Determinantal Calabi-Yau varieties in Grassmannians and the Givental $I$-functions

We examine a class of Calabi-Yau varieties of the determinantal type in Grassmannians and clarify what kind of examples can be constructed explicitly. We also demonstrate how to compute their genus-0 Gromov-Witten invariants from the analysis of the Givental $I$-functions. By constructing $I$-functions from the supersymmetric localization formula for the two dimensional gauged linear sigma models, we describe an algorithm to evaluate the genus-0 A-model correlation functions appropriately. We also check that our results for the Gromov-Witten invariants are consistent with previous results for known examples included in our construction.


Introduction
String compactifications to lower dimensions preserving supersymmetry motivate us to study Calabi-Yau varieties as initiated in [1]. To analyze various properties of Calabi-Yau backgrounds, it is useful to consider the two dimensional gauged linear sigma model (GLSM) [2]. This model corresponds to the UV description of the non-linear sigma models on Calabi-Yau backgrounds and has provided a powerful technique to compute the correlation functions exactly (see for example [3]). Utilizing the duality property of the model, a physical understanding about mirror symmetry has also been developed [4] (see also [5,6] for several recent developments).
On the other hand, by using the supersymmetric localization techniques [7], exact formula for the GLSM partition functions [8,9] (see also [10,11]) and correlation functions [12,13] on the 2-sphere backgrounds has been clarified. This means that one can evaluate the genus-0 Gromov-Witten invariants from the GLSM calculation in a direct fashion. This methodology has also been applied to the GLSMs with non-abelian gauge groups and the study of the complete intersection Calabi-Yau varieties in Grassmannians has progressed in the last few years [14,15].
Several notable aspects of the GLSM correlation functions have also been clarified in [16,17].
While the toric complete intersection varieties described by abelian GLSMs have been thoroughly investigated in various contexts, a comprehensive understanding about the non-complete intersection varieties requires further efforts. In [18], as an example of a class of non-complete intersections, the determinantal varieties [19,20] and the associated non-abelian GLSMs have been investigated. The aim of our work is to make advances in the study of the determinantal varieties and provides a further step toward the comprehensive understanding of general Calabi-Yau backgrounds with non-abelian GLSM descriptions.
In this paper we explicitly clarify what kind of determinantal Calabi-Yau varieties in Grassmannians can be constructed while satisfying several requirements. We mainly focus on the 3-fold examples and the analysis for the determinantal Calabi-Yau 2-folds and 4-folds is summarized in Appendix. We will also study the quantum aspects of the GLSM associated with determinantal varieties. Our method is based on the analysis of the so-called Givental I-functions [21,22,23] which can be extracted from the localization formula for the GLSM on a supersymmetric 2-sphere. In particular, we will compute the genus-0 Gromov-Witten invariants of determinantal Calabi-Yau varieties by using a conjectural handy formula, and check that our results coincide with previous results for known examples included in our classification. This paper is organized as follows. First we examine a class of determinantal Calabi-Yau varieties in Grassmannians satisfying several requirements and specify the possible examples in Section 2. In Section 3, we briefly review the computation of the genus-0 invariants of complete intersection Calabi-Yau varieties in complex projective spaces and Grassmannians utilizing the I-functions, and provide a conjectural formula for Grassmannian Calabi-Yau varieties. In Section 4, we compute the genus-0 invariants of determinantal Calabi-Yau varieties with GLSM realizations by using the algorithm described in Section 3. Section 5 is devoted to conclusions and discussions. In Appendix A we summarize our results of the analysis for determinantal Calabi-Yau 2-folds and 4-folds. In Appendix B, we take a brief look at the computation of Hodge numbers using the Koszul complex and demonstrate several computations explicitly. In Appendix C we summarize the data of the genus-0 invariants of several determinantal Calabi-Yau 4-folds.

Determinantal Calabi-Yau varieties in Grassmannians
In this section, we first review a minimal ingredient of determinantal varieties, following [18] (see also [19,20]). Afterwards we classify a class of determinantal Calabi-Yau 3-folds in Grassmannians satisfying appropriate conditions.

