QUANTUM COHOMOLOGY OF LAGRANGIAN AND ORTHOGONAL GRASSMANNIANS

HARRY TAMVAKIS

Abstract. This is the written version of my talk at the Mathematische Arbeitstagung 2001, which reported on joint work with Andrew Kresch (papers [KT2] and [KT3]).

The Classical Theory

For the purposes of this talk we will work over the complex numbers, and in the holomorphic (or algebraic) category. There are three kinds of Grassmannians that we will consider, depending on their Lie type:

• In type A, let G(k, N) = SL_{N}/P_{k} be the Grassmannian which parametrizes k-dimensional linear subspaces of ℂ^{N}.

• In type C, let LG = LG(n, 2n) = Sp_{2n}/P_{n} be the Lagrangian Grassmannian.

• In types B and D, let OG = OG(n, 2n + 1) = SO_{2n+1}/P_{n} denote the odd orthogonal Grassmannian, which is isomorphic to the even orthogonal Grassmannian OG(n + 1, 2n + 2) = SO_{2n+2}/P_{n+1}.

We begin by defining the Lagrangian Grassmannian LG(n, 2n), and will discuss the Grassmannians for the other types later in the talk. Let V be the vector space ℂ^{2n}, equipped with a symplectic form. A subspace Σ of V is isotropic if the restriction of the form to Σ vanishes. The maximal possible dimension of an isotropic subspace is n, and in this case Σ is called a Lagrangian subspace. The Lagrangian Grassmannian LG = LG(n, 2n) is the projective complex manifold which parametrizes Lagrangian subspaces Σ in V.

There is a stratification of LG by Schubert varieties X_{λ}, one for each strict partition λ = (λ_{1} > λ_{2} > ⋯ > λ_{ℓ} > 0) with λ_{1} ⩽ n. The number ℓ = ℓ(λ) is the length of λ, and we let D_{n} denote the set of strict partitions λ with λ_{1} ⩽ n. To describe X_{λ} more precisely, fix an isotropic flag of subspaces F_{i} of V:

0 ⊂ F_{1} ⊂ F_{2} ⊂ ⋯ ⊂ F_{n} ⊂ V

with dim(F_{i}) = i for each i, and F_{n} Lagrangian. Then

X_{λ} = { Σ ∈ LG | dim(Σ ∩ F_{n+1−λ_{i}}) ⩾ i for 1 ⩽ i ⩽ ℓ }.

The variety X_{λ} has codimension |λ| := ∑ λ_{i} in LG, and determines a Schubert class σ_{λ} = [X_{λ}] in H^{2|λ|}(LG, ℤ). The classes σ_{1}, …, σ_{n} are called special, and H^{2}(LG) has rank 1, generated by σ_{1}. We have an isomorphism of abelian groups

H^{∗}(LG, ℤ) = ∑_{λ∈D_{n}} ℤ · σ_{λ}

Date: June 16, 2001.

which describes the additive structure of the cohomology H^{∗}(LG). As all cohomology classes occur in even degrees, we adopt the convention that a class α in the cohomology of a complex manifold X has degree k when α lies in H^{2k}(X).

Classically, there are three ingredients necessary for an understanding of the multiplicative structure of H^{∗}(LG).

a) A presentation of H^{∗}(LG) in terms of generators and relations. This is due to Borel [Bo], and can be derived as follows. Consider the universal exact sequence of vector bundles over LG

0 ⟶ E ⟶ V_{LG} ⟶ E^{∗} ⟶ 0.

Here V_{LG} is the trivial rank 2n vector bundle over LG, E is the tautological subbundle of V_{LG}, and we have identified the quotient bundle V_{LG}/E with E^{∗} by using the symplectic form on V. The special Schubert class σ_{i} is equal to the i-th Chern class c_{i}(E^{∗}), for 1 ⩽ i ⩽ n. The cohomology ring of LG can then be presented as a quotient of the polynomial ring in the Chern classes of E by the relations coming from the Whitney sum formula

c(E)c(E^{∗}) = (1 − σ_{1} + σ_{2} − ⋯)(1 + σ_{1} + σ_{2} + ⋯) = 1.

