A natural explicit condition is given ensuring that an action of the multiplicative monoid of non-negative reals on a manifold F comes from homotheties of a vector bundle structure on F, or, equivalently, from an Euler vector field. This is used in showing that double (or higher) vector bundles present in the literature can be equivalently defined as manifolds with a family of commuting Euler vector fields. Higher vector bundles can be therefore defined as manifolds admitting certain $\N^n$-grading in the structure sheaf. Consequently, multi-graded (super)manifolds are canonically associated with higher vector bundles that is an equivalence of categories. Of particular interest are symplectic multi-graded manifolds which are proven to be associated with cotangent bundles. Duality for higher vector bundles is then explained by means of the cotangent bundles as they contain the collection of all possible duals. This gives, moreover, higher generalizations of the known `universal Legendre transformation' T*E->T*E*, identifying the cotangent bundles of all higher vector bundles in duality. The symplectic multi-graded manifolds, equipped with certain homological Hamiltonian vector fields, lead to an alternative to Roytenberg's picture generalization of Lie bialgebroids, Courant brackets, Drinfeld doubles and can be viewed as geometrical base for higher BRST and Batalin-Vilkovisky formalisms. This is also a natural framework for studying n-fold Lie algebroids and related structures.
We introduce the concept of a graded bundle which is a natural generalization of the concept of a vector bundle and whose standard examples are higher tangent bundles T^nQ playing a fundamental role in higher order Lagrangian formalisms. Graded bundles are graded manifolds in the sense that we can choose an atlas whose local coordinates are homogeneous functions of degrees 0,1,...,n. We prove that graded bundles have a convenient equivalent description as homogeneity structures, i.e. manifolds with a smooth action of the multiplicative monoid of non-negative reals. The main result states that …
We present a construction (and classification) of certain invariant 2-forms on the real symplectic group. They are used to define a symplectic form on the quotient by a maximal torus and to "lift" a symplectic structure from a symplectic manifold to the bundle of frames. This is a by-product of a failed attempt to prove certain finiteness theorems for basic symplectic cohomologies. In the last part of the paper we include a valid proof.
We investigate G-invariant symplectic structures on the cotangent bundle T*P of a principal G-bundle P(M,G) which are canonically related to automorphisms of the tangent bundle TP covering the identity map of P and commuting with the action of TG on TP. The symplectic structures corresponding to connections on P(M,G) are also investigated. The Marsden-Weinstein reduction procedure for these symplectic structures is discussed.
We define an integer graded symplectic Floer cohomology and a spectral sequence which are new invariants for monotone Lagrangian sub-manifolds and exact isotopies. Such an integer graded Floer cohomology is an integral lifting of the usual Floer-Oh cohomology with $Z_{\Si (L)}$ grading. As one of applications of the spectral sequence, we offer an affirmative answer to an Audin's question for oriented, embedded, monotone Lagrangian tori, i.e. $\Si (L) = 2$.
In the main part of this paper a connection is just a fiber projection onto a (not necessarily integrable) distribution or sub vector bundle of the tangent bundle. Here curvature is computed via the Froelicher-Nijenhuis bracket, and it is complemented by cocurvature and the Bianchi identity still holds. In this situation we determine the graded Lie algebra of all graded derivations over the horizontal projection of a connection and we determine their commutation relations. Finally, for a principal connection on a principal bundle and the induced connection on an associated bundle we show how o…
We study symplectic (contact) structures on nilmanifolds that correspond to the filiform Lie algebras - nilpotent Lie algebras of the maximal length of the descending central sequence. We give a complete classification of filiform Lie algebras that possess a basis e_1, ..., e_n, [e_i,e_j]=c_{ij}e_{i{+}j} (N-graded Lie algebras). In particular we describe the spaces of symplectic cohomology classes for all even-dimensional algebras of the list. It is proved that a symplectic filiform Lie algebra is a filtered deformation of some N-graded symplectic filiform Lie algebra. But this condition is no…
We review the concept of a graded bundle as a natural generalisation of a vector bundle. Such geometries are particularly nice examples of more general graded manifolds. With hindsight there are many examples of graded bundles that appear in the existing literature. We start with a discussion of graded spaces, passing through graded bundles and their linearisation to end with weighted (graded) Lie groupoids/algebroids.
