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A Hamiltonian $\coprod\limits_n BO(n)$-action, stratified Morse theory and the $J$-homomorphismFeb 18 2019We use sheaves of spectra to quantize a Hamiltonian $\coprod\limits_n BO(n)$-action on $\varinjlim\limits_{N}T^*\mathbf{R}^N$ that naturally arises from Bott periodicity. We employ the category of correspondences developed in [GaRo] to give an enrichment ... More

A Hamiltonian $\coprod\limits_n BO(n)$-action, stratified Morse theory and the $J$-homomorphismFeb 18 2019Feb 19 2019We use sheaves of spectra to quantize a Hamiltonian $\coprod\limits_n BO(n)$-action on $\varinjlim\limits_{N}T^*\mathbf{R}^N$ that naturally arises from Bott periodicity. We employ the category of correspondences developed in [GaRo] to give an enrichment ... More

Morse-Bott Cohomology from Homological Perturbation TheoryFeb 18 2019In this paper, we construct cochain complexes generated by cohomology of critical manifolds for Morse-Bott theory under minimum transversality assumptions. We discuss the relations between different constructions of cochain complexes for Morse-Bott theory. ... More

Moduli spaces of framed $G$--Higgs bundles and symplectic geometryFeb 18 2019Let $X$ be a compact connected Riemann surface, $D\, \subset\, X$ a reduced effective divisor, $G$ a connected complex reductive affine algebraic group and $H_x\, \subsetneq\, G_x$ a Zariski closed subgroup for every $x\, \in\, D$. A framed principal ... More

On the iterated Hamiltonian Floer homologyFeb 18 2019The focus of the paper is the behavior under iterations of the filtered and local Floer homology of a Hamiltonian on a symplectically aspherical manifold. The Floer homology of an iterated Hamiltonian comes with a natural cyclic group action. In the filtered ... More

On the Cohomology Ring of Symplectic FillingsFeb 18 2019Let $Y$ be the contact boundary of a $2n$-dimensional flexible Weinstein domain $W$ with vanishing first Chern class. We show that for any Liouville filling $W'$ of $Y$ with vanishing first Chern class, there is a linear isomorphism $\phi:H^*(W;\mathbb{R}) ... More

Physicists' $d=3+1$, $N=1$ superspace-time and supersymmetric QFTs from a tower construction in complexified ${\Bbb Z}/2$-graded $C^\infty$-Algebraic Geometry and a purge-evaluation/index-contracting mapFeb 17 2019The complexified ${\Bbb Z}/2$-graded $C^\infty$-Algebraic Geometry aspect of a superspace(-time) $\widehat{X}$ in Sec.\,1 of D(14.1) (arXiv:1808.05011 [math.DG]) together with the Spin-Statistics Theorem in Quantum Field Theory, which requires fermionic ... More

Basic Kirwan injectivity and its applicationsFeb 17 2019Feb 19 2019Consider the Hamiltonian action of a torus on a transversely symplectic foliation that is also Riemannian. When the transverse hard Lefschetz property is satisfied, we establish a foliated version of the Kirwan injectivity theorem, and use it to study ... More

Basic Kirwan infectivity and its applicationsFeb 17 2019Consider the Hamiltonian action of a torus on a transversely symplectic foliation that is also Riemannian. When the transverse hard Lefschetz property is satisfied, we establish a foliated version of the Kirwan injectivity theorem, and use it to study ... More

An application of spherical geometry to hyperkähler slicesFeb 14 2019This work is concerned with Bielawski's hyperk\"ahler slices in the cotangent bundles of homogeneous affine varieties. One can associate such a slice to the data of a complex semisimple Lie group $G$, a reductive subgroup $H\subseteq G$, and a Slodowy ... More

On Lagrangian embeddings of closed non-orientable 3-manifoldsFeb 14 2019We prove that for any compact orientable connected 3-manifold with torus boundary, a concatenation of it and the direct product of the circle and the Klein bottle with an open 2-disk removed admits a Lagrangian embedding into the standard symplectic 6-space. ... More