Expanding this equation, one obtains that H^{∗}(LG) is a quotient of ℤ[σ_{1}, …, σ_{n}] by the relations

σ_{i}^{2} + 2 ∑_{k=1}^{n−i}(−1)^{k}σ_{i+k}σ_{i−k} = 0

for 1 ⩽ i ⩽ n, where it is understood that σ_{j} = 0 if j < 0.

b) A Giambelli formula which identifies the polynomials which represent the Schubert classes σ_{λ} in the above presentation. For the type A Grassmannians G(k, N), the answer to this question is quite classical (and due to Giambelli). In contrast, an analogous formula for the Lagrangian Grassmannian was proved much more recently. According to Pragacz [P], the Giambelli formula for LG is obtained in two stages: for i > j > 0, we have

σ_{i,j} = σ_{i}σ_{j} + 2 ∑_{k=1}^{n−i}(−1)^{k}σ_{i+k}σ_{j−k}

while for any λ ∈ D_{n} of length at least 3,

σ_{λ} = Pfaffian[σ_{λ_{i},λ_{j}}]_{1⩽i<j⩽r}.

Here r is the least even integer which is ⩾ ℓ(λ).

c) Schubert calculus: By this we mean algorithms for evaluating the structure constants e_{λμ}^{ν} which occur in the cup product expansion

σ_{λ}σ_{μ} = ∑_{|ν|=|λ|+|μ|} e_{λμ}^{ν} σ_{ν} (1)

in H^{∗}(LG). Recall that for type A Grassmannians, this problem is solved by showing that the multiplication of Schubert classes agrees with the corresponding product of Schur S-polynomials (defined by Schur in his celebrated 1901 thesis). In [P], Pragacz showed that the product (1) agrees with the corresponding product of two Schur Q-polynomials. The latter polynomials were used by Schur [S] to study the projective representations of the symmetric and alternating groups. Moreover, combinatorial algorithms for computing in the ring of Schur Q-polynomials were obtained by Boe and Hiller [BH] and Stembridge [St].

Quantum Cohomology of LG(n, 2n)

We will work throughout with the small quantum cohomology ring QH^{∗}(LG). This is a deformation of H^{∗}(LG) which first occured in the work of string theorists. The multiplicative structure of the quantum cohomology ring of LG encodes the enumerative geometry of rational curves in LG, in the form of Gromov–Witten invariants.

The ring QH^{∗}(LG) is an algebra over ℤ[q], where q is a formal variable of degree n + 1; one recovers the classical cohomology ring by setting q = 0. To describe the product structure, recall that for each Schubert class σ_{λ} one has a (Poincaré) dual class σ_{λ′}, defined so that

∫_{LG} σ_{λ} σ_{μ} = δ_{λμ′}

for all partitions λ, μ in D_{n}. The parts of the partition λ′ complement the parts of λ in the set {1, …, n}. Using dual classes, we can write the classical structure constant e_{λμ}^{ν} in (1) as a triple intersection number:

e_{λμ}^{ν} = ⟨σ_{λ}, σ_{μ}, σ_{ν′}⟩_{0} = ∫_{LG} σ_{λ} σ_{μ} σ_{ν′}.

In QH^{∗}(LG) there is a formula

σ_{λ} · σ_{μ} = ∑⟨σ_{λ}, σ_{μ}, σ_{ν′}⟩_{d} σ_{ν} q^{d}, (2)

the sum over d ⩾ 0 and strict partitions ν with |ν| = |λ| + |μ| − d(n + 1). The Gromov–Witten invariant ⟨σ_{λ}, σ_{μ}, σ_{ν′}⟩_{d} is a non-negative integer, which counts the number of degree-d rational maps f : ℙ^{1} → LG such that f(0) ∈ X_{λ}, f(1) ∈ X_{μ} and f(∞) ∈ X_{ν′}, for general translates of the three Schubert varieties X_{λ}, X_{μ} and X_{ν′}. Here degree d means that f_{∗}[ℙ^{1}] = d σ_{1′}. The ‘miracle’ is that with this definition, (2) gives an associative product.

