Periodic Reporting for period 3 - ReNewQuantum (Recursive and Exact New Quantum Theory) Reporting period: 2021-05-01 to 2022-10-31 Summary of the context and overall objectives of the project The overarching goal of ReNewQuantum is to renew the mathematical foundation behind quantum phenomena.We aim to construct a recursive and exact new approach to quantum theory. Quantum theory is one of the pillars of modern science. Its success stretches from elementary quantum mechanical models, developed a century ago by quantization of classical mechanics, to advanced quantum field theories such as the standard model of particle physics, which is the quantization of a gauge theory. However, a precise and universal mathematical formulation of the quantization procedure is still lacking. In addition, there are very few analytic methods in Quantum Mechanics and in Quantum Field Theory. They are typically based on approximation schemes which often lead to quantitative and even qualitative failures in our descriptions.In response to these shortcomings, the main objective of ReNewQuantum is to construct a completely new mathematical approach to quantization and to quantum systems. This quantum theory will provide:- a global, explicit and recursive description of the series of quantum corrections,- access to exact quantum regimes beyond perturbation theory,- a well founded mathematical theory underlying the quantization procedure, based on geometric structures, and applicable to quantum field theory and string theory.ReNewQuantum will take the lead among the world scientific community in building this new theory of quantum physics. The researchers behind ReNewQuantum have already made important contributions along these directions. The construction of a recursive and exact new approach to quantum theory with the stated properties will only be possible through their joint synergetic effort and a combination of their deep mathematical and physical expertises, including geometry, topology and the mathematical theory of quantization (Andersen, Kontsevich), and quantum mechanics, quantum field theory, random matrix theory and string theory (Eynard, Mariño). Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far The developments in this period have been mostly of two types. A first development has concerned the general theory of perturbative expansions and their resummation, also known as the theory of resurgence. According to the theory of resurgence, quantum theories are characterized by a (possibly infinite) set of formal power series and relations between them, encoded in complex numbers called Stokes constants. We have developed a general mathematical framework for this theory, making contact with recent developments in geometry and the theory of so-called wall-crossing structures. At the same time we have used this framework to study concrete examples: spectral problems in quantum mechanics, special functions appearing in quantum topology, and perturbative series in a quantum field theory which, albeit simple, has deep mathematical implications: Chern-Simons theory. This has led to a complete understanding of the general resurgent structures in Chern-Simons theory, in various non-trivial examples. This opens the way to further developments in quantum field theory, and has led to new results in quantum topology. Another direction which has been pursued vigorously is the theory of topological recursion. This theory was born in the context of large N expansions in random matrix theory, but it has applications in many different fields and it governs many problems in algebraic and enumerative geometry. Recent extensions of this framework involve the theory of geometric recursion and the theory of quantum Airy structures. We have developed the framework of geometric recursion significantly further and the applications now include Masur-Veech volumes, Kontsevich combinatorial Teichmüller spaces, and solutions to the Quantum Master Equation. Several further applications are anticipated. Topological Recursion is also extensively used in combinatorics of maps, as it allows to count maps of any topology, knowing only the generating function of planar maps. The combination of topological recursion and resurgence techniques is expected to yield results on the large genus asymptotics of these enumerative problems.A recent series of works have related the physics of JT-black holes to the enumerative geometry of hyperbolic surfaces and matrix models, which puts topological recursion and resurgence on the foreground in black hole physics. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) The successful resurgent analysis of meromorphic transformations by J. Andersen was unexpected progress. It will very likely lead to wide generalizations and we expect it to have a deep impact on the resurgent analysis of quantum invariants.The development of the notion of analytic stability structures by M.Kontsevich and Y.Soibelman is a very important theoretical breakthrough, giving a clean geometrical understanding of the wide class of resurgent series relevant to the project, in particular to WKB expansions in quantum mechanics and asymptotic expansions in Chern-Simons theory. The new paradigm changes even the way to reconstruct analytic functions in sectors from factorially divergent series, and it gives an alternative to the usual inverse Borel transform. This development was not expected a priori. Another important development is the much better understanding of the structure of Stokes constants for Chern-Simons theory. Here 3 PIs (Andersen, Kontsevich and Mariño) actively participated in the progress. The resulting picture is that Stokes constants 1) correspond to the count of holomorphic discs in a complex algebraic symplectic torus with boundaries on the union of two K2 Lagrangian subvarieties, and that 2) their generating series is naturally a solution of a q-difference equation. Again, this development was not anticipated at the beginning of the program, and the determination of the full resurgent structure of quantum knot invariants goes significantly beyond the state of the art and constitutes important progress. The theory of quantum curves has become an important unifying subject at the interface between geometry and mathematical physics. One of the main problems in this area is the following: given an algebraic curve, let us consider the perturbative wavefunction obtained by applying topological recursion to it. Can one find a quantum operator which reduces to the original curve in the classical limit and annihilates the perturbative wavefunction? This problem is in a sense the reverse of the WKB method. An affirmative answer to this question existed until now only for rational spectral curves, but recently we have been able to extend this result to more general curves (including hyperelliptic ones). This is a major step towards a general theory. The above developments indicate that a first level of synthesis of the geometric structures underlying quantum mechanics has been achieved. Let us recall that the three basic entities in quantum mechanics - spectrum, wave functions, amplitudes- all give rise hypothetically to resurgent series in the Planck constant. The current developments will likely lead to a proof of the resurgence property for all three problems. Indeed, in the case of wave functions, the resurgent structure of the WKB approximation should be covered by the wall-crossing formalism of Kontsevich and Soibelman. The Stokes indices are given by counting pseudo-holomorphic discs with boundaries on Lagrangian submanifolds in the complexified phase space. In the case of a cotangent bundle, this count is equivalent to the the count of spectral networks of Gaiotto, Moore and Neitzke. From a general perspective, one should be able to derive the structure of Stokes indices in a purely algebraic way from coordinate changes between different cluster coordinates in character varieties. Similarly, in the case of complex symplectic algebraic tori, the count of pseudo-holomorphic discs should give rise to a holonomic q-difference module. The partition function of complex Chern-Simons theory should be an example of such a resurgent structure, via an equivalence with the quantum mechanics on complex tori/character varieties. An outstanding question concerning the relation of wall-crossing structures to topological recursion is expected to be resolved, presumably during the next year. The questions about resurgent structures for low-energy eigenvalues (spectra) and for heat kernels (amplitudes) have another level of complexity, but we expect this question to be completely clarified at the end of the project. Also, we expect that, with the knowledge of Stokes indices, the solution of these problems could be treated with efficient numerical algorithms. Finally, we plan to address the more involved case of quantum field theories with non-trivial renormalization properties. We expect to be able to understand their resurgent structure at least in the case of integrable massive theories in two space-time dimensions, where significant progress has also been made in this period. More ambitiously, one can hope to understand resurgent structures for matrix integrals and string partition functions.
