Wonderful Loop Group Embeddings and Applications to the Moduli of G-bundles on Curves
- Author(s): Solis, Pablo
- Advisor(s): Teleman, Constantin
- et al.
Moduli problems have become a central area of interest in a wide range of mathematical fields such as representation theory and topology but particularly in the geometries (differential, complex, symplectic, algebraic). In addition, studying moduli problems often requires utilizing tools from other mathematical fields and creates unexpected bridges within mathematics and between mathematics and other fields.
A notable example came in 1991 when the mathematical physicists Edward Witten made a conjecture connecting the partition function for quantum gravity in two dimensions with numbers associated to the cohomology of the moduli space of stable curves, a space that was already of independent interest to algebraic geometers.
We study a related moduli problem MG of principal G-bundle on stable curves for G a simple algebraic group. A defect of MG over singular curves is that it is not compact and thus more difficult to study. We focus specifically on nodal singularities and examine how to compactify MG over nodal curves.
The approach we present relies on two main mathematical objects: the loop group and the wonderful compactification of a semisimple adjoint group. For an algebraic group G the loop group LG is the group of maps Dx → G where x is a punctured formal disk, see 2.2 for a precise definition. The connection between LG and MG is that G-bundles can be described by transition functions and roughly speaking any such transition function comes from an element of LG.
The wonderful compactification is a particularly nice way of comapactifying a semi simple group. Then in a sentence, the aim this dissertation is to (1) extend the construction of the wonderful compactification for semi simple group to LG and (2) use this compactification to compactify MG over nodal curves.
We give a brief introduction in Chapter 1. Chapter 2 addresses (1) and Chapter 3 addresses (2).
We begin in Chapter 2 with a discussion of the classical wonderful compactification of an adjoint group given by De Concini and Procesi in [DCP83]. Because the group LG is infinite dimensional many of the elements in De Concini and Procesi's construction do not immediately extend or have more than one possible generalization. The technical heart of the paper is developing the appropriate analogs of all the elements needed to make the construction possible for LG. Also building on work of Brion and Kumar we give an enhancement of the compactificaiton from schemes to stacks that we utilize in Chapter 3.
Chapter 3 returns to the problem of compactifying MG over nodal curves. We begin by carefully studying the points in the boundary of the compactification of LG and relating them to moduli problems over nodal curves. The moduli problems that appear in this way are closely related to flag varieties for the loop group and can be identified as moduli of torsors for a particular group scheme determined by parabolic subgroups of the loop group.
We go on to show that the moduli problem of torsors on nodal curves is isomorphic to a moduli problem of G-bundles on twisted nodal curve; these are orbifolds that are isomorphic to the original nodal curve on the smooth locus. Finally, building on related work of Kausz [Kau00,Kau05a] and Thaddeus and Martens [MaT] and the results of Chapter 2 we introduce a larger moduli problem XG of G bundles on twisted curves which compactifies MG.