Painlev\'e Kernels and Surface Defects at Strong Coupling

It is well established that the spectral analysis of canonically quantized four-dimensional Seiberg-Witten curves can be systematically studied 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 associated with Painlev\'e equations and whose spectral traces are related to 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 $\mathrm{O}(2)$ matrix model. We then show that these eigenfunctions are computed by surface defects in $\mathrm{SU}(2)$ 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.


Introduction
Building upon the work of Seiberg and Witten [1,2], important results have been obtained for 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 Ω-background [3][4][5][6].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 N = 2 supersymmetric gauge theories in four dimensions to the spectral theory of quantum mechanical operators on the space of square-integrable functions L2 (R).
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 N = 2 gauge theories in the self-dual phase of the Ω-background (ϵ 1 = −ϵ 2 = ϵ), 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 [42][43][44][45][46]. Their relevance in the context of four-dimensional supersymmetric gauge theories and topological string was pointed out in [47][48][49][50] in close connection with the TS/ST duality [51][52][53][54] and the isomonodromy/CFT/gauge theory correspondence [55][56][57][58][59][60].
In this paper we focus on a specific operator associated to the Painlevé III 3 equation and whose spectral traces compute correlation functions in the 2d Ising model [42,45,61].Its integral kernel on R reads 2 ρ(x, y) = e −4t 1/4 cosh x e −4t 1/4 cosh y cosh x−y 2 . (1.1) For t > 0 the kernel (1.1) is positive and of trace class on L 2 (R), hence the corresponding operator has a discrete positive spectrum {E n } n⩾0 with square-integrable eigenfunctions {φ n (x, t)} n⩾0 , R dy ρ(x, y)φ n (y, t) = E n φ n (x, t). (1.2) As we will review in subsection 3.2, the spectrum is computed by the Nekrasov function of 4d, N = 2, SU(2) super Yang-Mills (SYM) in the self-dual phase of the Ω-background [47], see (3.7) and (3.8).The purpose of this paper is to study the eigenfunctions of (1.1) and relate them to surface defects in 4d, N = 2, SU(2) SYM in the self-dual phase of the Ω-background.In addition, we use this relation to obtain a strong coupling expression for such defects which resums both the instanton expansion and the ϵ expansion.

Results
The paper can be summarized as follows.Adopting the approach of [46,62] we construct eigenfunctions of (1.1) from expectation values of a determinant-like expression, Ξ ± (x, t, E) = e −4t 1/4 cosh x e ±x/2 N ⩾0 (±κ More precisely, (2.1) are square-integrable eigenfunctions φ n of (1.1) if we set κ = −E −1 n , where E n is an eigenvalue of the operator (1.1), In section 6 we show that (2.1) and (2.2) are explicitly related to surface defects in 4d, SU (2), N = 2 SYM in the self-dual phase (ϵ 1 = −ϵ 2 = ϵ) of the Ω-background 3 .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 [17, Sec.2.3.3] 4 .Hence we denote its partition function by Z II tot (q, t, σ).The explicit expression is given in (6.1) and (6.4).In the gauge theory we typically use where y is related to the position of the defect, ϵ = ϵ 1 = −ϵ 2 is the Ω-background parameter, Λ ∼ e −1/g 2 YM is the instanton counting parameter and a is the Coulomb branch parameter.The relation between the determinant like expression (2.1)The quantization condition for the energy spectrum of (1.1) was derived in [47], see (3.7) and (3.8).By evaluating the defect partition function on the lhs of (2.5) at the corresponding quantized values of σ = 1/2 + iσ n , we obtain the eigenfunctions φ n of (1.1), where σ n ∈ R \ {0} are solutions to (3.8).An example is shown in Figure 1.In section 6 and Appendix B we show that we can equivalently write (2.5) as where σ * is such that 0 < σ * < |Re(q)| if Re(q) ̸ = 0 and simply σ * > 0 if Re(q) = 0.This choice of σ * guarantees that the integration over σ in (2.7) avoids the poles of the integrand.Let us elaborate more on the meaning of (2.7).
-The Fourier transform on the rhs of (2.7) relates two types of defects [17,Sec. 2.3.3]or, more precisely, two phases of the same defect [19,Sec. 4.2]5 .In particular, while Z II tot (q, t, σ) 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 [19], see also [63,[71][72][73] 6 .Following [17,Sec. 2.3.3],we refer to the Fourier transform of a type II defect as a type I defect 7 .Via the AGT correspondence [74], the latter is realized in Liouville CFT by considering the five point function of four primaries with one degenerate field, the so called Φ 2,1 field [24,25,68,[75][76][77].One can equivalently realize this defect by coupling the four-dimensional theory to a two-dimensional theory, see for instance [15][16][17][18][19][20][78][79][80][81] and references therein.
This also means that we could get rid of the Fourier transform on the lhs of (2.5) by replacing the partition function of the type II defect Z II with the partition function of the type I defect Z I .The instanton counting like-expression of type I defect can be found for instance in [15], however we will not use such expression here as we will mainly focus on type II defects.
-The integral over σ on the lhs of (2.7) is responsable for the change of frame: it brings us from the weakly coupled electric frame, where Z II tot is defined, to the magnetic frame which is the suitable frame to describe the monopole point of SYM, see section 5.
In summary, (2.7) means that the matrix model average (2.2) computes the type I surface defect partition function of 4d, SU(2), N = 2 SYM in the self-dual phase (ϵ 1 = −ϵ 2 = ϵ) of the Ω-background and in the magnetic frame.In this identification z = exp(x) is the position of the defect and the 't Hooft parameter of the matrix model is identified with the dual period, N ϵ = a D .
Note also that (2.2) is exact both in Λ and in ϵ; 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/Λ expansion can be obtained in a straitforward way from (2.2) since it corresponds to expanding the matrix model around its Gaussian point, see [50,Sec. 5] and references therein.

