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 non-Abelian gauged linear sigma models. We also discuss a Seiberg-like duality interchanging Gr(n,m) and Gr(m-n,m).

R l and E l are R l = n * ←→ E l = S, (2.1)

· · ·
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 anti-fundamental 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 W (P, Φ) = s l=1 P l G l (Φ). (2.5) 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 m i=1 φ i (φ i ) † = ξ1 n , G l (φ) = 0. (2.8) 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); X n,m ⊕ s l=1 E * l := {[φ i ] ∈ Gr(n, m)|G l (φ) = 0, l = 1, 2, · · · , s}. (2.9) 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 (n + q + 1)! q!(n − 1)! R l = Sym q n * , n! (n − q)!q! R l = Λ q n * . (2.11) Since we are interested in the cases where the target spaces are CY manifolds, the axial anomaly has to be canceled: Tr R l F 12 = 0. (2.12) 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 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 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 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.
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, The period integrals and the mirror map can be extracted from the equivariant Atwisted GLSM as follows. Motivated by the factorization property of the physical S 2 partition function [20,21,22], we rewrite correlation functions as follows. After redefinitions of integration variables, the expectation values of (Trσ) M inserted at north and south can are rewritten as Here {j} = {j 1 , j 1 , · · · , j n } and {j} = 1≤j 1 <j 2 <···<jn≤m . Then the generating function of the correlation functions (Trσ) M | N (Trσ) N | S can be written as 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.
Here λ {j} = diag(λ j 1 , λ j 2 , · · · , λ jn ). From the view point of Higgs branch localization, ) 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 , (3.21) 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 . (3.31) Here the sign of z is fixed by requiring the first instanton number to be positive. If the I-function for a Calabi-Yau 3-fold is expanded as 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 (3.33) 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 Hence the contributions from the one-loop determinants of the chiral multiplets are for P 2 , . . . , P 5 .
Our second example is X 3,5 (Λ 2 S * )⊗O (1) , which is known by [11] to be deformation-equivalent to the complete intersection of two copies of Gr (2,5) in P 9 . This is a U(3) GLSM with five fundamental chiral multiplets and one chiral multiplet P in the gauge representation (Λ 2 3 * ) ⊗ det −1 as shown in Table 3. The set of weights ρ(σ) for the chiral multiplet P is given by (4.12)   The superpotential is From (3.7), the Yukawa coupling of this model with the sign change z → −z is given by (4.14) The function (3.33) for this model is given by  21888 5283 Table 6: The triple intersection number and instanton numbers for CY 3-folds in Gr (5, 7) and Gr (6,8).

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 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.