Search for: All records

Abstract We relate the Fukaya category of the symmetric power of a genus zero surface to deformed category$$\mathcal {O}$$ $O$of a cyclic hypertoric variety by establishing an isomorphism between algebras defined by Ozsváth–Szabó in Heegaard–Floer theory and Braden–Licata–Proudfoot–Webster in hypertoric geometry. The proof extends work of Karp–Williams on sign variation and the combinatorics of the$$m=1$$ $m=1$amplituhedron. We then use the algebras associated to cyclic arrangements to construct categorical actions of$$\mathfrak {gl}(1|1)$$ $\mathrm{gl}\left(1\right|1)$, and generalize our isomorphism to give a conjectural algebraic description of the Fukaya category of a complexified hyperplane complement. 

Abstract A previous result about the decategorified bordered (sutured) Heegaard Floer invariants of surfaces glued together along intervals, generalizing the decategorified content of Rouquier and the author’s higher-tensor-product-based gluing theorem in cornered Heegaard Floer homology, was proved only over $${\mathbb{F}}_2$$ and without gradings. In this paper we add signs and prove a graded version of the interval gluing theorem over $${\mathbb{Z}}$$, enabling a more detailed comparison of these aspects of decategorified Heegaard Floer theory with modern work on non-semisimple 3d TQFTs in mathematics and physics. 

We define larger variants of the vector spaces one obtains by decategorifying bordered (sutured) Heegaard Floer invariants of surfaces. We also define bimodule structures on these larger spaces that are similar to, but more elaborate than, the bimodule structures that arise from decategorifying the higher actions in bordered Heegaard Floer theory introduced by Rouquier and the author. In particular, these new bimodule structures involve actions of both odd generatorsEandFof\mathfrak{gl}(1|1), whereas the previous ones only involved actions ofE. Over\mathbb{F}_2, we show that the new bimodules satisfy the necessary gluing properties to give a 1+1 open-closed TQFT valued in graded algebras and bimodules up to isomorphism; in particular, unlike in previous related work, we have a gluing theorem when gluing surfaces along circles as well as intervals. Over the integers, we show that a similar construction gives two partially defined open-closed TQFTs with two different domains of definition depending on how parities are chosen for the bimodules. We formulate conjectures relating these open-closed TQFTs with the\mathfrak{psl}(1|1)Chern–Simons TQFT recently studied by Mikhaylov and Geer–Young.