The work performed has addressed the three issues listed above.
1) Concerning the global and recursive description of quantum corrections, three main results have emerged. First of all, the procedure of geometric quantization has been generalized to a large class of classical phase spaces and provides an explicit recursive quantization algorithm. A second development concerns topological recursion. This procedure associates an infinite series of quantum corrections to an algebraic curve, and it can be regarded as a powerful method to quantize curves, with multiple applications in geometry and physics. However, the meaning is obscure and the main issue is known as the “quantum curve conjecture”, which resisted a general proof for a long time. One of the main results of the project is an understanding of the connection between topological recursion and the conventional WKB method of quantum mechanics in a wide class of examples, and therefore a proof of the quantum curve conjecture in those cases. Finally, the approach based on geometric structures has led to new technical tools for the WKB method.
2) For access to exact quantum regimes beyond perturbation theory, the project based its approach on Ecalle’s theory of resurgence. According to this theory, the factorially divergent perturbative series appearing in quantum theory have a hidden structure which gives access to non-perturbative sectors. This structure consists of new perturbative series with exponential corrections, weigthed by numerical constants called Stokes constants. The set of these series and their Stokes constants is called the resurgent structure of the original perturbative series. Exact results in the theory can then be obtained by resuming the original perturbative series and the new series in the resurgent structure. Finding the resurgent structure associated to a given series is a well-defined but challenging problem. One of the main results of the project has been the determination of these resurgent structures in a variety of situations relevant to quantum theory and to quantum geometry including:
1) in complex Chern--Simons theory, the Ohtsuki series of Seifert manifolds and the perturbative invariants of hyperbolic knots;
2) topological string theory on Calabi-Yau threefolds;
3) intersection numbers on the moduli space of Riemann surfaces (topological gravity) and Weil—Petersson volumes;
4) the free energy of integrable asymptotically free theories in two dimensions.
In addition, it was found that, in perturbative series associated to quantum invariants, the Stokes constants in the resurgent structure are actually new, different quantum invariants. For example, in the case of topological string theory on Calabi-Yau threefolds, the perturbative series gives Gromov-Witten invariants, which count holomorphic curves, while the corresponding Stokes constants are Donaldson-Thomas invariants, which count sheaves.
3) Concerning the development of a mathematical theory underlying the quantization procedure, based on geometric structures, two different approaches have been developed. The first one follows an axiomatic approach, inspired by topological field theory, and gives in principle direct access to the resurgent structures. Another direction has been the development of a new mathematical foundation for the theory of resurgence based on the theory of sheaves and on Floer theory. In addition, the geometric approach provides an intimate relation between perturbative series and the classical Riemann—Hilbert problem. This approach, although rather abstract, can be made concrete in examples. In particular, it leads to explicit results for the Stokes constants in the case of the perturbative series in quantum topology.
The program for quantizing curved phase spaces has further made it possible to devise brand new quantum algorithm with proven advantage over classical methods for the non-universal quantum platform called Gaussian Boson Sampling (GBS). New techniques for solving Gaussian weighted integral using GBS samples have been proposed and it has been proven, using the new ReNewQuantum technology that there are many non-empty open subsets of the problem space where GBS can outperform Monte Carlo Sampling exponentially. This work has paved the way for concrete applications of the ReNewQuantum program within the financial, defense and pharma industries.
