arXiv research

We define an equivariant index of Spin$^c$-Dirac operators on possibly noncompact manifolds, acted on by compact, connected Lie groups. The main result in this paper is that the index decomposes into irreducible representations according to the quantisation commutes with reduction principle.

Study quantisation of geometric operators on manifolds with group actions.

Paradan and Vergne generalised the quantisation commutes with reduction principle of Guillemin and Sternberg from symplectic to Spin$^c$-manifolds. We extend their result to noncompact groups and manifolds. This leads to a result for cocompact actions, and a result for non-cocompact actions for reduction at zero. The r…

Let $G$ be a connected, linear, real reductive Lie group with compact centre. Let $K<G$ be maximal compact. For a tempered representation $π$ of $G$, we realise the restriction $π|_K$ as the $K$-equivariant index of a Dirac operator on a homogeneous space of the form $G/H$, for a Cartan subgroup $H<G$. (The result in f…

We study two special cases of the equivariant index defined in part I of this series. We apply this index to deformations of Spin$^c$-Dirac operators, invariant under actions by possibly noncompact groups, with possibly noncompact orbit spaces. One special case is an index defined in terms of multiplicities of discrete…

We provide a Geometric Quantisation formulation of the AJ-conjecture for the Teichmüller TQFT, and we prove it in detail in the case of the knot complements of $4_{1}$ and $5_2$. The conjecture states that the level-$N$ Andersen-Kashaev invariant, $J^{(\mathrm{b},N)}_{M,K}$, is annihilated by the non-homogeneous $\hat{…

In the first part of this series, we defined an equivariant index without assuming the group acting or the orbit space of the action to be compact. This allowed us to generalise an index of deformed Dirac operators, defined for compact groups by Braverman. In this paper, we investigate properties and applications of th…

Let $G$ be a connected, linear, real reductive Lie group with compact centre. Let $K<G$ be compact. Under a condition on $K$, which holds in particular if $K$ is maximal compact, we give a geometric expression for the multiplicities of the $K$-types of any tempered representation (in fact, any standard representation) …

A new algorithm improves Bayesian federated learning by reducing communication overhead.

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. Donald…

Reinterprets quantization commutes with reduction using KK-theory.

In this paper, we prove that the "quantization commutes with reduction" phenomenon of Guillemin-Sternberg applies in the context of the metaplectic correction.

In quantum physics, the operators associated with the position and the momentum of a particle are unbounded operators and $C^*$-algebraic quantisation does therefore not deal with such operators. In the present article, I propose a quantisation of the Lie-Poisson structure of the dual of a Lie algebroid which deals wit…

For spherically symmetric distributions, efficient quantisation can be achieved with moderate sample sizes.

We study algebro-geometric consequences of the quantised extremal Kähler metrics, introduced in the previous work of the author. We prove that the existence of quantised extremal metrics implies weak relative Chow polystability. As a consequence, we obtain asymptotic weak relative Chow polystability and $K$-semistabili…

Paper explores pruning and quantisation to compress neural networks.

The topological recursion of Eynard and Orantin governs a variety of problems in enumerative geometry and mathematical physics. The recursion uses the data of a spectral curve to define an infinite family of multidifferentials. It has been conjectured that, under certain conditions, the spectral curve possesses a non-c…

New geometric quantisation scheme for hyper-Kähler manifolds.

We compare the covariant formulation of Quantum Mechanics on a curved spacetime fibred on absolute time with the standard Geometric Quantisation.

New construction provides non-trivial representations for geometric quantisation.

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

Kasparov defined a distinguished K-homology fundamental class, so called the Dirac element. We prove a localization formula for the Dirac element in K-homology of crossed product of C^{*}-algebras. Then we define the quantization of Hamiltonian G-spaces as a push-forward of the Dirac element. With this, we develop a K-…

An analogue of geometric quantization of Poisson algebras obtained by algebraic reduction of symmetries is developed. Interpretation of the obtained results and their application to the problem of commutativity of quantization and reduction are given

We explain how to translate several recent results in derived algebraic geometry to derived differential geometry. These concern shifted Poisson structures on NQ-manifolds, Lie groupoids, smooth stacks and derived generalisations, and include existence and classification of various deformation quantisations.

Study the commutativity of reduction and symplectification in contact Hamiltonian systems.

Quantized BNNs maintain uncertainty estimation quality despite reduced precision.

New method shows unitarity in quantization for toric manifolds.

Efficiently calibrates Bergomi models to VIX derivatives using vector quantization.

We represent algebraic curves via commuting matrix polynomials. This allows us to show that the Hilbert scheme of cohomologically stable twisted rational curves of degree $d$ in ${\Bbb P}^3\backslash {\Bbb P}^1$ is isomorphic to a complexified hyperkähler quotient of an open subset of a vector space by a non-reductive …

Survey of bundle gerbes in geometry, field theory, and quantization.

We introduce and study the notion of Sasaki--Weyl manifold, which is a natural generalization of the notion of Sasaki manifold. We construct a reduction of Sasaki--Weyl manifolds and we show that it commutes with several reductions already existing in the literature.

Classifies star products on Lie algebroid duals and extends to projectable quantizations.

A refined expression for the Faddeev-Popov determinant is derived for gauge theories quantised around a reducible classical solution. We apply this result to Chern-Simons perturbation theory on compact spacetime 3-manifolds with quantisation around an arbitrary flat gauge field isolated up to gauge transformations, poi…

The version of Marsden-Ratiu reduction theorem for Nambu-Poisson manifolds by a regular distribution has been studied by Ib$\acute{\text{a}}\tilde{\text{n}}$ez et al. In this paper we show that the reduction is always ensured unless the distribution is zero. Next we extend the more general Falceto-Zambon Poisson reduct…

Study on special Lie groups with Lorentzian metrics.

Neural networks help auditors efficiently assess financial statements by learning underlying data patterns.

Improved hierarchical discrete VAEs for better stability and performance.

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.…

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…

Geometric derivation of quantum dynamics from Lie group actions.

Formula for sectional curvatures on matrix groups.

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 extends symplectic reduction to cosymplectic groupoids and explores their properties.

The geometric approach [1312.1262] to iterated variations of local functionals -- e.g., of the (master-)action functional -- resulted in an extension of the deformation quantisation technique to the set-up of Poisson models of field theory [IHES/M/15/13]. It also allowed of a rigorous proof ([1312.1262],[1210.0726]) fo…

In this paper we show that the `quantization commutes with reduction' principle of Guillemin-Sternberg holds for the coadjoint orbits that parametrize the discrete series of a real connected semi-simple Lie group.

Global analysis of Dixmier traces and Wodzicki residues on compact Lie groups.

We study noncommutative bundles and Riemannian geometry at the semiclassical level of first order in a deformation parameter $λ$, using a functorial approach. The data for quantisation of the cotangent bundle is known to be a Poisson structure and Poisson preconnection and we now show that this data defines to a functo…

In this paper we explore the idea of looking at the Dirac quantisation conditions as $\hbar$-dependent constraints on the tangent bundle to phase-space. Starting from the path-integral version of classical mechanics and using the natural Poisson brackets structure present in the cotangent bundle to the tangent bundle o…