Arboreal singularities and loose Legendrians IFeb 13 2019Arboreal singularities are an important class of Lagrangian singularities. They are conical, meaning that they can be understood by studying their links, which are singular Legendrian spaces in $S^{2n-1}_{\text{std}}$. Loose Legendrians are a class of ... More

Categorical Saito theory, I: A comparison resultFeb 12 2019In this paper, we present an explicit cyclic minimal $A_\infty$ model for the category of matrix factorizations $\MF(W)$ of an isolated hypersurface singularity. The key observation is to use Kontsevich's deformation quantization technique. Pushing this ... More

Categorification of Legendrian knotsFeb 12 2019Feb 17 2019Perverse schober defined by Kapranov--Schechtman is a categorification of the notion of perverse sheaf. In their definition, a key ingredient is certain purity property of perverse sheaves. In this short note, we attempt to describe a real analogue of ... More

Categorification of Legendrian knotsFeb 12 2019Perverse schober defined by Kapranov--Schechtman is a categorification of the notion of perverse sheaf. In their definition, a key ingredient is certain purity property of perverse sheaves. In this short note, we attempt to describe a real analogue of ... More

Embedded contact knot homology and a surgery formulaFeb 11 2019Embedded contact knot homology (ECK) is a variation on Embedded contact homology (ECH), defined with respect to an open book decomposition compatible with a contact structure on some 3-manifold, M. The knot in question is given by the (null-homologous) ... More

Holomorphic curves in the presence of holomorphic hypersurface foliationsFeb 07 2019We prove a result which establishes restrictions on the pseudoholomorphic curves which can exist in a stable Hamiltonian manifold in the presence of certain $\mathbb{R}$-invariant foliations of the symplectization by holomorphic hypersurfaces. This result ... More

Quasi-morphisms on contactomorphism groups and Grassmannians of 2-planesFeb 06 2019We construct a natural prequantization space over a monotone product of a toric manifold and an arbitrary number of complex Grassmannians of 2-planes in even-dimensional complex spaces, and prove that the universal cover of the identity component of the ... More

Flat affine symplectic Lie groupsFeb 05 2019We give a new characterization of flat affine manifolds in terms of an action of the Lie algebra of classical infinitesimal affine transformations on the bundle of linear frames. We prove that a left invariant flat affine symplectic connection on a connected ... More

Spine removal surgery and the geography of symplectic fillingsFeb 04 2019We prove that for any contact 3-manifold supported by a spinal open book decomposition with planar pages, there is a universal bound on the Euler characteristic and signature of its minimal symplectic fillings. The proof is an application of the spine ... More

Uniqueness of real Lagrangians up to cobordismFeb 04 2019Feb 13 2019We prove that a real Lagrangian submanifold in a closed symplectic manifold is unique up to cobordism. We then discuss the classification of real Lagrangians in $\mathbb{C} P^2$ and $S^2\times S^2$. In particular, we show that a real Lagrangian in $\mathbb{C} ... More

Fukaya categories of Lagrangian cobordisms and dualityFeb 03 2019We introduce a new type of duality structure for $A_\infty$-categories called a relative weak Calabi-Yau pairing which generalizes Kontsevich and Soibelman's notion of a weak (proper) Calabi-Yau structure. We prove the existence of a relative weak Calabi-Yau ... More

A simple construction of an action selector on aspherical symplectic manifoldsFeb 02 2019We construct an action selector on aspherical symplectic manifolds that are closed or convex. Such selectors have been constructed by Matthias Schwarz using Floer homology. The construction we present here is simpler and uses only Gromov compactness.