Definitions
Let V be a compact algebraic variety, and A : E p → F q be a linear map from a rank p vector bundle E p on V to a rank q vector bundle F q on V . Here we assume that the linear map A is a global holomorphic section of the rank pq-bundle Hom(E p , F q ) ∼ = E * p ⊗ F q with maximal rank at a generic point of V . By representing the linear map A locally as a q × p matrix A(φ) of the holomorphic sections, a determinantal variety is defined as Here (ℓ + 1) × (ℓ + 1) minors of A(φ) generate the ideal I(Z(A, ℓ)). Since codim Z(A, ℓ) = (p − ℓ)(q − ℓ) < p ℓ+1 q ℓ+1 for ℓ ≥ 1, the ideal I(Z(A, ℓ)) has non-trivial relations called syzygies and the determinantal variety Z(A, ℓ) for ℓ ≥ 1 is not a complete intersection. As argued in [18], a simple analysis of the Jacobian matrix implies that Z(A, ℓ) for ℓ ≥ 1 has singular loci along Z(A, ℓ − 1) ⊂ Z(A, ℓ) only. One can resolve these singularities by the so-called incidence correspondence [18,19], where V Ep,p−ℓ denotes the fibration with Grassmannian fibers G(p − ℓ, E p ) of (p − ℓ)-planes with respect to the p-dimensional fibers of E p . It is worth noting that the codimension of the singular loci in V is codim Z(A, ℓ − 1) = codim Z(A, ℓ) + p + q − 2ℓ + 1, and then the determinantal variety Z(A, ℓ) with the dimension less than p + q − 2ℓ + 1 does not have singular loci [18]. 1 Remark 2.1 ( [18]). Since ℓ < min(p, q), the determinantal varieties with dimension less than 2 do not have singular loci. The determinantal 3-folds have singular points only when p = q = ℓ+1.
In this paper we only consider the square (p = q) determinantal varieties with V = G(k, n), where G(k, n) is the complex Grassmannian defined by the set of k-planes in C n , and O V is the structure sheaf of V . Then the variety V Ep,p−ℓ can be described by a product variety V Ep,p−ℓ ∼ = G(k, n) × G(p − ℓ, p) and the incidence correspondence (2.3) becomes In addition we require n = ℓ ∨ p c 1 (F p ) derived from Calabi-Yau condition [18]. Here is the first Chern class of F p and σ 1 = c 1 (Q) is the Schubert class of G(k, n). Q is the universal quotient bundle on G(k, n). The dimension of X A is given by dim By taking the duality G(k, n) ∼ = G(n − k, n) into consideration, here we only consider the case with 2k ≤ n. Furthermore, the rank condition 0 ≤ ℓ < p can be rephrased as 0 < ℓ ∨ p ≤ p = rank F p . In summary, we have seen that the following conditions 2 must be satisfied in order to realize the determinantal varieties appropriately.

Dimensional condition:
2. Calabi-Yau condition: 4. Rank condition: In the following, we will classify the determinantal Calabi-Yau 3-folds satisfying the above four conditions. Although we consider the desingularized determinantal varieties X A , the following analysis also gives a classification of Z(A, ℓ). 3 1 As noted in [18], one can also use the incidence correspondence (2.3) to describe the determinantal varieties without singular loci. 2 Note that we do not impose irreducibility or the conditions H i (XA, OX A ) = 0 in our analysis. 3 See Appendix A for the analysis of determinantal Calabi-Yau 2-folds and 4-folds.

General dimensions
Before moving on to the discussion about the determinantal Calabi-Yau 3-folds, let us consider general implications of the above requirements. In general dimensions, obviously the following two ansatz always satisfy the dimensional condition (2.7).
In the following, we will illustrate what kind of setups for k, n; ℓ ∨ p , c 1 (F p ) satisfy all the above requirements if we start from the Ansatz (I) or (II). 4

Ansatz (I)
In this case, the Calabi-Yau condition (2.8) becomes Then the duality condition (2.9) implies which means that examples with k ≥ 3 provide determinantal varieties with dim X A ≥ 8.
When k = 1, V is given by G(1, n) ∼ = P n−1 and one obtains the solutions with k, n; ℓ ∨ p , c 1 (F p ) = (1, dim X A + 2; 1, dim X A + 2) . (2.15) In this case the rank condition (2.10) is trivially satisfied, and appropriate F p on V are given by the following vector bundles associated with the integer partitions of dim X A + 2 : When k = 2, V becomes G(2, n) and one finds the solutions with Since n has to be an integer, this type of solution can exist only when the dimension of X A is odd. Moreover, (2.14) requires dim X A ≥ 3.

