Equivariant A-twisted GLSM and Gromov-Witten invariants of CY 3-folds in Grassmannians

We compute genus-zero Gromov-Witten invariants of Calabi-Yau complete intersection 3-folds in Grassmannians using supersymmetric localization in A-twisted nonAbelian gauged linear sigma models. We also discuss a Seiberg-like duality interchanging Gr(n, m) and Gr(m − n, m).


Introduction
In a Calabi-Yau (CY) compactifications of E 8 ×E 8 heterotic string theory with the standard embedding, there are four types of the Yukawa couplings (1 3 , 1 · 27 · 27 * , 27 * 3 , 27 3 ) in the 4-dimensional low energy effective theory. It is known that 27 3 -type Yukawa couplings do not receive either loop or world-sheet instanton corrections. On the other hand, the 27 * 3 -type Yukawa couplings receive corrections coming from world-sheet instantons. It was conjectured in [1] that the world-sheet instanton corrected 27 * 3 -type Yukawa couplings can be explicitly computed from the 27 3 -type Yukawa couplings of the mirror manifold.
A generalization of A-twisted gauged linear sigma models (GLSMs) with one omegabackground parameter on a 2-sphere S 2 has been constructed in [2]. Recently, the supersymmetric localization computations on this geometry have been performed in [3] (See also [4]). It gives an explicit formula for cubic correlation functions of scalars in the vector multiplet, which conjecturally give the Yukawa couplings of the mirror when the omegabackground parameter is set to zero. An interesting point here is that one can compute the 27 3 -type Yukawa couplings without knowing the mirror manifold. This formula goes back to [5] when the gauge group is Abelian. A mathematical conjecture in the Abelian case, called toric residue mirror conjecture, is formulated in [6,7] and proved in [8][9][10]. The formula obtained in [3] works also for non-Abelian gauge theories, and can be regarded as a generalization of toric residue mirror conjecture to CY manifolds in non-toric manifolds. We also give an explicit computation of the mirror map in a framework of A-twisted GLSMs with omega-background parameter. This allows us to give a conjectural computation of the genus-zero Gromov-Witten invariants of the CY manifolds.
CY 3-folds with one-dimensional Kähler moduli defined as complete intersections of zero loci of sections of equivariant vector bundles on Grassmannians are classified in [11]. In this paper, we realize some of them as phases of GLSMs, and compute the Yukawa JHEP09(2017)128 couplings in terms of A-twisted GLSM. This allows us to give a conjectural computation of genus-zero Gromov-Witten invariants of such CY 3-folds. The result agrees with a mathematically rigorous treatment obtained earlier in [12] based on Abelian/non-Abelian correspondence [13][14][15].
This paper is organized as follows: in section 2, we briefly review GLSMs with unitary gauge groups and their relation to CY 3-folds in Grassmannians. In section 3, we discuss the mirror map for CY 3-folds in Grassmannians in terms of A-twisted GLSMs with omega background. In section 4, we use the method in section 3 to compute the genus-zero Gromov-Witten invariants of some CY 3-folds in Grassmannians. In section 5, we discuss a Seiberg-like dual description of the Yukawa coupling. The last section is devoted to a summary.

GLSMs and CY 3-folds in Grassmannians
In this section, we consider 2d N = (2, 2) GLSMs [16] with gauge group G = U(n) which flow to infrared non-linear sigma models (NLSMs) with large positive Fayet-Iliopoulos parameter (FI-parameter) ξ 0. The target spaces are given by the Higgs branch moduli of GLSMs. In this paper, we study the case where the target spaces are CY 3-folds defined as complete intersections of zeros of sections of vector bundles constructed from the dual of the universal subbundle S on Grassmannians.
The matter multiplets consist of m fundamental chiral multiplets Φ i for i = 1, · · · , m and chiral multiplets P l for l = 1, · · · s in the gauge representation R l . When all the P l are absent, the D-term vacuum condition for Φ i in the Higgs branch defines the Grassmannian Gr(n, m) in the positive FI-parameter region. The introduction of P l modifies the D-term vacuum condition to the total space of the vector bundle on the Grassmannian ⊕ s l=1 E R l → Gr(n, m) with appropriate choices of gauge representations R l . The vector bundle E R l is determined by the gauge representation R l . For examples, relations between R l and E l are Here n * , det −1 , Sym q n * , and Λ q n * represent the anti-fundamental representation, the inverse of the determinant representation, the q-th symmetric products of the antifundamental representation, and the q-th anti-symmetric products of the anti-fundamental representation of the gauge group U(n), respectively. The bundle S is the universal subbundle and O(−1) = Λ n S is the inverse of the determinant line bundle. We introduce the following superpotential term

