GLSMs for exotic Grassmannians

In this paper we explore nonabelian gauged linear sigma models (GLSMs) for symplectic and orthogonal Grassmannians and flag manifolds, checking e.g. global symmetries, Witten indices, and Calabi-Yau conditions, following up a proposal in the math community. For symplectic Grassmannians, we check that Coulomb branch vacua of the GLSM are consistent with ordinary and equivariant quantum cohomology of the space.


JHEP10(2020)200
Gauged linear sigma models (GLSMs) [1] have proven to be extraordinary physical tools to examine a wide range of questions in string theory and string compactifications, ranging from global properties of moduli spaces of SCFTs for Calabi-Yau compactifications to representations of quantum cohomology rings. The bulk of that work has focused on abelian two-dimensional theories, but in recent years technology has developed to the point where we can make inroads on understanding nonabelian theories.
To further that program, one of the tasks one must accomplish is to find physical descriptions of more geometries. For example, one can write down nonabelian GLSMs which have nontrivial IR fixed points, but to efficiently compute e.g. chiral rings, it helps enormously if one can interpret the resulting phases geometrically. To this end, in this paper we will explore (nonabelian) GLSMs for some additional spaces, namely symplectic and orthogonal Grassmannians, following up a brief proposal in [2]. We will check that description by e.g. comparing physically-derived quantum cohomology rings against known mathematics results, and study the phases and other properties of the GLSMs.
Ordinary Grassmannians G(k, n) can be described with GLSMs using methods that have been known for a long time, going back to [3]. They have played an important role in many papers. However, they are not the only notion of Grassmannians known to mathematicians. There are other Grassmannians in the mathematics literature, notably the symplectic and orthogonal Grassmannians. These also occasionally arise in physics, see e.g. [4] and references therein, but aside from a brief proposal in [2], their GLSM realizations have not been studied at all. The purpose of this paper is to fill this gap, following up the proposal of [2] by comparing GLSM predictions for quantum cohomology rings, Witten indices, Calabi-Yau conditions, and studying the GLSM phases.
Possible Grassmannians and flag manifolds 1 are given mathematically as cosets G/P , with P a parabolic subgroup of G, and G describing the symmetries of the space, which also correspond to global symmetries of the corresponding physical theory. We list below some examples from [5, section 23.3]: • A n : these are the Grassmannians G(k, n + 1) = SL(n + 1)/P , which have global symmetry U (n + 1) U (1) = P SU (n + 1) = SU (n + 1) (1.1) • B n : these are the orthogonal Grassmannians OG(n, 2n + 1) = SO(2n + 1, C)/P .
(The various Grassmannians above are sometimes referred to as type A, B, C, D Grassmannians respectively, in reference to their symmetries.) In each of these cases, the global JHEP10(2020)200 symmetry group of the GLSM for G/P is given by G (up to finite quotients), and has Lie algebra indicated by the classification above. (This is simply the subgroup of the group P SU (2n) or P SU (2n + 1) of rotations of the chiral primaries that preserves the superpotential, which requires that either a metric or symplectic form be preserved.) For example, for the B n series, we will see the GLSM has 2n + 1 chiral primaries, and a superpotential defined by a metric on those chiral primaries. The resulting symmetry group is the subgroup of P SU (2n + 1) that preserves that metric -hence, some finite group quotient of SO(2n + 1), corresponding to B n .
In addition to the A, B, C, and D type Grassmannians and flag manifolds, one can also obtain Grassmannians and flag manifolds from exceptional groups. We will leave the development of their GLSMs to future work.
Let us also remark on mirrors. In mathematics, there is a notion of mirrors to homogeneous spaces, see e.g. [6][7][8][9]. When we speak about mirrors to nonabelian theories, we will be using a slightly different mirror symmetry construction, described in [10]. These two constructions were compared in [10]; briefly, although in general they give different Landau-Ginzburg models, all we really are concerned with in a Landau-Ginzburg model is its IR behavior, encoded in its critical loci, and at least on the face of it, these differentlooking constructions of Landau-Ginzburg mirrors seem to encode the same IR physics, at least so far as we are aware.
We begin in section 2 by describing GLSMs for symplectic Grassmannians SG(k, 2n) and flag manifolds, the type C spaces listed above. These can be understood as submanifolds of ordinary Grassmannians G(k, 2n) and flag manifolds satisfying an isotropy condition, which is the key to the GLSM we present. We check our description by comparing ordinary and equivariant quantum cohomology rings arising in the GLSM to those arising in mathematics, as well as by comparing Witten indices across different phases. We also check that the Calabi-Yau condition arising physically matches that in mathematics. Finally, we discuss mirrors of the GLSMs for symplectic Grassmannians.
In section 3 we perform the analogous analyses for orthogonal Grassmannians OG(k, n) and flag manifolds, the type B and D spaces listed above. After proposing GLSMs for these spaces, we study the mixed Higgs-Coulomb phases arising for r 0, and compare the Calabi-Yau condition arising physically in these GLSMs to that arising mathematically. Finally, we discuss mirrors of the GLSMs for orthogonal Grassmannians.
In several appendices we collect various technical computations which supplement and clarify the computations in the text.
To be clear, our paper is not the first to describe GLSMs for all of these cases: GLSMs for symplectic and orthogonal Grassmannians were proposed at the end of [2]. The purpose of this paper is to more systematically analyze the physics of these theories, carefully checking that the GLSMs have the desired properties (quantum cohomology relations, Witten indices, global symmetries, and so forth), as well as to explore novel physical aspects of these theories.

Background and GLSM realization
The symplectic Grassmannian SG(k, 2n) is a space parameterizing k-dimensional subspaces of C 2n which are isotropic with respect to a symplectic form on C 2n . This can be described more explicitly as a subvariety of an ordinary Grassmannian G(k, 2n), a fact that will be used in GLSM realizations. The dimension of SG(k, 2n) is given by [11], [12, When k = n, SG(n, 2n) is the space of maximal isotropic subspaces of C 2n , which is also known as the Lagrangian Grassmannian, and often denoted LG(n, 2n). In the case k = 1, the isotropy condition trivializes, and [11] as we shall see explicitly in GLSMs in a moment. Another common special case is SG(2, 4) = LG(2, 4) ∼ = P 4 [2], as is mentioned in appendix F. The Euler characteristic of SG(k, 2n), relevant for Witten index computations, is given by [13] 2 k n k .
The GLSM for a symplectic Grassmannian implements an isotropy condition on top of an ordinary Grassmannian, so let us first quickly review the GLSM for an ordinary Grassmannian G(k, N ), following [3]. This is a U (k) gauge theory with N chirals in the fundamental representation, and no superpotential. The D-terms of the theory can be interpreted as the statement that the N fundamental-valued chirals form a set of k orthogonal, normalized vectors in C N , and gauging the U (k) effectively quotients out the rotations, hence this describes k-dimensional subspaces of C N . The GLSM for a symplectic Grassmannian implements an isotropy condition on top of an ordinary Grassmannian.