Ansatz (II)
In this case, the Calabi-Yau condition (2.8) becomes 18) and the duality condition (2.9) is trivially satisfied. Thus we only need to consider the rank condition given by For example, when k = 1, we obtain the following solutions (2.20) Note that the rank condition (2.19) strongly constrain the possible vector bundles.

Determinantal Calabi-Yau 3-folds
Here we will focus on the square determinantal Calabi-Yau 3-folds and determine what kind of setups satisfy the above four conditions. Let us start with the dimensional condition given by Then we will find out which type of choices for k, n; ℓ ∨ p , c 1 (F p ) can be possible while changing the parameter k.

k = 1
In this case we have V = G(1, n) ∼ = P n−1 . From (2.15) and (2.16) one finds that there exists a "quintic family" (see for example [24]) given by (2.22) The example constructed from F p = O V (5) with p = 1 (i.e. ℓ = 0) is the well-known quintic Calabi-Yau 3-fold, which is the zero locus of a holomorphic section of O P 4 (5).
Since ℓ ∨ p = 1 (i.e. p = ℓ + 1), according to the Remark 2.1, generically the above quintic families have singular points. The determinantal Calabi-Yau 3-folds in this class are connected by the deformations of complex structures, and it is known that the desingularized 3-folds are related by the so-called extremal transitions. 5 5 The comparison of topological invariants in Section 4.3.1 makes this point clearly understandable.
Apart from the above quintic family, one can also find the following solutions k, n; ℓ ∨ p , c 1 (F p ) = (1, 8; 2, 4) Here the first two examples with p = 2 (i.e. ℓ = 0) in (2.23) can be identified with the well-known complete intersection Calabi-Yau 3-folds as where X d 1 ,...,dr ⊂ P n−1 denotes the complete intersection variety defined by the zero locus of a holomorphic section of the vector bundle ⊕ r a=1 O P n−1 (d a ). The last two examples in (2.23) are Gulliksen-Negård type 3-folds studied in [25].
Moreover, (2.20) provides another type of solutions given by ℓ = 0) has the following isomorphism: The other example constructed from F p = O V (1) ⊕5 has been studied in [18].

k = 2
In this case, V becomes the Grassmannians V = G(2, n). Compared with the complex projective spaces, there exist additional components for the vector bundles on the Grassmannians, as explained in the followings.
When k ≥ 2, beside the line bundle O V (d) on V = G(k, n), one can also consider vector bundles with rank greater than one denoted by S * and Q.
These are known as the dual of the universal subbundle and the universal quotient bundle on G(k, n), respectively. Note that they fulfill the relation ∧ k S * ∼ = O V (1) and satisfy rank S * = k, c 1 (S * ) = 1, rank Q = n − k, c 1 (Q) = 1.
Accordingly, general irreducible vector bundles can be constructed as Returning to the main subject of the classification, (2.17) with the rank condition (2.10) implies that the following setups are possible k, n; ℓ ∨ p , c 1 (F p ) = (2, 4; 1, 4) with  Another class of solutions can be obtained by the ansatz (II) in (2.12) and the result is k, n; ℓ ∨ p , c 1 (F p ) = (2, 28; 7, 4) with

k ≥ 3
In this case, we have V = G(k, n). Interestingly, there exist four "infinite families" given by (2.27) By using mathematical induction, one can check that the duality condition (2.9), the rank condition (2.10), and in particular ℓ ∨ p < rank F p , are maintained. Since we do not impose the irreducibility condition in our analysis, it is still possible that the above infinite families can be reduced to other trivial or non-trivial examples. In any case, it is required to thoroughly investigate various topological invariants of these higher rank examples, and this would require a considerable effort and we leave this issue as an open problem.

