The B-model connection and mirror symmetry for Grassmannians

Abstract

We consider the Grassmannian X = Gr n−k (C n ) and describe a 'mirror dual' Landau-Ginzburg model (X • , Wq :X • → C), whereX • is the complement of a particular anti-canonical divisor in a Langlands dual GrassmannianX, and we express W succinctly in terms of Plücker coordinates. First of all, we show this Landau-Ginzburg model to be isomorphic to the one proposed by the second author in [73]. Secondly we show it to be a partial compactification of the Landau-Ginzburg model defined earlier 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. We conjecture that the free submodule above is isomorphic to the entire Gauss-Manin system (a tameness property for Wq). Our results imply a special case of [73, Conjecture 8.1]. They also imply [6, Conjecture 5.2.3] and resolve [67, Problem 13].

Moment curves and cyclic symmetry for positive Grassmannians

“…The inverse of the right twist τ is the analogously defined left twist. We remark that Marsh and Rietsch employed the twist map to study the superpotential F q [30,Section 12]. This uses an elegant expression of Marsh and Scott [31,Proposition 3.5] for the twist of certain Plücker coordinates; these coordinates happen to include those appearing † Π • k,n is the top-dimensional stratum of the positroid stratification of Gr k,n (C) studied by Knutson, Lam, and Speyer [24].…”

“…There is a connection to the superpotential The function L q was introduced by Eguchi, Hori, and Xiong [9, Appendix B] in constructing a Landau-Ginzburg model for Gr k,n (C). Marsh and Rietsch [30,Proposition 6.10] showed that L q equals the pullback of F q to (C × ) k×(n−k) , under the embedding ι : (C × ) k×(n−k) → Gr k,n (C) given by (This gives the expression for F q which we used in Example 3.13.) As observed by Galashin and Pylyavskyy [16,Section 5], the critical points of L q in (C × ) k×(n−k) are precisely the fixed points of birational rowmotion.…”

Pseudotoric structures and special Lagrangian torus fibrations on certain flag varieties

“…For complex Grassmannians, Marsh and Rietsch constructed a mirror superpotential in terms of the Plücker coordinates in [17], which is isomorphic to the Lietheoretic mirror superpotential for a general homogeneous variety G/P constructed earlier by Rietsch [24]. There should be a construction of the mirror superpotential for F ℓ 1,n−1;n in terms of the Plücker coordinates and similar to that in [17], which we refer to as Rietsch's mirror in the above table. However, the precise expression of such a superpotential, though should be known to experts, is still missing in the literature.…”

“…In the case of Gr(2, n), it has already been shown by Nohara and Ueda [21] that potential functions of the Lagrangian torus fibers of different Gelfand-Cetlin-type fibrations (which correspond to triangulations of the regular n-gon) can be glued (via cluster transformations) to give an open dense subset of Marsh-Rietsch's mirror [17]. But this is still insufficient as the open dense subset misses some of the critical points of Marsh-Rietsch's mirrors.…”

“…

We construct pseudotoric structures (à la Tyurin [26]) on the twostep flag variety F ℓ 1,n−1;n , and explain a general relation between pseudotoric structures and special Lagrangian torus fibrations, the latter of which are important in the study of SYZ mirror symmetry [25]. As an application, we speculate how our constructions can explain the number of terms in the superpotential of Rietsch's Landau-Ginzburg mirror [24,17,22,23].

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