Homological mirror symmetry for higher dimensional pairs of pants

Abstract

Using Auroux's description of Fukaya categories of symmetric products of punctured surfaces, we compute the partially wrapped Fukaya category of the complement of k + 1 generic hyperplanes in CP n , for k ≥ n, with respect to certain stops in terms of the endomorphism algebra of a generating set of objects. The stops are chosen so that the resulting algebra is formal. In the case of the complement of (n + 2)-generic hyperplanes in CP n (n-dimensional pair-of-pants), we show that our partial wrapped Fukaya category is equivalent to a certain categorical resolution of the derived category of the singular affine variety x 1 x 2 ..x n+1 = 0. By localizing, we deduce that the (fully) wrapped Fukaya category of the n-dimensional pair-of-pants is equivalent to the derived category of x 1 x 2 ...x n+1 = 0. We also prove similar equivalences for finite abelian covers of the n-dimensional pair-of-pants.

From hypertoric geometry to bordered Floer homology via the m=1 amplituhedron

“…Recall that the algebras B l (n, k), B r (n, k), B(n, k), and B ′ (n, k) have interpretations as the homology of (formal) dg strands algebras A(Z, k) of the form appearing in bordered Floer homology [LP18,MMW19b]. Here Z is an arc diagram as considered by Zarev [Zar11], except that instead of matchings on a collection of oriented intervals, we allow matchings on a collection of oriented intervals and circles.…”

“…The case of interest. When discussing bimodules from Heegaard diagrams in this section, we note that based on the most literal extensions of the bordered sutured theory to this case, the Heegaard diagrams would actually produce bimodules over the dg strands versions of the algebras A(Z l ) [LP18,MMW19b], which are formal with homology B l (n, k). Ozsváth-Szabó's methods skip the dg step and interpret holomorphic disk counts directly in terms of algebras like B l (n, k); we follow their approach here.…”

“…We prove our conjecture in an interesting special case, namely when the arrangement V is cyclic, by identifying B(V) with a known description of the Fukaya category in this case [OSz18,LP18,MMW19b,Aur10]. Cyclic arrangements are of independent interest for at least three reasons, each of which we discuss later in this section:…”

“…In particular, by [Aur10] the wrapped Fukaya categories of these symmetric products underlie bordered Floer homology [LOT18], the extended TQFT approach to Heegaard Floer homology. When the surface is a multiply-punctured disc, the results of [LP18,MMW19b] together with those of [Aur10] imply that the Fukaya category for a certain sutured structure on the surface (determining stops for the wrapping) is described by an algebra used recently by Ozsváth-Szabó for very fast knot Floer homology (HFK) computations in their theory of bordered HFK [OSz18,OSz19b]. Conjecture 1.1 for cyclic arrangements is then implied by the following theorem, which we prove in Section 4.…”

Strands algebras and the affine highest weight property for equivariant hypertoric categories

“…In fact, the Ozsváth-Szabó algebras we have been studying are also related to strands algebras A(Z), this time for the arc diagram Z can n shown in Figure 4. As shown in [LP18] and independently in [MMW19b], Ozsváth-Szabó's algebra B l (n, k) is the homology of A(Z can n ), which is a formal dg algebra (similar results hold for the other variants of the Ozsváth-Szabó algebra). 8.1.2.…”