# Matrix Field Theory

May 15, 2020

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.

Keywords:

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