††institutetext: Section de Mathématiques, Université de Genève, 1211 Genève
4, Switzerland††institutetext: Theoretical Physics Department, CERN, 1211
Geneva 23, Switzerland

# Painlevé kernels and surface defects at strong coupling

Matijn François and Alba Grassi<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

It is well established that the spectral analysis of canonically quantized
four-dimensional Seiberg-Witten curves can be systematically addressed via the
Nekrasov-Shatashvili functions. In this paper we explore another aspect of the
relation between $\mathcal{N}=2$ supersymmetric gauge theories in four
dimensions and operator theory. Specifically, we study an example of an
integral operator connected to Painlevé equations and whose spectral traces
compute correlation functions of the 2d Ising model. This operator does not
correspond to a canonically quantized Seiberg-Witten curve, but its kernel can
nevertheless be interpreted as the density matrix of an ideal Fermi gas.
Adopting the approach of Tracy and Widom, we provide an explicit expression
for its eigenfunctions via an $\operatorname{O(2)}2$ matrix model. We then
show that these eigenfunctions are computed by surface defects in
$\operatorname{SU(2)}2$ super Yang-Mills in the self-dual phase of the
$\Omega$-background. Our result also yields a strong coupling expression for
such defects which resums the instanton expansion. Even though we focus on one
concrete example, we expect these results to hold for a larger class of
operators arising in the context of isomonodromic deformation equations.

††preprint: CERN-TH-2023-169

##  1 Introduction

Building upon the work of Seiberg and Witten sw1 ; sw2 , important results
have been obtained for $\mathcal{N}=2$ supersymmetric gauge theories in four
dimensions. One remarkable achievement is the exact evaluation of the path
integral, made possible thanks to localization techniques and the introduction
of the $\Omega$-background Moore:1997dj ; Losev:1997tp ; n ; no2 . This led to
the discovery of a new class of special functions, so-called Nekrasov
functions, which today have found a wide range of applications in various
fields of mathematics and theoretical physics. Despite the exceptional control
these functions grant us over the weak coupling region, a strong coupling
expansion requires alternative methods. This is one of the motivations behind
the present work. In addition, we explore a particular extension of the
correspondence relating $\mathcal{N}=2$ supersymmetric gauge theories in four
dimensions to the spectral theory of quantum mechanical operators on the space
of square-integrable functions $L^{2}({\mathbb{R}})$.

The first concrete example of such a correspondence was obtained in ns , where
the authors showed how to use $\mathcal{N}=2$ supersymmetric gauge theories in
the four-dimensional Nekrasov-Shatasvili (NS) phase of the $\Omega$-background
$(\epsilon_{2}=0)$ to solve the spectral theory of a certain class of ordinary
differential equations. For example, the quantization condition for the
operator spectrum corresponds to the quantization of the twisted
superpotential ns ; mirmor ; mirmor2 ; Zenkevich2011 ; Nekrasov:2011bc , while
the eigenfunctions are computed from the surface defect partition functions
Kozlowski:2010tv ; Alday:2010vg ; Gaiotto:2014ina ; Kanno:2011fw ; gmn ;
Sciarappa:2017hds ; Jeong:2021rll ; Jeong:2023qdr ; Jeong:2018qpc ;
Jeong:2017pai ; Bonelli:2022ten ; Piatek:2017fyn ; Alday:2009fs ;
Drukker:2009id . More recently, explicit expressions for the Fredholm
determinants ggm ; ghn  and the connection coefficients Bonelli:2022ten  of
such differential equations have been derived in closed form.111A rigorous
derivation of some of these results can be found in Lisovyy:2022flm . A
q-perturbative approach to connection coefficients is also discussed in
Hollands:2021itj ; Hollands:2017ahy , see also Jeong:2018qpc . See also
Gaiotto:2014bza ; Ito:2017iba ; Yan:2020kkb ; Imaizumi:2021cxf ;
Hollands:2013qza ; kpt ; Fioravanti:2019vxi ; Hollands:2019wbr ; Ito:2021sjo
for related development in the context of WKB and four-dimensional gauge
theories and Grassi:2018bci  for a different quantization scheme in higher
rank gauge theories. All the operators appearing in the above setup can be
obtained via the standard quantization of four-dimensional Seiberg-Witten (SW)
curves or, equivalently, by considering the semicalssical limit of BPZ
equations Belavin:1984vu .

In this paper we explore another facet of the interplay between spectral
theory and supersymmetric gauge theories. On the gauge theory side we focus on
four-dimensional $\mathcal{N}=2$ gauge theories in the self-dual phase of the
$\Omega$-background ($\epsilon_{1}=-\epsilon_{2}=\epsilon$), while on the
operator theory side we study a class of operators which do not correspond to
canonically quantized four-dimensional SW curves. These operators originally
appeared in the framework of isomonodromic deformations equations wu1 ;
Widom:1996ff ; Tracy:1998aa ; zamo ; Tracy:1995ax . Their relevance in the
context of four-dimensional supersymmetric gauge theories and topological
string was pointed out in bgt ; bgt2 ; Bonelli:2017gdk ; Gavrylenko:2023ewx
in close connection with the TS/ST duality ghm ; cgm2 ; kama ; kmz  and the
isomonodromy/CFT/gauge theory correspondence gil1 ; gil ; bsu ; Iorgov:2014vla
; Bershtein:2018srt ; Bershtein:2014yia .

In this paper we focus on a specific operator associated to the Painlevé $\rm
III_{3}$ equation and whose spectral traces compute correlation functions in
the 2d Ising model wu1 ; zamo ; gm . Its integral kernel on ${\mathbb{R}}$
reads222We sometimes refer to (1.1) as a Fermi gas operator, since it
corresponds to the density matrix for an ideal Fermi gas in the external
potential $8t^{1/4}\cosh\mathopen{}\mathclose{{}\left(x}\right)$.

$\rho(x,y)={{\rm e}^{-4t^{1/4}\cosh x}{\rm e}^{-4t^{1/4}\cosh
y}\over\cosh\mathopen{}\mathclose{{}\left(x-y\over 2}\right)}~{}.$ (1.1)

For $t>0$ the kernel (1.1) is positive and of trace class on
$L^{2}({\mathbb{R}})$, hence the corresponding operator has a discrete
positive spectrum $\mathopen{}\mathclose{{}\left\\{E_{n}}\right\\}_{n\geqslant
0}$ with square-integrable eigenfunctions
$\mathopen{}\mathclose{{}\left\\{\varphi_{n}(x,t)}\right\\}_{n\geqslant 0}$ ,

$\int_{\mathbb{R}}\mathrm{d}y\
\rho\mathopen{}\mathclose{{}\left(x,y}\right)\varphi_{n}\mathopen{}\mathclose{{}\left(y,t}\right)=E_{n}\varphi_{n}\mathopen{}\mathclose{{}\left(x,t}\right).$
(1.2)

As we will review in subsection 3.2, the spectrum is computed by the Nekrasov
function of 4d, $\mathcal{N}=2$, $\operatorname{SU(2)}2$ super Yang-Mills
(SYM) in the self-dual phase of the $\Omega$-background bgt , see (3.16) and
(3.17). The purpose of this paper is to study the eigenfunctions of (1.1) and
relate them to surface defects in 4d, $\mathcal{N}=2$, $\operatorname{SU(2)}2$
SYM in the self-dual phase of the $\Omega$-background. In addition, we use
this relation to obtain a strong coupling expression for such defects which
resums both the instanton expansion and the $\epsilon$ expansion.

##  2 Summary

###  2.1 Results

The paper can be summarized as follows. Adopting the approach of Tracy:1995ax
; Marino:2016rsq  we construct eigenfunctions of (1.1) from expectation values
of a determinant-like expression,

$\Xi_{\pm}\mathopen{}\mathclose{{}\left(x,t,E}\right)={\rm e}^{-4t^{1/4}\cosh
x}{\rm e}^{\pm x/2}\sum_{N\geqslant
0}(\pm\kappa)^{N}\Psi_{N}\mathopen{}\mathclose{{}\left({\rm
e}^{x},t}\right)\,,\qquad\qquad x\in{\mathbb{R}}\,,$ (2.3)
$\displaystyle\Psi_{N}(z,t)={1\over
N!}\int_{{\mathbb{R}}^{N}_{>0}}\mathopen{}\mathclose{{}\left(\prod_{i=1}^{N}{{\rm
d}z_{i}\over z_{i}}{z-z_{i}\over
z+z_{i}}~{}\exp\mathopen{}\mathclose{{}\left(-{4t^{1/4}}\mathopen{}\mathclose{{}\left(z_{i}+z_{i}^{-1}}\right)}\right)\prod_{j=i+1}^{N}\mathopen{}\mathclose{{}\left({z_{i}-z_{j}\over
z_{i}+z_{j}}}\right)^{2}}\right)\,.$ (2.4)

More precisely, (2.3) are square-integrable eigenfunctions $\varphi_{n}$ of
(1.1) if we set $\kappa=-E_{n}^{-1}$, where $E_{n}$ is an eigenvalue of the
operator (1.1),

$\varphi_{n}(x,t)=\Xi_{+}\mathopen{}\mathclose{{}\left(x,t,-{1\over
E_{n}}}\right)=(-1)^{n}\Xi_{-}\mathopen{}\mathclose{{}\left(x,t,-{1\over
E_{n}}}\right)\,.$ (2.5)

In section 6 we show that (2.3) and (2.4) are explicitly related to surface
defects in 4d, $\operatorname{SU(2)}2$, $\mathcal{N}=2$ SYM in the self-dual
phase ($\epsilon_{1}=-\epsilon_{2}=\epsilon$) of the
$\Omega$-background333This is in line with the generic expectation that matrix
model averages are related to partitions functions in the presence of D-branes
adkmv ; em ; Gukov:2011qp ; Maldacena:2004sn ; Borot2012 ; Kozcaz:2010af ;
Gaiotto:2013sma .. We first consider the surface defect which is engineered
using the open topological vertex with a D-brane on the external leg, see
Appendix A for details. Using the explicit vertex expression of Appendix A it
is easy to see that this correspond to the special case of a 4d/2d defect
called type II defect in (Sciarappa:2017hds, , Sec. 2.3.3)444Following
(Sciarappa:2017hds, , Sec. 2.3.3) the two-dimensional $\mathcal{N}=(2,2)$
theory here consists of 2 free chiral multiplets living on a disc. See also
Bullimore:2014awa . . Hence we denote its partition function by $Z^{\rm
II}_{\rm tot}(q,t,\sigma)$. The explicit expression is given in (6.110) and
(6.113). In the gauge theory we typically use

$q={y\over 2\epsilon}\,,\qquad\qquad
t=\mathopen{}\mathclose{{}\left({\Lambda\over\epsilon}}\right)^{4}\,,\qquad\qquad\sigma=\mathrm{i}{a\over
2\epsilon}\,,$ (2.6)

where $y$ is related to the position of the defect,
$\epsilon=\epsilon_{1}=-\epsilon_{2}$ is the $\Omega$-background parameter,
$\Lambda\sim{\rm e}^{-1/g_{\mathrm{YM}}^{2}}$ is the instanton counting
parameter and $a$ is the Coulomb branch parameter. The relation between the
determinant like expression (2.3) and the defect partition function $Z^{\rm
II}_{\rm tot}$ is given in (6.116) and reads

$\Xi_{\pm}\mathopen{}\mathclose{{}\left(x,t,\frac{\cos(2\pi\sigma)}{2\pi}}\right)=\\\
{{\rm e}^{3\zeta^{\prime}(-1)}{\rm e}^{4\sqrt{t}}\over
2^{11/12}\pi^{3/2}t^{3/16}}\int_{{\mathbb{R}}}{\rm d}q~{}{\rm e}^{{\rm
i}{2}qx}\sum_{k\in{\mathbb{Z}}}\mathopen{}\mathclose{{}\left(Z^{\rm II}_{\rm
tot}\mathopen{}\mathclose{{}\left(\pm q,t,\sigma+k}\right)+Z^{\rm II}_{\rm
tot}\mathopen{}\mathclose{{}\left(\mp q-{{\rm i}\over 2},t,\sigma+k+{1\over
2}}\right)}\right)\,.$ (2.7)

The quantization condition for the energy spectrum of (1.1) was derived in bgt
, see (3.16) and (3.17). By evaluating the defect partition function on the
lhs of (2.7) at the corresponding quantized values of $\sigma=1/2+{\rm
i}\sigma_{n}$, we obtain the eigenfunctions $\varphi_{n}$ of (1.1),

$\varphi_{n}(x,t)=\\\ {{\rm e}^{3\zeta^{\prime}(-1)}{\rm e}^{4\sqrt{t}}\over
2^{11/12}\pi^{3/2}t^{3/16}}\int_{{\mathbb{R}}}{\rm d}q~{}{\rm e}^{{\rm
i}{2}qx}\sum_{k\in{\mathbb{Z}}}\mathopen{}\mathclose{{}\left(Z^{\rm II}_{\rm
tot}\mathopen{}\mathclose{{}\left(q,t,k+\frac{1}{2}+{\rm
i}\sigma_{n}}\right)+Z^{\rm II}_{\rm
tot}\mathopen{}\mathclose{{}\left(-q-{{\rm i}\over 2},t,k+{\rm
i}\sigma_{n}}\right)}\right),$ (2.8)

where
$\sigma_{n}\in\mathbb{R}\setminus\mathopen{}\mathclose{{}\left\\{0}\right\\}$
are solutions to (3.17). An example is shown in Figure 1.

Figure 1: The first (blue) and second (red) eigenfunction of (1.1) computed
using the surface defect expression in (2.8) for $t=1/100\pi^{8}$.

In section 6 and Appendix B we show that we can equivalently write (2.7) as

$\int_{{\mathbb{R}}+{\rm i}\sigma_{*}}{{\rm
d}\sigma}{\tan\mathopen{}\mathclose{{}\left(2\pi\sigma}\right)\over\mathopen{}\mathclose{{}\left(2\cos(2\pi\sigma)}\right)^{N}}\mathopen{}\mathclose{{}\left(Z^{\rm
II}_{\rm tot}\mathopen{}\mathclose{{}\left(q,t,\sigma}\right)+Z^{\rm II}_{\rm
tot}\mathopen{}\mathclose{{}\left(-q-{{\rm i}\over 2},t,\sigma+{1\over
2}}\right)}\right)\\\ =\mathrm{i}{2^{11/12}\sqrt{\pi}{t^{3/16}}\over{\rm
e}^{3\zeta^{\prime}(-1)}{\rm
e}^{4\sqrt{t}}\mathopen{}\mathclose{{}\left(4\pi}\right)^{N}}\int_{\mathbb{R}}{{\rm
d}x}~{}{\rm e}^{-{\rm i}{2}qx}{\rm e}^{-4t^{1/4}\cosh x}{\rm
e}^{x/2}\Psi_{N}\mathopen{}\mathclose{{}\left({\rm e}^{{x}},t}\right)\,,$
(2.9)

where $\sigma_{*}$ is such that $0<\sigma_{*}<|{\rm Re}(q)|$ if ${\rm
Re}(q)\neq 0$ and simply $\sigma_{*}>0$ if ${\rm Re}(q)=0$. This choice of
$\sigma^{*}$ guarantees that the integration over $\sigma$ in (2.9) avoids the
poles of the integrand. Let us elaborate more on the meaning of (2.9).

  * -

The Fourier transform on the rhs of (2.9) relates two types of defects
(Sciarappa:2017hds, , Sec. 2.3.3) or, more precisely, two phases of the same
defect (Jeong:2023qdr, , Sec. 4.2)555Both surface defects are defined by
coupling the 4d theory to the same 2d gauged linear sigma model, but in
different phases of the 2d theory. The transition between these phases
explains the Fourier transform relation between their vacuum expectation
values as explained in Jeong:2023qdr . We thank Saebyeok Jeong for pointing
that out.. In particular, while $Z^{\rm II}_{\rm tot}(q,t,\sigma)$ is
geometrically engineered in topological string theory by inserting a brane on
the external leg of the toric diagram, its Fourier transform with respect to
the defect variable $q$, naturally makes contact with a brane in the inner
edge of the toric diagram Jeong:2023qdr , see also Aganagic:2001nx ;
Aganagic:2000gs ; adkmv ; Kashani-Poor:2006puz 666We would like to thank A.
Neitzke for useful discussions on this point.. Following (Sciarappa:2017hds, ,
Sec. 2.3.3), we refer to the Fourier transform of a type II defect as a type I
defect777Since we are in rank 2 theories this is equivalent to type III defect
defined in (Sciarappa:2017hds, , Sec. 2.3.3) . Via the AGT correspondence agt
, the latter is realized in Liouville CFT by considering the five point
function of four primaries with one degenerate field, the so called
$\Phi_{2,1}$ field Drukker:2009id ; Alday:2009fs ; Kozcaz:2010af ;
Dimofte:2010tz ; Awata:2010bz ; Taki:2010bj . One can equivalently realize
this defect by coupling the four-dimensional theory to a two-dimensional
theory, see for instance Jeong:2021rll ; Jeong:2023qdr ; gmn ; Jeong:2018qpc ;
Kanno:2011fw ; Ashok:2017odt ; Ashok:2017lko ; Sciarappa:2017hds ; Pan:2016fbl
; Gukov2016  and references therein.

This also means that we could get rid of the Fourier transform on the lhs of
(2.7) by replacing the partition function of the type II defect $Z^{\rm II}$
with the partition function of the type I defect $Z^{\rm I}$. The instanton
counting like-expression of type I defect can be found for instance in
Kanno:2011fw , however we will not use such expression here as we will mainly
focus on type II defects.

  * -

The integral over $\sigma$ on the lhs of (2.9) is responsable for the change
of frame: it brings us from the weakly coupled electric frame, where $Z^{\rm
II}_{\rm tot}$ is defined, to the magnetic frame which is the suitable frame
to describe the monopole point of SYM, see section 5.

In summary, (2.9) means that the matrix model average (2.4) computes the type
I surface defect partition function of 4d, $\operatorname{SU(2)}2$,
$\mathcal{N}=2$ SYM in the self-dual phase
($\epsilon_{1}=-\epsilon_{2}=\epsilon$) of the $\Omega$-background and in the
magnetic frame. In this identification
$z=\exp\mathopen{}\mathclose{{}\left(x}\right)$ is the position of the defect
and the ’t Hooft parameter of the matrix model is identified with the dual
period, $N\epsilon=a_{D}$.

Note also that (2.4) is exact both in $\Lambda$ and in $\epsilon$; it resums
the instanton expansion of the defect partition function and provides an
explicit interpolation from the weak to the strong coupling region. The
$1/\Lambda$ expansion can be obtained in a straitforward way from (2.4) since
it corresponds to expanding the matrix model around its Gaussian point, see
(Gavrylenko:2023ewx, , Sec. 5) and references therein.

###  2.2 Derivation

Let us briefly comment on the derivation of equations (2.7), (2.8) and (2.9).
Firstly, we obtained these results by analyzing the large $N$ expansion of the
matrix models (2.4) and then extrapolating to finite $N$. Secondly, part of
the intuition also comes from the open version of the TS/ST correspondence
Marino:2016rsq ; Marino:2017gyg , see subsection 3.4 and section 7. By
combining these two approaches we obtained (2.7)-(2.9), which we further
tested numerically. However we do not have a rigorous mathematical proof of
these results.

This paper is structured as follows. In section 3 we give an overview of the
well-established relationship between the modified Mathieu operator and the
four-dimensional, $\operatorname{SU(2)}2$, $\mathcal{N}=2$ SYM in the NS phase
of the $\Omega$-background. We then present the connection between the
operator (1.1) and the same gauge theory but in the self-dual phase of the
$\Omega$-background. In section 4 we compute the planar resolvent of (2.4) as
well as the planar two-point function and show how the Seiberg-Witten geometry
emerges from it. In section 5 we show that the ’t Hooft expansion of (2.4)
reproduce the $\epsilon$ expansion of the type I self-dual surface defect in
the magnetic frame. To establish this connection, we rely on two crucial
findings. Firstly, according to the results presented in Kozcaz:2010af , the
$\epsilon$ expansion of the self-dual type I surface defect in the electric
frame is determined by topological recursion eo . Secondly, the self-dual
surface defect (or, more generally, the open topological string partition
function) behaves as a wave functions under a change of frame Grassi:2013qva .
In section 6 we test (2.9) numerically for finite $N$ and analytically in a
$1/N$ expansion, and we verify (2.8) numerically.

##  3 Preparation: spectral theory and 4d, $\mathcal{N}=2$ gauge theory

###  3.1 Well known: differential operators and the NS phase of the
$\Omega$-background

Let us start by reviewing the well-known correspondence relating ordinary
differential equations to four-dimensional $\mathcal{N}=2$ gauge theories in
the NS phase of the $\Omega$-background, i.e. $\epsilon_{2}=0,$
$\epsilon_{1}=\epsilon\neq 0$. In this work we focus on
$\operatorname{SU(2)}2$ SYM. The corresponding operator is the modified
Mathieu operator ${\rm O}_{\rm Ma}$ acting as

${\rm O}_{\rm
Ma}~{}\varphi(x)=\mathopen{}\mathclose{{}\left(-\partial_{x}^{2}+\sqrt{t}\cosh
x}\right)\varphi(x)~{}.$ (3.10)

If $t>0$, then the operator (3.10) has a positive discrete spectrum with
square-integrable eigenfunctions. The result of ns  is that we can obtain the
spectrum by using the so-called NS functions.

More precisely, the standard four-dimensional NS partition function computes
the spectrum of (3.10) via the quantization condition of the twisted
superpotential and the Matone relation. The first condition leads to

$\partial_{\sigma}F^{\rm NS}(t,\sigma)=\mathrm{i}2\pi
n~{},\qquad\qquad\qquad\qquad n\in\mathbb{N}~{},$ (3.11)

where $F^{\rm NS}$ is the Nekrasov-Shatashvili (NS) free energy. Its small $t$
expansion reads888We have
$t=\mathopen{}\mathclose{{}\left(\Lambda/\epsilon}\right)^{4}$ and
$\sigma=\mathrm{i}a/2\epsilon$ as in (2.6).