Euler flows and singular geometric structuresJan 31 2019Tichler proved that a manifold admitting a smooth closed one-form fibers over a circle. More generally a manifold admitting $k$ independent closed one-forms fibers over a torus $T^k$. In this article we explain a version of this construction for manifolds ... More

Toric generalized Kaehler structures. IIIJan 30 2019The paper clarifies some subtle points surrounding the definition of scalar curvature in generalized K$\ddot{a}$hler (GK) geometry. We have solved an open problem in GK geometry of symplectic type posed by R. Goto \cite{Go1} on relating the scalar curvature ... More

Lie 2-algebra moment maps in multisymplectic geometryJan 30 2019Consider a closed non-degenerate 3-form $\omega$ with an infinitesimal action of a Lie algebra $\mathfrak{g}$. Motivated by the fact that the observables associated to $\omega$ form a Lie 2-algebra, we introduce homotopy moment maps defined on a Lie 2-algebra ... More

Deformation of Singular Fibers of Genus $2$ Fibrations and Small Exotic Symplectic $4$-ManifoldsJan 30 2019We introduce the $2$-nodal spherical deformation of certain singular fibers of genus $2$ fibrations, and use such deformations to construct various examples of simply connected minimal symplectic $4$-manifolds with small topology. More specifically, we ... More

The action spectrum characterizes closed contact 3-manifolds all of whose Reeb orbits are closedJan 29 2019A classical theorem due to Wadsley implies that, on a contact manifold all of whose Reeb orbits are closed, there is a common period for the Reeb orbits. In this paper we show that, for any Reeb flow on a closed 3-manifold, the following conditions are ... More

On coupled constant scalar curvature Kähler metricsJan 29 2019We provide a moment map interpretation for the coupled K\"ahler-Einstein equations introduced by Hultgren and Witt Nystr\"om, and in the process introduce a more general system of equations, which we call coupled cscK equations. A differentio-geometric ... More

Survey on recent developments in semitoric systemsJan 29 2019Semitoric systems are a special class of completely integrable systems in four dimensions for which one of the first integrals generates an $\mathbb{S}^1$-action. They were classified by Pelayo & Vu Ngoc in terms of five symplectic invariants about a ... More

Open WDVV equations and Virasoro constraintsJan 29 2019In their fundamental work, B. Dubrovin and Y. Zhang, generalizing the Virasoro equations for the genus $0$ Gromov--Witten invariants, proved the Virasoro equations for a descendent potential in genus $0$ of an arbitrary conformal Frobenius manifold. More ... More

Fukaya's conjecture on $S^1$-equivariant de Rham complexJan 28 2019Getzler-Jones-Petrack introduced $A_\infty$ structures on the equivariant complex for manifold $M$ with smooth $\mathbb{S}^1$ action, motivated by geometry of loop spaces. Applying Witten's deformation by Morse functions followed by homological perturbation ... More

Sheaf Quantization of Lagrangians and Floer cohomologyJan 27 2019Given an exact Lagrangian submanifold $L$ in $T^*N$, we want to construct a complex of sheaves in the derived category of sheaves on $N\times {\mathbb R} $, such that its singular support, $SS({\mathcal F}^\bullet_L)$, is equal to $\widehat L$, the cone ... More

Floer homotopy theory, revisitedJan 24 2019In 1995 the author, Jones, and Segal introduced the notion of "Floer homotopy theory". The proposal was to attach a (stable) homotopy type to the geometric data given in a version of Floer homology. More to the point, the question was asked, "When is ... More

Homotopy $4$-spheres associated to an infinite order loose corkJan 24 2019We show the homotopy spheres $\Sigma_{n} = -W\smile_{f^{n}}W$, formed by doubling the infinite order loose-cork $(W,f)$ by iterates of the cork diffeomorphism $f: \partial W \to \partial W$, is obtained by Gluck twistings of $S^4$; then by this we show ... More

Spectral numbers and manifolds with boundaryJan 23 2019We consider a smooth submanifold $N$ with a smooth boundary in an ambient closed manifold $M$ and assign a spectral invariant $c(\alpha,H)$ to every singular homological class $\alpha\in H_*(N)$ and a Hamiltonian $H$ defined on the cotangent bundle $T^*M$. ... More

All 3-manifolds are the boundary of exotic 4-manifoldsJan 23 2019In this note we show that any closed, oriented 3-manifold is the boundary of a simply connected 4-manifold that admits infinitely many distinct smooth structures. We also show that any fillable contact 3-manifold is the boundary of a simply connected ... More

