Stringy Instanton Counting and Topological Strings

We study the stringy instanton partition function of four dimensional ${\cal N}=2$ $U(N)$ supersymmetric gauge theory which was obtained by Bonelli et al in 2013. In type IIB string theory on $\mathbb{C}^2\times T^*\mathbb{P}^1\times \mathbb{C}$, the stringy $U(N)$ instantons of charge $k$ are described by $k$ D1-branes wrapping around the $\mathbb{P}^1$ bound to $N$ D5-branes on $\mathbb{C}^2\times \mathbb{P}^1$. The KK corrections induced by compactification of the $\mathbb{P}^1$ 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 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 [8,9,6]. 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 A-model 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 [21,22], and the parameter is identified with the topological string coupling constant g s . In [19,20], refinements of the topological vertex formalism were proposed so that the SU(N) 1 The instanton partition function Z Nek (ǫ 1 , ǫ 2 ) also depends on the Counlomb moduli described by the Cartan subalgebra for gauge group and the dynamical scales in four dimensional gauge theory. In this introduction we abbreviate these arguments for simplicity.
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 in 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.
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 (2.18) 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 2 The stringy instanton counting in this paper means the "gauge theoretic" instanton counting with the stringy corrections. This is different from "stringy" (or "exotic") instantons discussed in e.g. [ stringy corrections, and shows that quantization of the Kähler modulus ζ of the ADHM moduli space 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 U(1), as discussed in [27], we confirm agreements with the 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 . In Section 5 we give our conclusions and discuss some future directions. Appendix A is a note on the multiple gamma function. 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 discussed in [47]. In Appendix C, we review the Fock space description of H * T Hilb k (C 2 ), Q [48], and compute some equivariant correlators for comparing with the stringy U(1) instanton partition function. In Appendix D, 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 [49]).

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 and α = 1, . . . , N. By restricting the U(1) V R-charges to be non-negative, these are constrained as 0 < p < q < 1.
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 [50] (see also e.g. [51]). 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 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 W = Tr k χ([B 1 , B 2 ] + IJ) whose total U(1) V R-charge is two. The Fayet- 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 [55], one obtains the stringy U(N) k-instanton partition function [27]: 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, 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] 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 the simple poles are classified by ten types of two-tuple of Young diagrams µ 1,...,10 as , (2.11) and the contours are enclosing the real axes counterclockwise [8,9,6]. The index of the leading behavior r −2kN coincides with dim C M k,N = 2kN. It is useful to express the Nekrasov partition function by using N-tuple of Young diagrams µ = (µ 1 , . . . , µ N ) as [6,56,57,10] , (2.13) Here a αβ = a α − a β , and a µ (s) and ℓ µ (s) are the arm-and leg-length, respectively. In

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 Especially it was proposed that Z V (z) gives the Givental's I-function [45,46] of the ADHM moduli space M k,N [27,58]: Here λ 1 , . . . , λ k are identified with the Chern roots of the tautological bundle on M k,N (for N = 1, see [59]). r −1 is identified with the equivariant parameter which gives the generator of S 1 acting on the two sphere 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 6 The N + 2 dimensional torus T N +2 = U (1) N × U (1) 2 acts on M k,N as U (1) N : 2 . This action introduces the equivariant parameters a α , α = 1, . . . , N and ǫ 1,2 corresponding to the twisted masses in the GLSM as described in Table 1.
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 it needs the equivariant mirror map [59]: (2.16) Let us decompose the stringy instanton partition function into two parts with the classical and quantum stringy (α ′ ) corrections is the stringy instanton partition function except the α ′ corrections. Here n(µ) = s∈µ ℓ µ (s) and n(µ t ) = s∈µ a µ (s). 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 [60]), 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 (2.18) 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 except 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 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 [11,61] (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 (2.19), 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 a (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 [62] is given by [39,40] where m ∈ Z is the Chern-Simons coefficient, and Let β be the radius of the five dimensional circle. After 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 Chern-Simons coefficient m to the Kähler modulus ζ. The K-theoretic "perturbative" (in the gauge theoretic sense) partition function

