In the framework of Euclidean quantum fields, the Yang--Mills field models electromagnetism and the other fundamental forces. It can roughly be described as a random field over a smooth Riemannian manifold modeling a Euclidean space-time. More rigorously, the Yang--Mills field is a random  connection, understood in a distributional sense, on a fiber bundle over this manifold.

Gaussian multiplicative chaos, which arise for example as scaling limits of several models of random planar maps, is a random measure on a planar domain which describes space-time at small scale in the model of Liouville quantum gravity. Thus, it is reasonable to try and define the Yang--Mills field over a Gaussian multiplicative chaos. 

Unfortunately, for a typical realisation of a Gaussian multiplicative chaos, it is extremely irregular and therefore not the measure associated with a smooth Riemannian metric: the usual construction of the Yang--Mills field fails indeed in this framework. 

We propose to adapt this construction in order to define the Yang--Mills field over a Gaussian multiplicative chaos, relying crucially on the fact that both theories, in a 2-dimensional setting, naturally belong in a framework of measured spaces rather than metric spaces. To this end, we have identified a class of loops and a distance on them such that the holonomy along these loops can be used as test functions for a distributional connection. The difficulties involved include proving that this family is rich enough that it describes entirely the connection, but small enough that it can be approximated by a subclass for which the holonomy of the Yang--Mills field can be defined naturally and then extended by continuity. The distance on the space of loops is indeed inherited from the measure on the planar domain, without relying on an underlying metric. 

The second aspect of the project would involve the study of continuity properties with respect to the measure, and approximability by discrete models. In particular, there are natural models of decorated planar maps which can be expected to converge, in the small mesh limit, toward the Yang--Mills field over a Gaussian multiplicative chaos that we would have defined in the first part of the project. We intend to find an appropriate topology for which this convergence would hold indeed.