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Deformed planar topological open string amplitudes on Seiberg-Witten curve

We study refined B-model via the beta ensemble of matrix models. Especially, for four dimensional N=2 SU(2) supersymmetric gauge theories with N_f=0,1 and 2 fundamental flavors, we discuss the correspondence between deformed disk amplitudes on each Seiberg-Witten curve and the Nekrasov-Shatashvili limit of the corresponding irregular one point block of a degenerate operator via the AGT correspondence. We also discuss the relation between deformed annulus amplitudes and the irregular two point block of the degenerate operator, and check a desired agreement for N_f=0 and 1 cases.


Introduction
The AGT correspondence between four dimensional N = 2 supersymmetric gauge theory and two dimensional conformal field theory gives us several new insights into both theories. This correspondence was originally found, or conjectured by Alday, Gaiotto and Tachikawa in 2009 [1] as SU(2) superconformal quiver gauge theory in the Omega background / Liouville field theory (Virasoro algebra) correspondence by compactifying the six dimensional A 1 (2, 0) theory on a Riemann surface with punctures [2]. The Omega background has two deformation parameters ǫ 1 and ǫ 2 which generate the rotation of [3] (see also [4]): (z 1 , z 2 ) → (e iǫ 1 z 1 , e iǫ 2 z 2 ). (1.1) After that, various extensions and generalizations have been discussed. For instance, • The extension to SU(N) superconformal quiver gauge theory / A N −1 Toda field theory (W N −1 -algebra) correspondence [5], and the generalization to the non-conformal cases by decoupling the flavors [6,7,8]. We correctively call these correspondences "AGT correspondence".
• The extension to a half BPS "simple type" surface operator which brakes the gauge symmetry as SU(N) to SU(N − 1) × U(1) / a degenerate operator in Toda field theory correspondence [9,10,11]. This correspondence is called "AGGTV correspondence".
In [12], Dijkgraaf and Vafa explained the AGT correspondence by topological B-model / matrix model correspondence, and by free field representation of the matrix model. In this paper, we only concentrate on SU(2) cases. Although the corresponding Penner type matrix model which has logarithmic potential plays an important role in their explanation, in section 2 we consider the beta ensemble of matrix models with a polynomial potential V (z) for avoiding some subtleties: (1.2) When β = 1, this model reduces to the hermitian one-matrix model. From this model, one can obtain "refined" topological recursion (2.5) as the loop equation [13,14,15] which reduces to the Eynard-Orantin recursion [16] in β = 1 case.
In [12], they also proposed a refinement of topological strings by the beta deformation of matrix models. After forgetting the matrix model origin of the "refined" topological recursion, we utilize this recursion as the refinement of topological B-model on the Seiberg-Witten curve of Gaiotto form [1,6]. And then we study the relation between refined open topological string free energy (amplitude) and chiral block with a degenerate operator Φ 1,2 corresponding to the "simple type" surface operator. This work is a refined version of the B side computation discussed in [10,17] for SU (2) cases. 1 In section 3, we study an operator in the beta ensemble corresponding to the degenerate operator on the CFT side as [18], and formally discuss a relation between its correlator and "deformed free energy" with a deformation index ℓ obtained from the "refined" topological recursion. Especially, by taking the Nekrasov-Shatashvili (NS) limit ǫ 2 → 0 [19], we obtain the simple relation (3.11) which claims that the NS limit of the m point block of the degenerate operator equals to the summation of deformed disk amplitudes on Seiberg-Witten curve obtained from the "refined" topological recursion. 2 By direct computation of the deformed disk amplitudes, in section 4, we check the relation for SU (2) theories with N f = 0, 1 and 2 fundamental flavors. In section 5, we discuss the deformed annulus amplitudes, and obtain the expected relation (5.5) to the two point block of the degenerate operator, by taking the ambiguity independent parts on both sides. For SU (2) theories with N f = 0 and 1 fundamental flavor, we check this relation only up to ℓ = 1. Section 6 is devoted to the conclusion of this paper, and a future direction. In appendix A, we summarize the computation on the CFT side [17]. In appendix B, we discuss that the Bergman kernel which defines the annulus amplitude on genus one Seiberg-Witten curve can be written as a functional of the period of the curve. Appendix C is the summary of some detailed computation on the B-model side in section 4 and 5.