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Here G l (Φ) is a homogeneous polynomial in Φ i which belongs to the complex conjugate representation of R l . The polynomial G l (φ) defines a section of the bundle E * l on Gr(n, m), where E * l is the dual bundle of E l . The F-term equations of (2.5) are given by If the equation (2.6) defines a smooth complete intersection in the Grassmannian, then it follows from the Jacobian criterion for smoothness that the rank of the matrix ∂G l ∂φ i appearing in (2.7) is equal to the sum s l=1 dim R l of the dimensions of p l , so that the only solution to (2.7) is p l = 0 for all l. The F-term and D-term equations reduce to Then the GLSM flows to non-linear sigma model whose target space is given by the complete intersection of zero section of E * l in the Grassmannian Gr(n, m); The complex dimension of X n,m ⊕ s l=1 E l is given by Recall that the dimensions of the symmetric and anti-symmetric representations are given by (2.11) Since we are interested in the cases where the target spaces are CY manifolds, the axial anomaly has to be canceled: We will give a computation which conjecturally gives genus-zero Gromov-Witten invariants of CY 3-folds realized as axial anomaly free non-Abelian GLSMs.
3 Equivariant A-twisted GLSM on two sphere and mirror symmetry Supersymmetric backgrounds in two dimensions have been studied from a rigid limit of linearized new minimal supergravity [2]. There exists a new supersymmetric background JHEP09(2017)128 on S 2 which is an extension of the topological A-twist by one omega-background parameter (equivariant A-twisted GLSM). If is set to zero, the theory reduces to the ordinary A-twisted GLSM on S 2 .
In [3,4], the correlation functions of gauge invariant operators coming from the vector multiplet scalar σ inserted at the north and the south poles of S 2 have been evaluated by supersymmetric localization. For G = U(n), the saddle point value of the a-th diagonal component σ at the north and the south pole are given by Here k a for a = 1, · · · , n are the magnetic charges for the diagonal elements of the gauge fields. The correlation function of f (σ)| N g(σ)| S for G = U(n) is given by Here f (σ)| N and g(σ)| S are gauge invariant operators constructed from σ and inserted on the north and south pole respectively. The variable z is the exponential of the complexified FI-parameter defined as z := e 2π 2π ξ) with theta angle θ. The factors Z vec k and Z chiral k are the contributions from the one-loop determinants of the U(n) vector multiplet and chiral multiplets with magnetic charge k = diag(k 1 , · · · , k n ) respectively, and have the following forms: Here ∆(R) is the set of weights of R. λ is the twisted mass. JK-Res(Q(σ * ), η) is the Jeffrey-Kirwan residue operation determined by a charge vector η at a singular locus σ * . The variable r is an integer R-charge of the lowest component scalar in the chiral multiplet. We assign r = 0 for the fundamental chiral multiplets which parametrize the coordinates of the target space in the low energy NLSM. This assignment is compatible with the R-charge assignment in an A-twisted NLSM which is relevant to the Yukawa coupling computation [19]. Then, according to (2.5), the R-charge of the lowest component scalar p l in the chiral multiplet P l is determined to be r = 2.
For all the GLSMs that we consider in the next section, the weights ρ of the chiral multiplet P l satisfy the condition ρ(k) ≤ 0 for k ∈ Z n ≥0 . Then with the choice of η = (1, · · · , 1), the Jeffrey-Kirwan residue is the sum of residues at the poles coming from the JHEP09(2017)128 fundamental chiral multiplets, and has following simple form: .

(3.6)
Here λ i is the twisted mass of Φ i and λ l is the twisted mass of P l . The contour integrals enclose all the poles σ a = −j − λ i with i = 1, · · · , m, j = − ka 2 , · · · , ka 2 and k a = 0, 1, · · · . If we set = 0, equivariant A-twist reduces to ordinary A-twist on S 2 . In this case, σ(x) is invariant under the supersymmetric transformation at any point on S 2 , and the saddle point value is simply given by a constant configuration σ a . Then the correlation function of (Trσ) M with λ i = λ l = 0 is given by If the target space of the low energy NLSM is a CY 3-fold, the expectation values (Trσ) M =0 except for M = 3 are zero, and it was conjectured that the expectation value (Trσ) 3 =0 gives the 27 3 -type Yukawa coupling in four dimensions compactified by the mirror manifoldX.
Here Ω(z) is the holomorphic (3, 0)-form onX and z is the complex structure moduli ofX. We comment on the over all sign ambiguity and sign difference between exponentiated FI-parameter and the complex structure moduli in (3.7) and (3.8). Since the first coefficient of (3.8) in series expansion of z agrees with the triple intersection number of X which is positive, the overall sign of (3.7) is fixed by requiring the positivity of the triple intersection number. In general, the exponentiated FI-parameter z has a different sign from the complex structure. In the computation of Gromov-Witten invariants, this sign ambiguity is fixed by requiring the Gromov-Witten invariant to be positive. In the models treated in the next section, the sign shift z → (−1) m z in the GLSM gives the correct sign of the mirror Yukawa coupling. When the gauge group is U(1), the ambient space of the target space is the complex projective space P m−1 , and (3.8) was first proposed by [5]. A mathematical interpretation of this formula was proposed by [6]. It has been shown in [3] that (3.7) for GLSMs studied in [18] gives correct mirror Yukawa coupling given in [17]. So we expect that (3.8) also works for the non-Abelian GLSMs considered in the next section.