I-functions and Gromov-Witten invariants
In this section, we briefly overview the computation of genus-0 Gromov-Witten invariants using the Givental I-functions [21,22,23] (see also [26]). We will also provide a handy formula for the computations of the Gromov-Witten invariants of Grassmannian Calabi-Yau varieties, which is also applicable to the determinantal varieties.

Building blocks of I-functions
When a Fano or a Calabi-Yau variety X has a GLSM realization with gauge group G, one can easily construct the Givental I-function of X by using the supersymmetric localization formula (see [27,14,15,16,17]). Here we clarify the building blocks of the I-function of X associated with such a GLSM on the Ω-deformed 2-sphere S 2 which has a vector multiplet and chiral matter multiplets with R-charge 0 or 2 under U (1) R . The deformation parameter is identified with an equivariant parameter. Let x = (x 1 , . . . , x rk(g) ) ∈ h ⊗ R C be Coulomb branch parameters and q = (q 1 , . . . , q rk(g) ) ∈ Z rk(g) ⊂ ih be magnetic charges for Cartan subalgebra h of a Lie algebra g associated with G, where rk(g) denotes the rank of g. Here the parameters x are identified with the Chern roots of X which give the total Chern class of X as To construct the I-function of X, first we need a "classical block" associated with the subgroup where c is the number of the central. The Fayet-Iliopoulos (FI) parameters ξ a and theta angles θ a , a = 1, . . . , c, are associated with each U (1) c factor, and the classical block of the I-function is given by Other contributions come from the 1-loop determinants of multiplets of the GLSM. The vector multiplet provides a block given by where ∆ + is the set of positive roots of g. In general, the GLSM also has chiral matter multiplets Φ with R-charge 0 and P with R-charge 2 in a certain representation R. Note that one can turn on a twisted mass parameter λ while preserving supersymmetry, which is identified with an equivariant parameter. Their contributions are given as follows: and As we will see next, the genus-0 Gromov-Witten invariants can be extracted from this function.

Examples
Here we will demonstrate how to compute genus-0 Gromov-Witten invariants via the I-functions for well-studied examples, and find out a useful formula for treating Grassmannian Calabi-Yau varieties.

Complete intersections in P n−1
Let us consider a complete intersection variety X 1 = X d 1 ,...,dr ⊂ P n−1 defined by the zero locus of a holomorphic section of a vector bundle E = ⊕ r a=1 O V (d a ) on V = P n−1 satisfying Fano or Calabi-Yau condition r a=1 d a ≤ n. Note that rank E = r and c 1 (E) = r a=1 d a . This variety has a complex dimension Table 1: Matter content of the U (1) GLSM for the complete intersection variety X 1 in P n−1 .
and is described by a U (1) GLSM whose matter content is shown in Table 1. This model has For each matter multiplet we assign twisted masses and U (1) R-charges as described in Table 1. Combining the building blocks (3.2), (3.4) and (3.5) with the assignment in Table 1, the I-function in the geometric large volume phase with FI parameter ξ > 0 is constructed as [21,22,23] Geometrically z = e −2πξ+ √ −1θ provides the Kähler moduli parameter of X 1 , and x is identified with the equivariant second cohomology element of X 1 satisfying n−1 i=0 (x − w i ) = 0, where the twisted masses w i give the equivariant parameters acting on P n−1 . The twisted masses λ a correspond to the equivariant parameters acting on E = ⊕ r a=1 O V (d a ). Then it can be shown that the I-function (3.7) obeys the ordinary differential equation In the Calabi-Yau case r a=1 d a = n with vanishing equivariant parameters w i = λ a = 0, the differential equation (3.8) yields the Picard-Fuchs equation for the periods of the holomorphic (n − r − 1)-form on X 1 given by [28,29,30,31] where ). If we expand the I-function around = ∞ as the coefficients I k (z) precisely give the solutions to the Picard-Fuchs equation. One can also obtain the flat coordinate q on the Kähler moduli space of X 1 through the relation called the mirror map. It has been shown in [32] that the genus-0 3-point A-model correlators . . , n − r − 1, which enumerate the number of rational curves are given by where I k (z) are inductively constructed from the I-functions as Here the observable O h p is associated with the hyperplane class h ∈ H 1,1 (X 1 ), and is the classical intersection number of X 1 . I k (z) have relations Note that there is a selection rule n i=1 p i = dim X 1 + n − 3 to realize non-trivial genus-0 n-point in (3.12) is an integer and enumerates the number of degree d holomorphic maps intersecting with the cycles dual to h, h k , and h n−r−k−2 [30,31,33] (see also [34,35]).
In a special case with k = 1, the relation Θ z = I 1 (z)Θ q and the so-called divisor equation an integer which enumerates the number of degree d holomorphic maps intersecting with the cycle dual to h n−r−3 . When n − r = 4 (i.e. dim X 1 = 3), by the divisor equation, (3.16) yields [36,37] Table 2: Matter content of the U (k) GLSM for the complete intersection variety X 2 in G(k, n).

