Stringy instanton counting and topological strings

We study the stringy instanton partition function of four dimensional N=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=2 $$\end{document} U(N) supersymmetric gauge theory which was obtained by Bonelli et al. in 2013. In type IIB string theory on ℂ2×T*ℙ1×ℂ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathrm{\mathbb{C}}}^2\times {T}^{*}{\mathrm{\mathbb{P}}}^1\times \mathrm{\mathbb{C}} $$\end{document}, the stringy U(N) instantons of charge k are described by k D1-branes wrapping around the ℙ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathrm{\mathbb{P}}}^1 $$\end{document} bound to N D5-branes on ℂ2×ℙ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathrm{\mathbb{C}}}^2\times {\mathrm{\mathbb{P}}}^1 $$\end{document}. The KK corrections induced by compactification of the ℙ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathrm{\mathbb{P}}}^1 $$\end{document} give the stringy corrections. We find a relation between the stringy instanton partition function whose quantum stringy corrections have been removed and the K-theoretic instanton partition function, or by geometric engineering, the refined topological A-model partition function on a local toric Calabi-Yau threefold. We also study the quantum stringy corrections in the stringy instanton partition function which is not captured by the refined topological strings.


Introduction
In 1994, by Seiberg and Witten the exact prepotential F 0 = F pert 0 +F inst 0 in four dimensional N = 2 SU(2) supersymmetric gauge theory was obtained [1,2], where F pert 0 and F inst 0 are the perturbative and instanton part of the prepotential, respectively. The prepotential is computed from a period of a two dimensional algebraic curve on the Coulomb branch which is called Seiberg-Witten curve. This exact result can be generalized to other gauge theories with ADE gauge symmetries. By the compactification of type IIA strings on JHEP07(2015)097 a local Calabi-Yau threefold given by ALE space fibration of ADE type over P 1 , one obtains N = 2 supersymmetry with ADE gauge symmetry in four dimensions. This realization is known as geometric engineering of gauge theory [3][4][5]. Then the Seiberg-Witten curve is embedded into the mirror dual of the local Calabi-Yau threefold. In 2002, by Nekrasov the prepotential F 0 was directly derived from path integral formulation using the localization technique [6]. A necessary ingredient of this computation is to introduce the Omega background described by R 4 ∼ = C 2 fibration over two dimensional torus T 2 . The Omega background has two generators ǫ 1 and ǫ 2 of T 2 . The instanton moduli space of the gauge theory can be described by the ADHM moduli space whose dynamical variables are given by matrices [7] (see (2.1)). The computation of the path integral which gives the instanton partition function Z Nek (ǫ 1 , ǫ 2 ) on the Omega background reduces to the computation of the equivariant volume of the ADHM moduli space whose IR behavior was regularized by ǫ 1,2 [6,8,9]. 1 By the localization, the instanton partition function Z Nek (ǫ 1 , ǫ 2 ) can be exactly computed, and one obtains the asymptotic expansion of the form log Z Nek (ǫ 1 , ǫ 2 ) = ∞ g,ℓ=0 (ǫ 1 ǫ 2 ) g−1 (ǫ 1 + ǫ 2 ) ℓ F Nek g,ℓ .
Here the leading term coincides with the instanton part of the prepotential F Nek 0,0 = F inst 0 [6,[10][11][12]. The SU(N ) instanton partition function on the anti-self-dual Omega background ǫ 1 = −ǫ 2 = can be also obtained from a geometric engineering limit of the topological Amodel partition function on a local toric Calabi-Yau threefold given by ALE space fibration of A N −1 type over P 1 [13][14][15][16][17][18]. Here the topological A-model partition function is computed by the topological vertex formalism [19,20], and the parameter is identified with the topological string coupling constant g s . In [21,22], refinements of the topological vertex formalism were proposed so that the SU(N ) instanton partition function on the general Omega background is obtained [22][23][24].
In 2012, the partition function of N = (2, 2) gauged linear sigma model (GLSM) on S 2 was exactly computed [25,26]. By using this result, the stringy instanton partition function of four dimensional N = 2 U(N ) supersymmetric gauge theory was given in [27]. The GLSM, which gives the stringy instanton partition function, flows in the IR fixed point to an N = (2, 2) non-linear sigma model (NLSM) whose target space is given by the ADHM moduli space. In [27], it was shown that in the degenerate limit of the worldsheet S 2 the stringy instanton partition function yields the U(N ) instanton partition function Z Nek N (ǫ 1 , ǫ 2 ) (see (2.12) for the k-instanton sector). 2 Here Z Nek N (ǫ 1 , ǫ 2 ) does not depend on the Kähler modulus ζ of the ADHM moduli space, whereas the stringy instanton partition function depends on ζ. In section 2 we review the results in [27], and discuss the formal structures of the stringy U(N ) instanton partition function.

