Floer Theory of Higher Rank Quiver 3-folds

Abstract

We study threefolds Y fibred by $$A_m$$ A m -surfaces over a curve S of positive genus. An ideal triangulation of S defines, for each rank m, a quiver $$Q(\Delta _m)$$ Q ( Δ m ) , hence a $$CY_3$$ C Y 3 -category $$\mathcal {C}(W)$$ C ( W ) for any potential W on $$Q(\Delta _m)$$ Q ( Δ m ) . We show that for $$\omega $$ ω in an open subset of the Kähler cone, a subcategory of a sign-twisted Fukaya category of $$(Y,\omega )$$ ( Y , ω ) is quasi-isomorphic to $$(\mathcal {C},W_{[\omega ]})$$ ( C , W [ ω ] ) for a certain generic potential $$W_{[\omega ]}$$ W [ ω ] . This partially establishes a conjecture of Goncharov (in: Algebra, geometry, and physics in the 21st century, Birkhäuser/Springer, Cham, 2017) concerning ‘categorifications’ of cluster varieties of framed $${\mathbb {P}}GL_{m+1}$$ P G L m + 1 -local systems on S, and gives a symplectic geometric viewpoint on results of Gaiotto et al. (Ann Henri Poincaré 15(1):61–141, 2014) on ‘theories of class $${\mathcal {S}}$$ S ’.

Joyce structures on spaces of quadratic differentials I

“…This is a wellknown problem in higher Teichmüller theory. The expectation is then that the stability space of a CY 3 category of the type discussed in [34,57] should be an open subset of the cotangent bundle of this space.…”

“…Rather, we should consider the holonomy groupoid, which leads to the analytic analogue of a Deligne-Mumford stack [48]. More-or-less by definition, the tangent bundle of Z is then the quotient of that of X × P 1 by the distribution (57). We will ignore this point here, since we only really use Z as a convenient language to describe objects which can easily be defined directly on X.…”