The B-model connection and mirror symmetry for Grassmannians

This paper is dedicated to the memory of Andrei Zelevinsky
https://doi.org/10.1016/j.aim.2020.107027Get rights and content

Abstract

We consider the Grassmannian X=Grnk(Cn) and describe a ‘mirror dual’ Landau-Ginzburg model (Xˇ,Wq:XˇC), where Xˇ is the complement of a particular anti-canonical divisor in a Langlands dual Grassmannian Xˇ, and we express W succinctly in terms of Plücker coordinates. First of all, we show this Landau-Ginzburg model to be isomorphic to one proposed for homogeneous spaces in a previous work by the second author. Secondly we show it to be a partial compactification of the Landau-Ginzburg model defined in the 1990's by Eguchi, Hori, and Xiong. Finally we construct inside the Gauss-Manin system associated to Wq a free submodule which recovers the trivial vector bundle with small Dubrovin connection defined out of Gromov-Witten invariants of X. We also prove a T-equivariant version of this isomorphism of connections. Our results imply in the case of Grassmannians an integral formula for a solution to the quantum cohomology D-module of a homogeneous space, which was conjectured by the second author. They also imply a series expansion of the top term in Givental's J-function, which was conjectured in a 1998 paper by Batyrev, Ciocan-Fontanine, Kim and van Straten.

Introduction

The genus 0 Gromov-Witten invariants of a Grassmannian X answer enumerative questions about rational curves in X and are put together in various ways to define rich mathematical structures such quantum cohomology rings, flat pencils of connections on Frobenius manifolds [9]. These structures are part of the so-called ‘A-model’ of X. Mirror symmetry in the sense we consider here seeks to describe these structures in terms a mirror dual ‘B-model’ or Landau-Ginzburg (LG) model associated to X. Explicitly, the data from the A-model should be encoded by singularity theory or oscillating integrals of a regular function Wq (the superpotential) which is defined on a ‘mirror dual’ affine Calabi-Yau variety Xˇ with a holomorphic volume form ω. In the present paper we construct such a mirror datum in canonical and concrete terms for Grassmannians and prove associated mirror conjectures. Our results in particular imply and enhance a conjecture formulated in the 1990s by Batyrev, Ciocan-Fontanine, Kim and van Straten [3, Conjecture 5.2.3] concerning a series expansion for a coefficient of Givental's J-function. This conjecture was restated in a paper of Bertram, Ciocan-Fontanine and Kim [7, Section 3], where the special case of Grassmannians of 2-planes was proved by an entirely different method. The conjecture was again restated as a problem of interest in [60, Problem 14] some 10 years later.

To give a flavour of our results, considerS(q)=λ(e1ħWqpλω)PD(σλ), which is an H(X,C)-valued function depending on a single variable q (and a parameter ħ). The integrand involves the superpotential Wq. This is a particular rational function introduced in this paper via an explicit formula in terms of Plücker coordinates pλ on an isomorphic (but Langlands dual) Grassmannian Xˇ; see for example (3.1). The integration is over a natural compact torus in an open subvariety Xˇ of Xˇ; see Theorem 4.2. Note that the function S(q) in (1.1) is expressed as a linear combination of Schubert classes σλ permuted by Poincaré duality, denoted by PD. By Theorem 4.2 (one of our main results) the function S(q) satisfies the flat section equation of the Dubrovin connection, i.e. we have

. Here
denotes the hyperplane class of X, and q denotes the quantum cup product on the small quantum cohomology ring of X.

A conjecture of Batyrev, Ciocan-Fontanine, Kim and van Straten proposes an integral formula e1ħLqω for the coefficient of the top class PD(σ) of a flat section of the Dubrovin connection, and employs a Laurent polynomial Lq introduced by Eguchi, Hori and Xiong [14] in place of our Wq. We prove this conjecture by recovering the Laurent polynomial superpotential Lq as a restriction of Wq to a particular open dense torus inside Xˇ and applying our more general Theorem 4.2.

