Floer theory of higher rank quiver 3-folds

Abstract

We study 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 $C{Y}_3$-category $(C,W)$ for any potential $W$ on $Q({\Delta }_m)$. We show 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 concerning `categorifications' of cluster varieties of framed $PG{L}_{m+1}$-local systems on $S$, and gives a symplectic geometric viewpoint on results of Gaiotto, Moore and Neitzke.

23 pages, 14 figures

References in corpus (3)

• The representation type of Jacobian algebras
• Quiver algebras as Fukaya categories
• Algebraicity and Asymptotics: An explosion of BPS indices from algebraic generating series

Cited by in corpus (1)

• A-branes, foliations and localization