Local B-model Yukawa couplings from A-twisted correlators

Using the exact formula for the A-twisted correlation functions of the two dimensional $\mathcal{N}=(2,2)$ gauged linear sigma model, we reconsider the computation of the B-model Yukawa couplings of the local toric Calabi-Yau varieties. Our analysis is based on an exact result that has been evaluated from the supersymmetric localization technique and careful treatment of its application. We provide a detailed description of a procedure to investigate the local B-model Yukawa couplings and also test our prescription by comparing the results with known expressions evaluated from the local mirror symmetry approach. In particular, we find that the ambiguities of classical intersection numbers of a certain class of local toric Calabi-Yau varieties discovered previously can be interpreted as degrees of freedom of the twisted mass deformations.


Introduction
Two dimensional N = (2, 2) gauged linear sigma model (GLSM) [1] has played significant roles in the study of the topological quantum field theories and the mirror symmetry. A physical proof of mirror symmetry in [2] makes use of the duality properties of this model. Shortly after the work of [1], the topological A-twisted version of the GLSM was studied in [3] and the genus zero correlation functions were exactly computed for the A-models with target space a toric variety or a toric Calabi-Yau hypersurface. These developments led to a conjecture called toric residue mirror conjecture [4,5] (later proven in [6,7,8,9]), which states the existence of an equality between an integration over the toric compactification 1 of the moduli space of genus zero maps in the A-model and the toric residue [12] over the space of a holomorphic section in a toric variety in the B-model (see also [13]). The toric residue mirror conjecture provides an efficient method to compute the B-model Yukawa couplings exactly for the mirror varieties of toric Calabi-Yau complete intersections.
On the other hand, by using the supersymmetric localization technique [14], the exact formula about the A-twisted N = (2, 2) GLSM correlation functions on a supersymmetric two-sphere background has been clarified in [15,16]. 2 This approach does not depend on a particular choice of gauge group and enables us to calculate the Yukawa couplings for the GLSMs with non-abelian gauge groups efficiently [15,19] (see also [20]). Meanwhile, the application for the models with non-compact target spaces has not been thoroughly investigated. The aim of this work is to fill this gap and establish the explicit formalism to investigate the local B-model Yukawa couplings from the A-twisted GLSM correlation functions on local toric Calabi-Yau varieties.
In this paper, we will explicitly demonstrate how to apply the exact formula about Atwisted GLSM correlators in [15] for the explicit computation of the B-model Yukawa couplings of local toric Calabi-Yau varieties. We remind that the application of the formula to the local toric Calabi-Yau varieties requires a careful treatment of the mass parameters of the chiral multiplets. 3 Then we propose how to determine the proper mass deformations to conduct the exact calculation of the A-twisted GLSM correlators and check that the resulting A-twisted correlators coincide with known results for the B-model Yukawa couplings evaluated by the local mirror symmetry approach [22,23,24]. To our best knowledge, this relationship has not been thoroughly investigated before. Moreover, as a by-product, we also find that an ambiguity of the classical intersection numbers of a certain class of local toric Calabi-Yau varieties argued before in [23] can admit a new interpretation. This paper is organized as follows. First we will take a brief look at the exact results about the GLSM correlation functions in Section 2. In Section 3, we explain the details about the topological properties of local toric Calabi-Yau varieties and propose the rules of mass assignment 1 Toric compactifications can be generalized to stable toric quasimaps [10] allowing some degeneracies of moduli space of maps and further to stable quasimaps [11] which also include non-abelian varieties. 2 It is worth noting that the supersymmetric localization of N = (2, 2) gauge theories has been also performed on a different two-sphere background, which corresponds to a fusion of the A-andĀ-twists on two hemispheres [17,18]. 3 We refer the reader to [21] where several new aspects of twisted mass deformations have been clarified.
for applying the formula into the non-compact backgrounds. Then in Section 4 we will evaluate the GLSM correlators for several examples of local toric Calabi-Yau threefolds and fourfolds, and demonstrate that the resulting expressions are completely consistent with known results obtained by the local mirror symmetry approach. Finally we will conclude and propose several future directions. In Appendix A, we collect the building blocks of the localization formula, which are useful to understand the mirror transformation. The list of the local B-model Yukawa couplings for the local del Pezzo surface K dP 2 and the local A 2 geometry are summarized in Appendix B and C, respectively.
2 N = (2, 2) GLSM on the Ω-deformed two-sphere Here we will take a brief look at the two dimensional N = (2, 2) GLSM on the Ω-deformed two-sphere S 2 which is a one-parameter deformation of the A-twisted sphere. As performed in [15], supersymmetric localization formula considerably simplifies the exact calculations of the A-twisted correlation functions and realizes the quantum cohomology relations appropriately.