Derivation
Let us briefly comment on the derivation of equations (2.5), (2.6) and (2.7).Firstly, we obtained these results by analyzing the large N expansion of the matrix models (2.2) and then extrapolating to finite N .Secondly, part of the intuition also comes from the open version of the TS/ST correspondence [62,82], see subsection 3.4 and section 7. By combining these two approaches we obtained (2.5)-(2.7),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 wellestablished relationship between the modified Mathieu operator and the four-dimensional, SU(2), N = 2 SYM in the NS phase of the Ω-background.We then present the connection between the operator (1.1) and the same gauge theory but in the self-dual phase of the Ωbackground.In section 4 we compute the planar resolvent of (2.2) as well as the planar twopoint function and show how the Seiberg-Witten geometry emerges from it.In section 5 we show that the 't Hooft expansion of (2.2) reproduce the ϵ expansion of the type I selfdual surface defect in the magnetic frame.To establish this connection, we rely on two crucial findings.Firstly, according to the results presented in [68], the ϵ expansion of the self-dual type I surface defect in the electric frame is determined by topological recursion [83].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 [84].In section 6 we test (2.7) numerically for finite N and analytically in a 1/N expansion, and we verify (2.6) numerically.
If t > 0, then the operator (3.1) has a positive discrete spectrum with square-integrable eigenfunctions.The result of [7] 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 (3.1) via the quantization condition of the twisted superpotential and the Matone relation.The first condition leads to where F NS is the Nekrasov-Shatashvili (NS) free energy.Its small t expansion reads8 where ψ is the polygamma function of order -2.Higher orders in (3.3) can be computed by using combinatorics of Young diagrams, see [85] for a review and list of references.
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 defect10 in the NS phase of the Ω-background, see [12][13][14][15][16][17][18][19] and reference there.

New: Painlevé kernels and the self-dual phase of the Ω-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 Ω-background (ϵ 1 + ϵ 2 = 0).We focus on the four-dimensional, N = 2, SU(2) SYM.In this case the relevant operator is denoted by ρ and its kernel on R is which corresponds to the density matrix for an ideal Fermi gas in an external potential − log[v(x)] [61].We therefore refer to (3.5) as a Fermi gas operator.For t > 0 (3.5) is a trace class operator on L 2 (R) with a positive discrete spectrum It was demonstrated in [47] that the spectrum is given by where σ n ∈ R are solutions to k∈Z and Z Nek (t, σ) is the Nekrasov function in the self-dual phase of the Ω-background: (3.9) The convergence of the series (3.9) was proven in [86] for any t > 0 and fixed 2σ / ∈ Z. Often the Nekrasov function is expressed using Λ, a and ϵ, which are related to t and σ via (2.4) As shown in [47], the quantization condition (3.8) follows from the identity which was demonstrated using the theory of Painlevé equations.Even though Z Nek (t, σ) has poles when 2σ ∈ Z, the sum on the rhs of (3.11) removes these poles and the resulting expression is well-defined also for these values of σ [86].
It is useful to write the Fredholm determinant on the lhs of (3.11) by using the spectral traces, det(1 ) S N being the permutation group of N elements.The Cauchy identity allows us to write (3.13) as [61] It was found in [47] that the matrix model (3.14)The equality (3.15) was demonstrated in [47,48].Finally, we emphasise that (3.14) is exact with respect to both the instanton counting parameter Λ and the Ω-background parameter ϵ.When we expand (3.14) at large Λ, while keeping ϵ and a D fixed, we obtain an analogous expansion to that found when performing a large-time expansion in isomonodromic deformation equations [50,86,[91][92][93][94][95].On the matrix model side this is an expansion around the Gaussian point.Similarly, if we expand at small ϵ while keeping Λ and a D fixed, we recover the expansion resulting from the holomorphic anomaly algorithm [96,97].
The goal of this work is to extend these results to the eigenfunctions of (3.5), which on the gauge theory side corresponds to inserting surface defects.As a first observation, we note that the kernel (3.5) falls in the class of operators studied in [46], and more recently in [62,Sec. 2].In particular, following [46,62] we can construct eigenfunctions of (3.5) using the matrix model (3.14).Let us define Ξ ± (x, t, κ) = e −4t 1/4 cosh(x) e ±x/2 N ⩾0 (±κ) N Ψ N (e x , t), x ∈ R , (3.17) This can be verified by using [62, eqs.2.46, 2.59] and φ n (x, t) = (−1) n φ n (−x, t).We will argue in the forthcoming sections that the matrix model with insertion Ψ N (z, t) corresponds to a surface defect in the four-dimensional, N = 2, SU(2) SYM in the self-dual phase of the Ω-background and in the magnetic frame.