Fluxes in Exceptional Field Theory and Threebrane Sigma-ModelsJan 23 2019Feb 06 2019Starting from a higher Courant bracket associated to exceptional generalized geometry, we provide a systematic derivation of all types of fluxes and their Bianchi identities for four-dimensional compactifications of M-theory. We show that these fluxes ... More

Symplectic Coarse-Grained Dynamics: Chalkboard Motion in Classical and Quantum MechanicsJan 19 2019In the usual approaches to mechanics (classical or quantum) the primary object of interest is the Hamiltonian, from which one tries to deduce the solutions of the equations of motion (Hamilton or Schr\"odinger). In the present work we reverse this paradigm ... More

Lagrangian cobordism groups of higher genus surfacesJan 17 2019We study Lagrangian cobordism groups of oriented surfaces of genus greater than two. We compute the immersed oriented Lagrangian cobordism group of these surfaces. We show that a variant of this group, with relations given by unobstructed immersed Lagrangian ... More

Hamiltonian spectral invariants, symplectic spinors and Frobenius structures IIJan 17 2019Feb 02 2019In this article, we continue our study of 'Frobenius structures' and symplectic spectral invariants in the context of symplectic spinors. By studying the case of $C^1$-small Hamiltonian mappings on symplectic manifolds $M$ admitting a metaplectic structure ... More

Ancient solutions in Lagrangian mean curvature flowJan 16 2019Ancient solutions of Lagrangian mean curvature flow in C^n naturally arise as Type II blow-ups. In this extended note we give structural and classification results for such ancient solutions in terms of their blow-down and, motivated by the Thomas-Yau ... More

Duality of gauges and symplectic forms in vector spacesJan 10 2019A gauge $\gamma$ in a vector space $X$ is a distance function given by the Minkowski functional associated to a convex body $K$ containing the origin in its interior. Thus, the outcoming concept of gauge spaces $(X, \gamma)$ extends that of finite dimensional ... More

A surgery formula for knot Floer homologyJan 08 2019Let $K$ be a rationally null-homologous knot in a $3$-manifold $Y$, equipped with a nonzero framing $\lambda$, and let $Y_\lambda(K)$ denote the result of $\lambda$-framed surgery on $Y$. Ozsv\'ath and Szab\'o gave a formula for the Heegaard Floer homology ... More

A wrapped Fukaya category of knot complementJan 08 2019This is the first of a series of two articles where we construct a version of wrapped Fukaya category $\mathcal W\mathcal F(M\setminus K;H_{g_0})$ of the cotangent bundle $T^*(M \setminus K)$ of the knot complement $M \setminus K$ of a compact 3-manifold ... More

Knot concordances in $S^1\times S^2$ and exotic smooth $4$-manifoldsJan 03 2019It is known that there is a unique concordance class in the free homotopy class of $S^1\times pt \subset S^1 \times S^2$. The constructive proof of this fact is given by the second author. It turns out that all the concordances in this construction are ... More

On contact invariants of non-simply connected Gorenstein toric contact manifoldsDec 26 2018The first two authors showed in~\cite{AM1} how the Conley-Zehnder index of any contractible periodic Reeb orbit of a non-degenerate toric contact form on a good toric contact manifold with zero first Chern class, i.e. a Gorenstein toric contact manifold, ... More

Classification of six dimensional monotone symplectic manifolds admitting semifree circle actions IDec 24 2018Let $(M,\omega_M)$ be a six dimensional closed monotone symplectic manifold admitting an effective semifree Hamiltonian $S^1$-action. We show that if the minimal (or maximal) fixed component of the action is an isolated point, then $(M,\omega_M)$ is $S^1$-equivariant ... More

Equivariant Lagrangian Floer cohomology via semi-global Kuranishi structuresDec 23 2018Using a simplified version of Kuranishi perturbation theory that we call semi-global Kuranishi structures, we give a definition of the equivariant Lagrangian Floer cohomology of a pair of Lagrangian submanifolds that are fixed under a finite symplectic ... More