Exact formula about GLSM correlation functions
Following the approach to curved space rigid supersymmetry advocated in [25], supersymmetric backgrounds in two dimensions were studied in detail in [26]. A remarkable result is that the topological A-twist on the two-sphere admits an interesting U (1) equivariant deformation called the Ω-deformation [27]. This deformation can be characterized by an equivariant parameter which corresponds to a non-trivial expectation value for the graviphoton field. The ordinary topological A-twist is obtained by setting = 0. Let G be a rank rk(G) gauge group with Lie algebra g and Cartan subalgebra h ⊂ g. In [15], an exact formula about correlation functions of N = (2, 2) GLSM with gauge group G on the Ω-deformed two-sphere S 2 was clarified as by using supersymmetric localization technique (see also [16]). Here O 1 (u)| N and O 2 (u)| S are gauge invariant operators constructed from the complex scalars u = u 1 , . . . , u rk(G) ∈ h ⊗ R C in the vector multiplet inserted at the north and south poles, respectively. The order of the Weyl group of G is denoted by |W|, and where θ i and ξ i are theta angles and Fayet-Iliopoulos parameters for the central U (1) r ⊂ G. The parameters d = (d 1 , . . . , d r ) represent magnetic charges for the central U (1) r called GNO charges [28]. The factor I d (O 1 O 2 ) is a differential form given by which consists of the one-loop determinants for the vector multiplet and for the chiral matter multiplets Φ a in a representation R a with R-charge r a and the twisted mass λ a . Here ∆ + is the set of positive roots and ρ a denote the weights of R a . The products α( * ) and ρ a ( * ) are defined by the canonical pairing. The one-loop determinant for chiral multiplets (2.5) has poles along hyperplanes defined from the Γ-function in the denominators. For later convenience, we will use a collective form for the gauge charges as Q = Q i ∈ Z rk(G) ⊂ h * where i labels all the components of the multiplets of the GLSM. The hyperplanes can intersect at a point u = u * = (u * 1 , . . . , u * rkG ) and realize a codimension rk(G) pole. Such intersecting hyperplanes simultaneously specify a subset of charge vectors with at least rk(G) elements and we denote it by Q * ⊂ Q.
A crucial ingredient of the exact formula (2.1), whose treatment will be described in the next section, is the Jeffrey-Kirwan residue operation [29] 4 JK-Res [Q * , η] at u = u * depending on a choice of a covector η ∈ h * . As adopted in [15], we always choose η to be parallel and pointing in the same direction with ξ i and therefore η specifies the phase of the model.

The Jeffrey-Kirwan residue operation
In this section, we will briefly review the Jeffrey-Kirwan residue operation. For more details, we refer the reader to [7,32]. Definition 2.1. When Q * ⊂ Q lies within an open half-space of h * , the associated intersection point u = u * is called projective point. Such an arrangement of hyperplanes is called projective arrangement.

