Papers

Non-degenerate periodic orbits in three-dimensional Reeb flows are classified into three types: positive hyperbolic, negative hyperbolic and elliptic. In the present paper, we consider a closed connected contact three-manifold with a non-degenerate contact form. We show that its Reeb flow has a simple positive hyperbolic orbit if it has at least three simple periodic orbits. We mainly study the case in which no elliptic orbit exists. We prove that there is no non-degenerate contact form on a closed connected three-manifold with $b_1=0$ such that all simple periodic orbits are negative hyperbolic. As a corollary, by combining the author's previous result in the presence of an elliptic orbit and the known result for $b_1>0$, we obtain the main result. The proof uses the Weyl law for ECH spectral invariants. We also use compactness for genus zero $J$-holomorphic curves counted by the $U$-map. Under the contrary assumption, the number of simple orbits in each action interval $[L,2L]$ is uniformly bounded with respect to $L$. We use this property to study ECH generators and genus zero $U$-curves.

On a symplectic manifold, any differential two-form has a natural decomposition into two components: a primitive part and a non-primitive one. Applying this decomposition to the curvature two-form of a principal bundle over a symplectic manifold, we obtain a natural splitting of the Yang-Mills (YM) functional into two functionals that intrinsically depend on the symplectic structure: the primitive Yang-Mills (PYM) functional and the trace Yang-Mills (TYM) functional. We work out the basic properties of the critical solutions of these two functionals. The PYM functional in particular exhibits many of the desirable properties of the YM functional, including the ellipticity of its Euler-Lagrange equations and an algebraic classification of its flat solutions on G-bundles. We also prove a monotonicity formula for the PYM functional as a first step towards characterizing its moduli space of solutions.

We study a cut-and-paste operation in which torus neighborhoods in the author's symplectic building blocks $Y_K$ and $X_K$, associated to a genus-one fibered knot $K$, are replaced by marked exteriors of tori in $S^4$ with infinite cyclic complement group and two null peripheral slopes. We also recall the exact Luttinger-surgery realization of $M_K\times S^1$ from $\Sigma_g\times T^2$, keeping the knot-surgery/fiber-sum and Luttinger-surgery viewpoints in the same framework. For the trefoil block $Y_K$, two marked replacements give a simply connected manifold with intersection form $H$, hence a manifold homeomorphic to $S^2\times S^2$. A one-exterior gluing gives a smooth homotopy $4$-sphere. For the rank-six construction we use the identity double of two copies of $Y_K\setminus\nu\Sigma_2$, rather than the involutive gluing defining the original $X_K$. Two marked replacements along the surviving rim tori give a simply connected manifold with $e=8$ and $\sigma=0$. An explicit geometric basis has intersection form $3H$, so the resulting manifold is homeomorphic to $\#_3(S^2\times S^2)$. The Case II small-perturbation Seiberg–Witten invariant vanishes. The Seiberg–Witten invariant of the identity-glued rank-six family also vanishes; in particular, these manifolds are nonsymplectic.

We propose a new class of sigma models that extend AKSZ sigma models, which we refer to as gauged AKSZ sigma models. These models are constructed using QP-manifolds. The compatibility between the AKSZ structure and gauging is characterized by geometric conditions on the target graded manifold. As an example, we explicitly analyze the gauging of higher Courant sigma models in higher dimensions. When fluxes and boundaries are introduced into gauged higher Courant sigma models, generalizations of momentum maps and Hamiltonian $G$-spaces arise as consistency conditions for the models.

A $\mu$-constant deformation of isolated complex hypersurface singularities of dimensions different from 2 preserves the diffeomorphism class of the Milnor fiber. If it would be known that $\mu$-constant deformations have no vanishing folds as defined in by O'Shea, this result is even true in all dimensions since the members in the family would have a common Milnor ball-radius. In this note a symplectic version of this is shown which in particular applies to weighted homogeneous polynomial singularities.

