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Seiberg-Witten theory as a Fermi gas

We explore a new connection between Seiberg-Witten theory and quantum statistical systems by relating the dual partition function of SU(2) Super Yang-Mills theory in a self-dual Omega-background to the spectral determinant of an ideal Fermi gas. We show that the spectrum of this gas is encoded in the zeroes of the Painleve III tau function. In addition we find that the Nekrasov partition function on this background can be expressed as an O(2) matrix model. Our construction arises as a four-dimensional limit of a recently proposed conjecture relating topological strings and spectral theory. In this limit, we provide a mathematical proof of the conjecture for the local P1xP1 geometry.

In this paper we explore a new aspect of this web of connections, which concerns quantum statistical systems. More precisely, we show that the dual partition function of pure N = 2 SU (2) Super Yang-Mills in four dimensions in a self-dual Ω-background is the spectral determinant of a quantum system given by an ideal Fermi gas. This in turn is also related to a statistical model describing self-avoiding polymers in two-dimensions.
The core of our work relies on the merging of two topics. The first one is the connection between four-dimensional N = 2 gauge theories and Painlevé transcendants found in [18][19][20][21][22][23] and the other is a recently proposed conjecture relating topological strings and spectral theory [24].
More precisely, for the pure N = 2 gauge theory it has been shown that the NO partition function, defined in terms of the standard Nekrasov partition function as Z NO (η, a/ , T ) = n∈Z e 4πinη Z Nek ( , a + n , T ) , (1.1) computes the general τ function of the Painlevé III 3 equation. The gauge theory parameters a and η are related to the initial conditions of the specific solution. A thorough discussion of this relation including new long distance expansions of solutions to Painlevé equations and their relation with Argyres-Douglas points will be presented elsewhere [25].
On the other hand, in the conjecture of [24], one associates to any toric Calabi-Yau threefold X a trace class operator ρ X whose spectral determinant computes the non-perturbative topological string partition function. More precisely one has det (1 + κρ X ) = n∈Z e J(µ+2πin) κ = e µ (1.2) where J is the non-perturbative topological string free energy (or grand potential). It is well known that a class of N = 2 gauge theories in four dimensions can be engineered by using toric Calabi-Yau threefolds [17,26]. For instance, by taking the so-called geometric engineering limit of local P 1 × P 1 one generates the N = 2 supersymmetric SU (2) gauge theory out of string theory. More precisely, different phases of the four-dimensional theory in the gravitational background parameters can be obtained by different scaling limits of the refined topological string. In [27] the authors specified the conjecture of [24] in the four-dimensional limit corresponding to the Nekrasov-Shatashvili (NS) regime ( 2 = 0) and the resulting spectral problem was found to coincide with the quantum Toda system found in [4].
In this paper we study a different regime in the Ω-background parameters of the fourdimensional gauge theory, namely 1 = − 2 = . Moreover, we scale the Kähler parameters of the Calabi-Yau in terms of , therefore implementing a limit which is slightly different from the usual geometric engineering one, see Section 3 for details. In this limit we find that the conjecture (1.2) connects to the NO partition function (1.1) with η = 0.
Our result can be interpreted in a twofold way. On one side it is a proof of the conjecture (1.2) in the four-dimensional limit. On the other, it is a new connection between Seiberg-Witten theory on self-dual background and the spectral theory of an ideal Fermi gas. This paper is organized as follows. In the next section we recall the content of the conjecture of [24] in a form suitable for the subsequent manipulations.
In Section 3 we give a proof for the conjecture of [24] in the four-dimensional limit by using the well known results on spectral determinants [28] together with the recent developments in the context of Painlevé equations [18][19][20][21][22][23]. More precisely we show that both the l.h.s and the r.h.s of (1.2) satisfy the Painlevé III 3 equation in the τ form with the same initial conditions.