Definition 2.2. Consider an integrand
The cases with generic u * can be realized by shifting the coordinates appropriately. Then the Jeffrey-Kirwan residue JK-Res u=u * [Q * , η] is given by where Cone(Q 1 , . . . , Q rk(G) ) is the closed cone spanned by Q 1 , . . . , Q rk(G) .
More constructive definition can be represented as follows. Let FL(Q * ) be a finite set of flags such that Q * contains a basis of each of the flags F j , j = 1, . . . , rk(G). Then one can choose an ordered set Q F * = (Q i 1 , . . . , Q i rk(G) ) such that the first j elements {Q im } j m=1 give a basis of F j . Definition 2.3. For the above defined ordered basis Q F * of a flag F ∈ FL(Q * ), the iterated residue Res F of ω = ω 1...rk(G) dQ i 1 (u) ∧ · · · ∧ dQ i rk(G) (u) is defined by Here in each step of the residue operations on the right hand side, the higher variables are remained to be free parameters. Note that the iterated residue only depends on the flag F , and does not depend on the choice of the ordered basis.
Let us define the closed simplicial cone for a flag F ∈ FL(Q * ) as and denote by FL + (Q * , η) a set of flags such that the corresponding cone s + (F, Q * ) contains η.
Definition 2.4. By the partial sums of the elements of Q * = {Q 1 , . . . , Q n }, n ≥ rk(G), define a set An element v ∈ Cone(Q * ) is called regular with respect to ΣQ * if v ∈ Cone sing (ΣQ * ), where Cone sing (ΣQ * ) is the union of the closed cones spanned by rk(G) − 1 independent elements of ΣQ * . A regular v ∈ Cone sing (ΣQ * ) identifies one chamber in h * .
Then the following theorem can be proved [7].

GLSM correlators and local B-model Yukawa couplings
It has been clarified in [15] that the A-twisted correlators given by (2.1) with vanishing Ωdeformation precisely give the B-model Yukawa couplings of the mirror of compact Calabi-Yau manifolds and non-compact orbifolds. 5 Here, we consider local toric Calabi-Yau varieties. In order to perform the Jeffrey-Kirwan residue operation for these backgrounds appropriately, it is required to introduce the twisted masses to matter fields and realize a projective hyperplane arrangement of Definition 2.1. The main purpose of this paper is to provide a detailed description of this procedure to investigate the local B-model Yukawa couplings.

Local toric Calabi-Yau varieties and correlation functions
Consider a local toric Calabi-Yau m-fold X described by a symplectic quotient X = C n // ξ (C * ) r with m = n − r, m ≥ 3. 6 In terms of the GLSM, this background can be specified by n chiral matter multiples Φ i , i = 1, . . . , n with R-charge 0 and U (1) r charge vectors Q i ∈ Z r satisfying the Calabi-Yau condition n i=1 Q i = 0. One can also assign the twisted masses −λ i for the chiral multiplets Φ i . Then a model can be specified by a set of charge vectors and twisted masses as In the limit of the vanishing Ω-deformation = 0, the exact formula about correlation functions (2.1) in the geometric phase reduces to a simple expression where the moduli parameters z = (z 1 , . . . , z r ) are defined in (2.2) and the one-loop determinant for chiral multiplets in (2.5) or (A.5) takes a reduced form Choosing the monomials of u as the operators O(u), we focus on the following GLSM correlators: Remark 3.1. The phase structure of the Calabi-Yau variety X is described by the secondary fan which consists of the set of the charge vectors {Q 1 , . . . , Q n } in h * (see e.g. [33]). Although we will consider a particular choice of η ∈ h * inside the geometric phase of X for actual computation of the GLSM correlators (3.4), the result does not depend on the choice of η as long as the Jeffrey-Kirwan residue operation is carried out appropriately.

Criteria for the twisted mass deformations
Computations of the GLSM correlators (3.4) for local toric Calabi-Yau varieties require a careful treatment. This is because generically the associated hyperplane arrangement without twisted mass parameters λ i becomes non-projective and one should not use the Jeffrey-Kirwan residue operation directly. Therefore it is required to introduce twisted masses for chiral multiplets in a proper way, such that the hyperplane arrangement of the model becomes projective. 7 Although 6 We refer the reader to [33] for a detailed introduction on this subject. 7 Note that the relevance of twisted mass deformations has been also clarified in [21] from the other perspective.
the necessity of this kind of prescription has been argued before, up to the authors knowledge, there has been no satisfactory attempt to clarify the proper way of the twisted mass insertions.
Here we propose the following rules as a proper determination of the twisted mass deformations: 1. The twisted masses should not be inserted for chiral multiplets with Q i = 0 describing a non-compact fiber coordinate, or a blow up coordinate of a singularity.

