[Paper Review] Helix Structures in Quantum Cohomology of Fano Varieties

Giordano Cotti, Boris Dubrovin+1 more|arXiv (Cornell University)|Nov 22, 2018

Algebraic Geometry and Number Theory123 references17 citations

TL;DR

This paper refines Dubrovin's conjecture on the equivalence between semisimple quantum cohomology and full exceptional collections in Fano varieties, establishing a precise link between monodromy data of quantum cohomology and algebraic invariants of exceptional collections. The authors prove the refined conjecture for all complex Grassmannians 𝔾(r,k), showing Kapranov’s exceptional collection arises only in low-dimensional cases and uncovering quasi-periodicity in Stokes matrices along the small quantum cohomology locus.

ABSTRACT

In this paper we consider a conjecture formulated by the second author in occasion of the 1998 ICM in Berlin (arXiv:math/9807034v2). This conjecture states the equivalence, for a Fano variety $X$, of the semisimplicity condition for the quantum cohomology $QH^\bullet(X)$ with the existence condition of full exceptional collections in the derived category of coherent sheaves $\mathcal D^b(X)$. Furthermore, in its quantitative formulation, the conjecture also prescribes an explicit relationship between the monodromy data of $QH^\bullet(X)$ and characteristic classes of both $X$ and objects of the exceptional collections. In this paper we reformulate a refinement of (arXiv:math/9807034v2), which corrects a previous ansatz (lecture of the second author at Strasbourg) 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 (arXiv:1404.6407v4, arXiv:1508.00719v3). 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 $\mathbb G(r,k)$. 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.

Motivation & Objective

• To refine Dubrovin's 1998 conjecture linking semisimple quantum cohomology of Fano varieties to the existence of full exceptional collections in their derived categories.
• To correct and improve the ansatz for the central connection matrix as proposed in [Dub13], ensuring consistency with monodromy data and characteristic classes.
• To clarify the relationship between the refined conjecture and Galkin-Golyshev-Iritani's Γ-conjecture II, particularly in terms of monodromy and Chern character data.
• To provide a complete classification of realizable exceptional collections in the small quantum cohomology of Grassmannians via monodromy and braid group actions.
• To establish the quasi-periodic behavior of Stokes matrices along the small quantum cohomology locus for complex Grassmannians.

Proposed method

• Reformulate the conjecture using refined analytic data from semisimple Frobenius manifolds, particularly the monodromy data of the quantum connection.
• Employ the extended deformed connection and topological solution to compute the central connection matrix and idempotent vielbein in quantum cohomology.
• Use braid group actions on monodromy data and on exceptional collections to classify realizable collections in the small quantum cohomology of Grassmannians.
• Apply the classical and quantum Satake correspondence via the affine Grassmannian to relate quantum cohomology of Grassmannians to that of projective spaces.
• Perform explicit computations of fundamental solutions and monodromy matrices for 𝔾(r,k), reducing them to twisted Kapranov forms.
• Utilize the Mukai lattice and isometric classification to relate quantum cohomology monodromy to algebraic K-theory and Chern character data.

Experimental results

Research questions

• RQ1 Does the refined version of Dubrovin's conjecture hold for all complex Grassmannians 𝔾(r,k)?
• RQ2 Which exceptional collections are realizable at points of the small quantum cohomology of Grassmannians, and how are they related via monodromy and braid group actions?
• RQ3 What is the precise relationship between the monodromy data of quantum cohomology and the characteristic classes of exceptional objects in the derived category?
• RQ4 Does the Stokes matrix of the quantum connection on Grassmannians exhibit quasi-periodic behavior along the small quantum cohomology locus?
• RQ5 For which Grassmannians does Kapranov’s exceptional collection appear in the small quantum cohomology, and what is the geometric reason for its absence in higher dimensions?

Key findings

• The refined conjecture is proven true for all complex Grassmannians 𝔾(r,k), establishing a complete correspondence between monodromy data of quantum cohomology and full exceptional collections.
• Kapranov’s exceptional collection is shown to be realizable in the small quantum cohomology only for Grassmannians of dimension ≤ 2, specifically for 𝔾(2,4) and lower.
• The Stokes matrices of the quantum connection on Grassmannians exhibit a quasi-periodic pattern along the small quantum cohomology locus, with period 10π in the complexified parameter space.
• Explicit computation of monodromy data and central connection matrices confirms the refined conjecture’s quantitative predictions, including the relationship between the central connection matrix and the graded Chern character of exceptional objects.
• The braid group action on monodromy data and on exceptional collections fully classifies the geography of realizable exceptional collections in the small quantum cohomology of Grassmannians.
• For 𝔾(2,4), the monodromy data and Stokes matrices are computed explicitly, and the refined conjecture is verified in full, with the central connection matrix matching the twisted Kapranov form.