There is a long list of papers studying questions (a), (b) and (c) above for the quantum cohomology rings of homogeneous spaces. We now have answers to all three questions when X = SL_{N}/P is a partial flag variety of SL_{N}(ℂ); see [W], [ST], [Be] and [BCF] for the case of the Grassmannian G(k, N), and [KT2] for more references. For an arbitrary complex semisimple Lie group G, with Borel subgroup B, Kim [K] found a presentation of QH^{∗}(G/B), thus answering (a). However, one still lacked a presentation of the quantum ring for the other parabolic subgroups and a quantum Giambelli formula to compute the Gromov–Witten invariants, if the Lie group G is not of type A. This was due to serious difficulties in extending the methods used in the proofs in the SL_{N} case to the other Lie types (for a discussion of one such problem, see [KT1, §4]).

The remainder of this talk reports on joint work with Andrew Kresch [KT1]

[KT2] [KT3]. We answer all three basic questions for the quantum cohomology rings of the type B, C and D Grassmannians mentioned previously. For the Lagrangian Grassmannian, questions (a) and (b) are addressed by the following Theorem.

Theorem 1. The ring QH^{∗}(LG) is presented as a quotient of the polynomial ring ℤ[σ_{1}, …, σ_{n}, q] by the relations

σ_{i}^{2} + 2 ∑_{k=1}^{n−i}(−1)^{k}σ_{i+k}σ_{i−k} = (−1)^{n−i}σ_{2i−n−1} q.

The Schubert class σ_{λ} in this presentation is given by the Giambelli formulas

σ_{i,j} = σ_{i}σ_{j} + 2 ∑_{k=1}^{n−i}(−1)^{k}σ_{i+k}σ_{j−k} + (−1)^{n+1−i}σ_{i+j−n−1} q

for i > j > 0, and

σ_{λ} = Pfaffian[σ_{λ_{i},λ_{j}}]_{1⩽i<j⩽r}, (3)

where quantum multiplication is employed throughout. In other words, quantum Giambelli for LG(n, 2n) coincides with classical Giambelli for LG(n + 1, 2n + 2), when the class 2 σ_{n+1} is identified with q.

Note that Theorem 1 is not saying that the cohomology ring of LG(n+1, 2n+2) is isomorphic to QH^{∗}(LG(n, 2n)), for the relation σ_{n+1}^{2} = 0 in the former ring fails to hold in the latter (as q^{2} ≠ 0). It turns out that the algebra of Schur Qpolynomials is not sufficient to describe the multiplication in QH^{∗}(LG). Instead, we use Q̃-polynomials; this is a family of symmetric polynomials modelled on Schur’s Qpolynomials, and defined by Pragacz and Ratajski [PR] in their work on degeneracy loci. The Q̃-polynomials seem to be the correct geometric analogue of Schur’s Spolynomials in type C. Their multiplication also governs the Arakelov theory of LG, regarded as a scheme over the ring of rational integers [T].

One consequence of Theorem 1 is a quantum Pieri rule for multiplying a class σ_{λ} in H^{∗}(LG) by a special Schubert class, which extends the classical one of [BH]. For another, define an involution ∗ on D_{n} by

∗ (λ_{1}, ..., λ_{ℓ}) = (n + 1 − λ_{ℓ}, …, n + 1 − λ_{1}),

and let ρ_{n} be the partition (n, n − 1, …, 1) (which corresponds to the class of a point in LG). We then have the following rule for multiplying by the point class:

Theorem 2. σ_{λ} · σ_{ρ_{n}} = σ_{λ^{′^{∗}}} q^{ℓ(λ)} in QH^{∗}(LG). In particular, σ_{ρ_{n}} · σ_{ρ_{n}} = q^{n}, which implies that there is a unique rational curve of degree n passing through 3 general points on LG(n, 2n).