Complete intersections in G(k, n)
Let us consider a complete intersection variety X 2 defined by the zero locus of a holomorphic and can be described by a U (k) GLSM whose matter content is given in Table 2. This model [38]. For each matter multiplet we assign twisted masses and U (1) R-charges as described in Table 2. Combining the associated building blocks (3.2), (3.3), (3.4) and (3.5) for the U (k) vector multiplet and the chiral multiples in Table 2, we can construct the I-function for X 2 in the geometric phase with large FI parameter ξ > 0 as [39] . (

3.19)
Geometrically z = e −2πξ+ √ −1θ provides the Kähler moduli parameter of X 2 , and x i are identified with the degree 2 elements in the equivariant cohomology of X 2 . The twisted masses w I and λ a correspond to the equivariant parameters acting on G(k, n) and E = ⊕ r a=1 O V (d a ), respectively. Remark 3.1. For w I = 0, the cohomology ring of G(k, n) is given by [40] (see [41] for the equivariant quantum cohomology ring): where S k is the symmetric group on k elements and are the complete symmetric polynomials.
For the Calabi-Yau case r a=1 d a = n with vanishing equivariant parameters w I = λ a = 0, the I-function I X 2 (z; where s P (x) = s P (x 1 , . . . , x k ) is the Schur polynomial for a partition P = {p 1 , . . . , p k }, and note that s p,0,...,0 (x) = h p (x) and s 1,1,..., coordinate which provides the mirror map is given by As a non-abelian generalization of the formula (3.16), here we conjecture that the genus-0 1-point for the Grassmannian Calabi-Yau variety X 2 is given by Here is the classical intersection number associated with the Poincaré dual H of a codimension dim X 2 − 2 cycle in X 2 , where σ P denotes the Poincaré dual of a Schubert cycle of codimension |P | in G(k, n) [42]. The numbers n d (H) are integer invariants associated with H which are related to Gromov-Witten invariants of X 2 [43,44,11,45].
The Giambelli's formula and the definition of Schur polynomials yield Then one can reformulate the expression in (3.24) in terms of the classes σ 1 and σ 2 as where I X 2 [t] denotes the coefficient of t at = 1 in the expansion (3.21). Then, it is obvious that the first term in (3.23) is determined from the classical block. One can also see that for Now we claim that the conjectural formula (3.24) is also applicable not only for complete intersection Grassmannian Calabi-Yau varieties but also for the determinantal Calabi-Yau varieties, as we will see in the next Section.
Remark 3.2. Instead of Pieri's formula for Schubert cycles, the intersection numbers of Grassmannian G(k, n) can also be computed by Martin's formula [40]: Generic case can also be treated with a slight modification. Suppose that a Grassmannian Calabi-Yau variety X, defined by the zero locus of a holomorphic section of a vector bundle on G(k, n), has a GLSM realization with a massless matter multiplet P in a representation R of where Let us consider the dual of the universal subbundle E = S * on G(k, n). A Grassmannian Calabi-Yau variety defined by the zero locus of a holomorphic section of E = S * is described For the matter multiplet P with twisted mass λ, (3.5) becomes Similarly, for instance, for vector bundles E = Sym m S * (d) and E = ∧ m S * (d) we get with P in Sym m k ⊗ det −d and with P in ∧ m k ⊗ det −d , respectively. Using these building blocks with the help of our formula (3.24), one can obtain the genus-0 Gromov-Witten invariants of Grassmannian Calabi-Yau varieties computed e.g. in [14].
To consider a Grassmannian Calabi-Yau variety associated with the universal quotient bundle one can realize a corresponding GLSM for the vector bundle The resulting model consists of n matter multiplets P i in det −d , R-charge 0 [46]. The associated building block of the I-function without twisted mass is then given by (3.34)