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In order to extract genus-zero Gromov-Witten invariants from the mirror Yukawa coupling, we have to rewrite the mirror Yukawa coupling in terms of the flat coordinate t. The relation between z and t is given by the mirror map, defined as the ratio of two period integrals I 0 (z) and I 1 (z) ofX; Here I 0 (z) is the fundamental period normalized as I 0 (0) = 1, and I 1 (z) is the period with a logarithmic monodromy I 1 (z) = I 0 (z) log z +Ĩ 1 (z) withĨ 1 (0) = 0. By solving (3.9) recursively and expressing z as a function of q := e t , the Yukawa coupling in the flat coordinate is written as Here n d for d = 0, 1, 2, . . . are the instanton numbers of the target space X of genus 0 and degree d. N d for d = 0, 1, 2, . . . are genus zero Gromov-Witten invariants. In particular, n 0 = X H 3 is the triple intersection number of the hyperplane class H in X.
When the twisted masses are distinct, we can explicitly perform the contour integrals in (3.17) and obtain the following vortex factorization form of the generating function. (3.20) Here λ {j} = diag(λ j 1 , λ j 2 , · · · , λ jn ). From the view point of Higgs branch localization, Z {j} (λ, λ ) is interpreted as the 1-loop determinant and the vortex partition function Z v,{j} (z, λ, λ , − ) (Z v,{j} (z, λ, λ , )) is the point like vortex contribution on north (south) pole of S 2 at a root of Higgs branch specified by twisted masses λ {j} . Now we discuss the relation between (3.6), (3.18) and Givental's work [23,24]. For clarity of exposition, we restrict ourselves to the case when the gauge group is G = U(1), the gauge charge of Φ i is 1 for i = 1, . . . , m, and the gauge charge of P l is −q l with m = s l=1 q l . The target space of the low-energy NLSM is a CY complete intersection in P m−1 . Then (3.6) with f (σ) = e ασ and g(σ) = 1 is expressed as which agrees with the function which agrees with the Abelian case Givental's I-function I(z, x, ) for the complete intersection in P m−1 is given by The period integrals I i (z) appear in the expansion of the I-function as When the gauge group is non-Abelian, we expect that Z(z, x, λ, λ , ) is again related to Givental's I-function I(z, x, ) of X n,m φ i , (i = 1, · · · , 7) p 1 p l , (l = 2, · · · , 5) then the mirror map is again given by the ratio of two periods as (3.9). If we define Z 0 (z) and Z 1 (z) as the first two coefficients of expansion Then I 0 and I 1 are related to Z 0 and Z 1 by

Computation of Gromov-Witten invariants
In this section, we compute the Yukawa couplings and genus-zero Gromov-Witten invariants of some examples of compact CY 3-folds in Grassmannians which are obtained as complete intersections of equivariant vector bundles. Our first example is X 2,7 Sym 2 S * ⊕O(1) ⊕4 , which is a complete intersection CY 3-fold of Sym 2 S * and four copies of O(1) in Gr(2, 7). The gauge group is U(2), and there are seven fundamental chiral multiplets Φ i , (i = 1, · · · , 7), one chiral multiplet P 1 in the gauge representation Sym 2 2 * , and four chiral multiplet P i (i = 2, · · · , 5) in the gauge representation det −1 . The charge assignment for the lowest component scalars in the chiral multiplets is listed in table 1. The superpotential is given by Here a, b denote color indices, and the notations (ij) and [i, j] show that the indices i and j are symmetric and anti-symmetric under the permutation respectively. The F-term equation for P l gives Let us compute the mirror Yukawa coupling and instanton numbers. The set of weights of the chiral multiplet P l evaluated at diag(σ 1 , σ 2 ) is given by {−σ 1 − σ 2 } for P 2 , . . . , P 5 .

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Hence the contributions from the one-loop determinants of the chiral multiplets are for P 2 , . . . , P 5 .