Comment on blowup equations
It was first pointed out in [98] that the five-dimensional NS and sefl-dual partition functions are closely connected, which was subsequently demonstrated using Nakajima-Yoshioka blowup equations in [99].The interplay between these two phases of the Ω-background was extended to surface defects in four dimensions in [100,101].Applications in the context of Painlevé equations are discussed in [95,[100][101][102][103][104].The relevance of blowup equations in the context of resurgence was also recently investigated in [105].
Given such results, it is natural to wonder whether blowup equations can be used to relate the spectrum and eigenfunctions of (3.1) and (3.5).Regarding the spectrum, the blowup formula presented in [103, eq. 5.7] reveals a one-to-one correspondence between the solutions {σ n } n⩾0 of (3.2) and the solutions {σ n } n⩾0 of (3.8).However, to obtain the spectrum we further need the quantum Matone relation (3.4) on the Mathieu side, and the relation (3.7) on the Fermi gas side.These two relations are very different, and therefore the spectrum of (3.1) and (3.5) is related in a highly non-trivial way.It would be interesting to see if blowup equations in presence of defects [100,101], could be used to establish a map between the eigenfunctions of these two operators.

Two limits of quantum mirror curves
Both operators (3.1) and (3.5) 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 N = 2 supersymmetric theories can be engineered by using topological string theory on toric CY manifolds [106][107][108].The partition function of refined topological string theory is then identified with the partition function of a five-dimensional N = 1 theory on R 4 × S 1 [109,110].If we shrink the S 1 circle we get the 4d theory we are interested in, we refer to [85] for a review and more references.For N = 2, SU(2) SYM the relevant setup is topological string theory on local F 0 .The mirror curve of local F 0 is where κ and m F 0 are the complex moduli.The quantization of this curve [63,111] leads to the operator is of trace class with a positive discrete spectrum 13 [51, 53, 114].Hence a natural object to consider is its Fredholm determinant The operator (3.1) can be obtained from (3.21) by implementing the usual geometric engineering limit [106,108] where we scale and take β → 0. In this limit we obtain the modified Mathieu operator (3.1), Likewise the Fredholm determinant becomes and we have an explicit expression for this determinant via the NS functions [26,Sec. 5] where C(t) is a normalization constant and the relation σ ≡ σ(t, E) is obtained from (3.4).The Fermi gas operator (3.5) on the other hand can be obtained from ρ F 0 by implementing a rescaled limit [47], and β → 0. This is called "the dual 4d limit" in [47].The scaling (3.28) 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 where ρ is the operator (3.5).This determinant can also be written as the Zak transform of the self-dual Nekrasov function (3.11).
Let us conclude this section by emphasizing that (3.1) has a natural interpretation directly within the four-dimensional theory, independently of the five-dimensional quantum curve.In particular, (3.1) is the standard quantization of the four-dimensional SW curve for SU(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 (3.5) 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 [115][116][117][118] 14 .