I-functions of determinantal Calabi-Yau varieties
In this section, we describe how to utilize our formula (3.24) to compute genus-0 Gromov-Witten invariants of the determinantal Calabi-Yau varieties. Here we focus on the desingularized determinantal Calabi-Yau variety X A in (2.6) which can be described by a U (k) × U (ℓ ∨ p ) GLSM with matter content in the left of Table 3. This GLSM is called a PAX model and has a superpotential [18] (4.1) The PAX model has several distinct phases. Let ξ 1 and ξ 2 be the FI parameters associated with the central U (1) factors of U (k) and U (ℓ ∨ p ), respectively. For example, a geometric phase called a "X A phase" with ξ 1 > 0 and ξ 2 < 0 of the PAX model in the IR describes the variety X A in (2.6), and another geometric phase "X A T phase" with kξ 1 + ℓ ∨ p ξ 2 > 0 and ξ 2 > 0 corresponds to an incidence correspondence constructed from the transposed matrix A(φ) T . The chiral matter multiplet P in the fundamental representation ℓ ∨ p under the U (ℓ ∨ p ) factor corresponds to the vector bundle S * on G(ℓ ∨ p , p). By taking the Seiberg-like duality with respect to the gauge group U (ℓ ∨ p ), S * is mapped to Q on G(ℓ, p) and as indicated by the short exact sequence (3.32), the chiral matter multiplet P is mapped to the dual chiral matter multiplets Y and P i in the right of Table 3. This dualized GLSM is called a PAXY model with gauge group U (k) × U (ℓ) and has a superpotential given by [18] (4.2)

I-functions and A-model correlators
Let us consider the PAX model with massless matter multiplets shown in Table 3. FI parameters ξ 1 > 0 and ξ 2 < 0 is given by where , , and in particular w parametrizes the blowing up in (2.3). x (resp. y) are identified with the degree 2 elements in the cohomology of G(k, n) (resp. G(ℓ ∨ p , p)). As performed in (3.21), the I-function (4.3) can be expanded around = ∞ in terms of Schur polynomials as The flat coordinates q z and q w , which provide the exponentiated Kähler moduli parameters of X A , are given by log q z = I 1;0 (z, w) I 0;0 (z, w) = log z + O(z, w), log q w = I 0;1 (z, w) I 0;0 (z, w) = log w + O(z, w). (4.7) Here are the classical intersection numbers associated with the Poincaré dual H of a codimension dim X A − 2 cycle in X A , where σ P (resp. τ P ) is the Poincaré dual of a Schubert cycle of codimension |P | in G(k, n) (resp. G(ℓ ∨ p , p)). The genus-0 invariants n d 1 ,d 2 (H) associated with H, which are related to Gromov-Witten invariants, are conjecturally integers.

An algorithm to compute genus-0 invariants
In a similar fashion to the computation (3.26), by taking the classes σ 1 , σ 2 , τ 1 and τ 2 for the special Schubert cycles, the 1-point correlator (4.7) can be evaluated with where I X A [t] denotes the coefficient of t at = 1 in the expansion (4.5). From the coefficients I X A [t] and the classical intersection numbers one can compute the 1-point A-model correlator (4.7) and obtain the integer invariants. The classical intersection numbers can be computed by Martin's formula (3.28) as where

Illustrative examples of the computations
Here we will consider several examples of the desingularized determinantal Calabi-Yau 3-folds investigated in Section 2.3 and compute their genus-0 invariants n d 1 ,d 2 defined in (4.9). 6 6 We only focus on the determinantal varieties described by U (k) × U (ℓ ∨ p ) PAX models with k ≤ 2, ℓ ∨ p ≤ 2. In Appendix C we summarize our computational results for several determinantal Calabi-Yau 4-folds.