JHEP10(2020)200
on C 2n , following from the fact that (These GLSMs were also briefly described in [2, section 2.4.3, example 3].) The isotropy condition resulting from the superpotential above takes the simple form In the special case k = 1, this condition is satisfied trivially. In this special case, there are no q ab (since ∧ 2 1 is empty) and the superpotential is not present. As a result, SG(1, 2n) coincides with the ambient G(1, 2n) = P 2n−1 .
The global symmetries of this theory are rotations of the chiral superfields that are compatible with the superpotential. Specifically, rotations of the chiral superfields themselves are represented by the group The rotations that preserve the superpotential are precisely those which preserve the symplectic form, hence the global symmetry group is Sp(2n, C). Clearly, symplectic Grassmannians SG(k, 2n) can be embedded into ordinary Grassmannians G(k, 2n). The Plücker embedding of ordinary Grassmannians, realized physically as SU (k)-invariant baryons for α = ±i, also is relevant for symplectic Grassmannians. Just as for ordinary Grassmannians, these define a map into a projective space of dimension 2n k − 1. (2.10) In the special case that k = n, there is a second class of SU (k)-invariant operators, that also define a map. These operators are given bỹ and they define a map from LG(n, 2n) into a projective space of dimension 2 n −1. (See [15, section 3.3] for more information.) For later use, since one of the matter representations is slightly unusual, we give here the D-terms: In the special case that k is odd, the GLSM for SG(k, 2n) has no Higgs branch for r 0. This follows from the diagonal terms in the D terms above, and the fact that since

JHEP10(2020)200
q is an antisymmetric matrix, it can only have an even number of eigenvalues. The matrix q can therefore be put in the form with one vanishing row and one vanishing column. In this basis, the diagonal D term corresponding to the bottom right entry has the form 15) with no sum over a and no q fields. This D term has no vanishing solutions for r 0, hence there can be no Higgs phase for r 0 if k is odd. The cases of k even are different. Here, there does appear to be a Landau-Ginzburg phase, which contributes to the Witten index. Consider the case SG(2, 2n). Here, there is one q field, q 12 , which transforms only under the overall U (1) = det U (2). In the r 0 phase, it Higgses the gauge group U (2) to SU (2) ⊂ U (2). From [16, section 3.2], [17, section 4.7], the Witten index of an SU (2) gauge theory with N fundamentals is (1/2)(N − 2) N even, (2.16) so we see that the Higgs branch of the r 0 phase of the SG(2, 2n) GLSM has Witten index (1/2)(2n − 2) = n − 1. (2.17) For example, the r 0 phase of the GLSM for LG(2, 4) has Witten index 2 − 1 = 1. For SG(k, 2n) for even k > 2, we expect a similar story. For r 0, the q fields get a vev, which Higgses 2 U (k) to Sp(k) (see e.g. [18]). Beyond that, we do not have a complete understanding. Based on the fact that there are k(k − 1) off-diagonal D terms generating relations, we (naively) suspect that this theory has the same Witten index as a two-dimension Sp(k) gauge theory with 2nk − k(k − 1) chirals, in 2n − k + 1 copies of the fundamental representation, which from [19, equ'

JHEP10(2020)200
We will check this conjecture numerically in a pair of examples in table 1 in section 2.4, and leave a more detailed analysis for future work. We have argued that for r 0, for odd k, there is no Higgs branch contribution to the GLSMs for symplectic Grassmannians, and for even k, there is a Higgs branch. In the full quantum theory, there is a Coulomb branch contribution, just as in GLSMs for ordinary Grassmannians, and we will see explicitly that those Coulomb vacua provide an explicit representation of the quantum cohomology ring. Furthermore, the Higgs branch (if it exists) will also supply the difference between the Witten index of the r 0 phase and the number of Coulomb vacua (corresponding to roots of the ring relations).
In the next section we will describe how the Coulomb branch realizes both ordinary and equivariant quantum cohomology of general Lagrangian Grassmannians, and also check in special cases that the quantum cohomology ring of other symplectic Grassmannians is also realized by these GLSMs.

Ordinary quantum cohomology
In this section, we will argue that the physical chiral ring of this theory coincides with known results for the quantum cohomology of SG(n, 2n), which serves as a consistency check on the GLSM description above.
We begin by studying the Coulomb branch. The effective twisted superpotential is where ρ a ib = δ a b , ρ a µν = −δ a µ − δ a ν and α a µν = −δ a µ + δ a ν . Here, the weights ρ a µν correspond to q ab in the representation ∧ 2 V * and can be obtained from the weights for V * following appendix C. One can check that the ρ a µν make the superpotential (2.4) gauge invariant. Note that the sum of the W-boson contributions contributes a i(n − 1)π-shift to t. Computing the critical locus, we have which gives the chiral ring relations with (from the ambient theory) excluded locus σ a = σ b if a = b. Mathematically, the quantum cohomology ring relations for SG(n, 2n) are known, and can be found in e.g. [20, equ'n (3)]: (To avoid symbol abuse, we have used e i 's andq here, instead of σ i 's and q as in [20].) In equation (2.22), e i (x) is the i-th elementary symmetric polynomial in the Chern roots {x 1 , . . . , x n } of the tautological 3 bundle S, which can be understood as the restriction of the tautological bundle on the ambient G(n, 2n) to SG(n, 2n), and so fits in the short exact sequence We will argue in this section that the physical chiral ring relations (2.21) reproduce the quantum cohomology ring relations (2.22) known in the mathematics literature for (2.24) These two identifications make sense in the following ways. First, as in [3,21,22], we can interpret the σ a as the Chern roots of S * , hence σ a = −x a . From naively counting degrees, one finds that both q andq have degree n + 1, and they should match up to a constant factor, which one can show is (−) n−1 . Before giving a general argument that the physical chiral ring relations match the ring relations known in mathematics, we demonstrate how this works in several examples. The mathematical quantum cohomology ring relations in this case are To see how the mathematical relations follow from the physical relations, we subtract the two physical chiral ring relations to get Taking into account the excluded locus σ 1 = σ 2 , σ 2 1 − σ 2 2 can be factored out and the above equation becomes σ 2 1 + σ 2 2 = 0, (2.28) which is the same as e 1 (x) 2 = 2e 2 (x) with σ a = −x a .