The Seiberg-Witten geometry from the matrix model
In this section we study the 't Hooft expansion of the matrix model (3.18) and show how the Seiberg-Witten geometry emerges from it.For this purpose it is useful to parameterise t = (Λ/ϵ) 4 as before in (3.10) and to introduce the potential 15 and we take Λ, ϵ > 0 for convenience.The matrix models (3.14) and (3.18) can then be studied in a 't Hooft limit where with the defect insertion parameter z, the instanton counting parameter Λ, and the 't Hooft parameter λ all kept fixed.This limit was implemented on the matrix model without insertions (3.14) in [47,61].In particular, in this limit the eigenvalues distribute along and the 't Hooft parameter λ is given by ) 14 We would like to thank M. Mariño and N. Orantin for useful discussions on this point. 15The potential of the one-dimensional ideal Fermi gas is − log(v(x)) = V (e x )/ϵ.
where K and E are the complete elliptic integrals of the first (C.1) and second (C.2) kind respectively 16 .Later we will use the inversion of this relation for small λ, In the 't Hooft limit (4.2) we have the following behaviour where ≃ stands for asymptotic equality.The first two terms read ) and higher order terms can also be computed systematically [47].
Let us now consider the model with insertions (3.18).In the 't Hooft limit (4.2) we have the following behaviour [119][120][121][122][123][124] The leading order term T 0 is related to the even part of the planar resolvent 17 ω 0 + [62, eq.3.35], and the subleading order term T 1 is given by [62,123,124] where W 0 ++ is the even part of the planar two-point correlator.It can be expressed explicitly in terms of g as [124], [62, eq. 3.43] . (4.12) 16 See Appendix C for our conventions on elliptic integrals. 17The planar resolvent is defined and computed explicitly further on in subsection 4.1.

The planar resolvent
The planar resolvent is where the normalized expectation value is with respect to matrix model without insertions Z(t, N ) (3.14), and the z n are the eigenvalues over which one integrates in (3.14).At large z one finds and we refer to ⟨W ⟩ as a Wilson loop by analogy with [125].It is useful to split the the planar resolvent in an even and an odd part, where ω 0 ± (z) are both even in z.The even part of the planar resolvent ω 0 + for the model (3.18) has the following integral form [124, eq. 4.16], where C is an anticlockwise contour around the branch cut from g to g −1 , which does not include the two poles at y = ±z.In the matrix model Ψ N (z, t) (3.18) we naturally have z > 0. However, it is useful to consider more generally z ∈ C from now on, and (4.16) makes indeed sense for complex values of z as well [124].
we can write write (4.16) as where we used the form of the potential given in (4.1).The integral in (4.17) can be decomposed in partial fractions, where 0 < b < a and we defined Using [126, eqs. 256.39, 257.39] one finds for z / ∈ [b, a], where k is the elliptic modulus given by and sn(v|k 2 ) is the Jacobi elliptic function known as the sine amplitude (C.5).From [126, eq. 340.01] where F and Π are the incomplete elliptic integrals of the first (C.1) and third (C.3) kind respectively.It is useful to note that v = 0 corresponds to ϕ = 0 and v = K(k 2 ) corresponds to ϕ = π/2.At the end this gives as well as These particular combinations of elliptic integrals can be reduced to square roots by making use of the following addition formula for 0 < k < 1 [126, eqs. 117.02] 18 , where Combining (4.25) and (4.26) gives and the combination of (4.24) and (4.26) leads to where Taking a −1 = b = g and using everything above we finally find for the even planar resolvent Even though we derive (4.30) for z 2 ∈ C \ [g 2 , g −2 ] one can verify that (4.30) holds on the whole complex plane.As a consistency check we compared the analytical result (4.30) against the numerical evaluation of (4.16), and found perfect agreement.One can also see that (4.30) has the correct asymptotic behaviour, In addition, from the coefficient of the z −2 -term in the z → ∞ expansion we get a closed form expression for the Wilson loop (4.14), Using (4.5) we obtain Using (4.30) gives for the leading order T 0 (4.10) of the matrix model (4.9) in the 't Hooft limit (4.2) An important point of (4.34) is that the Seiberg-Witten curve of N = 2, SU(2) SYM, emerges in the planar limit, provided we identify the following quadratic differentials and at the same time relate g to u by In equations (4.35) and (4.37) u denotes as usual the vacuum expectation value of the scalar in the vector multiplet of SU(2) SYM.

The planar two-point function
In the previous section we showed that the Seiberg-Witten curve (4.35) 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 [76] B q 1 ,q 2 ,q 3 (z with and where q i are the branch points of σ(z) = −z(z 2 − (u/4Λ 2 )z + 1), The choice of the order fixes the choice of frame.What we find is that the relevant order here is (4.42) 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 (4.12) is related to the Bergman kernel (4.38) by Hence the subleading order T 1 (4.11) of the matrix model (4.9) in the 't Hooft limit (4.2) becomes (4.44) 5 Testing the ϵ 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 [127].Consequently, the genus g free energies behave as almost modular forms under a change of frame [128].The wavefunction behaviour was generalized to the open topological string sector in [84].
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 [106][107][108].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 A cycle and the corresponding A period on the SW curve (4.35) are chosen to be where y(z) is given in (4.35) and E is the complete elliptic integral of second kind (C.2).We usually denote a ≡ Π A .Likewise the B cycle and the corresponding period are where K is the complete elliptic integral of the first kind (C.1).The g ±1 are roots of the SW curve, y(g ±1 ) = 0, and are given in (4.37).We usually denote a D ≡ iΠ 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 A-and B-cycles.For the SU(2) SYM that we study in this paper, the transformation properties of the partition function under a change of frame were studied in [97].The ϵ expansion of (3.15) leads exactly to such transformations, as we discuss below.

