arXiv research

In the paper "Direct Images, Fields of Hilbert Spaces, and Geometric Quantization", Lempert and Szőke proved that any flat analytic Hilbert field will induce a hermitian Hilbert bundle and gave an example of a flat Hilbert field that does not induce any Hilbert bundle. In this paper, we will provide an example of an an…

New Hilbert bundles with ends defined from indexed bases.

Holonomies match for higher local systems and principal 2-bundles.

Establishes a framework for stringor bundles, proving their canonical isomorphism to Stolz-Teichner's.

Proves elliptic operator images are closed on Hilbert bundles.

Constructs stable Hilbert bundles on curves using Diophantine approximation.

Explains historical connections between vector bundle splitting and Riemann-Hilbert problems.

Study proves existence of precotangent bundles for Grassmannians.

Ph.D. thesis on complex Brunn-Minkowski theory using Hilbert bundles.

We construct a prequantum 2-Hilbert space for any line bundle gerbe whose Dixmier-Douady class is torsion. Analogously to usual prequantisation, this 2-Hilbert space has the category of sections of the line bundle gerbe as its underlying 2-vector space. These sections are obtained as certain morphism categories in Wald…

Defines complex structure for families of Hilbert spaces with reasonable curvature.

We present a construction of a 2-Hilbert space of sections of a bundle gerbe, a suitable candidate for a prequantum 2-Hilbert space in higher geometric quantisation. We introduce a direct sum on the morphism categories in the 2-category of bundle gerbes and show that these categories are cartesian monoidal and abelian.…

For a symplectic manifold admitting a metaplectic structure and for a Kuiper map, we construct a complex of differential operators acting on exterior differential forms with values in the dual of the Kostant's symplectic spinor bundle. Defining a Hilbert $C^*$-structure on this bundle for a suitable $C^*$-algebra, we o…

We prove an arithmetic Hilbert-Samuel type theorem for semi-positive singular hermitian line bundles of finite height. In particular, the theorem applies to the log-singular metrics of Burgos-Kramer-Kühn. Our theorem is thus suitable for application to some non-compact Shimura varieties with their bundles of cusp forms…

We derive a general obstruction to the existence of Riemannian metrics of positive scalar curvature on closed spin manifolds in terms of hypersurfaces of codimension two. The proof is based on coarse index theory for Dirac operators that are twisted with Hilbert C*-module bundles. Along the way we give a complete and s…

Paper generalizes complex Brunn-Minkowski theory and proves new extension theorems.

GCNNs on homogeneous spaces use vector bundles and Hilbert spaces.

Geometric quantization often produces not one Hilbert space to represent the quantum states of a classical system but a whole family $H_s$ of Hilbert spaces, and the question arises if the spaces $H_s$ are canonically isomorphic. [ADW] and [Hi] suggest to view $H_s$ as fibers of a Hilbert bundle $H$, introduce a connec…

We study a problem of the geometric quantization for the quaternion projective space. First we explain a Kaehler structure on the punctured cotangent bundle of the quaternion projective space, whose Kaehler form coincides with the natural symplectic form on the cotangent bundle and show that the canonical line bundle o…

When there is a family of complex structures on the phase space, parametrized by a set $S$, the prequantum Hilbert spaces produced by geometric quantization, using the half-form correction, also depends on these parameters. This way we obtain a field of Hilbert spaces $p:H^{pr Q}\rightarrow S$. We show that this field …

In this paper, we use the parametrised strict deformation quantization of C*-bundles obtained in a previous paper, and give more examples and applications of this theory. In particular, it is used here to classify H_3-twisted noncommutative torus bundles over a locally compact space. This is extended to the case of gen…

The paper studies Einstein-Hilbert functional and its relation to K-semistability.

We investigate when the idempotent barycenter map restricted to the points with no-trivial fibers is a trivial bundle with the fiber Hilbert cube.

Dirac operator invertibility proven for specific manifolds.

Introduces non-abelian Hodge correspondence linking algebraic structures to geometry.