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The stringy instanton partition function has the classical stringy corrections and quantum stringy (α ′ ) corrections. In the anti-self-dual Omega background ǫ 1 = −ǫ 2 = , the stringy instanton partition function does not have the quantum stringy corrections, and only have the classical stringy corrections [27]. In section 3, we study the classical part Z cSI N (ǫ 1 , ǫ 2 ; ζ) of the stringy U(N ) instanton partition function (see (2.17) for the k-instanton sector), and show that Z cSI N (ǫ 1 , ǫ 2 ; ζ) is reduced from a four dimensional limit of the K-theoretic ("five dimensional R 4 × S 1 ") U(N ) instanton partition function Z K−Nek N,m (ǫ 1 , ǫ 2 ) with five dimensional Chern-Simons coefficient m ∈ Z [39,40]. By geometric engineering, the classical stringy instanton partition function Z cSI N (ǫ 1 , ǫ 2 ; ζ) is also reduced from the refined topological A-model partition function Z refA N,m (ǫ 1 , ǫ 2 ) on a family of local toric Calabi-Yau threefolds with resolved A N −1 singularity labeled by m ∈ Z described in figure 2 (section 3.2). Then we obtain a relation Here the four dimensional (4d) limit is given by where β is the radius of the five dimensional circle S 1 , and r is the radius of the worldsheet S 2 . The limit r → 0 corresponds to the degenerate limit of the S 2 . In section 3.3, we give a physical explanation of the above relation by revisiting string dualities discussed in [41]. Then this relation claims geometric engineering of the instantons with the classical stringy corrections, and shows that the Kähler modulus ζ of the ADHM moduli space is obtained from the strong coupling limit of the five dimensional Chern-Simons term, and quantization of ζ can be interpreted as a five dimensional Chern-Simons coefficient m ∈ Z.
It was conjectured in [42] and proved in [43] that the N = (2, 2) GLSM partition function Z GLSM X on S 2 which flows in the IR fixed point to an N = (2, 2) NLSM on S 2 whose target space is a Calabi-Yau geometry X gives the quantum-corrected Kähler potential K X on the Kähler moduli space of X: Therefore the stringy U(N ) instanton partition function gives the Kähler potential for the ADHM moduli space. Note that the ADHM moduli space has a hyper-Kähler structure [44], and thus satisfies the Calabi-Yau condition. In [27], it was proposed that the quantum stringy corrections gives us the Givental's I-function [45] (see also [46]) of the ADHM moduli space. In section 4, for U(1), U(2), and U(3) we study the full stringy instanton partition functions with the quantum stringy corrections. For each N , we find a universal structure of the quantum stringy corrections for arbitrary instanton charge k. For U(1), as discussed in [27], we confirm agreements with quantum correlators in the T -equivariant cohomology ring H * T Hilb k (C 2 ), Q (with the equivariant parameters ǫ 1,2 ) of the Hilbert scheme Hilb k (C 2 ) of points on C 2 . For U(2) and U(3), using the formulation of Maulik and Okounkov [47], we also check agreements with the quantum parts of quantum correlators in the equivariant cohomology of the ADHM moduli space. In section 5 we give JHEP07(2015)097 our conclusions and discuss some future directions. Appendix A is a note on the multiple gamma function which appears in this paper. In appendix B, we discuss relations between the stringy U(1) instanton partition function, the simple Hurwitz theory, and the topological A-model on a local toric Calabi-Yau threefold. Here a relation between the latter two theories was previously discussed in [48]. In appendix C, we review the Fock space description of H * T Hilb k (C 2 ), Q [44,[49][50][51], and compute some equivariant correlators for comparing with the stringy U(1) instanton partition function. In appendix D, we review and discuss the higher rank generalization of the formulation in appendix C [47,52]. In appendix E, we summarize the formulas of the exact Kähler potentials on quantum Kähler moduli spaces of Calabi-Yau threefolds (e.g. [42]) and fourfolds (conjectured in [53]).