The electric frame
We consider a type I defect in the self-dual phase of the Ω-background (ϵ 1 = −ϵ 2 = ϵ), and we denote the partition function of this surface defect by if we are in the electric frame.As pointed out in [68], based on [129,130], we can compute these defects via the Eynard-Orantin topological recursion [83].More precisely we have where W g,h (z 1 , ...z h )dz 1 . . .dz h is an infinite sequence of meromorphic differentials constructed via the topological recursion [83] and whose starting point is the underling SW geometry (4.35).Note that we are implicitly using the dictionary (3.10) and the SW relation (5.1) to express σ = ia/2ϵ as a function of the SW parameter u.
For the so-called disk amplitude W 0,1 we have19 and we note (5.6) The annulus amplitude W 0,2 is given by where B q 1 , q 2 , q 3 is defined as in (4.38), but the choice of q i 's is different.Here we have (5.8) so that q 1 = q 3 , q 2 = q 2 and q 3 = q 1 (4.42).We denote (5.9) Hence to subleading order (5.4) reads (5.10) Given the spectral curve (4.35) with W 0,1 and W 0,2 , higher order terms in the ϵ expansion (5.4) can be computed recursively by using the topological recursion [83].

The magnetic frame
Our proposal is that the matrix model (3.18) computes the type I surface defect (5.3) in the magnetic frame.In this section we test this proposal in the 't Hooft expansion (4.2).

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 ϵ expansion of (3.15).Using the dictionary (3.10) the ϵ expansion of the Nekrasov function reads where F g are the genus g free energies of SU(2) SYM in the electric frame.(5.12) It was found in [47] that the F g 's in (5.12) are the SYM free energies in the magnetic frame.More precisely e g⩾0 ϵ 2g−2 Fg(Λ,λ) ∼ iR da e −πaN/ϵ e g⩾0 ϵ 2g−2 Fg(Λ,a) , ( where ∼ indicates a proportionality between two (divergent) series 20 .The integral on the rhs of (5.13) should be understood as a saddle point expansion.This saddle point expansion characterizes the change of frame in SW theory and topological string [127], and it has a direct interpretation from the point of view of modular transformations [128].It allows us to make the transition from the weak coupling electric frame, where the Nekrasov function (3.9) is defined, to the strong coupling magnetic frame, where the matrix model (3.14) naturally emerges.By writing the saddle point expansion on the rhs of (5.13) explicitly we get iR da e −πaN/ϵ e g⩾0 ϵ 2g−2 Fg(Λ,a) = exp 1 where λ = N ϵ and a(λ) is determined by the saddle point equation (5.15) By using (3.9) and the dictionary (3.10) we get we find that λ in (5.15) agrees with (4.4) as it should.The matching between the two sides of (5.13) was discussed in [47].We also note that the classical Matone relation (5.17) can be inverted and one finds the usual expression for the A-period of the SW curve (4.35) given in (5.1).Likewise ∂ a F 0 is identified with the B-period of the SW curve21 where a D is given in (5.2).

The partition function with defects
We are interested in extending the analysis to the 't Hooft expansion (4.9) of the matrix model with insertion Ψ N (z, t) (3.18).More precisely we claim that Ψ N (z, t) gives the selfdual type I surface defect (5.3) in the magnetic frame.As we reviewed above, the change of frame for the partition function is encoded in an integral transform (5.13).As first shown in [84], 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 [78].
At the level of the ϵ expansion our conjecture reads ∼ iR da e −πaN/ϵ e g⩾0 ϵ 2g−2 Fg(Λ,a) e g⩾0 h⩾1 ϵ 2g−2+h z where W g,h (z 1 , . . ., z h )dz 1 • • • dz h are the electric differentials appearing in the topological recursion setup (5.4); whereas T n are the magnetic matrix model coefficients appearing in (4.9), (5.20) Parallel to (5.13), the integral on the rhs of (5.19) should be understood as a saddle point expansion which characterizes the change of frame.Equation (5.19) reads to subleading order in ϵ +W I 1 (z, a(λ))+O(ϵ) , (5.21) where a and λ are again related by the saddle point equation (5.15).In (5.21) we already used (5.13) and (5.14) to get rid of terms involving only the free energies F g and F g .We show below that the equality in (5.21) indeed holds order by order in ϵ.
At the leading order ϵ −1 , the matching on the two sides of (5.21) follows directly from (4.34) and (5.5).For the subleading order ϵ 0 , we first note that the Bergman kernel entering in T 1 (4.44), can be written as where we used (4.37) and (5.8).Hence we can rewrite (4.44), which leads then to where we used (5.1).From (5.18) we have and by combining (5.25) with the identity (5.26) we find which is precisely what we wanted to prove.
To summarize, we have tested (5.19) at leading and subleading order22 in ϵ.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 (5.3) satisfy the topological recursion was conjectured in [68], based on [129,130] which was recently demonstrated in [131].On the matrix model side instead, the inclusion of topological recursion in our matrix model can be derived from [83,132,133].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.