Rotation numbers and the Euler class in open booksDec 14 2018This paper introduces techniques for computing a variety of numerical invariants associated to a Legendrian knot in a contact manifold presented by an open book with a Morse structure. Such a Legendrian knot admits a front projection to the boundary of ... More

Extensions in groupoidsDec 13 2018As groupoids generalize groups, motivated by group extension we propose a concept of groupoid extension for Lie groupoids, i.e, \[ \sf 1\to A \to G \to K \to 1 \] where $\sf A,G$ and $\sf K$ are Lie groupoids. Similar to the theory of group extension, ... More

Clebsch-Lagrange variational principle and geometric constraint analysis of relativistic field theoriesDec 11 2018Inspired by the Clebsch optimal control problem, we introduce a new variational principle that is suitable for capturing the geometry of relativistic field theories with constraints related to a gauge symmetry. Its special feature is that the Lagrange ... More

On the volume elements of a manifold with transverse zeroesDec 10 2018Dec 18 2018Moser proved in 1965 in his seminal paper that two volume forms on a compact manifold can be conjugated by a diffeomorphism, that is to say they are equivalent, if and only if their associated cohomology classes in the top cohomology group of a manifold ... More

Removing a ray from a noncompact symplectic manifoldDec 02 2018We prove that any noncompact symplectic manifold which admits a properly embedded ray with a wide neighborhood is symplectomorphic to the complement of the ray by constructing an explicit symplectomorphism between the standard Euclidean space and the ... More

Cotangent models for group actions on $b$-Poisson manifoldsNov 29 2018Dec 13 2018In this article we give a normal form of a $b$-symplectic form in the neighborhood of a compact orbit of a Lie group action on a $b$-symplectic manifold. We establish cotangent models for Poisson actions on $b$-Poisson manifolds and a $b$-symplectic slice ... More

Equivariant commutative stringy cohomology rings on almost complex manifoldsNov 28 2018Dec 05 2018In this paper, motivated by Chen--Ruan's stringy orbifold theory on almost complex orbifolds, we construct a new cohomology ring $\mathscr H^\ast_{G,cs}(X)$ for an equivariant almost complex pair $(X,G)$, where $X$ is a compact connected almost complex ... More

Hamiltonian Lie algebroidsNov 27 2018In previous work with M.C. Fernandes, we found a Lie algebroid symmetry for the Einstein evolution equations. The present work allows us to explain the coisotropic structure of the constraint subset for the initial value problem by extending the notion ... More

Relative stability conditions on Fukaya categories of surfacesNov 26 2018In this paper, we introduce the novel notion of a relative Bridgeland stability condition, in the context of a wrapped Fukaya category of a marked surface with respect to part of its boundary. This construction can be shown to have nice functorial properties ... More

Contact between Lagrangian manifoldsNov 26 2018Tangential intersections of Lagrangian manifolds up to contact equivalence correspond to smooth function germs (generating functions) up to right equivalence locally around the intersection point. We extend this result of Golubitsky and Guillemin for ... More

Split Canonical RelationsNov 25 2018A Lagrangian subspace $L$ of a weak symplectic vector space is called \emph{split Lagrangian} if it has an isotropic (hence Lagrangian) complement. When the symplectic structure is strong, it is sufficient for $L$ to have a closed complement, which can ... More

Translated points for prequantization spaces over monotone toric manifoldsNov 25 2018We prove a version of Sandon's conjecture on the number of translated points of contactomorphisms for the case of a prequantization bundle over a closed monotone toric manifold. Namely we show that any contactomorphism of this prequantization bundle lying ... More

Hypertoric manifolds of infinite topological typeNov 24 2018We analyse properties of hypertoric manifolds of infinite topological type, including their topology and complex structures. We show that our manifolds have the homotopy type of an infinite union of compact toric varieties. We also discuss hypertoric ... More

