Floer Theory Research Articles

Poisson geometry, monoidal Fukaya categories, and commutative Floer cohomology rings

We describe connections between concepts arising in Poisson geometry and the theory of Fukaya categories. The key concept is that of a symplectic groupoid, which is an integration of a Poisson manifold. The Fukaya category of a symplectic groupoid is monoidal, and it acts on the Fukaya categories of the symplectic leaves of the Poisson structure. Conversely, we consider a wide range of known monoidal structures on Fukaya categories and observe that they all arise from symplectic groupoids. We also use the picture developed to resolve a conundrum in Floer theory: why are some Lagrangian Floer cohomology rings commutative?

On definite lattices bounded by a homology 3–sphere and Yang–Mills instanton Floer theory

On definite lattices bounded by a homology 3–sphere and Yang–Mills instanton Floer theory

On the tau invariants in instanton and monopole Floer theories

AbstractWe unify two existing approaches to the tau invariants in instanton and monopole Floer theories, by identifying , defined by the second author via the minus flavors and of the knot homologies, with , defined by Baldwin and Sivek via cobordism maps of the 3‐manifold homologies induced by knot surgeries. We exhibit several consequences, including a relationship with Heegaard Floer theory, and use our result to compute and for twist knots.

Naturality and functoriality in involutive Heegaard Floer homology

We prove first-order naturality of involutive Heegaard Floer homology, and furthermore, construct well-defined maps on involutive Heegaard Floer homology associated to cobordisms between three-manifolds. We also prove analogous naturality and functoriality results for involutive Floer theory for knots and links. The proof relies on the doubling model for the involution, as well as several variations.

Isolated hypersurface singularities, spectral invariants, and quantum cohomology

Abstract We study the relation between isolated hypersurface singularities (e.g., ADE) and the quantum cohomology ring by using spectral invariants, which are symplectic measurements coming from Floer theory. We prove, under the assumption that the quantum cohomology ring is semi-simple, that (1) if the smooth Fano variety degenerates to a Fano variety with an isolated hypersurface singularity, then the singularity has to be an A m {A_{m}} -singularity, (2) if the symplectic manifold contains an A m {A_{m}} -configuration of Lagrangian spheres, then there are consequences for the Hofer geometry, and that (3) the Dehn twist reduces spectral invariants.

On fillings of contact links of quotient singularities

AbstractWe study several aspects of fillings for links of general isolated quotient singularities using Floer theory, including co‐fillings, Weinstein fillings, strong fillings, exact fillings and exact orbifold fillings, focusing on the non‐existence of exact fillings of contact links of isolated terminal quotient singularities. We provide an extensive list of isolated terminal quotient singularities whose contact links are not exactly fillable, including for , which settles a conjecture of Eliashberg, quotient singularities from general cyclic group actions and finite subgroups of , and all terminal quotient singularities in complex dimension 3. We also obtain uniqueness of the orbifold diffeomorphism type of exact orbifold fillings of contact links of some isolated terminal quotient singularities.

Enumerative geometry of del Pezzo surfaces

We prove an equivalence between the superpotential defined via tropical geometry and Lagrangian Floer theory for special Lagrangian torus fibres in del Pezzo surfaces constructed by Collins-Jacob-Lin [Duke Math. J. 170 (2021), pp. 1291–1375]. We also include some explicit calculations for the projective plane, which confirm some folklore conjectures in this case.

Twisted Sectors for Lagrangian Floer Theory on Symplectic Orbifolds

The notion of twisted sectors play a crucial role in orbifold Gromov-Witten theory. We introduce the notion of dihedral twisted sectors in order to construct Lagrangian Floer theory on symplectic orbifolds and discuss related issues.

Cuplength estimates for time-periodic measures of Hamiltonian systems with diffusion

We show how methods from Hamiltonian Floer theory can be used to establish lower bounds for the number of different time-periodic measures of time-periodic Hamiltonian systems with diffusion. After proving the existence of closed random periodic solutions and of the corresponding Floer curves for Hamiltonian systems with random walks with step width 1/n for every n∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\in \\mathbb {N}$$\\end{document}, we show that, after passing to a subsequence, they converge in probability distribution as n→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\rightarrow \\infty $$\\end{document}. Besides using standard results from Hamiltonian Floer theory and about convergence of tame probability measures, we crucially use that sample paths of Brownian motion are almost surely Hölder continuous with Hölder exponent 0<α<12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\alpha <\\frac{1}{2}$$\\end{document}.

Decategorified Heegaard Floer theory and actions of both $E$ and $F$

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 generators E and F of \mathfrak{gl}(1|1) , whereas the previous ones only involved actions of E . 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.

Involutive knot Floer homology and bordered modules

We prove that, up to local equivalences, a suitable truncation of the involutive knot Floer homology of a knot in S^3 and the involutive bordered Heegaard Floer theory of its complement determine each other. In particular, given two knots K_1 and K_2 , we prove that the \mathbb{F}_2[U,V]/(UV) -coefficient involutive knot Floer homology of K_1 \sharp -K_2 is \iota_K -locally trivial if \widehat{CFD}(S^3 \backslash K_1) and \widehat{CFD}(S^2 \backslash K_2) satisfy a certain condition which can be seen as the bordered counterpart of \iota_K -local equivalence. We further establish an explicit algebraic formula that computes the hat-flavored truncation of the involutive knot Floer homology of a knot from the involutive bordered Floer homology of its complement. It follows that there exists an algebraic satellite operator defined on the local equivalence group of knot Floer chain complexes, which can be computed explicitly up to a suitable truncation.

