Research Project:
Quantum field Theory led years ago to many beautiful ideas and results, but the goal to construct a mathematical consistent four dimensional model has not been reached.

In recent years, a modification of the space-time structure led to a new treatment of matrix-type models. Within these models we formulated in earlier work Schwinger-Dyson equations and Ward identities. Combining them allowed us to reduce the solution to solving Riemann-Hilbert problems for the N-point correlation functions.

As a next step we treated a Φ3 model in dimensions d = 2, 4, 6, performed the neces- sary renormalization and obtained all correlation function explicitly. The model is closely related to the Kontsevich model. For this model a direct correspondence between topolog- ical sectors of the moments of the measure and a procedure called topological recursion, is known. In this spirit, the non-linear Schwinger-Dyson equation for the second moment corresponds to the spectral curve in topological recursion. The combinatorics of the linear integral equations for the higher moments was solved.

As a byproduct we obtained an easy algorithm to generated intersection numbers.
In previous work it was possible by one of us (Raimar Wulkenhaar) together with Eric Panzer to solve the integral equation for the two point function of one sector of the Φ4- model in two dimensions. We intend to use the algebraic structure behind the topological recursion to get more information about the other sectors.

Research Team: Harald Grosse (U Vienna), Raimar Wulkenhaar (U of Münster)

First stay: April 28 - May 3, 2019
Second stay: June 13 - 18, 2019 
Third stay: July 28 - August 2, 2019