Moment Lagrangian $L_\mu$ is a Lagrangian in $T^*G^- \times Y^- \times Y$ associated to a Hamiltonian $G$-space $Y$ with a moment map $\mu$. In this paper, we prove that $L_\mu$ is tautologically unobstructed under mild assumptions on $Y$. As a key ingredient in the proof, we constructed a new symplectic groupoid structure on $T^*G^- \times Y^- \times Y$ over $G\times Y$ for which $L_\mu$ is simultaneously the unit and the fixed locus of the inversion, which might be of independent interest.

We extend the sphere-summing construction of \cite{AZ} from torus sums of spheres to symplectic sums along surfaces of arbitrary genus. The relative surfaces may have arbitrary positive intersection number with the summing surface. We give formulas for the genus and self-intersection of the resulting surface, together with graph and simultaneous versions of the construction. The original elliptic-surface case is recovered as a special case, and higher-genus examples are obtained from hyperelliptic Lefschetz fibrations and braided symplectic surfaces in $\Sigma_h\times S^2$.

We construct symplectic four-manifolds by taking mixed fiber sums along explicit cyclic multisections in ruled surfaces. For a connected unbranched degree-$p$ multisection in $\Sigma_g\times S^2$, we determine the first homology and fundamental group of the complement and prove that its boundary is incompressible. It follows that no direct gluing of two such complements can be simply connected; moreover, the first homology of every direct sum retains finite quotients determined by the covering degrees. We classify the mixed sums having Euler characteristic $4$ and signature $0$. Up to interchanging the two summands, exactly three possibilities occur, corresponding to the degree pairs $(2,3)$, $(2,4)$, and $(3,3)$. For each of these cases, suitable adapted product-framed symplectic gluings have the rational cohomology ring of $S^2\times S^2$. Varying the gluing by symplectic transvections produces infinitely many pairwise nondiffeomorphic examples, distinguished by the unbounded orders of their finite first homology groups. We also construct the twisted ruled analogue of the $(2,4)$ case. Explicit finite-holonomy multisections give connected square-zero symplectic surfaces in the classes $2S_3-F_3$ and $4S_2-2F_2$ in the nontrivial $S^2$-bundles over $\Sigma_3$ and $\Sigma_2$, respectively. More generally, for a square-zero degree-$p$ multisection in the nontrivial bundle the complement has first homology $\mathbb Z^{2g}\oplus\mathbb Z/(p/2)$. Suitable gluings in the twisted $(2,4)$ case have $b_1=0$, $b_2=2$, and signature zero, and every such sum is non-spin. Hence they have the rational cohomology ring of $\mathbb CP^2\#\overline{\mathbb CP}^{\,2}$. We compare these constructions with the author's 2006 construction of minimal symplectic four-manifolds having the integral cohomology $S^2\times S^2$, obtained via knot surgery and twisted fiber sums.

We ask how much of the geometry a charged particle moves in is fixed by its motion, and how much a particle must bring. We ask of a symplectic structure only that it relate velocity to momentum as Hamilton's equations do, and we ask it of every energy at once. In particular, we show that the structures meeting that demand are the canonical one and its twists by a closed two-form of the base. A field provides the two-form, a particle the multiplier before it, which we identify constitutively with its charge. Thus, a single energy governs a family of structures, and each particle takes the one its charge fixes. We then ask what a particle must bring to be given a momentum, and we show that the degree of that map settles the degree at which a field enters Newton's Second Law. An antisymmetric bilinear form returns no Lorentz force, whilst a Randers metric returns one — a length whose difference from a Riemannian one is linear in the velocity. Moreover, we find that metric already within the twisted structure, as its primitive over a level of the free energy, and its law of transport to be nonlinear, no affine connection being known to serve. Under an indefinite signature the length parts from the dynamics, and the extremals turn from shortest to longest. On the round sphere a monopole flux leaves no such metric, whilst the transport remains and prequantisation, given a unit of action, restricts the charge to a lattice. In this manner, we conclude that each charge-to-mass ratio receives a geometry of its own, so that by a functionalist criterion none of them is the spacetime of a charged particle.