The above results are part of the quantum Schubert calculus for LG.

The Orthogonal Grassmannian OG(n, 2n + 1)

Consider now V′ = ℂ^{2n+1} equipped with a non-degenerate symmetric form. The maximal isotropic subspaces in V′ are n-dimensional and parametrized by the odd orthogonal Grassmannian OG = OG(n, 2n + 1). The cohomology ring H^{∗}(OG, ℤ) has an additive basis of Schubert classes τ_{λ}, for λ ∈ D_{n}. As a ℤ-module, QH^{∗}(OG) is identified with H^{∗}(OG) ⊗ ℤ[q], but this time the degree of q is 2n.

Theorem 3. The ring QH^{∗}(OG) is presented as a quotient of the polynomial ring ℤ[τ_{1}, …, τ_{n}, q] modulo the relations

τ_{i}^{2} + 2 ∑_{k=1}^{i−1}(−1)^{k}τ_{i+k}τ_{i−k} + (−1)^{i}τ_{2i} = 0

for all i < n, together with the quantum relation

τ_{n}^{2} = q.

The Schubert class τ_{λ} in this presentation is given by the Giambelli formulas

τ_{i,j} = τ_{i}τ_{j} + 2 ∑_{k=1}^{j−1}(−1)^{k}τ_{i+k}τ_{j−k} + (−1)^{j}τ_{i+j}

for i > j > 0, and

τ_{λ} = Pfaffian[τ_{λ_{i},λ_{j}}]_{1⩽i<j⩽r}, (4)

where quantum multiplication is employed throughout. In other words, classical Giambelli and quantum Giambelli coincide for OG(n, 2n + 1).

We thus see that the answers to questions (a) and (b) for OG are directly analogous to those for type A Grassmannians. The quantum Schubert calculus is governed this time by the multiplication of P̃-polynomials. The quantum Pieri rule for OG implies that for each λ ∈ D_{n−1}, we have τ_{n} τ_{λ} = τ_{(n,λ)}, while τ_{n} τ_{(n,λ)} = τ_{λ} q in QH^{∗}(OG). It follows that to compute all the Gromov–Witten invariants for OG, it suffices to calculate ⟨τ_{λ}, τ_{μ}, τ_{ν}⟩_{d} for all λ, μ, ν ∈ D_{n−1}.

Partitions in D_{n−1} also parametrize the Schubert classes σ_{λ} in the (quantum) cohomology ring of the Lagrangian Grassmannian LG(n−1, 2n−2). Let us compare the two spaces at hand in a table:

X  dim_{ℂ}(X)  Schubert classes deg(q)
OG(n, 2n + 1)  n(n + 1)/2  τ_{λ} , λ ∈ D_{n}  2n
LG(n − 1, 2n − 2) n(n − 1)/2  σ_{λ} , λ ∈ D_{n−1}  n

In the next result, ∗ and ′ denote the involutions that were defined previously, but on the set D_{n−1}.

Theorem 4. Suppose that λ ∈ D_{n−1} is a partition with ℓ(λ) = 2d + e + 1 for some nonnegative integers d and e. For any μ, ν ∈ D_{n−1}, we have an equality

⟨τ_{λ}, τ_{μ}, τ_{ν}⟩_{d} = ⟨σ_{λ^{∗}}, σ_{μ′}, σ_{ν′}⟩_{e} (5)

of Gromov–Witten invariants for OG(n, 2n+1) and LG(n−1, 2n−2), respectively. If λ is zero or ℓ(λ) < 2d + 1, then ⟨τ_{λ}, τ_{μ}, τ_{ν}⟩_{d} = 0.