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 [20,23,24] (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 Coulomb moduli a α in the SU(N) gauge theory [4]: (3.10) 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 in 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 (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 in Section 2, the instantons with the stringy corrections are described by the intersecting D1-and D5-branes in type IIB string theory on C 2 × T * P 1 × C. 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 [50]. 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.
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 As shown in [64], 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 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 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 [66,67,68]. Therefore we argue that the coefficient ζg 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). Then as in [39], the "scaled" intersection number ζ is identified with the FI parameter, and 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 (2.18): 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.
By lifting to the M-theory, 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 is physically derived. 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 also given by (3.6).
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 where f (z) 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 partition function is invariant under simultaneous constant shift a α → a α + c, α = 1, . . . , N − 1. By (2.14), the second normalization factor removes the dependence of the Euler constant γ [27], and fixes the precoefficients of the simple zeta values ζ(s) which receive the perturbative α ′ corrections. These normalizations fix the regularization scheme in the perturbative α ′ corrections described in Section 2.2. In this section, we study the structure of this partition function for N = 1, 2, 3 cases.

Conclusion and discussions
In this paper, we have studied the stringy instanton partition function in 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 of the K-theoretic instanton partition function. 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.
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 stringy corrections have the universal structure, which does not depend on k, for each N as in (4.4), (4.27), and (4.55). 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 computation 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.
It would be interesting to further study the quantum structure of the U(N) stringy instanton partition function, and to compare the structure with the theory of the quantum multiplication for the ADHM moduli space M k,N [60]. The refined topological vertex [19,20] do not capture the quantum stringy corrections, and so it would be also interesting to formulate "quantum refined topological vertex" which captures such quantum corrections.
A six dimensional analogue of the four dimensional instanton partition function was discussed in [69,70,71]. It would be interesting to study the stringy generalization of the six dimensional instanton partition function. 1 (x + n 1 ω 1 + · · · + n r ω r ) s , Re(s) > r. (A.2)
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 [72] (see also e.g. [73]). 10 Then the connected simple Hurwitz numbers H P 1 • g,k are obtained from the genus expansion of the free energy where = i log zz and x = (iq log zz) 2 . We see that the perturbative free energies F g (x) are given as [74,75] where c g,i are constants, 11 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 k! x k . The relation between the Hurwitz numbers of P 1 and the intersection numbers on Hilb k (C 2 ) was discussed in [76].
As described 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. 11 For example these are given by F 2 (x) = y 2 1440(1 − y) 5 (6y + 1), F 3 (x) = y 2 725760(1 − y) 10 (720y 4 + 3816y 3 + 3482y 2 + 548y + 9). 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 [47]. As pointed out in [47], the simple Hurwitz theory is related with the U(1) instanton counting with the first and second Casimir operators (see also [77]). 12 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 : Hilb k (C 2 ) = {J ⊂ C[x, y]|J is an ideal, dim C C[x, y]/J = k}. (C.1) 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 [78,44,48]. The Fock module over the Heisenberg algebra {α ±n , n ∈ N | [α m , α n ] = mδ m+n } 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 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 [48] 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 the equivariant classical intersection numbers of the divisor class are computed as where c µ (s) = (i,j)∈µ (j − i) and h µ (s) = a µ (s) + ℓ µ (s) + 1 is the hook length defined for a Young diagram µ as described in Figure 1 of Section 2.1.
The (quantum) equivariant three-point functions in H * T Hilb k (C 2 ), Q are also computed as [27]: where we have changed the equivariant parameters as ǫ 1,2 → −ǫ 1,2 . Here |Y 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 [49]). 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 associated with J i ∈ H 1,1 (X), then the prepotential F (t) can be obtained from the three- For Calabi-Yau fourfold, it was conjectured that the Kähler potential K around a large radius point is given by [49] e where i, j, k, ℓ = 1, . . . , h 1,1 (X) and m, n = 1, . . . , h 2,2 prim (X). 13 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 13 h 2,2 prim (X) is the dimension of the primary subspace H 2,2 prim (X) ⊂ H 2,2 (X) whose elements are given by the wedge products of the elements of H 1,1 (X).
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 (D.4), the generating functions G kℓ (t) = 1 (2πi) 2 β∈H 2 (X,Z)\{0} n β,kℓ Li 2 (q β ) (D.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 (D.4) are defined by