The B-model connection and mirror symmetry for Grassmannians☆
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 (the superpotential) which is defined on a ‘mirror dual’ affine Calabi-Yau variety 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, consider which is an -valued function depending on a single variable q (and a parameter ħ). The integrand involves the superpotential . This is a particular rational function introduced in this paper via an explicit formula in terms of Plücker coordinates on an isomorphic (but Langlands dual) Grassmannian ; see for example (3.1). The integration is over a natural compact torus in an open subvariety of ; see Theorem 4.2. Note that the function in (1.1) is expressed as a linear combination of Schubert classes permuted by Poincaré duality, denoted by PD. By Theorem 4.2 (one of our main results) the function satisfies the flat section equation of the Dubrovin connection, i.e. we have . Here denotes the hyperplane class of X, and 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 for the coefficient of the top class of a flat section of the Dubrovin connection, and employs a Laurent polynomial introduced by Eguchi, Hori and Xiong [14] in place of our . We prove this conjecture by recovering the Laurent polynomial superpotential as a restriction of to a particular open dense torus inside 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 identifying Schubert classes of X with homogeneous coordinates of . This identification comes from the fact that both left-hand and right-hand sides agree with the k-th fundamental representation of , the special linear group which acts on . 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 via intersection cohomology of Schubert varieties of the affine Grassmannian for the Langlands dual group . (The Schubert variety which arises in the construction of 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 , which are precisely the homogeneous spaces which in the geometric Satake correspondence appear as ‘minimal’ Schubert varieties of the affine Grassmannian . 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 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 by an isomorphism which identifies the Schubert class with the element of the Jacobi ring represented by the Plücker coordinate . 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 introduced in this paper, together with the cluster structure on , 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 .
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 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 .
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 . We focus on the example and 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 we denote the Schubert basis of by Here is in and denotes the number of boxes in the Young diagram λ. Furthermore 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 we need to introduce a new Grassmannian . Both X and have dimension . To be more precise, if then we think of as , which is embedded by a Plücker embedding in . Here denotes the vector space dual to , with an action of the Langlands dual from the right. The Plücker coordinates for correspond in a natural way to the Schubert classes in
Isomorphism of -modules
In the B-model, we consider the Gauss-Manin system on . The first main theorem shows that the A-model datum, consisting of together with its small Dubrovin connection, can be recovered inside .
Theorem 4.1 The -module is a –submodule of , and the map is an isomorphism of -modules. Under this isomorphism is identified with and is identified with .
The proof of this theorem hinges on verifying the following formulas for the
Equivariant results
The Grassmannian is a homogeneous space for . In particular, the maximal torus of -diagonal matrices and the Borel subgroups of upper-triangular and lower-triangular matrices, denoted by and respectively, all act on X. In the A-model this means that we can consider the -equivariant cohomology of X and the -equivariant quantum cohomology of X, and we also have a -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 . 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 as a cluster algebra
By [69, Thm. 3], the homogeneous coordinate ring of the Grassmannian 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 . We now recall these constructions.
A skew-symmetric cluster algebra with frozen variables is defined as follows [17, §5]. Let and consider the field of rational functions in indeterminates. A seed in is a pair where
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, and , of the three superpotentials and . Recall that these domains are related by maps see Section 6.2. We begin by recalling how each of the holomorphic volume forms is constructed.
Definition 8.1 The domain of the Laurent polynomial superpotential is naturally a torus.The holomorphic volume form on [27]
Schubert classes and proof of the free basis lemma
The first aim of this section is to show that the isomorphism between and identifies the classes of the Plücker coordinate with the Schubert class . 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 : where μ, ν are exactly as in the quantum Monk's rule for . 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 , we denote by any partition obtained from λ by adding a single box. Our aim in this section is, given , to define a Postnikov diagram containing and all as labels. Moreover, the faces labelled should be adjacent to the face labelled . This diagram will be used later in explicitly computing the action of on . We start by constructing special, symmetric Postnikov diagrams in the case and ; the diagrams for arbitrary can then be
The superpotential written in terms of an arbitrary Plücker extended cluster
Restricting the regular function on to a cluster torus gives a Laurent polynomial in the associated cluster variables. We will need an explicit formula for this in the case where 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 in terms of an arbitrary Postnikov extended cluster. The Laurent polynomial expansion of is given in
Construction of a perfect matching
Suppose and let be the Postnikov diagram constructed by Theorem 11.1. For example, Fig. 4 shows the Postnikov diagram for , noting that . Let be the associated boundary-adjusted Postnikov diagrams and be the corresponding dual bipartite graphs, associated to D in Section 12. We also have corresponding quivers (see Definition 12.1, Definition 12.2). For example, Fig. 16 shows , and for the case .
Our aim in this
How to obtain all perfect matchings from
If or , is the unique perfect matching on , so we assume in this section that . Our main aim is to show that every other perfect matching on can be obtained from 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 of 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
Our main aim in this section is to compute the matching monomial associated to , i.e. its contribution towards the matching polynomial of . This, combined with Theorem 14.20, will be used in Section 17 in order to compute the action of on . We assume in this section that . Recall the definition (Definition 13.3) of the weighting on . We make the following useful definition.
Definition 15.1 Let D be a Postnikov diagram with closure . Let I be a vertex of . For any minimal cycle
The vector fields
In this section we define a family of vector fields , , , on , denoting also by . We allow the cases . We start by defining a vector field in a fixed Postnikov extended cluster containing via an explicit formula involving combinatorially defined coefficients . We then prove that the coefficients satisfy an additivity property which in particular will imply that the vector field extends to a regular vector field on the
Action of the vector field on
Fix a partition . Our main aim in this section is to compute the action of on W for each (Theorem 17.3). We include the cases . Let be the Postnikov diagram associated to λ constructed in Theorem 11.1. Recall that D has a face labelled . Let denote the Postnikov extended cluster associated to D.
We collect together the information we will need. Recall first that, on the cluster torus , the vector field 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: where μ, ν are exactly as in the quantum Monk's rule for .
We first of all note a corollary to Theorem 17.3.
Corollary 18.1 Let λ be an arbitrary Young diagram in . Then we have: where μ, ν are exactly as in the quantum Monk's rule for .
Proof Part (a) is the case 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 in [65]) and describe it in the case of the Grassmannian in terms of Plücker coordinates.
The -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 , and was shown to recover the equivariant quantum cohomology rings in their presentation due to Dale
Action of the vector field: -equivariant case
In this section we prove the formulas needed to complete the proof of Theorem 5.5.
Theorem 21.1 Let . Then we have: and
where, in each case, μ, ν are exactly as in the quantum Monk's rule for .
To prove Theorem 21.1, we compute the action of the vector field on . Note that , where Recall that In
Declaration of Competing Interest
None.
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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].