$F^{\rm
NS}(t,\sigma)=-\psi^{(-2)}(1-2\sigma)-\psi^{(-2)}(1+2\sigma)+\sigma^{2}\log(t)\\\
+\mathopen{}\mathclose{{}\left[\frac{2}{4\sigma^{2}-1}}\right]t+\mathopen{}\mathclose{{}\left[\frac{\mathopen{}\mathclose{{}\left(20\sigma^{2}+7}\right)}{\mathopen{}\mathclose{{}\left(4\sigma^{2}-1}\right)^{3}\mathopen{}\mathclose{{}\left(4\sigma^{2}-4}\right)}}\right]t^{2}+\mathcal{O}\mathopen{}\mathclose{{}\left(t^{3}}\right),$
(3.12)

where $\psi$ is the polygamma function of order -2. Higher orders in (3.12)
can be computed by using combinatorics of Young diagrams, see
Tachikawa:2013kta  for a review and list of references. The resulting
expansion in (3.12) is convergent999Beside numerical and physical evidence, we
are not aware of a rigorous mathematical proof of this fact, see ilt ;
Arnaudo:2022ivo  for related discussions and proofs in other phases of the
$\Omega$-background. when $2\sigma\notin{\mathbb{Z}}$ . Equation (3.11) has
then a discrete set of solutions
$\mathopen{}\mathclose{{}\left\\{\sigma_{n}}\right\\}_{n\geqslant 0}$ for a
fixed value of $t$, and applying the quantum Matone relation matone ;
francisco ,

$E=-t\partial_{t}F^{\rm NS}\mathopen{}\mathclose{{}\left(t,\sigma}\right).$
(3.13)

gives the spectrum
$\mathopen{}\mathclose{{}\left\\{E_{n}}\right\\}_{n\geqslant 0}$ of the
modified Mathieu operator (3.10).

For the eigenfunctions there is a parallel developpement, but one has to
consider the four-dimensional partition function with the insertion of a type
I defect101010As before we are following the terminology of
(Sciarappa:2017hds, , Sec. 2.3.3). We also note that, since we are considering
only rank 1 theories, type I and type III defects as in (Sciarappa:2017hds, ,
Sec. 2.3.3) are identified. in the NS phase of the $\Omega$-background, see
Kozlowski:2010tv ; Alday:2010vg ; Kanno:2011fw ; Gaiotto:2014ina ; gmn ;
Sciarappa:2017hds ; Jeong:2021rll ; Jeong:2023qdr  and reference there.

###  3.2 New: Painlevé kernels and the self-dual phase of the
$\Omega$-background

In this work we consider another class of operators whose spectral properties
are encoded in the gauge theory partition functions in the self-dual phase of
the $\Omega$-background ($\epsilon_{1}+\epsilon_{2}=0$). We focus on the four-
dimensional, $\mathcal{N}=2$, $\operatorname{SU(2)}2$ SYM. In this case the
relevant operator is denoted by $\rho$ and its kernel on ${\mathbb{R}}$ is

${\rm\rho}(x,y)={\sqrt{v(x)}\sqrt{v(y)}\over\cosh\mathopen{}\mathclose{{}\left({x-y\over
2}}\right)}~{},\qquad\qquad\qquad v(x)={\rm e}^{-{8t^{1/4}}\cosh(\rm x)}~{},$
(3.14)

which corresponds to the density matrix for an ideal Fermi gas in an external
potential
$-\log\mathopen{}\mathclose{{}\left[v\mathopen{}\mathclose{{}\left(x}\right)}\right]$
gm . We therefore refer to (3.14) as a Fermi gas operator. For $t>0$ (3.14) is
a trace class operator on $L^{2}({\mathbb{R}})$ with a positive discrete
spectrum $\mathopen{}\mathclose{{}\left\\{E_{n}}\right\\}_{n\geqslant 0}$ ,

$\int_{\mathbb{R}}\mathrm{d}y\
\rho\mathopen{}\mathclose{{}\left(x,y}\right)\varphi_{n}\mathopen{}\mathclose{{}\left(y,t}\right)=E_{n}\varphi_{n}\mathopen{}\mathclose{{}\left(x,t}\right).$
(3.15)

It was demonstrated in bgt  that the spectrum is given by

$E_{n}^{-1}=-{1\over
2\pi}\cos\mathopen{}\mathclose{{}\left(2\pi\mathopen{}\mathclose{{}\left({1\over
2}+{\rm i}{\sigma_{n}}}\right)}\right),$ (3.16)

where $\sigma_{n}\in{\mathbb{R}}$ are solutions to

$\sum_{k\in{\mathbb{Z}}}Z^{\rm Nek}\mathopen{}\mathclose{{}\left(t,k+{1\over
2}+{\rm i}\sigma_{n}}\right)=0$ (3.17)

and $Z^{\rm Nek}(t,\sigma)$ is the Nekrasov function in the self-dual phase of
the $\Omega$-background:

$\displaystyle Z^{\rm Nek}(t,\sigma)={t^{\sigma^{2}}\over
G(1-2\sigma)G(1+2\sigma)}\mathopen{}\mathclose{{}\left(1+\frac{t}{2\sigma^{2}}+\frac{\mathopen{}\mathclose{{}\left(8\sigma^{2}+1}\right)t^{2}}{4\sigma^{2}\mathopen{}\mathclose{{}\left(4\sigma^{2}-1}\right)^{2}}+\mathcal{O}(t^{3})}\right).$
(3.18)

The convergence of the series (3.18) was proven in ilt  for any $t>0$ and
fixed $2\sigma\notin{\mathbb{Z}}$. Often the Nekrasov function is expressed
using $\Lambda$, $a$ and $\epsilon$, which are related to $t$ and $\sigma$ via
(2.6)

$t=\mathopen{}\mathclose{{}\left(\frac{\Lambda}{\epsilon}}\right)^{4}\,,\qquad\qquad\qquad\sigma={{\rm
i}}\frac{a}{2\epsilon}\,.\qquad$ (3.19)

As shown in bgt , the quantization condition (3.17) follows from the identity

$\det\mathopen{}\mathclose{{}\left(1+{\cos(2\pi\sigma)\over
2\pi}\rho}\right)=2^{1/12}{\rm e}^{3\zeta^{\prime}(-1)}t^{-1/16}{\rm
e}^{4\sqrt{t}}\sum_{k\in{\mathbb{Z}}}Z^{\rm Nek}(t,\sigma+k)\,,$ (3.20)

which was demonstrated using the theory of Painlevé equations. Even though
$Z^{\rm Nek}(t,\sigma)$ has poles when $2\sigma\in{\mathbb{Z}}$, the sum on
the rhs of (3.20) removes these poles and the resulting expression is well-
defined also for these values of $\sigma$ ilt .

It is useful to write the Fredholm determinant on the lhs of (3.20) by using
the spectral traces,

$\det\mathopen{}\mathclose{{}\left(1+\kappa\rho}\right)=\sum_{N\geqslant
0}\kappa^{N}Z(t,N)~{},$ (3.21) $Z(t,N)={1\over N!}\sum_{s\in S_{N}}(-1)^{{\rm
sgn}(s)}\int_{{\mathbb{R}}}{\rm
d}^{N}x\prod_{i=1}^{N}\rho(x_{i},x_{s(i)})~{},$ (3.22)

$S_{N}$ being the permutation group of $N$ elements. The Cauchy identity
allows us to write (3.22) as gm

$Z(t,N)={1\over
N!}\int_{{\mathbb{R}}^{N}_{>0}}\mathopen{}\mathclose{{}\left(\prod_{i=1}^{N}{{\rm
d}z_{i}\over
z_{i}}v\mathopen{}\mathclose{{}\left(\log\mathopen{}\mathclose{{}\left(z_{i}}\right)}\right)\prod_{j=i+1}^{N}\mathopen{}\mathclose{{}\left({z_{i}-z_{j}\over
z_{i}+z_{j}}}\right)^{2}}\right)\,.$ (3.23)

It was found in bgt  that the matrix model (3.23) computes the partition
function of $\operatorname{SU(2)}2$ SYM in the self-dual phase of the
$\Omega$-background and in the magnetic frame. The relation to the Nekrasov
function (3.18) can be obtained from (3.20) and reads (bgt2, , eq. 2.28)

$Z(t,N)=-\mathrm{i}(4\pi)^{N}2^{1/12}{\rm
e}^{3\zeta^{\prime}(-1)}t^{-1/16}{\rm e}^{4\sqrt{t}}\int_{{\mathbb{R}}+{\rm
i}\sigma_{*}}{{\rm
d}\sigma}\frac{\tan\mathopen{}\mathclose{{}\left(2\pi\sigma}\right)}{\mathopen{}\mathclose{{}\left(2\cos(2\pi\sigma)}\right)^{N}}Z^{\rm
Nek}\mathopen{}\mathclose{{}\left(t,\sigma}\right),$ (3.24)

where111111 One can take $\sigma_{*}=\mathrm{arccosh}(2\pi)/2\pi$ as in bgt2
or any other $\sigma^{*}>0$ as long as the integral over $\sigma$ in (3.24)
does not hit the poles of the integrand. $\sigma_{*}>0$. Therefore the matrix
model (3.23) provides a resumation of the instanton expansion and allows to
probe the strong coupling regime. In this context we identify the ’t Hooft
parameter of the matrix model, $N\epsilon$, with the dual or magnetic period
in SW theory121212It is possible to analytically continue the results to non-
integer values of $N$ bgt2 ; cgm .

$a_{D}=N\epsilon~{}.$ (3.25)

The equality (3.24) was demonstrated in bgt ; bgt2 . Finally, we emphasise
that (3.23) is exact with respect to both the instanton counting parameter
$\Lambda$ and the $\Omega$-background parameter $\epsilon$. When we expand
(3.23) at large $\Lambda$, while keeping $\epsilon$ and $a_{D}$ fixed, we
obtain an analogous expansion to that found when performing a large-time
expansion in isomonodromic deformation equations ilt ; Bonelli:2016qwg ;
Nagoya:2015cja ; Nagoya:2018pgp ; Lisovyy:2016qig ; Gavrylenko:2020gjb ;
Gavrylenko:2023ewx . On the matrix model side this is an expansion around the
Gaussian point. Similarly, if we expand at small $\epsilon$ while keeping
$\Lambda$ and $a_{D}$ fixed, we recover the expansion resulting from the
holomorphic anomaly algorithm bcov ; hk06 .

The goal of this work is to extend these results to the eigenfunctions of
(3.14), which on the gauge theory side corresponds to inserting surface
defects. As a first observation, we note that the kernel (3.14) falls in the
class of operators studied in Tracy:1995ax , and more recently in
(Marino:2016rsq, , Sec. 2). In particular, following Tracy:1995ax ;
Marino:2016rsq  we can construct eigenfunctions of (3.14) using the matrix
model (3.23). Let us define

$\Xi_{\pm}\mathopen{}\mathclose{{}\left(x,t,\kappa}\right)={\rm
e}^{-{4t^{1/4}}\cosh(x)}{\rm e}^{\pm x/2}\sum_{N\geqslant
0}(\pm\kappa)^{N}\Psi_{N}\mathopen{}\mathclose{{}\left({\rm
e}^{x},t}\right),\qquad\qquad x\in{\mathbb{R}}~{},$ (3.26)
$\displaystyle\Psi_{N}(z,t)={1\over
N!}\int_{{\mathbb{R}}^{N}_{>0}}\mathopen{}\mathclose{{}\left(\prod_{i=1}^{N}{{\rm
d}z_{i}\over z_{i}}{z-z_{i}\over
z+z_{i}}~{}\exp\mathopen{}\mathclose{{}\left(-{4t^{1/4}}\mathopen{}\mathclose{{}\left(z_{i}+z_{i}^{-1}}\right)}\right)\prod_{j=i+1}^{N}\mathopen{}\mathclose{{}\left({z_{i}-z_{j}\over
z_{i}+z_{j}}}\right)^{2}}\right).$ (3.27)

Then $\Xi_{\pm}(x,t,\kappa)$ in (3.26) are square-integrable eigenfunctions
$\varphi_{n}(x,t)$ of (3.14) if we evaluate them at $\kappa=-E_{n}^{-1}$,

$\varphi_{n}(x,t)=\Xi_{+}\mathopen{}\mathclose{{}\left(x,t,-{1\over
E_{n}}}\right)=\mathopen{}\mathclose{{}\left(-1}\right)^{n}\Xi_{-}\mathopen{}\mathclose{{}\left(x,t,-{1\over
E_{n}}}\right).$ (3.28)

This can be verified by using (Marino:2016rsq, , eqs. 2.46, 2.59) and
$\varphi_{n}(x,t)=(-1)^{n}\varphi_{n}(-x,t)$. We will argue in the forthcoming
sections that the matrix model with insertion $\Psi_{N}(z,t)$ corresponds to a
surface defect in the four-dimensional, $\mathcal{N}=2$,
$\operatorname{SU(2)}2$ SYM in the self-dual phase of the $\Omega$-background
and in the magnetic frame.

###  3.3 Comment on blowup equations

It was first pointed out in huang1606  that the five-dimensional NS and sefl-
dual partition functions are closely connected, which was subsequently
demonstrated using Nakajima–Yoshioka blowup equations in ggu . The interplay
between these two phases of the $\Omega$-background was extended to surface
defects in four dimensions in Jeong:2020uxz ; Nekrasov:2020qcq . Applications
in the context of Painlevé equations are discussed in Lencses:2017dgf ;
Gavrylenko:2020gjb ; Jeong:2020uxz ; Nekrasov:2020qcq ; Bershtein:2021uts ;
daCunha:2022ewy . The relevance of blowup equations in the context of
resurgence was also recently investigated in Gu:2023wum .

Given such results, it is natural to wonder whether blowup equations can be
used to relate the spectrum and eigenfunctions of (3.10) and (3.14). Regarding
the spectrum, the blowup formula presented in (Bershtein:2021uts, , eq. 5.7)
reveals a one-to-one correspondence between the solutions
$\\{\sigma_{n}\\}_{n\geqslant 0}$ of (3.11) and the solutions
$\\{\sigma_{n}\\}_{n\geqslant 0}$ of (3.17). However, to obtain the spectrum
we further need the quantum Matone relation (3.13) on the Mathieu side, and
the relation (3.16) on the Fermi gas side. These two relations are very
different, and therefore the spectrum of (3.10) and (3.14) is related in a
highly non-trivial way. It would be interesting to see if blowup equations in
presence of defects Jeong:2020uxz ; Nekrasov:2020qcq , could be used to
establish a map between the eigenfunctions of these two operators.

###  3.4 Two limits of quantum mirror curves

Both operators (3.10) and (3.14) have a common origin from the point of view
of quantum mirror curves in toric Calabi-Yau (CY) manifolds (i.e. five-
dimensional quantum SW curves), which we review here briefly.

It is well known that four-dimensional $\mathcal{N}=2$ supersymmetric theories
can be engineered by using topological string theory on toric CY manifolds kkv
; Iqbal:2003zz ; Klemm:1996bj . The partition function of refined topological
string theory is then identified with the partition function of a five-
dimensional $\mathcal{N}=1$ theory on ${\mathbb{R}}^{4}\times S^{1}$ ikv ; ta
. If we shrink the $S^{1}$ circle we get the 4d theory we are interested in,
we refer to Tachikawa:2013kta  for a review and more references. For
$\mathcal{N}=2$, $\operatorname{SU(2)}2$ SYM the relevant setup is topological
string theory on local ${\mathbb{F}}_{0}$. The mirror curve of local
${\mathbb{F}}_{0}$ is

${\rm e}^{x}+{\rm e}^{-x}+{1\over m_{{\mathbb{F}}_{0}}}{\rm e}^{y}+{\rm
e}^{-y}+{\widehat{\kappa}\over\sqrt{m_{{\mathbb{F}}_{0}}}}=0~{},$ (3.29)

where $\widehat{\kappa}$ and $m_{{\mathbb{F}}_{0}}$ are the complex moduli.
The quantization of this curve adkmv ; acdkv  leads to the operator

$\mathrm{O}_{{\mathbb{F}}_{0}}={\rm e}^{\rm x}+{\rm e}^{-{\rm x}}+{1\over
m_{{\mathbb{F}}_{0}}}{\rm e}^{{\rm y}}+{\rm e}^{-{\rm y}}~{},\qquad\qquad[{\rm
x},{\rm y}]={\rm i}\hbar~{}.$ (3.30)

If $\hbar,m_{{\mathbb{F}}_{0}}>0$ the inverse operator

$\rho_{{\mathbb{F}}_{0}}=\mathrm{O}_{{\mathbb{F}}_{0}}^{-1}$ (3.31)

is of trace class with a positive discrete spectrum131313It is also possible
to take $\hbar$ and $m_{{\mathbb{F}}_{0}}$ imaginary within some range. In
this case one still has the trace class property but the spectrum may no
longer be real Codesido:2016ixn ; gmcomp . ghm ; kama ; Laptev:2015loa . Hence
a natural object to consider is its Fredholm determinant

$\det\mathopen{}\mathclose{{}\left(1+\widehat{\kappa}\rho_{{\mathbb{F}}_{0}}}\right).$
(3.32)

The operator (3.10) can be obtained from (3.30) by implementing the usual
geometric engineering limit kkv ; Klemm:1996bj  where we scale

$m_{{\mathbb{F}}_{0}}={1\over
t\beta^{4}}~{},\qquad\qquad\hbar=\beta~{},\qquad\qquad{\widehat{\kappa}\over\sqrt{m_{{\mathbb{F}}_{0}}}}=-{2\over\sqrt{t}\beta^{2}}+{E\over\sqrt{t}}~{},$
(3.33)

and take $\beta\to 0$. In this limit we obtain the modified Mathieu operator
(3.10),

$\mathopen{}\mathclose{{}\left(\mathrm{O}_{{\mathbb{F}}_{0}}+\widehat{\kappa}}\right)\psi(x)=0~{}\to~{}\mathopen{}\mathclose{{}\left({\rm{O}_{\rm
Ma}}+E}\right)\psi(x)=0~{}.$ (3.34)

Likewise the Fredholm determinant becomes

$\det\mathopen{}\mathclose{{}\left(1+\widehat{\kappa}\rho_{{\mathbb{F}}_{0}}}\right)\to\det\mathopen{}\mathclose{{}\left(1+E\,{\rm
O}_{\rm Ma}^{-1}}\right),$ (3.35)

and we have an explicit expression for this determinant via the NS functions
(ggm, , Sec. 5)

$\det\mathopen{}\mathclose{{}\left(1+E\,{\rm O}_{\rm
Ma}^{-1}}\right)=C(t){\sinh\mathopen{}\mathclose{{}\left(\partial_{\sigma}F^{\rm
NS}(t,\sigma)}\right)\over\sin\mathopen{}\mathclose{{}\left(2\pi\sigma}\right)}~{},$
(3.36)

where $C(t)$ is a normalization constant and the relation
$\sigma\equiv\sigma(t,E)$ is obtained from (3.13).

The Fermi gas operator (3.14) on the other hand can be obtained from
$\rho_{{\mathbb{F}}_{0}}$ by implementing a rescaled limit bgt ,

$\log\mathopen{}\mathclose{{}\left(m_{{\mathbb{F}}_{0}}}\right)={1\over
2\pi\beta}\mathopen{}\mathclose{{}\left(-\log(\beta^{4}t)+2{\rm
i}\beta\sigma}\right),\quad\hbar={1\over\beta}~{},\quad\log\mathopen{}\mathclose{{}\left(\widehat{\kappa}}\right)=\log(1+{\rm
e}^{4\pi{\rm
i}\sigma})-{4\pi\over\beta}\log\mathopen{}\mathclose{{}\left(\beta^{4}t}\right),$
(3.37)

and $\beta\to 0$. This is called “the dual 4d limit” in bgt . The scaling
(3.37) may seem strange at first sight, but it is a natural limit from the
point of view of the TS/ST correspondence. In the dual 4d limit we have

$\det\mathopen{}\mathclose{{}\left(1+\widehat{\kappa}\rho_{{\mathbb{F}}_{0}}}\right)\to\det\mathopen{}\mathclose{{}\left(1+\mathopen{}\mathclose{{}\left({\cos(2\pi\sigma)\over
2\pi}}\right)\rho}\right),$ (3.38)

where $\rho$ is the operator (3.14). This determinant can also be written as
the Zak transform of the self-dual Nekrasov function (3.20).

Let us conclude this section by emphasizing that (3.10) has a natural
interpretation directly within the four-dimensional theory, independently of
the five-dimensional quantum curve. In particular, (3.10) is the standard
quantization of the four-dimensional SW curve for $\operatorname{SU(2)}2$ SYM,
which is related to the semiclassical limit of BPZ equations via the AGT
correspondence. On the other hand for the Fermi gas operator (3.14) we do not
have a parallel interpretation at the moment. It may be possible to relate
this operator to some different quantizatization scheme of the four-
dimensional SW curve. Probably a scheme similar to the one used in the context
of topological recursion Iwaki:2015xnr ; Iwaki:2019zeq ; Marchal:2019nsq ;
Marchal:2019bia 141414We would like to thank M. Mariño and N. Orantin for
useful discussions on this point..