In Section 4 we analyze our results from the four-dimensional gauge theory perspective. We show that the four-dimensional SU (2) Nekrasov-Okounkov partition function can be expressed by using the grand partition function of an ideal Fermi gas. This gas is characterized by the following density operator This means that we have a one-dimensional ideal Fermi gas in an external confining potential In this way we provide a Fermi gas picture of gravitational corrections to the Seiberg-Witten theory. Notice that this picture is different from the one of [29], where the authors consider free chiral fermions lying on the Seiberg-Witten curve. As a consequence of our computation we can write the Nekrasov partition function as an O(2) matrix model [30,31] which reads The 't Hooft expansion of this model reproduces the standard perturbative expansion of the gauge theory in a self-dual Ω-background in the magnetic frame. As we will see in Section 4.2 the 't Hooft parameter N T −1/4 of the model has an explicit meaning in the gauge theory as the dual period of the Seiberg-Witten curve. Moreover (1.5) is manifestly convergent, well-defined also for finite N and it leads to an exact expression for the Nekrasov partition function which resums the instanton expansion. We would like also to stress that our model is derived without using Nekrasov's results but simply by quantizing the mirror curve and by taking a particular limit. Therefore the fact that (1.5) reproduces Nekrasov's partition function is a highly non-trivial fact.
2 The spectral theory/topological string conjecture In this section we briefly review the conjecture of [24]. For a pedagogical review see [32].
In [24], based on previous works [33][34][35][36][37], it has been conjectured that there is a precise and exact duality between spectral theory and enumerative geometry. In this construction one associates a non-pertubative trace class operator ρ X to any toric Calabi-Yau X whose mirror curve has genus one 1 . This operator has a discrete spectrum, denoted by e −En , which can be organized into a spectral determinant: (2.1) We also refer to (2.1) as Fredholm determinant. In this paper the relevant geometry is the anti-canonical bundle of P 1 × P 1 and the corresponding mirror curve is e x + e p + e −p + m F 0 e −x − κ = 0, (2.2) where κ = e µ is the so-called complex modulus of P 1 × P 1 while m F 0 is the so-called mass parameter. In the following we will often use the "renormalized" mass parameter introduced in A detailed description of this geometry and its enumerative content can be found in [40][41][42]. The trace class operator arising in the quantization of the mirror curve (2.2) is The conjecture of [24] states that the spectral determinant (2.1) can be written explicitly in terms of (refined) GV invariants of X as where the grand potential, J(µ, ) can be expressed in terms of enumerative invariants of X. This quantity was first introduced in the context of ABJM in [43] and subsequently studied in a series of works [24,33,34,[44][45][46][47][48][49]. More precisely we have The WKB part is given by where F NS is the Nekrasov-Shatashvili free energy [4], A(m, ) the so-called constant map contribution [50] and b i are real numbers related to the genus one free energy (see [38] for a detailed explanation). We denote by t i (µ, ) the quantum Kähler parameters of X and by µ the complex modulus of the mirror geometry. In addition J(µ, ) can depend on a set of mass parameters. However, for sake of notation, we do not explicitly write mass dependence. The "worldsheet" part in (2.6) is given by where F GV denotes the (unrefined) Gopakumar-Vafa [51] free energy and B is the so-called B field. Both F GV and F NS are expressed explicitly in terms of (refined) enumerative invariants [51][52][53] as explained in appendix B . For the local P 1 × P 1 geometry we have [24] where t(µ, ) is the quantum A-period of the curve (2.2), it is computed in appendix A. Moreover the B field for this geometry is B = 2 and the b i coefficients in (2.7) are b = {1/24, 1/24}. (2.10) The explicit expression for the constant map contribution A(m, ) of local P 1 × P 1 has been computed in [54] and it reads and F CS (k, M ) is the U (M ) Chern-Simons (CS) free energy. In the context of CS theory M and k are integers. However in the expression for the constant map (2.11) they are complex.
Therefore we need to analytically continue the CS partition function. This analytic continuation is not unique, however the spectral problem associated to local P 1 × P 1 fixes this ambiguity [48] and the relevant analytic continuation turns out to be the one proposed in [54], namely (2.13) Even though the conjecture of [24] has been successfully tested in many examples [24,35,39,48,[54][55][56][57][58], a proof is still missing. In this paper we perform a first step in this direction by proving the conjecture in a four-dimensional limit of local P 1 × P 1 . More precisely we show that, in this limit, the spectral determinant (2.1) and its conjectural expression (2.5) both fulfill the Painlevé III 3 equation with the same asymptotic behavior.

