arXiv subjects

Hanwool Bae

Publications and source records attributed to Hanwool Bae.

11 recordsLinked to original sources

arXiv2026

Rabinowitz Floer homology for Legendrian submanifolds in prequantization bundles

Let $Y$ be a prequantization bundle over an integral symplectic manifold $(\Sigma,\omega)$. Let $L$ be a closed monotone Lagrangian submanifold that admits a Legendrian lift $\mathcal{L}$ in $Y$. Under the assumption that the minimal Maslov number $N_L$ of $L$ is greater than 2, we define the Rabinowitz Floer homology of $\mathcal{L}$. We then establish an isomorphism between the $\mathbb{Z}_d$-equivariant Rabinowitz Floer homology of $\mathcal{L}$ and the quantum homology of $L$, where $d$ is the degree of the covering map $\mathcal{L}\to L$. Under a more restrictive condition on $N_L$, we show that this map is a ring isomorphism. Using this isomorphism, we compute the quantum homology ring of Lagrangian spheres in quadrics and two-step flag manifolds. Furthermore, we investigate the implications of the quantum invertibility of $\omega$ for the vanishing of the quantum homology of $L$ and the obstructions to topologically simple fillings of $\mathcal{L}$. We also show that if $(\Sigma,\omega)$ admits a polarization and $L$ is disjoint from the Lagrangian trace, the quantum homology of $L$ vanishes.

math.SG↗

arXiv2025

Vanishing arcs for isolated plane curve singularities

The variation operator associated with an isolated hypersurface singularity is a classical topological invariant that relates relative and absolute homologies of the Milnor fiber via a non trivial isomorphism. Here we work with a topological version of this operator that deals with proper arcs and closed curves instead of homology cycles. Building on the classical framework of geometric vanishing cycles, we introduce the concept of vanishing arcsets as their counterpart using this geometric variation operator. We characterize which properly embedded arcs are sent to geometric vanishing cycles by the geometric variation operator in terms of intersections numbers of the arcs and their images by the geometric monodromy. Furthermore, we prove that for any distinguished collection of vanishing cycles arising from an A'Campo's divide, there exists a topological exceptional collection of arcsets whose variation images match this collection.

math.GT↗

arXiv2023

Pseudo-Anosov autoquivalances arising from Symplectic topology and their hyperbolic actions on stability conditions

Within $N$-Calabi-Yau categories associated with quivers whose base graphs form trees, we delve into the study of the asymptotic behaviors of autoequivalences of a specific type. These autoequivalences, which we call "Penner type," exhibit straightforward asymptotic characteristics, making them noteworthy exemplars of "pseudo-Anosov" autoequivalences in the sense of \cite{Fan-Filip-Haiden-Katzarkov-Liu21}, and also in a stronger sense that we define in the present paper. In addition, we provide a practical methodology for calculating the stretching factors of Penner type autoequivalences. We expect that this computational approach can have applications. As an example, we establish a positive lower bound on the translation length of the induced action these autoequivalences have on the space of stability conditions. Our anticipation is that this lower bound is, in fact, exact. Notably, we have observed instances of Penner type $\Phi$ where the induced actions align precisely with this lower bound. In other words, these examples induce hyperbolic actions on the space of stability conditions.

math.SG↗

arXiv2023

Floer theory for the variation operator of an isolated singularity

The variation operator in singularity theory maps relative homology cycles to compact cycles in the Milnor fiber using the monodromy. We construct its symplectic analogue for an isolated singularity. We define the monodromy Lagrangian Floer cohomology, which provides categorifications of the standard theorems on the variation operator and the Seifert form. The key ingredients are a special class $\Gamma$ in the symplectic cohomology of the inverse of the monodromy and its closed-open images. For isolated plane curve singularities whose A'Campo divide has depth zero, we find an exceptional collection consisting of non-compact Lagrangians in the Milnor fiber corresponding to a distinguished collection of vanishing cycles under the variation operator.

math.SG↗

arXiv2023

Calabi-Yau structures on Rabinowitz Fukaya categories

In this paper, we prove that the derived Rabinowitz Fukaya category of a Liouville domain $M$ of dimension $2n$ is $(n-1)$-Calabi--Yau assuming the wrapped Fukaya category of $M$ admits an at most countable set of Lagrangians that generate it and satisfy some finiteness condition on morphism spaces between them.

math.SG↗

arXiv2022

Cluster categories from Fukaya categories

We show that the derived wrapped Fukaya category $D^\pi\mathcal{W}(X_{Q}^{d+1})$, the derived compact Fukaya category $D^\pi\mathcal{F}(X_{Q}^{d+1})$ and the cocore disks $L_{Q}$ of the plumbing space $X_{Q}^{d+1}$ form a Calabi--Yau triple. As a consequence, the quotient category $D^\pi\mathcal{W}(X_{Q}^{d+1})/D^\pi\mathcal{F}(X_{Q}^{d+1})$ becomes the cluster category associated to $Q$. One of its properties is a Calabi--Yau structure. Also it is known that this quotient category is quasi-equivalent to the Rabinowitz Fukaya category due to the work of Ganatra--Gao--Venkatesh. We compute the morphism space of $L_{Q}$ in $D^\pi\mathcal{W}(X_{Q}^{d+1})/D^\pi\mathcal{F}(X_{Q}^{d+1})$ using the Calabi--Yau structure, which is isomorphic to the Rabinowitz Floer cohomology of $L_{Q}$.