Let $\Lambda$ be a compact Bohr–Sommerfeld Lagrangian submanifold of a compact Kähler manifold equipped with a holomorphic prequantum line bundle. We study the asymptotic expansion of the Lagrangian states associated with $\Lambda$. In particular, we compute explicitly the first nontrivial correction term and show that it is expressed in terms of geometric invariants of the ambient Kähler manifold and the Lagrangian submanifold, including their scalar curvatures, the second fundamental form, and the mean curvature. As a consequence, we obtain the corresponding second-order asymptotic formula for the $L^2$-norm of the Lagrangian states.

This paper is a sequel to the authors' paper on rigidity of higher dimensional contact Anosov flows~[GRH]. At the time the authors were unaware of much earlier work of Hamenstädt~[Ham] devoted to the same problem. The methods of~[GRH] and~[Ham] are overlapping but not entirely the same. In this paper we strengthen some the results of Hamenstädt, by further bootstrapping the regularity of the conjugacy of 2-pinched contact Anosov flows. We also introduce a notion of symplecto-rigidity for symplectomorphisms and prove that some Anosov diffeomorphisms are symplecto-rigid. We further combine symplecto-rigidity with various rigidity techniques to establish a number of smooth rigidity results of Anosov symplectomorphisms. Some new phenomena are present in the realm of Anosov symplectomorphism. For example, while the well-known de la Llave example on the 4-torus demonstrates absence of rigidity in the space of smooth Anosov diffeomorphisms, however, we prove that it is rigid in the space of Anosov sympelctomorphisms.

Let $\Omega$ be the Lorentz future cone in $\mathbb{R}^{d+1}$ with respect to the Lorentz product and let $T^M$ be the $M$-fold product of the future tube $T=\mathbb{R}^{d+1}+i\Omega$. The Lorentz group $\mathrm{SO}_0(1,d)$ acts diagonally on $T^M$, and its complexification $\mathrm{SO}(1,d)^\mathbb{C}$ acts on $\mathbb{C}^{(d+1)\times M}$. We prove that the domain $G^\mathbb{C}\cdot T^M$ is a Stein manifold for any connected unipotent subgroup $G$ of $\mathrm{SO}_0(1,d)$.

The heavy top is a classic example of Euler-Poincare reduction on a semidirect product group, demonstrating how an advected quantity-the direction of gravity-couples to rotational motion. While the description of relativistic spinning particles via Lie group configurations is well established in Hamiltonian (symplectic) mechanics, we demonstrate that an explicit Lagrangian Euler-Poincare reduction provides a direct, transparent parallel to the heavy top. Specifically, we show that the free relativistic spinning particle on the Poincare group is the precise Lorentzian analog of this system, with spacetime position being the advected quantity. This geometric unification not only illuminates the underlying symmetry of both systems but also clarifies why the introduction of external electromagnetic fields leads to a partial reduction. Finally, we establish that the transition to left-trivialized coordinates acts as a geometric selection mechanism, naturally favoring a class of interaction Lagrangians where spin-field coupling manifests as a dynamical redefinition of the particle's mass.

We define a derived stack $\mathscr{Rep}_n(X)$ which generalizes the assignment $A\leadsto \operatorname{Rep}_n(A)$ to an algebra of its derived $GL_n$-representation variety of arXiv:1112.1449 from algebras $A$ to perfect stacks $X$ over characteristic 0 fields. In the case of a quasi-projective classical scheme $X$, we show that $\mathscr{Rep}_n(X)$ admits a subfunctor $\operatorname{Quot}^{n,\text{fr}}_{\mathscr{O}_X^n}(X)\subset \mathscr{Rep}_n(X)$, which is in fact represented by a derived scheme almost of finite type. We further construct a Fourier-Mukai integral transform between the derived categories of quasi-coherent sheaves on $X$ and $\operatorname{Quot}^{n,\text{fr}}_{\mathscr{O}_X^n}(X)$ which induces a trace map at the level of Hochschild homology generalizing the trace morphism constructed in arXiv:1112.1449 (for finitely presented commutative algebras). We show that the subfunctor $\operatorname{Quot}^{n,\text{fr}}_{\mathscr{O}_X^n}(X)$ is a derived enhancement of the framed locus of the Quot scheme of points and that this derived scheme is a $GL_n$-torsor over the stack of coherent length $n$ torsion sheaves. Hence, this stack is an analog of the derived character stack for quasi-projective schemes, and we show that for smooth, Calabi-Yau $X$, it inherits a shifted symplectic structure in the sense of arXiv:1111.3209 from the one constructed in arXiv:1812.11913 on the moduli stack of perfect complexes with proper support. This structure is shown to give a generalization of the classical symplectic structure on character varieties of surfaces of genus 1 constructed by Goldman \cite{gold}.