The geometry of graded principal bundles is discussed in the framework of graded manifold theory of Kostant-Berezin-Leites. In particular, we prove that a graded principal bundle is globally trivial if and only if it admits a global graded section and, further, that the sheaf of vertical derivations on such a bundle coincides with the graded distribution induced by the action of the structure graded Lie group. This result leads to a natural definition of the graded connection in terms of graded distributions; its relation with Lie superalgebra-valued graded differential forms is also exhibited…
A theory of dg schemes is developed so that it becomes a homotopy site, and the corresponding infinity category of stacks is equivalent to the infinity category of stacks, as constructed by Toen and Vezzosi, on the site of dg algebras whose cohomologies have finitely many generators in each degree. Stacks represented by dg schemes are shown to be derived schemes under this correspondence.
In a previous article, we introduced a reduction procedure for locally conformally symplectic manifolds at any regular value of the momentum mapping. We use this construction to prove an analogue of a well-known theorem in the symplectic setting about the reduction of cotangent bundles.
We give a method to lift $(2,0)$-tensors fields on a manifold $M$ to build symplectic forms on $TM$. Conversely, we show that any symplectic form $\Om$ on $TM$ is symplectomorphic, in a neighborhood of the zero section, to a symplectic form built naturally from three $(2,0)$-tensor fields associated to $\Om$.
Graded 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 graded bundle. In a natural sense weighted $A$-connections are adapted to the basic geometric structure of a graded bundle in the same way as linear $A$-connections are adapted to the structure of a vector bundle. This notion generalises directly to multi-graded bundles and in particular we present the notion of a bi-weighted $A$-connection on a d…
Let X be a smooth complex projective curve and S a finite subset of X. We show that an orthogonal or symplectic parabolic Higgs bundle on X with parabolic structure over S admits a Hermitian-Einstein connection if and only if it is polystable.
We study the notion of duality in the context of graded manifolds. For graded bundles, somehow like in the case of Gelfand representation and the duality: points vs. functions, we obtain natural dual objects which belongs to a different category than the initial ones, namely graded polynomial (co)algebra bundles and free graded Weil (co)algebra bundles. Our results are then applied to obtain elegant characterizations of double vector bundles and graded bundles of degree 2. All these results have their supergeometric counterparts. For instance, we give a simple proof of a nice characterisation …
Graded bundles are a class of graded manifolds which represent a natural generalisation of vector bundles and include the higher order tangent bundles as canonical examples. We present and study the concept of the linearisation of graded bundle which allows us to define the notion of the linear dual of a graded bundle. They are examples of double structures, graded-linear ($\mathcal{GL}$) bundles, including double vector bundles as a particular case. On $\mathcal{GL}$-bundles we define what we shall call weighted algebroids, which are to be understood as algebroids in the category of graded bu…
We construct the full linearisation functor which takes a graded bundle of degree $k$ (a particular kind of graded manifold) and produces a $k$-fold vector bundle. We fully characterise the image of the full linearisation functor and show that we obtain a subcategory of $k$-fold vector bundles consisting of symmetric $k$-fold vector bundles equipped with a family of morphisms indexed by the symmetric group ${\mathbb S}_k$. Interestingly, for the degree 2 case this additional structure gives rise to the notion of a symplectical double vector bundle, which is the skew-symmetric analogue of a met…
In the previous paper \cite{Goto_2017}, the notion of an Einstein-Hermitian metric of a generalized holomorphic vector bundle over a generalized Kahler manifold of symplectic type was introduced from the moment map framework. In this paper we establish a Kobayashi-Hitchin correspondence, that is, the equivalence of the existence of an Einstein-Hermitian metric and $ψ$-polystability of a generalized holomorphic vector bundle over a compact generalized Kahler manifold of symplectic type. Poisson modules provide intriguing generalized holomorphic vector bundles and we obtain $ψ$-stable Poisson mo…
Given a projective sequence of Banach bundles, each one provided with a of weak symplectic form, we look for conditions under which, the corresponding sequence of weak symplectic forms gives rise to weak symplectic form on the projective limit bundle. Then we apply this results to the tangent bundle of a projective limit of Banach manifolds. This naturally leads to ask about conditions under which the Darboux Theorem is also true on the projective limit of Banach manifolds. We will give some necessary and some sufficient conditions so that such a result is true. Then we discuss why, in general…