Note that the Schubert basis and the Plücker coordinates above are indexed by the same set. This has its explanation in the geometric Satake correspondence. The key point is that there is a natural identification H0(Xˇ,O(1))=H(X,C) identifying Schubert classes of X with homogeneous coordinates of Xˇ. This identification comes from the fact that both left-hand and right-hand sides agree with the k-th fundamental representation kCn of SLn(C), the special linear group which acts on Xˇ. For the left-hand side this is by the Borel-Weil theorem, and for the right-hand side it is by a very special case of the geometric Satake correspondence [24], [35], [43], [49] which constructs representations of G=SLn(C) via intersection cohomology of Schubert varieties of the affine Grassmannian GrG for the Langlands dual group G=PSLn(C). (The Schubert variety which arises in the construction of kCn turns out to be homogeneous for G and coincides with the Grassmannian X.)

The methods developed here are useful and adaptable to other co-minuscule G/P, which are precisely the homogeneous spaces which in the geometric Satake correspondence appear as ‘minimal’ Schubert varieties of the affine Grassmannian GrG. In particular, since this paper appeared, the methods we use have been employed to obtain results analogous to Theorem 4.2 in the case of even and odd-dimensional quadrics; see [54], [55]. Moreover, in the case of Lagrangian Grassmannians, a partial result in this direction, the ‘canonical’ description of the superpotential in terms of Plücker coordinates, has been obtained in [53].

The main work in this paper is concerned with constructing the A-model Dubrovin connection in terms of the Gauss-Manin system associated to a mirror LG model, in the most natural way possible, in the case where X is a Grassmannian. Our main theorem, Theorem 4.1, explicitly describes the Dubrovin connection of X in terms of the Gauss-Manin system of the mirror LG model we introduce. This underlies the formula for the global flat section obtained in Theorem 4.2. We note that the superpotential is additionally shown (see Theorem 4.10) to be isomorphic to the Lie-theoretic superpotential of X that was defined for general G/P in [65, Section 4.2]. Our result therefore also implies a version of the mirror conjecture concerning solutions to the quantum differential equations in terms of the superpotential considered there, namely [65, Conjecture 8.1], in the Grassmannian case.

Our results also shed new light on the (small) quantum cohomology ring of X, which is shown to agree with the Jacobi ring of Wq by an isomorphism which identifies the Schubert class σλ with the element of the Jacobi ring represented by the Plücker coordinate pλ. All of our results, including this one, are also stated and proved in the T-equivariant setting; see Proposition 5.2 and Theorem 5.5.

In more recent work of the second author together with L. Williams [66], the superpotential Wq introduced in this paper, together with the cluster structure on C[Xˇ], is shown by tropicalisation to define a set of polytopes which can be identified as Newton-Okounkov convex bodies of X. This can be understood as another form of mirror symmetry relating X and (Xˇ,Wq).

A version of the present paper was placed on the arXiv in December 2015 (arXiv:1307.1085v2 [math.AG]). In May 2017 a preprint of Lam and Templier [39] appeared on the minuscule G/P case of the mirror conjecture [65, Conjecture 8.1] for the Lie-theoretic superpotential (see Section 6.2). Their approach, which covers the Grassmannian case, employs completely different methods and uses the minuscule property. From this approach they can in their setting deduce that the Gauss-Manin system has the expected rank. On the other hand, our approach, using that the Grassmannian is co-minuscule, gives rise to explicit formulas for all of the components of a flat section of the Dubrovin connection in certain canonical coordinates. As mentioned above, the proof of the mirror conjecture along the same lines as in the present paper has already been carried out for even and odd-dimensional quadrics in [54], [55], the latter of which are not minuscule and hence not covered by the work in [39]. We expect our approach to be applicable, with analogous canonical coordinates, to all co-minuscule G/P.

The outline of the paper is as follows. We begin with a concrete introduction of the A-model structures. Then we give definitions of the B-model structures, as well as a careful statement of the main results. In Section 6.4 we show how our Plücker formula for the superpotential relates to the formulations in [3], [14], [65]. The proof of the main theorem begins in Section 10 and takes up the remainder of the paper. It makes use of deep properties of the cluster algebra structure of the homogeneous coordinate ring of a Grassmannian. In the final sections we prove a version of the main theorem in the torus-equivariant setting.