JHEP10(2020)200
Similarly, the sum of the two physical chiral ring relations is where the last equality follows from This is the same as e 2 (x) 2 = −qe 1 (x) with σ a = −x a andq = −q.
General case. In the cases above, we used algebraic tricks to construct Weyl invariant polynomials of the σ a , which led to the quantum cohomology ring relations. We will next use more general methods to study the cases n ≥ 3. First, note that the left-hand side of equation (2.21) can be expanded in terms of Weyl invariant polynomials of the σ a . For example, when n = 2k + 1, it can be expanded as: where e i (σ) is the i-th elementary symmetric polynomial. Similarly, for n = 2k, it can be rewritten as Let us consider the n = 2k + 1 case first. Since the e i (σ) are Weyl invariant, the e i (σ) are constant on Weyl orbits. Rewrite equation (2.32) as for a = 1, . . . , n. Then the σ 2 a satisfy relations determined by the coefficients of P (σ 2 a ) according to Vieta's formula [23, theorem 2], which says that any nth order polynomial p(z) with roots a 1 , · · · , a n can be written in the form p(z) ∝ z n − e 1 (a)z n−1 + e 2 (a)z n−2 + · · · + (−) n e n (a), (2.35) where the e i (a) are elementary symmetric polynomials in the roots a j . If we let x a denote a root of equation (2.34), a solution for σ 2 a , then from Vieta's formula and the coefficients of the σ 2 terms in (2.34), we have 1≤a≤n x a = 0, x a 1 . . . x a k+1 = (−1) k q, The reader should note that the number of possible roots (2k + 1, the degree of equation (2.34), matches the rank (n) of the gauge group. The equations above (derived from the coefficients of equation (2.34) can be written more compactly for ≤ n = 2k + 1 as in conventions in which e i = 0 for i < 0 and e 0 = 1. Applying identity (A.2), which we repeat below, we recover the quantum cohomology ring relations for SG(2k + 1, 4k + 2) known in the math community, equation (2.22), with σ a = −x a and q = (−) 2kq =q.
The same argument works for n = 2k. Rewrite equation (2.33) as then Vieta's formula and the coefficients of the polynomial above give the relations 1≤a≤n σ 2 a = 0,

JHEP10(2020)200
and where here we have abused notation by labeling the possible solutions for σ 2 by σ 2 instead of x. Just as before, these can be summarized compactly as in equation (2.38), and applying identity (A.2), we recover the quantum cohomology ring relations for SG(2k, 4k), equation (2.22). So far we have checked our previous claim that the chiral ring relations (2.21) in the Coulomb branch do reproduce the quantum cohomology ring relations (2.22).

Equivariant quantum cohomology
The above story can be generalized to the equivariant case, equivariant with respect to a maximal torus of the flavor symmetry group, which physically corresponds to turning on twisted masses [24]. We will verify that the equivariant quantum cohomology ring predicted by the physics of this GLSM matches that known in mathematics for Lagrangian Grassmannians.
Due to the global symmetry Sp(2n), which preserves the isotropy condition (2.7), we have m −i = −m i . To keep the superpotential invariant, the twisted masses for q ab is (2.43) Therefore, the effective twisted superpotential becomes and the critical locus of this effective twisted superpotential gives the following physical chiral ring relations

JHEP10(2020)200
We will show that these reproduce the equivariant quantum cohomology ring relations (D.9) in appendix D. We repeat them here for convenience: Note that the right-hand-side of equation (2.44) can be expanded as while the left-hand-side of equation (2.44) can be expanded in the same way as before. To establish that the mathematical ring is a consequence of the physical chiral ring, we can use the same methods as in our analysis of the ordinary quantum cohomology. Here for brevity we will only give the details for the n = 2k + 1 case. First, we rewrite equation (2.44) as Following the same reasoning as before, we choose to work on one vacuum, corresponding to one Weyl orbit of solutions for the σ a , and on this orbit the e i (σ) are constant due to Weyl invariance. Therefore, the equation above is of degree (2k + 1) in σ 2 a . From Vieta's formula, we have following sets of relations:

Quantum cohomology for general symplectic Grassmannians
In this section, we will compare quantum cohomology of general symplectic Grassmannians SG(k, 2n) to the predictions of the physical chiral ring. We will not give a general proof that they always match, but instead will merely check several families of examples. The analysis in the previous section can be applied to SG(k, 2n). The physical chiral ring relations have the same form as before, but now there are only k < n relations: These chiral ring relations will reproduce the quantum cohomology ring relations [25], which are where c r is the r-th Chern class of the quotient bundle Q over SG(k, 2n), defined by restricting the universal quotient bundle over the ambient G(k, 2n). It obeys If we interpret the σ a as the Chern roots of S * , then we have c i (S) = (−) i e i (σ), and where h r (σ) is the rth complete homogeneous symmetric polynomial in the σ a . We will check in a series of examples that physics correctly reproduces the quantum cohomology ring relations above.

JHEP10(2020)200
which is indeed the chiral ring relation for P 2n−1 . At the same time, equation (2.50) for k = 1 is just where c n = h n (σ) = σ n . Therefore, the ring relations match if we identify This identification of q andq is generally true as we will see in the following examples.
Next we consider the case SG(2, 6). The physical chiral ring relations in this case are Some algebraic manipulations give and these two Weyl invariant equations indeed reproduce equation (2.50) for quantum cohomology when k = 2: with q = (−) 3−1q =q. Note that we have used equation (A.5) in appendix A. Our next example is SG (2,8). The physical chiral ring relations are which yield We can check that these two equations reproduce equation This calculation can be generalized to SG(2, 2n) and, due to the equation (A.5), the chiral ring relations for SG(2, 2n) reproduce the quantum cohomology ring relations.