A locally conformally symplectic (LCS) form is an almost symplectic form $ω$ such that a closed one-form $θ$ exists with $dω=θ\wedgeω$. A fiber bundle with LCS fiber $(F, ω,θ)$ is called LCS if the transition maps are diffeomorphisms of $F$ preserving $ω$ (and hence $θ$). In this paper, we find conditions for the total space of an LCS fiber bundle to admit an LCS form which restricts to the LCS form of the fibers. This is done by using the coupling form introduced by Sternberg and Weinstein, \cite{gls}, in the symplectic case. The construction is related to an adapted Hamiltonian action called…
Vector bundles and double vector bundles, or $2$-fold vector bundles, arise naturally for instance as base spaces for algebraic structures such as Lie algebroids, Courant algebroids and double Lie algebroids. It is known that all these structures possess a unified description using the language of super\-geometry and graded manifolds of degree $\leq 2$. Indeed, a link has been established between the super and classical pictures by the geometrization process, leading to an equivalence of the category of graded manifolds of degree $\leq 2$ and the category of (double) vector bundles with additi…
We give a systematic derivation of the local expressions of the NS H-flux, geometric F- as well as non-geometric Q- and R-fluxes in terms of bivector beta- and two-form B-potentials including vielbeins. They are obtained using a supergeometric method on QP-manifolds by twist of the standard Courant algebroid on the generalized tangent space without flux. Bianchi identities of the fluxes are easily deduced. We extend the discussion to the case of the double space and present a formulation of T-duality in terms of canonical transformations between graded symplectic manifolds. Finally, the constr…
Branched covers of orbit cylinders are the basic examples of holomorphic curves studied in symplectic field theory. Since all curves with Fredholm index one can never be regular for any choice of cylindrical almost complex structure, we generalize the obstruction bundle technique of Taubes for determining multiple cover contributions from Gromov-Witten theory to the case of moduli spaces with boundary. Our result proves that the differential in rational symplectic field theory and contact homology is strictly decreasing with respect to the natural action filtration.
We describe a reduction process for symplectic principal $\mathbb{R}$-bundles in the presence of a momentum map. This type of structures plays an important role in the geometric formulation of non-autonomous Hamiltonian systems. We apply this procedure to the standard symplectic principal $\mathbb{R}$-bundle associated with a fibration $π:M\to\mathbb{R}$. When $π$ is a principal $G$-bundle and $G_ν$ denotes the isotropy group associated with an element $ν$ in the dual to the Lie algebra of $G$, we use the reduction process in order to describe a Poisson structure on the quotient manifold $M/G_…
We show that, for any two orientable smooth open 4-manifolds $X_0,X_1$ which are homeomorphic, their cotangent bundles $T^*X_0,T^*X_1$ are symplectomorphic with their canonical symplectic structure. In particular, for any smooth manifold $R$ homeomorphic to $\mathbb{R}^4$, the standard Stein structure on $T^*R$ is Stein homotopic to the standard Stein structure on $T^*\mathbb{R}^4 = \mathbb{R}^8$. We use this to show that any exotic $\mathbb{R}^4$ embeds in the standard symplectic $\mathbb{R}^8$ as a Lagrangian submanifold. As a corollary, we show that $\mathbb{R}^8$ has uncountably many smoot…
We prove that a compact 4-manifold which supports a circle-invariant fat SO(3)-bundle is diffeomorphic to either S^4 or CP^2-bar. The proof involves studying the resulting Hamiltonian circle action on an associated symplectic 6-manifold. Applying our result to the twistor bundle of Riemannian 4-manifolds shows that S^4 and CP^2-bar are the only 4-manifolds admitting circle-invariant metrics solving a certain curvature inequality. This can be seen as an analogue of Hsiang-Kliener's theorem that only S^4 and CP^2 admit circle-invariant metrics of positive sectional curvature.
Recently, Tsai-Tseng-Yau constructed new invariants of symplectic manifolds: a sequence of Aoo-algebras built of differential forms on the symplectic manifold. We show that these symplectic Aoo-algebras have a simple topological interpretation. Namely, when the cohomology class of the symplectic form is integral, these Aoo-algebras are equivalent to the standard de Rham differential graded algebra on certain odd-dimensional sphere bundles over the symplectic manifold. From this equivalence, we deduce for a closed symplectic manifold that Tsai-Tseng-Yau's symplectic Aoo-algebras satisfy the Cal…
We give an overview of the work of Corlette, Donaldson, Hitchin and Simpson leading to the non-abelian Hodge theory correspondence between representations of the fundamental group of a surface and the moduli space of Higgs bundles. We then explain how this can be generalized to a correspondence between character varieties for representations of surface groups in real Lie groups G and the moduli space of G-Higgs bundles. Finally we survey recent joint work with Bradlow, García-Prada and Mundet i Riera on the moduli space of maximal Sp(2n,R)-Higgs bundles.
