in \(\textrm_}\) and \(\nu \), \(\mathcal _^]}\mathcal _^]}\) coincides with the prefactor of the \(\textrm_}^\) series in [5, (A.45)]. Finally, for the first two terms of the instanton expansion we obtain [5, (A.46)] with (4.24)

pagination

$$\begin & \mathcal _^]}(-2\epsilon _2,2\epsilon _2)= 1+\Bigg ( \frac}+\left( 2\text _2\frac\right) \text \nu +4\text _3\Bigg )\cdot \frac_}}\nonumber \\ & +\Bigg ( -\frac+\frac\left( \text _2\frac\right) \nu ^4-\text _3\text \nu ^3 -\left( 2\text _2^2\frac\text _2\frac\right) \nu ^2\nonumber \\ & +\text _3\left( 8\text _2\frac\right) \text \nu +8\text _3^22\text _4\frac_2}\frac\Bigg ) \cdot \frac_}^2}+O\left( \frac_}^3}\right) , \end$$

(4.37)

where, to match the \(\nu \)-independent part of \(\textrm^_}\) in [5, (A.46)] we imposed condition \(\check_4=w_4|_\).

4.3 QPIII\(_1\)/\(N_\textrm=2\)

Here we present the quantum deformation of the Painlevé III\(_1\) large time expansion on the real canonical rays from [5, Sec. A.3]. For the (quantum) Painlevé III\(_1\) we have \(d=1/2\) and we set \(s=8\textrm\, t^\).

4.3.1 Classical Part

For this expansion we have \(\beta \ne 0\) and \(N_\textrm=2\). The first few leading \(s^\) relations for (3.1) and (3.4) imply that the classical part can be taken as

$$\begin&\mathcal _^}]}(a_}, \phi ;\epsilon _1,\epsilon _2|s)=\mathcal ^}]}_(\phi ;\epsilon _1,\epsilon _2|s) \mathcal ^}]}_(a_};\epsilon _1,\epsilon _2|s) \quad \text\nonumber \\ &\mathcal ^}]}_(\phi ;\epsilon _1,\epsilon _2|s)=e^} \, s^}, \quad \mathcal ^}]}_(a_};\epsilon _1,\epsilon _2|s)=e^} s}} \, s^}^2}}, \end$$

(4.38)

where \(\phi \) is a dimension 2 integration constant that is not fixed by the further \(s^\) relations, such that \(\phi (2\epsilon _1,\epsilon _2\epsilon _1)=\phi (\epsilon _1\epsilon _2,2\epsilon _2)\). So we consider that \(\phi \) depends on \(\epsilon \) but not on \(\epsilon _1\epsilon _2\) as in the previous cases. Note that the quantum Painlevé III\(_1\) (3.1) and (3.4) depends only on \(e_2=m_1m_2\) but not on \(w_2=m_1^2m_2^2\) (c.f. Appendix  D.2 for the classical case).

4.3.2 Expansion

We then solve the successive \(s^\) relations, obtaining step by step the leading terms of \(\mathcal _^]}\) together with blowup factors \(\mathfrak _n^]}\). In this way we computed \(\mathfrak _n^]}\) for \(|n|\le 2\)

$$\begin & \mathfrak _0^]}=\mathfrak _}^]}=1, \quad \mathfrak _^]}= \prod \limits _ \left( a_}+\lambda \frac_1\tilde_2}\pm \frac\right) ,\nonumber \\ & \mathfrak _}^]}= \prod \limits _ \prod \limits _ (i,j)=\\ \scriptscriptstyle (1,3),(3,1) \end}\left( a_}+\lambda \frac_1\tilde_2}\pm \fracj\epsilon _2}\right) \nonumber \\ & \mathfrak _^]}= \prod \limits _ \prod \limits _ (i,j)=\\ \scriptscriptstyle (5,1),(3,3),(1,5) \end}\left( a_}+\lambda \frac_1\tilde_2}\pm \fracj\epsilon _2}\right) , \end$$

(4.39)

$$\begin & \text \text \qquad \phi (e_2,\tilde_2)=\frac(\epsilon ^2\tilde_2)-\frac, \nonumber \\ & e_2=\tilde_1\tilde_2, \quad \tilde_2=\tilde_1^2+\tilde_2^2 \,.\end$$

(4.40)

We computed the terms of \(\mathcal _^]}\) up to \(s^\), up to \(s^\) in the logarithmic expansion they are

$$\begin & -\epsilon _1\epsilon _2 \ln \mathcal _^}]}(a_},\tilde_2,e_2;\epsilon _1,\epsilon _2|s) =\left( 2a_}^3-\frac9\tilde_22\epsilon _1\epsilon _23\epsilon ^2}a_}\right) \cdot \frac\nonumber \\ & +\left( 5 a_}^4-\frac51\tilde_214\epsilon _1\epsilon _217\epsilon ^2} a_}^2 -\frac33\tilde_212\epsilon _1\epsilon _29\epsilon ^2}(2e_2\tilde_2\epsilon ^2)\right) \cdot \frac\nonumber \\ & \qquad +\left( 22a_}^5-\frac615\tilde_2186\epsilon _1\epsilon _2205\epsilon ^2}a_}^3+c_1 a_}\right) \cdot \frac\nonumber \\ & \qquad +\bigg ( 126a_}^6-2(1790e_2945\tilde_2303\epsilon _1\epsilon _2315\epsilon ^2)a_}^4+c_2 a_}^2\nonumber \\ & \qquad +c_3(2e_2\tilde_2\epsilon ^2)\bigg )\cdot \frac+\cdots +O\left( \frac\right) , \end$$

(4.41)

where

$$\begin c_1&=\frac\tilde_2^2+\frac282\epsilon _1\epsilon _2861\epsilon ^2}\tilde_2-\frace_2^2+\frac643\epsilon ^2}e_2\nonumber \\&\quad +\frac220\epsilon _1\epsilon _2\epsilon ^2135\epsilon ^4}, \end$$

(4.42)

$$\begin c_2&=\frac\tilde_2^2+\frac878\epsilon _1\epsilon _22025\epsilon ^2)}\tilde_2-\frace_2^2+\frac1025\epsilon ^2)}e_2 \nonumber \\&\quad +\frac5428\epsilon _1\epsilon _2\epsilon ^22943\epsilon ^4}, \end$$

(4.43)

$$\begin c_3&=\frac\tilde_2^2+\frac73\epsilon _1\epsilon _287\epsilon ^2)}\tilde_2+\frace_2^2+\frac243\epsilon ^2}e_2\nonumber \\&\quad +\frac204\epsilon _1\epsilon _2\epsilon ^281\epsilon ^4)}. \end$$

(4.44)

Above \(\mathfrak _n^]}\) are of form (4.7), so our ansatz (4.4) gives