2.
Suppose a GLSM of a local toric Calabi-Yau variety of interest has a chiral multiplet Φ 0 which is neutral under the j-th U (1) gauge symmetry and describes a non-compact fiber coordinate X 0 by its scalar component. 8 Let λ i , i = 1, . . . , s be twisted masses for other chiral multiplets Φ i with non-zero charges with respect to the j-th U (1) gauge symmetry. If a divisor defined by X 0 = 0 contains a blow up coordinate of a singularity, one need to impose for the twisted mass λ 0 of Φ 0 as a "Calabi-Yau condition on the divisor".
3. As long as the above requirements are fulfilled, one can turn on generic twisted masses for the remaining chiral multiplets while respecting the symmetries of the model such as the permutation of the homogeneous coordinates of the toric variety.
In Section 4, by using the above prescription for various examples, we will explicitly check that the GLSM correlators (3.4) give the same results predicted by the mirror symmetry approach. Moreover, we find that the ambiguities of the intersection numbers for a certain class of varieties argued in [23] can be reinterpreted as degrees of freedom of the proper twisted mass deformations.
Remark 3.2. In this paper, we do not consider the so-called non-nef toric varieties such as [34,35,36]. It would be interesting to investigate such kind of varieties and try to extend our prescription.

Local Yukawa couplings and the mirror map
The topological B-model m-point Yukawa couplings of a Calabi-Yau m-fold X ∨ are defined by where ∇ is a flat connection called Gauss-Manin connection (see e.g. [33]) and T * M is the cotangent bundle of the complex structure moduli space M of the Calabi-Yau m-fold X ∨ . L is a holomorphic line bundle over M whose section is given by the nowhere vanishing holomorphic m-form Ω(z) on X ∨ . The topological A-model m-point Yukawa couplings for the mirror X of X ∨ can be obtained from (3.5) by changing the complex structure moduli parameters {log z i } into the flat coordinates {log q i } parametrizing h i ∈ H 1,1 (X) as This transformation map is called mirror map. Note that the quantity X 0 (z) 2 is a square of the monodromy invariant fundamental period of X ∨ and is equal to 1 for local toric Calabi-Yau varieties, while for compact Calabi-Yau m-folds it is a non-trivial function of the moduli parameters z.
Generically the A-model three-point Yukawa coupling of a Calabi-Yau threefold, say X 3 , takes the following form [37,38,39,40,22]: This provides a generating function of integer invariants n d which give the Gromov-Witten invariants and enumerates the number of holomorphic maps φ : intersecting with cycles dual to h i , h j , and h k . Here κ h i h j h k is called classical triple intersection number and the summation with respect to d is taken only over non-negative elements.
For Calabi-Yau m-folds, the m-point Yukawa couplings can be factorized into three-point Yukawa couplings in accordance with the fusion rules of the underlying Frobenius algebra (see [41,42,43] for details). The so obtained A-model three-point Yukawa couplings similarly provide generating functions of genus zero Gromov-Witten invariants for m-folds. In the case of a Calabi-Yau fourfold X 4 , the A-model four-point Yukawa couplings are factorized into threepoint Yukawa couplings as where H α , H β ∈ H 2,2 (X 4 ) represent the so-called primary elements generated by the wedge product of the elements in H 1,1 (X 4 ) and η αβ is the inverse matrix of the intersection matrix associated with H α and H β . The A-model three-point Yukawa couplings Y h i h j H (q) have a generic form [41,42,43,44]: where κ h i h j H is the classical intersection number associated with cycles dual to h i , h j and H.