In this paper we develop a geometric approach to higher order mechanics on graded bundles in both, the Lagrangian and Hamiltonian formalism, via the recently discovered weighted algebroids. We present the corresponding Tulczyjew triple for this higher order situation and derive in this framework the phase equations from an arbitrary (also singular) Lagrangian or Hamiltonian, as well as the Euler-Lagrange equations. As important examples, we geometrically derive the classical higher order Euler-Lagrange equations and analogous reduced equations for invariant higher order Lagrangians on Lie grou…
Generalized geometry finds many applications in the mathematical description of some aspects of string theory. In a nutshell, it explores various structures on a generalized tangent bundle associated to a given manifold. In particular, several integrability conditions can be formulated in terms of a canonical Dorfman bracket, an example of Courant algebroid. On the other hand, smooth manifolds can be generalized to involve functions of $\mathbb{Z}$-graded variables which do not necessarily commute. This leads to a mathematical theory of graded manifolds. It is only natural to combine the two t…
We study noncommutative generalizations of such notions of the classical symplectic geometry as degenerate Poisson structure, Poisson submanifold and quotient manifold, symplectic foliation and symplectic leaf for associative Poisson algebras. We consider these structures for the case of the endomorphism algebra of a vector bundle, and give the full description of the family of Poisson structures for this algebra.
We extend the AKSZ formulation of the Poisson sigma model to more general target spaces, and we develop the general theory of graded geometry for poly-symplectic and poly-Poisson structures. In particular we prove a Schwarz-type theorem and transgression for graded poly-symplectic structures, recovering the action functional and the poly-symplectic structure of the reduced phase space of the poly-Poisson sigma model, from the AKSZ construction.
Let $X$ be a compact connected Riemann surface and $D$ an effective divisor on $X$. Let ${\mathcal N}_H(r,d)$ denote the moduli space of $D$-twisted stable Higgs bundles (a special class of Hitchin pairs) on $X$ of rank $r$ and degree $d$. It is known that ${\mathcal N}_H(r,d)$ has a natural holomorphic Poisson structure which is in fact symplectic if and only if $D$ is the zero divisor. We prove that ${\mathcal N}_H(r,d)$ admits a natural enhancement to a holomorphic symplectic manifold which is called here ${\mathcal M}_H(r,d)$. This ${\mathcal M}_H(r,d)$ is constructed by trivializing, over…
In this paper, we compute contact homology of some quasi-regular contact structures, which admit Hamiltonian actions of Reeb type of Lie groups. We will discuss the toric contact case, (where the torus is of Reeb type), and the case of homogeneous contact manifolds. In both of these cases the quotients by the Reeb action are Kähler and admit perfect Morse-Bott functions via the moment map. It turns out that the contact homology depends only on the homology of the symplectic base and the bundle data of the contact manifolds as a circle bundle over the base. Moreover we can identify the relevant…
These notes give an introduction to Geometric Invariant Theory and symplectic reduction, with lots of pictures and simple examples. We describe their applications to moduli of bundles and varieties, and their infinite dimensional analogues in gauge theory and the theory of special metrics on algebraic varieties. Donaldson's "quantisation" link between the infinite and finite dimensional situations is described, as are surprisingly strong connections between the bundle and variety cases.
We consider the Bochner Laplacian on high tensor powers of a positive line bundle on a closed symplectic manifold (or, equivalently, the semiclassical magnetic Schrödinger operator with the non-degenerate magnetic field). We assume that the operator has discrete wells. The main result of the paper states asymptotic expansions for its low-lying eigenvalues.
We show that if K: P \to R is an autonomous Hamiltonian on a symplectic manifold (P,Ω) which attains 0 as a Morse-Bott nondegenerate minimum along a symplectic submanifold M, and if c_1(TP)|_M vanishes in real cohomology, then the Hamiltonian flow of K has contractible periodic orbits with bounded period on all sufficiently small energy levels. As a special case, if the geodesic flow on the cotangent bundle of M is twisted by a symplectic magnetic field form, then the resulting flow has contractible periodic orbits on all low energy levels. These results were proven by Ginzburg and Gürel when …
We prove a theorem on singular symplectic cotangent bundle reduction in the Fréchet setting and apply it to Yang-Mills-Higgs theory with special emphasis on the Higgs sector of the Glashow-Weinberg-Salam model. For the latter model we give a detailed description of the reduced phase space and show that the singular structure is encoded in a finite-dimensional Lie group action.
We give a comparative description of the Poisson structures on the moduli spaces of flat connections on real surfaces and holomorphic Poisson structures on the moduli spaces of holomorphic bundles on complex surfaces. The symplectic leaves of the latter are classified by restrictions of the bundles to certain divisors. This can be regarded as fixing a "complex analogue of the holonomy" of a connection along a "complex analogue of the boundary" in analogy with the real case.
