Towards random noncommutative geometry
30.04.2025, 11:00
– Campus Golm, Building 9, Room 2.22 and via Zoom
Arbeitsgruppenseminar Analysis
Carlos Perez-Sanchez (Heidelberg)
The well-known question whether one can 'hear the shape of a drum', posed by Marek Kac, has, also famously, a negative answer constructed by Gordon, Webb and Wolpert. In noncommutative geometry, classical dynamics depends only on the spectrum, in that case, of an operator of Dirac type. If, additionally, algebraic data are provided and some axioms verified --building what is known as spectral triple-- this structure does allow to reconstruct a manifold, thus answering a weaker version of Kac's question positively.
Spectral triples are relevant in Connes' noncommutative geometric setting, whose path integral quantisation that "averages over noncommutative geometries" shall rely on the concept of ensembles of Dirac operators. This is to be contrasted with a path integral over Riemannian metrics in quantum gravity. In this talk I first explore what an ensemble of noncommutative geometries on a fixed graph is (gauge fields are on, while gravity is still off). Using elements of quiver representation theory
- - we associate a Dirac operator to a quiver representation (in a category that emerges in noncommutative geometry);
- - we derive the constraints that the set of Wilson loops satisfies (generalised Makeenko-Migdal equations):
- - and explore the consequences of the positivity of a certain matrix of Wilson loops ('bootstrap')
In the special case that our graph is a rectangular lattice and our physical action quartic, we obtain Wilsonian lattice Yang-Mills theory, hence the terminology. Unsurprisingly, our ensembles (for an arbitrary graph) boil down to integrating noncommutative polynomials against a product Haar measure on unitary groups. The classical aspects of this theory were constructed in [2401.03705], and the loop equations in [2409.03705].
For more information and log in details please contact Christian Molle.