Acknowledgments

The first author thanks Gregg Musiker and Jeanne Scott for useful conversations. The second author thanks Dale Peterson for his inspiring work and lectures, Clelia Pech for useful conversations and Lauren Williams for helpful comments. We would like to thank the referee for many helpful comments.

Section snippets

The A-model introduction

Let us suppose X is the Grassmannian Grnk(Cn). We focus on the example k=3 and n=5 to illustrate our results in the introduction. The cohomology of X has a basis called the Schubert basis, which is indexed by partitions or Young diagrams (see e.g. [20]). In the example of X=Gr2(C5) we denote the Schubert basis of H(X)=H(X,C) by Here σλ is in H2|λ|(X) and |λ| denotes the number of boxes in the Young diagram λ. Furthermore Xλ will be a Schubert variety associated to λ, representing the

The mirror LG model

To give our presentation of the mirror LG model of the Grassmannian X=Grnk(n) we need to introduce a new Grassmannian Xˇ:=Grk(n). Both X and Xˇ have dimension N=k(nk). To be more precise, if X=Grnk(Cn) then we think of Xˇ as Grk((Cn)), which is embedded by a Plücker embedding in P(k(Cn)). Here (Cn) denotes the vector space dual to Cn, with an action of the Langlands dual GLn(C) from the right. The Plücker coordinates pλ for Xˇ correspond in a natural way to the Schubert classes σλ in H(X

Isomorphism of DP-modules

In the B-model, we consider the Gauss-Manin system on GWq. The first main theorem shows that the A-model datum, consisting of HA,0=H(X,C[z,q]) together with its small Dubrovin connection, can be recovered inside GWq.

Theorem 4.1

The C[z±1,q±1]-module HB is a DP–submodule of GWq, and the mapΦ:HAHBσλ[pλω] is an isomorphism of DP-modules. Under this isomorphism HA,0=H(X,C[z,q]) is identified with HB,0 and HA is identified with HB.

The proof of this theorem hinges on verifying the following formulas for the

Equivariant results

The Grassmannian X=Grnk(Cn) is a homogeneous space for G=GLn(C). In particular, the maximal torus T of n×n-diagonal matrices and the Borel subgroups of upper-triangular and lower-triangular matrices, denoted by B+ and B respectively, all act on X. In the A-model this means that we can consider the T-equivariant cohomology of X and the T-equivariant quantum cohomology of X, and we also have a T-equivariant version of the Dubrovin connection; see in particular [28], [42], [48]. Our final

The three versions of the superpotential

We have already mentioned the three different versions of a Landau-Ginzburg model dual to the A-model Grassmannian X=Grnk(Cn). In this section their superpotentials are defined in detail and we show how they are related to one another. We proceed in reverse chronological order, beginning with the superpotential introduced in Section 3.

The coordinate ring C[Xˇ] as a cluster algebra

By [69, Thm. 3], the homogeneous coordinate ring of the Grassmannian Xˇ has a cluster algebra structure (see also [21, §3], [22, Thm. 4.17]). In the latter this cluster algebra structure is shown to induce a cluster algebra structure on C[Xˇ]. We now recall these constructions.

A skew-symmetric cluster algebra with frozen variables is defined as follows [17, §5]. Let r,mN and consider the field F=C(u1,,ur+m) of rational functions in r+m indeterminates. A seed in F is a pair (x˜,Q˜) where x˜={x

The three versions of the holomorphic volume form

In this section we conclude the proof of the comparison Theorem 4.7, Theorem 4.10 which was begun in Section 6. To do this it remains to compare the holomorphic volume forms on the domains, T,Xˇ and R, of the three superpotentials Lq,Wq and Fq. Recall that these domains are related by mapsTιXˇπL1BU+w˙Pw˙01U+πRR, see Section 6.2. We begin by recalling how each of the holomorphic volume forms is constructed.

