arXiv research

In [3], the authors showed the existence and the uniqueness of a sl(m+1,\R)-equivariant quantization in the non-critical situations. The curved generalization of the sl(m+1,\R)-equivariant quantization is the natural and projectively equivariant quantization. In [1] and [7], the existence of such a quantization was pro…

The paper extends equivariant quantization to super circles $S^{1|n}$ for $n\geqslant 3$.

The existence of a natural and projectively equivariant quantization in the sense of Lecomte [20] was proved recently by M. Bordemann [4], using the framework of Thomas-Whitehead connections. We give a new proof of existence using the notion of Cartan projective connections and we obtain an explicit formula in terms of…

Equivariant quantization is a new theory that highlights the role of symmetries in the relationship between classical and quantum dynamical systems. These symmetries are also one of the reasons for the recent interest in quantization of singular spaces, orbifolds, stratified spaces... In this work, we prove existence o…

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…

The concept of conformally equivariant quantizations was introduced by Duval, Lecomte and Ovsienko in \cite{DLO} for manifolds endowed with flat conformal structures. They obtained results of existence and uniqueness (up to normalization) of such a quantization procedure. A natural generalization of this concept is to …

Conformally equivariant quantization is a peculiar map between symbols of real weight $δ$ and differential operators acting on tensor densities, whose real weights are designed by $λ$ and $λ+δ$. The existence and uniqueness of such a map has been proved by Duval, Lecomte and Ovsienko for a generic weight $δ$. Later, Si…

Study non-commutative function algebras using contact geometry.

We investigate the concept of projectively equivariant quantization in the framework of super projective geometry. When the projective superalgebra pgl(p+1|q) is simple, our result is similar to the classical one in the purely even case: we prove the existence and uniqueness of the quantization except in some critical …

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…

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

We investigate the concept of equivariant quantization over the superspace R^{p+q|2r}, with respect to the orthosymplectic algebra osp(p+1,q+1|2r). Our methods and results vary upon the superdimension p+q-2r. When the superdimension is nonzero, we manage to obtain a result which is similar to the classical theorem of D…

We extend projectively equivariant quantization and symbol calculus to symbols of pseudo-differential operators. An explicit expression in terms of hypergeometric functions with noncommutative arguments is given. Some examples are worked out, one of them yielding a quantum length element on $S^3$.

We construct an explicit scheme to associate to any potential symbol an operator acting between sections of natural bundles (associated to irreducible representations) for a so-called AHS-structure. Outside of a finite set of critical (or resonant) weights, this procedure gives rise to a quantization, which is intrinsi…

Smooth approximations of Kähler-Ricci solitons found using quantized metrics and Futaki invariants.

Let G be a compact, simply connected Lie group. We develop a `quantization functor' from pre-quantized quasi-Hamiltonian G-spaces at level k to the fusion ring (Verlinde algebra) R_k(G). The quantization Q(M) is defined as a push-forward in twisted equivariant K-homology. It may be computed by a fixed point formula, si…

We study the existence of natural and projectively equivariant quantizations for differential operators acting between order 1 vector bundles over a smooth manifold M. To that aim, we make use of the Thomas-Whitehead approach of projective structures and construct a Casimir operator depending on a projective Cartan con…

Proves formality conjecture for Hamiltonian actions using equivariant Hochschild-Kostant-Rosenberg quasi-isomorphism.

In [8], P. Lecomte conjectured the existence of a natural and projectively equivariant quantization. In [1], M. Bordemann proved this existence using the framework of Thomas-Whitehead connections. In [9], we gave a new proof of the same theorem thanks to the Cartan connections. After these works there was no explicit f…

The spaces of higher-order differential operators (in Dimension 1|2), which are modules over the stringy Lie superalgebra K(2), are isomorphic to the corresponding spaces of symbols as orthosymplectic modules in non resonant cases. Such an osp (2|2)-equivariant quantization, which has been given in second-order differe…

This work takes place over a conformally flat spin manifold (M,g). We prove existence and uniqueness of the conformally equivariant quantization valued in spinor differential operators, and provide an explicit formula for it when restricted to first order operators. The Poisson algebra of symbols is realized as a space…

Study of $n$-ary differential operators on weighted densities with canonical symbol and quantization maps.