Mirror map from the localization formula
Starting from the factors (2.4) and (2.5) in the localization formula, one can find building blocks of the Givental I-function [45,46,47] from which the mirror map can be properly derived. Here we go straight to the point and some technical details are relegated to Appendix A. For a local toric Calabi-Yau m-fold X with GLSM description (3.1), one can construct the Givental I-function for X with the following form which consists of the building blocks defined in (A.8). To find the mirror map, it is sufficient to consider the case without twisted mass deformations. Expanding the I-function with λ i = 0 around = ∞, we obtain Then the inverse mirror map log q i , i = 1, . . . , r can be obtained by log q i = I 1,i (z) [22,48] and the result is where we put 1/n! = 0 for n < 0 and ′ I means that the summation is taken over all I satisfying For a compact Calabi-Yau manifold, the I-function is generically annihilated by the Picard-Fuchs equation for the mirror. 9 Then the higher order terms of the expansion in (3.11) are expected to possess the information about the genus zero Gromov-Witten invariants (see e.g. [49]). But for a local toric Calabi-Yau variety, say X local , the information obtained from the I-function without twisted mass parameters (i.e. λ i = 0 for all i) is sometimes not enough to get Gromov-Witten invariants. For instance, if dim H 4 (X local , Z) = 0, the higher order terms O( −2 ) in (3.11) vanish and cannot say anything about the Gromov-Witten invariants.
It is worth noting that by using the I-function with twisted mass parameters and the Birkhoff factorization [47], the equivariant local mirror symmetry for local toric Calabi-Yau threefolds was developed in [34,35,36] and the Gromov-Witten invariants have been computed.

Examples
Based on the criteria for the twisted mass deformations mentioned in Section 3.2, here we will compute the A-twisted GLSM correlators (3.4) for various local nef toric Calabi-Yau varieties in the geometric phase and demonstrate that the corresponding local B-model Yukawa couplings can be calculated appropriately. A novel aspect of the previously observed ambiguities for certain class of varieties [23] will be also discussed.