(4.13)
The quintic family can be described by GLSMs with U (1) × U (1) gauge group. Following Section 4.2 and Appendix B, one can compute topological invariants of the quintic family as summarized in Table 4, which is consistent with the previous works. Here one can also check that h 1,0 = 0. By comparing (4.13) with the entries n d 1 ,d 2 in Table 4 of each determinantal 3-fold, we see that they exhibit a behavior of the extremal transition [47] (see also [10]): where N is a certain finite positive integer.   Table 5. Here one can also check that h 1,0 = 0.   Table 6, where one can check that they exhibit the behavior (4.14) of the extremal transition and h 1,0 = 0. Note that, via the incidence correspondence (2.6), a geometric phase of the determinantal Calabi-Yau variety with F p = O V (1) ⊕4 on V = P 7 in (2.23) can be identified with a geometric phase of the variety with F p = (S * ) ⊕4 on V = G(2, 4) in (2.25) [18]. Indeed, by taking d 1 ↔ d 2 , the genus-0 invariants n d 1 ,d 2 of the former coincide with the genus-0 invariants of the latter [10].

Conclusions
In this paper we have examined a class of square determinantal Calabi-Yau varieties in Grassmannians satisfying appropriate conditions about dimension, a Calabi-Yau definition, duality G(k, n) ∼ = G(n − k, n), and rank of the vector bundles. We found that infinite families of examples associated with non-abelian quiver GLSMs might be possible. Furthermore, we explicitly demonstrated how to compute genus-0 integer invariants of the determinantal Calabi-Yau varieties via the Givental I-functions. By constructing the I-functions from the supersymmetric localization formula for the GLSM on a supersymmetric 2-sphere, we provided a guideline for the evaluation of the genus-0 A-model correlators. We also found the handy formula for the 1-point correlators for Grassmannian Calabi-Yau varieties, which turned out to be generalized into the cases with the determinantal varieties. We hope that our results would give a clue to understand various properties of the less studied GLSMs with non-abelian gauge groups.
Finally, we comment on possible future research directions.
• Since we have not imposed irreducibility as a requirement, to make our classification more rigorous, a comprehensive study of topological invariants such as Hodge numbers • We have classified the square determinantal varieties based on the requirement (2.5). It would be interesting to examine determinantal varieties with general vector bundles E p such as E p = L ⊗ O ⊕p V , where L is a line bundle, as studied in [25].
• We conjectured the formula (3.24) for the genus-0 1-point A-model correlators for Grassmannian Calabi-Yau varieties, which generalizes the formula (3.16). It would be interesting to find out the 3-point extension of our formula (3.24) as a natural generalization of the formula (3.12) studied in [32], and give a proof of it.
• In [52], GLSM realizations of the so-called Veronese embeddings and the Segre embeddings were proposed, and it opened up the possibility of more broad class of Calabi-Yau varieties.
Various exotic Calabi-Yau examples including the constructions in [53] have also been discussed, and it would be interesting to consider these examples and discuss their I-

functions.
A Determinantal Calabi-Yau 2-folds and 4-folds In Section 2.3, we have focused on the realization of a class of determinantal Calabi-Yau 3-folds of square type. In a similar spirit, here we discuss the classification of determinantal Calabi-Yau 2-folds and 4-folds satisfying the requirements (2.7) -(2.10).