Witten indices
As a consistency check of our description and analysis, here we will check that Witten indices are preserved across various phases of these GLSMs. (Although the target is not Calabi-Yau, and so the axial R-symmetry is anomalous, nevertheless, there can be JHEP10(2020)200 nonanomalous finite subgroups, and so one expects continuous paths connecting the r 0 and r 0 phases of these GLSMs, hence the Witten indices should match.) For simplicity, let us begin with the case LG(2, 4). As previously described, the Euler characteristic of this space is 2 2 = 4, which should match the Witten index of the r 0 phase, if our GLSM is correct. This should also be the Witten index of the r 0 phase, which we will now check.
The r 0 phase is a mixed Higgs/Coulomb branch. As previously discussed in section 2.1, the Higgs branch at low energies is an SU (2) gauge theory with 4 fundamentals, which has Witten index 1. The Coulomb branch is defined by solutions to the equations to the quantum cohomology ring relations 57) or after simplification, It is straightforward to check that there are 3 distinct unordered pairs (x 1 , x 2 ) which solve these equations (and are consistent with the excluded locus), hence there are 3 Coulomb vacua. The sum of the number of Coulomb vacua and the Witten index of the Higgs branch is 4, matching the Witten index of the r 0 phase, as expected. Next, consider SG(2, 2n). The Euler characteristic of this space, the Witten index of the phase r 0, is From the general analysis of section 2.1, the Higgs branch of the r 0 phase has Witten index n − 1, so to be consistent, there should be We count the Coulomb vacua of the r 0 phase of the GLSM for SG(2, 2n) as follows. The chiral ring relations are σ 2n 1 = q(σ 1 + σ 2 ) = σ 2n 2 , (2.61) and the (unordered) roots of these equations are the Coulomb vacua (subject to the excluded locus σ 1 = ±σ 2 , which requires σ a = 0). Since σ 2n 1 = σ 2n 2 , we see that σ 1 and σ 2 differ by a 2n-th root of unity (excluding −1, as that is on the excluded locus). Excluding ±1, we see that σ 1 differs from σ 2 by 2n − 2 possible phases. Plugging in, the chiral ring relations reduce to a degree 2n − 1 polynomial in either of the σ a , hence 2n − 1 solutions. Dividing by 2 to account for ordering, we have a total of possible Coulomb vacua, which is precisely right for the Witten index of the r 0, r 0 phases of SG(2, 2n) to match.

JHEP10(2020)200
To the same end, let us now discuss SG(k, 2n) with k odd. Here, as previously argued, there is no Higgs branch, only a Coulomb branch, hence for consistency the number of Coulomb vacua should match the Euler characteristic of the space, which is We can see this as follows. The chiral ring relations with k odd are As we are counting vacua away from the excluded locus (σ a = 0, σ a = ±σ b for a = b) the solutions should always have this Z 2 symmetry. Note that this is not true for cases in which k is even, because elementary symmetric polynomials of odd degrees appear.
In SG(k, 2n) for k odd, as previously discussed, there is no Higgs branch at r 0, only Coulomb vacua. For k even, there can be a nontrivial Higgs branch, which contributes to the Witten index, as discussed in greater detail previously.
In table 1 we have summarized results for a number of cases, comparing Euler characteristics of large-radius phases, number of Coulomb vacua, and Witten indices of Higgs branches at r 0. In each case, we find that the large-radius Euler characteristic matches the sum of the number of Coulomb vacua and the Witten index of the Higgs branch.

Calabi-Yau condition
As another consistency test, we briefly mention Calabi-Yau conditions. Mathematically, the intersection of the Plücker embedding of SG(k, 2n) with a hypersurface of degree 2n − k + 1 hypersurface is Calabi-Yau. We reproduce the same condition physically as the condition for the sum of the charges under any U (1) subgroup of the gauge group to vanish.
The GLSM for SG(k, 2n) is a U (k) gauge theory with 2n chirals in the fundamental V , and one chiral in ∧ 2 V * . Under any U (1) ⊂ U (k), the 2n chirals in the fundamental contribute a total of 2n to the sum of the U (1) charges, and the one chiral in ∧ 2 V * contributes −(k − 1), so that the sum of the U (1) charges is 2n − k + 1. (2.67)

JHEP10(2020)200
Geometry χ(geometry) Num. Under the same U (1), any element of the Plücker embedding has charge 1, so we see that intersecting the image of SG(k, 2n) with a hypersurface of degree 2n − k + 1 should be Calabi-Yau, reproducing the mathematics result. Let us consider two special cases as explicit confirmations.
• SG(1, 2n) = P 2n−1 . The condition for a hypersurface in P 2n−1 to be Calabi-Yau is that it has degree 2n, which is reproduced by the condition above.
• SG(2, 4) = P 4 [2]. Here, the Calabi-Yau condition is that a hypersurface should have degree 3, which is reproduced by the condition above.

Symplectic flag manifolds
In addition to symplectic Grassmannians, there also exist symplectic flag manifolds. At the level of group cosets, these are of the form Sp(2n, C)/P for suitable parabolic subgroups P . We can describe them as submanifolds of ordinary flag manifolds F (k 1 , · · · , k p , 2n) (k 1 < k 2 < · · · ) satisfying an isotropy condition on the maximal vector space. Let us briefly describe GLSMs for these flag manifolds. We begin with the GLSM for an ordinary flag manifold F (k 1 , · · · , k p , 2n) [26]. This is a gauge theory with bifundamentals (k 1 , k 2 ), (k 2 , k 3 ), and so forth to (k p−1 , k p ), along with 2n chirals in representation k p of U (k p ), following [26]. To build the GLSM for a symplectic JHEP10(2020)200 flag manifold SF (k 1 , · · · , k p , 2n), we add a chiral superfield q ab transforming in the ∧ 2 k p representation of U (k p ), along with a superpotential We only impose an isotropy condition on the last, maximal flag: as all other vector spaces in the flag are subspaces of the maximal flag, this suffices to guarantee that all subspaces satisfy the isotropy condition. We will not compute quantum cohomology rings from the GLSM here, but mathematical discussions of quantum cohomology rings for symplectic flag manifolds can be found in [27].
with superpotential where ρ a µν = −δ a µ −δ a ν . Here we are considering general symplectic Grassmannians SG(k, 2n) with k ≤ n, which includes Lagrangian Grassmannians as a special case when k = n.
Let us check explicitly that the chiral ring relations match. First, integrate out the σ a to get the constraints

JHEP10(2020)200
We define then the superpotential reduces to On the critical locus, we have Therefore, the chiral ring relations are Now we can see that the mirror reproduces the chiral ring relation of the original theory via the operator mirror map Π a ↔ σ a . (2.77) 3 Orthogonal Grassmannians OG(k, n)