Stringy U(N ) instanton partition function
In this section, we review the stringy U(N ) instanton partition function given in [27], and discuss its formal structures.
2.1 N = (2, 2) GLSM on S 2 for ADHM moduli space Let us consider type IIB strings on C 2 × T * P 1 × C, and introduce D1-D5 brane system consisting of N D5-branes on C 2 × P 1 and k D1-branes wrapping around the P 1 . By embedding the U(1) spin connection on P 1 into the SO(4) R-symmetry in the world volume theory of the N D5-branes, 3 and by compactifying this theory on C 2 the four dimensional N = 2 U(N ) supersymmetric gauge theory is obtained [54] (see also e.g. [55]). Here the k D1-branes describe the instantons of charge k in the four dimensional gauge theory. 4 As the world volume theory on the k D1-branes, these instantons are described by an N = (2, 2) NLSM on P 1 whose target space is the ADHM (framed k-instanton) moduli space 5 Here B 1,2 : C k → C k , I: C N → C k , J: C k → C N , and ζ > 0 defines the Kähler modulus. The gauge transformation is given by where R ∈ U(k). This NLSM is obtained in the IR fixed point of an N = (2, 2) U(k) GLSM on S 2 ∼ = P 1 with the matter content described in table 1 [27]. The twisted masses ǫ 1,2 ∈ R give the generators of T 2 = U(1) 2 which rotates the C 2 in C 2 × P 1 , and these masses induce the Omega background. The twisted masses a α ∈ R give the generators of the Cartan subalgebra of U(N ). These chiral fields interact each other through a superpotential Table 1. Matter content of the GLSM for the ADHM moduli space M k,N . Here ǫ = ǫ 1 + ǫ 2 and α = 1, . . . , N . By restricting the U(1) V R-charges to be non-negative, these are constrained as 0 < p < q < 1.
parameter for central U(1) ⊂ U(k) gives the Kähler modulus ζ in the ADHM moduli space (2.1). By using the formula of the GLSM partition function on S 2 obtained by the supersymmetric localization [25,26], after taking the limit p, q → 0 + due to the non-compactness of the M k,N [59], one obtains the stringy U(N ) k-instanton partition function [27]: 2) where z = e −2πζ+iθ with the theta angle θ, m ab = m a − m b , and σ ab = σ a − σ b . Here is the one loop determinant of the chiral multiplets including the chiral fields I α , J α , and is the one loop determinant of the chiral multiplets including the chiral fields χ, B 1,2 , where r is the radius of S 2 , and ǫ = ǫ 1 + ǫ 2 .
In the large radius phase ζ ≫ 0, the above partition function can be written as [27] Z k,N (ǫ 1 , ǫ 2 , a, z) where is the Pochhammer symbol. Here by shifting the theta angle θ, we have scaled z as z → (−1) N +k−1 z for simplicity. The contours in (2.5) are enclosing the imaginary axes counterclockwise, and the simple poles which give residues only come from the numerators in (2.6) given by λ a = −ia α , λ ab = −iǫ 1 , and λ ab = −iǫ 2 . As the result the simple poles are labeled by an N -tuple of Young diagrams µ = (µ 1 , . . . , µ N ) with k = | µ| = N α=1 |µ α |, where |µ α | is the number of boxes of the Young diagram µ α .
As an example, in the N = 2 and k = 3 case, we see that the simple poles are classified by ten types of two-tuple of Young diagrams µ 1,...,10 as and using Γ(r) = r −1 + O(r 0 ) to estimate Z L , one finds that the stringy U(N ) kinstanton partition function (2.5) yields the U(N ) k-instanton (Nekrasov) partition function Z Nek k,N [27]: , (2.11) and the contours are enclosing the real axes counterclockwise [6,8,9]. The exponent of the leading behavior r −2kN coincides with dim C M k,N = 2kN . It is useful to express , Here a αβ = a α − a β , and a µ (s) and ℓ µ (s) are the arm-and leg-length, respectively. In figure 1 we describe an example of Young diagram µ with |µ| = 20. The expansion (2.10) contains the stringy corrections O(r −2kN +1 ) to the Nekrasov partition function, and the main subject of this paper is to study these corrections.

Stringy corrections
As discussed in [27], Z L in (2.6) contains the perturbative α ′ corrections. In the supersymmetric localization [25,26], these corrections depends on the regularization scheme of the infinite products in the one loop determinant of the chiral multiplets. In [25,26] this ambiguity was fixed by the zeta function regularization we see that the perturbative α ′ corrections corrections in (2.10) contain the Euler constant γ and the simple zeta values ζ(s) = ∞ n=1 1 n s . On the other hand, Z V (z) in (2.7) contains the non-perturbative α ′ corrections and does not depend on the regularization scheme. Especially it was proposed that Z V (z) gives the Givental's I-function [45,46] of the ADHM moduli space M k,N [27,62]: Here λ 1 , . . . , λ k are identified with the Chern roots of the tautological bundle on M k,N (for N = 1, see [63]). The inverse radius r −1 of the two sphere S 2 is identified with JHEP07(2015)097 the equivariant parameter which gives the generator of S 1 acting on the S 2 . By the equivariant mirror map, the I-function is related with the small J -function J (z). Then by expanding the J -function around r = 0, one can obtain the T N +2 -equivariant Gromov-Witten invariants. 6 It is known that the coefficient of r 1 in the expansion around r = 0 of the I-function gives the equivariant mirror map. As discussed in [27], by (2.15) and the behavior (2.9) of Z V (z), one finds that the equivariant mirror maps for the N ≥ 2 cases are trivial: J (z) = I(z) = Z V (z), but for the N = 1 case the equivariant mirror map is needed as [63] J In (2.5), by taking the leading terms of Z L ∼ O(r −2kN ) and Z V (z) ∼ 1 around r = 0, we obtain the stringy instanton partition function without the quantum stringy (α ′ ) corrections: , where n(µ) = s∈µ ℓ µ (s) and n(µ t ) = s∈µ a µ (s). Especially in the anti-self-dual case ǫ 1 = −ǫ 2 = , one finds that the stringy instanton partition function does not have the α ′ corrections [27] (see also [47]), and thus In section 3 and section 4 we study the classical and quantum stringy corrections, respectively.