##  4 The Seiberg-Witten geometry from the matrix model

In this section we study the ’t Hooft expansion of the matrix model (3.27) and
show how the Seiberg-Witten geometry emerges from it. For this purpose it is
useful to parameterise
$t=\mathopen{}\mathclose{{}\left(\Lambda/\epsilon}\right)^{4}$ as before in
(3.19) and to introduce the potential151515The potential of the one-
dimensional ideal Fermi gas is
$-\log\mathopen{}\mathclose{{}\left(v\mathopen{}\mathclose{{}\left(x}\right)}\right)=V\mathopen{}\mathclose{{}\left(\mathrm{e}^{x}}\right)/\epsilon$.
$V$ such that

$v\mathopen{}\mathclose{{}\left(\log\mathopen{}\mathclose{{}\left(z}\right)}\right)=\exp\mathopen{}\mathclose{{}\left(-\frac{V\mathopen{}\mathclose{{}\left(z}\right)}{\epsilon}}\right)\,,\qquad\qquad\qquad
V\mathopen{}\mathclose{{}\left(z}\right)=4\Lambda\mathopen{}\mathclose{{}\left(z+\frac{1}{z}}\right)\,,$
(4.39)

and we take $\Lambda,\epsilon>0$ for convenience. The matrix models (3.23) and
(3.27) can then be studied in a ’t Hooft limit where

$N\to+\infty\,,\qquad\qquad\epsilon\to 0\,,\qquad\qquad\lambda=N\epsilon\,,$
(4.40)

with the defect insertion parameter $z$, the instanton counting parameter
$\Lambda$, and the ’t Hooft parameter $\lambda$ all kept fixed.

This limit was implemented on the matrix model without insertions (3.23) in
bgt ; gm . In particular, in this limit the eigenvalues distribute along

$\mathopen{}\mathclose{{}\left[{\tt g},{\tt
g}^{-1}}\right]\subset{\mathbb{R}}_{>0}\,,\qquad\qquad\qquad 0<{\tt g}<1\,,$
(4.41)

and the ’t Hooft parameter $\lambda$ is given by

$\lambda=2\Lambda\frac{\mathrm{K}\mathopen{}\mathclose{{}\left(1-{\tt
g}^{4}}\right)\mathopen{}\mathclose{{}\left[2\mathrm{E}\mathopen{}\mathclose{{}\left({\tt
g}^{4}}\right)-\mathopen{}\mathclose{{}\left(1-{\tt
g}^{4}}\right)\mathrm{K}\mathopen{}\mathclose{{}\left({\tt
g}^{4}}\right)}\right]-\pi}{\pi{\tt
g}\mathrm{K}\mathopen{}\mathclose{{}\left({\tt g}^{4}}\right)}\,,$ (4.42)

where $\mathrm{K}$ and $\mathrm{E}$ are the complete elliptic integrals of the
first (C.201) and second (C.202) kind respectively161616See Appendix C for our
conventions on elliptic integrals.. Later we will use the inversion of this
relation for small $\lambda$,

${\tt
g}=1-\frac{\sqrt{\lambda\over\Lambda}}{\sqrt{2}}+\frac{\lambda}{4\Lambda}-\frac{\lambda^{3/2}}{32\sqrt{2}\Lambda^{3/2}}-\frac{\lambda^{2}}{64\Lambda^{2}}+\mathcal{O}\mathopen{}\mathclose{{}\left(\lambda^{5/2}}\right)\,.$
(4.43)

In the ’t Hooft limit (4.40) we have the following behaviour

$\log Z(t,N)\simeq\sum_{g\geqslant
0}\epsilon^{2g-2}F_{g}\mathopen{}\mathclose{{}\left(\Lambda,\lambda}\right)\,,$
(4.44)

where $\simeq$ stands for asymptotic equality. The first two terms read

${{\rm d}^{2}\over{\rm d}\lambda^{2}}F_{0}=-2\pi{\mathrm{K}({\tt
g}^{4})\over\mathrm{K}(1-{{\tt g}^{4}})}\,,$ (4.45)
$F_{1}=-\frac{1}{4}\log\mathopen{}\mathclose{{}\left(\mathrm{K}\mathopen{}\mathclose{{}\left(1-\frac{1}{{\tt
g}^{4}}}\right)\mathrm{K}\mathopen{}\mathclose{{}\left(1-{\tt
g}^{4}}\right)}\right)-\frac{1}{6}\log\mathopen{}\mathclose{{}\left(\frac{1}{{\tt
g}^{2}}-{\tt g}^{2}}\right)+{\rm constant}\,,$ (4.46)

and higher order terms can also be computed systematically bgt .

Let us now consider the model with insertions (3.27). In the ’t Hooft limit
(4.40) we have the following behaviour Kostovo2 ; kos39 ; kos40 ; kos ; ek1 ;
Eynard:1995nv

$\log\mathopen{}\mathclose{{}\left({{\sqrt{v(z)}}\Psi_{N}(z,t)\over
Z(t,N)}}\right)\simeq\sum_{n\geqslant 0}\epsilon^{n-1}{\cal
T}_{n}\mathopen{}\mathclose{{}\left(z}\right)\,.$ (4.47)

The leading order term ${\cal T}_{0}$ is related to the even part of the
planar resolvent171717The planar resolvent is defined and computed explicitly
further on in subsection 4.1. $\omega_{+}^{0}$ (Marino:2016rsq, , eq. 3.35),

$\mathcal{T}_{0}\mathopen{}\mathclose{{}\left(z}\right)=-2\Lambda\mathopen{}\mathclose{{}\left(z+\frac{1}{z}}\right)+2\lambda\int_{\infty}^{z}{\rm
d}z_{1}~{}\omega_{+}^{0}\mathopen{}\mathclose{{}\left(z_{1}}\right)\,,$ (4.48)

and the subleading order term ${\cal T}_{1}$ is given by Marino:2016rsq ;
Eynard:1995nv ; ek1

${\cal
T}_{1}\mathopen{}\mathclose{{}\left(z}\right)={2}\int^{{z}}_{\infty}\int^{{z}}_{\infty}{\rm
d}z_{1}{\rm d}z_{2}\
W_{++}^{0}\mathopen{}\mathclose{{}\left(z_{1},z_{2}}\right)\,,$ (4.49)

where $W_{++}^{0}$ is the even part of the planar two-point correlator. It can
be expressed explicitly in terms of ${\tt g}$ as Eynard:1995nv ,
(Marino:2016rsq, , eq. 3.43)

$W_{++}^{0}\mathopen{}\mathclose{{}\left(z_{1},z_{2}}\right)=\\\ \frac{{\tt
g}^{2}+{\tt g}^{-2}-2{\tt
g}^{-2}\frac{\mathrm{E}\mathopen{}\mathclose{{}\left(1-{\tt
g}^{4}}\right)}{\mathrm{K}\mathopen{}\mathclose{{}\left(1-{\tt
g}^{4}}\right)}-\mathopen{}\mathclose{{}\left(z_{1}^{2}+z_{2}^{2}}\right)\mathopen{}\mathclose{{}\left(1-\frac{\mathopen{}\mathclose{{}\left(\sqrt{\mathopen{}\mathclose{{}\left(z_{1}^{2}-{\tt
g}^{-2}}\right)\mathopen{}\mathclose{{}\left(z_{1}^{2}-{\tt
g}^{2}}\right)}-\sqrt{\mathopen{}\mathclose{{}\left(z_{2}^{2}-{\tt
g}^{-2}}\right)\mathopen{}\mathclose{{}\left(z_{2}^{2}-{\tt
g}^{2}}\right)}}\right)^{2}}{\mathopen{}\mathclose{{}\left(z_{1}^{2}-z_{2}^{2}}\right)^{2}}}\right)}{8\sqrt{\mathopen{}\mathclose{{}\left(z_{1}^{2}-{\tt
g}^{-2}}\right)\mathopen{}\mathclose{{}\left(z_{1}^{2}-{\tt
g}^{2}}\right)}\sqrt{\mathopen{}\mathclose{{}\left(z_{2}^{2}-{\tt
g}^{-2}}\right)\mathopen{}\mathclose{{}\left(z_{2}^{2}-{\tt
g}^{2}}\right)}}\,.$ (4.50)

###  4.1 The planar resolvent

The planar resolvent is

$\omega^{0}(z)=\lim_{N\to+\infty}{1\over
N}\mathopen{}\mathclose{{}\left\langle\sum_{n=1}^{N}\mathopen{}\mathclose{{}\left(\frac{1}{z-z_{n}}}\right)}\right\rangle\,,$
(4.51)

where the normalized expectation value is with respect to matrix model without
insertions $Z\mathopen{}\mathclose{{}\left(t,N}\right)$ (3.23), and the
$z_{n}$ are the eigenvalues over which one integrates in (3.23). At large $z$
one finds

$\omega^{0}(z)=\frac{1}{z}+\frac{\langle
W\rangle}{z^{2}}+\mathcal{O}\mathopen{}\mathclose{{}\left(\frac{1}{z^{3}}}\right)\,,\qquad\qquad\qquad\langle
W\rangle=\lim_{N\to+\infty}\frac{1}{N}\mathopen{}\mathclose{{}\left\langle\sum_{n=1}^{N}z_{n}}\right\rangle\,,$
(4.52)

and we refer to $\langle W\rangle$ as a Wilson loop by analogy with
Suyama:2012uu . It is useful to split the the planar resolvent in an even and
an odd part,

$\omega^{0}(z)=\omega_{+}^{0}(z)+{z}\omega_{-}^{0}(z)\,,$ (4.53)

where $\omega_{\pm}^{0}\mathopen{}\mathclose{{}\left(z}\right)$ are both even
in $z$.

The even part of the planar resolvent $\omega_{+}^{0}$ for the model (3.27)
has the following integral form (Eynard:1995nv, , eq. 4.16),

$\lambda\omega_{+}^{0}(z)=-\frac{\mathrm{i}}{2}\oint_{\mathcal{C}}\frac{\mathrm{d}y}{2\pi\mathrm{i}}\mathopen{}\mathclose{{}\left(\frac{V^{\prime}(y)y}{z^{2}-y^{2}}}\right)\frac{\sqrt{z^{2}-{\tt
g}^{-2}}\sqrt{z^{2}-{\tt g}^{2}}}{\sqrt{{\tt g}^{-2}-y^{2}}\sqrt{y^{2}-{\tt
g}^{2}}}\,,$ (4.54)

where $\mathcal{C}$ is an anticlockwise contour around the branch cut from
${\tt g}$ to ${\tt g}^{-1}$, which does not include the two poles at $y=\pm
z$. In the matrix model $\Psi_{N}\mathopen{}\mathclose{{}\left(z,t}\right)$
(3.27) we naturally have $z>0$. However, it is useful to consider more
generally $z\in\mathbb{C}$ from now on, and (4.54) makes indeed sense for
complex values of $z$ as well Eynard:1995nv .

If $z^{2}\in\mathbb{C}\setminus[{\tt g}^{2},{\tt g}^{-2}]$ we can write write
(4.54) as

$\lambda\omega_{+}^{0}(z)=\mathopen{}\mathclose{{}\left(\frac{2\Lambda}{\pi}}\right)\sqrt{z^{2}-{\tt
g}^{-2}}\sqrt{z^{2}-{\tt g}^{2}}\int_{\tt g}^{{\tt
g}^{-1}}\mathrm{d}y\,\mathopen{}\mathclose{{}\left(\frac{y-1/y}{z^{2}-y^{2}}}\right)\frac{1}{\sqrt{\mathopen{}\mathclose{{}\left({\tt
g}^{-2}-y^{2}}\right)\mathopen{}\mathclose{{}\left(y^{2}-{\tt
g}^{2}}\right)}}\,.$ (4.55)

where we used the form of the potential given in (4.39). The integral in
(4.55) can be decomposed in partial fractions,

$\int_{b}^{a}\mathrm{d}y\mathopen{}\mathclose{{}\left(y-\frac{1}{y}}\right)\mathopen{}\mathclose{{}\left(\frac{1}{z^{2}-y^{2}}}\right)\frac{1}{\sqrt{\mathopen{}\mathclose{{}\left(a^{2}-y^{2}}\right)\mathopen{}\mathclose{{}\left(y^{2}-b^{2}}\right)}}=\\\
\frac{1}{2z^{2}}\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(1-z^{2}}\right)(\mathcal{I}(z,a,b)+\mathcal{I}(-z,a,b))-2\,\mathcal{I}(0,a,b)}\right]\,,$
(4.56)

where $0<b<a$ and we defined

$\mathcal{I}\mathopen{}\mathclose{{}\left(z,a,b}\right)=\int_{b}^{a}\mathrm{d}y\mathopen{}\mathclose{{}\left(\frac{1}{y-z}}\right)\frac{1}{\sqrt{\mathopen{}\mathclose{{}\left(a^{2}-y^{2}}\right)\mathopen{}\mathclose{{}\left(y^{2}-b^{2}}\right)}}\,.$
(4.57)

Using (byrd1971handbook, , eqs. 256.39, 257.39) one finds for
$z\notin\mathopen{}\mathclose{{}\left[b,a}\right]$,

$\displaystyle\mathcal{I}\mathopen{}\mathclose{{}\left(z,a,b}\right)$
$\displaystyle=\mathopen{}\mathclose{{}\left(\frac{2}{a+b}}\right)\mathopen{}\mathclose{{}\left(\frac{1}{b-z}}\right)\int_{0}^{\mathrm{K(k^{2})}}\mathrm{d}v\mathopen{}\mathclose{{}\left(\frac{1-k\,\mathrm{sn}^{2}\mathopen{}\mathclose{{}\left(v\middle|k^{2}}\right)}{1-k\mathopen{}\mathclose{{}\left(\frac{z+b}{z-b}}\right)\mathrm{sn}^{2}\mathopen{}\mathclose{{}\left(v\middle|k^{2}}\right)}}\right)$
(4.58)
$\displaystyle=\mathopen{}\mathclose{{}\left(\frac{2}{a+b}}\right)\mathopen{}\mathclose{{}\left(\frac{1}{a-z}}\right)\int_{0}^{\mathrm{K(k^{2})}}\mathrm{d}v\mathopen{}\mathclose{{}\left(\frac{1-\mathopen{}\mathclose{{}\left(-k}\right)\,\mathrm{sn}^{2}\mathopen{}\mathclose{{}\left(v\middle|k^{2}}\right)}{1-k\mathopen{}\mathclose{{}\left(\frac{a+z}{a-z}}\right)\mathrm{sn}^{2}\mathopen{}\mathclose{{}\left(v\middle|k^{2}}\right)}}\right)$

where $k$ is the elliptic modulus given by

$k=\frac{a-b}{a+b}\,,$ (4.59)

and $\mathrm{sn}\mathopen{}\mathclose{{}\left(v\middle|k^{2}}\right)$ is the
Jacobi elliptic function known as the sine amplitude (C.205). From
(byrd1971handbook, , eq. 340.01)

$\int\mathrm{d}v\mathopen{}\mathclose{{}\left(\frac{1-\alpha_{1}^{2}\mathrm{sn}^{2}\mathopen{}\mathclose{{}\left(v\middle|k^{2}}\right)}{1-\alpha^{2}\mathrm{sn}^{2}\mathopen{}\mathclose{{}\left(v\middle|k^{2}}\right)}}\right)=\frac{1}{\alpha^{2}}\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(\alpha^{2}-\alpha_{1}^{2}}\right)\Pi\mathopen{}\mathclose{{}\left(\alpha^{2},\phi\middle|k^{2}}\right)+\alpha_{1}^{2}v}\right]\,,\qquad
v=\mathrm{F}\mathopen{}\mathclose{{}\left(\phi\middle|k^{2}}\right)\,,$ (4.60)

where $\mathrm{F}$ and $\Pi$ are the incomplete elliptic integrals of the
first (C.201) and third (C.203) kind respectively. It is useful to note that
$v=0$ corresponds to $\phi=0$ and
$v=\mathrm{K}\mathopen{}\mathclose{{}\left(k^{2}}\right)$ corresponds to
$\phi=\pi/2$. At the end this gives

$\displaystyle\mathcal{I}\mathopen{}\mathclose{{}\left(z,a,b}\right)$
$\displaystyle=\mathopen{}\mathclose{{}\left(\frac{2}{a+b}}\right)\mathopen{}\mathclose{{}\left(\frac{1}{b^{2}-z^{2}}}\right)\mathopen{}\mathclose{{}\left[2b\Pi\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\frac{z+b}{z-b}}\right)k\middle|k^{2}}\right)+(z-b)\mathrm{K}\mathopen{}\mathclose{{}\left(k^{2}}\right)}\right]$
(4.61)
$\displaystyle=\mathopen{}\mathclose{{}\left(\frac{2}{a+b}}\right)\mathopen{}\mathclose{{}\left(\frac{1}{a^{2}-z^{2}}}\right)\mathopen{}\mathclose{{}\left[2a\Pi\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\frac{a+z}{a-z}}\right)k\middle|k^{2}}\right)+(z-a)\mathrm{K}\mathopen{}\mathclose{{}\left(k^{2}}\right)}\right]$

and hence we have in (4.56) for
$z^{2}\notin\mathopen{}\mathclose{{}\left[b^{2},a^{2}}\right]$

$\displaystyle\mathcal{I}\mathopen{}\mathclose{{}\left(z,a,b}\right)+\mathcal{I}\mathopen{}\mathclose{{}\left(-z,a,b}\right)$
(4.62)
$\displaystyle\qquad=\mathopen{}\mathclose{{}\left(\frac{4b}{a+b}}\right)\mathopen{}\mathclose{{}\left(\frac{1}{b^{2}-z^{2}}}\right)\mathopen{}\mathclose{{}\left[\Pi\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\frac{z+b}{z-b}}\right)k\middle|k^{2}}\right)+\Pi\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\frac{z-b}{z+b}}\right)k\middle|k^{2}}\right)-\mathrm{K}\mathopen{}\mathclose{{}\left(k^{2}}\right)}\right]$
$\displaystyle\qquad=\mathopen{}\mathclose{{}\left(\frac{4a}{a+b}}\right)\mathopen{}\mathclose{{}\left(\frac{1}{a^{2}-z^{2}}}\right)\mathopen{}\mathclose{{}\left[\Pi\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\frac{a+z}{a-z}}\right)k\middle|k^{2}}\right)+\Pi\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\frac{a-z}{a+z}}\right)k\middle|k^{2}}\right)-\mathrm{K}\mathopen{}\mathclose{{}\left(k^{2}}\right)}\right]$

as well as

$\begin{split}\mathcal{I}\mathopen{}\mathclose{{}\left(0,a,b}\right)&=\mathopen{}\mathclose{{}\left(\frac{2}{a+b}}\right)\mathopen{}\mathclose{{}\left(\frac{1}{b}}\right)\mathopen{}\mathclose{{}\left[2\Pi\mathopen{}\mathclose{{}\left(-k\middle|k^{2}}\right)-\mathrm{K}\mathopen{}\mathclose{{}\left(k^{2}}\right)}\right]\\\
&=\mathopen{}\mathclose{{}\left(\frac{2}{a+b}}\right)\mathopen{}\mathclose{{}\left(\frac{1}{a}}\right)\mathopen{}\mathclose{{}\left[2\Pi\mathopen{}\mathclose{{}\left(k\middle|k^{2}}\right)-\mathrm{K}\mathopen{}\mathclose{{}\left(k^{2}}\right)}\right]\,.\end{split}$
(4.63)

These particular combinations of elliptic integrals can be reduced to square
roots by making use of the following addition formula for $0<k<1$
(byrd1971handbook, , eqs. 117.02)181818The statement in (byrd1971handbook, ,
eq. 117.02) is for $\alpha^{2}\in\mathbb{R}\setminus\mathcal{H}_{k^{2}}$, but
when $0<k<1$ this can be extended to
$\alpha^{2}\in\mathbb{C}\setminus\mathcal{H}_{k^{2}}$ by making use of the
identity theorem.,

$\Pi\mathopen{}\mathclose{{}\left(\alpha^{2}\middle|k^{2}}\right)+\Pi\mathopen{}\mathclose{{}\left(\frac{k^{2}}{\alpha^{2}}\middle|k^{2}}\right)-\mathrm{K}\mathopen{}\mathclose{{}\left(k^{2}}\right)=\frac{\pi}{2}\sqrt{\frac{\alpha^{2}}{\mathopen{}\mathclose{{}\left(1-\alpha^{2}}\right)\mathopen{}\mathclose{{}\left(\alpha^{2}-k^{2}}\right)}}\,,\qquad\alpha^{2}\in\mathbb{C}\setminus\mathcal{H}_{k^{2}}~{},$
(4.64)

where ${\mathcal{H}_{k^{2}}}$ is

${\mathcal{H}_{k^{2}}}=\mathopen{}\mathclose{{}\left[0,k^{2}}\right]\cup\mathopen{}\mathclose{{}\left[1,\infty}\right]\,.$
(4.65)

Combining (4.63) and (4.64) gives

$\begin{split}\mathcal{I}\mathopen{}\mathclose{{}\left(0,a,b}\right)&=\mathopen{}\mathclose{{}\left(\frac{2}{a+b}}\right)\mathopen{}\mathclose{{}\left(\frac{1}{b}}\right)\mathopen{}\mathclose{{}\left(\frac{\pi}{2}}\right)\mathopen{}\mathclose{{}\left(\frac{1}{1+k}}\right)=\mathopen{}\mathclose{{}\left(\frac{\pi}{2}}\right)\mathopen{}\mathclose{{}\left(\frac{1}{ab}}\right)\\\
&=\mathopen{}\mathclose{{}\left(\frac{2}{a+b}}\right)\mathopen{}\mathclose{{}\left(\frac{1}{a}}\right)\mathopen{}\mathclose{{}\left(\frac{\pi}{2}}\right)\mathopen{}\mathclose{{}\left(\frac{1}{1-k}}\right)=\mathopen{}\mathclose{{}\left(\frac{\pi}{2}}\right)\mathopen{}\mathclose{{}\left(\frac{1}{ab}}\right)\,,\end{split}$
(4.66)

and the combination of (4.62) and (4.64) leads to

$\begin{split}\mathcal{I}&\mathopen{}\mathclose{{}\left(z,a,b}\right)+\mathcal{I}\mathopen{}\mathclose{{}\left(-z,a,b}\right)\\\
&\qquad\qquad=\mathopen{}\mathclose{{}\left(\frac{4b}{a+b}}\right)\mathopen{}\mathclose{{}\left(\frac{1}{b^{2}-z^{2}}}\right)\mathopen{}\mathclose{{}\left[\frac{\pi}{4}\mathopen{}\mathclose{{}\left(\frac{a+b}{b}}\right)\sqrt{\frac{b^{2}-z^{2}}{a^{2}-z^{2}}}}\right]=-\frac{\pi}{\sqrt{z^{2}-a^{2}}\sqrt{z^{2}-b^{2}}}\\\
&\qquad\qquad=\mathopen{}\mathclose{{}\left(\frac{4a}{a+b}}\right)\mathopen{}\mathclose{{}\left(\frac{1}{a^{2}-z^{2}}}\right)\mathopen{}\mathclose{{}\left[\frac{\pi}{4}\mathopen{}\mathclose{{}\left(\frac{a+b}{a}}\right)\sqrt{\frac{a^{2}-z^{2}}{b^{2}-z^{2}}}}\right]=-\frac{\pi}{\sqrt{z^{2}-a^{2}}\sqrt{z^{2}-b^{2}}}\end{split}$
(4.67)

where
$z^{2}\in\mathbb{C}\setminus\mathopen{}\mathclose{{}\left[b^{2},a^{2}}\right]$.

