Half-sided modular inclusions and relative commutants

You will join an international team of collaborators spanning UK, US, Japan and Denmark.

Von Neumann algebras appear in many different areas of mathematics, ranging from Hilbert space operators over non-commutative geometry to quantum field theory.

Whereas von Neumann algebras as such have been largely classified in the seminal works of von Neumann/Murray, and later Connes, the understanding of their inclusions (pairs) is much less developed.

In applications to quantum field theory, it is however often much more interesting to analyse the relative commutant of an inclusion (which carries information about localised operators) rather than the algebras themselves (which carry hardly any information).

In recent years, the special class of so-called split inclusions (which have an accessible relative commutant) have been applied successfully in the context of inverse scattering problems and constructions of integrable models.

The split assumption is however fairly restrictive, and in various applications, one is faced with different types of inclusions. A prominent example is the class of half-sided modular inclusions, which are defined in terms of a specific action of the modular data of the large algebra on the smaller one. Such inclusions lie at the heart of chiral conformal field theory, which provides many examples.

Whereas a split inclusion can be characterised by certain operators related to modular theory being compact, or even nuclear, a similar characterisation is not known in the half-sided case.

The project will deal with such questions, with one aim being the development of tools, potentially in the form of generalised spectral density criteria for modular operators, for the analysis of the relative commutant of a half-sided inclusion.

These ideas should then be tested in concrete examples as they appear in massless (conformal) field theories. A potential application could be an operator-algebraic analysis of asymptotic freedom, which is a central property in quantum field theory.

We are interested in pursuing this project and welcome applications if you are self-funded or have funding from other sources, including government sponsorships or your employer.

Please contact the supervisor when you want to pursue this project, citing the project title in your email, or find out more about our PhD programme in Mathematics.

Supervisors

Photograph of Dr Gandalf Lechner

Dr Gandalf Lechner

Lecturer

Email:
lechnerg@cardiff.ac.uk
Telephone:
+44(0)29 208 75553

Programme information

For programme structure, entry requirements and how to apply, visit the Mathematics programme.

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