Let $(M,g)$ be a pseudo-Riemannian manifold and $F_λ(M)$ the space of densities of degree $λ$ on $M$. We study the space $D^2_{λ,μ}(M)$ of second-order differential operators from $F_λ(M)$ to $F_μ(M)$. If $(M,g)$ is conformally flat with signature $p-q$, then $D^2_{λ,μ}(M)$ is viewed as a module over the group of confo…

We prove the existence and the uniqueness of a conformally equivariant symbol calculus and quantization on any conformally flat pseudo-Riemannian manifold $(M,\rg)$. In other words, we establish a canonical isomorphism between the spaces of polynomials on $T^*M$ and of differential operators on tensor densities over $M…

Defines geometric quantization for non-compact Hamiltonian torus manifolds using index theory.

This article is a survey of recent work of the authors developing a new approach to quantization based on the equivariance with respect to some Lie group of symmetries. Examples are provided by conformal and projective differential geometry: given a smooth manifold M endowed with a flat conformal/projective structure, …

New framework solves string theory's RR-field tadpole cancellation problems.

Efficient equivariant MobileNetV2 for medical applications on mobile devices.

The elaboration of new quantization methods has recently developed the interest in the study of subalgebras of the Lie algebra of polynomial vector fields over a Euclidean space. In this framework, these subalgebras define maximal equivariance conditions that one can impose on a linear bijection between observables tha…

We develop a Chern character map for twisted equivariant non-abelian cohomology.

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…

We consider the space of differential operators $\mathcal{D}_{λμ}$ acting between $λ$- and $μ$-densities defined on $S^{1|2}$ endowed with its standard contact structure. This contact structure allows one to define a filtration on $\mathcal{D}_{λμ}$ which is finer than the classical one, obtained by writting a differen…

Let $G$ be a compact connected Lie group acting on a stable complex manifold $M$ with equivariant vector bundle $E$. Besides, suppose $φ$ is an equivariant map from $M$ to the Lie algebra $\mathfrak{g}$. We can define some equivalence relation on the triples $(M, E, φ)$ such that the set of equivalence classes form an …

The existence of a natural and projectively invariant quantization in the sense of P. Lecomte [Progr. Theoret. Phys. Suppl. (2001), no. 144, 125-132] was proved by M. Bordemann [math.DG/0208171], using the framework of Thomas-Whitehead connections. We extend the problem to the context of supermanifolds and adapt M. Bor…

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 $\dirac$, constructed using a Hermitian metric $h$ and connection $\nabla$ on the symplectic vector bun…

Consider a compact prequantizable symplectic manifold M on which a compact Lie group G acts in a Hamiltonian fashion. The ``quantization commutes with reduction'' theorem asserts that the G-invariant part of the equivariant index of M is equal to the Riemann-Roch number of the symplectic quotient of M, provided the quo…

This is my Habilitation a Diriger des Recherches (French thing). In this paper I summarize the work I have done to realize the program of Witten called -non abelian localization-. This work deals first with problems of localization in equivariant cohomology. The second part of this paper concerns the Guillemin-Sternber…

A symplectic fibration is a fibre bundle in the symplectic category. We find the relation between deformation quantization of the base and the fibre, and the total space. We use the weak coupling form of Guillemin, Lerman, Sternberg and find the characteristic class of deformation of symplectic fibration. We also prove…

Quantization and reduction studied for CR manifolds with group actions.

In this paper, we analyse the question of existence of a natural and projectively equivariant symbol calculus, using the theory of projective Cartan connections. We establish a close relationship between the existence of such a natural symbol calculus and the existence of an \sl(m+1,\R)-equivariant calculus over \R^{m}…

GLAD improves latent graph generation by quantizing discrete latent space.

Data-free quantization method improves model performance without fine-tuning.

We develop a new approach, based on quantization methods, to study higher symmetries of invariant differential operators. We focus here on conformally invariant powers of the Laplacian over a conformally flat manifold and recover results of Eastwood, Leistner, Gover and Šilhan. In particular, conformally equivariant qu…

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…

Let $G$ be a compact connected Lie group, and $M$ a compact Hamiltonian $G$-space, with moment map $J$. For each $G$-equivariant Hermitian vector bundle $E$ over $M$, one has an associated twisted Spin-C Dirac operator, whose equivariant index is a symplectic invariant of $E$. In the present paper, we study gluing prop…

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…

Let M be a manifold endowed with a symmetric affine connection $Γ.$ The aim of this paper is to describe a quantization map between the space of second-order polynomials on the cotangent bundle T^{*} M and the space of second-order linear differential operators, both viewed as modules over the group of diffeomorphisms …

In order to facilitate the comparison of Riemannian homogeneous spaces of compact Lie groups with noncommutative geometries ("quantizations") that approximate them, we develop here the basic facts concerning equivariant vector bundles and Dirac operators over them in a way that uses only global constructions and argume…