Matrix models, eigenfunctions and the type II defect
In this section we consider the Fourier transform of the matrix model with insertion Ψ N (e x , t) (3.18).The corresponding defect in four-dimensional, N = 2, SU(2) SYM can be geometrically engineered using the open topological string partition function of local 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 Ω-background is t + q( q + 1) 2 − q(10 q 2 + 19 q + 10)σ 2 + (8 where we defined for the sake of readability q = iq + 1/2.The variables q, t, σ can be expressed in terms of y, Λ, a, ϵ as in (2.4).The relation between Z II and the matrix model where σ * is chosen such that 0 < σ * < |Re(q)| if Re(q) ̸ = 0, and simply σ * > 0 if Re(q) = 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 s 3) It is convenient to introduce the total partition function as so that (6.2) can be written in a compact form as This equality can be equivalently written as Following subsection 3.2, we get the square-integrable eigenfunctions of (3.5) when we evaluate (6.7) at the values of σ which satisfy the quantization condition (3.8).That is where σ n are solutions of (3.8).In Figure 1 we plot the rhs of (6.8) for the two smallest values of σ n which satisfy the quantization condition (3.8).As a cross-check we also verified this result by a purely numerical analysis of the operator (3.5), see subsection 6.4.Let us make a few comments on the analytic properties of the gauge theoretic functions.
-The function Z Nek (t, σ) has poles when 2σ ∈ Z and Z II tot (q, t, σ) has additional poles when q and σ satisfy q = i 2 ± iσ + iℓ with ℓ ∈ Z.
-If we are strictly interested only in the spectral problem associated to the integral kernel (3.5), then q ∈ R and σ ∈ 1 2 + iR >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 Nek (t, σ) appearing on the lhs of (3.11), then this has no longer poles in σ: the summation over k in (3.11) 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 (6.7) has also the effect of removing the poles.
In the forthcoming subsections we test (6.5) and (6.8) in several ways.

Testing N = 0
As a first check of (6.5) we test the N = 0 case.From (3.18) one can see that Ψ 0 (e x , t) = 1 so that the rhs of (6.5) is 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, for some function F (q, t).For instance we have when i2q Hence we already see the structure of the lhs of (6.5) appearing.On the gauge theory side we can perform the integral at small t by using Cauchy's residue theorem, (6.12) To get the last line in (6.12) we have included the first instanton correction in Z II tot (6.4), and higher instanton corrections can be treated similarly.The poles contributing to the integral in (6.12) are By employing the series expansions (6.11) and (6.12) we can systematically verify (6.5) for N = 0, order by order in t.

Testing N = 1
As a second consistency check of (6.5) 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 (6.5) as a one-dimensional integral.Let us define  1.Comparison between the two sides of (6.5) for N = 1, t = 1/55π 4 with q = 1/9 + i2/ √ 3 (upper left), q = 1/π (upper right), and q = i/3 (lower).I 1 is the integral (6.15) appearing on the rhs of (6.5); n inst refers to the number of instanons we include in the defect partition function appearing on the lhs of (6.5).
After some algebra we get One useful observation is that the above integral vanishes when q = −i/4, which is in perfect agreement with the lhs of (6.5).Unfortunately we can not compute the integral (6.15) analytically.Hence for N = 1 the test of (6.5) is done numerically and we find perfect agreement.One such test is given in Table 1.

Testing large N with a 't Hooft limit
Another analytical test of the identity (6.5) consists of comparing both sides in the 't Hooft limit where as in (4.2) and with the 't Hooft coupling λ fixed.We will need to use that q and t scale as in (2.4), with both the position of the defect y ∈ C and the instanton counting parameter Λ > 0 kept fixed.
The computation of the 't Hooft limit of (6.5) is simplified by using the corresponding statement for the theory without defects, which is given in (3.15) and was obtained in [47,48].In particular one can divide both sides of (6.5) by (3.15) to get Note that (6.18) is by (3.15) equivalent to (6.5), but rewritten in a form suitable and convenient for the 't Hooft limit (6.16).