Classical stringy corrections
In this section we study the classical stringy corrections. Let us define the generating function of the classical stringy U(N ) instanton partition functions (2.17) by where Λ is the dynamical scale in four dimensional gauge theory. This partition function gives the instanton ("non-perturbative" in the gauge theoretic sense) partition function without the α ′ corrections of the N D5-world volume theory on C 2 × P 1 that was dimensionally reduced to the C 2 . In [27], the "perturbative" (in the gauge theoretic sense) partition function TaJ) where Ta = diag(e ia 1 , . . . , e ia N ), and U(1) 2 : (B1, B2, I, J) → (Tǫ 1 B1, Tǫ 2 B2, I, Tǫ 1 Tǫ 2 J) where Tǫ 1,2 = e iǫ 1,2 . This action introduces the equivariant parameters aα, α = 1, . . . , N and ǫ1,2 corresponding to the twisted masses in the GLSM as described in table 1.

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was also computed. Here gives the well-known perturbative partition function of the four dimensional N = 2 U(N ) gauge theory on the Omega background [11,64] (see also appendix A.2). The quantum stringy part contains the perturbative α ′ corrections, where Γ r (z|ω 1 , . . . , ω r ) is the multiple gamma function (see appendix A), and one finds Z qD5 N ( , − , a) = 1. Then in the anti-self-dual case ǫ 1 = −ǫ 2 = , there are no α ′ corrections as in the instanton partition function (2.18), In the following we show that the classical stringy partition functions (3.1) (and (3.3)) are reduced from a four dimensional limit of the K-theoretic Nekrasov partition function with five dimensional Chern-Simons term. We also find that the instantons with the classical stringy corrections can be embedded into (refined) topological string theory.

Relation with K-theoretic instanton partition function
The K-theoretic U(N ) instanton (Nekrasov) partition function with five dimensional Chern-Simons term TrA ∧ F ∧ F [65] is given by [39,40] where m ∈ Z is the Chern-Simons coefficient, and we see that the K-theoretic Nekrasov partition function yields the classical stringy instanton partition function (3.1): The four dimensional limit (3.8) relates (the strong coupling limit of) the Chern-Simons coefficient m to the Kähler modulus ζ. The K-theoretic "perturbative" (in the gauge JHEP07(2015)097  .18) and (3.5) the stringy partition functions do not receive the α ′ corrections, and the limit (3.8) of the K-theoretic perturbative/non-perturbative (instanton) partition functions completely coincide with the stringy partition functions.

Relation with topological strings
By geometric engineering [3][4][5], it is known that the K-theoretic SU(N ) Nekrasov partition function with a Chern-Simons coefficient m coincides with the partition function Z refA N,m of the refined topological A-model on a local toric Calabi-Yau threefold X N,m given by ALE space fibration of A N −1 type over P 1 [22][23][24] (see [13][14][15][16][17][18] for the unrefined case). Topological type of this SU(N ) geometry X N,m described in figure 2 is labeled by an integer m which is identified with the Chern-Simons coefficient [39]. The SU(N ) geometry X N,m has one modulus T b of the base P 1 and N − 1 moduli T fα , α = 1, . . . , N − 1 of the fiber consisting of N − 1 resolved P 1 's. Here T b is identified with the dynamical scale Λ, and T fα are identified with N − 1 independent Coulomb moduli a α in the SU(N ) gauge theory [4]: Then by taking the limit (3.8), we find that the partition function Z refA N,m of the refined A-model on the family of the local toric Calabi-Yau threefolds X N,m labeled by m yields the classical stringy partition function of the four dimensional SU(N ) gauge theory: The anti-self-dual case ǫ 1 = −ǫ 2 = corresponds to the topological string (unrefined) limit, and is identified with the topological string coupling constant g s . Therefore the JHEP07(2015)097 (unrefined) A-model partition function under the four dimensional limit (3.8) completely coincides with the stringy partition function on the anti-self-dual Omega background. 7

