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The Yang-Mills measure on surfaces and the master field on the plane

Speaker: Antoine Dahlqvist (University of Sussex)
Speaker: Antoine Dahlqvist (University of Sussex)
When Jan 30, 2024
from 04:00 PM to 05:00 PM
Where Online via Zoom
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Abstract: This talk will present some recent progress about a model of random matrices, called the Yang-Mills measure in two dimension, and its large N limit.
The Yang-Mills measure appears in physics at the core of the standard model of elementary particles, as a quantum field theory.  It gives a model for each type of interaction and each geometry of space-time.  The interaction field can be described by a process, called holonomy field, associating a matrix to each loop in space-time.  For a fixed type of interaction, these matrices are constraint to belong to a fixed compact group of matrices, called structure group.  For symmetry reasons, it is enough to focus on the trace of these matrices, they are called Wilson loops.  Physicists realised in the  70’s and 80’s, that considering matrices of size N, with N going to infinity, in place of a fixed structure group, allows to reasonably approximate the Wilson loops, and to compute, sometimes explicitly, their limit called master field.   

As a Euclidean quantum field theories,  the Yang-Mills measure can be formally understood as a probability measure, like the Wiener measure,  on a very large probability space. Making rigorous mathematical sense of this formal point of view is a challenging problem, that is most often open, in particular for the Yang-Mills measure with a non-abelian structure group and a four dimensional space-time. Nonetheless, for a two dimensional space-time, this problem has been solved recently in different ways, leading to a well-defined notion of Yang-Mills measure, as a specific law of a random holonomy field. We will recall a construction due to T. Lévy and discuss its large N limit. We will present partial results and a conjecture, based on  joint works with T. Lemoine, according to which, in negative Euler characteristics, the topology of space-time do not impact the master field.


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