The paper solves Riemann-Hilbert problems using framed holomorphic bundles.

Formulates Hilbert reciprocity law on 3-manifolds.

We consider the construction of the basic bundle gerbe on SU(n) introduced by Meinrenken and show that it extends to a range of groups with unitary actions on a Hilbert space including U(n), diagonal tori and the Banach Lie group of unitary operators differing from the identity by an element of a Schatten ideal. In all…

Abstract: Study of metrics on line bundles over complex varieties.

Parseval frames can be thought of as redundant or linearly dependent coordinate systems for Hilbert spaces, and have important applications in such areas as signal processing, data compression, and sampling theory. We extend the notion of a Parseval frame for a fixed Hilbert space to that of a moving Parseval frame for…

In this article we calculate the dimension of the Hilbert space of Kahler quantization of the moduli space of vortices on a Riemann surface. This dimension is given by the holomorphic Euler characteristic of the quantum line bundle.

Derives the derivative of the Riemann-Hilbert map for surface connections.

New construction of isoparametric submanifolds in Hilbert spaces.

In this paper, we consider the similarity and quasi-affinity problems for Hilbert modules in the Cowen-Douglas class associated with the complex geometric objects, the hermitian anti-holomorphic vector bundles and curvatures. Given a "simple" rank one Cowen-Douglas Hilbert module $\mathcal{M}$, we find necessary and su…

Develops noncommutative Cowen-Douglas theory for noncommuting operators.

The paper studies the topology of hyperspaces of k-dimensional convex sets.

We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one but it is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it a pure state of some quantum s…

The Serre-Swan theorem in differential geometry establishes an equivalence between the category of smooth vector bundles over a smooth compact manifold and the category of finitely generated projective modules over the unital ring of smooth functions. This theorem is here generalized to manifolds of bounded geometry. I…

New criterion for solving inverse Hessian equations, including J-equation.

For a certain class of complexes of pre-Hilbert $A$-modules, we prove that their cohomology groups equipped with a canonical quotient structure are again pre-Hilbert $A$-modules and derive the Hodge decomposition for them. We call these complexes self-adjoint parametrix possessing. We show that $A$-elliptic complexes o…

A local Riemann-Hilbert correspondence for tame meromorphic connections on a curve compatible with a parahoric level structure will be established. Special cases include logarithmic connections on G-bundles and on parabolic G-bundles, where G is a complex reductive group. The corresponding Betti data involves pairs (M,…

In this paper we interpret the integrability of the Dirac structures on some Hilbert C*-modules in terms of an automorphism group. This is the group of orthogonal transformations on the Hilbert C*-module of sections of a Hermitian vector bundle over an smooth manifold M. Some topological properties of the group of inte…

We sharpen the construction of representation space in the paper "Principal Series Representations of Infinite Dimensional Lie Groups II: Construction of Induced Representations". We show that the principal series representation spaces constructed there, are completions of spaces of sections of Hilbert bundles rather t…

Constructs hyperkähler metrics on Higgs bundle moduli spaces using Gaiotto coordinates.

Under appropriate assumptions, we generalize the concept of linear almost Poisson struc- tures, almost Lie algebroids, almost differentials in the framework of Banach anchored bundles and the relation between these objects. We then obtain an adapted formalism for mechanical systems which is illustrated by the evolution…

We give an infinite dimensional description of the differential K-theory of a manifold $M$. The generators are triples $[H, A, ω]$ where $H$ is a ${\bf Z}_2$-graded Hilbert bundle on $M$, $A$ is a superconnection on $H$ and $ω$ is a differential form on $M$. The relations involve eta forms. We show that the ensuing gro…

Study of real and quaternionic Lie algebroid connections on manifolds.

We introduce a Hilbert $A$-module structure on the higher oscillatory module, where $A$ denotes the $C^*$-algebra of bounded endomorphisms of the basic oscillatory module. We also define the notion of an exterior covariant derivative in an $A$-Hilbert bundle and use it for a construction of an $A$-elliptic complex of d…