Within the framework of classical contact geometry, the first author introduced the notion of contact symplectic structure –the data of an exact symplectic fibration with a contact base– and proved, under mild conditions, that the total space of such a fibration inherits a contact structure compatible with the fibration map, thereby establishing the contact Thurston theorem. This paper provides a derived contact version of that result by incorporating our prior work on the derived symplectic Thurston theorem. In this paper, we prove, under certain conditions, that if a morphism $\pi: X \rightarrow S$ of derived stacks has a shifted exact symplectic fibration structure and the target stack $S$ admits a shifted contact structure, then one can construct a shifted contact structure on the source stack $X$, compatible with $\pi$ in a sense similar to the smooth case. Our framework relies on the theory of relative shifted structures; hence our result, called the derived contact Thurston theorem, in fact establishes a relative-to-absolute type construction. As an application, we present examples of our relative-to-absolute construction formalism in the derived contact setting, including conormal stacks, quotient mapping stacks, and affine exact symplectic fibrations.

We define a construction of Legendrians inside contact manifolds that arise by doubling an exact Lagrangian filling in the page of an open book decomposition. This can be seen as a generalization of a previous construction by Courte and Ekholm to arbitrary open books. These Legendrians, called Legendrian doubles, are shown to admit regular flexible exact Lagrangian fillings, and they are thus classified up to Legendrian isotopy by classical data. Finally, we show that the Legendrian suspension construction, as defined by Arikan and the author in previous work,-this is a Legendrian contained inside a page of an open book that is obtained by using Seidel's suspension of Lefschetz fibrations- is a Legendrian double.

We prove a version of genus-0 mirror symmetry for toric Fano surfaces at the level of big quantum cohomology. The mirror Landau-Ginzburg model $(\check{X}, W_{\infty})$ of such a surface $X$ can be defined in terms of counts of Maslov index two tropical or holomorphic discs with bulk insertions. While these mirror LG models are not well-defined, the Jacobian ring $\operatorname{\mathsf{Jac}}(W_{\infty})$ of the superpotential $W_{\infty}$ is well-defined. We show that $\operatorname{\mathsf{Jac}}(W_{\infty})$ is isomorphic over the Novikov base to the big quantum cohomology of $X$ using tropical methods, providing a purely algebraic analogue of the cobordism argument used in works of Fukaya-Oh-Ohta-Ono.

Line operators of 4d $\mathcal{N}=2$ theories of class $\mathcal{S}$ of type $A_{N-1}$ in a half Omega background provide a physical model for $SL_N$ skein algebras. In this paper we extend this correspondence to $GL_N$ skein algebras and prove that the $GL_2$ skein algebra of the punctured torus is isomorphic to the $GL_2$ spherical double affine Hecke algebra ${{\mathbf{S}\ddot{\mathbf{H}}}}^{q,t}_2$. The construction is based on $\mathfrak{q}$-nonabelianization for the $SU(2)$ $\mathcal{N}=2^*$ gauge theory, which realizes the skein algebra as a quantum torus algebra associated with the Seiberg-Witten curve. Different regions of the Coulomb branch yield distinct presentations of ${{\mathbf{S}\ddot{\mathbf{H}}}}^{q,t}_2$. These range from the Macdonald $q$-difference module in Fenchel-Nielsen (weak coupling) charts to cluster-type realizations in Fock-Goncharov (strong coupling) charts. Transitions between these descriptions are governed by the vanilla BPS spectrum of the $\mathcal{N}=2^*$ theory via framed wall-crossing, which provides a unified physical framework for several representations of spherical DAHA.

We prove a Legendrian Reidemeister theorem for Legendrian knots in thickened convex surfaces.