The 't Hooft limit on the gauge theory side
The general pattern of the 't Hooft expansion of the left hand side in (6.18) is the same as in subsection 5.2.Using that the integration variable σ can be related to the Coulomb branch parameter a by (3.10), one expands the logarithm of the Nekrasov partition function Z Nek in even powers of ϵ with the leading order being ϵ −2 .On the other hand, the expansion of the logarithm of the defect partition function Z II contains all integer powers of ϵ starting from ϵ −1 , (2π Hence the saddles of both integrals on the left hand side of (6.18) are determined by the same equation (5.15).This gives the functional relation a(Λ, λ), but for us it will be convenient to rather invert this to λ(Λ, a) and keep a explicitly.Keeping this in mind the 't Hooft limit of the left hand side of (6.18) leads eventually to where we suppressed the functional dependence on Λ and a in the notation.

The 't Hooft limit on the matrix model side
Consider the inverse Fourier transform on the right hand side in (6.18),The sum over s is a sum over the saddles and x s = x s (y) is determined by the saddle point equation, where we used (4.34) and z = exp(x).Taking the square of this equation gives the Seiberg-Witten curve (4.35) if we take as before (4.37), .26)This leads to the following two solutions, Let us take a moment to consider the behaviour of z ± (y) as a function of y.One can check that z ± (y) is real and outside the branch cut region of the matrix model if and only if iy ∈ R \ {0}.It is important to note that with this choice of iy ∈ R \ {0} one has z 2 − (y) > 1/g 2 and 0 ⩽ z 2 + (y) < g 2 .Moreover there are no possible choices of y ∈ C such that 0 ⩽ z 2 − (y) < g 2 or z 2 + (y) > 1/g 2 .One finds on the other hand that z ± (y) is real and inside the branch cut region if and only if 0 ⩽ y 2 ⩽ u − 8Λ 2 , and also that z ± (y) is purely imaginary if and only if y 2 ⩾ u + 8Λ 2 .For all other choices of y ∈ C one will find generic complex z ± (y).

Comparing the gauge theory and the matrix model
To analyze the leading order of the 't Hooft expansion in (6.24) with the saddles (6.27) 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 (6.25).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 (6.24) one gets by making use of the saddle point equation (6.25) and its solutions d dy T ±,0 (y) = −ix ± (y) .(6.28) Comparing this to the leading order of the gauge theory (6.20) we can check that24 d dy W II 0 (±y) − T ±S,0 (y) = 0 , ( where S = sgn[arg(i(y 2 − a 2 ))] with the convention that sgn(0) = −1 .
Let us then look at the constant term for y = 0.At the gauge theory side we have for the leading order (6.20) From (6.27) one can see that z ± (0) = g ± > 0, with g ± as in (6.26).Using (4.10) gives for the leading order of the matrix model (6.24) where the even planar resolvent ω 0 + (z) is given in (4.30).Note that the difference between the leading terms of the two saddles is The last equality can be obtained in an Λ → 0 expansion or exactly using [124, eq. 4.18], which shows that this relation does not depend on the particular form of the potential.
We have furthermore that with K and E the complete elliptic integrals of the first (C.1) and second (C.2) kind respectively.The last equality was found in an Λ → 0 or equivalently g → 0 expansion using (6.26) and the Matone relation (5.17).Hence from (6.30), (6.32) and (6.33) So the constant parts of the leading order terms agree and together with (6.29) this proves the equality in (6.18) and hence (6.5) to leading order in the 't Hooft limit (6.16).
The subleading order can be checked in analogy with section 5. Matching at higher order in ϵ can then be inferred from topological recursion, as we discussed near the end of section 5.

Numerical eigenfunctions
The numerical analysis of the spectrum and the eigenfunctions for the integral kernel ρ (3.5) is done exactly as in [134, sec.2.2].To make the presentation self-contained let us review the strategy of [134, sec.2.2].We are interested in studying numerically the eigenvalue equation R dy ρ(x, y)φ n (y, t) = E n φ n (x, t) , (6.35) where the kernel ρ(x, y) is defined in (3.5).It is convenient to decompose ρ(x, y) as and to define v Then (6.35) reads which we can also write as with H the infinite dimensional Hankel matrix defined by .40)This means that the eigenvalues of H coincide with those of ρ(x, y) and the eigenvectors of H give the eigenfunctions of ρ(x, y) via (6.38).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 th eigenvector of the Hankel matrix H (6.40) truncated at size M .Defining φ we recover the true eigenfunctions of the kernel ρ in the M → +∞ limit, lim where the proportionality factor is a numerical constant and φ n is the n th eigenfunction of (6.35) in the normalization of (6.8).We computed the lhs of (6.42) numerically and checked that this numerical expression agrees with the eigenfunctions computed by using the defect expression on the rhs of (6.8) with high precision.For instance for t = 1/(100π 8 ), by including 0 instantons in (6.8) 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 respectively25 .