Brane construction and geometric engineering
In this section by revisiting a correspondence discussed in [41] between the world volume theory of N D5-branes on C 2 × {vanishing P 1 } and the geometrically engineered quantum field theory by local Calabi-Yau threefold with resolved A N −1 singularity, we give a physical explanation of the relations (3.9) and (3.11). As described in section 2, the intersecting D1and D5-branes in type IIB string theory on C 2 × T * P 1 × C lead to the instantons with the stringy corrections. In the D5-world volume theory, after the compactification of the P 1 , to preserve eight supercharges in the four dimensions, the U(1) spin connection on the P 1 needs to be embedded into the SO(4) R-symmetry [54]. This topological twist breaks the R-symmetry as SO(4) ∼ = SU(2) × SU(2) to U(1) × SU (2), and one obtains six dimensional N = (1, 0) U(N ) supersymmetric gauge theory on C 2 × P 1 which leads to four dimensional N = 2 supersymmetric gauge theory.
Type IIB theory → Type IIB theory: firstly, by taking the S-duality in type IIB string theory, the N D5-and k D1-branes are turned into NS5-branes and the fundamental strings wrapping around the compactified P 1 .
Type IIB theory → Type IIA theory: as was shown in [67], the N NS5-branes in type IIB (IIA) string theory is equivalent to type IIA (IIB) string theory on an A N −1 ALE space. By this duality, we obtain type IIA string theory compactified on a local Calabi-Yau threefold given by ALE space fibration of A N −1 type over P 1 , and then four dimensional N = 2 SU(N ) supersymmetric gauge theory is geometrically engineered [3][4][5]. The low energy type IIA supergravity theory has the Chern-Simons term is the NS-NS B field and C (3) is the R-R 3-form field. From this Chern-Simons term, by integration on the local Calabi-Yau threefold, we obtain Here φ α = 1 ζ b Cα B (2) and A α = Cα C (3) defined for the two cycles C α , α = 1, . . . , N − 1 on the ALE space give scalar fields and gauge fields on C 2 , respectively, and g αβγ is given by a classical triple intersection number of divisors on the local Calabi-Yau threefold. 8 It is known that the NS-NS B field introduces a noncommutative parameter in the four dimensions, and it also introduces the FI parameter in the ADHM moduli space [69][70][71]. Therefore we argue that the coefficient ζ b g αβγ is identified with the FI parameter.
Let X N (ζ) be the local Calabi-Yau threefold obtained from the toric Calabi-Yau threefold X N,m in figure 2 at the phase under the limit (3.8). 9 By considering this Calabi-Yau JHEP07(2015)097 threefold X N (ζ), as discussed above, we find an interaction term like (3.12). As discussed in [39], the parameter ζ obtained by the scaling of the intersection number m in X N,m is related with the coefficient of (3.12), and so identified with the FI parameter. Then we see that the instanton partition function with the term (3.12) on the Omega background is given by the classical stringy instanton partition functions (2.17): where e iζφ is an element of the T = U(1) N × U(1) 2 -equivariant cohomology H * T M k,N [8]. Note that we also have the fundamental strings wrapping around the base P 1 which give the degree k worldsheet instanton for this base whose Kähler modulus gives the dynamical scale Λ.
Type IIA theory → M-theory: lifting to the M-theory leads to the relation (3.9) between the K-theoretic SU(N ) instanton partition function with five dimensional Chern-Simons term and the classical stringy SU(N ) instanton partition function in four dimensions. We see that by this five dimensional lift, the FI parameter ζ is quantized, and it gives a five dimensional Chern-Simons coefficient m ∈ Z. As the result, we argue that the K-theoretic classical stringy instanton partition function is given by the K-theoretic instanton partition function (3.6) with five dimensional Chern-Simons coefficient.
The K-theoretic instanton partition function and the refined topological A-model partition function do not capture the quantum stringy corrections. In the next section, we study these quantum corrections.

Quantum stringy corrections
In the general Omega background, the stringy instanton partition function (2.5) has the quantum stringy (α ′ ) corrections, and it gives the quantum-corrected Kähler potential K k,N on the Kähler moduli space of the ADHM moduli space M k,N [27,42,43,62]: up to an ambiguity by Kähler transformations is a holomorphic function of z. In the following, let us fix this ambiguity by a normalization Here the first normalization factor (zz) −ir k N N α=1 aα shifts the classical stringy corrections, and then we see that this normalized partition function is invariant under simultaneous constant shift a α → a α + c, α = 1, . . . , N . As the result, the U(1) factor is decoupled from the partition function: U(N ) → SU(N ). By (2.14), the second normalization factor removes the dependence of the Euler constant γ [27], and fixes the precoefficients of the JHEP07(2015)097 simple zeta values ζ(s) which receive the perturbative α ′ corrections. These normalizations fix the regularization scheme in the perturbative α ′ corrections mentioned in section 2.2.
In this section, we study the structure of this partition function for N = 1, 2, 3 cases, and find a universal structure of the quantum stringy corrections for arbitrary instanton charge k as in (4.4), (4.28), and (4.55).