Local toric Calabi-Yau threefolds
First we will focus on the local nef toric Calabi-Yau threefolds whose exact properties have been studied by using the local mirror symmetry [22,23,34,35,36,50]. We confirm that the exact formula about the A-twisted GLSM correlation functions match the previous results of the local B-model Yukawa couplings with the aid of our prescriptions. We also see that the ambiguities of the intersection numbers in [23] can be interpreted as degrees of freedom of the twisted mass deformations. Let us consider the resolved conifold described by a U (1) GLSM with the following charge vectors and the twisted masses 10 : where the inclusion of the twisted mass parameters λ = 0 has been determined in accordance with the rules in Section 3.2, such that the pole u * = λ of the exact formula associated with Figure 1 becomes a projective point. In other words, without turning on the twisted masses, the hyperplane arrangement of the model remains to be non-projective and the usage of the Jeffrey-Kirwan residue operation cannot be justified. By taking η ∈ Cone(Q * ), namely in the geometric phase, the projective point u * = λ contributes to the Jeffrey-Kirwan residue in (3.4) and we obtain In this example the mirror map (3.12) becomes trivial, i.e. log q = log z, and the A-model Yukawa coupling (3.6) is given by Remark 4.1. Turning on the twisted masses in a symmetric manner for the base and fiber directions as λ 1 = λ 2 = λ 3 = λ 4 = λ to maintain the flop symmetry z ↔ z −1 of the resolved conifold interchanging the compact and the non-compact directions, we obtain Interestingly, this expression has a different classical triple intersection number and consistent with the result argued in [23].
Next, we consider the local toric Calabi-Yau threefold O(0) ⊕ O(−2) → P 1 described by a U (1) GLSM with Following the criteria in Section 3.2, a twisted mass parameter λ = 0 has been introduced such that u * = λ associated with Q * = {Q 1 , Q 2 } = 1 gives a projective point (see Figure 1). Especially, according to the rule 2, we have assigned a twisted mass λ 3 = −2λ for the neutral  . The point u * = λ is a projective point associated with Q * = Q 1,2 (resp. Q * = Q 1,2,3 ). The orbifold phase of the middle (resp. lower) left diagram is given by the orbifold geometry C × C 2 /Z 2 (resp. C 3 /Z 3 ). chiral multiplet Φ 3 corresponding to O(0) in order to satisfy the "Calabi-Yau condition on the divisor" λ 1 + λ 2 + λ 3 = 0. For η ∈ Cone(1) in the geometric phase, the projective point u * = λ contributes to the Jeffrey-Kirwan residue in (3.4) and we obtain (4.6) This agrees with the Yukawa coupling evaluated in [23]. The mirror map (3.12) becomes and the A-model Yukawa coupling (3.6) is given by As an explicit example with dim H 4 (X, Z) = 1, we consider the local Calabi-Yau threefold K P 2 described by a U (1) GLSM with Here we have introduced a twisted mass parameter λ = 0 for the base space P 2 such that u * = λ associated with Q * = {Q 1 , Q 2 , Q 3 } = 1 provides a projective point (see Figure 1). For η ∈ Cone(1) in the geometric phase, the projective point u * = λ gives a contribution to the Jeffrey-Kirwan residue in the GLSM correlator (3.4) and we obtain . (4.10) The mirror map (3.12) is given by and by using (3.7), the Gromov-Witten invariants computed in [22] can be precisely reproduced.
Local F 0 : O(−2, −2) → P 1 × P 1 As a second example with dim H 4 (X, Z) = 1, let us consider the local Hirzebruch surface K F 0 = K P 1 ×P 1 described by a U (1) 2 GLSM with Here, based on the rules in Section 3.2, we have introduced a twisted mass parameter λ = 0 in a symmetric way for z 1 ↔ z 2 in the base space P 1 × P 1 , such that u * = (λ, λ) associated with Q * = {Q 1 , Q 2 , Q 3 , Q 4 } provides a projective point (see Figure 2). For η ∈ Cone(Q 1 , Q 1 + Q 3 ) inside the geometric phase, the projective point u * = (λ, λ) gives a contribution to the Jeffrey-Kirwan residue in the GLSM correlation functions in (3.4) as and each component has the following exact form: where The above result is in agreement with the Yukawa couplings evaluated from the local mirror symmetry approach in [23,24]. The mirror map for K F 0 is obtained by (3.12) as and then the formula (3.7) correctly reproduces the Gromov-Witten invariants studied in [22].
Remark 4.2. In addition to the twisted masses represented in (4.17), we can also turn on a nonzero value for λ 2 as λ 1 = pλ, λ 2 = qλ, λ 3 = λ 4 = λ, λ 5 = 0 to make the poles associated with Then we obtain the same result with (4.19) while x is defined by This means that the additional mass deformation is possible in this case, but does not introduce an independent moduli parameter.
From (3.12), one can easily show that the mirror map for K F 1 is given by where c 1 = 2 and c 2 = 1. Combining with the formula in (3.7), the Gromov-Witten invariants studied in [22] can be appropriately reproduced. Local F 2 As a fourth example with dim H 4 (X, Z) = 1, let us study the local Hirzebruch surface K F 2 (a local A 1 geometry) described by a U (1) 2 GLSM with Here we have introduced twisted mass parameters ǫ, λ = 0 with ǫ = λ for compact directions such that u * = (λ, ǫ) and (2ǫ, ǫ) associated with Q * = {Q 1 , Q 3 , Q 4 } and {Q 2 , Q 3 , Q 4 } provide the projective points respectively (see Figure 4). In this model, the chiral multiplet Φ 5 which is neutral under the second U (1) gauge symmetry describes the non-compact fiber coordinate X 5 , and the compact divisor {X 5 = 0} (i.e. F 2 ) contains the blow up mode of the Z 2 singularity. Therefore, following the rule 2 in Section 3.2, we impose the "Calabi-Yau condition on the divisor" as λ 2 + λ 3 + λ 4 + λ 5 = 0, which implies to take a limit ǫ → 0 in the final step. For η ∈ Cone(Q 1 , Q 1 + Q 3 ) in the geometric phase, the projective points u * = (λ, ǫ) and (2ǫ, ǫ) contribute to the Jeffrey-Kirwan residue in (3.4) as Res and we obtain , The above expressions indeed agree with the result obtained in [23]. The mirror map (3.12) for K F 2 is given by where i = 1, 2, and (c 1 , c 2 ) = (1, −2). The Gromov-Witten invariants computed in [22] from the local mirror symmetry approach can be also reproduced by using (3.7).