The four-dimensional limit
In this section we prove the conjecture [24] in a particular double-scaling limit of local P 1 × P 1 . This double-scaling limit is defined as follows. Let us denote by We parametrize the above quantities by 2) The four-dimensional limit we are going to study consists in taking β → 0 while keeping the other parameters fixed. In particular this means that therefore we are probing the non-perturbative regime of [24]. In this limit one of the two Kähler classes of P 1 × P 1 goes to infinity while the other is keep fixed. In [24] m, are positive numbers, therefore i and iσ are real and different from zero. Notice that the scaling (3.2) is slightly different from the typical four-dimensional scaling of geometric engineering [17,26] where one defines In the scaling limit (3.4) the conjecture of [24] makes contact with the SU (2) quantum Toda as shown in [27]. However to make contact with Painlevé III 3 equation we have to consider the scaling (3.2). In the following we compute the four-dimensional limit (3.2) of the spectral determinant (2.1), see Subsection 3.2, and of its conjectural expression (2.5), see Subsection 3.1, and we show that they both satisfy Painlevé III 3 equation in the τ form with the same initial conditions.

The topological string computation
In this section we focus on the four-dimensional limit of (2.5) for the local P 1 × P 1 geometry.
The standard and NS free energies determining the grand potential of this geometry are computed in appendix B. By using these results we can write the grand potential (2.6) as (3.6) We denote by J inst the instantons part of the Gopakumar-Vafa free energy (B.8), namely (3.8) In our notation J oneloop (Q f , ) contains the one-loop contribution both of standard and NS free energies. By using the techniques of [27,54], illustrated in appendix C, we can write (3.8) in closed form as Notice that the one-loop term in the gauge theory typically requires a certain regularization scheme and there exists a scheme in which one recovers the one-loop term of the topological vertex [59,60]. This is the same scheme used in the context of AGT correspondence (see appendix B.2 of [7]). As was pointed out there, this is different from the scheme used in the NO partition function [29]. Remarkably, the grand potential (2.6) automatically implements the correct scheme to make contact with the NO partition function and the Painlevé III 3 equation. Similarly the analysis of [27] shows that (2.6) also implements the correct scheme in the context of SU (2) Toda. In (3.6) we denote by P(t(µ, ), m, ) the polynomial part of the grand potential, which reads The last term in (3.6) denotes the instanton part of the WKB grand potential (2.7) which is completely determined by the instanton part of the NS free energy (B.9). This term will not be important here since it vanishes in the four-dimensional limit (3.2) where we have It is straightforward to see that (B.9), and as a consequence the instanton part of the WKB grand potential (3.11), vanishes as we take β → 0. Therefore we are studying precisely the opposite regime with respect to the one considered in the standard geometric engineering limit (3.4) and in the context of Toda systems [4,27].
To compute the spectral determinant (2.5) we have to perform the shift By using the expression for the quantum A-period given in appendix A we have Similarly the quantities in (3.1) shift according to We are now going to compute the four-dimensional limit of each quantity in (3.6).

Instanton contribution
By using the refined topological vertex [59], together with (3.2) and (3.15) we have where c k (σ, n) are the well-known coefficients of the Nekrasov partition function [61][62][63]. The first few coefficients read (3.17) Therefore in the four-dimensional limit, the original shift in the chemical potential (2.5) translates into a shift of the parameter σ by σ → σ + n. where This is the well known instanton contribution to the Nekrasov partition function as computed in [61][62][63] and it defines a convergent series in T . As mentioned earlier, in [24] m and are positive numbers, therefore σ is purely imaginary. It follows that B(T, σ) has no poles in the domain of interest.

