Matrix Field Theory

TL;DR

This thesis explores matrix field theories, focusing on their applications in quantum field theory, quantum gravity, and algebraic geometry, and investigates their renormalisation, correlation functions, and connections to topological recursion.

Contribution

It introduces generalized matrix field models with new potentials and spectral dimensions, and demonstrates their compatibility with topological recursion and exact solvability in certain cases.

Findings

Renormalisation is compatible with topological recursion for the cubic model.

Exact 2-point function computed for the quartic model.

Structural similarities identified between the quartic model and hermitian 2-matrix models.

Abstract

This thesis studies matrix field theories, which are a special type of matrix models. First, the different types of applications are pointed out, from (noncommutative) quantum field theory over 2-dimensional quantum gravity up to algebraic geometry with explicit computation of intersection numbers on the moduli space of complex curves. The Kontsevich model, which has proved the Witten conjecture, is the simplest example of a matrix field theory. Generalisations of this model will be studied, where different potentials and the spectral dimension (defined by the asymptotics of the external matrix) are introduced. Because they are naturally embedded into a Riemann surface, the correlation functions are graded by the genus and the number of boundary components. The renormalisation procedure of quantum field theory leads to finite UV-limit. We provide a method to determine closed Schwinger-Dyson equations with the usage of Ward-Takahashi identities in the continuum limit. The cubic (Kontsevich model) and the quartic (Grosse-Wulkenhaar model) potentials are studied separately. For the cubic potential, we show that the renormalisation procedure is compatible with topological recursion (TR). This means that the exact results computed by TR coincide perturbatively with the graph expansion renormalised by Zimmermann's forest formula. For the quartic model, the first correlation function (2-point function) is computed exactly. We give hints that the quartic model has structurally the same properties as the hermitian 2-matrix model with genus zero spectral curve.