Symplectic Vector Bundle Research Articles

On lifts of symplectic vector bundles and connections to Weil bundles

On lifts of symplectic vector bundles and connections to Weil bundles

The geometry and DSZ quantization four-dimensional supergravity

We implement the Dirac-Schwinger-Zwanziger integrality condition on four-dimensional classical ungauged supergravity and use it to obtain its duality-covariant, gauge-theoretic, differential-geometric model on an oriented four-manifold $M$ of arbitrary topology. Classical bosonic supergravity is completely determined by a submersion $\pi$ over $M$ equipped with a complete Ehresmann connection, a vertical euclidean metric, and a vertically-polarized flat symplectic vector bundle $\Xi$. Building on these structures, we implement the Dirac-Schwinger-Zwanziger integrality condition through the choice of an element in the degree-two sheaf cohomology group with coefficients in a locally constant sheaf $\mathcal{L}\subset \Xi$ valued in the groupoid of integral symplectic spaces. We show that this data determines a Siegel principal bundle $P_{\mathfrak{t}}$ of fixed type $\mathfrak{t}\in \mathbb{Z}^{n_v}$ whose connections provide the global geometric description of the local electromagnetic gauge potentials of the theory. Furthermore, we prove that the Maxwell gauge equations of the theory reduce to the polarized self-duality condition determined by $\Xi$ on the connections of $P_{\mathfrak{t}}$. In addition, we investigate the continuous and discrete U-duality groups of the theory, characterizing them through short exact sequences and realizing the latter through the gauge group of $P_{\mathfrak{t}}$ acting on its adjoint bundle. This elucidates the geometric origin of U-duality, which we explore in several examples, illustrating its dependence on the topology of the fiber bundles $\pi$ and $P_{\mathfrak{t}}$ as well as on the isomorphism type of $\mathcal{L}$.

Irreducibility of Lagrangian Quot schemes over an algebraic curve

Let C be a complex projective smooth curve and W a symplectic vector bundle of rank 2n over C. The Lagrangian Quot scheme LQ_{-e}(W) parameterizes subsheaves of rank n and degree -e which are isotropic with respect to the symplectic form. We prove that LQ_{-e}(W) is irreducible and generically smooth of the expected dimension for all large e, and that a generic element is saturated and stable.

Enumerating Non-Stable Vector Bundles

Abstract In this article, we establish a motivic analog of an enumeration result of James–Thomas [ 28] on non-stable vector bundles in topological setting. Using this, we obtain results on enumeration of projective modules of rank $d$ over a smooth affine $k$-algebra $A$ of dimension $d$, recovering in particular results of Suslin and Bhatwadekar on cancellation of such vector bundles. Admitting a conjecture of Asok and Fasel, we prove cancellation of such vector bundles of rank $d-1$ if the base field $k$ is algebraically closed. We also explore the cancellation properties of symplectic vector bundles.

Construction of symplectic vector bundles on projective space [formula omitted

The moduli spaces of symplectic vector bundles of arbitrary rank on projective space P3 are far from being well-understood. By now the only type of such bundles having satisfactory description are the so-called tame symplectic instantons. It is shown by U. Bruzzo, D. Markushevich and the first author in two papers from 2012 and 2016 that the moduli spaces of tame symplectic instantons are irreducible generically reduced algebraic spaces of dimension prescribed by the deformation theory. In the present paper we construct an infinite series of smooth irreducible moduli components of symplectic vector bundles of an arbitrary even rank 2r,r≥1, obtained by an iterative use of the monad construction applied to tame symplectic instantons. As a particular case we obtain an infinite series of irreducible moduli components of stable rank 2 vector bundles on P3. We show that this series contains as a subseries a large part of an infinite series of moduli components constructed by the authors and S. Tikhomirov in 2019. We also prove that, for any integers n,r, where r≥1 and n≥r+147, there exists a moduli component, not necessarily unique, of our series such that symplectic bundles from this component have rank 2r and second Chern class n.