Definition 8.1

The holomorphic volume form on T [27]

The domain T of the Laurent polynomial superpotential Lq is naturally a torus.

Schubert classes and proof of the free basis lemma

The first aim of this section is to show that the isomorphism between C[Xˇ×Cq]/(XˇWq) and qH(X)[q1] identifies the classes of the Plücker coordinate pλ with the Schubert class σλ. After that we will prove Lemma 3.3.

Outline of the proof of Theorem 4.1

The remainder of the paper will be devoted to proving Theorem 4.1 and its equivariant counterpart. Here we summarize the proof. We keep all of the notation of the previous sections. We need to show that the following hold for all λPk,n:[qWqpλω]=μ[pμω]+qν[pνω];1z[Wpλω]=nz(μ[pμω]+qν[pνω])|λ|[pλω], where μ, ν are exactly as in the quantum Monk's rule for σqσλ. This will be shown in Theorem 18.5.

In Section 7, we have recalled the cluster structure on the Grassmannian, following Scott [69]

A special Postnikov diagram associated to a partition λ

Given λPk,n, we denote by λ any partition obtained from λ by adding a single box. Our aim in this section is, given λPk,n, to define a Postnikov diagram containing Jλ and all Jλ as labels. Moreover, the faces labelled Jλ should be adjacent to the face labelled Jλ. This diagram will be used later in explicitly computing the action of Xλ on Wq. We start by constructing special, symmetric Postnikov diagrams in the case n=2k and Jλ={1,3,,2k1}; the diagrams for arbitrary (k,n) can then be

The superpotential written in terms of an arbitrary Plücker extended cluster

Restricting the regular function Wq on Xˇ to a cluster torus XˇC˜ gives a Laurent polynomial in the associated cluster variables. We will need an explicit formula for this in the case where C˜ is a Postnikov extended cluster associated to a Postnikov diagram D. To obtain such a formula, we shall use [47, Thm. 1.1] which expresses a twisted version of an arbitrary Plücker coordinate pμ in terms of an arbitrary Postnikov extended cluster. The Laurent polynomial expansion of pμ|XˇC˜ is given in

Construction of a perfect matching

Suppose λPk,n and let D=Dλ be the Postnikov diagram constructed by Theorem 11.1. For example, Fig. 4 shows the Postnikov diagram

for G(3,6), noting that
. Let D1,D2,,Dn be the associated boundary-adjusted Postnikov diagrams and G1,G2,,Gn be the corresponding dual bipartite graphs, associated to D in Section 12. We also have corresponding quivers Q1,Q2,,Qn (see Definition 12.1, Definition 12.2). For example, Fig. 16 shows D3, G3 and Q3 for the case
.

Our aim in this

How to obtain all perfect matchings from Mi

If k=1 or n1, Mi is the unique perfect matching on Gi, so we assume in this section that k1,n1. Our main aim is to show that every other perfect matching on Gi can be obtained from Mi by performing a sequence of face flips, in the sense of Definition 14.1 below, on interior faces. We will also show that the sequences of face flips can be constructed in such a way that a distinguished face Fi of Gi always appears at the beginning of the sequence and then never appears again.

Definition 14.1

Let M be a perfect

The matching monomial associated to Mi

Our main aim in this section is to compute the matching monomial associated to Mi, i.e. its contribution towards the matching polynomial of Gi. This, combined with Theorem 14.20, will be used in Section 17 in order to compute the action of Xλ on Wq. We assume in this section that k1,n1. Recall the definition (Definition 13.3) of the weighting on Gi. We make the following useful definition.