Stringy U(1) instanton counting
In the N = 1 case, the ADHM moduli space M k,1 is isomorphic to the Hilbert scheme of points Hilb k (C 2 ) on C 2 . In this case, an equivariant mirror map described in (2.16) is needed for reproducing the correct Gromov-Witten invariants of Hilb k (C 2 ), and thus we study [27] Z norm Note that the Coulomb modulus a has been removed by the first normalization factor in (4.2). We find that its asymptotic expansion around r = 0 is given by where ε 1,2 = irǫ 1,2 , ε = ε 1 + ε 2 = irǫ, and l n (z) ≡ 1 n! log n zz. The coefficients e −K (n) 1 (z) , n = 3, . . . , 6, and e −K (n) 1;i (z) , n = 5, 6, i = 1, 2, 3 are given below. We have checked this expansion up to k = 5(-instanton), and for the classical stringy corrections up to k = 7(-instanton). Here where c µ (s) = s∈µ (a µ (s) − ℓ µ (s)) = (i,j)∈µ (j − i), and h µ (s) = a µ (s) + ℓ µ (s) + 1 is the hook length. Some of these numbers are given by One finds that G k,ℓ coincides with the equivariant classical intersection number D 2ℓ cl in (C.8), where T = U(1) 2 acts on C 2 . As discussed in appendix B, these numbers also give the disconnected simple Hurwitz numbers of P 1 which count degree k covers of P 1 only with simple branch points (see (B.4)). The normalized stringy U(1) instanton partition function Z norm k,1 gives the Kähler potential on the quantum Kähler moduli space of Hilb k (C 2 ), and by (2.15) the non-perturbative α ′ corrections should be given by the equivariant Gromov-Witten invariants. In the following, we explicitly describe the expansion coefficients of (4.4), and confirm that such corrections are given by the equivariant Gromov-Witten invariants computed in appendix C.

(4.23)
We find that the coefficients K 1;3 (z) and K 1;3 (z) are written in terms of the other coefficients as The stringy k = 2-instanton partition function in (4.4) receives these non-perturbative α ′ corrections, whereas the equivariant three-point functions D, Y 2 , Y computed in (C.11), (C.12) are trivial for k = 2, and thus we argue that the nonperturbative α ′ corrections in each coefficient are determined by the equivariant three-point functions as K 1;3 (z) : ǫ 1 ǫ 2 D, D, D , K 1;3 (z) : ǫ 1 ǫ 2 ǫD, D, D . We also find that the classical intersection numbers obtained by the above computation completely coincide with the equivariant classical intersection numbers (C.8) of the divisor class D in H * T Hilb k (C 2 ), Q .

(4.58)
Up to the perturbative α ′ corrections we find e −K (5) 3;2 (z) ≡ e −K (5) 1;2 (z) ≡ e −K (5) 2;2 (z) , and the coefficient e −K (6) 3;2 (z) is given by As in the N = 2 case, we find that there exist the observable D for the divisor class and an observable Y where c is a constant. Here H * T M k,3 is the T -equivariant cohomology of M k,3 with the 3 + 2 dimensional torus action T mentioned in footnote 6. By the expansion (4.55), we see that the three-point function D, D, D does not have quantum corrections.
As same as the N = 2 case, we see that the above result (4.60) is consistent with the result (D.12) in appendix D.

JHEP07(2015)097 5 Conclusion and discussions
In this paper, we have studied the stringy instanton partition function of four dimensional N = 2 U(N ) supersymmetric gauge theory given in [27]. In section 3.1, we found that the stringy instanton partition function whose α ′ corrections have been removed coincides with the four dimensional limit (3.8) of the K-theoretic instanton partition function with a five dimensional Chern-Simons coefficient. Here the Kähler modulus ζ of the ADHM moduli space was obtained from the strong coupling limit of the five dimensional Chern-Simons term. We also discussed that the classical stringy instanton partition function is embedded to (refined) topological string theory on the local toric Calabi-Yau threefolds labeled by m ∈ Z in section 3.2. This gives geometric engineering of the instantons with classical stringy corrections, and we provided a physical explanation of this realization in section 3. 3 We further studied the stringy instanton partition function with the α ′ corrections for U(1), U(2), and U(3) cases. We found that the quantum stringy corrections have the universal structure for arbitrary instanton charge k, as in (4.4), (4.28), and (4.55). This universal structure shows that the stringy corrections for an instanton charge k contain the stringy corrections for instanton charges l ≤ k. From the viewpoint of the Fock space description of the equivariant cohomology ring of the ADHM moduli space M k,N in appendix C and D, such structure is clear from the construction of the elements of the equivariant cohomology by the generators of the Heisenberg algebra. Note that this structure seems to be related with a compactification of the UV non-compactness by pointlike instantons, which is called the Uhlenbeck compactification of the framed moduli space M where S l C 2 is the l-th symmetric product of C 2 . After resolving the orbifold singularities in M k,N , one obtains M k,N .
For U(1) case, as discussed in [27] we read off some equivariant three-point functions on M k,1 ∼ = Hilb k (C 2 ), and confirmed the agreement with the computations in appendix C. Using this result, we extracted some equivariant three-point functions on M k,2 and M k,3 from the stringy instanton partition functions, and also checked that these results are consistent with the computations in appendix D.
It would be interesting to further study the classical and the quantum structure of the U(N ) stringy instanton partition function, and to compare the structure with the discussion of the quantum multiplication for the ADHM moduli space M k,N (as described in appendix D) [47]. The refined topological vertex [21,22] do not capture the quantum stringy corrections, and thus it would be also interesting to formulate "quantum refined topological vertex" which captures such quantum corrections.