Bifurcations on contact manifolds

Consider a 1-parameter compactly supported family of Legendrian submanifolds of the 1-jet bundle of a compact manifold with its natural contact structure and a path of intersection points of the Legendrian family with the 1-jet of a constant function. Since the contact distribution is a symplectic vector bundle, it is possible to assign a Maslov-type index to the intersection path. We show that the non-vanishing of the Maslov intersection index implies that there exists at least one point of bifurcation from the given path of intersection points. This result can be viewed as a kind of analogue in bifurcation theory of the Arnold-Sandon conjecture on intersections of Legendrian submanifols. The proof is based on the technique of generating functions that relates the properties of Hamiltonian diffeomorphisms to the Morse theory of the associated functions.

Generalized Einstein-Scalar-Maxwell theories and locally geometric U-folds

We give the global mathematical formulation of the coupling of four-dimensional scalar sigma models to Abelian gauge fields on a Lorentzian four-manifold, for the generalized situation when the duality structure of the Abelian gauge theory is described by a flat symplectic vector bundle [Formula: see text] defined over the scalar manifold [Formula: see text]. The construction uses a taming of [Formula: see text], which we find to be the correct mathematical object globally encoding the inverse gauge couplings and theta angles of the “twisted” Abelian gauge theory in a manner that makes no use of duality frames. We show that global solutions of the equations of motion of such models give classical locally geometric U-folds. We also describe the groups of duality transformations and scalar-electromagnetic symmetries arising in such models, which involve lifting isometries of [Formula: see text] to the bundle [Formula: see text] and hence differ from expectations based on local analysis. The appropriate version of the Dirac quantization condition involves a discrete local system defined over [Formula: see text] and gives rise to a smooth bundle of polarized Abelian varieties, endowed with a flat symplectic connection. This shows, in particular, that a generalization of part of the mathematical structure familiar from [Formula: see text] supergravity is already present in such purely bosonic models, without any coupling to fermions and hence without any supersymmetry.

Section sigma models coupled to symplectic duality bundles on Lorentzian four-manifolds

We give the global mathematical formulation of a class of generalized four-dimensional theories of gravity coupled to scalar matter and to Abelian gauge fields. In such theories, the scalar fields are described by a section of a surjective pseudo-Riemannian submersion π over space–time, whose total space carries a Lorentzian metric making the fibers into totally-geodesic connected Riemannian submanifolds. In particular, π is a fiber bundle endowed with a complete Ehresmann connection whose transport acts through isometries between the fibers. In turn, the Abelian gauge fields are “twisted” by a flat symplectic vector bundle defined over the total space of π. This vector bundle is endowed with a vertical taming which locally encodes the gauge couplings and theta angles of the theory and gives rise to the notion of twisted self-duality, of crucial importance to construct the theory. When the Ehresmann connection of π is integrable, we show that our theories are locally equivalent to ordinary Einstein-Scalar-Maxwell theories and hence provide a global non-trivial extension of the universal bosonic sector of four-dimensional supergravity. In this case, we show using a special trivializing atlas of π that global solutions of such models can be interpreted as classical “locally-geometric” U-folds. In the non-integrable case, our theories differ locally from ordinary Einstein-Scalar-Maxwell theories and may provide a geometric description of classical U-folds which are “locally non-geometric”.

Spinors, Lagrangians and rank 2 Higgs bundles

The paper considers the Dirac operator on a Riemann surface coupled to a symplectic holomorphic vector bundle W. Each spinor in the null-space generates through the moment map a Higgs bundle, and varying W one obtains a holomorphic Lagrangian subvariety in the moduli space of Higgs bundles. Applying this to the irreducible symplectic representations of SL(2) we obtain Lagrangian submanifolds of the rank 2 moduli space which link up with m-period points on the Prym variety of the spectral curve as well as Brill-Noether loci on the moduli space of semistable bundles. The case of genus 2 is investigated in some detail.

Non-defectivity of Grassmannian bundles over a curve

Let [Formula: see text] be the Grassmann bundle of two-planes associated to a general bundle [Formula: see text] over a curve [Formula: see text]. We prove that an embedding of [Formula: see text] by a certain twist of the relative Plücker map is not secant defective. This yields a new and more geometric proof of the Hirschowitz-type bound on the isotropic Segre invariant for maximal isotropic sub-bundles of orthogonal bundles over [Formula: see text], analogous to those given for vector bundles and symplectic bundles in [I. Choe and G. H. Hitching, Secant varieties and Hirschowitz bound on vector bundles over a curve, Manuscripta Math. 133 (2010) 465–477, I. Choe and G. H. Hitching, Lagrangian sub-bundles of symplectic vector bundles over a curve, Math. Proc. Cambridge Phil. Soc. 153 (2012) 193–214]. From the non-defectivity, we also deduce an interesting feature of a general orthogonal bundle of even rank over [Formula: see text], contrasting with the classical and symplectic cases: a general maximal isotropic sub-bundle of maximal degree intersects at least one other such sub-bundle in positive rank.