Outlook
In this paper we have shown that the eigenfunctions of the operator (1.1) are computed by surface defects in N = 2, SU(2) SYM in the self-dual phase of the four-dimensional Ω-background (ϵ 1 + ϵ 2 = 0).This result, together with [47,48,50], extends the correspondence between 4d 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 (2.2).This provides a representation for the surface defect partition function which resums both the instanton and the ϵ 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/Λ (exact in ϵ 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, N = 2, SU(2) SYM and the operator (1.1).It would be interesting to extend our results in a systematic way to all 4d N = 2 theories.For example in the case of N = 2, SU(N) SYM we have N − 1 non-commuting Fermi gas operators as discussed in [48,50].We expect their eigenfunctions to be computed by surface defects in SU(N) SYM in the self-dual phase of the Ω background.
-Our results should follow from the open version of the TS/ST correspondence [62,82] by implementing the dual four-dimensional limit, see subsection 3.4.However the formulation of [62,82] only holds when the mass parameters of the underlying CY geometry are set to their most symmetric values (for local F 0 this is equivalent to setting m F 0 = 1 in (3.20)).To derive (2.7) one need a formulation of the open TS/ST correspondence for generic values of the parameters.We will report on this somewhere else [135].
-The Fredholm determinant of (1.1) computes the tau function of the Painlevé 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 [136].
-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 [45,137].
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 Ω-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 [138][139][140][141].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 SU(2) SYM leads to the Mathieu operator (3.1).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 Ωbackground, a natural quantization scheme to investigate would be the one arizing in the context of the topological recursion [115][116][117][118].
The product of the pertubative (A.38) and instanton (A.29) (A.30) parts give us then the complete partition function for the type II defect in 4d, N = 2, SU (2)  B From the matrix model identity to the eigenfunction identity In this appendix we argue for the equivalence between the identities (6.5) and (6.6)32 .
Our strategy is similar to the one used in the context of ABJM theory, see e.g.[153]   where we have absorbed the (−1) N into a shift of σ, and σ * is a strictly positive number which guarantees that the integration contour on the left hand side of (B.1) does not hit the poles of Z II tot .This is the case if 0 < σ * < |Re(q)| ̸ = 0.If Re(q) = 0 one can take σ * to be any strictly positive number as in footnote 11.For the sake of notation let us define f (N ) = i 2 11/12 √ πt 3/16 e 3ζ ′ (−1) e 4 Note that the second equality in (B.4) assumes some good analytic properties of g, for instance g is such that the sum over n on the right hand side of (B.4) is convergent.This is the case for (B.2).Furthermore it is part of our conjecture that the function k∈Z g(σ + iσ * + k) is not only well-defined but also an entire function of σ.Hence we are free to deform the integration path in (B.4) to any path C {−1/2,1/2} , beginning at σ = −1/2 and ending at σ = 1/2, as long as we don't cross the poles coming from the tangent when σ + iσ * ∈ Z/2 + 1/2.Consider then the change of variables given by where the last equality holds whenever the Fourier series on the previous line is convergent.
We expect this to be true in our case even tough we do not have a rigorous proof.We also used that f (−N ) = 0 for N ∈ N \ {0}.
To go in the opposite direction from
The notation in [126, pp. 8-10] is slightly different and we denote their elliptic integrals with a tilde.In particular we have the normal or incomplete elliptic integral of the first kind for k 2 ∈ R, −π/2 < ϕ < π/2 and k 2 sin 2 (ϕ) < 1, [155], [126, eq. 110.02],The complete elliptic integrals are obtained by taking ϕ = π/2, The complete elliptic integrals of the first and second kind are analytic on C apart from a branch cut along the positive real line for k 2 ⩾ 1, and the complete elliptic integral of the third kind is analytic on C 2 apart from similar branch cuts for k 2 , α 2 ⩾ 1 [154,156,157].
We also need a Jacobi elliptic function sn which is an inverse of the incomplete elliptic integral of the first kind [126, p. 18], sn v k 2 , sn F ϕ k 2 k 2 = sin(ϕ) , (C.5) which is sometimes called the sine amplitude.

3
Preparation: spectral theory and 4d, N = 2 gauge theory 3.1 Well known: differential operators and the NS phase of the Ω-background Let us start by reviewing the well-known correspondence relating ordinary differential equations to four-dimensional N = 2 gauge theories in the NS phase of the Ω-background, i.e. ϵ 2 = 0, ϵ 1 = ϵ ̸ = 0.In this work we focus on SU(2) SYM.The corresponding operator is the modified Mathieu operator O Ma acting as

. 33 )
We cross-checked(4.33)by expanding the matrix model around its Gaussian point, similar to what was done in [61, app.B].
super Yang-Mills31, Note that we use a slightly different notation compared to the main text: what we call x here is called q elsewhere.
and reference therein.Let us start by writing (6.5) as