arXiv research

This paper provides theoretical foundations for using quantized actions in behavior cloning.

We construct Hermitian representations of Lie algebroids and associated unitary representations of Lie groupoids by a geometric quantization procedure. For this purpose we introduce a new notion of Hamiltonian Lie algebroid actions. The first step of our procedure consists of the construction of a prequantization line …

Geometric quantization for specific symplectic structures proved.

We present explicit universal strict deformation quantization formulae for actions of Iwasawa subgroups AN of SU(1,n). This answers a question raised by Rieffel.

Quantization and reduction studied for CR manifolds with group actions.

We prove several versions of "quantization commutes with reduction" for circle actions on manifolds that are not symplectic. Instead, these manifolds possess a weaker structure, such as a spin^c structure. Our theorems work whenever the quantization data and the reduction data are compatible; this condition always hold…

We study a notion of pre-quantization for $b$-symplectic manifolds. We use it to construct a formal geometric quantization of $b$-symplectic manifolds equipped with Hamiltonian torus actions with nonzero modular weight. We show that these quantizations are finite dimensional $T$-modules.

Symmetries of Poisson manifolds are in general quantized just to symmetries up to homotopy of the quantized algebra of functions. It is therefore interesting to study symmetries up to homotopy of Poisson manifolds. We notice that they are equivalent to Poisson principal bundles and describe their quantization to symmet…

We first introduce an invariant index for G-equivariant elliptic differential operators on a locally compact manifold M admitting a proper cocompact action of a locally compact group G. It generalizes the Kawasaki index for orbifolds to the case of proper cocompact actions. Our invariant index is used to show that an a…

Quantizes $b$-symplectic toric manifolds using $T$-modules.

We establish a geometric quantization formula for a Hamiltonian action of a compact Lie group acting on a noncompact symplectic manifold with proper moment map.

By computing certain cohomology of Vect(M) of smooth vector fields we prove that on 1-dimensional manifolds M there is no quantization map intertwining the action of non-projective embeddings of the Lie algebra sl(2) into the Lie algebra Vect(M). Contrariwise, for projective embeddings sl(2)-equivariant quantization ex…

For any Lie groupoid $G$, the vector bundle $g^*$ dual to the associated Lie algebroid $g$ is canonically a Poisson manifold. The (reduced) C*-algebra of $G$ (as defined by A. Connes) is shown to be a strict quantization (in the sense of M. Rieffel) of $g^*$. This is proved using a generalization of Weyl's quantization…

New method shows unitarity in quantization for toric manifolds.

A G-equivariant spin^c structure on a manifold gives rise to a virtual representation of the group G, called the spin^c quantization of the manifold. We present a cutting construction for S^1-equivariant spin^c manifolds, and show that the quantization of the original manifold is isomorphic to the direct sum of the qua…

Many mathematical models of physical phenomena that have been proposed in recent years require more general spaces than manifolds. When taking into account the symmetry group of the model, we get a reduced model on the (singular) orbit space of the symmetry group action. We investigate quantization of singular spaces o…

Quantizes contact structures using dynamical methods.

We construct an action of the braid group B_N on the twisted quantized enveloping algebra U'_q(o_N) where the elements of B_N act as automorphisms. In the classical limit q -> 1 we recover the action of B_N on the polynomial functions on the space of upper triangular matrices with ones on the diagonal. The action prese…

This work is a contribution to the area of Strict Quantization (in the sense of Rieffel) in the presence of curvature and non-Abelian group actions. More precisely, we use geometry to obtain explicit oscillatory integral formulae for strongly invariant strict deformation quantizations of a class of solvable symplectic …

Geometric quantization shows compatibility of symmetries on coadjoint orbits and Kähler-Einstein manifolds.

One of the most significant bottleneck in training large scale machine learning models on parameter server (PS) is the communication overhead, because it needs to frequently exchange the model gradients between the workers and servers during the training iterations. Gradient quantization has been proposed as an effecti…

Geometric quantization on hyperKähler manifolds via brane quantization.

Study quantized SL2-character variety of a once-punctured torus, finding three Coulomb branch isomorphisms.

Quantizes Kähler manifolds using sheaves and differential operators.