One-loop contribution
The four-dimensional limit of the one-loop contribution can be computed straightforwardly from the integral representation (3.9). We have where G(z) is the Barnes G-function. Notice that in our convention σ is purely imaginary. We get . (3.23)

Constant map contribution
By using the explicit expression (2.11) it is easy to see that (3.25)

The four-dimensional limit of topological string
By combining together (3.24) , . (3.26) This is the four-dimensional limit (3.2) of the conjectural expression (2.5) for the spectral determinant (2.1). We add the subscript S in (3.26) to stress that this is the spectral determinant computed from the string theory side of the conjecture.
Using these results, we conclude that the conjectural expression (2.5) in the four-dimensional limit reproduces the τ function of Painlevé III 3 , namely To completely fix the τ function we also have to specify the small T expansion. We have . (3.33) Notice that in our set up iσ ∈ R/{0}, hence we do not have any problems with discontinuities or poles.

The operator theory computation
In this section we focus on the operator side of the conjecture [24]. We would like to show that the spectral determinant (2.1) for the operator is related to the τ function of Painlevé III 3 as in (3.31). We do this without using the conjectural expression (2.5). The strategy relies on the work of [28] which is summarized in appendix D.
The kernel of the operator (3.34) has been computed in [39] and it reads We denote by Φ b the Faddeev's quantum dilogarithm [66,67]. The parameter γ in (3.35) is related to the mass m in (2.3), more precisely we have The small κ expansion of the spectral determinant describing the operator (3.34) is given by where Z(N, ) can be written as a matrix model of the form [39] Z . (3.39) The potential V (z, ) is given by (3.40) We are interested in computing the four-dimensional limit of It is useful to write the quantum dilogarithm in terms of double sine function s b as in [68]. We have where we used . (3.43) The integral form of the double sine is [54] i log s b (z) = πz 2 2 + Li 2 (−e 2πbz ) 2πb 2 + The results of appendix A together with the dictionary (3.2) lead to Hence we obtain e V tot ( ,m,µ)

(3.48)
Therefore in the four-dimensional limit we can write the spectral determinant (3.38) as We add the subscript O in (3.49) to stress that this is the spectral determinant computed in the operator theory side of the conjecture. This is precisely a spectral determinant of the Zamolodchikov form (D.2). It follows that the four-dimensional limit of (2.1), for the operator (3.34), computes the τ function of Painlevé III 3 according to It is well known that the Barnes G-function is related to the polygamma function of negative order ψ (−2) as By using (D.10) together with (3.53) it is straightforward to see that the small T asymptotic of (3.52) is given precisely by (3.33). Summarizing, we have shown that, in the four-dimensional limit, both the spectral determinant (2.1) and its conjectural expression (2.5) satisfy the Painlevé III 3 equation in the τ form with the same monodromy conditions given by (σ, η = 0), (3.54) and the same asymptotic expansion (3.33). Therefore, within the dictionary (3.2) and in particular for σ ∈ iR/{0}, and T > 0, we have which concludes our proof.

Gauge theory and quantum gas
In this section we focus on the gauge theory interpretation of the above results.