Taking $a^{-1}=b={\tt g}$ and using everything above we finally find for the
even planar resolvent

$\boxed{\lambda\omega_{+}^{0}(z)=\Lambda\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(1-\frac{1}{z^{2}}}\right)-\frac{\sqrt{z^{2}-{\tt
g}^{2}}\sqrt{z^{2}-{\tt g}^{-2}}}{z^{2}}}\right)\,.}$ (4.68)

Even though we derive (4.68) for $z^{2}\in{\mathbb{C}}\setminus[{\tt
g}^{2},{\tt g}^{-2}]$ one can verify that (4.68) holds on the whole complex
plane. As a consistency check we compared the analytical result (4.68) against
the numerical evaluation of (4.54), and found perfect agreement. One can also
see that (4.68) has the correct asymptotic behaviour,

$\displaystyle\omega_{+}^{0}(z)=$ $\displaystyle\
\mathcal{O}\mathopen{}\mathclose{{}\left(1}\right)\,,\qquad\qquad$
$\displaystyle\text{for}\ z\to 0\,,$ (4.69) $\displaystyle\omega_{+}^{0}(z)=$
$\displaystyle\
\mathcal{O}\mathopen{}\mathclose{{}\left(z^{-2}}\right)\,,\qquad\qquad$
$\displaystyle\text{for}\ z\to\infty\,.$

In addition, from the coefficient of the $z^{-2}$-term in the $\ z\to\infty$
expansion we get a closed form expression for the Wilson loop (4.52),

$\langle W\rangle=\frac{\mathopen{}\mathclose{{}\left({\tt
g}^{2}-1}\right)^{2}\Lambda}{2{\tt g}^{2}\lambda}\,.$ (4.70)

Using (4.43) we obtain

$\frac{\mathopen{}\mathclose{{}\left({\tt g}^{2}-1}\right)^{2}\Lambda}{2{\tt
g}^{2}\lambda}=1+\frac{\lambda}{16\Lambda}-\frac{\lambda^{2}}{256\Lambda^{2}}+\frac{5\lambda^{3}}{8192\Lambda^{3}}-\frac{33\lambda^{4}}{262144\Lambda^{4}}+\mathcal{O}\mathopen{}\mathclose{{}\left(\lambda^{5}}\right).$
(4.71)

We cross-checked (4.71) by expanding the matrix model around its Gaussian
point, similar to what was done in (gm, , app. B).

Using (4.68) gives for the leading order $\mathcal{T}_{0}$ (4.48) of the
matrix model (4.47) in the ’t Hooft limit (4.40)

$\partial_{z}\mathcal{T}_{0}\mathopen{}\mathclose{{}\left(z}\right)=-2\Lambda\mathopen{}\mathclose{{}\left(1-\frac{1}{z^{2}}}\right)+2\lambda\omega_{+}^{0}(z)=-2\Lambda\frac{\sqrt{z^{2}-{\tt
g}^{2}}\sqrt{z^{2}-{\tt g}^{-2}}}{z^{2}}~{}.$ (4.72)

An important point of (4.72) is that the Seiberg-Witten curve of
$\mathcal{N}=2$, $\operatorname{SU(2)}2$ SYM,

$y^{2}(z)=-4\Lambda^{2}\mathopen{}\mathclose{{}\left(z^{2}+z^{-2}}\right)+{u}\,,$
(4.73)

emerges in the planar limit, provided we identify the following quadratic
differentials

$\mathopen{}\mathclose{{}\left[\partial_{z}\mathcal{T}_{0}\mathopen{}\mathclose{{}\left(z}\right)\mathrm{d}z}\right]^{2}=-\mathopen{}\mathclose{{}\left[\frac{y\mathopen{}\mathclose{{}\left(z}\right)}{z}\mathrm{d}z}\right]^{2},$
(4.74)

and at the same time relate ${\tt g}$ to $u$ by

${\tt g}^{2}+{\tt g}^{-2}={u\over 4\Lambda^{2}}\,,\qquad\qquad\qquad{\tt
g}^{\pm
2}=\mathopen{}\mathclose{{}\left(\frac{u}{8\Lambda^{2}}}\right)\mp\sqrt{\mathopen{}\mathclose{{}\left(\frac{u}{8\Lambda^{2}}}\right)^{2}-1}\,.$
(4.75)

In equations (4.73) and (4.75) $u$ denotes as usual the vacuum expectation
value of the scalar in the vector multiplet of $\operatorname{SU(2)}2$ SYM.

###  4.2 The planar two-point function

In the previous section we showed that the Seiberg-Witten curve (4.73)
naturally emerges when considering the planar resolvent. Here we will see that
similarly the Bergman kernel emerges when considering the even part of the
planar two-point function. We will see later that this characterises the
annulus amplitude in the surface defect. The Bergman kernel is defined as
Awata:2010bz

$B_{q_{1},q_{2},q_{3}}(z_{1},z_{2})={1\over
2(z_{1}-z_{2})^{2}}\mathopen{}\mathclose{{}\left({2f(z_{1},z_{2})+G_{q_{1},q_{2},q_{3}}\mathopen{}\mathclose{{}\left(k^{2}}\right)(z_{1}-z_{2})^{2}\over
2\sqrt{\sigma(z_{1})\sigma(z_{2})}}+1}\right)\,,$ (4.76)

with

$f(z_{1},z_{2})=\frac{u\mathopen{}\mathclose{{}\left(z_{1}^{2}+4z_{2}z_{1}+z_{2}^{2}}\right)}{24\Lambda^{2}}+\frac{1}{2}\mathopen{}\mathclose{{}\left(-z_{1}-z_{2}}\right)-\frac{1}{2}z_{1}z_{2}\mathopen{}\mathclose{{}\left(z_{1}+z_{2}}\right)\,,$
(4.77)
$G_{q_{1},q_{2},q_{3}}\mathopen{}\mathclose{{}\left(k^{2}}\right)=\mathopen{}\mathclose{{}\left(q_{1}-q_{3}}\right)\mathopen{}\mathclose{{}\left[\frac{\mathrm{E}\mathopen{}\mathclose{{}\left(k^{2}}\right)}{\mathrm{K}\mathopen{}\mathclose{{}\left(k^{2}}\right)}}\right]+q_{3}-\frac{u}{12\Lambda^{2}}\,,\qquad\qquad
k^{2}=\frac{q_{1}-q_{2}}{q_{1}-q_{3}}\,,$ (4.78)

and where $q_{i}$ are the branch points of
$\sigma(z)=-z\mathopen{}\mathclose{{}\left(z^{2}-\mathopen{}\mathclose{{}\left(u/4\Lambda^{2}}\right)z+1}\right)$,

$q_{i}\in\mathopen{}\mathclose{{}\left\\{0,\mathopen{}\mathclose{{}\left(\frac{u}{8\Lambda^{2}}}\right)\mp\sqrt{\mathopen{}\mathclose{{}\left(\frac{u}{8\Lambda^{2}}}\right)^{2}-1}}\right\\}=\mathopen{}\mathclose{{}\left\\{0,{\tt
g}^{2},{\tt g}^{-2}}\right\\}\,.$ (4.79)

The choice of the order fixes the choice of frame. What we find is that the
relevant order here is

$q_{3}=0\,,\qquad
q_{2}=\mathopen{}\mathclose{{}\left(\frac{u}{8\Lambda^{2}}}\right)-\sqrt{\mathopen{}\mathclose{{}\left(\frac{u}{8\Lambda^{2}}}\right)^{2}-1}={\tt
g}^{2}\,,\qquad
q_{1}=\mathopen{}\mathclose{{}\left(\frac{u}{8\Lambda^{2}}}\right)+\sqrt{\mathopen{}\mathclose{{}\left(\frac{u}{8\Lambda^{2}}}\right)^{2}-1}={\tt
g}^{-2}\,.$ (4.80)

As we will discuss later this choice makes contact with the magnetic frame.
One can check that the even part of the planar two-point function (4.50) is
related to the Bergman kernel (4.76) by

$W_{++}^{0}\mathopen{}\mathclose{{}\left(z_{1},z_{2}}\right)=-z_{1}z_{2}B_{{q}_{1},{q}_{2},{q}_{3}}\mathopen{}\mathclose{{}\left(z_{1}^{2},z_{2}^{2}}\right)-\frac{1}{4\mathopen{}\mathclose{{}\left(z_{1}+z_{2}}\right)^{2}}\,,$
(4.81)

Hence the subleading order $\mathcal{T}_{1}$ (4.49) of the matrix model (4.47)
in the ’t Hooft limit (4.40) becomes

$\boxed{\mathcal{T}_{1}\mathopen{}\mathclose{{}\left(z}\right)=2\int_{\infty}^{z}\int_{\infty}^{z}{\rm
d}z_{1}{\rm
d}z_{2}\mathopen{}\mathclose{{}\left(-z_{1}z_{2}B_{{q}_{1},{q}_{2},{q}_{3}}\mathopen{}\mathclose{{}\left(z_{1}^{2},z_{2}^{2}}\right)-\frac{1}{4\mathopen{}\mathclose{{}\left(z_{1}+z_{2}}\right)^{2}}}\right)}\;.$
(4.82)

##  5 Testing the $\epsilon$ expansion for the type I defect

From the perspective of the B-model, the partition functions of open and
closed topological strings can be defined as objects associated to an
algebraic curve, and thus, they depend on a choice frame, namely a choice of a
symplectic basis for the homology of the algebraic curve. The transformation
properties of the closed string partition function under a change of frame can
be derived from the observation that such a partition function behaves like a
wavefunction Witten:1993ed . Consequently, the genus $g$ free energies behave
as almost modular forms under a change of frame abk . The wavefunction
behaviour was generalized to the open topological string sector in
Grassi:2013qva .

Recall that the partition functions of the four-dimensional gauge theories
under consideration are derived from the topological string partition
functions via the geometric engineering construction kkv ; Iqbal:2003zz ;
Klemm:1996bj . As a result the same transformation properties hold.

At the level of terminology, the large radius frame in topological string
theory is mapped to the electric frame in the four-dimensional theory. In this
frame the $\mathcal{A}$ cycle and the corresponding A period on the SW curve
(4.73) are chosen to be

$\Pi_{A}={1\over 2\pi{\rm i}}\oint_{\mathcal{A}}{y(z)\over z}{\rm
d}z={1\over\mathrm{i}\pi}\int_{-{\rm i}\pi}^{{\rm i}\pi}{y(z)\over z}{\rm
d}z=\frac{2\sqrt{8\Lambda^{2}+u}~{}{\mathrm{E}}\mathopen{}\mathclose{{}\left(\frac{16\Lambda^{2}}{8\Lambda^{2}+u}}\right)}{\pi}\,,$
(5.83)

where $y(z)$ is given in (4.73) and $\mathrm{E}$ is the complete elliptic
integral of second kind (C.202). We usually denote $a\equiv\Pi_{A}$. Likewise
the $\mathcal{B}$ cycle and the corresponding period are

$\Pi_{B}={1\over 2\pi{\rm i}}\oint_{\mathcal{B}}{y(z)\over z}{\rm
d}z={1\over{\rm i}\pi}\int_{{\tt g}}^{{\tt g}^{-1}}{y(z)\over z}{\rm d}z\\\
={\sqrt{8\Lambda^{2}+u}\over{\rm
i}\pi}\mathopen{}\mathclose{{}\left({\mathrm{K}}\mathopen{}\mathclose{{}\left(\frac{2u}{8\Lambda^{2}+u}-1}\right)-{\mathrm{E}}\mathopen{}\mathclose{{}\left(\frac{2u}{8\Lambda^{2}+u}-1}\right)}\right)$
(5.84)

where $\mathrm{K}$ is the complete elliptic integral of the first kind
(C.201). The ${\tt g}^{\pm 1}$ are roots of the SW curve,
$y\mathopen{}\mathclose{{}\left({\tt g}^{\pm 1}}\right)=0$, and are given in
(4.75). We usually denote $a_{D}\equiv{\rm i}\Pi_{B}$.

On the other hand, the conifold frame in topological strings corresponds to
the magnetic frame in the four-dimensional theory. This frame is related to
the electric frame by an S-duality which exchanges the $\mathcal{A}$\- and
$\mathcal{B}$-cycles. For the $\operatorname{SU(2)}2$ SYM that we study in
this paper, the transformation properties of the partition function under a
change of frame were studied in hk06 . The $\epsilon$ expansion of (3.24)
leads exactly to such transformations, as we discuss below.

###  5.1 The electric frame

We consider a type I defect in the self-dual phase of the $\Omega$-background
($\epsilon_{1}=-\epsilon_{2}=\epsilon$), and we denote the partition function
of this surface defect by

$Z^{\rm I}(z,t,\sigma)$ (5.85)

if we are in the electric frame. As pointed out in Kozcaz:2010af , based on
bkmp ; Marino:2006hs , we can compute these defects via the Eynard-Orantin
topological recursion eo . More precisely we have

$\log\mathopen{}\mathclose{{}\left(Z^{\rm
I}(z,t,\sigma)}\right)\simeq\sum_{g\geqslant 0}\sum_{h\geqslant
1}{\epsilon^{2g-2+h}}\int^{z}_{\infty}\cdots\int^{z}_{\infty}W_{g,h}(z_{1},\ldots,z_{h})\
{\rm d}z_{1}\cdots{\rm d}z_{h}$ (5.86)

where $W_{g,h}(z_{1},...z_{h}){\rm d}z_{1}\dots\rm dz_{h}$ is an infinite
sequence of meromorphic differentials constructed via the topological
recursion eo  and whose starting point is the underling SW geometry (4.73).
Note that we are implicitly using the dictionary (3.19) and the SW relation
(5.83) to express $\sigma=\mathrm{i}a/2\epsilon$ as a function of the SW
parameter $u$.

For the so-called disk amplitude ${W}_{0,1}$ we have191919Here we take $z>{\tt
g}^{-1}>0$ to simplify the discussion and to make contact with the standard
formulas in the literature Kozcaz:2010af ; Awata:2010bz . Hence on the matrix
model side we are outside the branch cut.

${W}_{0,1}(z)\ {\rm
d}z=-\frac{2\Lambda}{z}\sqrt{\mathopen{}\mathclose{{}\left(z^{2}+\frac{1}{z^{2}}}\right)-{u\over
4\Lambda^{2}}}\ {\rm d}z\,,$ (5.87)

and we note

$\mathcal{W}_{0}^{\rm I}(z,\Lambda,u)=\int^{z}_{\infty}{W}_{0,1}(z_{1})\ {\rm
d}z_{1}\,.$ (5.88)

The annulus amplitude ${W}_{0,2}$ is given by

${W}_{0,2}(z_{1},z_{2})\ {\rm d}z_{1}{\rm
d}z_{2}=-2z_{1}z_{2}\mathopen{}\mathclose{{}\left(B_{\widetilde{q}_{1},\widetilde{q}_{2},\widetilde{q}_{3}}(z_{1}^{2},z_{2}^{2})-{1\over(z_{1}^{2}-z_{2}^{2})^{2}}}\right)\
{\rm d}z_{1}{\rm d}z_{2}\,,$ (5.89)

where $B_{\widetilde{q}_{1},\widetilde{q}_{2},\widetilde{q}_{3}}$ is defined
as in (4.76), but the choice of $q_{i}$’s is different. Here we have

$\widetilde{q_{1}}=0\,,\qquad\widetilde{q_{2}}=\mathopen{}\mathclose{{}\left(\frac{u}{8\Lambda^{2}}}\right)-\sqrt{\mathopen{}\mathclose{{}\left(\frac{u}{8\Lambda^{2}}}\right)^{2}-1}={\tt
g}^{2}\,,\qquad\widetilde{q_{3}}=\mathopen{}\mathclose{{}\left(\frac{u}{8\Lambda^{2}}}\right)+\sqrt{\mathopen{}\mathclose{{}\left(\frac{u}{8\Lambda^{2}}}\right)^{2}-1}={\tt
g}^{-2}\,,$ (5.90)

so that $\widetilde{q_{1}}=q_{3}$, $\widetilde{q_{2}}=q_{2}$ and
$\widetilde{q_{3}}=q_{1}$ (4.80). We denote

$\mathcal{W}_{1}^{\rm
I}(z,\Lambda,u)=\int^{z}_{\infty}\int^{z}_{\infty}{W}_{0,2}(z_{1},z_{2})\ {\rm
d}z_{1}{\rm d}z_{2}\,.$ (5.91)

Hence to subleading order (5.86) reads

$\log\mathopen{}\mathclose{{}\left(Z^{\rm
I}(z,t,\sigma)}\right)={{1\over\epsilon}\mathcal{W}_{0}^{\rm
I}(z,\Lambda,u)+\mathcal{W}_{1}^{\rm
I}(z,\Lambda,u)+\mathcal{O}(\epsilon)}\,.$ (5.92)

Given the spectral curve (4.73) with $W_{0,1}$ and $W_{0,2}\,$, higher order
terms in the $\epsilon$ expansion (5.86) can be computed recursively by using
the topological recursion eo .

###  5.2 The magnetic frame

Our proposal is that the matrix model (3.27) computes the type I surface
defect (5.85) in the magnetic frame. In this section we test this proposal in
the ’t Hooft expansion (4.40).

####  5.2.1 The partition function without defects

It is useful to start by reviewing the change of frame in the partition
function without defect, which follows from the $\epsilon$ expansion of
(3.24). Using the dictionary (3.19) the $\epsilon$ expansion of the Nekrasov
function reads

$\log\mathopen{}\mathclose{{}\left(Z^{\rm
Nek}(t,\sigma)}\right)\simeq\sum_{g\geqslant
0}\epsilon^{2g-2}\mathcal{F}_{g}(\Lambda,a)\,,$ (5.93)

where $\mathcal{F}_{g}$ are the genus $g$ free energies of
$\operatorname{SU(2)}2$ SYM in the electric frame. Thanks to (3.24) we can
relate (5.93) to the ’t Hooft expansion of the matrix model (3.23),

$\log\mathopen{}\mathclose{{}\left(Z(t,N)}\right)\simeq\sum_{g\geqslant
0}\epsilon^{2g-2}F_{g}(\Lambda,\lambda)\,.$ (5.94)

It was found in bgt  that the $F_{g}$’s in (5.94) are the SYM free energies in
the magnetic frame. More precisely

${\rm e}^{\sum_{g\geqslant
0}\epsilon^{2g-2}F_{g}(\Lambda,\lambda)}\sim\int_{{\rm i}{\mathbb{R}}}{{\rm
d}a}\ {\rm e}^{-\pi aN/\epsilon}\ {\rm e}^{\sum_{g\geqslant
0}\epsilon^{2g-2}\mathcal{F}_{g}(\Lambda,a)}\,,$ (5.95)

where $\sim$ indicates a proportionality between two (divergent)
series202020For sake of notation we omitted the proportionality factor
$2^{1/12}{\rm
e}^{3\zeta^{\prime}(-1)}\mathopen{}\mathclose{{}\left(\Lambda/\epsilon}\right)^{-1/4}{\rm
e}^{4\mathopen{}\mathclose{{}\left(\Lambda/\epsilon}\right)^{2}}$. The
integral on the rhs of (5.95) should be understood as a saddle point
expansion. This saddle point expansion characterizes the change of frame in SW
theory and topological string Witten:1993ed , and it has a direct
interpretation from the point of view of modular transformations abk . It
allows us to make the transition from the weak coupling electric frame, where
the Nekrasov function (3.18) is defined, to the strong coupling magnetic
frame, where the matrix model (3.23) naturally emerges. By writing the saddle
point expansion on the rhs of (5.95) explicitly we get

$\int_{{\rm i}{\mathbb{R}}}{{\rm d}a}\ {\rm e}^{-\pi aN/\epsilon}\ {\rm
e}^{\sum_{g\geqslant
0}\epsilon^{2g-2}\mathcal{F}_{g}(\Lambda,a)}=\exp\mathopen{}\mathclose{{}\left[\frac{1}{\epsilon^{2}}\mathopen{}\mathclose{{}\left({\mathcal{F}}_{0}\mathopen{}\mathclose{{}\left(\Lambda,a(\lambda)}\right)-{\pi
a(\lambda)\lambda}}\right)+\mathcal{O}(1)}\right]\,,$ (5.96)

where $\lambda=N\epsilon$ and $a(\lambda)$ is determined by the saddle point
equation

$\partial_{a}{\mathcal{F}}_{0}\mathopen{}\mathclose{{}\left(\Lambda,a}\right)=\pi\lambda\,.$
(5.97)

By using (3.18) and the dictionary (3.19) we get

$\partial_{a}{\mathcal{F}}_{0}(\Lambda,a)=2a(\log(a)-\log(\Lambda)-1)\\\
+\frac{4\Lambda^{4}}{a^{3}}+\frac{30\Lambda^{8}}{a^{7}}+\frac{480\Lambda^{12}}{a^{11}}+\frac{10283\Lambda^{16}}{a^{15}}+\frac{1287648\Lambda^{20}}{5a^{19}}+\mathcal{O}\mathopen{}\mathclose{{}\left(\Lambda^{24}}\right)\,,$
(5.98)

and if we make use of (4.75) together with the classical Matone relation,

$u=-\Lambda\partial_{\Lambda}\mathcal{F}_{0}\mathopen{}\mathclose{{}\left(\Lambda,a}\right)=a^{2}+\frac{8\Lambda^{4}}{a^{2}}+\frac{40\Lambda^{8}}{a^{6}}+\frac{576\Lambda^{12}}{a^{10}}+\frac{11752\Lambda^{16}}{a^{14}}+\frac{286144\Lambda^{20}}{a^{18}}+\mathcal{O}\mathopen{}\mathclose{{}\left(\Lambda^{24}}\right)\,,$
(5.99)

we find that $\lambda$ in (5.97) agrees with (4.42) as it should. The matching
between the two sides of (5.95) was discussed in bgt . We also note that the
classical Matone relation (5.99) can be inverted and one finds the usual
expression for the $A$-period of the SW curve (4.73) given in (5.83). Likewise
$\partial_{a}\mathcal{F}_{0}$ is identified with the $B$-period of the SW
curve212121Hence we have $a_{D}=\lambda$.