Background and GLSM realization
Orthogonal Grassmannians, denoted OG(k, n), are submanifolds of an ordinary Grassmannian G(k, n), satisfying an isotropy condition with respect to a nondegenerate quadratic form. (Specifically, an isotropic subspace W of a vector space V has the property that for all vectors x, y ∈ W , x · y = 0 for the dot product defined by the quadratic form; nondegeneracy simply means that the orthogonal complement of the entire vector space V is just 0.) They can be realized by GLSMs for Grassmannians with a superpotential realizing the isotropy condition. The resulting GLSMs look very similar to those for symplectic Grassmannians, except that one has a field coupling to a symmetric-tensor-square representation rather than in an antisymmetric tensor representation. We have two slightly different GLSMs depending upon whether n is even or odd. First consider the case of n odd. Write n = 2m + 1. The GLSM is a U (k) gauge theory with n chirals φ a i in the fundamental representation V (a ∈ {1, · · · , k}, i ∈ {−m, −m + 1, · · · , 0, · · · , +m}), and one chiral q ab in the representation Sym 2 V * , with superpotential We interpret φ a α as defining k vectors in C 2m+1 , and the F terms imply the isotropy condition x · y = 0 for each of k vectors in C 2m+1 , with a dot product defined by the symmetric matrix   

JHEP10(2020)200
corresponding to the quadratic form 3) The case of n even is similar. Write n = 2m. The GLSM is a U (k) gauge theory with n chirals φ a i in the fundamental representation V (a ∈ {1, · · · , k}, i ∈ {±1, ±2, · · · , ±m}), and one chiral q ab in the representation Sym 2 V * , with superpotential We interpret φ a α as the components of k vectors in C 2m , and the F terms imply the isotropy condition x·y = 0 for each of k vectors in C 2m , with a dot product defined by the symmetric matrix 0 I m I m 0 , (3.5) corresponding to the quadratic form These GLSMs were also briefly described in [2, section 2.4.3, example 2]. The dimension of OG(k, n) (or rather, the dimension of one component, in the case n = 2k) is 4 [11], [12] For later use, note that in the special case n = 2k, the complex dimension of OG ± (k, 2k) is (1/2)k(k − 1). In the special case of OG(m, 2m), the orthogonal Grassmannian decomposes into a disjoint union of two spaces, denoted OG ± (m, 2m): OG(m, 2m) = OG + (m, 2m) OG − (m, 2m).

JHEP10(2020)200
When k = 1, OG(k, n) is a quadric hypersurface in P n−1 . This follows immediately from the GLSM. For k = 1, the symmetric tensor representation has only one component, so there is only one q field, of charge −2. It multiplies a quadric polynomial in the φ fields, and hence coincides with the GLSM for the vanishing locus of that quadric polynomial in P n−1 . For example, a quadric in P 2 is 5 P 1 , and so we recover the standard result that OG(1, 3) = P 1 . The simplest example in which to see the decomposition of OG(m, 2m) explicitly is OG(1, 2). As described above, this is a quadric hypersurface in P 1 given by φ 1 φ −1 = 0. This equation is reducible, and in any event any hypersurface in P 1 will describe a collection of points. In this case, we see OG(1, 2) is two points, so OG ± (1, 2) are each a single point. Geometrically, we can think of this in terms of isotropic subspaces of C 2 with quadratic form defined by the symmetric matrix There are two isotropic subspaces, each one-dimensional, one generated by (1, 0) T , the other by (0, 1) T . Since the subspaces are unique, there are no deformations, and so OG ± (1, 2) should each be a single point. We can understand the decomposition of OG(2, 4) similarly: think of this in terms of isotropic subspaces of two copies of the vector space above, C 4 with a quadratic form defined by two copies of the matrix (3.13) along the diagonal. Then, for example, the vectors (1, 0, 1, 0) T , (0, 1, 0, 1) T each lie in distinct isotropic subspaces. In this case, however, these vectors lie in larger families, which we can visualize by moving the two choices of C 2 's inside C 4 . Such a choice is equivalent to choosing a one-dimensional subspace of C 2 , which is P 1 , hence we see OG ± (2, 4) = P 1 .
Global symmetries of this QFT follow the same pattern discussed earlier. We can rotate the chiral superfields into one another, preserving the superpotential. For OG(k, n), we have n chiral superfields, and chiral superfield rotations are described by U (n)/U (1) = P SU (n); restricting to those preserving the superpotential -in particular, those preserving the metric -restrict to SO(n, C) (up to possible finite group quotients).
Just as in symplectic Grassmannians, orthogonal Grassmannians OG(k, n) can be trivially embedded into G(k, n), and share the Plücker map defined by SU (k)-invariant baryons 5 If we didn't projectivize, the reader will note that this is an equation for C 2 /Z2, or explicitly (φ0) 2 + φ1φ−1 = 0.
However, after projectivization, corresponding to gauging the U (1) symmetry, this becomes a curve, and codimension-one quotient singularities do not exist on curves as varieties.

Mixed Higgs-Coulomb phases at r 0
In this section we will study the r 0 phases of the GLSMs for orthogonal Grassmannians. Here, the r 0 phases will be mixed Higgs-Coulomb branches, containing both Higgs and Coulomb vacua. (In appendix E we discuss such phases in the simpler context of hypersurfaces in projective spaces.) One would be tempted to try to analyze the resulting theories using the methods of Hori-Tong [16]. There, one had U (k) gauge theories with fundamentals as well as fields charged only under det U (k). These theories were analyzed in a Born-Oppenheimer approximation, 'fibering' the SU (k) gauge theories over the space of vacua defined by the fields charged only under det U (k). Here, however, there are no fields charged solely under det U (k); all the fields are charged nontrivially under SU (k), so no analogous Born-Oppenheimer analysis is pertinent.
OG(n, 2n). As described above, the orthogonal Grassmannian OG(n, 2n) is described as a GLSM with a U (n) gauge group with matter • 2n chiral fields φ a i in the fundamental representation V , • 1 chiral field q ab in the symmetric tensor product representation Sym 2 V * , where a = 1, · · · , n, i = ±1, · · · , ±n, and superpotential Next, we consider the phase r 0. Here, from the D-terms, the q ab cannot all vanish simultaneously, while the F -terms imply that the vevs of all φ a i should vanish. In particular, there can be a nontrivial Higgs branch when r 0, which must be taken into account when computing vacua. This Landau-Ginzburg phase will play a role in the next analysis.