Topological recursion of the beta ensemble
In this section, we summarize the beta ensemble of matrix models defined by (1.2) and its topological recursion. The connected correlator of the "trace" type operator, can be expanded under g s → 0, N → ∞, and fixed g s N as On topological string theory side, g and h are interpreted as "genus" and "boundary" of the world sheet respectively, and in this beta ensemble one has new index ℓ. In the β = 1 case, the perturbative correlators W (g,h) ℓ≥1 (p 1 , . . . , p h ) vanish, and we call these quantities "deformed correlators" with the deformation index ℓ. It is well known that the (undeformed) disk and annulus correlators are given by [40,41,42,43], . When the potential V (z) is a polynomial, we see that M(p) is a rational function, and σ(p) = p 2s−1 + · · · , or p 2s + · · · is a polynomial. B(p 1 , p 2 ) is the Bergman kernel on the spectral curve. The other correlators W (g,h) ℓ (p H ) = W (g,h) ℓ (p H ) for (g, h, ℓ) = (0, 1, 0), (0, 2, 0) are obtained from the "refined" topological recursion [15,13,14]:  [16,44,43]. The third type differential dE q (p) is given as [43], If q is contained in the cycle A i , then C i (q) must be replaced with C reg where the sign depends on the branch. L i (p) are the (normalized) bases of the holomorphic differentials on the spectral curve: In the two-cut case, the differential is concretely written by the complete elliptic integral of the first and third kind as [45], where q 1 < q 2 < q 3 < q 4 are the branch points on the spectral curve. The differential (2.8) is defined by C reg (q) (C(q)) for q inside (outside) the cycle A 1 around the cut C 1 = [q 1 , q 2 ].

The Nekrasov-Shatashvili limit of refined B-model
Here, we discuss the NS limit of the beta ensemble (1.2). We consider the theory which is defined by the topological recursion (2.5) on spectral curve as refined topological B-model on spectral curve.
To discuss the NS limit of the beta ensemble, let us introduce the parameters These parameters are identified with the Omega background parameters in (1.1). And then two types of NS limit are defined [19]: limit B : ǫ 1 → 0, ǫ 2 : finite, (g s → 0, β → 0).
We now consider the two "determinant" type operators [38], By using the free chiral boson representation for the beta ensemble [46,47,18,38]: the operator O α (p) is represented as free fermion O α (p) = exp gs ǫα φ(p) , and from the correspondence to the B-brane in the topological B-model this operator is called "ǫ αbrane" [38]. Let us consider the m point correlator of the "ǫ α -branes": . By (3.5), one can find that the right hand side of (3.6) is rewritten as and then the following perturbative expansion of (3.6) is obtained [18]: where S (m) α;i,j (p) are the expansion coefficients of the left hand side of (3.6). On the right hand side of (3.8), we have used the expansion (2.2), and defined the perturbative free energies F (g,h) ℓ (p 1 , . . . , p h ) by (3.9). By this definition, by absorbing the potential term V (p) of (2.3) we redefine the disk correlator as [18], (3.10) Actually if V ′′ (p) is an analytic function inside A, the higher order correlators are not changed under this redefinition as seen from the recursion (2.5).
Let us consider the "ǫ 1 , (or ǫ 2 )-brane", and by taking the limit A (3.2), (or the limit B (3.3)), we obtain the relation: as the coefficients of ǫ ℓ−1 1 , (or ǫ ℓ−1 2 ). From this relation, we see that under the NS limit, the m point correlator of the "ǫ α -brane" is factorized into the one point correlators: In the next section, we compute the right hand side of (3.11). By the recursion (2.5) and the redefinition (3.10), the ℓ-th order deformed disk correlator W (0,1) is obtained from the planar "refined" topological recursion: Using this correspondence, we compute the left hand side in the relation (3.11) as the perturbative expansion of the correlation function of the degenerate operator Φ 1,2 . By taking the appropriate CFT vacuum |G(a) [6] corresponding to each SU(2) gauge theory with the Coulomb moduli parameter a via the AGT correspondence, we consider the perturbative expansion of the chiral one point block: where after introducing the mass scale g s on the CFT side as the scaling a → a/g s , we have introduced the parameters For the charge conservation, the momentum a of the left hand side in (4.2) is shifted.
These parameters correspond to (3.1) introduced on the matrix model side, and then the free energies G 1;i,j (p). In this correspondence, we have identified parameters as ε 1 = ǫ 1 and ε 2 = ǫ 2 . Therefore, one can expect the relation: Note that the left hand side of this relation gives us the momentum shift invariant solutions in (4.2). The computations on these CFT side are summarized in appendix A.
The "disk" part of the relation (4.4): is generally proved on the CFT side [9]. Note that for the cases of N f = 0 and 1, the matrix model, or beta ensemble realizations as (1.2) are not known, but we can consider the refined B-model on each corresponding Seiberg-Witten curve by the recursion (2.5). 3 In case that the spectral curve y(p) = M(p) σ(p) has two cuts C 1 = [q 1 , q 2 ] and (p) is given by [15], where A 1 and A 2 are the cycles surrounding the cuts C 1 and C 2 respectively. In this section, we check the relation (4.4) up to the second order deformed disk correlator (4.7) The detailed computations are summarized in appendix C.