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A six dimensional analogue of the four dimensional instanton partition function was discussed in [72][73][74]. It would be interesting to discuss the stringy generalization of the six dimensional instanton partition function.

Acknowledgments
I would like to thank Rajesh Gopakumar, Yoshinori Honma, Kentaro Hori, and Piotr Su lkowski for useful discussions and comments. I would also like to thank the organizers of National Strings Meeting 2013 held at IIT Kharagpur, where a part of this work was presented. Many of the results presented here were obtained during the period of postdoctoral fellow at Harish-Chandra Research Institute (HRI). I would like to thank HRI for supporting my work. I would also like to thank Kavli IPMU for hospitality during the final stage of this work. This work has been supported by the ERC Starting Grant no. 335739 "Quantum fields and knot homologies" funded by the European Research Council under the European Union's Seventh Framework Programme.
B Stringy U(1) instanton counting, simple Hurwitz theory, and topological strings In this appendix, we discuss relations between the stringy U(1) instanton counting, the simple Hurwitz theory, and the topological A-model on local toric curve. Let us consider the generating function of the stringy U(1) k-instanton partition functions (2.5) for N = 1 on the (r-scaled) anti-self-dual Omega background ǫ 1 = −ǫ 2 = /r: 11 where c µ (s) = (i,j)∈µ (j −i) and h µ (s) = a µ (s)+ℓ µ (s)+1 is the hook length. By expanding this partition function around = 0: one finds that the coefficients coincide with the equivariant classical intersection numbers D 2ℓ cl ǫ=0 of the divisor class on the Hilbert scheme of points Hilb k (C 2 ) on C 2 as computed in (C.8). One also finds that gives the disconnected simple Hurwitz number of P 1 which counts the degree k ramified cover f : Σ g → P 1 with m = 2g − 2 + 2k simple branch points, where Σ g is a genus g Riemann surface [75] (see also e.g. [76]). 12 Then the connected simple Hurwitz numbers H P 1 • g,k are obtained from the genus expansion of the free energy (B.5) where = i log zz and x = (iq log zz) 2 . We see that the perturbative free energies F g (x) are given as [77,78] where c g,i are constants, 13 and y = −W (−x) is the Lambert W function defined by the inverse function of the spectral curve x = ye −y which has the series expansion y = ∞ k=1 k! x k . The relation between the Hurwitz numbers of P 1 and the intersection numbers on Hilb k (C 2 ) was discussed in [79].
As mentioned in section 3.2, the stringy U(1) instanton partition function (B.1) is obtained from the A-model partition function Z topA X 1,m on the local curve X 1,m = O(m − 2) ⊕ O(−m) → P 1 described in figure 3. By the geometric engineering e −T b ∼ (βq) 2 , g s ∼ , and the four dimensional limit (3.8): β → 0, m → ∞ with fixed βm = r log zz, we have The simple branch point is a branch point such that the branching number is one, and the number m of simple branch points is determined by the Riemann-Hurwitz formula 2g is a profile over an i-th branch point. Here the profile of the simple branch point is given by µ = (2, 1 k−2 ). Note that the genus g of the disconnected simple Hurwitz numbers can be a negative integer. 13 For example these are given by F2(x) = y 2 1440(1 − y) 5 (6y + 1), F3(x) = y 2 725760(1 − y) 10 (720y 4 + 3816y 3 + 3482y 2 + 548y + 9).

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Therefore one also obtains the relation between the topological A-model on X 1,m and the simple Hurwitz theory which was previously discussed in [48]. As pointed out in [48], the simple Hurwitz theory is related with the U(1) instanton counting with the first and second Casimir operators (see also [80]). 14 Then we see that the stringy U(1) instanton partition function (B.1) also coincides with the U(1) instanton partition function with the first and second Casimir operators.
C Equivariant correlators on Hilb k (C 2 ) Let Hilb k (C 2 ) be the Hilbert scheme of points on C 2 : The T -equivariant cohomology H * T Hilb k (C 2 ), Q of Hilb k (C 2 ) with the equivariant action T = U(1) 2 on C 2 has the Fock space description over Q as was constructed by Grojnowski and Nakajima [44,49,50] (see also [51]). The Fock module over the Heisenberg algebra {α ±n , n ∈ N | [α m , α n ] = mδ m+n,0 } is given as follows. The Fock vacuum |∅ is annihilated by α n>0 : α n |∅ = 0 for n > 0, and the basis of the Fock space F is created by α n<0 : Here Y is a partition with Y 1 ≥ Y 2 ≥ · · · ≥ Y ℓ Y > 0, and Aut(Y ) is the order of the automorphism group of the partition. Then a Fock module |Y with Hilb k (C 2 ), Q , and a canonical isomorphism is obtained, where ǫ 1,2 are the equivariant parameters. The inner product on the Fock space which gives the equivariant two-point functions on Hilb k (C 2 ) is normalized as The Poincaré dual of the divisor class in H * T Hilb k (C 2 ), Q is given by |D = −|2, 1 k−2 . The operator of small quantum multiplication by D is given by the q-deformed Calogero-Sutherland Hamiltonian [51] 14 The (simple) Hurwitz theory is also related with the topological A-model on P 1 [76,79].