JHEP10(2020)200
In general, since the r 0 phase is Fano, one would ordinarily expect that the r 0 phase is accompanied by discrete Coulomb vacua [44]. However, in these theories, describing OG(n, 2n) specifically, there are no discrete Coulomb vacua for n > 1, as we will establish next.
The one-loop corrected twisted superpotential is and α a µν = −δ a µ + δ a ν , and where we have assumed n > 1. Simplifying, we get with q = exp(−t). However, this has no solutions that are not contained inside the excluded locus {σ a = σ b }. We can see this as follows. Our analysis follows the same form as in section 2. Suppose for the moment that n = 2k + 1. We write equation (3.17) as 4q (σ a ) 2k + · · · + σ 2 a e 2k−2 (σ) + e 2k (σ) = (σ a ) 2k , (3.18) where the e i are elementary symmetric polynomials in all of the σ a . Broadly speaking, this equation should have 2k roots for the value of (σ a ) 2 , but since n = 2k + 1, there are 2k + 1 different values of σ a that must be assigned. (In particular, the excluded locus condition requires that the σ a must all be distinct.) There are two possible ways to assign values.
• One option is if one σ a = 0, and the others are distinct and nonzero. In this case, the Coulomb branch relation (3.17) reduces to a =b σ a = 0, (3.19) so at least one other value of σ a must vanish, giving σ a = σ b (= 0) for some a = b, a contradiction.
The analysis for the case of n even is nearly identical. As a result, we see that for n > 1 there are not enough distinct solutions to the Coulomb branch relations to satisfy the excluded locus condition, hence there is no Coulomb branch, the r 0 phase is pure Higgs.
Now, let us consider the Higgs branch in the phase r 0. In the case n = 1, the representation defining q is one-dimensional, so the GLSM is that for a quadric hypersurface in P 1 , with the r 0 phase a Z 2 orbifold of a Landau-Ginzburg model with superpotential i.e. a mass term. Since there is an even number of massive fields, there are two vacua [19,[45][46][47], [48, section 4.2]. The r 0 phase is OG(1, 2), which is two points, so we see that the Euler characteristics match, trivially.
For r 0 for more general cases, D terms imply that the q ab are not all zero, which Higgses the gauge group U (n) to Z 2 × SO(n) (see e.g. [18]), and gives a mass to the φ fields. The Z 2 subgroup arises as a subgroup of det U (n) that acts trivially on the q ab , and so, from decomposition [19,[45][46][47][48][49], we expect that the Landau-Ginzburg geometry is a disjoint union of two spaces, just as the r 0 geometry. We shall not pursue the geometry of this Landau-Ginzburg phase further in this paper, but it would be interesting to do so, especially to compare to the predictions for this phase from homological projective duality [50][51][52][53].
OG(n, 2n + 1). First, we shall look for discrete Coulomb vacua. We shall find that, unlike the case of OG(n, 2n), this theory does have nontrivial discrete Coulomb vacua, as well as a nontrivial Landau-Ginzburg model.
The effective twisted superpotential is We will see in examples that this admits nontrivial solutions.

JHEP10(2020)200
Let us first consider OG(1, 3). Specializing our previous discussion, the corresponding GLSM is a U (1) gauge theory with one q field of charge −2 and three φ fields φ 0,±1 of charge +1, with superpotential As discussed earlier, for r 0, this describes P 2 [2] = P 1 . Now, let us turn to the r 0 phase. This is a Z 2 orbifold of a Landau-Ginzburg model with superpotential describing three massive fields. Since there is an odd number of massive fields in this Z 2 orbifold, there is a single vacuum [19,[45][46][47], [48, section 4.2]. However, we also need to take into account discrete Coulomb vacua [44]. The chiral ring relation is 4q = σ, (3.26) so we see we have one discrete Coulomb vacuum, for a total of two vacua, matching the Euler characteristic of the large-radius phase OG(1, 3) = P 1 , as expected. Unfortunately, since the vacua live on a combination of Coulomb and Higgs vacua, we do not know of a method to directly compute the product relations, as we have done previously for theories in which all of the vacua arise on a Coulomb branch.

Calabi-Yau condition
As another consistency test, we briefly mention Calabi-Yau conditions. Mathematically, the intersection of the Plücker embedding of OG(k, n) with a hypersurface of degree n−k−1 is Calabi-Yau. We reproduce the same condition physically as the condition for the sum of the charges under any U (1) subgroup of the gauge group to vanish.
The GLSM for OG(k, n) is a U (k) gauge theory with n chirals in the fundamental V , and one chiral in Sym 2 V * . Under any U (1) ⊂ U (k), the n chirals in the fundamental contribute a total of n to the sum of the U (1) charges, and the one chiral in Sym 2 V * contributes −2 − (k − 1) = −k − 1, so that the sum of the U (1) charges is Under the same U (1), any element of the Plücker embedding has charge 1, so we see that intersecting the image of OG(k, n) with a hypersurface of degree n − k − 1 should be Calabi-Yau, reproducing the mathematics result. Let us consider a set of special cases to explicitly check this result. Recall OG(1, n) = P n−1 [2]. The Calabi-Yau condition is that an additional hypersurface should have degee n − 2, which matches n − k − 1.

Orthogonal flag manifolds
In addition to orthogonal Grassmannians, there also exist orthogonal flag manifolds. At the level of group cosets, these are of the form SO(n, C)/P for suitable parabolic subgroups P . We can describe them as submanifolds of ordinary flag manifolds F (k 1 , · · · , k p , 2n) satisfying an isotropy condition on the maximal vector space. Let us briefly describe GLSMs for these flag manifolds. We begin with the GLSM for an ordinary flag manifold F (k 1 , · · · , k p , 2n) [26]. This is a gauge theory with bifundamentals (k 1 , k 2 ), (k 2 , k 3 ), and so forth to (k p−1 , k p ), along with 2n chirals in representation k p of U (k p ), following [26]. To build the GLSM for an orthogonal flag manifold OF (k 1 , · · · , k p , 2n), we add a chiral superfield q ab transforming in the Sym 2 k p representation of U (k p ), along with a superpotential of the form (depending upon whether n is even or odd). As for symplectic flag manifolds, we only impose an isotropy condition on the last, maximal, flag, as all other vector spaces in the flag are subspaces. We will not compute quantum cohomology rings from the GLSM here, but mathematical discussions of quantum cohomology rings for orthogonal flag manifolds can be found in [27].