The spectral problem in four dimensions
The computation of Section 3 shows that the spectrum of the operator (1.3) is determined by the zeros of the tau function (3.27). This in turn determines the spectrum of an ideal Fermi gas. Let us write the four-dimensional matrix model (3.50) as This is the so-called polymer matrix model studied in [28,[69][70][71] (see chapter 20 of [72] for a survey on the subject). It was found in [69] that this matrix model describes an ideal Fermi gas whose density matrix is Similarly to what we did in Section (3.2) we find that, in the four-dimensional limit, the kernel (3.35) reproduces the kernel (4.2). The operator corresponding to (4.2) is As discussed in [69] this corresponds to an ideal Fermi gas in an external potential and with a non-standard kinetic term given by To get some physical intuition into this system it is useful to consider the large energy limit of this gas, i.e. p, x large. In this regime the Hamiltonian can be approximated by (4.6) Therefore we have an ultra-relativistic Fermi gas in a confining potential. The operator (4.3) is of trace class and has a positive, discrete spectrum e −En which is determined by the vanishing locus of In statistical mechanics one typically refers to Ξ 4d O (κ, T ) as the grand canonical partition function of the gas. To make contact with the Painlevé III 3 dictionary we have 2πκ = cos(2πσ). (4.8) Order  Table 1. The first two energy levels for the operator (4.2), obtained from the vanishing locus of the τ function (3.26) as explained in equation (4.10). The expression of Ξ 4d S (σ, T ) is given as a convergent series at small T . As we keep more terms in the series expansion we quickly approach the energy obtained by using the numerical methods of [37] applied to the operator (4.2). We take T 1/4 = π 21 .
Therefore the region which is interesting from the spectral theory point of view is parametrized by 5 Using this dictionary, we can compute numerically the zeros of (3.26) which give the spectrum of the Fermi gas described by (4.2). More precisely we have In Table 1 we compare the numerical spectrum of the operator (4.2) with the zeros of (3.26): we find perfect agreement. Therefore the vanishing locus of the τ function which solves the Painlevé III 3 equation gives the spectrum of the operator (4.2), as we showed analytically in Section 3. If we consider the gauge theory perspective, the results of Section 3 show that the spectrum of (4.2) is computed by the four-dimensional Nekrasov partition function [61] Z Nek ( , a, Λ) (4.11) where the equivariant parameters are set to 1 = − 2 = . Indeed we have In [29] the authors introduced the dual partition function as Z NO (η, σ, T ) = n∈Z e 4πinη Z(σ + n, T ). (4.14) Our analysis shows that this dual partition function, at η = 0, corresponds to the grand canonical partition function of an ideal Fermi gas whose density matrix is given by (4.2). In this correspondence the Seiberg-Witten period σ and the instanton counting parameter T correspond to the chemical potential of the gas and the strength of the external potential respectively.