N f = 0 case
For the pure SU(2) case with the quantum Coulomb moduli parameter u, and the dynamical scale parameter Λ, the Seiberg-Witten curve is given by [6], The branch points of this curve are By computing the period of this curve, one obtains the weak coupling expansion of the quantum modulus u, By computing the disk amplitude F (0,1) 0 (p), we have directly checked that this coincides [17], and then the relation (4.5) surely holds. Next, let us compute the first order deformed disk amplitude F is obtained. By taking c = 1 2 log a, we have checked that this result coincides with G 1,0 (p) computed in (A.8) up to Λ 12 . The second order deformed disk amplitude F (0,1) 2 (p) is also computed by (4.7), and we have checked that this coincides with −G 2,0 (p) computed in (A.9) up to Λ 12 .

N f = 1 case
The Seiberg-Witten curve for the SU(2) gauge theory with one fundamental matter of mass m is [6], (4.14) The branch points of this curve are perturbatively obtained as where for the double signs, q 1 takes the plus signs, and q 2 takes the minus signs. Because q 1 and q 2 are expanded around ∞, we need to take the cycle A 1 as containing the point at infinity. By the period computation 1,B (p) = 0 is proved in appendix C.1, and we checked the same relation to the N f = 0 case: The branch points of this curve are where for the double signs, q 1 and q 3 take the plus signs, and q 2 and q 4 take the minus signs. Note that as the N f = 1 case, because q 1 and q 2 (q 3 and q 4 ) are expanded around ∞ (0), we need to take the cycles A 1 and A 2 as containing ∞ and 0 respectively. The computation of the A 1 -period gives us

Deformed annulus amplitudes
Here we discuss the deformed annulus amplitudes on Seiberg-Witten curve for SU (2) gauge theory. For the m point correlator of the "ǫ 1 -branes", by comparing the coefficients We now consider the case of m = 2, and as in section 4, let us identify S 1;i,j (p 1 , p 2 ) with G (2) i,j (p 1 , p 2 ) defined by the perturbative expansion of the chiral two point block: As discussed in appendix A, we can check the factorization property (3.12): G i,0 (p 2 ). Because the free energies G (2) i,1 (p 1 , p 2 ) depend on the momentum shift of in-and out-states, we need to take the shift invariant free energies G (2) i,1 (p 1 , p 2 ) by considering the invariant parts under the shift For the cases in appendix A, we see that the free energies G i,1 (p 1 , p 2 ) have the expansion: a (i,1) n 1 ,n 2 p n 1 1 p n 2 2 + a (i,1) by neglecting the singular term at p 1 = p 2 . And then by (5.1), we obtain the relation: where on the right hand side, we have only considered the "universal terms" which do not depend on the constants of integration in (3.9).
In the genus one with two sheeted cases, the Bergman kernel is given by the Akemann B(p 1 , p 2 ) = dp 1 dp 2 where S d are the degree d elementary symmetric polynomials of the branch points q i obtained from f (p, p) = σ(p) = 4 i=1 (p − q i ). From the recursion (2.5), the first order deformed annulus amplitude F (0,2) 1 (0,1) 1 (p 1 ) + dp 1 d dp 1 B(p 1 , p 2 ) − dp 1 dp 2 2(p 1 − p 2 ) 2 . (5.7) In the following, for N f = 0 and 1 cases, we check the relation (5.5) up to ℓ = 1 by taking the "universal terms" on the right hand side.
For N f = 0 case (4.8), the annulus amplitude is obtained as computed in [17]. In this computation, because the Seiberg-Witten curve (4.8) has cubic form σ(p) = −p 3 + S 2 p 2 − S 3 p + S 4 , we need to replace f (p 1 , p 2 ) and G(k) in the formula (5.6) with By comparing this result with (A.15), we have checked the relation (5.5) for ℓ = 0 up to Λ 8 [17]. By (5.7), the first order deformed annulus amplitude F For N f = 1 case (4.14), as above, the (deformed) annulus amplitudes F

