ReNewQuantum

The theory of resurgence was developed to provide a deeper understanding of asymptotic series in various fields of mathematics. More recently, this theory has been applied to problems in geometry and topology, leading to a new perspective on enumerative invariants in three-manifold topology and in algebraic geometry. A remarkable synthesis is emerging, relating analytic invariants in the theory of resurgence to enumerative invariants, as well as their wall-crossing behavior. The physical framework for these phenomena is non-perturbative quantum field theory and string theory, and beautiful results and conjectures are being obtained both in physics and in mathematics.

Topological recursion is an algebra-geometric construction which takes the spectral curve and makes out of it a recursive definition of infinite sequences of symmetric meromorphic n-forms with poles. It was first discovered for random matrices. A main goal of random matrix theory is to find the large size asymptotic expansion of n-point correlation functions, and in some cases, the asymptotic expansion takes the form of an infinite series. The n-form ω(g,n) is than the gth coefficient in the asymptotic expansion of the n-point correlation function. It was found that the coefficients ω(g,n) always obey the same recursion on 2 g -2 +n. The idea is to consider this universal recursion relation beyond random matrix theory, and to promote it as a definition of algebraic curve invariants, was first introduced by Eynard-Orantin in 2007, where they studied the main properties of these invariants.