Given a convex lattice polygon $\Delta\subset \mathbb R^2$, let $N_\Delta$ be the count of rational, nodal curves in the linear system of the ample line bundle $L_\Delta$ on the toric variety $\mathbb P_\Delta$ defined by $\Delta$, having fixed intersection with the toric boundary. We show that the relative Jacobian of the linear system defines an integrable system that has a mirror dual in the language of constructible sheaves on a two-torus microsupported on a Legendrian link. In this setting, we define the dual counting problem using an analogue of rulings of Legendrian links in three-space, then prove equivalence with $N_\Delta$. We perform calculations of $N_\Delta$ in several examples using constructible methods, tropical curve counting, and localization in logarithmic Gromov-Witten theory to demonstrate the equality of the different approaches. Moreover, the rulings give rise to a stratification of the moduli of constructible sheaves. We conjecture that this ruling decomposition recovers the refined tropical invariants of Block-Göttsche, and hence encodes higher-genus logarithmic Gromov–Witten invariants, by a theorem of Bousseau.

The Lusternik–Schnirelmann category of a closed symplectic manifold admits various estimates. Locally conformally symplectic (LCS) geometry is a generalization of symplectic geometry involving a closed one-form, called the Lee form. In this paper, we introduce Farber's Lusternik–Schnirelmann category for closed one-forms into LCS geometry by applying it to the Lee class. In particular, this provides a new topological approach to LCS structures of the second kind. We then obtain lower bounds for this invariant under LCS blow-ups.

Motivated by Smale's contraction of the compactly supported diffeomorphism group of the two-disc, we study contact forms on an open rotationally symmetric Darboux ball that agree with the standard contact form outside a compact set and whose Reeb flows have no trapped orbits. Such forms are called vertically convex. In dimensions three and five, we identify the quotient of their space by compactly supported diffeomorphisms with a contractible monodromy space and prove an equivariant product splitting. Consequently, the orbit of the standard contact form is a strong deformation retract. It follows that the space of vertically convex contact forms is contractible in dimension three and homotopy equivalent to the compactly supported diffeomorphism group, hence connected, in dimension five. The analogous subspace of forms defining the standard contact structure has the homotopy type of the corresponding compactly supported contactomorphism group in dimension five and is contractible in dimension three.

We introduce, motivated by noncommutative Carrollian geometry, the notion of a Carrollian–Poisson bundle understood as a Carrollian manifold with a compatible Poisson structure. Compatibility is expressed as the inclusion of the Carroll distribution in the image of the Poisson anchor. We explore the geometric consequences of this compatibility and present several simple examples of Carrollian–Poisson bundles, including the (outer) Kerr horizon, the BTZ horizon, and future null infinity. Furthermore, we establish that every Carrollian–Poisson bundle can be equipped with a Vaisman (contravariant) connection that is both metric-compatible and clock-compatible, and thus geometric kinematics can be defined and studied.

In dimension 3, following Eliashberg-Thurston and Mitsumatsu, one can associate a transverse pair of oppositely oriented contact structures to any Anosov flow. We show that when the ambient manifold is hyperbolic, the isotopy classes of these contact structures completely determine the flow up to isotopy orbit equivalence. This then implies that on any closed hyperbolic 3-manifold, the number of Anosov flows modulo isotopy orbit equivalence is finite. This approach also yields an explicit bound on the number of orbit equivalence classes of Anosov flows in terms of the number of universally tight contact structures. In addition, we show finiteness of pseudo-Anosov flows for which the complement of the singular orbits is atoroidal and, in the non- hyperbolic case, we obtain a control on the dynamics of Anosov flows on the hyperbolic pieces of the JSJ-decomposition.

We settle two questions of Serge Tabachnikov regarding the trapping of light rays by mirrors. We show (i) that the largest dimension of a trapped family of rays in $\mathbb{R}^n$ is $2n-3$, and (ii) that in $\mathbb{R}^3$ one can trap a 2-parameter family of rays that is not locally normal to any smooth surface. We first establish that no $2n-2$-parameter family can be trapped by showing that no open subset of $L_n$ can be trapped due to Poincaré's recurrence theorem, yielding an upper bound of $2n-3$. We then show that a system of mirrors traps a $2n-3$ parameter family of rays by noting that a compact subset of the set of states of rays in the central "waist" of the trap is a normally hyperbolic invariant manifold, making it amenable to a stable manifold theorem for NHIMs, which implies that the set of rays asymptotic to the waist of the trap has dimension $2n-3$. We then truncate our mirror system, which permits the entrance of rays from arbitrarily far outside of the trap, attaining the upper bound $2n-3$ and resolving the first question. We then show that in $\mathbb{R}^3$, there exists within the trapped family of rays a 2-parameter subfamily on which the canonical symplectic form on the space of oriented lines is nowhere vanishing, implying that the subfamily is not locally normal to any smooth surface, which settles the second question.

