Floer theory of higher rank quiver \(3\)-folds. (English) Zbl 1490.53101
The author studies threefolds \(Y\) fibred by \(A_m\)-surfaces over a curve \(S\) of positive genus. An ideal triangulation of \(S\) defines, for each rank \(m\), a quiver \(Q(\Delta_m)\), hence a \(CY_3\)-category \(C(W)\) for any potential \(W\) on \(Q(\Delta_m)\). The author shows that for \(\omega\) in an open subset of the Kähler cone, a subcategory of a sign-twisted Fukaya category of \((Y,\omega)\) is quasi-isomorphic to \((C,W_{[\omega]})\) for a certain generic potential \(W_{[\omega]}\). This partially establishes a conjecture of Goncharov. A. B. Goncharov conjectured in [Prog. Math. 324, 31–97 (2017; Zbl 1392.13007)] that the \(CY_3\)-category associated to \(Q(\Delta_m)\) and the ‘canonical’ potential \(W = W(\Delta_m)\) on the underlying bipartite graph should be realised as a subcategory of a Fukaya category. Goncharov’s conjecture, stemming from general expectations around ‘categorifications’ of cluster varieties, was further elaborated by E. Abrikosov [“Potentials for moduli spaces of \(A_m\)-local systems on surfaces”, Preprint, arXiv:1803.06353]; the result in the paper under review proves the formulation given there. In addition, it gives a symplectic geometric viewpoint on results of D. Gaiotto et al. [Ann. Henri Poincaré 15, No. 1, 61–141 (2014; Zbl 1301.81262)] on ‘theories of class \(\mathcal{S}\)’.
Reviewer: Roman Golovko (Praha)
MSC:
| 53D40 | Symplectic aspects of Floer homology and cohomology |
| 53D12 | Lagrangian submanifolds; Maslov index |
| 81T60 | Supersymmetric field theories in quantum mechanics |
| 16G20 | Representations of quivers and partially ordered sets |
| 13F60 | Cluster algebras |
| 32Q28 | Stein manifolds |
| 53D37 | Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category |
| 14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
| 32B25 | Triangulation and topological properties of semi-analytic and subanalytic sets, and related questions |
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