$a_{D}={\pi^{-1}}\partial_{a}\mathcal{F}_{0}\mathopen{}\mathclose{{}\left(\Lambda,a}\right),$
(5.100)

where $a_{D}$ is given in (5.84).

####  5.2.2 The partition function with defects

We are interested in extending the analysis to the ’t Hooft expansion (4.47)
of the matrix model with insertion $\Psi_{N}(z,t)$ (3.27). More precisely we
claim that $\Psi_{N}(z,t)$ gives the self-dual type I surface defect (5.85) in
the magnetic frame. As we reviewed above, the change of frame for the
partition function is encoded in an integral transform (5.95). As first shown
in Grassi:2013qva , this is still the case if one considers the partition
function in the presence of surface defects which are engineered via the open
topological string partition function, see also Ashok:2017odt .

At the level of the $\epsilon$ expansion our conjecture reads

$\exp\mathopen{}\mathclose{{}\left[\sum_{g\geqslant
0}\epsilon^{2g-2}F_{g}(\Lambda,\lambda)+\sum_{n\geqslant 0}\epsilon^{n-1}{\cal
T}_{n}(z)}\right]\\\ \sim\int_{{\rm i}{\mathbb{R}}}{{\rm d}a}\ {\rm e}^{-\pi
aN/\epsilon}\ {\rm e}^{\sum_{g\geqslant
0}\epsilon^{2g-2}\mathcal{F}_{g}(\Lambda,a)}{\rm e}^{\sum_{g\geqslant
0}\sum_{h\geqslant
1}{\epsilon^{2g-2+h}}\int^{z}_{\infty}\cdots\int^{z}_{\infty}W_{g,h}(z_{1},\ldots,z_{h}){\rm
d}z_{1}\cdots{\rm d}z_{h}}\,,$ (5.101)

where $W_{g,h}(z_{1},\ldots,z_{h}){\rm d}z_{1}\cdots{\rm d}z_{h}$ are the
electric differentials appearing in the topological recursion setup (5.86);
whereas ${\cal T}_{n}$ are the magnetic matrix model coefficients appearing in
(4.47),

$\mathopen{}\mathclose{{}\left({{\sqrt{v(z)}}\Psi_{N}(z,t)\over
Z(t,N)}}\right)\simeq\exp\mathopen{}\mathclose{{}\left[\sum_{n\geqslant
0}\epsilon^{n-1}{\cal T}_{n}(z)}\right]\,.$ (5.102)

Parallel to (5.95), the integral on the rhs of (5.101) should be understood as
a saddle point expansion which characterizes the change of frame. Equation
(5.101) reads to subleading order in $\epsilon$

$\epsilon^{-1}\mathcal{T}_{0}(z)+\mathcal{T}_{1}(z)+\mathcal{O}\mathopen{}\mathclose{{}\left(\epsilon}\right)=\epsilon^{-1}{\mathcal{W}}_{0}^{\rm
I}\mathopen{}\mathclose{{}\left(z,a(\lambda)}\right)-\frac{\mathopen{}\mathclose{{}\left[\partial_{a}{\mathcal{W}}_{0}^{\rm
I}\mathopen{}\mathclose{{}\left(z,a(\lambda)}\right)}\right]^{2}}{2\partial_{a}^{2}\mathcal{F}_{0}\mathopen{}\mathclose{{}\left(a(\lambda)}\right)}+{\mathcal{W}}_{1}^{\rm
I}(z,a(\lambda))+\mathcal{O}\mathopen{}\mathclose{{}\left(\epsilon}\right)\,,$
(5.103)

where $a$ and $\lambda$ are again related by the saddle point equation (5.97).
In (5.103) we already used (5.95) and (5.96) to get rid of terms involving
only the free energies $F_{g}$ and $\mathcal{F}_{g}$. We show below that the
equality in (5.103) indeed holds order by order in $\epsilon$.

At the leading order $\epsilon^{-1}$, the matching on the two sides of (5.103)
follows directly from (4.72) and (5.87). For the subleading order
$\epsilon^{0}$, we first note that the Bergman kernel entering in
$\mathcal{T}_{1}$ (4.82), can be written as

$B_{q_{1},q_{2},q_{3}}(z_{1},z_{2})=\\\
B_{\widetilde{q}_{1},\widetilde{q}_{2},\widetilde{q}_{3}}(z_{1},z_{2})+\frac{\pi\mathopen{}\mathclose{{}\left[{z_{1}z_{2}\mathopen{}\mathclose{{}\left({\tt
g}^{4}z_{1}-{\tt
g}^{2}\mathopen{}\mathclose{{}\left(z_{1}^{2}+1}\right)+z_{1}}\right)\mathopen{}\mathclose{{}\left({\tt
g}^{4}z_{2}-{\tt
g}^{2}\mathopen{}\mathclose{{}\left(z_{2}^{2}+1}\right)+z_{2}}\right)}}\right]^{-1/2}}{8\,\mathrm{K}\mathopen{}\mathclose{{}\left({\tt
g}^{4}}\right)\mathrm{K}\mathopen{}\mathclose{{}\left(1-{\tt g}^{4}}\right)}$
(5.104)

where we used (4.75) and (5.90). Hence we can rewrite (4.82),

$\begin{split}\mathcal{T}_{1}\mathopen{}\mathclose{{}\left(z}\right)&=2\int_{\infty}^{z}\int_{\infty}^{z}\mathopen{}\mathclose{{}\left(-z_{1}z_{2}B_{{q}_{1}{q}_{2}{q}_{3}}\mathopen{}\mathclose{{}\left(z_{1}^{2},z_{2}^{2}}\right)-\frac{1}{4\mathopen{}\mathclose{{}\left(z_{1}+z_{2}}\right)^{2}}}\right){\rm
d}z_{1}{\rm d}z_{2}\\\ &=\mathcal{W}_{1}^{\rm
I}\mathopen{}\mathclose{{}\left(z}\right)-\int^{z}_{\infty}\int^{z}_{\infty}\mathopen{}\mathclose{{}\left(\frac{\pi{\tt
g}^{4}}{16\mathopen{}\mathclose{{}\left({\tt
g}^{4}-1}\right)^{2}{\Lambda\mathstrut}^{2}}\frac{\partial_{{\tt
g}}\partial_{z_{1}}\mathcal{T}_{0}(z_{1})\partial_{{\tt
g}}\partial_{z_{2}}\mathcal{T}_{0}(z_{2})}{\mathrm{K}\mathopen{}\mathclose{{}\left({\tt
g}^{4}}\right)\mathrm{K}\mathopen{}\mathclose{{}\left(1-{\tt
g}^{4}}\right)}}\right){\rm d}z_{1}{\rm d}z_{2}\,,\end{split}$ (5.105)

which leads then to

$\displaystyle\mathcal{T}_{1}\mathopen{}\mathclose{{}\left(z}\right)=$
$\displaystyle\,\mathcal{W}_{1}^{\rm
I}\mathopen{}\mathclose{{}\left(z}\right)-\frac{\pi{\tt
g}^{4}}{16\mathopen{}\mathclose{{}\left({\tt
g}^{4}-1}\right)^{2}{\Lambda\mathstrut}^{2}}\frac{\mathopen{}\mathclose{{}\left(\partial_{{\tt
g}}\mathcal{T}_{0}(z)}\right)^{2}}{\mathrm{K}\mathopen{}\mathclose{{}\left({\tt
g}^{4}}\right)\mathrm{K}\mathopen{}\mathclose{{}\left(1-{\tt g}^{4}}\right)}$
(5.106) $\displaystyle=$ $\displaystyle\,\mathcal{W}_{1}^{\rm
I}\mathopen{}\mathclose{{}\left(z}\right)-\mathopen{}\mathclose{{}\left(\partial_{a}\mathcal{T}_{0}(z)}\right)^{2}\frac{\mathrm{K}\mathopen{}\mathclose{{}\left(\frac{4{\tt
g}^{2}}{\mathopen{}\mathclose{{}\left({\tt
g}^{2}+1}\right)^{2}}}\right)^{2}}{\pi\mathopen{}\mathclose{{}\left({\tt
g}^{2}+1}\right)^{2}\mathrm{K}\mathopen{}\mathclose{{}\left({\tt
g}^{4}}\right)\mathrm{K}\mathopen{}\mathclose{{}\left(1-{\tt
g}^{4}}\right)}\,,$

where we used (5.83). From (5.100) we have

$\partial_{a}^{2}\mathcal{F}_{0}\mathopen{}\mathclose{{}\left(a}\right)=\frac{\pi{\mathrm{K}}\mathopen{}\mathclose{{}\left(\frac{2u}{8\Lambda^{2}+u}-1}\right)}{{\mathrm{K}}\mathopen{}\mathclose{{}\left(\frac{16\Lambda^{2}}{8\Lambda^{2}+u}}\right)}\,,$
(5.107)

and by combining (5.107) with the identity

$\frac{\mathrm{K}\mathopen{}\mathclose{{}\left(\frac{4{\tt
g}^{2}}{\mathopen{}\mathclose{{}\left({\tt
g}^{2}+1}\right)^{2}}}\right)}{\mathopen{}\mathclose{{}\left({\tt
g}^{2}+1}\right)^{2}\mathrm{K}\mathopen{}\mathclose{{}\left({\tt
g}^{4}}\right)\mathrm{K}\mathopen{}\mathclose{{}\left(1-{\tt
g}^{4}}\right)}=\frac{1}{2\mathrm{K}\mathopen{}\mathclose{{}\left(\frac{\mathopen{}\mathclose{{}\left({\tt
g}^{2}-1}\right)^{2}}{\mathopen{}\mathclose{{}\left({\tt
g}^{2}+1}\right)^{2}}}\right)}$ (5.108)

we find

$\boxed{\mathcal{T}_{1}\mathopen{}\mathclose{{}\left(z}\right)=\mathcal{W}_{1}^{\rm
I}\mathopen{}\mathclose{{}\left(z}\right)-{\mathopen{}\mathclose{{}\left(\partial_{a}\mathcal{T}_{0}(z)}\right)^{2}\over
2\partial_{a}^{2}\mathcal{F}_{0}}}\,,$ (5.109)

which is precisely what we wanted to prove.

To summarize, we have tested (5.101) at leading and subleading order222222For
the part involving only the free energies the all order results follows from
bgt ; bgt2 . in $\epsilon$. The matching of higher orders can be inferred from
the application of topological recursion. On the canonical defect side, the
fact that higher orders in (5.85) satisfy the topological recursion was
conjectured in Kozcaz:2010af , based on bkmp ; Marino:2006hs  which was
recently demonstrated in Eynard:2012nj . On the matrix model side instead, the
inclusion of topological recursion in our matrix model can be derived from eo
; Borot:2009ia ; Borot:2013lpa . Our computations above shows that the initial
data for such recursion are the same on both sides, therefore matching at all
orders is also expected.

##  6 Matrix models, eigenfunctions and the type II defect

In this section we consider the Fourier transform of the matrix model with
insertion $\Psi_{N}\mathopen{}\mathclose{{}\left(\mathrm{e}^{x},t}\right)$
(3.27). The corresponding defect in four-dimensional, $\mathcal{N}=2$,
$\operatorname{SU(2)}2$ SYM can be geometrically engineered using the open
topological string partition function of local ${\mathbb{F}}_{0}$, where we
insert a D-brane on the external leg, see Appendix A. The partition function
of the resulting type II defect in the self-dual phase of the
$\Omega$-background is

$\begin{split}Z^{\rm
II}(q,t,\sigma)&=\exp\mathopen{}\mathclose{{}\left(\frac{\mathrm{i}}{2}q\log\mathopen{}\mathclose{{}\left(t}\right)}\right)\Gamma\mathopen{}\mathclose{{}\left(-\mathrm{i}q-\sigma+\frac{1}{2}}\right)\Gamma\mathopen{}\mathclose{{}\left(-\mathrm{i}q+\sigma+\frac{1}{2}}\right)Z_{\rm
inst}^{\rm II}(q,t,\sigma)\,,\\\ Z_{\rm
inst}^{\text{II}}\mathopen{}\mathclose{{}\left(q,t,\sigma}\right)&=1-\mathopen{}\mathclose{{}\left[\frac{\widetilde{q}}{2\sigma^{2}\mathopen{}\mathclose{{}\left(\widetilde{q}^{2}-\sigma^{2}}\right)}}\right]t\\\
&+\mathopen{}\mathclose{{}\left[\frac{\widetilde{q}\mathopen{}\mathclose{{}\left(\widetilde{q}+1}\right)^{2}-\widetilde{q}\mathopen{}\mathclose{{}\left(10\widetilde{q}^{2}+19\widetilde{q}+10}\right)\sigma^{2}+\mathopen{}\mathclose{{}\left(8\widetilde{q}^{2}+30\widetilde{q}+9}\right)\sigma^{4}}{4\sigma^{4}\mathopen{}\mathclose{{}\left(4\sigma^{2}-1}\right)^{2}\mathopen{}\mathclose{{}\left(\widetilde{q}^{2}-\sigma^{2}}\right)\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\widetilde{q}+1}\right)^{2}-\sigma^{2}}\right)}}\right]t^{2}+\mathcal{O}\mathopen{}\mathclose{{}\left(t^{3}}\right)\,,\end{split}$
(6.110)

where we defined for the sake of readability $\widetilde{q}=\mathrm{i}q+1/2$.
The variables $q,t,\sigma$ can be expressed in terms of $y,\Lambda,a,\epsilon$
as in (2.6). The relation between $Z^{\rm II}$ and the matrix model (3.27)
reads232323This also suggests that the defect partition function $Z^{\rm
II}\mathopen{}\mathclose{{}\left(q,t,\sigma}\right)$ for this theory has an
infinite radius of convergence in $t$ for fixed $\sigma$ and $q$ which are
away from the poles. For the self-dual partition function without defect this
was rigorously proven in ilt , see also Arnaudo:2022ivo  for more recent
developments.

$\begin{split}-\mathrm{i}\,2^{1/12}{\rm e}^{3\zeta^{\prime}(-1)}t^{-1/16}{\rm
e}^{4\sqrt{t}}&\int_{{\mathbb{R}}+{\rm i}\sigma_{*}}{{\rm
d}\sigma}{\tan\mathopen{}\mathclose{{}\left(2\pi\sigma}\right)\over\mathopen{}\mathclose{{}\left(2\cos(2\pi\sigma)}\right)^{N}}Z^{\rm
Nek}(t,\sigma)\mathopen{}\mathclose{{}\left(\sum_{s}Z^{\rm
II}_{s}(q,t,\sigma)}\right)\\\
&={2\sqrt{\pi}{t^{1/8}}\over(4\pi)^{N}}\int_{\mathbb{R}}{{\rm d}x}\,{\rm
e}^{-{\rm i}{2}qx}{\rm
e}^{-4t^{1/4}\cosh\mathopen{}\mathclose{{}\left(x}\right)+\frac{x}{2}}\Psi_{N}({\rm
e}^{{x}},t)\,,\end{split}$ (6.111)

where $\sigma_{*}$ is chosen such that $0<\sigma_{*}<|{\rm Re}(q)|$ if
$\mathrm{Re}\mathopen{}\mathclose{{}\left(q}\right)\neq 0$, and simply
$\sigma_{*}>0$ if $\mathrm{Re}\mathopen{}\mathclose{{}\left(q}\right)=0$. This
guarantees that the integral on the lhs does not hits the poles of the
integrand. The sum over $s$ can be seen as a sum over saddle points of the
integral over $x$. We find that

$\mathopen{}\mathclose{{}\left(\sum_{s}Z^{\rm
II}_{s}(q,t,\sigma)}\right)=Z^{\rm
II}\mathopen{}\mathclose{{}\left(q,t,\sigma}\right)+\mathopen{}\mathclose{{}\left(-1}\right)^{N}Z^{\rm
II}\mathopen{}\mathclose{{}\left(-q-{{\rm i}\over 2},t,\sigma}\right)\,.$
(6.112)

It is convenient to introduce the total partition function as

$Z^{\rm II}_{\rm tot}(q,t,\sigma)=Z^{\rm Nek}(t,\sigma)Z^{\rm
II}(q,t,\sigma)\,,$ (6.113)

so that (6.111) can be written in a compact form as

$\boxed{\begin{aligned} &\int_{{\mathbb{R}}+{\rm i}\sigma_{*}}{{\rm
d}\sigma}{\tan\mathopen{}\mathclose{{}\left(2\pi\sigma}\right)\over\mathopen{}\mathclose{{}\left(2\cos(2\pi\sigma)}\right)^{N}}\mathopen{}\mathclose{{}\left(Z^{\rm
II}_{\rm tot}(q,t,\sigma)+(-1)^{N}Z^{\rm II}_{\rm
tot}\mathopen{}\mathclose{{}\left(-q-{{\rm i}\over
2},t,\sigma}\right)}\right)\\\ &={\rm
i}{2^{11/12}\sqrt{\pi}{t^{3/16}}\over{\rm e}^{3\zeta^{\prime}(-1)}{\rm
e}^{4\sqrt{t}}(4\pi)^{N}}\int_{\mathbb{R}}{{\rm d}x}\,{\rm e}^{-{\rm
i}{2}qx}{\rm e}^{-4t^{1/4}\cosh x+{x\over 2}}\Psi_{N}({\rm
e}^{{x}},t)\,.\end{aligned}}$ (6.114)

This equality can be equivalently written as

$\displaystyle\sum_{k\in{\mathbb{Z}}}\mathopen{}\mathclose{{}\left(Z^{\rm
II}_{\rm tot}(q,t,\sigma+k)+Z^{\rm II}_{\rm
tot}\mathopen{}\mathclose{{}\left(-q-{{\rm i}\over 2},t,\sigma+k+{1\over
2}}\right)}\right)$ (6.115)
$\displaystyle={2^{11/12}\sqrt{\pi}{t^{3/16}}\over{\rm
e}^{3\zeta^{\prime}(-1)}{\rm e}^{4\sqrt{t}}}\int_{\mathbb{R}}{{\rm d}x}\,{\rm
e}^{-{\rm i}{2}qx}{\rm e}^{-4t^{1/4}\cosh x+{x\over
2}}\mathopen{}\mathclose{{}\left[\sum_{N\geqslant 0}\Psi_{N}({\rm
e}^{{x}},t)\mathopen{}\mathclose{{}\left(\frac{\cos(2\pi\sigma)}{2\pi}}\right)^{N}}\right]\,.$

One can use an argument based on Fourier series to get from (6.114) to
(6.115), while the other direction uses Cauchy’s residue theorem. See Appendix
B for details. By inverting the Fourier transform (6.115) we have

$\boxed{\begin{aligned} &\int_{{\mathbb{R}}}{\rm d}q\ {\rm e}^{{\rm
i}{2}qx}\sum_{k\in{\mathbb{Z}}}\mathopen{}\mathclose{{}\left(Z^{\rm II}_{\rm
tot}\mathopen{}\mathclose{{}\left(q,t,\sigma+k}\right)+Z^{\rm II}_{\rm
tot}\mathopen{}\mathclose{{}\left(-q-{{\rm i}\over 2},t,\sigma+k+{1\over
2}}\right)}\right)\\\ &={2^{11/12}\pi^{3/2}t^{3/16}\over{\rm
e}^{3\zeta^{\prime}(-1)}{\rm e}^{4\sqrt{t}}}{\rm e}^{-4t^{1/4}\cosh x+{x\over
2}}\sum_{N\geqslant 0}\Psi_{N}({\rm
e}^{{x}},t)\mathopen{}\mathclose{{}\left({\cos(2\pi\sigma)\over
2\pi}}\right)^{N}\end{aligned}}$ (6.116)

Following subsection 3.2, we get the square-integrable eigenfunctions of
(3.14) when we evaluate (6.116) at the values of $\sigma$ which satisfy the
quantization condition (3.17). That is

$\boxed{\begin{aligned} \varphi_{n}(x,t)=&{{\rm e}^{3\zeta^{\prime}(-1)}{\rm
e}^{4\sqrt{t}}\over 2^{11/12}\pi^{3/2}t^{3/16}}\\\ &\int_{{\mathbb{R}}}{\rm
d}q\,{\rm e}^{{\rm
i}{2}qx}\sum_{k\in{\mathbb{Z}}}\mathopen{}\mathclose{{}\left(Z^{\rm II}_{\rm
tot}\mathopen{}\mathclose{{}\left(q,t,k+{1\over 2}+{\rm
i}\sigma_{n}}\right)+Z^{\rm II}_{\rm
tot}\mathopen{}\mathclose{{}\left(-q-{{\rm i}\over 2},t,k+{\rm
i}\sigma_{n}}\right)}\right)\end{aligned}}$ (6.117)

where $\sigma_{n}$ are solutions of (3.17). In Figure 1 we plot the rhs of
(6.117) for the two smallest values of $\sigma_{n}$ which satisfy the
quantization condition (3.17). As a cross-check we also verified this result
by a purely numerical analysis of the operator (3.14), see subsection 6.4.

