Quantization of Character varieties as a Model for quantum chaos – QCM

The goal of Quantum Chaos is to understand the quantum counterparts of various notions of ergodicity of classical dynamical systems. In the context of geodesic flows acting on the cotangent bundle of a Riemannian manifold, this consists in understanding the large-scale distribution of the eigenvalues and eigenfunctions of the Laplace-Beltrami operator in terms of ergodicity properties of the geodesic flow.

One can also consider the quantization of a discrete dynamical system acting on a compact phase space. The basic example of interest in this context is Arnold’s Cat Map acting on the two-dimensional torus, whose quantization has then been intensively studied as a toy model for Quantum Chaos. The Cat Map acting on the two-dimensional torus is the simplest example of a very rich class of discrete dynamical systems of great interest in geometry and topology, given by the action of Mapping Class Groups on character varieties and whose quantization bears the name of Quantum Representations. Several different models for these quantizations are now available, including topological models on one hand and differential-geometric models on the other.

The goal of this project is to use these various models of Quantum Representations as a pool of example to deepen our understanding of the typical phenomena in Quantum Chaos. This is achieved by bringing together a team of young researchers, half of which consists of experts in the topological models and the other half of experts in the analytic and differential-geometric tools of quantization. This project contains three tightly related Topics :

The first Topic consists in the study of the quantum ergodicity properties which are well known in the case of the two-dimensional torus, but in the case of more general character varieties. In first place, we plan to study the Quantum Egrodicity Principle in specific examples of well-studied character varieties. In second place, we plan to study more refined phenomena, such as exceptional sequence contradiction the Unique Quantum Ergodicity principle and the Fractal Uncertainty Principle, in cases when the character variety is naturally identified with the quotient of a higher dimensional torus by a finite group. In last place, we plan to study the dynamical applications of the semiclassical trace formula in this context.

The second Topic naturally fits in the continuation of the first, since it concerns Witten's asymptotic formula, seen as a semiclassical formula in the context of non-abelian character varieties. In first place, we plan to extend this formula, which is known in particular cases only, to more general cases using tools of local index theory over surfaces with cusps. In second place, we plan to extend this study to the character varieties associated with 3-manifolds obtained by gluing, using existing models linking the topological models with the differential-geometric models of Quantum Representations. In last place, we plan to link the topological models with the differential-geometric ones via the study of the Curve Operators and quantum integrable systems.

The third Topic naturally fits in the continuation of the first two, since it concerns the extension of the differential-geometric intepretation of the topological models for Quantum Representations in cases when they are not Hermitian spaces, but are endowed instead with a non-definite positive product. In first place, we plan to construct differential-geometric models for such Quantum Representations using an axiomatic approach. In second place, we plan to interpret the Volume conjectures of Kashaev and Chen-Yang as semiclassical trace formulas in this context, in order to reduce these conjectures to an equivalence problem between the topological models and our own differential-geometric models. In last place, we will study the so-called homological models in this context in order link the topological models with our own differential-geometric models.