JHEP10(2020)200
where ρ a µν = −δ a µ − δ a ν . From the definition of this mirror Landau-Ginzburg model, the dimension can be counted as 2nk − k(k − 1) − 1 2 k(k + 1) − k = 1 2 k(4n − 3k − 1), which matches the dimension of OG(k, 2n). Now let us compute the chiral ring relation. First, integrate out σ a 's and we will get namely, we have therefore the superpotential becomes Now let us look at the critical locus defined by exp ∂W ∂φ = 1, for φ an arbitrary field.
In components, for each a, we have where in the third equation, we have used U aµ = U µa . Therefore, on the critical locus, we Then plugging back into equation (3.32), we have The chiral ring relations obtained from mirror models are equivalent to each other given that Π a ↔ σ a .
First integrate out σ a 's, and define The superpotential becomes Using the same calculations as in case of OG(k, 2n), we have which is the same as the chiral ring relations for OG(k, 2n + 1) by

JHEP10(2020)200 4 Conclusions
In this paper we have studied GLSM realizations of symplectic and orthogonal Grassmannians and flag manifolds, which is to say, spaces of the form SO(n)/P and Sp(n)/P for suitable subgroups P , generalizing GLSMs for ordinary Grassmannians G(k, n) which are of the form U (n)/P . We have checked our descriptions by comparing ordinary and equivariant quantum cohomology rings predicted by GLSMs with those derived mathematically, compared Witten indices of different phases. We have also discussed mirrors of these GLSMs. One future direction is to generalize to GLSMs for Grassmannians and flag manifolds derived from exceptional groups. Another direction is to understand how to interpret the various phases in terms of homological projective duality [50].

A Symmetric polynomials
In this section, we briefly define two classes of symmetric polynomials and list some identities which are used extensively in this paper.
Elementary symmetric polynomials. The kth elementary symmetric polynomial in n variables x 1 , · · · , x n , denoted e k (x), is defined by with e 0 (x) = 1 and e k (x) = 0 for k < 0. For example, the elementary symmetric polynomials in 3 variables x 1 , x 2 and x 3 include It can be shown that for elementary symmetric polynomials in n variables x 1 , · · · , x n ,

JHEP10(2020)200
Complete homogeneous symmetric polynomials. The k-th complete homogeneous symmetric polynomial in n variables x 1 , · · · , x n , denoted h k (x), is defined by with h 0 (x) = 1 and h k (x) = 0 for k < 0. They can also be defined as where i 1 , . . . , i n are non-negative integers. For example, the complete homogeneous symmetric polynomials in 3 variables x 1 , x 2 and x 3 include For complete homogeneous symmetric polynomials in two variables x 1,2 , define for a given integer n ≥ 2. We will use the following identity in section 2.3: As we do not know a reference where this is written explicitly, we briefly outline an argument for this identity here. First, it is straightforward to check that it is true for n ≤ 3, so we will use induction to argue it for general n. Assume it is true for n, n + 1, then we will argue it is true for n + 2. Now, for polynomials in two indeterminates,

B Equivariant quantum cohomology
Equivariant quantum cohomology can be obtained from gauged linear sigma models by turning on twisted masses for global symmetries [24]. In this section, we will review how this works in detail for projective spaces and Grassmannians, comparing to known math results [54]. Mathematically, many results on equivariant cohomology on these spaces follow from the universal sequence over any Grassmannian where S is the universal subbundle, Q the universal quotient bundle, and V a trivial bundle. For G(k, N ), S has rank k, Q has rank N − k, and the trivial bundle V has rank N . If we turn on equivariant parameters with respect to the maximal torus in GL(N ), then we write V as a sum of eigenspaces for the action: for generic equivariant parameters t 1 , . . . , t n . The total Chern class in equivariant cohomology is given by The equivariant cohomology ring of the Grassmannian can be expressed in terms of the equivariant Chern classes of S. For later use, the resulting expressions can often be efficiently written in terms of functions h N (x|t), known as the factorial complete homogeneous Schur functions, which are defined as for p an integer N − k + 1 ≤ p ≤ N .

B.1 Projective spaces
From the GLSM for the projective space P N −1 , the chiral ring relation after turning on twisted masses is given by Mathematically, for P N −1 , the equivariant quantum cohomology ring relation is 6 [54] h N (x|t) =q. (B.6) In this case, k = 1, p can only be N and t = (t 1 , . . . , t N ), so the mathematical relation (B.6) for equivariant quantum cohomology matches the physical chiral ring relation (B.5) if we identify x = σ, t i = m i , andq = q.
In terms of the universal subbundle S and its equivariant Chern classes, from (B.1) we have Classically, c N (Q) = 0. In the quantum theory, c N (Q) = (−1) Nq , which yields the equivariant cohomology ring relations h N (x|t) =q in [54], for x = c 1 (S).

B.2 Grassmannians
Let us consider the general Grassmannians, G(k, N ). It can be realized in the U (k) GLSM with N fundamentals. The chiral ring relations are

JHEP10(2020)200
First, consider the special case G (2,4). Define x 1 , x 2 as the Chern roots of the universal subbundle S. The equivariant quantum cohomology ring relations are given by [54] c 3 (Q) = −h 3 (x|t) = 0, We claim the GLSM predictions match. The physical chiral ring relations for G(2, 4) are Subtracting these equations and factoring out σ 1 − σ 2 (since the excluded locus forbids σ 1 = σ 2 ), we have Since the left-hand side of the equation above is h 3 (σ|m), we recover the first equivariant quantum cohomology ring relation after identifying t i = m i and x a = σ a . The sum of the physical chiral ring relations is Adding (σ 1 + σ 2 )h 3 (σ|m) to the left-hand side gives 2h 4 (σ|m). Therefore, we end up with h 4 (σ|m) = −q, (B.13) which matches the second equivariant quantum cohomology ring relation if we also identifỹ q = q. Next, consider G(2, N ). The equivariant quantum cohomology ring relations are given by [54] h N −1 (x|t) = 0, We follow the same pattern to show that these ring relations follow from the physical chiral ring relations which matches the first mathematical ring relation. Next, summing the two chiral ring relations gives Since we have h N −1 (σ|m) = 0, we can add (σ 1 + σ 2 )h N −1 (m|t) to the left-hand side of the above equation to get (B.17) from which the second mathematical relation follows. Therefore, we see that for σ a = x a , m i = t i and q =q, the physical chiral ring relations reproduce the mathematical equivariant cohomology relations.
To derive the mathematical ring relations from the physical chiral ring relations, we proceed as follows. First, subtract each two of the three chiral ring relations and factor out (σ 1 − σ 2 ), (σ 1 − σ 3 ), and (σ 2 − σ 3 ), to get.
Subtracting any two of the equations above gives a relation of the form We simplify this using which is the second equivariant quantum cohomology ring relation.
The last equivariant quantum cohomology ring relation can be obtained by summing the three physical chiral ring relations: Thus, we see that all the equivariant quantum cohomology ring relations can be derived from the physical chiral ring relations for G (3, N ).
These methods are straightforward to generalize to G(k, N ) for k > 3, and so we do not give further details here.

