Luuk Verhoeven >> Research

Current main interests

  1. Random fuzzy geometries
  2. Functonial integrals over fuzzy geometries (with extra structure)
  3. Factorization of Dirac operators

Non-commutative geometry

Within NCG I am interested in the study of compositions. Let \((A, E, S)\) be an unbounded \(A\)-\(B\)-cycle (essentially a spectral triple with the algebra \(B\) replacing \(\mathbb{C}\)) and \((B, F, T)\) a \(B\)-\(C\)-cycle, then they can be composed to \((A, E \otimes_B F, S \otimes 1 + 1 \otimes_\nabla T)\) which will then be an \(A\)-\(C\)-cycle. Putting \(C = \mathbb{C}\) the triple \((A, E, S)\) essentially turns one spectral triple into another, forming a sort of "map between manifolds".

This construction requires a connection, \(\nabla\) on \(E\) in some sense compatible with \(T\) as well as more technical compatibility requirements between \(S, T\) and \(A, B\). But if those requirements are met, there are, for example, notions of curvature for these cycles. I am interested in further investigating this construction and understanding the structure of maps between spectral triples, as well as the idea that in this picture manifolds are encoded at the same level as maps between manifolds.

Random matrix theory

My interest in random matrix theory mostly comes from attempts to quantize the metric, and thus gravity, on spectral triples. In Barrett's fuzzy geometry framework the space of Dirac operators is parametrized by a set of matrices. Attaching an action, say \(S(D) = \text{Tr}(g_2 D^2 + g_4 D^4)\), to a Dirac operator allows us to then formulate a well-defined path-integral over this space of Dirac operators which then can be translated to a random matrix model.

I want to investigate if this framework can be used to establish a theory of quantum gravity based on fuzzy (as opposed to lattice) geometries. As well as investigate other questions, such as the effect of adding in a fermionic action and the nature of dimension in this setting.