Complementing the previous paper in the series, this paper classifies $|2|$-graded parabolic geometries, listing their important properties: the group $G_0$, the graded tangent bundle $gr(T)$ and its algebraïc bracket, the relevant cohomology spaces and the standard Tractor bundle $\mc{T}$. Several of these geometries are then explored in more detail, and the paper ends with a case study that that partially solves the equivalence problem for generic six distributions on nine dimensional manifolds.
We continue our study of symplectically flat bundles. We broaden the notion of symplectically flat connections on symplectic manifolds to $ζ$-flat connections on smooth manifolds. These connections on principal bundles can be represented by maps from an extension of the base's fundamental group to the structure group. We introduce functionals with zeroes being symplectically flat connections and study their critical points. Such functionals lead to novel geometric flows. We also describe some characteristic classes of the $ζ$-flat bundles.
We define and make initial study of Lie groupoids equipped with a compatible homogeneity (or graded bundle) structure, such objects we will refer to as weighted Lie groupoids. One can think of weighted Lie groupoids as graded manifolds in the category of Lie groupoids. This is a very rich geometrical theory with numerous natural examples. Note that $\mathcal{VB}$-groupoids, extensively studied in the recent literature, form just the particular case of weighted Lie groupoids of degree one. We examine the Lie theory related to weighted groupoids and weighted Lie algebroids, objects defined in a …
Let $S$ be a spinor bundle of a pseudo-Euclidean vector bundle $(E,\mathrm{g})$ of even rank. We introduce a new filtration on the algebra $\mathcal{D}(M,S)$ of differential operators on $S$. As main property, the associated graded algebra $\mathrm{gr}\mathcal{D}(M,S)$ is isomorphic to the algebra $\mathcal{O}(\mathcal{M})$ of functions on $\mathcal{M}$, where $\mathcal{M}$ is the symplectic graded manifold of degree $2$ canonically associated to $(E,\mathrm{g})$. Accordingly, we define the Weyl quantization on $\mathcal{M}$ as a map $\mathcal{WQ}_\hbar:\mathcal{O}(\mathcal{M})\to\mathcal{D}(M…
We prove a version of the Arnol'd conjecture for Lagrangian submanifolds of conformal symplectic manifolds: a Lagrangian $L$ which has non-zero Morse-Novikov homology for the restriction of the Lee form $β$ cannot be disjoined from itself by a $C^0$-small Hamiltonian isotopy. Furthermore for generic such isotopies the number of intersection points equals at least the sum of the free Betti numbers of the Morse-Novikov homology of $β$. We also give a short exposition of conformal symplectic geometry, aimed at readers who are familiar with (standard) symplectic or contact geometry.
We generalize the Donagi and Witten construction of a first obstruction class for splitting of a supermanifold via differential operators using the theory of $n$-fold vector bundles and graded manifolds. Applying the generalized Donagi--Witten construction we obtain a family of embeddings of the category of supermanifolds into the category of $n$-fold vector bundles and into the category of graded manifolds. This leads to a realization of any non-split supermanifold in terms of a collection of vector bundles and some morphism between them. Further we study the images of these embeddings into t…
Roughly speaking, $\mathbb{Z}_2^n$-manifolds are `manifolds' equipped with $\mathbb{Z}_2^n$-graded commutative coordinates with the sign rule being determined by the scalar product of their $\mathbb{Z}_2^n$-degrees. We examine the notion of a symplectic $\mathbb{Z}_2^n$-manifold, i.e., a $\mathbb{Z}_2^n$-manifold equipped with a symplectic two-form that may carry non-zero $\mathbb{Z}_2^n$-degree. We show that the basic notions and results of symplectic geometry generalise to the `higher graded' setting, including a generalisation of Darboux's theorem.
In this paper we prequantize the moduli space of non-abelian vortices. We explicitly calculate the symplectic form arising from the $L^2$ metric and we construct a prequantum line bundle whose curvature is proportional to this symplectic form. The prequantum line bundle turns out to be Quillen's determinant line bundle with a modified Quillen metric. Next, as in the case of abelian vortices, we construct Quillen line bundles over the moduli space whose curvatures form a family of symplectic forms which are parametrised by $Ψ_0$, a section of a certain bundle.
Geometric structures on $\mathbb N Q$-manifolds, i.e.~non-negatively graded manifolds with an homological vector field, encode non-graded geometric data on Lie algebroids and their higher analogues. A particularly relevant class of structures consists of vector bundle valued differential forms. Symplectic forms, contact structures and, more generally, distributions are in this class. We describe vector bundle valued differential forms on non-negatively graded manifolds in terms of non-graded geometric data. Moreover, we use this description to present, in a unified way, novel proofs of known r…