Engel Manifolds and Contact 3-OrbifoldsNov 22 2018In early study of Engel manifolds from R. Montgomery, the Cartan prolongation and the development map are central figures. However, the development map can be globally defined only if the characteristic foliation is "nice". In this paper, we introduce ... More

SYZ mirror symmetry from Witten-Morse theoryNov 22 2018This is a survey article on the recent progress in understanding the Strominger-Yau-Zaslow (SYZ) mirror symmetry conjecture, especially on the effect of quantum corrections, via Witten-Morse theory using the program first depicted by Fukaya to obtain ... More

Asymptotic velocity for four celestial bodiesNov 20 2018Asymptotic velocity is defined as the Ces\`aro limit of velocity. As such, its existence has been proven for bounded interaction potentials. This is known to be wrong in celestial mechanics with four or more bodies. Here we show for a class of pair potentials ... More

The orientation morphism: from graph cocycles to deformations of Poisson structuresNov 19 2018We recall the construction of the Kontsevich graph orientation morphism $\gamma \mapsto {\rm O\vec{r}}(\gamma)$ which maps cocycles $\gamma$ in the non-oriented graph complex to infinitesimal symmetries $\dot{\mathcal{P}} = {\rm O\vec{r}}(\gamma)(\mathcal{P})$ ... More

Graded Poisson AlgebrasNov 18 2018This note is an expanded and updated version of our entry with the same title for the 2006 Encyclopedia of Mathematical Physics. We give a brief overview of graded Poisson algebras, their main properties and their main applications, in the contexts of ... More

Almost Kaehler geometry of adjoint orbits of semisimple Lie groupsNov 16 2018Nov 26 2018We study the almost Kaehler geometry of adjoint orbits of non-compact real semisimple Lie groups endowed with the Kirillov-Kostant-Souriau symplectic form and a canonically defined almost complex structure. We give explicit formulas for the Chern-Ricci ... More

Complex symplectic structures on Lie algebrasNov 14 2018We investigate Lie algebras endowed with a complex symplectic structure and develop a method, called \emph{complex symplectic oxidation}, to construct certain complex symplectic Lie algebras of dimension $4n+4$ from those of dimension $4n$. We specialize ... More

Newton's Second Law in Field TheoryNov 13 2018In this article we present a natural generalization of Newton's Second Law valid in field theory, i.e., when the parameterized curves are replaced by parameterized submanifolds of higher dimension. For it we introduce what we have called the geodesic ... More

The rigid-flexible value for symplectic embeddings of four-dimensional ellipsoids into polydiscsNov 09 2018In previous work of Cristofaro-Gardiner, Frenkel, and Schlenk, the embedding function $c_b(a)$ describing the problem of symplectically embedding an ellipsoid $E(1,a)$ into the smallest scaling of the polydisc $P(1,b)$ was determined for all integers ... More

Symplectic cohomology rings of affine varieties in the topological limitNov 08 2018We construct a multiplicative spectral sequence converging to the symplectic cohomology ring of any affine variety $X$, with first page built out of topological invariants associated to strata of any fixed normal crossings compactification $(M,\mathbf{D})$ ... More

Contact Hamiltonian SystemsNov 08 2018In this paper we study Hamiltonian systems on contact manifolds, which is an appropriate scenario to discuss dissipative systems. We prove a coisotropic reduction theorem similar to the one in symplectic mechanics.

Sub-leading asymptotics of ECH capacitiesNov 01 2018In previous work, the first author and collaborators showed that the leading asymptotics of the embedded contact homology (ECH) spectrum recovers the contact volume. Our main theorem here is a new bound on the sub-leading asymptotics.