Let us make a few comments on the analytic properties of the gauge theoretic
functions.

  * -

The function $Z^{\rm Nek}(t,\sigma)$ has poles when $2\sigma\in{\mathbb{Z}}$
and $Z^{\rm II}_{\rm tot}(q,t,\sigma)$ has additional poles when $q$ and
$\sigma$ satisfy $q=\frac{\mathrm{i}}{2}\pm{\rm i}\sigma+{\rm i}\ell$ with
$\ell\in\mathbb{Z}$.

  * -

If we are strictly interested only in the spectral problem associated to the
integral kernel (3.14), then $q\in{\mathbb{R}}$ and
$\sigma\in\frac{1}{2}+{{\rm i}}{\mathbb{R}}_{>0}$. So these poles are not
realized.

  * -

However we can go beyond this special domain. For example if we consider the
Zak transform of $Z^{\rm Nek}(t,\sigma)$ appearing on the lhs of (3.20), then
this has no longer poles in $\sigma$: the summation over $k$ in (3.20) removes
the poles. Likewise it seems that the summation over integers and the
particular combination of defect partition functions appearing in the
integrand on the lhs of (6.116) has also the effect of removing the poles.

In the forthcoming subsections we test (6.114) and (6.117) in several ways.

###  6.1 Testing $N=0$

As a first check of (6.114) we test the $N=0$ case. From (3.27) one can see
that $\Psi_{0}({\rm e}^{{x}},t)=1$ so that the rhs of (6.114) is

$\mathopen{}\mathclose{{}\left({\rm i}\frac{2^{11/12}\sqrt{\pi}t^{3/16}}{{\rm
e}^{3\zeta^{\prime}(-1)}{\rm e}^{4\sqrt{t}}}}\right)\int_{\mathbb{R}}{{\rm
d}x}\ {\rm e}^{-{\rm i}{2}qx}{\rm e}^{-4t^{1/4}\cosh x+{x\over
2}}=\mathopen{}\mathclose{{}\left({\rm
i}\frac{2^{11/12}\sqrt{\pi}t^{1/16}}{{\rm e}^{3\zeta^{\prime}(-1)}{\rm
e}^{4\sqrt{t}}}}\right)2t^{1/8}K_{\mathrm{i}2q-\frac{1}{2}}\mathopen{}\mathclose{{}\left(4t^{1/4}}\right)\,,$
(6.118)

where $K$ is the modified Bessel function of the second kind. By expanding at
small $t$ we find that the Bessel function has the following structure,

$2t^{1/8}K_{\mathrm{i}2q-\frac{1}{2}}\mathopen{}\mathclose{{}\left(4t^{1/4}}\right)=F(q,t)+F\mathopen{}\mathclose{{}\left(-q-{{\rm
i}\over 2},t}\right)$ (6.119)

for some function $F(q,t)$. For instance we have when ${\rm
i}2q-\frac{1}{2}\notin{\mathbb{Z}}$

$F(q,t)={\rm e}^{{\rm
i}\frac{q}{2}\log(t)}\mathopen{}\mathclose{{}\left[2^{\mathrm{i}2q-\frac{1}{2}}\Gamma\mathopen{}\mathclose{{}\left(-\mathrm{i}2q+\frac{1}{2}}\right)-2^{\mathrm{i}2q+\frac{3}{2}}\Gamma\mathopen{}\mathclose{{}\left(-\mathrm{i}2q-\frac{1}{2}}\right)\sqrt{t}+\mathcal{O}\mathopen{}\mathclose{{}\left(t}\right)}\right]\,,$
(6.120)

Hence we already see the structure of the lhs of (6.114) appearing. On the
gauge theory side we can perform the integral at small $t$ by using Cauchy’s
residue theorem,

$\displaystyle\int_{{\mathbb{R}}+{\rm i}\sigma_{*}}{{\rm
d}\sigma}\tan\mathopen{}\mathclose{{}\left(2\pi\sigma}\right)Z_{\mathrm{tot}}^{\rm
II}(q,t,\sigma)$ (6.121) $\displaystyle=\int_{{\mathbb{R}}+{\rm
i}\sigma_{*}}{{\rm d}\sigma}\tan(2\pi\sigma){\rm e}^{{\rm
i}\frac{q}{2}\log(t)}t^{\sigma^{2}}\frac{\Gamma\mathopen{}\mathclose{{}\left(-{\rm
i}q-\sigma+\frac{1}{2}}\right)\Gamma\mathopen{}\mathclose{{}\left(-{\rm
i}q+\sigma+\frac{1}{2}}\right)}{G(1-2\sigma)G(1+2\sigma)}\mathopen{}\mathclose{{}\left(1+\mathcal{O}(t)}\right)$
$\displaystyle=\mathopen{}\mathclose{{}\left({\rm
i}\frac{2^{11/12}\sqrt{\pi}t^{1/16}}{{\rm e}^{3\zeta^{\prime}(-1)}{\rm
e}^{4\sqrt{t}}}}\right){\rm e}^{{\rm
i}\frac{q}{2}\log(t)}\mathopen{}\mathclose{{}\left[2^{\mathrm{i}2q-\frac{1}{2}}\Gamma\mathopen{}\mathclose{{}\left(-\mathrm{i}2q+\frac{1}{2}}\right)-2^{\mathrm{i}2q+\frac{3}{2}}\Gamma\mathopen{}\mathclose{{}\left(-\mathrm{i}2q-\frac{1}{2}}\right)\sqrt{t}}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\mathopen{}\mathclose{{}\left.+2^{\mathrm{i}2q+\frac{5}{2}}\Gamma\mathopen{}\mathclose{{}\left(-\mathrm{i}2q-\frac{3}{2}}\right)\,t+\mathcal{O}\mathopen{}\mathclose{{}\left(t^{3/2}}\right)}\right]\,.$

To get the last line in (6.121) we have included the first instanton
correction in $Z_{\mathrm{tot}}^{\rm II}$ (6.113), and higher instanton
corrections can be treated similarly. The poles contributing to the integral
in (6.121) are

$\sigma={\ell\over 2}\qquad\text{and}\qquad\sigma={1\over 4}+{\ell\over
2}\qquad\text{with}\qquad\ell\in{\mathbb{Z}}~{}.$ (6.122)

By employing the series expansions (6.120) and (6.121) we can systematically
verify (6.114) for $N=0$, order by order in $t$.

###  6.2 Testing $N=1$

$n^{\text{inst}}$ |
---|---
0 |  2.8372834788 + 4.7204648771 i
1 |  2.8312289304 + 4.7137559136 i
2 |  2.8313227416 + 4.7137434937 i
3 |  2.8313226948 \+ 4.7137435320 i
$I_{1}$ | 2.8313226948 + 4.7137435320 i

$n^{\text{inst}}$ |
---|---
0 |  0.050235280369 + 0.018141366757 i
1 |  0.050242014312 + 0.018111611547 i
2 |  0.050241915710 + 0.018111648299 i
3 |  0.050241915600 \+ 0.018111648316 i
$I_{1}$ | 0.050241915600 + 0.018111648316 i

$n^{\text{inst}}$ |
---|---
0 |  0.1587865901507390 i
1 |  0.1587680111233355 i
2 |  0.1587680126408577 i
3 |  0.1587680126408951 i
$I_{1}$ | 0.1587680126408951 i

Table 1: Comparison between the two sides of (6.114) for $N=1$,
$t=1/55\pi^{4}$ with $q=1/9+\mathrm{i}2/\sqrt{3}$ (upper left), $q=1/\pi$
(upper right), and $q=\mathrm{i}/3$ (lower). $I_{1}$ is the integral (6.124)
appearing on the rhs of (6.114); $n^{\text{inst}}$ refers to the number of
instanons we include in the defect partition function appearing on the lhs of
(6.114).

As a second consistency check of (6.114) we test the $N=1$ case. First we note
that by a change of variables we can rewrite the double integral appearing on
the rhs of (6.114) as a one-dimensional integral. Let us define

$\displaystyle I_{1}(q,t)=\mathopen{}\mathclose{{}\left({\rm
i}{2^{11/12}\sqrt{\pi}{t}^{3/16}\over{\rm e}^{3\zeta^{\prime}(-1)}{\rm
e}^{4\sqrt{{t}}}\mathopen{}\mathclose{{}\left(4\pi}\right)}}\right)\int_{\mathbb{R}}{{\rm
d}x}\ {\rm e}^{-{\rm i}{2}qx}{\rm e}^{-4t^{1/4}\cosh
x+\frac{x}{2}}\Psi_{1}({\rm e}^{{x}},t).$ (6.123)

After some algebra we get

$\displaystyle I_{1}(q,t)=\mathopen{}\mathclose{{}\left({\rm
i}\frac{2^{11/12}t^{3/16}}{\sqrt{\pi}{\rm e}^{3\zeta^{\prime}(-1)}{\rm
e}^{4\sqrt{t}}}}\right)\times$ (6.124)
$\displaystyle\int_{3}^{+\infty}\mathrm{d}U\frac{\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\frac{U^{2}+\sqrt{U^{4}-10U^{2}+9}-3}{2U}}\right)^{-\mathrm{i}2q+\frac{1}{2}}-\mathopen{}\mathclose{{}\left(\frac{U^{2}+\sqrt{U^{4}-10U^{2}+9}-3}{2U}}\right)^{\mathrm{i}2q-\frac{1}{2}}}\right)K_{\mathrm{i}2q-\frac{1}{2}}\mathopen{}\mathclose{{}\left(4\sqrt[4]{t}U}\right)}{U-U^{-1}}\,.$

One useful observation is that the above integral vanishes when $q=-{\rm
i}/4$, which is in perfect agreement with the lhs of (6.114). Unfortunately we
can not compute the integral (6.124) analytically. Hence for $N=1$ the test of
(6.114) is done numerically and we find perfect agreement. One such test is
given in Table 1.

###  6.3 Testing large $N$ with a ’t Hooft limit

Another analytical test of the identity (6.114) consists of comparing both
sides in the ’t Hooft limit where as in (4.40)

$N\to+\infty\,,\qquad\qquad\epsilon\to 0\,,\qquad\qquad\lambda=N\epsilon>0\,,$
(6.125)

and with the ’t Hooft coupling $\lambda$ fixed. We will need to use that $q$
and $t$ scale as in (2.6),

$q=\frac{y}{2\epsilon}\,,\qquad\qquad\qquad
t=\mathopen{}\mathclose{{}\left(\frac{\Lambda}{\epsilon}}\right)^{4}\,,$
(6.126)

with both the position of the defect $y\in\mathbb{C}$ and the instanton
counting parameter $\Lambda>0$ kept fixed.

The computation of the ’t Hooft limit of (6.114) is simplified by using the
corresponding statement for the theory without defects, which is given in
(3.24) and was obtained in bgt ; bgt2 . In particular one can divide both
sides of (6.114) by (3.24) to get

$\frac{\int_{\mathbb{R}+\mathrm{i}\sigma_{*}}\mathrm{d}\sigma\frac{\tan\mathopen{}\mathclose{{}\left(2\pi\sigma}\right)}{\mathopen{}\mathclose{{}\left[2\cos\mathopen{}\mathclose{{}\left(2\pi\sigma}\right)}\right]^{N}}Z^{\text{Nek}}\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\frac{\Lambda}{\epsilon}}\right)^{4},\sigma}\right)\mathopen{}\mathclose{{}\left[Z^{\mathrm{II}}\mathopen{}\mathclose{{}\left(\frac{y}{2\epsilon},\mathopen{}\mathclose{{}\left(\frac{\Lambda}{\epsilon}}\right)^{4},\sigma}\right)+\mathopen{}\mathclose{{}\left(-1}\right)^{N}Z^{\mathrm{II}}\mathopen{}\mathclose{{}\left(\frac{-y-\mathrm{i}\epsilon}{2\epsilon},\mathopen{}\mathclose{{}\left(\frac{\Lambda}{\epsilon}}\right)^{4},\sigma}\right)}\right]}{\mathopen{}\mathclose{{}\left(2\pi}\right)\int_{\mathbb{R}+\mathrm{i}\sigma_{*}}\mathrm{d}\sigma\frac{\tan\mathopen{}\mathclose{{}\left(2\pi\sigma}\right)}{\mathopen{}\mathclose{{}\left[2\cos\mathopen{}\mathclose{{}\left(2\pi\sigma}\right)}\right]^{N}}Z^{\text{Nek}}\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\frac{\Lambda}{\epsilon}}\right)^{4},\sigma}\right)}\\\
=\frac{\sqrt{\Lambda}}{\sqrt{\pi\epsilon}}\int_{\mathbb{R}}\mathrm{d}x\,\exp\mathopen{}\mathclose{{}\left(-\frac{\mathrm{i}}{\epsilon}xy}\right)\frac{\exp\mathopen{}\mathclose{{}\left(-4\mathopen{}\mathclose{{}\left(\frac{\Lambda}{\epsilon}}\right)\cosh\mathopen{}\mathclose{{}\left(x}\right)+\frac{x}{2}}\right)\Psi_{N}\mathopen{}\mathclose{{}\left(\mathrm{e}^{x},\mathopen{}\mathclose{{}\left(\frac{\Lambda}{\epsilon}}\right)^{4}}\right)}{Z\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\frac{\Lambda}{\epsilon}}\right)^{4},N}\right)}\,.$
(6.127)

Note that (6.127) is by (3.24) equivalent to (6.114), but rewritten in a form
suitable and convenient for the ’t Hooft limit (6.125).

####  6.3.1 The ’t Hooft limit on the gauge theory side

The general pattern of the ’t Hooft expansion of the left hand side in (6.127)
is the same as in subsection 5.2. Using that the integration variable $\sigma$
can be related to the Coulomb branch parameter $a$ by (3.19),

$\sigma=\mathrm{i}\frac{a}{2\epsilon}\,,$ (6.128)

one expands the logarithm of the Nekrasov partition function
$Z^{\mathrm{Nek}}$ in even powers of $\epsilon$ with the leading order being
$\epsilon^{-2}$. On the other hand, the expansion of the logarithm of the
defect partition function $Z^{\mathrm{II}}$ contains all integer powers of
$\epsilon$ starting from $\epsilon^{-1}$,

$(2\pi)^{-1}Z^{\rm
II}\mathopen{}\mathclose{{}\left(\frac{y}{2\epsilon},\mathopen{}\mathclose{{}\left(\frac{\Lambda}{\epsilon}}\right)^{4},\mathrm{i}\frac{a}{2\epsilon}}\right)\simeq\exp\mathopen{}\mathclose{{}\left(\sum_{n\geqslant
0}\mathcal{W}^{\rm II}_{n}(y)\epsilon^{n-1}}\right)\,.$ (6.129)

Hence the saddles of both integrals on the left hand side of (6.127) are
determined by the same equation (5.97). This gives the functional relation
$a\mathopen{}\mathclose{{}\left(\Lambda,\lambda}\right)$, but for us it will
be convenient to rather invert this to
$\lambda\mathopen{}\mathclose{{}\left(\Lambda,a}\right)$ and keep $a$
explicitly. Keeping this in mind the ’t Hooft limit of the left hand side of
(6.127) leads eventually to

$\exp\mathopen{}\mathclose{{}\left\\{\frac{1}{\epsilon}\mathcal{W}_{0}^{\mathrm{II}}\mathopen{}\mathclose{{}\left(y}\right)+\mathopen{}\mathclose{{}\left[-\frac{\mathopen{}\mathclose{{}\left(\partial_{a}{\mathcal{W}}_{0}^{\mathrm{II}}\mathopen{}\mathclose{{}\left(y}\right)}\right)^{2}}{2\partial_{a}^{2}\mathcal{F}_{0}}+\mathcal{W}_{1}^{\mathrm{II}}\mathopen{}\mathclose{{}\left(y}\right)}\right]+\mathcal{O}\mathopen{}\mathclose{{}\left(\epsilon}\right)}\right\\}+\\\
\exp\mathopen{}\mathclose{{}\left\\{\frac{1}{\epsilon}\mathopen{}\mathclose{{}\left[\mathrm{i}\pi\lambda+\mathcal{W}_{0}^{\mathrm{II}}\mathopen{}\mathclose{{}\left(-y}\right)}\right]+\mathopen{}\mathclose{{}\left[\mathrm{i}\partial_{y}\mathcal{W}_{0}^{\mathrm{II}}\mathopen{}\mathclose{{}\left(-y}\right)-\frac{\mathopen{}\mathclose{{}\left(\partial_{a}{\mathcal{W}}_{0}^{\mathrm{II}}\mathopen{}\mathclose{{}\left(-y}\right)}\right)^{2}}{2\partial_{a}^{2}\mathcal{F}_{0}}+\mathcal{W}_{1}^{\mathrm{II}}\mathopen{}\mathclose{{}\left(-y}\right)}\right]+\mathcal{O}\mathopen{}\mathclose{{}\left(\epsilon}\right)}\right\\},$
(6.130)

where we suppressed the functional dependence on $\Lambda$ and $a$ in the
notation.

####  6.3.2 The ’t Hooft limit on the matrix model side

Consider the inverse Fourier transform on the right hand side in (6.127),

$\frac{\sqrt{2\Lambda}}{\sqrt{2\pi\epsilon}}\int_{\mathbb{R}}\mathrm{d}x\exp\mathopen{}\mathclose{{}\left(-\frac{\mathrm{i}}{\epsilon}xy}\right)\frac{\exp\mathopen{}\mathclose{{}\left(-4\mathopen{}\mathclose{{}\left(\frac{\Lambda}{\epsilon}}\right)\cosh\mathopen{}\mathclose{{}\left(x}\right)+\frac{x}{2}}\right)\Psi_{N}\mathopen{}\mathclose{{}\left(\mathrm{e}^{x},\mathopen{}\mathclose{{}\left(\frac{\Lambda}{\epsilon}}\right)^{4}}\right)}{Z\mathopen{}\mathclose{{}\left(t,N}\right)},$
(6.131)

where the ’t Hooft expansion of the integrand is,

$\frac{\exp\mathopen{}\mathclose{{}\left(-4\mathopen{}\mathclose{{}\left(\frac{\Lambda}{\epsilon}}\right)\cosh\mathopen{}\mathclose{{}\left(x}\right)+\frac{x}{2}}\right)\Psi_{N}\mathopen{}\mathclose{{}\left(\mathrm{e}^{x},\mathopen{}\mathclose{{}\left(\frac{\Lambda}{\epsilon}}\right)^{4}}\right)}{Z\mathopen{}\mathclose{{}\left(t,N}\right)}=\exp\mathopen{}\mathclose{{}\left[\frac{1}{\epsilon}\mathcal{T}_{0}\mathopen{}\mathclose{{}\left(\mathrm{e}^{x}}\right)+\frac{x}{2}+\mathcal{T}_{1}\mathopen{}\mathclose{{}\left(\mathrm{e}^{x}}\right)+\mathcal{O}\mathopen{}\mathclose{{}\left(\epsilon}\right)}\right]\,,$
(6.132)

by using (4.47) and again suppressing the functional dependence on the other
variables. In the limit $\epsilon\to 0$ the inverse Fourier transform in
(6.131) is dominated by its saddles and becomes

$\sum_{s}\exp\mathopen{}\mathclose{{}\left\\{\frac{1}{\epsilon}\widehat{\mathcal{T}_{s,0}}\mathopen{}\mathclose{{}\left(y}\right)+\widehat{\mathcal{T}_{s,1}}\mathopen{}\mathclose{{}\left(y}\right)+\mathcal{O}\mathopen{}\mathclose{{}\left(\epsilon}\right)}\right\\}=\\\
\sum_{s}\exp\mathopen{}\mathclose{{}\left\\{\frac{1}{\epsilon}\mathopen{}\mathclose{{}\left[-\mathrm{i}x_{s}y+\mathcal{T}_{0}\mathopen{}\mathclose{{}\left(\mathrm{e}^{x_{s}}}\right)}\right]+\mathopen{}\mathclose{{}\left[\frac{\log\mathopen{}\mathclose{{}\left(2\Lambda}\right)}{2}-\frac{\log\mathopen{}\mathclose{{}\left(-\partial_{x}^{2}\mathcal{T}_{0}\mathopen{}\mathclose{{}\left(\mathrm{e}^{x_{s}}}\right)}\right)}{2}+\frac{x_{s}}{2}+\mathcal{T}_{1}\mathopen{}\mathclose{{}\left(\mathrm{e}^{x_{s}}}\right)}\right]+\mathcal{O}\mathopen{}\mathclose{{}\left(\epsilon}\right)}\right\\}.$
(6.133)

The sum over $s$ is a sum over the saddles and
$x_{s}=x_{s}\mathopen{}\mathclose{{}\left(y}\right)$ is determined by the
saddle point equation,

$y+\mathrm{i}\partial_{x}\mathcal{T}_{0}\mathopen{}\mathclose{{}\left(\mathrm{e}^{x}}\right)=y-\mathrm{i}2\Lambda\frac{\sqrt{\mathrm{e}^{2x}-{\tt
g}^{2}}\sqrt{\mathrm{e}^{2x}-{\tt g}^{-2}}}{\mathrm{e}^{x}}=0\,,$ (6.134)

where we used (4.72) and $z=\exp\mathopen{}\mathclose{{}\left(x}\right)$.
Taking the square of this equation gives the Seiberg-Witten curve (4.73) if we
take as before (4.75),

${\tt g}^{2}+{\tt g}^{-2}={u\over 4\Lambda^{2}}\,,\qquad\qquad\qquad{\tt
g}^{\pm
2}=\mathopen{}\mathclose{{}\left(\frac{u}{8\Lambda^{2}}}\right)\mp\sqrt{\mathopen{}\mathclose{{}\left(\frac{u}{8\Lambda^{2}}}\right)^{2}-1}\,.$
(6.135)

This leads to the following two solutions,

$\mathrm{e}^{2x_{\pm}\mathopen{}\mathclose{{}\left(y}\right)}=z_{\pm}^{2}\mathopen{}\mathclose{{}\left(y}\right)={-\mathopen{}\mathclose{{}\left[\frac{\mathopen{}\mathclose{{}\left(y^{2}-u}\right)\pm\sqrt{\mathopen{}\mathclose{{}\left(y^{2}-u}\right)^{2}-64\Lambda^{4}}}{8\Lambda^{2}}}\right]}.$
(6.136)

Let us take a moment to consider the behaviour of
$z_{\pm}\mathopen{}\mathclose{{}\left(y}\right)$ as a function of $y$. One can
check that $z_{\pm}\mathopen{}\mathclose{{}\left(y}\right)$ is real and
outside the branch cut region of the matrix model if and only if
$\mathrm{i}y\in\mathbb{R}\setminus\mathopen{}\mathclose{{}\left\\{0}\right\\}$.
It is important to note that with this choice of
$\mathrm{i}y\in\mathbb{R}\setminus\mathopen{}\mathclose{{}\left\\{0}\right\\}$
one has $z_{-}^{2}\mathopen{}\mathclose{{}\left(y}\right)>1/{\tt g}^{2}$ and
$0\leqslant z_{+}^{2}\mathopen{}\mathclose{{}\left(y}\right)<{\tt g}^{2}$.
Moreover there are no possible choices of $y\in\mathbb{C}$ such that
$0\leqslant z_{-}^{2}\mathopen{}\mathclose{{}\left(y}\right)<{\tt g}^{2}$ or
$z_{+}^{2}\mathopen{}\mathclose{{}\left(y}\right)>1/{\tt g}^{2}$. One finds on
the other hand that $z_{\pm}\mathopen{}\mathclose{{}\left(y}\right)$ is real
and inside the branch cut region if and only if $0\leqslant y^{2}\leqslant
u-8\Lambda^{2}$, and also that
$z_{\pm}\mathopen{}\mathclose{{}\left(y}\right)$ is purely imaginary if and
only if $y^{2}\geqslant u+8\Lambda^{2}$. For all other choices of
$y\in\mathbb{C}$ one will find generic complex
$z_{\pm}\mathopen{}\mathclose{{}\left(y}\right)$.