A matrix model for Nekrasov's partition function
In the above subsection we focused on the grand canonical ensemble in which the grand canonical partition function Ξ 4d of the gas makes contact with the dual NO partition function. In this section, instead, we study the canonical ensemble, in particular we show that the canonical partition function Z 4d can be identified with the Nekrasov partition function in the magnetic frame.
It is well known [29,61,73,74] that the 4-dimensional Nekrasov partition function reproduces the Seiberg-Witten prepotential F 0 (a, Λ) and its gravitational corrections 6 F g (a, Λ) in the expansion where is the vacuum expectation value of the self-dual graviphoton field strength. In this section we show that the free energies F g emerge when we study the so-call 't Hooft expansion of (3.50).
The 't Hooft regime of (3.50) is defined as [69] N, T → ∞, N 2T 1/4 = λ fixed, (4.16) and In general it is difficult to compute the genus expansion (4.17) directly in a matrix model, i.e. without using tools such as the holomorphic anomaly. However, for the O(2) matrix models, exact formulae for F 0 and F 1 exist in the literature [69,[79][80][81]. For the particular potential (3.51) one can show that (3.50) is a one cut matrix model with endpoints (a, a −1 ) where a is related to λ as [69] λ(a) = −π + 2E(k) + −1 + k 2 K(k) K (k) and we denote by K, E the elliptic integral of first and second kinds. For small values of λ one has which leads to This is precisely the genus zero free energy of Seiberg-Witten theory F 0 in the magnetic frame [82] where we identify the 't Hooft parameter λ with the dual period 7ã D of the Seiberg-Witten curve λ = −ã D . Moreover, as in [82], we set the Seiberg-Witten scale to be Λ = 1. Therefore in this correspondence the parameters describing the canonical ensemble of the gas, namely the number of particles N and the strength of the potential T , get mapped to the dual periodã D and the graviphoton strength . Similarly by specializing the general formula for F 1 of [69,79] to the potential (3.51) we find log 1 a(λ) 2 − a(λ) 2 + constant, (4.23) where the constant is fixed by the Gaussian behavior of the matrix model (3.50) at small λ. By using (4.19) we obtain This reproduces the genus one free energy of Seiberg-Witten theory F 1 in the magnetic frame [82] with the identification (4.22). Even though we do not have a direct all order proof, we expect this identification to hold also for higher F 4d g 's. Therefore in our formalism the perturbative expansion (4.15) of the Nekrasov partition function in the magnetic frame emerges in the 't Hooft limit (4.17) of the gas. In particular this implies we can now solve the O(2) matrix model (3.50) at all order in the genus expansion by using the holomorphic anomaly equations of the gauge theory.
Notice that the matrix model (3.50) provides an expression for the spectral determinant (3.26) which resums the expansion in T . Moreover the 't Hooft expansion of the spectral determinant (3.49) naturally reproduces the long distance solution to Painlevé III 3 proposed in [19]. A detailed discussion of this regime will appear in [25].
Other proposals for a matrix model description of the Nekrasov partition function appeared before in the literature as for instance [83][84][85][86][87][88]. These are constructed by expressing the conformal blocks as matrix models and are different from our which arises from the quantization of mirror curves to toric CYs. In particular the perturbative expansion of [86] reproduces the SU (2) gauge theory in the NS background ( 2 = 0) while in our model we recover the perturbative expansion in the self-dual background. Similarly in [83][84][85] a Penner-like matrix model was proposed to describe SU (2) gauge theory with hypermultiplets in a self-dual background. It would be interesting to extend our construction to the case of gauge theory with hypermultiplets and compare with these proposals more in detail.
From a pure mathematical point of view, the main difference between previous proposals and the one we are considering here consists in the interaction term. In our model the interaction is the one of an O(2) model while in previous proposals it was given by the Vandermonde determinant 8 . Moreover the standard 't Hooft expansion of (3.50) describes Seiberg-Witten theory in the magnetic frame and not the electric one as it was the case in previous proposals. In turns, these differences are related to the fact that our model arises as a four-dimensional limit of the matrix model describing topological string on toric CYs [24,39]. Other proposals instead are more related to topological string theory on the Dijkgraaf-Vafa types of manifold [89] which can also be used to engineer Seiberg-Witten theory in four dimension [17,26,[75][76][77][78].