On symplectic fillings of spinal open book decompositions I: Geometric constructionsOct 29 2018A spinal open book decomposition on a contact manifold is a generalization of a supporting open book which exists naturally e.g. on the boundary of a symplectic filling with a Lefschetz fibration over any compact oriented surface with boundary. In this ... More

Pairings in mirror symmetry between a symplectic manifold and a Landau-Ginzburg $B$-modelOct 26 2018We find a relation between Lagrangian Floer pairing of a symplectic manifold and Kapustin-Li pairing of the mirror Landau-Ginzburg model under localized mirror functor. They are conformally equivalent with an interesting conformal factor $(vol^{Floer}/vol)^2$, ... More

On the existence of closed $C^{1,1}$ curves of constant curvatureOct 22 2018Jan 27 2019We show that on any Riemannian surface for each $0<c<\infty$ there exists an immersed $C^{1,1}$ curve that is smooth and with curvature equal to $\pm c$ away from a point. We give examples showing that, in general, the regularity of the curve obtained ... More

Semitoric familiesOct 16 2018Jan 25 2019Semitoric systems are a type of four-dimensional integrable system for which one of the integrals generates a global $S^1$-action; these systems were classified by Pelayo and Vu Ngoc in terms of five symplectic invariants. We introduce and study semitoric ... More

On symplectic lifts of actions for complete Lagrangian fibrationsOct 12 2018In this note we discuss symplectic lifts of actions for a complete Lagrangian fibration. Firstly, we describe the symplectic cotangent lifts of a G-action on a manifold Q in terms of 1-cocycles in the cohomology of G induced by the action with values ... More

Categorical primitive forms and Gromov-Witten invariants of $A_n$ singularitiesOct 11 2018Nov 15 2018We introduce a categorical analogue of Saito's notion of primitive forms. Let $W$ denote the potential $\frac{1}{n+1} x^{n+1}$. For the category $MF(W)$ of matrix factorizations of $W$ we prove that there exists a unique, up to non-zero constant, categorical ... More

Connections Adapted to Non-Negatively Graded StructuresOct 10 2018Nov 15 2018Graded bundles are a particularly nice class of graded manifolds and represent a natural generalisation of vector bundles. By exploiting the formalism of supermanifolds to describe Lie algebroids we define the notion of a weighted $A$-connection on a ... More

Gauge equivalences for foliations and pre-symplectic structuresOct 09 2018We consider the deformation theory of two kinds of geometric objects: foliations on one hand, pre-symplectic forms on the other. For each of them, we prove that the geometric notion of equivalence given by isotopies agrees with the algebraic notion of ... More

Moment Maps, Strict Linear Precision, and Maximum Likelihood Degree OneOct 08 2018We study the moment maps of a smooth projective toric variety. In particular, we characterize when the moment map coming from the quotient construction is equal to a weighted Fubini-Study moment map. This leads to an investigation into polytopes with ... More

Interval topology in contact geometryOct 03 2018A topology is introduced on spaces of Legendrian submanifolds and groups of contactomorphisms. The definition is motivated by the Alexandrov topology in Lorentz geometry.

Orbit spaces of maximal torus actions on oriented Grassmannians of planesOct 01 2018Motivated by Buchstaber's and Terzic' work on the complex Grassmannians G(2,4) and G(2,5) we describe the moment map and the orbit space of oriented Grassmannians of planes under the action of a maximal compact torus. Our main tool is the realisation ... More

Heegaard Floer invariants of contact structures on links of surface singularitiesSep 28 2018Let a contact 3-manifold $(Y, \xi_0)$ be the link of a normal surface singularity equipped with its canonical contact structure $\xi_0$. We prove a special property of such contact 3-manifolds of "algebraic" origin: the Heegaard Floer invariant $c^+(\xi_0)\in ... More

Equivariant deformation quantization and coadjoint orbit methodSep 24 2018The purpose of this paper is to apply deformation quantization to the study of the coadjoint orbit method in the case of real reductive groups. We first prove some general results on the existence of equivariant deformation quantization of vector bundles ... More

Hilbert series associated to symplectic quotients by $\operatorname{SU}_2$Sep 20 2018We compute the Hilbert series of the graded algebra of real regular functions on the symplectic quotient associated to an $\operatorname{SU}_2$-module and give an explicit expression for the first nonzero coefficient of the Laurent expansion of the Hilbert ... More