+ Res
(4.30) Note that the number of intersecting hyperplanes at the point u * = ((p + 1)λ, pλ, pλ) is larger than r = 3. To deal with this "degenerated point" which requires a careful treatment of the order of the iterated residue, we have applied Theorem 2.5 for a flag F with κ F = (Q 5 , Q 4 + Q 5 , Q 1 + Q 3 + Q 4 + Q 5 ) and ν(F ) = 1 as the only constituent in FL As a result, we finally obtain and with z 2 ↔ z 3 symmetry. Here 33) and the analytic expressions of Y z i z j z k (z 1 , z 2 , z 3 ) p=q=0 are summarized in Appendix B. Note that p = q = 0 corresponds to (x, y) = (1/4, −1/4).
The above results completely agree with the Yukawa couplings with the "moduli parameters" x and y given in [23]. The mirror map (3.12) for K dP 2 is given by where i = 1, 2, 3, and the associated Gromov-Witten invariants studied in [22] are correctly reproduced by using (3.7).

Local A 2 geometry
Let us consider a local A 2 geometry with dim H 4 (X, Z) = 2, which is a fibered A 2 geometry over P 1 represented in Figure 5. This geometry engineers the four dimensional pure SU (3) gauge theory and can be described by a U (1) 3 GLSM with [51,52,22]: (4.35) Figure 5: The toric web diagram for the local A 2 geometry. X i , i = 1, . . . , 6 are homogeneous local coordinates corresponding to the chiral multiplets Φ i , and {X 4 = 0} or {X 5 = 0} gives a compact divisor. The parameters q 1 , q 2 and q 3 are exponentiated Kähler moduli associated with the U (1) 3 gauge group.
Here we have introduced the twisted mass parameters ǫ, λ = 0 with ǫ = λ for a projective hyperplane arrangement in the geometric phase with Following the rule 1 in Section 3.2, we have not included the mass parameters for the chiral multiplets Φ 3 , Φ 4 and Φ 5 with charge vectors Q 3 , Q 4 and Q 5 which blow up the singularities. Furthermore, the chiral multiplet Φ 4 which is neutral with respect to the first U (1) gauge symmetry describes a non-compact fiber coordinate X 4 and the compact divisor {X 4 = 0} contains the blow up mode of the Z 2 singularity. Therefore, by following the rule 2, we impose the "Calabi-Yau condition on the divisor" as λ 1 + λ 2 + λ 3 + λ 4 = 0, which implies to take a limit ǫ → 0 in the final step.

Local toric Calabi-Yau fourfolds
Finally we focus on the local nef toric Calabi-Yau fourfolds whose exact properties have been studied in [44]. We again confirm that the localization formula about the A-twisted GLSM correlation functions provides the exact expressions for local B-model Yukawa couplings appropriately, with the aid of our formalism.
Let us consider the local toric Calabi-Yau fourfold O(−1) ⊕ O(−2) → P 2 described by a U (1) GLSM with where, following the rules in Section 3.2, we have introduced a twisted mass parameter λ = 0 for the base space P 2 such that u * = λ associated with Q * = {Q 1 , Q 2 , Q 3 } = 1 gives a projective point (see Figure 6). For η ∈ Cone(1) in the geometric phase, the projective point u * = λ contributes to the Jeffrey-Kirwan residue in the GLSM correlation function (3.4) as . (4.40) In this example, the mirror map (3.12) is trivial, i.e. log q = log z, and the A-model Yukawa coupling (3.6) takes a form Y hhhh (q) = 1 2(1 + 4q) . (4.41) One finds that the identity is equivariant to the factorization (3.8) of the four-point Yukawa coupling Y hhhh (q) into the three-point Yukawa coupling Y hhh 2 (q) in [44]: Next, let us consider the local toric Calabi-Yau fourfold K P 3 described by a U (1) GLSM with where a twisted mass parameter λ = 0 has been introduced such that u * = λ associated with Q * = {Q 1 , Q 2 , Q 3 , Q 4 } = 1 provides a projective point (see Figure 6). For η ∈ Cone(1) in the geometric phase, the projective point u * = λ contributes to the Jeffrey-Kirwan residue in (3.4) and we obtain . (4.45) The mirror map (3.12) for K P 3 is given by and one can check that the factorization (3.8) into the three-point Yukawa couplings correctly reproduces the previous results about the Gromov-Witten invariants in [44].