We formulate a quantization commutes with reduction principle in the setting where the Lie group $G$, the symplectic manifold it acts on, and the orbit space of the action may all be noncompact. It is assumed that the action is proper, and the zero set of a deformation vector field, associated to the momentum map and a…

This paper connects symplectic and Kähler manifolds via brane quantization.

We prove a bubble-neck decomposition together with an energy quantization result for sequences of Willmore surfaces into an arbitrary euclidian space with uniformly bounded energy and non-degenerating conformal type. We deduce the strong compactness of Willmore closed surfaces of a given genus modulo the Möbius group a…

We study the dependence of geometric quantization of the standard symplectic torus on the choice of invariant polarization. Real and mixed polarizations are interpreted as degenerate complex structures. Using a weak version of the equations of covariant constancy, and the Weil-Brezin expansion to describe distributiona…

In this note, we will show one example of hamiltonian Lie algebra action which has no invariant star product.

Conditions for pre-quantizability of G-invariant forms are derived using moment maps.

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…

We adapt the framework of geometric quantization to the polysymplectic setting. Considering prequantization as the extension of symmetries from an underlying polysymplectic manifold to the space of sections of a Hermitian vector bundle, a natural definition of prequantum vector bundle is obtained which incorporates in …

Mathematical framework for brane quantization using SYZ mirror symmetry.

The full 6d Hopf-Wess-Zumino term in the action functional for the M5-brane is anomalous as traditionally defined. What has been missing is a condition implying the higher analogue of level quantization familiar from the 2d Wess-Zumino term. We prove that the anomaly cancellation condition is implied by the hypothesis …

It is postulated that quantum gravity is a sum over causal structures coupled to matter via scale evolution. Quantized causal structures can be described by studying simple matrix models where matrices are replaced by an algebra of quantum mechanical observables. In particular, previous studies constructed quantum grav…

Let $X$ be a compact connected orientable CR manifold with the action of a connected compact Lie group $G$. Under natural pseudoconvexity assumptions we show that the CR Guillemin-Strernberg map is Fredholm at the level of Sobolev spaces of CR functions. As application we study this map for holomorphic line bundles whi…

Formally equates two quantization methods and constructs non-commutative algebras.

The paper provides formulas linking knot invariants to deformation quantization.

Inverse metric matrices on Siegel-Jacobi spaces are calculated for Berezin quantization.

Deep Neural Networks (DNNs) typically require massive amount of computation resource in inference tasks for computer vision applications. Quantization can significantly reduce DNN computation and storage by decreasing the bitwidth of network encodings. Recent research affirms that carefully selecting the quantization l…

The paper improves quantization error estimates on Riemannian manifolds.

In this article we develop tools to compute the Geometric Quantization of a symplectic manifold with respect to a regular Lagrangian foliation via sheaf cohomology and obtain important new applications in the case of real polarizations. The starting point is the definition of representation spaces due to Kostant. Besid…

Motivated by topology, we develop a general theory of traces and shadows for an endobicategory, which is a~pair: bicategory $\mathbf{C}$ and endobifunctor $Σ\colon \mathbf C \to\mathbf C$. For a graded linear bicategory and a fixed invertible parameter $q$, we quantize this theory by using the endofunctor $Σ_q$ such th…

Using the results and techniques of a previous paper where we proved the quantization of gravity we extend the former result by adding a Yang-Mills functional and a Higgs term to the Einstein-Hilbert action.

In the case of a compact real analytic symplectic manifold M we describe an approach to the complexification of Hamiltonian flows [Se, Do1, Th1] and corresponding geodesics on the space of Kahler metrics. In this approach, motivated by recent work on quantization, the complexified Hamiltonian flows act, through the Gro…

We give a formulation of a deformation of Dirac operator along orbits of a group action on a possibly non-compact manifold to get an equivariant index and a K-homology cycle representing the index. We apply this framework to non-compact Hamiltonian torus manifolds to define geometric quantization from the view point of…

On every split supermanifold equipped with the Rothstein even super-Poisson bracket we construct a deformation quantization by means of a Fedosov-type procedure. In other words, the supercommutative algebra of all smooth sections of the dual Grassmann algebra bundle of an arbitrarily given vector bundle E (equipped wit…

Quantization and reduction for coisotropic A-branes on Hamiltonian manifolds.