From asymptotics to exact results: Painlevé I and Large-N phase transitions
Mathematical Physics Seminar
20th September 2019, 2:00 pm – 3:00 pm
Fry Building, 2.04
Matrix models offer non-perturbative descriptions of quantum gravity in simple settings, allowing us to study important dualities, the so-called large-N dualities, between the seemingly very different gauge and string theories. However, the large-N expansions of matrix models lead divergent series, only defined as asymptotic series. Furthermore, by fine-tuning the couplings of the matrix model we obtain models of pure gravity coupled to minimal conformal field theories. The simplest of these "minimal models" is 2d gravity, and the perturbative expansion of its partition function is asymptotic and formally satisfies the Painlevé I equation.
These asymptotic properties are connected to the existence of exponentially small contributions not captured by a perturbative analysis. The emerging structure can be accurately described by means of a resurgent transseries, capturing this perturbative/non-perturbative connection and its consequences. In this talk, I will analyse essential role of this resurgent transseries for the cases of Painlevé I and the quartic matrix model: together with exponentially accurate numerical and summation methods, I will show how to go beyond the asymptotic results and obtain (analytically and numerically) non-perturbative data.