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Using this Fock space description, let us compute the equivariant classical intersection numbers of the divisor class in H * T Hilb k (C 2 ), Q . Let H cl D = H D | q=0 be the classical part of the operator H D . By α ℓ n α k −n |∅ = k(k − 1) · · · (k − ℓ + 1)n ℓ α k−ℓ −n |∅ for ℓ ≤ k, one obtains and Then on Hilb k (C 2 ), the equivariant classical intersection numbers of the divisor class are computed as By ǫ 1,2 → −ǫ 1,2 , these results coincide with the classical part of the stringy U(1) instanton counting computed in (4.4). Especially from (4.6), one finds a formula of the classical intersection numbers at ǫ = ǫ 1 + ǫ 2 = 0: where ǫ = ǫ 1 + ǫ 2 . In [47], it was also shown that the divisor generates the quantum cohomology ring of M k,N . Note that in the N = 1 case, we need to modify the above operator H D by adding ǫq 1 − q ∞ n=1 α −n α n , (D.9)
In what follows, using the last term in (D.8): q n 1 − q n nβ −n β n , (D. 10) and β n |1 k = 0, we compute the quantum parts of two three-point functions related with the computations in section 4: where c 1,2 are constants, and L 0 (q) is defined in (C.13). It would be interesting to determine the classical parts and fix the normalization constants.

E Exact Kähler potential for Calabi-Yau threefolds and fourfolds
In this appendix, we summarize the exact Kähler potentials on quantum Kähler moduli spaces of Calabi-Yau threefolds (e.g. [42]) and fourfolds (conjectured in [53]). The Kähler moduli space M Kähler (X) of Calabi-Yau d-fold X is defined by H 1 (∧ 1 T * X), where T * X is the holomorphic cotangent bundle on X. By considering the NLSM propagating on X, the Kähler moduli space M Kähler (X) is quantized by the α ′ corrections. For Calabi-Yau threefold, it is known that around a large radius point the quantumcorrected Kähler potential K on M Kähler (X) is given by where i, j, k = 1, . . . , h 1,1 (X). Here t ℓ are the complexified Kähler parameters, κ ℓmn are the classical triple intersection numbers of divisors on X, χ(X) is the Euler characteristic of X, and is the prepotential which gives the Gromov-Witten invariants n β defined by the holomorphic maps ∂φ = 0 in the (A-twisted) NLSM φ : P 1 → X. Let O J i be the observables JHEP07(2015)097 associated with J i ∈ H 1,1 (X), then the prepotential F (t) can be obtained from the threepoint function O J i O J j O J k on P 1 in the topological A-model via the relation For Calabi-Yau fourfold, it was conjectured that the Kähler potential K around a large radius point is given by [53] where i, j, k, ℓ = 1, . . . , h 1,1 (X) and m, n = 1, . . . , h 2,2 prim (X). 17 Here κ ijkℓ are the classical quadruple intersection numbers of divisors on X, C ℓ = X c 3 (X) ∧ J ℓ defined by the third Chern class c 3 (X) of X and J ℓ ∈ H 1,1 (X), η mn is the inverse matrix of the intersection matrix η mn = X H m ∧ H n on H 2,2 prim (X). Similar to the case of Calabi-Yau threefold, the quantum corrections are given by the generating functions F n (t) = 1 (2πi) 2 β∈H 2 (X,Z)\{0} n β,n Li 2 (q β ) (E.5) which give the Gromov-Witten invariants n β,n defined by the holomorphic maps ∂φ = 0 intersecting with the cycle dual to H n ∈ H 2,2 prim (X) in the (A-twisted) NLSM φ : P 1 → X. The generating function F n (t) is obtained from the three-point function O J i O J j O Hn on P 1 in the topological A-model via the relation where O Hn is the observable associated with H n ∈ H 2,2 prim (X), and κ ijn = X J i ∧ J j ∧ H n is the classical intersection number. In the conjectural formula (E.4), the generating functions G kℓ (t) = 1 (2πi) 2 β∈H 2 (X,Z)\{0} n β,kℓ Li 2 (q β ) (E.7) count the Gromov-Witten invariants n β,kℓ defined by the holomorphic maps ∂φ = 0 intersecting with the cycle dual to J k ∧ J ℓ ∈ H 2,2 prim (X), and by definition these generating functions are written by a linear combination of F n (t). Other quantities in (E.4) are defined by 17 h 2,2 prim (X) is the dimension of the primary subspace H 2,2 prim (X) ⊂ H 2,2 (X) whose elements are obtained from the wedge products of the elements of H 1,1 (X).

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