Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2407.16358

Partially wrapped Fukaya categories of orbifold surfaces

Severin Barmeier·Sibylle Schroll·Zhengfang Wang
View original source ↗Read source document ↗Export metadata ↓

Abstract

We give a complete description of partially wrapped Fukaya categories of graded orbifold surfaces with stops. We show that a construction via global sections of a natural cosheaf of A$_\infty$ categories on a Lagrangian core of the surface is equivalent to a global construction via the (equivariant) orbit category of a smooth cover. We therefore establish the local-to-global properties of partially wrapped Fukaya categories of orbifold surfaces closely paralleling a proposal by Kontsevich for Fukaya categories of smooth Weinstein manifolds. From the viewpoint of Weinstein sectorial descent in the sense of Ganatra, Pardon and Shende, our results show that orbifold surfaces also have Weinstein sectors of type $\mathrm D$ besides the type $\mathrm A$ or type $\widetilde{\mathrm A}$ sectors on smooth surfaces. We describe the global sections of the cosheaf explicitly for any generator given by an admissible dissection of the orbifold surface and we give a full classification of the formal generators which arise in this way. This shows in particular that the partially wrapped Fukaya category of an orbifold surface can always be described as the perfect derived category of a graded associative algebra. We conjecture that associative algebras obtained from dissections of orbifold surfaces form a new class of associative algebras closed under derived equivalence.

Explore related subjects

math.SGmath.RAmath.RT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Severin Barmeier, Sibylle Schroll, Zhengfang Wang. 2024-07-23. Partially wrapped Fukaya categories of orbifold surfaces. https://arxiv.org/abs/2407.16358

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Tightness of Chekanov's bound on displacement energy for some Lagrangian knots

By a classical theorem of Chekanov, the displacement energy, $e$, of a Lagrangian submanifold is bounded from below by the minimal area, $\hbar$, of pseudo-holomorphic disks with boundary on the Lagrangian. We compute $e$ and $\hbar$ for displaceable Chekanov tori in $\mathbb{C}P^n$, and for an infinite family of exotic tori in $\mathbb{C}^3$ constructed by Brendel. In these families, $e=\hbar$. We compare continuity properties of $e$ and $\hbar$ on the space of Lagrangians. This provides an example (suggested by Fukaya, Oh, Ohta, and Ono) where $e>\hbar$. Our calculations have further applications such as a new proof, inspired by work of Auroux, that Brendel's family of exotic tori consists of infinitely many distinct Lagrangians.

math.SG↗
arXiv2026

On intrinsic homological mirror symmetry for toric degenerations

This paper studies the Floer-theoretic aspects of homological mirror symmetry inspired by proposals of Perutz and Siebert and the Gross--Siebert intrinsic mirror symmetry program. Given a maximally unipotent degeneration of smooth projective Calabi--Yau manifolds over the punctured disk, we construct a ring using the fixed point Floer cohomology groups of the iterates of the monodromy of the degeneration equipped with the pair of pants product. Under the assumption that this ring is commutative, we can consider a candidate mirror family defined by the relative Proj construction. Further assuming that a smooth fiber $X_t$ contains a so-called tropical Lagrangian section, we construct a fully faithful embedding from the derived category of perfect complexes on our candidate mirror family into the Fukaya category of $X_t$. We verify both of these assumptions for certain Batyrev--Borisov toric degenerations, as well as some toric degenerations of Calabi--Yau threefolds coming from the Gross--Siebert reconstruction algorithm. These two geometric hypotheses are both phrased to support the general study of mirror symmetry for maximally unipotent degenerations of Calabi--Yau manifolds, largely reducing the symplectic inputs for proving homological mirror symmetry to the problem of constructing tropical Lagrangian sections.

math.SG↗
arXiv2026

$b^k$-Symplectic Manifolds and $[Q,R]=0$

We study the geometric quantization of $b^k$-symplectic manifolds using the integrability of Lie algebroids. Using a groupoid index, we define a quantization for $b^k$-symplectic manifolds whose singular locus is a normal crossing divisor and which carry a Hamiltonian action of a compact connected Lie group, generalizing Guillemin--Miranda--Weitsman in a few directions. Firstly, we show this quantization is the index of a $\spinc$-Dirac operator, answering a question of theirs. In particular, it is a finite-dimensional virtual representation for every $k$, whereas their formal quantization is infinite-dimensional when the modular degrees are even. Secondly, our symplectic form can have singularities along hypersurfaces which can have normal crossings. Finally, we prove that quantization commutes with reduction for the Hamiltonian action of a possibly non-abelian compact connected Lie group, when the modular degrees are odd. In the case when the modular degrees are not odd, we give an example when $[Q,R]=0$ fails.

math.SG↗