Universal Moduli of Bundles on Curves

Over the last few decades the concept of a moduli space has become central in modern geometry. Roughly speaking, a moduli space is a geometric space that parameterizes instances of various kinds of objects in geometry, algebra, and physics. Somewhat surprisingly, the geometry of these moduli spaces themselves can be very rich, and encode the answers to all kinds of questions one can ask about the original objects.

Perhaps one of the most venerable moduli spaces, but still a central topic of study in modern mathematics, is Riemann's moduli space that parameterizes Riemann surfaces, also known as smooth complex algebraic curves. Riemann surfaces are two-dimensional surfaces that have a notion of angle, but not of length. Mathematician's caught the first glimpses of this moduli space in the 19th century, but it took almost a hundred years to place its construction on sound footing. Typical for such an important object is that it can be understood from a number of different viewpoints: in algebraic geometry, topology or differential geometry.

An unfortunate aspect of the naïve approach to the moduli space is that it yields a space that is not compact, which means that one can "run out of it". This defect was resolved however in a very beautiful way by Deligne and Mumford in the late 1960s: one can compactify (or complete) this space by generalizing the kind of objects a little bit, in particular by allowing the Riemann surfaces to develop certain mild singularities. Not only does this overcome the non-compactness, but the resulting larger moduli space is extremely well-behaved in all sorts of ways.

Another classical moduli space is the Jacobian variety of a fixed Riemann surface - which in modern parlance parameterizes line bundles on that Riemann surface. A special kind of abelian variety, it is the home of the Theta-functions, central objects in number theory and string theory alike. A natural generalization of the Jacobian variety seeks to classify non-abelian analogues of line bundles, a question that is strongly motivated by gauge theories such as the Standard Model in theoretical particle physics.

This proposal centers around the following basic question: what happens to these moduli spaces of non-abelian bundles when one allows the Riemann surface to degenerate, as in the Deligne-Mumford compactification? This is a very natural thing to ask, which has a surprising number of links with representation theory, quantum mechanics and conformal field theory. To be more precise, one would like to construct a modular compactification of the "universal" moduli space of bundles on smooth curves, where both are allowed to move simultaneously, that behaves in a similarly nice way as the Deligne-Mumford compactification.

We are proposing a new line of attack on this problem, based on recent developments initiated by the PI and others in the last few years, and aim to investigate its consequences and links with other disciplines.

As the bulk of the requested funding would cover the salary for the PDRA, it is fair to say that the biggest impact would be on the people pipeline and skills base of the UK mathematical community. As described in the Case for Support, the proposed project has big intra-disciplinary aspects, connecting algebraic geometry to symplectic geometry, gauge theory, representation theory and mathematical physics. The PDRA would therefore be trained in a broad research area, which would develop him or her into an independent research mathematician, who will contribute to the knowledge base of mathematics in the UK.

When it comes to the research results, this work will be of immediate interest to researchers working in algebraic geometry, symplectic geo- metry, topology and mathematical physics. It will be particularly relevant for people working on moduli spaces in these disciplines. Furthermore, the results can also be expected to impact people working in topological quantum field theory (through the role that the moduli spaces of bundles and their quantizations play in Chern-Simons theory) and representation theory - in particular the representation theory of affine Kac-Moody groups. Finally, both through the links with conformal field theory and geometric quantizations should be of relevance to researchers in theoretical physics.

Researchers in these fields would be reached through publications, and seminar and conference talks. All publications would be made publicly available on the arXiv and the institutional repository of the University. When it comes to seminars and conference talks, besides those given by myself and the PDRA, also talks by the Visiting Researchers would be of significant interest to the research community. For this it is worth mentioning that I am the coordinator of the GLEN seminar series in algebraic geometry, which is a joint endeavor between the Universities of Edinburgh, Glasgow, Liverpool, Manchester and Sheffield, with a target audience of essentially all researchers working in related fields in northern England and Scotland. It is my aim to ensure that the Visiting Researchers will each time speak in one of the GLEN workshops.

Finally, through the outreach component we would aim to reach the general public, and in particular secondary school pupils. Further details of this are described in the Pathways to Impact.

The expected impact of this is three-fold: first it builds knowledge and appreciation of modern mathematical research in the general audience. Secondly it raises the public profile of the School of Mathematics of the University of Edinburgh and its research activities. Thirdly, in particular aimed at secondary school children, it exhibits mathematics as a living endeavour, thereby warming them for possible careers in mathematical research or science in general.