Conclusion
In this paper we proposed a new double-scaling limit in which the spectral determinant of [24] makes contact with the dual four-dimensional SU (2) Nekrasov-Okounkov partition function and with the Painlevé III 3 equation. We used the recent developments in the context of Painlevé III 3 [18][19][20][21][22][23], together with previous works on spectral determinants [28], to prove the conjecture of [24] in this new double-scaling limit of local P 1 × P 1 . Within this construction, we showed that the vanishing locus of the τ function computes the spectrum of an ideal Fermi gas.
In addition, the partition function of this gas can be written as an O(2) matrix model which provides an exact expression for the Nekrasov partition function in the magnetic frame and in the self-dual Ω-background. Interestingly, this gas appears as well in the computation of twopoint functions for the 2d Ising model and in the study of 2d polymers [28,[70][71][72]90]. As a consequence, one can compute Nekrasov's partition function exactly by using the TBA techniques of [28,70,71]. It would be interesting to explore this relation in more detail in view of the results presented here. We hope to report on this in the near future.
Furthermore, it was conjectured in [4] that in the NS background ( 2 = 0) the gauge theory is strictly related to quantum integrable models. Our results show that there is an underlying quantum mechanical system also in the self-dual Ω-background. Moreover, we proved that the spectral determinant, and therefore the spectrum, of this system is computed by the NO partition function.
In this work, we focused on the pure SU (2) N = 2 four-dimensional gauge theory. However, it would be interesting to combine the techniques developed here with the work of [25] to study the four-dimensional gauge theory with matter. In particular, our approach can be used to construct explicit Fredholm determinant solutions to other Painlevé equations. To do this, the only task is to compute the kernel for the operators associated with the mirror curves that engineer gauge theories with matter by following the prescription in [24,55]. The four-dimensional limit (3.2) of these operators should provide the desired Fredholm determinant solution.
Most importantly, the gauge theory/Fermi gas correspondence presented here can be understood from a purely four-dimensional perspective by using lattice quantization. Indeed, the operator (1.3) can be constructed directly from the Seiberg-Witten curve by using an unusual quantization scheme: the so-called lattice quantization scheme [91]. This opens the road for an immediate generalization of the Fermi gas formalism of [43] to other gauge theories with matter multiplets and it allows to construct explicit Fredholm determinant solutions to others Painlevé equations. A detailed discussion of this generalization will be presented elsewhere.
Moreover, the case of higher rank gauge theory should also be investigated. In this context, one can deform the theory to the generating function of local BPS observables TrΦ k , where Φ is the scalar component of the N = 2 vector multiplet. It would be interesting to understand the counterpart of these deformations from the spectral theory point of view.
Notice that the general solution to Painlevé III 3 equation contains an additional parameter η which is set to zero in our construction. This suggests that a generalization of the spectral determinant may exist, including this extra parameter. It would be interesting to investigate its role in the context of topological string.
Finally, a natural extension of this work is the generalization to the five-dimensional case describing the full topological string amplitude. We will report on this in the near future [92].

B Standard and NS free energies
The free energy of standard topological string at large radius was computed in [51,99] and it reads The variable t denotes the Kähler parameters of the geometry, g s the string coupling and n d g the Gopakumar-Vafa invariants. These can be easily computed with the topological vertex [99] formalism or the holomorphic anomaly equation [50]. For the local P 1 × P 1 geometry we have for instance Similarly the free energy of refined topological string in the NS limit reads [59] F NS (t, ) = 1 6 and N d j L ,j R denote the refined BPS invariants [52,53]. These can be computed by using the refined topological vertex [59] or the refined holomorphic anomaly [100,101]. The last term in (B.3) is often called the one-loop contribution to the NS free energy. For the local P 1 × P 1 geometry we have for instance However, thanks to the refined topological vertex formalism [26,59], it is possible to perform a partial resummation in t 2 to obtain an expression which is valid around As an example we consider the standard free energy of local P 1 × P 1 . By using the refined topological vertex we obtain q 2 e −2t 1 q 2 e −4t 1 + q 2 + 4q(q + 1) 2 e −3t 2 − 2(q(q + 1)(q(q + 3) + 4) + 1)e −2t 2 + 4q(q + 1) 2 e −t 2 (q − 1) 2 (q + 1) 2 (e −t 2 − 1) 4 (q − e −t 2 ) 2 (qe −t 2 − 1) 2 + O(e −3t 1 ), q = e igs , (B.8) where F ol (e −t 2 ) is what we call the one-loop contribution of the standard topological strings. In appendix C we show that, when this is appropriately combined with the one-loop contribution of the NS free energy, one can resum it by using the methods of [54] . Similarly, by using the refined topological vertex one has (C.1) With some algebraic manipulations and by following [54] we can write it as (C.2) which reproduces precisely (3.8). This means that the one-loop part of the standard topological string plus the one-loop part of the NS limit of topological string sum up to give the non perturbative free energy of topological string on the resolved conifold as given in [54].

D Spectral determinant and Painlevé III equation
In this section we briefly review the results of [28].These results will be relevant in Section 3.
We define the Zamolodchikov spectral determinant Ξ Z (κ, t) as From [28,102,103] it follows that Ξ Z (κ, t) satisfies is the τ function corresponding to the solution of (D.6). The small t expansion of Ξ Z (κ, t) was also computed in [28] where the author shows that Ξ Z (κ, t) ≈ t 8