####  6.3.3 Comparing the gauge theory and the matrix model

To analyze the leading order of the ’t Hooft expansion in (6.133) with the
saddles (6.136) it is convenient to separately look at the case $y=0$ and the
derivative with respect to $y$. The reason is that the later simplifies
considerably as a consequence of the saddle point equation (6.134). Setting
$y=0$ serves then as a check of the constant term.

Let us first look at the $y$ derivative of the leading order part. At the
matrix model side (6.133) one gets by making use of the saddle point equation
(6.134) and its solutions

$\frac{\mathrm{d}}{\mathrm{d}y}\widehat{\mathcal{T}_{\pm,0}}\mathopen{}\mathclose{{}\left(y}\right)=-\mathrm{i}x_{\pm}\mathopen{}\mathclose{{}\left(y}\right)\,.$
(6.137)

Comparing this to the leading order of the gauge theory (6.129) we can check
that242424Upon using the Matone relation (5.99) and a careful treatment of the
branches.

$\begin{split}\frac{\mathrm{d}}{\mathrm{d}y}\mathopen{}\mathclose{{}\left[\mathcal{W}_{0}^{\mathrm{II}}\mathopen{}\mathclose{{}\left(\pm
y}\right)-\widehat{\mathcal{T}_{\pm
S,0}}\mathopen{}\mathclose{{}\left(y}\right)}\right]=0\,,\end{split}$ (6.138)

where
$S=\mathrm{sgn}\mathopen{}\mathclose{{}\left[\mathrm{arg}\mathopen{}\mathclose{{}\left(\mathrm{i}\mathopen{}\mathclose{{}\left(y^{2}-a^{2}}\right)}\right)}\right]$
with the convention that
$\mathrm{sgn}\mathopen{}\mathclose{{}\left(0}\right)=-1\,$.

Let us then look at the constant term for $y=0$. At the gauge theory side we
have for the leading order (6.129)

$\mathcal{W}_{0}^{\mathrm{II}}\mathopen{}\mathclose{{}\left(0}\right)=-\frac{\pi}{2}\mathopen{}\mathclose{{}\left|a}\right|.$
(6.139)

From (6.136) one can see that
$z_{\pm}\mathopen{}\mathclose{{}\left(0}\right)={\tt g}^{\pm}>0$, with ${\tt
g}^{\pm}$ as in (6.135). Using (4.48) gives for the leading order of the
matrix model (6.133)

$\widehat{\mathcal{T}_{\pm,0}}\mathopen{}\mathclose{{}\left(0}\right)=\mathcal{T}_{0}\mathopen{}\mathclose{{}\left(z_{\pm}\mathopen{}\mathclose{{}\left(0}\right)}\right)=-2\Lambda\mathopen{}\mathclose{{}\left({\tt
g}+{\tt g}^{-1}}\right)+2\int_{+\infty}^{{\tt g}^{\pm
1}}\mathrm{d}z\,\lambda\omega_{+}^{0}\mathopen{}\mathclose{{}\left(z}\right)\,,$
(6.140)

where the even planar resolvent
$\omega_{+}^{0}\mathopen{}\mathclose{{}\left(z}\right)$ is given in (4.68).
Note that the difference between the leading terms of the two saddles is

$\mathopen{}\mathclose{{}\left[\widehat{\mathcal{T}_{+,0}}\mathopen{}\mathclose{{}\left(0}\right)-\widehat{\mathcal{T}_{-,0}}\mathopen{}\mathclose{{}\left(0}\right)}\right]=-2\int_{{\tt
g}}^{{\tt
g}^{-1}}\mathrm{d}z\,\lambda\omega_{+}^{0}\mathopen{}\mathclose{{}\left(z}\right)=\mathrm{i}\pi\lambda\,.$
(6.141)

The last equality can be obtained in an $\Lambda\to 0$ expansion or exactly
using (Eynard:1995nv, , eq. 4.18), which shows that this relation does not
depend on the particular form of the potential. We have furthermore that

$\widehat{\mathcal{T}_{-,0}}\mathopen{}\mathclose{{}\left(0}\right)=-2\Lambda\mathopen{}\mathclose{{}\left({\tt
g}+{\tt g}^{-1}}\right)+2\int_{+\infty}^{{\tt g}^{-1}}\mathrm{d}z\
\lambda\omega_{+}^{0}\mathopen{}\mathclose{{}\left(z}\right)\\\
=-2\Lambda\mathopen{}\mathclose{{}\left[\frac{2\mathrm{E}\mathopen{}\mathclose{{}\left({\tt
g}^{4}}\right)+\mathopen{}\mathclose{{}\left({\tt g}^{-2}-{\tt
g}^{2}}\right)\mathopen{}\mathclose{{}\left[-2{\tt
g}^{2}\mathrm{K}\mathopen{}\mathclose{{}\left({\tt
g}^{4}}\right)+\mathrm{K}\mathopen{}\mathclose{{}\left({\tt
g}^{-4}}\right)+\mathrm{i}\mathrm{K}\mathopen{}\mathclose{{}\left(1-{\tt
g}^{-4}}\right)}\right]}{{\tt
g}}}\right]=-\frac{\pi}{2}\mathopen{}\mathclose{{}\left|a}\right|\,,$ (6.142)

with $\mathrm{K}$ and $\mathrm{E}$ the complete elliptic integrals of the
first (C.201) and second (C.202) kind respectively. The last equality was
found in an $\Lambda\to 0$ or equivalently ${\tt g}\to 0$ expansion using
(6.135) and the Matone relation (5.99). Hence from (6.139), (6.141) and
(6.142)

$\begin{split}\mathcal{W}_{0}^{\mathrm{II}}\mathopen{}\mathclose{{}\left(0}\right)-\widehat{\mathcal{T}_{-,0}}\mathopen{}\mathclose{{}\left(0}\right)&=0\,,\\\
\mathrm{i}\pi\lambda+\mathcal{W}_{0}^{\mathrm{II}}\mathopen{}\mathclose{{}\left(0}\right)-\widehat{\mathcal{T}_{+,0}}\mathopen{}\mathclose{{}\left(0}\right)&=0\,.\end{split}$
(6.143)

So the constant parts of the leading order terms agree and together with
(6.138) this proves the equality in (6.127) and hence (6.114) to leading order
in the ’t Hooft limit (6.125).

The subleading order can be checked in analogy with section 5. Matching at
higher order in $\epsilon$ can then be inferred from topological recursion, as
we discussed near the end of section 5.

###  6.4 Numerical eigenfunctions

The numerical analysis of the spectrum and the eigenfunctions for the integral
kernel $\rho$ (3.14) is done exactly as in (Hatsuda:2012hm, , sec. 2.2). To
make the presentation self-contained let us review the strategy of
(Hatsuda:2012hm, , sec. 2.2). We are interested in studying numerically the
eigenvalue equation

$\int_{{\mathbb{R}}}{\rm
d}y\,\rho(x,y)\varphi_{n}(y,t)=E_{n}\varphi_{n}(x,t)\,,$ (6.144)

where the kernel $\rho(x,y)$ is defined in (3.14). It is convenient to
decompose $\rho(x,y)$ as

$\rho(x,y)=\sum_{k\geqslant
0}\rho_{k}(x)\rho_{k}(y)\,,\qquad\qquad\qquad\rho_{k}(x)=\frac{\tanh^{k}\mathopen{}\mathclose{{}\left(\frac{x}{2}}\right)\exp\mathopen{}\mathclose{{}\left(-4t^{1/4}\cosh(x)}\right)}{\cosh\mathopen{}\mathclose{{}\left(\frac{x}{2}}\right)}\,,$
(6.145)

and to define

$v_{k}^{(n)}(t)=\int_{{\mathbb{R}}}{\rm d}y\,\rho_{k}(y)\varphi_{n}(y,t)\,.$
(6.146)

Then (6.144) reads

$\sum_{k\geqslant 0}\rho_{k}(x)v_{k}^{(n)}(t)=E_{n}\varphi_{n}(x,t)\,,$
(6.147)

which we can also write as

$\sum_{k\geqslant 0}H_{\ell,k}v_{k}^{(n)}(t)=E_{n}v_{\ell}^{(n)}(t)~{},$
(6.148)

with $H$ the infinite dimensional Hankel matrix defined by

$H_{k,\ell}=\int_{\mathbb{R}}{\rm
d}x\frac{\tanh^{k+\ell}\mathopen{}\mathclose{{}\left(\frac{x}{2}}\right)\exp\mathopen{}\mathclose{{}\left(-8{t^{1/4}}\cosh(x)}\right)}{\cosh^{2}\mathopen{}\mathclose{{}\left(\frac{x}{2}}\right)}\,,\qquad\qquad
k,\ell\geqslant 0\,.$ (6.149)

This means that the eigenvalues of $H$ coincide with those of
$\rho\mathopen{}\mathclose{{}\left(x,y}\right)$ and the eigenvectors of $H$
give the eigenfunctions of $\rho\mathopen{}\mathclose{{}\left(x,y}\right)$ via
(6.147). The advantage of working with $H$ is that we can numerically compute
its eigenvalues and eigenfunctions by truncating the matrix to a finite size
while maintaining control over the numerical error due to the truncation. Let
$v^{(n,M)}(t)$ be the $n^{\rm th}$ eigenvector of the Hankel matrix $H$
(6.149) truncated at size $M$. Defining $\varphi_{n}^{(M)}(x,t)$ by

$\varphi_{n}^{(M)}(x,t)=\sum_{k=0}^{M}\rho_{k}(x)v^{(n,M)}_{k}\,,\qquad\qquad
v^{(n,M)}=\mathopen{}\mathclose{{}\left(\begin{array}[]{c}v^{(n,M)}_{0}\\\
\vdots\\\ v^{(n,M)}_{M}\end{array}}\right)\,,$ (6.150)

we recover the true eigenfunctions of the kernel $\rho$ in the $M\to+\infty$
limit,

$\lim_{M\to+\infty}\varphi_{n}^{(M)}(x,t)\propto\varphi_{n}(x,t)\,,$ (6.151)

where the proportionality factor is a numerical constant and $\varphi_{n}$ is
the $n^{\rm th}$ eigenfunction of (6.144) in the normalization of (6.117). We
computed the lhs of (6.151) numerically and checked that this numerical
expression agrees with the eigenfunctions computed by using the defect
expression on the rhs of (6.117) with high precision. For instance for
$t=1/(100\pi^{8})$, by including $0$ instantons in (6.117) we get a pointwise
agreement of the order $10^{-6}$. Likewise by including 1, 2 and 3 instantons
we get a pointwise agreement of the order $10^{-11}$, $10^{-16}$ and
$10^{-22}$ respectively252525One can reach this precision with Hankel matrices
of size $M=2^{9}=512$..

##  7 Outlook

In this paper we have shown that the eigenfunctions of the operator (1.1) are
computed by surface defects in $\mathcal{N}=2$, $\operatorname{SU(2)}2$ SYM in
the self-dual phase of the four-dimensional $\Omega$-background
($\epsilon_{1}+\epsilon_{2}=0$). This result, together with bgt ; bgt2 ;
Gavrylenko:2023ewx , extends the correspondence between 4d $\mathcal{N}=2$
theories and spectral theory to a new class of operators. In addition we have
expressed the eigenfunctions of these operators in closed form via a matrix
model average (2.4). This provides a representation for the surface defect
partition function which resums both the instanton and the $\epsilon$
expansions. In this way we have a manifest interpolation from the weak to the
strong coupling region. In particular, the strong coupling expansion in
$1/\Lambda$ (exact in $\epsilon$ and $a_{D}$) corresponds to the expansion of
the matrix model around its Gaussian point and hence it is obtained in a
straightforward way.

Some further comments and generalizations:

  * -

In this work we focused on the specific example of 4d, $\mathcal{N}=2$,
$\operatorname{SU(2)}2$ SYM and the operator (1.1). It would be interesting to
extend our results in a systematic way to all 4d $\mathcal{N}=2$ theories. For
example in the case of $\mathcal{N}=2$, $\operatorname{SU(N)}$ SYM we have
$N-1$ non-commuting Fermi gas operators as discussed in bgt2 ;
Gavrylenko:2023ewx . We expect their eigenfunctions to be computed by surface
defects in $\operatorname{SU(N)}$ SYM in the self-dual phase of the $\Omega$
background.

  * -

Our results should follow from the open version of the TS/ST correspondence
Marino:2016rsq ; Marino:2017gyg  by implementing the dual four-dimensional
limit, see subsection 3.4. However the formulation of Marino:2016rsq ;
Marino:2017gyg  only holds when the mass parameters of the underlying CY
geometry are set to their most symmetric values (for local ${\mathbb{F}}_{0}$
this is equivalent to setting $m_{{\mathbb{F}}_{0}}=1$ in (3.29)). To derive
(2.9) one need a formulation of the open TS/ST correspondence for generic
values of the parameters. We will report on this somewhere else matijn-alba .

  * -

The Fredholm determinant of (1.1) computes the tau function of the Painlevé
$\rm III_{3}$ equation at specific initial conditions. It would be interesting
to understand what is the role of the eigenfunctions of (1.1) in the context
of Painlevé equations. In particular the relation to the solution of the
linear system associated to Painlevé equations as well as to the work
Bonelli:2022iob .

  * -

The Fredholm determinant and the spectral traces of (1.1) can also be
expressed via a pair of coupled TBA equations closely related to two
dimensional theories Cecotti:1992qh ; zamo . It would be interesting to
understand this better since this may reveal an interesting 4d-2d interplay
characterizing directly the self-dual phase of the $\Omega$-background.

  * -

The operator (1.1) is a particular example of a Painlevé kernel whose Fredholm
determinant computes the tau function. A more general class of Fredholm
determinants was constructed in Gavrylenko:2016zlf ; Gavrylenko:2017lqz ;
Desiraju:2019vna ; DelMonte:2020wty . It would be interesting to see if also
in this case the corresponding (formal) eigenfunctions are related to surface
defects.

  * -

It is well known that the standard quantization of the SW curve for
$\operatorname{SU(2)}2$ SYM leads to the Mathieu operator (3.10). We expect a
different quantization scheme to produce the operator (1.1). It is important
to understand what this other quantization scheme is. Since the spectral
analysis of (1.1) is encoded in the self-dual phase of the
$\Omega$-background, a natural quantization scheme to investigate would be the
one arizing in the context of the topological recursion Iwaki:2015xnr ;
Iwaki:2019zeq ; Marchal:2019nsq ; Marchal:2019bia .

## Acknowledgments

We would like to thank Anton Nedelin for collaborating during the early stages
of this work. We are grateful to Severin Charbonnier, Qianyu Hao, Saebyeok
Jeong, Rinat Kashaev, Marcos Mariño, Andy Neitzke, Nicolas Orantin, François
Pagano, Sara Pasquetti, Lukas Schimmer, Yasuhiko Yamada and Szabolcs Zakany
for useful discussions and correspondence. This work is partially supported by
the Swiss National Science Foundation Grant No. 185723 and the NCCR SwissMAP.

##  Appendix A The refined open topological vertex and the type II defect

In this appendix we explain how the type II defect partition function of 4d,
$\mathcal{N}=2$, $\mathrm{SU}\mathopen{}\mathclose{{}\left(2}\right)$ super
Yang-Mills can be obtained from open topological strings on the toric Calabi-
Yau manifold known as local
$\mathbb{F}_{0}=\mathbb{P}^{1}\times\mathbb{P}^{1}$.

###  A.1 Open topological strings on local $\mathbb{F}_{0}$

To find the open string partition function on local $\mathbb{F}_{0}$ we use
Iqbal:2007ii  and Cheng:2021nex . We will closely follow the notation and
presentation of Cheng:2021nex .

####  A.1.1 Notation and conventions

The variables $Q_{B,F}$ are related to the Kähler parameters $T_{B,F}$ of the
base and fiber $\mathbb{P}^{1}$’s respectively,

$Q_{B,F}=\exp\mathopen{}\mathclose{{}\left(-T_{B,F}}\right)\,,\qquad\qquad
T_{B,F}=-\log\mathopen{}\mathclose{{}\left(Q_{B,F}}\right)\,,$ (A.152)

and $Q_{1}$ is similarly the open Kähler parameter of the brane
(Cheng:2021nex, , pp. 9-14, 50). The variables $t,q$ are similarly connected
to the $\Omega$-background parameters $\epsilon_{1,2}$ by

$t=\exp\mathopen{}\mathclose{{}\left(\mathrm{i}\epsilon_{1}}\right),\qquad\qquad\qquad
q=\exp\mathopen{}\mathclose{{}\left(-\mathrm{i}\epsilon_{2}}\right)~{}.$
(A.153)

It is important to note that these variables are not related the $t$ and $q$
appearing in the main text.