Conclusions
In this paper, we thoroughly investigated the relationship between the topological B-model Yukawa couplings for the backgrounds with non-compact directions and the exact localization formula about the A-twisted correlation functions of the two dimensional N = (2, 2) gauged linear sigma models. Starting from the exact results for the A-twisted correlators of N = (2, 2) gauged linear sigma models [15], we demonstrated how to extract appropriately the B-model Yukawa couplings for the local nef toric Calabi-Yau varieties. First we explained that the inclusion of the twisted masses for the chiral matter multiplets is indispensable to conduct the explicit calculations about the Jeffrey-Kirwan residue formalism. Although it has been argued in the literatures that the twisted mass deformation is required to deal with the backgrounds with non-compact directions, a comprehensive study has not been conducted before. We addressed this important issue and proposed an algorithm to compute the GLSM correlation functions for local toric Calabi-Yau varieties appropriately.
We have also checked that our prescription for the twisted mass deformations of the GLSM correlation functions is totally consistent with known results for the Yukawa couplings evaluated from the local mirror symmetry approach. Moreover, we found that the ambiguities of classical intersection numbers of certain class of local toric Calabi-Yau varieties argued before are identified with the degrees of freedom of the assignment of the proper twisted mass parameters. Combining the exact localization formula about the A-twisted GLSM correlation functions, our prescription would provide an alternative efficient formalism to compute the B-model Yukawa couplings for generic local nef toric Calabi-Yau varieties.
Finally we would like to comment on the possible future research directions. Throughout this paper, we have not considered the models whose target spaces are the non-nef varieties [34,35,36]. It would be interesting to extend our analysis for the GLSM correlation functions and twisted mass deformations into such intriguing examples.
In our framework, the twisted masses are introduced only for the compact directions of the target space. It would be interesting to clarify a physical meaning of this requirement. One possible explanation is the following. From the viewpoint of the SUSY algebra, inclusion of the twisted masses modify the central charges of the model and implies the existence of additional charged BPS particles in a specific vacuum. The appearance of such extra massless states can be naturally interpreted as the D-branes wrapping on compact directions as discussed in [53,54], and is known to be indispensable to regularize a singularity of the model appropriately.
A Building blocks of I-functions from GLSM correlators Let us consider a GLSM which flows in the IR to two dimensional non-linear sigma model with a Fano or a Calabi-Yau variety X as a target space. Following [19] (see also [55]), here we will show that the building blocks of the Givental I-function [45,46,47] for X can be derived from the factors (2.4) and (2.5) in the localization formula.
First, let us reparametrize u and d as Then the factor (2.4) can be decomposed as The factor (2.5) for Φ a = Φ with representation R a = R, R-charge r a = 0 and twisted mass λ a = λ can be decomposed as (ρ(x) + λ − p ) , for ρ(q) ≤ −1. (A.8) Similarly, the factor (2.5) for Φ a = P with representation R a = R, R-charge r a = 2, and twisted mass λ a = λ can be decomposed as (−ρ(x) − λ + p ) −1 , for ρ(q) ≥ 1. (A.12) After taking the above decomposition, the ingredients I vec q (x; ) in (A.4), I Φ q (x, λ; ) in (A.8) and I P q (x, λ; ) in (A.12) are known to provide the building blocks of the Givental I-function for the variety X. There x and λ are identified with the equivariant cohomology elements or the Chern roots of X and the equivariant parameter acting on X, respectively [55,19,56,20,21]. Starting from the I-function, one can find an associated quantum differential equation called Picard-Fuchs equation from which the mirror map and genus zero Gromov-Witten invariants of X can be evaluated [45,46,47]. X 1 = X 3 = X 4 = 0. Then x becomes the open string moduli parameter in the B-model. From (C.1), we obtain a mirror curve in X ∨ at ω + = 0 or ω + = 0 as y 2 + 1 + x + z 2 x 2 + z 2 2 z 3 x 3 y + z 1 = 0. (C. 2) The branch points can be obtained from 1 + x + z 2 x 2 + z 2 2 z 3 x 3 2 − 4z 1 = 0, and the discriminant for x of the above equation given by describes the degenerate points of the mirror curve (C.2).