Topological Recursion, Duality and Applications

• Applicant: Dr. Alexander Hock
• Subject Area: Mathematics ; Nuclear and Elementary Particle Physics, Quantum Mechanics, Relativity, Fields
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 551478549

The interplay between various branches of mathematics often finds its heuristic origins in theoretical or mathematical physics. One such example is Witten's conjecture concerning the equivalence of two different approaches of two-dimensional quantum gravity. Mathematically, this implies that the generating functions of certain topological invariants of the classical moduli space satisfy a hierarchy of partial differential equations, a result established by Kontsevich.Today, we understand that enumerative geometry, integrable systems, matrix models, Hurwitz theory, topological string theory, certain quantum field theories, and the theory of free probability exhibit similar recursive structures. Whether enumerative invariants, correlation functions or expectations values, these recursive structures contain a topological component determined by the genus of a Riemann surface. The structure is recursive in terms of Euler characteristics. The solution to these recursive equations can be formulated using the algorithm of Topological Recursion, which generates a family of multidifferentials for a given spectral curve.Two different families of these multidifferentials, related by the so-called $x-y$ transformation, give rise to a duality. This universal duality finds diverse applications, providing explicit formulae, for instance, for intersection numbers on the moduli space of complex curves or the functional relationship between higher-order free cumulants and moments in free probability theory.We aim to extend the application of this universal duality from algebraic to non-algebraic spectral curves, opening up new possibilities for instance in topological string theory. We anticipate explicit formulae for Gromov-Witten invariants of toric Calabi-Yau 3-folds, among other applications, including the interplay of the $A$-polynomial with the colored Jones polynomial in knot theory. The $x-y$ duality is expressed through formal, non-convergent series in an expansion involving $\hbar$. However, the asymptotic behavior can be analysed through resurgence and Borel summation, allowing us to draw conclusions about the asymptotics of topological invariants and correlation functions and its non-perturbative regime. This issue is closely related to the problem of perturbation theory in quantum field theory, which is mathematically formulated as a formal series, while nature reveals non-vanishing values for both the fine-structure constant and the Planck constant.This research project will unravel new algebraic structures across different areas of mathematics and mathematical physics, and provide a deeper insight of the surprising connections between them.

• DFG Programme: Research Grants
• International Connection: Netherlands
• Cooperation Partner: Professor Dr. Sergey Shadrin