Floer theory of higher rank quiver \(3\)-folds
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. \textit{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 \textit{E. Abrikosov} [``Potentials for moduli spaces of \(A_m\)-local systems on surfaces'', Preprint, \url{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 \textit{D. Gaiotto} et al. [Ann. Henri Poincaré 15, No. 1, 61--141 (2014; Zbl 1301.81262)] on `theories of class \(\mathcal{S}\)'.
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