math.SG↗

arXiv2022

A comparison of categorical and topological entropies on Weinstein manifolds

Let $W$ be a symplectic manifold, and let $\phi:W \to W$ be a symplectic automorphism. Then, $\phi$ induces an auto-equivalence $\Phi$ defined on the Fukaya category of $W$. In this paper, we prove that the categorical entropy of $\Phi$ bounds the topological entropy of $\phi$ from below where $W$ is a Weinstein manifold and $\phi$ is compactly supported. Moreover, being motivated by the work of Cineli, Ginzburg, and Gurel, we propose a conjecture which generalizes a result in dynamical system.

math.SG↗

arXiv2022

On Categorical Entropy from the viewpoint of Symplectic Topology

In this paper, motivated by symplectic topology, we explore categorical entropy and present two main results. The first result establishes a relation between categorical entropies of functors on a category and its localization. Additionally, it demonstrates analogies between the notions of topological and categorical entropy. This result is then applied to symplectic topology, where we provide a method for calculating the categorical entropy of a functor on a (partially) wrapped Fukaya category, assuming that the functor is induced by a compactly supported symplectic automorphism. For the second main result of the paper, we observe the existence of natural examples of symplectic manifolds whose Fukaya categories satisfy a type of Floer-theoretic duality. Motivated by this observation, we prove that categorical entropy can be computed from the morphism spaces under the assumption of duality. The formula is similar to the result of [DHKK14], which is proven for the case of smooth and proper categories.

math.SG↗

arXiv2021

Applications of the theory of Floer to symmetric spaces

We quantize the problem considered by Bott-Samelson who applied Morse theory to any compact symmetric space $G/K$ and the associated real flag manifold $G_{\mathbb{R}}/B$ which is a real locus of a complex partial flag variety $G_{\mathbb{C}}/P_{\sigma}$. We prove that the Pontryagin ring $H_{-*}(\Omega(G/K))$ of the based loop space $\Omega(G/K)$ is isomorphic to the Floer cohomology ring $HF^*(G_{\mathbb{R}}/B,G_{\mathbb{R}}/B)$ after localization. When $G/K$ is a Lie group, this is a conjecture of Peterson, proved combinatorially by Lam-Shimozono, in the context of quantum cohomologies of complex flag varieties. Our approach is geometric in nature: we construct a Lagrangian correspondence from $T^*(G/K)$ to $G_{\mathbb{C}}/P_{\sigma}$ which geometrically composes with a cotangent fiber to $G_{\mathbb{R}}/B$, and compute the linear part of the associated Ma'u-Wehrheim-Woodward's $A_{\infty}$ homomorphism from a Floer model of $\Omega(G/K)$ to $CF^*(G_{\mathbb{R}}/B,G_{\mathbb{R}}/B)$. The crux is to make use of the geometry of $G/K$ to construct specific perturbation data which enables us to reduce the computations to the case when $G/K$ is a torus.

math.SG↗

arXiv2021

Peterson conjecture via Lagrangian correspondences and wonderful compactifications

For a simply-connected compact semisimple Lie group $G$ and its maximal torus $T$, we study the $A_{\infty}$-functor associated to the moment Lagrangian correspondence from the cotangent bundle $T^*G$ to the square $G/T^{-} \times G/T$. In particular, we compute the leading term of the $A_{\infty}$-homomorphism from the wrapped Floer cohomology $HW^*(T^*_e G, T^*_e G)$ of the cotangent fiber $T_e^*G$ to the Floer cohomology $HF^*(\Delta, \Delta)$ of the diagonal $\Delta$ in the square $G/T^{-} \times G/T$ by determining the count of certain pseudo-holomorphic quilts. As a consequence, we prove that the Floer cohomologies $HW^*(T^*_e G, T^*_e G)$ and $HF^*(\Delta,\Delta)$ are isomorphic as rings after a localization.

math.SG↗

arXiv2019

A computation of the ring structure in wrapped Floer homology

We give an explicit computation of the ring structure in wrapped Floer homology of a class of real Lagrangians in $A_k$-type Milnor fibers. In the $A_k$-type plumbing description, those Lagrangians correspond to the cotangent fibers or the diagonal Lagrangians. The main ingredient of the computation is to apply a version of the Seidel representation. For a technical reason, we first carry out computations in v-shaped wrapped Floer homology, and this in turn gives the desired ring structure via the Viterbo transfer map.

math.SG↗