C Tensor product representation
In this appendix, we briefly review the weights of tensor products of representations. Consider first the case V ⊗ V , where V is a two-dimensional representation. Denote a basis of JHEP10(2020)200 V by {v 1 , v 2 }, then a basis for the tensor product is In the representation V , the weights ρ a i are defined by H a v i = ρ a i v i , where H a is a Cartan generator [56]. In the tensor product representation V ⊗ V , namely, the weights are ρ a ij = ρ a i + ρ a j . We can further restrict to the (anti-)symmetric case by (anti-)symmetrizing the basis. In this particular example, the symmetric tensor product representation has the basis and the weights are 2ρ a 1 , 2ρ a 2 , ρ a 1 + ρ a 2 . In the anti-symmetric case, the basis is and the weights are ρ a 1 + ρ a 2 . This story can be generalized easily.

D Equivariant quantum cohomology for SG(n, 2n)
Consider the universal sequence over SG(n, 2n) with equivariant parameters turned on In this sequence, the tautological bundle S and the quotient bundle Q both have rank n and they are dual to each other, i.e. S * ∼ = Q, which implies c i (S) = (−) i c i (Q) and the Chern roots of S and Q are up to a minus sign. This universal bundle gives the following relation c V t = c(S)c(Q) = (1 + c 1 (S) + · · · + c n (S))(1 + c 1 (Q) + · · · + c n (Q)), where x i can be the Chern roots of either S or Q. The total Chern class of c V t is where t i 's are equivariant parameters for Sp(n)-action and e i (t 2 ) is the i-th elementary symmetric polynomial of t 2 1 , · · · , t 2 n . To obtain the (quantum) cohomology ring relations, we need to modify equation (D.2) by adding c i (S) and c i (Q) for n < i ≤ 2n and it becomes c(V t ) = (1 + c 1 (S) + · · · + c 2n (S)) (1 + c 1 (Q) + · · · + c 2n (Q)) . While to obtain the quantum cohomology ring relations,q has degree n + 1 for SG(n, 2n) and we need to set 2c n+1 (Q) =q, c n+2 (Q) = 0, · · · , c 2n (Q) = 0. (D. 6) In the following, we only verify the quantum case as it will reproduce the classical case in the limitq → 0. It turns out we should consider two situations, n odd and n even, separately.

JHEP10(2020)200
This set of relations will be the same for both classical case and quantum case. For the second set, we first need to write down c i (Q), 2k + 2 ≤ i ≤ 4k + 2, in terms of {c i (S)|i = 1, · · · , n = 2k + 1} and then use equation We can summarize the equations for even n and odd n cases above in a more compact form as for i = 1, · · · , n.

E Simple examples of mixed Higgs-Coulomb branches
In this appendix, we will outline some simple examples of theories with mixed Higgs-Coulomb branches, and the limitations of computing quantum cohomology with σ fields in each case. First, consider the case of a hypersurface of degree d in P 4 .
• d = 0. In this case, there is no hypersurface, this is just the GLSM for P 4 itself. In this case, there is no Higgs branch for r 0, only a Coulomb branch, with σ fields obeying σ 5 ∝ q. (E.1) There are then five solutions for σ, matching the Euler characteristic of P 4 , and those σ fields can be used to reproduce the quantum cohomology ring of P 4 , using known methods [59].

JHEP10(2020)200
• d = 5. This is the Calabi-Yau case. In this case, there is no Coulomb branch, no discrete Coulomb vacua, only the Higgs branch, corresponding to the Landau-Ginzburg orbifold phase of this Calabi-Yau hypersurface.
• d = 2. (This GLSM provides an alternative physical realization of the space SG(2, 4), which as a variety coincides with P 4 [2].) This is an example of a mixed branch, with both Higgs and Coulomb vacua. The Coulomb vacua are solutions of or σ 3 ∝ q, which only has three solutions. In addition, there is a Landau-Ginzburg orbifold, a Z 2 orbifold of a theory with superpotential of the form Here, the x fields are clearly massive, and as there is an odd number of them, taking the Z 2 orbifold only results in a single vacuum [19,[45][46][47], [48, section 4.2]. Combining the Landau-Ginzburg and Coulomb vacua, we have a total of 4 vacua, matching the Euler characteristic of the hypersurface P 4 [2].
As a related example, consider the GLSM for a hypersurface of degree d in P 3 .
• The degree d = 0 and d = 4 cases follow the same form as above. In one case, one has the GLSM for P 3 , which only has a Coulomb branch, no Landau-Ginzburg phase. In the other case, one only has a Landau-Ginzburg phase, no discrete Coulomb vacua.
• The degree d = 2 case here is a bit more interesting. The Coulomb vacua are solutions of σ 4 = (−2σ) 2 q, (E. 4) or σ 2 ∝ q, and so there are 2 discrete Coulomb vacua. The Landau-Ginzburg phase is a Z 2 orbifold of a theory with superpotential of the form W = x 2 1 + · · · + x 2 4 . (E.5) Again, the x fields are all massive, but there is an even number of them, so now the Z 2 orbifold results in two vacua [19,[45][46][47], [48, section 4.2]. Combining the Landau-Ginzburg vacua and Coulomb vacua, we have a total of four vacua, which matches the Euler characteristic of P 3 [2].
In passing, the Coulomb branch relation for a hypersurface of degree k in a projective space P n , namely σ n+1 = q(−k) k σ k , (E. 6) appear in discussions of the quantum cohomology ring of hypersurfaces in [57, equ'n (16),