Motivated by the problem of finding obstructions to the existence of symplectic structures in dimensions higher than four, we introduce an elliptic system of equations on almost Hermitian manifolds that reduces to the Seiberg–Witten equations with Taubes' perturbation in dimension four. To define the equations, one needs to choose a spin$^{\mathbb C}$ structure on the manifold. We prove that the system has index zero for the canonical spin$^{\mathbb C}$ structure. Moreover, on almost Kähler (symplectic) manifolds, we construct a canonical solution and prove its transversality with large values of the perturbation parameter, while in dimension six we obtain uniqueness of the canonical solution under a certain assumption. We also show that on Kähler manifolds the equations reduce, under a natural ansatz, to the vortex equations. These results provide evidence toward a possible higher dimensional gauge theory capable of producing invariants of almost complex structures and ultimately, obstructions to compatible symplectic forms.

We classify lens spaces for which the canonical negative-definite plumbing has minimal $b_2$ among all symplectic fillings of the standard contact structure. The classification is given by 4 infinite families of "forbidden subgraphs" in a manner similar to Aceto-McCoy-Park's work on smooth negative-definite fillings of lens spaces. In particular, no classification of this form can be given using a finite list of forbidden configurations, highlighting a difference between symplectic fillings and smooth negative-definite fillings of lens spaces.

We consider three specific $q$-deformed matrix models (the Chern-Simons, $q$-Laguerre, and $q$-Gaussian matrix models) which can be solved explicitly using the property of superintegrability. We show that one- and two-point functions of inverse characteristic polynomials can be interpreted as quantizations of Lagrangian subvarieties in $(\mathbb{C}^*)^2$ and $(\mathbb{C}^*)^4$, respectively. While such a geometric picture is expected for the Chern-Simons matrix model, our results for the $q$-Laguerre and $q$-Gaussian models are new and exhibit additional features. In particular, specific anti-symplectic birational involutions play an essential role in the construction. This reformulation provides a geometric framework for analyzing the semiclassical, large-$N$ expansion of matrix-model correlators. This picture is suggestive of the geometry underlying open topological strings, although the two constructions are not identical and the precise relation between them remains to be understood.

We investigate the effect of the clasp move of Legendrian submanifolds of dimension at least two on the moduli spaces of pseudo-holomorphic curves that contribute to the differential of the Chekanov-Eliashberg algebra. As an application, we compute the differential of Chekanov-Eliashberg algebra of twist spheres $\Lambda_k$ of dimension at least two. Moreover, we construct an infinite family of Legendrians $\mathcal{T}(\Lambda,k)$ from any horizontally displaceable Legendrian $\Lambda$ in $P\times\mathbb{R}$, such that the Chekanov-Eliashberg algebra of $\mathcal{T}(\Lambda,k)$ admits an augmentation, while $\mathcal{T}(\Lambda,k)$ has no exact embedded Lagrangian filling of vanishing Maslov class in the symplectization of $P\times \mathbb{R}$ for sufficiently large $k$.

We outline the contours of an emerging theory of Kähler metrics in derived noncommutative geometry. This is a refinement of the theory of Bridgeland stability conditions encoding underlying differential-geometric structures. We propose elements of such a structure in both Archimedean and non-Archimedean settings, including metrized objects, mass measures satisfying a BPS inequality, harmonic metrics, minimizing flows, and complexified Kähler potentials. We develop the framework through examples and constructions involving Fukaya categories, quiver representations and associated C$^*$-algebras, spectral networks, and comonadic adjunctions of stable $\infty$-categories.