Conclusion and future direction
In this paper, we have studied a refinement of topological B-model via beta ensemble of matrix models. Especially, we confirmed the agreement between the deformed planar open B-model amplitudes on Seiberg-Witten curve and chiral blocks with the degenerate operator Φ 1,2 for some SU(2) cases which was inspired from the AGT correspondence.
Although there are some ambiguities on both theories: ambiguities of the constants of integration (3.9) on the B-model side, and ambiguities of the momentum sift of in-and out-states for the charge conservation on the CFT side, by taking the independent parts of these ambiguities, we checked the correspondence of the deformed disk and annulus amplitudes to the computation on the CFT side.
As a future direction of this work, it is interesting to study refinement of topological B-model on Calabi-Yau threefold. By geometric engineering [50], one obtains the correspondence between the topological A-model partition function on some local toric Calabi-Yau threefolds with A N −1 singularity and the five dimensional (K-theoretic) version of the Nekrasov partition function with the self-dual condition ǫ 1 + ǫ 2 = 0 in (1.1). 6 This was confirmed in [51,52,53,54,55] by the topological vertex [56]. Using the refined topological vertex proposed in [57] and [58], the extension to general Omega background was also discussed [57,58,59,60]. 7 In [63], open string version of geometric engineering was proposed. By this proposal, the "simple type" surface operator is embedded to topological open string theory on some local toric Calabi-Yau threefolds with toric brane [10,11,64,17]. To consider refinement on the B-model side, matrix model formulation is an important step. 8 In [70], for the unrefined A-model on the local toric Calabi-Yau threefold which is the large N dual [71] of the U(N) Chern-Simons gauge theory on the lens space S 3 /Z p , the Chern-Simons matrix model which has the unitary measure instead of the hermitian measure ∆(z) in (1.2) was derived. By the proposal in [72,45], it was conjectured that from the Eynard-Orantin recursion [16], one can compute not only the correlation functions with a Wilson loop in the U(N) Chern-Simons gauge theory on S 3 /Z p , but also the unrefined open A-model amplitudes on arbitrary local toric Calabi-Yau threefold with toric brane. In [15], for refining this proposal, as a simple guess, it 6 In this correspondence, ǫ 1 = −ǫ 2 = g s gives us the topological string coupling constant. On the matrix model side, as seen from (3.1), the self-dual condition corresponds to β = 1. 7 Several world sheet interpretations of refined topological strings are discussed in [61,62,39] was considered a beta deformed Chern-Simons matrix model: corresponding to the unrefined A-model on the resolved conifold if β = 1. They checked the coincidence between the first deformed disk correlator W which agrees with the open refined A-model computation. It may be important to study the structure of this matrix model for formulating refined B-model. 9 It is also interesting to approach the problem from the viewpoint of five dimensional version of the AGT correspondence [75,76,77].
Acknowledgements: I would like to thank Hidetoshi Awata, Hiroyuki Fuji, Hiroaki Kanno and Yasuhiko Yamada for the collaboration [17] and discussions at the initial stage of this work.