Chern–Weil Maslov index and its orbifold analogue

We give Chern-Weil definitions of the Maslov indices of bundle pairs over a Riemann surfacewith boundary, which consists of symplectic vector bundle onand a Lagrangian subbundle on @� as well as its gen- eralization for transversely intersecting Lagrangian boundary conditions. We discuss their properties and relations to the known topological definitions. As a main application, we extend Maslov index to the case with orbifold inte- rior singularites, via curvature integral, and find also an analogous topological definition in these cases.

Symplectic instanton bundles on $${\mathbb {P}}^3$$ P 3 and ’t Hooft instantons

We study the moduli space $$I_{n,r}$$ of rank-2r symplectic instanton vector bundles on $${\mathbb {P}}^3$$ with $$r\geqslant 2$$ and second Chern class $$n\geqslant r+1,\,n-r\equiv 1\,(\mathrm {mod}\,2)$$ . We introduce the notion of tame symplectic instantons by excluding a kind of pathological monads and show that the locus $$I^*_{n,r}$$ of tame symplectic instantons is irreducible and has the expected dimension equal to $$4n(r+1)-r(2r+1)$$ . The proof is inherently based on a relation between the spaces $$I^*_{n,r}$$ and the moduli spaces of ’t Hooft instantons.

Moduli of symplectic instanton vector bundles of higher rank on projective space ℙ3

Symplectic instanton vector bundles on the projective space ℙ3 constitute a natural generalization of mathematical instantons of rank-2. We study the moduli space I n;r of rank-2r symplectic instanton vector bundles on ℙ3 with r ≥ 2 and second Chern class n ≥ r, n ≡ r (mod 2). We introduce the notion of tame symplectic instantons by excluding a kind of pathological monads and show that the locus I * of tame symplectic instantons is irreducible and has the expected dimension, equal to 4n(r + 1) −r(2r + 1).

On the geometric quantization of contact manifolds

Suppose that ( M , E ) is a compact contact manifold, and that a compact Lie group G acts on M transverse to the contact distribution E . In an earlier paper, we defined a G -transversally elliptic Dirac operator ⁄ D b , constructed using a Hermitian metric h and connection ∇ on the symplectic vector bundle E → M , whose equivariant index is well-defined as a generalized function on G , and gave a formula for its index. By analogy with the geometric quantization of symplectic manifolds, the virtual G -representation Q ( M ) = [ ker ⁄ D b ] − [ ker ⁄ D b ∗ ] can be interpreted as the “quantization” of the contact manifold ( M , E ) ; the character of this representation is then given by the equivariant index of ⁄ D b . By defining contact analogues of the algebra of observables, prequantum line bundle and polarization, we further extend the analogy by giving a contact version of the Kostant–Souriau approach to quantization, and discussing the extent to which this approach is reproduced by the index-theoretic method.

Algebraic aspects of the Hirzebruch signature operator and applications to transitive Lie algebroids

The index of the classical Hirzebruch signature operator on a manifold M is equal to the signature of the manifold. The examples of Lusztig ((10), 1972) and Gromov ((4), 1985) present the Hirzebruch signature operator for the cohomology (of a manifold) with coefficients in a flat symmetric or symplectic vector bundle. In (6), we gave a signature operator for the cohomology of transitive Lie algebroids. In this paper, firstly, we present a general approach to the signature operator, and the above four examples become special cases of a single general theorem. Secondly, due to the spectral sequence point of view on the signature of the cohomology algebra of certain filtered DG-algebras, it turns out that the Lusztig and Gromov examples are important in the study of the signature of a Lie algebroid. Namely, under some natural and simple regularity assumptions on the DG-algebra with a decreasing filtration for which the second term lives in a finite rectangle, the signature of the second term of the spectral sequence is equal to the signature of the DG algebra. Considering the Hirzebruch-Serre spectral sequence for a transitive Lie algebroid A over a compact oriented manifold for which the top group of the real cohomology of A is nontrivial, we see that the second term is just identical to the Lusztig or Gromov example (depending on the dimension). Thus, we have a second signature operator for Lie algebroids.