Evaluations of Link Polynomials and Recent Constructions in Heegaard Floer Theory

Using a definition of Euler characteristic for fractionally-graded complexes based on roots of unity, we show that the Euler characteristics of Dowlin's "$\mathfrak{sl}(n)$-like" Heegaard Floer knot invariants $HFK_n$ recover both Alexander polynomial evaluations and $\mathfrak{sl}(n)$ polynomial evaluations at certain roots of unity for links in $S^3$. We show that the equality of these evaluations can be viewed as the decategorified content of the conjectured spectral sequences relating $\mathfrak{sl}(n)$ homology and $HFK_n$.

Surface Gluing with Signs and Gradings in Decategorified Heegaard Floer Theory

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.

T-equivariant disc potential and SYZ mirror construction

We develop a G-equivariant Lagrangian Floer theory and obtain a curved A∞ algebra, and in particular a G-equivariant disc potential. We construct a Morse model, which counts pearly trees in the Borel construction LG. When applied to a smooth moment map fiber of a semi-Fano toric manifold, our construction recovers the T-equivariant toric Landau-Ginzburg mirror of Givental. We also study the S1-equivariant Floer theory of a typical singular fiber of a Lagrangian torus fibration (i.e. a pinched torus) and compute its S1-equivariant disc potential via the gluing technique developed in [14,37].

Floer theory and reduced cohomology on open manifolds

Floer theory and reduced cohomology on open manifolds

Floer theory of disjointly supported Hamiltonians on symplectically aspherical manifolds

We study the Floer-theoretic interaction between disjointly supported Hamiltonians by comparing Floer-theoretic invariants of these Hamiltonians with the ones of their sum. These invariants include spectral invariants, boundary depth and Abbondandolo-Haug-Schlenk's action selector. Additionally, our method shows that in certain situations the spectral invariants of a Hamiltonian supported in an open subset of a symplectic manifold are independent of the ambient manifold.

The McKay correspondence for isolated singularities via Floer theory

We prove the generalised McKay correspondence for isolated singularities using Floer theory. Given an isolated singularity $\mathbb{C}^n / G$ for a finite subgroup $G \subset SL(n, \mathbb{C})$ and any crepant resolution $Y$, we prove that the rank of positive symplectic cohomology $SH^{\ast}_{+} (Y)$ is the number $\lvert \operatorname{Conj}(G) \rvert$ of conjugacy classes of $G$, and that twice the age grading on conjugacy classes is the $\mathbb{Z}$-grading on $SH^{\ast-1}_{+} (Y)$ by the Conley–Zehnder index. The generalized McKay correspondence follows as $SH^{\ast-1}_{+} (Y)$ is naturally isomorphic to ordinary cohomology $H^\ast (Y)$, due to a vanishing result for full symplectic cohomogy. In the Appendix we construct a novel filtration on the symplectic chain complex for any non-exact convex symplectic manifold, which yields both a Morse–Bott spectral sequence and a construction of positive symplectic cohomology.

Twisted Mazur Pattern Satellite Knots &amp; Bordered Floer Theory

We use bordered Floer theory to study properties of twisted Mazur pattern satellite knots Qn(K). We prove that Qn(K) is not Floer homologically thin, with two exceptions. We calculate the 3-genus of Qn(K) in terms of the twisting parameter n and the 3-genus of the companion K, and we determine when Qn(K) is fibered. As an application to our results on Floer thickness and 3-genus, we verify the Cosmetic Surgery Conjecture for many of these satellite knots.

Time-periodic solutions of Hamiltonian PDEs using pseudoholomorphic curves

We extend the pseudoholomorphic curve methods from Floer theory to infinite-dimensional phase spaces and use our results to prove the existence of a forced time-periodic solution to a general Hamiltonian PDE with regularizing nonlinearity. In particular, when the nonlinearity is sufficiently regularizing, bounded and time-periodic, we prove an infinite-dimensional version of Gromov-Floer compactness by using ideas from the theory of Diophantine approximations to overcome the small divisor problem. Furthermore, in the case when the infinite-dimensional phase space is a product of a finite-dimensional closed symplectic manifold with linear symplectic Hilbert space, we prove a cup-length estimate for the number of periodic solutions.

Wrapped sheaves

We construct a sheaf-theoretic analogue of the wrapped Fukaya category in Lagrangian Floer theory, by localizing a category of sheaves microsupported away from some given Λ⊂S⁎M along continuation maps constructed using the Guillermou-Kashiwara-Schapira sheaf quantization.When Λ is a subanalytic singular isotropic, we also construct a comparison map to the category of compact objects in the category of unbounded sheaves microsupported in Λ, and show that it is an equivalence. The last statement can be seen as a sheaf-theoretical incarnation of the sheaf-Fukaya comparison theorem of Ganatra-Pardon-Shende.