Monodromy of Fukaya-Seidel categories mirror to toric varietiesSep 17 2018Mirror symmetry for a toric variety involves Laurent polynomials whose symplectic topology is related to the algebraic geometry of the toric variety. We show that there is a monodromy action on the Fukaya-Seidel categories of these Laurent polynomials ... More

Fano-Mathieu correspondenceSep 08 2018We show that $G$-Fano threefolds are mirror-modular. 1. Mirror maps are inversed reversed Hauptmoduln for moonshine subgroups of $SL_2(\mathbb{R})$. 2. Quantum periods, shifted by an integer constant (eigenvalue of quantum operator on primitive cohomology) ... More

Explicit formulas in Lie theorySep 04 2018We provide explicit formulas for general multiplicative geometric structures on Lie groupoids in terms of the underlying infinitesimal data. Combined with our previous work [8] which constructs a local Lie groupoid out of a Lie algebroid with a compatible ... More

Integrable systems on $Fl_n \times Fl_n \times Fl_n //SU(n)$ and $SU(n)$ tensor product multiplicitiesSep 04 2018We construct a densely defined torus action on the symplectic quotient of the product of three complete flag varieties. The closure of the image of the corresponding moment map is a convex polytope. The dimension of the geometric quantization of this ... More

Heisenberg model in pseudo-Euclidean spaces IIAug 31 2018In the review we describe a relation between the Heisenberg spin chain model on pseudospheres and light--like cones in pseudo--Eucli\-dean spaces and virtual billiards. A geometrical interpretation of the integrals associated to a family of confocal quadrics ... More

Connected sums of almost complex manifoldsAug 22 2018Aug 24 2018In this paper, firstly, for some $4n$-dimensional almost complex manifolds $M_{i}, ~1\le i \le \alpha$, we prove that $\left(\sharp_{i=1}^{\alpha} M_{i}\right) \sharp (\alpha{-}1) \mathbb{C} P^{2n}$ must admits an almost complex structure, where $\alpha$ ... More

Sharp systolic inequalities for Riemannian and Finsler spheres of revolutionAug 21 2018We prove that the systolic ratio of a sphere of revolution $S$ does not exceed $\pi$ and equals $\pi$ if and only if $S$ is Zoll. More generally, we consider the rotationally symmetric Finsler metrics on a sphere of revolution which are defined by shifting ... More

Pseudolattices, del Pezzo surfaces, and Lefschetz fibrationsAug 20 2018Motivated by the relationship between numerical Grothendieck groups induced by the embedding of a smooth anticanonical elliptic curve into a del Pezzo surface, we define the notion of a quasi del Pezzo homomorphism between pseudolattices and establish ... More

Symplectic classification of coupled angular momentaAug 17 2018The coupled angular momenta are a family of completely integrable systems that depend on three parameters and have a compact phase space. They correspond to the classical version of the coupling of two quantum angular momenta and they constitute one of ... More

$N=1$ fermionic D3-branes in RNS formulation I. $C^\infty$-Algebrogeometric foundations of $d=4$, $N=1$ supersymmetry, SUSY-rep compatible hybrid connections, and $\widehat{D}$-chiral maps from a $d=4$ $N=1$ Azumaya/matrix superspaceAug 15 2018As the necessary background to construct from the aspect of Grothendieck's Algebraic Geometry dynamical fermionic D3-branes along the line of Ramond-Neveu-Schwarz superstrings in string theory, three pieces of the building blocks are given in the current ... More

Locally conformal symplectic structures on Lie algebras of type I and their solvmanifoldsAug 09 2018We study Lie algebras of type I, that is, a Lie algebra $\g$ where all the eigenvalues of the operator $\ad_X$ are imaginary for all $X\in\g$. We prove that the Morse-Novikov cohomology of a Lie algebra of type I is trivial for any closed $1$-form. We ... More

A constraint on Chern classes of strictly pseudoconvex CR manifoldsAug 07 2018This short note gives a constraint on Chern classes of closed strictly pseudoconvex CR manifolds (or equivalently, closed holomorphically fillable contact manifolds) of dimension at least five.