Definition 15.1

Let D be a Postnikov diagram with closure D. Let I be a vertex of Q(D). For any minimal cycleI=I0I1I2

The vector fields Xλ(m)

In this section we define a family of vector fields Xλ(m), λPk,n, m[1,n], on Xˇ, denoting Xλ(n) also by Xλ. We allow the cases k=1,n1. We start by defining a vector field Xλ,C˜(m) in a fixed Postnikov extended cluster C˜ containing pλ via an explicit formula involving combinatorially defined coefficients cλ(m)(μ). We then prove that the coefficients satisfy an additivity property which in particular will imply that the vector field Xλ,C˜(m) extends to a regular vector field Xλ(m) on the

Action of the vector field Xλ on Wq

Fix a partition λPk,n. Our main aim in this section is to compute the action of Xλ(m) on W for each m[1,n] (Theorem 17.3). We include the cases k=1,n1. Let D=Dλ be the Postnikov diagram associated to λ constructed in Theorem 11.1. Recall that D has a face F(λ) labelled Jλ. Let C˜ denote the Postnikov extended cluster associated to D.

We collect together the information we will need. Recall first that, on the cluster torus XˇC˜, the vector field Xλ(m) is given by the following formula (see

Completion of the proof of Theorem 4.1

In this section we complete the proof of Theorem 4.1. Namely, we will show that:[qWqpλω]=μ[pμω]+qν[pνω];1z[Wqpλω]=nz(μ[pμω]+qν[pνω])|λ|[pλω], where μ, ν are exactly as in the quantum Monk's rule for σqσλ.

We first of all note a corollary to Theorem 17.3.

Corollary 18.1

Let λ be an arbitrary Young diagram in Pk,n. Then we have:

  • (a)

    XλWq=(μpμ+qνpν)qWqpλ;

  • (b)

    m=1nXλ(m)Wq=n(μpμ+qνpν)Wqpλ, where μ, ν are exactly as in the quantum Monk's rule for σqσλ.

Proof

Part (a) is the case m=n in Theorem 17.3. To see part

Background for mirror symmetry in the torus-equivariant setting

We now turn to the torus-equivariant mirror theorem, Theorem 5.5. We begin in Section 19.2 by reviewing the structure of the small equivariant quantum cohomology ring of a Grassmannian. We refer to [1] for background on equivariant cohomology, and [48] for relevant background on equivariant quantum cohomology. In Section 20 we recall the equivariant version of the superpotential (introduced for general G/P in [65]) and describe it in the case of the Grassmannian in terms of Plücker coordinates.

The T-equivariant version of the superpotential

By the equivariant superpotential of the target space X we mean a deformation of the usual superpotential to a (multi-valued) map involving the equivariant parameters, which encodes structures from the equivariant quantum cohomology of X. A torus-equivariant version of the superpotential for general type partial flag varieties was introduced in [65, Section 4], where it was denoted FP+ln(ϕ), and was shown to recover the equivariant quantum cohomology rings in their presentation due to Dale

Action of the vector field: T-equivariant case

In this section we prove the formulas needed to complete the proof of Theorem 5.5.

Theorem 21.1

Let λPk,n. Then we have:

  • (a)

    [qWeqqpλω]=(μ[pμω]+qν[pνω])+xλ[pλω], and

  • (b)

    1z[Wpλω]=|λ|[pλω]nz(μ[pμω]+q[pνω])nzxλ[pλω]+(nk)z(j=1nxj)[pλω],

where, in each case, μ, ν are exactly as in the quantum Monk's rule for σqσλ.

To prove Theorem 21.1, we compute the action of the vector field Xλ on Wqeq. Note that Wqeq=Wq+W˜qeq, whereW˜qeq=ln(q)(x1+x2++xnk)+i=1n1(xi+1xi)ln(pμi). Recall thatxλ=iVert(λ)xi. In

Declaration of Competing Interest

None.

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      This Landau–Ginzburg mirror symmetry has been established by A. Givental for full flag varieties and projective spaces in [12–14]. The construction for Grassmannians has been performed by R.J. Marsh and K. Rietsch in [23]. A standard problem, related to generalizations of Givental's approach, is to check that the number of critical points of the superpotential equals the dimension of the cohomology algebra of V, and then check that the oscillating integrals, which are intrinsically labeled by the critical points, indeed generate a basis of solutions of the system of quantum differential equations.

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    This work was supported by the Engineering and Physical Sciences Research Council [grant numbers EP/G007497/1 and EP/D071305/1].

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