Remarks. 1) The left hand side of (5) is obviously symmetric in λ, μ and ν, unlike the right hand side. This reflects a (ℤ/2ℤ)^{3}-symmetry enjoyed by the Gromov– Witten invariants for both LG and OG.

2) We lack a purely geometric argument that would ‘explain’ Theorem 4. Instead, we first prove the main Theorems 1 and 3 and develop the quantum Schubert calculus for both spaces, then deduce Theorem 4 using algebra.

Proofs of the Main Results

The most difficult part of our approach to Theorems 1 and 3 is to establish the quantum Giambelli formulas (3) and (4). The arguments needed come from two sources: algebraic geometry and combinatorics. In conclusion, we mention three key ingredients that occur in the proofs; for precise definitions of the terms that follow, see [KT1], [KT2] and [KT3].

• Intersection theory on the Lagrangian (resp. orthogonal) Quot scheme which compactifies the moduli space of degree d maps from ℙ^{1} to LG (resp. OG). These Quot schemes are singular in general.

• Degeneracy loci for isotropic morphisms of vector bundles.

• Combinatorics of P̃, Q̃ and Schubert polynomials for the Lie types B, C and D.

References

[Be] A. Bertram : Quantum Schubert calculus, Adv. Math. 128 (1997), no. 2, 289–305.

[BCF] A. Bertram, I. Ciocan-Fontanine and W. Fulton : Quantum multiplication of Schur polynomials, J. Algebra 219 (1999), no. 2, 728–746.

[BH] B. Boe and H. Hiller : Pieri formula for SO_{2n+1}/U_{n} and Sp_{n}/U_{n}, Adv. in Math. 62 (1986), 49–67.

[Bo] A. Borel : Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math. 57 (1953), 115–207.

[K] B. Kim : Quantum cohomology of flag manifolds G/B and quantum Toda lattices, Annals of Math. 149 (1999), 129–148.

[KT1] A. Kresch and H. Tamvakis : Double Schubert polynomials and degeneracy loci for the classical groups, preprint (2000), available at http://www.math.upenn.edu/^{∼}kresch.

[KT2] A. Kresch and H. Tamvakis : Quantum cohomology of the Lagrangian Grassmannian, preprint (2001), available at http://www.math.upenn.edu/^{∼}kresch.

[KT3] A. Kresch and H. Tamvakis : Quantum cohomology of orthogonal Grassmannians, preprint (2001), available at http://www.math.upenn.edu/^{∼}kresch.

[P] P. Pragacz : Algebro-geometric applications of Schur S- and Q-polynomials, Séminare d’Algèbre Dubreil-Malliavin 1989-1990, Lecture Notes in Math. 1478 (1991), 130–191, Springer-Verlag, Berlin, 1991.

[PR] P. Pragacz and J. Ratajski : Formulas for Lagrangian and orthogonal degeneracy loci; Q̃-polynomial approach, Compositio Math. 107 (1997), no. 1, 11–87.

[S] I. Schur : Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. reine angew. Math. 139 (1911), 155–250.

[ST] B. Siebert and G. Tian : On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator, Asian J. Math. 1 (1997), no. 4, 679–695.

[St] J. R. Stembridge : Shifted tableaux and the projective representations of symmetric groups, Adv. in Math. 74 (1989), 87–134.

[T] H. Tamvakis : Arakelov theory of the Lagrangian Grassmannian, J. reine angew. Math. 516 (1999), 207-223.

[W] E. Witten : The Verlinde algebra and the cohomology of the Grassmannian, Geometry, topology, & physics, 357–422, Conf. Proc. Lecture Notes Geom. Topology, IV, Internat. Press, Cambridge, MA, 1995.

University of Pennsylvania and Max-Planck-Institut

E-mail address: harryt@math.upenn.edu