RANK FOUR SYMPLECTIC BUNDLES WITHOUT THETA DIVISORS OVER A CURVE OF GENUS TWO

The moduli space [Formula: see text] of rank four semistable symplectic vector bundles over a curve X of genus two is an irreducible projective variety of dimension ten. Its Picard group is generated by the determinantal line bundle Ξ. The base locus of the linear system |Ξ| consists of precisely those bundles without theta divisors, that is, admitting nonzero maps from every line bundle of degree -1 over X. We show that this base locus consists of six distinct points, which are in canonical bijection with the Weierstrass points of the curve. We relate our construction of these bundles to another of Raynaud and Beauville using Fourier–Mukai transforms. As an application, we prove that the map sending a symplectic vector bundle to its theta divisor is a surjective map from [Formula: see text] to the space of even 4Θ divisors on the Jacobian variety of the curve.

Dual Double EPW-sextics and Their Periods

Eisenbud, Popescu and Walter have constructed certain singular sextic hypersurfaces (EPW-sextics) in P (Example (9.3) of [4]) which come provided with a natural double cover: we have shown [13] that the generic such double cover is a deformation of the Hilbert square of a K3 and that the family of double EPWsextics is a locally versal family of projective deformations of (K3). Thus the family of double EPW-sextics is similar to the family of Fano varieties of lines on a cubic 4-fold (see [2]), with the following difference: the Plucker ample divisor on the Fano variety of lines has square 6 for the Beauville-Bogomolov quadratic form (see [1, 2]) while the natural polarization of a double EPWsextic has square 2 (see [13]). Let Y ⊂ P be a generic EPW-sextic: we proved in [13] that the dual Y ∨ ⊂ (P5)∨ is another generic EPW-sextic. Thus we may associate to the natural double cover X of Y a “dual”variety X∨ namely the natural double cover of Y ∨. This construction defines a (rational) involution on the moduli space of double EPW-sextics. In [13] we showed that a generic EPW-sextic is not self-dual and hence the involution on the moduli space of double EPW-sextics is not the identity; in this paper we determine the relation between the periods of a double EPW-sextic and its dual. Before stating the result we recall the definition of EPW-sextics. Let V be a 6-dimensional C-vector space. Choose an isomorphism vol : ∧ V ∼ −→ C and let ω be the symplectic form on ∧V defined by wedge product followed by vol. Let P(V ) be the projective space of 1-dimensional sub vector spaces of V ; then ω gives ∧V ⊗OP(V ) the structure of a symplectic vector-bundle of rank 20. Let F be the sub-vector-bundle of ∧V ⊗OP(V ) whose fiber F[v] over [v] ∈ P(V ) consists of tensors divisible by v:

Subbundles of symplectic and orthogonal vector bundles over curves

AbstractWe review the notions of symplectic and orthogonal vector bundles over curves, and the connection between principal parts and extensions of vector bundles. We give a criterion for a certain extension of rank 2n to be symplectic or orthogonal. We then describe almost all of its rank n vector subbundles using graphs of sheaf homomorphisms, and give criteria for the isotropy of these subbundles. Finally, we sketch the use of these ideas in moduli questions for symplectic vector bundles. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Moduli of rank 4 symplectic vector bundles over a curve of genus 2

Let X be a complex projective curve which is smooth and irreducible of genus 2. The moduli space 2 of semistable symplectic vector bundles of rank 4 over X is a variety of dimension 10. After assembling some results on vector bundles of rank 2 and odd degree over X, we construct a generically finite cover of 2 by a family of 5-dimensional projective spaces, and outline some applications

Obstruction à la transversalité de lagrangiens

In this Note, we give a necessary and sufficient condition for Lagrangians in a symplectic vector bundle to be deformed stably into transversal Lagrangians. In the case of three Lagrangians, we show that the associated Grothendieck group can be identified with a Hermitian K-theory group. To cite this article: M. Karoubi, M.L. Lapa de Souza, C. R. Acad. Sci. Paris, Ser. I 342 (2006).