####  A.1.2 The open refined topological vertex

Important building blocks of the partition function are the so-called Nekrasov
factors

$\begin{split}N_{\mu,\nu}\mathopen{}\mathclose{{}\left(Q;t,q}\right)&=\prod_{k,\ell=1}^{+\infty}\frac{1-Q\
q^{\nu_{k}-\ell}t^{\mu_{\ell}^{t}-k+1}}{1-Q\ q^{-\ell}t^{-k+1}}\\\
&=\prod_{\mathopen{}\mathclose{{}\left(k,\ell}\right)\in\nu}\mathopen{}\mathclose{{}\left(1-Q\
q^{\nu_{k}-\ell}t^{\mu_{\ell}^{t}-k+1}}\right)\prod_{\mathopen{}\mathclose{{}\left(k,\ell}\right)\in\mu}\mathopen{}\mathclose{{}\left(1-Q\
q^{-\mu_{k}+\ell-1}t^{-\nu_{\ell}^{t}+k}}\right),\end{split}$ (A.154)

where $\mu$ and $\nu$ are plane partitions:

$\mu=\mathopen{}\mathclose{{}\left\\{\mu_{1},\mu_{2},\mu_{3},\cdots\ |\
\forall\
k,\ell\in\mathbb{N}\setminus\mathopen{}\mathclose{{}\left\\{0}\right\\}:\mathopen{}\mathclose{{}\left(\mu_{k}\in\mathbb{N}}\right)\land\mathopen{}\mathclose{{}\left(k\leqslant\ell\Rightarrow\mu_{k}\geqslant\mu_{\ell}}\right)}\right\\}.$
(A.155)

Hence $\mathopen{}\mathclose{{}\left(k,\ell}\right)\in\mu$ is any pair
$k,\ell\in\mathbb{N}\setminus\mathopen{}\mathclose{{}\left\\{0}\right\\}$ such
that $1\leqslant\ell\leqslant\mu_{k}$. Partitions can be represented by a
Young diagram, and the transposed partition $\mu^{t}$ is obtained by
transposing the Young diagram

$\mu^{t}=\mathopen{}\mathclose{{}\left\\{\mu_{1}^{t},\mu_{2}^{t},\mu_{3}^{t},\cdots\
|\ \forall\
\ell\in\mathbb{N}\setminus\mathopen{}\mathclose{{}\left\\{0}\right\\}:\mu_{\ell}^{t}=\mathopen{}\mathclose{{}\left|\mathopen{}\mathclose{{}\left\\{k\in\mathbb{N}\setminus\mathopen{}\mathclose{{}\left\\{0}\right\\}\
|\ \mu_{k}\geqslant\ell}\right\\}}\right|}\right\\}.$ (A.156)

The key ingredient in the topological vertex computation is 262626The first
Nekrasov factor on the second line of (A.157) differs from what is given in
(Cheng:2021nex, , p. 50, eq. 5.4) where they have $Q_{F}\leftrightarrow
Q_{F}q$. However comparison with (Iqbal:2007ii, , p. 35, eq. 90) and
(Marino:2017gyg, , p. 27, eqs. 4.9, 4.11) leads us to (A.157).,

$C_{\mu,\nu}\mathopen{}\mathclose{{}\left(Q_{1},Q_{F},t,q}\right)=\frac{N_{\emptyset,\mu^{t}}\mathopen{}\mathclose{{}\left(Q_{1}\frac{t^{2}}{q};t^{-1},q^{-1}}\right)N_{\emptyset,\nu}\mathopen{}\mathclose{{}\left(Q_{1}Q_{F}\frac{t^{2}}{q};t^{-1},q^{-1}}\right)}{N_{\emptyset,\mu^{t}}\mathopen{}\mathclose{{}\left(Q_{1}\frac{t}{q};t^{-1},q^{-1}}\right)N_{\emptyset,\nu}\mathopen{}\mathclose{{}\left(Q_{1}Q_{F}\frac{t}{q};t^{-1},q^{-1}}\right)}\\\
\frac{1}{N_{\mu^{t},\nu}\mathopen{}\mathclose{{}\left(Q_{F};t^{-1},q^{-1}}\right)N_{\mu^{t},\nu}\mathopen{}\mathclose{{}\left(Q_{F}\frac{t}{q};t^{-1},q^{-1}}\right)}~{},$
(A.157)

where $\emptyset$ is the empty partition $\\{\\}$. This gives for the “open-
closed t-brane partition function” of local $\mathbb{F}_{0}$ (Cheng:2021nex, ,
p. 50, eq. 5.4)

$Z^{\text{open-closed,
t-brane}}_{\mathbb{F}_{0}}\mathopen{}\mathclose{{}\left(Q_{1},Q_{B},Q_{F},t,q}\right)=\\\
\sum_{\mu,\nu}Q_{B}^{\mathopen{}\mathclose{{}\left|\mu}\right|+\mathopen{}\mathclose{{}\left|\nu}\right|}t^{||\nu^{t}||^{2}}q^{||\mu^{t}||^{2}}||Z_{\mu}\mathopen{}\mathclose{{}\left(t,q}\right)||^{2}||Z_{\nu}\mathopen{}\mathclose{{}\left(q,t}\right)||^{2}C_{\mu,\nu}\mathopen{}\mathclose{{}\left(Q_{1},Q_{F},t,q}\right),$
(A.158)

where one defines

$|\mu|=\sum_{k=1}^{+\infty}\mu_{k},\qquad\qquad||\mu||^{2}=\sum_{k=1}^{+\infty}\mu_{k}^{2},$
(A.159)
$\mathopen{}\mathclose{{}\left|\mathopen{}\mathclose{{}\left|Z_{\mu}\mathopen{}\mathclose{{}\left(t,q}\right)}\right|}\right|^{2}=Z_{\mu^{t}}\mathopen{}\mathclose{{}\left(t,q}\right)Z_{\mu}\mathopen{}\mathclose{{}\left(q,t}\right),\qquad\qquad
Z_{\mu}\mathopen{}\mathclose{{}\left(t,q}\right)=\prod_{\mathopen{}\mathclose{{}\left(k,\ell}\right)\in\mu}\mathopen{}\mathclose{{}\left(1-q^{\mu_{k}-\ell}t^{\mu_{\ell}^{t}-k+1}}\right)^{-1}.$
(A.160)

The t-brane instanton partition function for local $\mathbb{F}_{0}$ is then
(Cheng:2021nex, , p. 50, eq. 5.7)

$Z^{\text{t-brane,
inst}}_{\mathbb{F}_{0}}\mathopen{}\mathclose{{}\left(Q_{1},Q_{B},Q_{F},t,q}\right)=\frac{Z^{\text{open-
closed,
t-brane}}_{\mathbb{F}_{0}}\mathopen{}\mathclose{{}\left(Q_{1},Q_{B},Q_{F},t,q}\right)}{Z^{\text{closed}}_{\mathbb{F}_{0}}\mathopen{}\mathclose{{}\left(Q_{B},Q_{F},t,q}\right)},$
(A.161)

where $Z^{\text{closed}}_{\mathbb{F}_{0}}$ is given by

$Z^{\text{closed}}_{\mathbb{F}_{0}}\mathopen{}\mathclose{{}\left(Q_{B},Q_{F},t,q}\right)=Z^{\text{open-
closed,
t-brane}}_{\mathbb{F}_{0}}\mathopen{}\mathclose{{}\left(Q_{B},Q_{F},0,t,q}\right).$
(A.162)

In addition one has also the 1-loop part (Cheng:2021nex, , pp. 50, 53, eqs.
5.6, A.15, A.22, A.25)

$Z^{\text{t-brane,
1-loop}}_{\mathbb{F}_{0}}\mathopen{}\mathclose{{}\left(Q_{1},Q_{F},t,q}\right)=\mathopen{}\mathclose{{}\left(\frac{Q_{1}t}{q};\frac{1}{q}}\right)_{\infty}\mathopen{}\mathclose{{}\left(\frac{Q_{1}Q_{F}t}{q};\frac{1}{q}}\right)_{\infty}=\frac{1}{\mathopen{}\mathclose{{}\left(Q_{1}t;q}\right)_{\infty}\mathopen{}\mathclose{{}\left(Q_{1}Q_{F}t;q}\right)_{\infty}},$
(A.163)

where $\mathopen{}\mathclose{{}\left(a;q}\right)_{\infty}$ is the
$q$-Pochhammer symbol. The complete partition function is then given by

$Z^{\text{t-brane}}_{\mathbb{F}_{0}}\mathopen{}\mathclose{{}\left(Q_{1},Q_{B},Q_{F},t,q}\right)=\\\
Z^{\text{t-brane,
1-loop}}_{\mathbb{F}_{0}}\mathopen{}\mathclose{{}\left(Q_{1},Q_{F},t,q}\right)Z^{\text{t-brane,
instanton}}_{\mathbb{F}_{0}}\mathopen{}\mathclose{{}\left(Q_{1},Q_{B},Q_{F},t,q}\right).$
(A.164)

Two particular cases of the refined topological vertex which are interesting
are: the self-dual or Gopakumar-Vafa (GV) phase $\epsilon_{1}+\epsilon_{2}=0$
or $t=q$, and the Nekrasov-Shatasvili (NS) phase $\epsilon_{1}=0$ or $t=1$.

####  A.1.3 Non-pertubative topological strings

If we set $g_{s}=-\epsilon_{1}=\epsilon_{2}$, then we obtain the unrefined
topological string amplitude. In addition it was found in hmmo ; ghm ; cgm2
that when one includes the NS phase of the refined topological vertex as well
one obtains a non-pertubative completion of the topological string partition
function on toric CY. This was generalized to some extend from the closed to
the open vertex in Marino:2016rsq ; Marino:2017gyg , see also Kashani-
Poor:2016edc ; Sciarappa:2017hds . In particular for local ${\mathbb{F}}_{0}$
one finds Marino:2016rsq ; Marino:2017gyg

$Z_{\mathbb{F}_{0}}^{\text{open,
np}}\mathopen{}\mathclose{{}\left(X,Q_{B},Q_{F},g_{s}}\right)=Z_{\mathbb{F}_{0}}^{\text{(p)}}\mathopen{}\mathclose{{}\left(X,g_{s}}\right)Z_{\mathbb{F}_{0}}^{\text{GV}}\mathopen{}\mathclose{{}\left(X,Q_{B},Q_{F},g_{s}}\right)Z_{\mathbb{F}_{0}}^{\text{NS}}\mathopen{}\mathclose{{}\left(X^{S},Q_{B}^{S},Q_{F}^{S},g_{s}}\right).$
(A.165)

We explain all functions and symbols involved below. The usual open
topological string part
$Z_{\mathbb{F}_{0}}^{\text{GV}}\mathopen{}\mathclose{{}\left(X,Q_{B},Q_{F},g_{s}}\right)$
is the self-dual phase of the open refined topological vertex (A.161),
(A.163), (A.164),

$\begin{split}Z_{\mathbb{F}_{0}}^{\text{GV}}\mathopen{}\mathclose{{}\left(X,Q_{B},Q_{F},g_{s}}\right)=Z^{\text{t-brane}}_{\mathbb{F}_{0}}\mathopen{}\mathclose{{}\left(\frac{\sqrt{q}}{\sqrt{Q_{F}}X},Q_{B},Q_{F},\frac{1}{q},\frac{1}{q}}\right),\quad\qquad
q=\exp\mathopen{}\mathclose{{}\left(\mathrm{i}g_{s}}\right)~{}.\end{split}$
(A.166)

On the other hand one has
$Z_{\mathbb{F}_{0}}^{\text{NS}}\mathopen{}\mathclose{{}\left(X^{S},Q_{B}^{S},Q_{F}^{S},g_{s}}\right)$
which is the NS phase of the refined topological vertex, (A.161), (A.163),
(A.164),

$\begin{split}Z_{\mathbb{F}_{0}}^{\text{NS}}\mathopen{}\mathclose{{}\left(X^{S},Q_{B}^{S},Q_{F}^{S},g_{s}}\right)&=\lim_{p^{S}\to
1}Z^{\text{t-brane}}_{\mathbb{F}_{0}}\mathopen{}\mathclose{{}\left(-\frac{\sqrt{p^{S}}}{\sqrt{Q_{F}^{S}}X^{S}},Q_{B}^{S},Q_{F}^{S},\frac{1}{p^{S}},\frac{1}{q^{S}}}\right),\\\
q^{S}&=\exp\mathopen{}\mathclose{{}\left(\mathrm{i}\frac{\mathopen{}\mathclose{{}\left(2\pi}\right)^{2}}{g_{s}}}\right),\end{split}$
(A.167)

which is perturbative in the inverse string coupling $g_{s}^{-1}$. The
exponentiated Kähler parameters $X,Q_{B},Q_{F}$ and their NS counter parts
$X^{S},Q_{B}^{S},Q_{F}^{S}$ are related by powers of $2\pi/g_{s}$,

$A^{S}=A^{2\pi/g_{s}},\qquad\qquad\qquad
A=\mathopen{}\mathclose{{}\left(A^{S}}\right)^{g_{s}/2\pi},$ (A.168)

where $A$ stands for either one of $X,Q_{B},Q_{F}$. The “polynomial” part
$Z^{\text{(p)}}\mathopen{}\mathclose{{}\left(X,g_{s}}\right)$ is not obtained
from the topological vertex but is given by (Marino:2017gyg, , eq.
4.10)272727In Marino:2017gyg  they focus on the special case where there is
only a single Kähler parameter, so $Q_{B}=Q_{F}$. One can check however that
(Marino:2017gyg, , eq. 4.10) is also correct for $Q_{B}\neq Q_{F}$. To get
from (Marino:2017gyg, , eq. 4.10) to (A.169) one needs to take
$x=\log\mathopen{}\mathclose{{}\left(X^{\mathrm{S}}}\right)$ and
$\hbar=\mathopen{}\mathclose{{}\left(2\pi}\right)^{2}/g_{s}$.

$\begin{split}Z_{\mathbb{F}_{0}}^{\text{(p)}}\mathopen{}\mathclose{{}\left(X,g_{s}}\right)&=\exp\mathopen{}\mathclose{{}\left\\{-\frac{\mathrm{i}}{2}\frac{1}{g_{s}}\mathopen{}\mathclose{{}\left[\log^{2}\mathopen{}\mathclose{{}\left(X}\right)-\mathrm{i}2\pi\log\mathopen{}\mathclose{{}\left(X}\right)}\right]+\frac{\log\mathopen{}\mathclose{{}\left(X}\right)}{2}}\right\\}\\\
&=\exp\mathopen{}\mathclose{{}\left\\{-\frac{\mathrm{i}}{2}\frac{g_{s}}{\mathopen{}\mathclose{{}\left(2\pi}\right)^{2}}\mathopen{}\mathclose{{}\left[\log^{2}\mathopen{}\mathclose{{}\left(X^{S}}\right)+\mathrm{i}2\pi\log\mathopen{}\mathclose{{}\left(X^{S}}\right)}\right]-\frac{\log\mathopen{}\mathclose{{}\left(X^{S}}\right)}{2}}\right\\},\end{split}$
(A.169)

which explains all of the functions and symbols appearing in (A.165).

One can see that the GV and NS instanton parts are given as series expansions
in $Q_{B}$ and $Q_{B}^{S}$ respectively,

$\begin{split}&Z_{\mathbb{F}_{0}}^{\text{GV,
inst}}\mathopen{}\mathclose{{}\left(X,Q_{B},Q_{F},g_{s}}\right)=\\\ &\ \
\qquad
1-\mathopen{}\mathclose{{}\left[\frac{Q_{1}\mathopen{}\mathclose{{}\left(2Q_{F}Q_{1}-Q_{F}-1}\right)}{\mathopen{}\mathclose{{}\left(q-1}\right)\mathopen{}\mathclose{{}\left(Q_{F}-1}\right)^{2}\mathopen{}\mathclose{{}\left(Q_{1}-1}\right)\mathopen{}\mathclose{{}\left(Q_{F}Q_{1}-1}\right)}}\right]Q_{B}+\mathcal{O}\mathopen{}\mathclose{{}\left(Q_{B}^{2}}\right)\\\
&Z_{\mathbb{F}_{0}}^{\text{NS,
inst}}\mathopen{}\mathclose{{}\left(X^{S},Q_{B}^{S},Q_{F}^{S},g_{s}}\right)=\\\
&\ \ \qquad
1-\mathopen{}\mathclose{{}\left[\frac{\mathopen{}\mathclose{{}\left(q^{S}}\right)^{2}Q_{1}^{S}\mathopen{}\mathclose{{}\left(q^{S}\mathopen{}\mathclose{{}\left(q^{S}+1}\right)Q_{F}^{S}Q_{1}^{S}-Q_{F}^{S}-1}\right)}{(q^{S}-1)(Q_{F}^{S}-q^{S})(q^{S}Q_{F}^{S}-1)(q^{S}Q_{1}^{S}-1)(q^{S}Q_{F}^{S}Q_{1}^{S}-1)}}\right]Q_{B}^{S}+\mathcal{O}\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(Q_{B}^{S}}\right)^{2}}\right)\end{split}$
(A.170)

where $Q_{1}=\sqrt{q}/\sqrt{Q_{F}}X$ and $Q_{1}^{S}=-1/\sqrt{Q_{F}^{S}}X^{S}$.

By looking at the full expression (A.165), we see that there are two distinct
four-dimensional limits that can be implemented: one in which $Z^{\rm
GV}_{{\mathbb{F}}_{0}}$ survives, and another in which $Z^{\rm
NS}_{{\mathbb{F}}_{0}}$ survives. Using the TS/ST correspondence one can show
that, at the level of the operator theory the first limit makes contact with
(3.37) and (3.38). The second limit instead gives (3.33), (3.34). A more
detailled explanation of this phenomenon can be found in bgt ; bgt2 ;
Gavrylenko:2023ewx . For the purposes of this paper, our primary focus is
directed towards the first limit.

####  A.1.4 The 1-loop part in terms of Faddeev’s quantum dilogarithm

To take the 4d limit of the partition function (A.165) we first need to
rewrite the 1-loop part and combine $Z_{\mathbb{F}_{0}}^{\text{GV}}$ and
$Z_{\mathbb{F}_{0}}^{\text{NS}}$ into a single function. From (A.163), (A.166)
and (A.167) we have

$\begin{split}Z^{\text{GV,
1-loop}}_{\mathbb{F}_{0}}\mathopen{}\mathclose{{}\left(X,Q_{F},q}\right)&=\mathopen{}\mathclose{{}\left(\frac{\sqrt{q}}{\sqrt{Q_{F}}X};q}\right)_{\infty}\mathopen{}\mathclose{{}\left(\frac{\sqrt{q}\sqrt{Q_{F}}}{X};q}\right)_{\infty}\\\
&=\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(\frac{1}{\sqrt{q}\sqrt{Q_{F}}X};\frac{1}{q}}\right)_{\infty}\mathopen{}\mathclose{{}\left(\frac{\sqrt{Q_{F}}}{\sqrt{q}X};\frac{1}{q}}\right)_{\infty}}\right]^{-1},\\\
Z^{\text{NS,
1-loop}}_{\mathbb{F}_{0}}\mathopen{}\mathclose{{}\left(X^{S},Q_{F}^{S},q^{S}}\right)&=\mathopen{}\mathclose{{}\left(\frac{q^{S}}{\sqrt{Q_{F}^{S}}X^{S}};q^{S}}\right)_{\infty}\mathopen{}\mathclose{{}\left(\frac{q^{S}\sqrt{Q_{F}^{S}}}{X^{S}};q^{S}}\right)_{\infty}\\\
&=\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(\frac{1}{\sqrt{Q_{F}^{S}}X^{S}};\frac{1}{q^{S}}}\right)_{\infty}\mathopen{}\mathclose{{}\left(\frac{\sqrt{Q_{F}^{S}}}{X^{S}};\frac{1}{q^{S}}}\right)_{\infty}}\right]^{-1},\end{split}$
(A.171)

where $\mathopen{}\mathclose{{}\left(a;q}\right)_{\infty}$ is the
$q$-Pochhammer symbol. The product of $Z^{\text{GV, 1-loop}}_{\mathbb{F}_{0}}$
and $Z^{\text{NS, 1-loop}}_{\mathbb{F}_{0}}$ can then be written as a product
of quantum dilogarithms.

Faddeev’s quantum dilogarithm can be defined by faddeev  (Kashaev:2000ku, ,
eq. 2) (kama, , eqs. 157, 158)

$\Phi_{b}\mathopen{}\mathclose{{}\left(z}\right)=\frac{\mathopen{}\mathclose{{}\left(-\mathrm{e}^{\pi\mathrm{i}b^{2}+2\pi
bz};\mathrm{e}^{2\pi\mathrm{i}b^{2}}}\right)_{\infty}}{\mathopen{}\mathclose{{}\left(-\mathrm{e}^{-\pi\mathrm{i}b^{-2}+2\pi
b^{-1}z};\mathrm{e}^{-2\pi\mathrm{i}b^{-2}}}\right)_{\infty}},\qquad\qquad\qquad\mathrm{Im}\mathopen{}\mathclose{{}\left(b^{2}}\right)>0,$
(A.172)

and it can be extended to all
$b^{2}\in\mathbb{C}\setminus\mathbb{R}_{\leqslant 0}$. The quantum
dilogarithm282828Faddeev’s quantum dilogarithm appears in many fields of
science under various names (Faddeev:2014hia, , pp. 5-6). It is also known as
“non-compact quantum dilogarithm” or simply “quantum dilogarithm”
(Kashaev:2000ku, , p. 2) and it is closely related to the “modular quantum
dilogarithm” from spectral theory, the “double sine function” from number
theory, and the “hyperbolic gamma function” from integrable systems
(Faddeev:2014hia, , pp. 5-6). is a meromorphic function of $z$ with zero’s for
$z\in-\mathopen{}\mathclose{{}\left(b+b^{-1}}\right)\mathopen{}\mathclose{{}\left(\mathrm{i}/2+\mathrm{i}\mathbb{N}}\right)$
and poles for
$z\in\mathopen{}\mathclose{{}\left(b+b^{-1}}\right)\mathopen{}\mathclose{{}\left(\mathrm{i}/2+\mathrm{i}\mathbb{N}}\right)$
(kama, , eq. 160) and with an essential singularity at infinity
(Kashaev:2000ku, , p. 2). If one takes $b^{2}=2\pi/g_{s}>0$ one has

$q=\mathrm{e}^{\mathrm{i}g_{s}}=\mathrm{e}^{\mathrm{i}2\pi
b^{-2}}\,,\qquad\qquad\qquad
q^{S}=\mathrm{e}^{\mathrm{i}\mathopen{}\mathclose{{}\left(2\pi}\right)^{2}/g_{s}}=\mathrm{e}^{\mathrm{i}2\pi
b^{2}},$ (A.173)

and defining $z_{+}$ and $z_{-}$ by

$\frac{1}{\sqrt{Q_{F}}X}=-\mathrm{e}^{2\pi
b^{-1}z_{+}}=\mathopen{}\mathclose{{}\left(\frac{1}{\sqrt{Q_{F}^{S}}X^{S}}}\right)^{1/b^{2}},\quad\qquad\frac{\sqrt{Q_{F}}}{X}=-\mathrm{e}^{2\pi
b^{-1}z_{-}}=\mathopen{}\mathclose{{}\left(\frac{\sqrt{Q_{F}^{S}}}{X^{S}}}\right)^{1/b^{2}},$
(A.174)

one gets for the 1-loop part of the partition function (A.165)

$Z_{\mathbb{F}_{0}}^{\text{1-loop}}=Z^{\text{GV,
1-loop}}_{\mathbb{F}_{0}}Z^{\text{NS,
1-loop}}_{\mathbb{F}_{0}}=\Phi_{b}\mathopen{}\mathclose{{}\left(z_{+}}\right)\Phi_{b}\mathopen{}\mathclose{{}\left(z_{-}}\right).$
(A.175)

Another useful property of Faddeev’s quantum dilogarithm