A Computation on the CFT side
In this appendix, we summarize the computation on the CFT side corresponding to the SU(2) gauge theories with N f = 0, 1 and 2 fundamental flavors [17].
A.1 N f = 0 case The corresponding CFT vacuum state |∆ a , Λ which is called the Gaiotto state is defined by [6], See also [74] for recent developments of the beta deformation related to this topic.
We now consider the one point block of the degenerate primary operator Φ 1,2 with the conformal dimension h 1,2 = − 1 2 − 3 4b 2 : where for the charge conservation, we need to shift the momentums of in-and out-states.
Here we consider the one point block as As in the pure case, After scaling a → a/g s , Λ → Λ/g s , m → m/g s , and introducing the parameters (4.3), one obtains a series solution Y (1) (p, a, m, n (p, a, m): (A.20) And then we get the free energy where the lower order free energies G i,0 (p) which are the momentum shift invariant are We also consider the two point block of Φ 1,2 : The differential equations for Ψ (2) (p i , a, m, Λ) = f (p i )Y (2) (p i , a, m, Λ) are given by [17], As same as N f = 0 case, the free energy i,j (p 1 , p 2 ) := log is obtained, and we can check the relation G i,0 (p 1 , p 2 ) = G (1) i,0 (p 1 ) + G i,0 (p 2 ) from the differential equations. The shift invariant free energies G (2) i,1 (p 1 , p 2 ) under the shift (5.3) are (A. 30) A.3 N f = 2 case The irregular CFT corresponding to SU(2) gauge theory with two (anti-)fundamental matters is constructed from the Gaiotto state |∆ a , Λ/2, m [6]. In this case by multiplying exp(−Λ 2 /2ǫ 1 ǫ 2 ) which is called the U(1) factor by the norm ∆ a , Λ/2, m 2 |∆ a , Λ/2, m 1 , this coincides with the Nekrasov partition function [6,78].
As same as the previous discussions, the one point block . After introducing the mass scale g s , ε 1 , ε 2 as before, one obtains a solution n (p, a, m i ): The free energy is where the lower order free energies G i,0 (p) which are the momentum shift invariant are

B Functional formula for the Bergman kernel
The formula (5.6) of the Bergman kernel for genus one Seiberg-Witten curve as depends on the branch points of the spectral curve because of the factor In this appendix, by the same discussion to [79,80] for local toric del Pezzo surfaces, we prove that this factor can be written as a branch points independent form.
At first, we note the formula where τ is the elliptic modulus of the curve (B.1), and by making use of the prepotential , this is given as where c 1 is a constant. By this formula, we can rewrite the factor G(k) as [79], At first, let us note that in Seiberg-Witten theory, the period ω 1 gives us where c 2 is a constant. In general, the second Eisenstein series E 2 (τ ) is written as [80], E 2 (τ ) = d dτ 12 log ω 1 + log ∆ , (B.7) by the period ω 1 , and the discriminant ∆ = i<j (q i − q j ) 2 = 4 i=1 σ ′ (q i ) of the curve (B.1). By (B.4), the right hand side of (B.7) is rewritten as Here, we prove that W where note that for N f = 1 and 2 cases, as noticed in section 4.2 and 4.3 we should take the cycles A 1 and A 2 as containing ∞ and 0 respectively. For that proof, we use the fact [15] that by the Möbius transformation satisfying g(g(q)) = q such as q → q = g(q) := αq + β γq − α , α := q 1 q 4 − q 2 q 3 , β := (q 1 + q 4 )q 2 q 3 − (q 2 + q 3 )q 1 q 4 , γ := q 1 + q 4 − q 2 − q 3 , (C.2) C reg (q) and C(q) in (2.9) are related each other as dg(q) dq LC reg ( q) = LC(q) − γ α − γq , dg(q) dq LC( q) = LC reg (q) − γ α − γq . (C.3) By this Möbius transformation, the branch points q 1 , q 2 , q 3 and q 4 are mapped to q 4 , q 3 , q 2 and q 1 respectively, and then the cycle A 1 is mapped to the cycle A 2 , and vice versa.