A.1 Determinantal Calabi-Yau 2-folds
When dim X A = 2, the dimensional condition (2.7) becomes Note that, as mentioned in Remark 2.1, the generic determinantal Calabi-Yau 2-folds X A do not have the singular loci. In the following, we clarify which type of choices for k, n; ℓ ∨ p , c 1 (F p ) can be possible while changing the parameter k.
We find that there exist another type of solutions satisfying all the requirements (A.5), (2.8), (2.9), and (2.10) given by k, n; ℓ ∨ p , c 1 (F p ) = (2, 12; 4, 3) with Here the Calabi-Yau 4-fold X A constructed from F p = S * ⊕ O V (1) ⊕2 with p = 4 (i.e. ℓ = 0) can be described by the complete intersection Grassmannian Calabi-Yau 4-fold in G (2,8) with When k ≥ 3 we have V = G(k, n), and there exist six "infinite families" of determinantal 4-folds given by where the third (resp. fourth) and the fifth (resp. sixth) families of 4-folds associated with F p = (S * ) ⊕3 (resp. Q ⊕3 ) are given by the same recurrence relation with the different initial conditions. By using mathematical induction, one can check that the duality condition (2.9), the rank condition (2.10), and in particular ℓ ∨ p < rank F p , are maintained.

B Hodge number calculations via the Koszul complex
Following [54] (see also e.g. [55,56,57,58]), here we briefly review how to compute cohomologies and Hodge numbers of Calabi-Yau varieties via the Koszul complex. We will demonstrate the explicit computations for several examples.

B.1 General algorithm
Let V be a complex manifold, E p be a rank p vector bundle over V and consider the locus X ⊂ V as a holomorphic section of E p . In this appendix we describe how to compute the cohomologies First we describe a method to compute bundle-valued cohomologies of X, by using the Koszul exact sequence.
The Koszul exact sequence gives the resolution of O X over V as For the Koszul exact sequence, the Koszul spectral sequence (see e.g. [42]), can be associated as follows. Starting from which is associated with differentials with d r • d r = 0. Here E i,q r = 0 for i, q < 0, i > dim V , and q > p. At finite r = r 0 , E i,q r converges to E i,q r 0 = E i,q r 0 +1 = . . . = E i,q ∞ and we obtain a cohomology of X as where the summation represents a formal sum.
Then, by considering the Koszul spectral sequence associated with (B.8), one can obtain the bundle-valued cohomologies H i (X, F V | X ) in (B.2).
Therefore, by using the Koszul spectral sequence, the cohomologies H i (X, F V | X ) can be computed from the cohomologies H i (V, ∧ q E * p ⊗ F V ). For computing these quantities, the Bott-Borel-Weil theorem B.5 is quite useful. To state the theorem, consider a flag manifold A holomorphic homogeneous vector bundle F V over V can be described by a representation of U (n 1 ) × · · · × U (n F ), where a representation of each U (n) is described by a Young diagram which is given by a monotonically increasing sequence with length n of integers as (a 1 , . . . , a n ), a i ≤ a i+1 . Then, a vector bundle F V is described by a representation of U (n 1 ) × · · · × U (n F ) as F V ∼ (a 1 , . . . , a n 1 |b 1 , . . . , b n 2 | · · · |r 1 , . . . , r n F ). where S is the universal subbundle on V = P n−1 . The representations of tensor product and wedge product are obtained as e.g., where S is the rank (1, 2) universal subbundle on V .
Using the above representations (B.10) for vector bundles, the Bott-Borel-Weil theorem is stated as follows.
2. If the above sequence contains any same number, the cohomologies H i (V, F V ) are trivial, if not; 3. Minimally swap the above sequence, with the minimal swapping number I, so that the resulting sequence gives a strictly increasing sequence (y ′ 1 , . . . , y ′ N ), y ′ i < y ′ i+1 .
Remark B.6. The dimension D of a representation (y 1 , . . . , y N ) of U (N ), which is given by a Young diagram Y with length y i for the i-th row, is computed by where H Y (s) is the hook length of s in Y , and F Y (s) = N − i + j for s = (i, j) (the box of i-th row and j-th column).
Remark B.7. When V is a product manifold V = V 1 × V 2 of two flag manifolds V 1 and V 2 , for computing the cohomologies of V one can use the Künneth formula where F V is a vector bundle over V and each F V | V i is the restricted vector bundle over V i .
Several genus-0 invariants of the sextic family (A.6) are summarized in Table 7, where one can check that (h 1,0 , h 2,0 ) = (0, 0) and there is a relation originated from the extremal transition: where N is a certain finite positive integer.