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The function (3.33) for this model is given by From this equation, we obtain the first two coefficients as Z 0 (z) = 1 + 9z + 361z 2 + 21609z 3 + 1565001z 4 + 126630009z 5 + · · · , (4.16) The complex structure moduli is expressed as function of q as  These values reproduce the Gromov-Witten invariants of the complete intersection of two copies of Gr (2,5) in P 9 calculated in [31]. Gromov-Witten invariants for some other CY 3-folds in Grassmannian Gr(n, m) is listed in tables 4, 5 and 6. The manifold X 3,8 in table 4 is an Abelian 3-fold, so that its Gromov-Witten invariants are zero except at degree 0.

Seiberg like description of the mirror Yukawa coupling
In this section, we study dual U(m − n) GLSM description of Yukawa couplings of U(n) GLSM with m fundamental chiral and chiral multiplet P l , (l = 1, · · · , s) in several examples. We start with the case where all P l belong to the gauge representation det −q l and the target space of low energy NLSM is X n,m ⊕ s l=1 O(q l ) in Gr(n, m). The Seiberg-like duality of this model is studied in [18]. The CY 3-fold X n,m Gr(m − n, m) . Then, in the dual side, the GLSM is U(m − n) gauge group with m fundamental chiral multiplets and chiral multiplet P l in the det −q l representation for l = 1, · · · , s. For example, one has X 2,6 (2) . In U(2) GLSM description, the Yukawa coupling of X 2,6 O(1) ⊕4 ⊕O(2) is given by 2 a=1 σ 6(ka+1) a = 28(1 + 104z + 11248z 2 + 1214720z 3 ) + · · · . (5.1) 21888 5283 Table 6. The triple intersection number and instanton numbers for CY 3-folds in Gr(5, 7) and Gr (6,8).
In the dual U(4) GLSM description, the Yukawa coupling of X 4,6 O(1) ⊕4 ⊕O(2) is given by From (5.1) and (5.2), we find that two A-twisted GLSMs give the same Yukawa coupling. Table 7. Field contents of U(4) GLSM for X 4,6 S * ⊗O(1)⊕O (1) Next, we study the dual U(m − n) GLSM description of the Yukawa coupling for a U(n) GLSM with m fundamental chiral multiplets, one chiral multiplet P 1 in the gauge representation n * ⊗ det −1 and P 2 , . . . , P s in the gauge representation det −q l , which flow to NLSM with target space X n,m
To find the dual description, let us first consider the dual U(m−n) gauge theory description of U(n) with m fundamental chiral multiplet Φ i and an anti-fundamental chiral multiplet P 1 [32]. . We expect that the matter context of U(m − n) GLSM is m fundamental chiral multiplets Φ, m chiral multiplets M i in the representation det −1 , a chiral multipletΦ in the representation (m − n) * ⊗ det and chiral multiplets P 2 , . . . , P s in the representation det −q l with the superpotential W = m i=1 M iΦ Φ i + s l=2 P l G l (Φ). We compute the Yukawa coupling in both sides and see the agreement. We first consider the U(4) GLSM description of X 4,6 S * ⊗O(1)⊕O (1) . The field content is listed in table 7. The Yukawa coupling is given by   Table 9. Field content of GLSM for X 2,5 S * ⊗O(1)⊕O (2) .
The matter content of the dual U(2) description of X 4,6 S * ⊗O(1)⊕O(1) is given in table 8. The Yukawa coupling is given by We find (5.4) agrees with (5.3) up to the change z → −z of signs. Note that (5.4) has the same Yukawa coupling as X 2,7 O(1) ⊕7 , which is known by [11] to be deformation-equivalent to X 4,6 S * ⊗O(1)⊕O (1) Next we study the dual U(3) description of X 2,5 S * ⊗O(1)⊕O (2) . The field content of U(2) GLSM is given in table 9. In the original U(2) GLSM, the Yukawa coupling of X 2,5 S * ⊗O(1)⊕O(2) is given by From our observation, we expect that the matter content of U(3) GLSM is given by table 10 with the superpotential Here G(Φ) is a homogeneous polynomial of degree 4.

Summary
We studied genus-zero Gromov-Witten invariants of CY 3-folds defined as complete intersections in Grassmannians by using equivariant A-twisted GLSM on S 2 . The Yukawa coupling can be calculated from the cubic correlation function of the scalar in the vector JHEP09(2017)128 multiplet. In order to obtain the Yukawa coupling in the flat coordinate, we have to compute the mirror map, which gives the complex structure moduli as a function of the flat coordinate. The mirror map can be computed from the Z-function appearing in the factorization of correlation functions. We have also studied Seiberg-like duality between GLSMs with different ranks. We studied only the cases when the gauge group is U(n), and it would be interesting to extend our analysis to other gauge groups and quiver gauge theories. Cohomological Yang-Mills theories on curved backgrounds have recently been studied by coupling to background topological gravity in [34], which includes supersymmetric background studied in [2]. It is also interesting to perform the supersymmetric localization computation for GLSMs on these backgrounds, and figure out their interpretation as low energy target space geometry.