Stokes matrices for the quantum cohomologies of a class of orbifold projective lines

Iwaki, Kohei; Takahashi, Atsushi

doi:10.1063/1.4823483

Cited by 6 publications

(3 citation statements)

References 17 publications

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“…, E m of D b (X) such that the Stokes matrix of (H X , ∇) at z = 0 is equal to the matrix (χ(E i , E j )) i,j , where χ(E, F ) := k (−1) k dim Hom(E, F [k]) for objects E, F ∈ D b (X). Conjecture 1.1 has been proved for some X with semisimple quantum cohomology rings [13], [21], [29], [54], [55], etc.…”

Section: Dubrovin's Conjecturementioning

confidence: 99%

An analogue of Dubrovin’s conjecture

Sanda¹,

Shamoto²

2020

Annales De l'Institut Fourier

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Section: Dubrovin's Conjecturementioning

confidence: 99%

An analogue of Dubrovin’s conjecture

Sanda¹,

Shamoto²

2020

Annales De l'Institut Fourier

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“…This is consistent with the Dubrovin's conjecture [39] for the Stokes matrix of the quantum cohomology of CP 1 . The conjecture was proved by D. Guzzetti [58] for projective spaces, K. Ueda [96,97] for Grassmannians and smooth cubic surfaces; see also [73,51,52,92] for related works on the conjecture.…”

Section: 2mentioning

confidence: 99%

Reconstructing GKZ via Topological Recursion

Fuji

Iwaki

Manabe

et al. 2019

Commun. Math. Phys.

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In this article, a novel description of the hypergeometric differential equation found from Gel'fand-Kapranov-Zelevinsky's system (referred to GKZ equation) for Givental's J-function in the Gromov-Witten theory will be proposed. The GKZ equation involves a parameter , and we will reconstruct it as the WKB expansion from the classical limit → 0 via the topological recursion. In this analysis, the spectral curve (referred to GKZ curve) plays a central role, and it can be defined as the critical point set of the mirror Landau-Ginzburg potential. Our novel description is derived via the duality relations of the string theories, and various physical interpretations suggest that the GKZ equation is identified with the quantum curve for the brane partition function in the cohomological limit. As an application of our novel picture for the GKZ equation, we will discuss the Stokes matrix for the equivariant CP 1 model and the wall-crossing formula for the total Stokes matrix will be examined. And as a byproduct of this analysis we will study Dubrovin's conjecture for this equivariant model.

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“…(10) In [IT13], K. Iwaki and A. Takahashi proved validity of point (2) of Conjecture 5.1 for a class of orbifold projective lines P 1 A whose quantum cohomology is known to be semisimple, being isomorphic to the Frobenius manifold constructed from the theory of primitive forms for the polynomial for the symplectic isotropic Grassmannian IG(2, 6). The importance of this result is due to the fact that it underlines the need of considering the whole big quantum cohomology for the formulation of the conjecture, the small quantum locus being contained in the caustic (recall Definition 2.16).…”

Section: The Main Conjecturementioning

confidence: 99%

Helix Structures in Quantum Cohomology of Fano Varieties

Cotti,

Dubrovin,

Guzzetti

2018

Preprint

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In this paper we consider a conjecture formulated by the second author in occasion of the 1998 ICM in Berlin ([Dub98]). This conjecture states the equivalence, for a Fano variety X, of the semisimplicity condition for the quantum cohomology QH • (X) with the existence condition of full exceptional collections in the derived category of coherent sheaves D b (X). Furthermore, in its quantitative formulation, the conjecture also prescribes an explicit relationship between the monodromy data of QH • (X) and characteristic classes of both X and objects of the exceptional collections. In this paper we reformulate a refinement of [Dub98], which corrects the ansatz of [Dub13] for what concerns the conjectural expression of the central connection matrix. We clarify the precise relationship between the refined conjecture presented in this paper and Γ-conjecture II of S. Galkin, V. Golyshev and H. Iritani ([GGI16, GI15]). Through an explicit computation of the monodromy data and a detailed analysis of the action of the braid group on both the monodromy data and the set of exceptional collections, we prove the validity of our refined conjecture for all complex Grassmannians G(r, k). From these results, it is outlined an explicit description of the "geography" of the exceptional collections realizable at points of the small quantum cohomology of Grassmannians, i.e. corresponding to the monodromy data at these points. In particular, it is proved that Kapranov's exceptional collection appears at points of the small quantum cohomology only for Grassmannians of small dimension (namely, less or equal than 2). Finally, a property of quasi-periodicity of the Stokes matrices of complex Grassmannians, along the locus of the small quantum cohomology, is described.

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Stokes matrices for the quantum cohomologies of a class of orbifold projective lines

Cited by 6 publications

References 17 publications

An analogue of Dubrovin’s conjecture

An analogue of Dubrovin’s conjecture

Reconstructing GKZ via Topological Recursion

Helix Structures in Quantum Cohomology of Fano Varieties

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