Quantum Geometry of Parabolic Moduli Spaces and the Hitchin-KZ Equivalence

This work studies the geometry and quantisation of moduli spaces of parabolic bundles. Two such moduli spaces are specifically studied: the moduli space of parabolic principal bundles over a Riemann surface and in particular over the sphere, as well as the moduli space of parabolic Higgs bundles over general Riemann surfaces, each of which is the subject of a theorem proven in this work.

The main theorem proven in this thesis is the projective equivalence of the Knizhnik–Zamolodchikov connection to the Hitchin connection in genus 0 with at least 3 marked points. The Hitchin connection is a connection defined on the Verlinde bundle, given fibrewise by the geometric quantisation of the moduli space of flat connections in Chern–Simons gauge theory, over a family of complex structures on the surface. The Knizhnik–Zamolodchikov connection is defined on the sheaf of conformal blocks in the Tsuchiya–Ueno–Yamada model of conformal field theory, defined over the same family of complex structures.

It has long been known that these two bundles are isomorphic, to mathematicians as Pauly’s Isomorphism and to physicists as CS/WZNW duality. The main theorem is then a statement that Pauly’s Isomorphism is projectively flat with respect to these two connections. This mathematical statement has the physical interpretation that the duality holds dynamically, although the physics will not be touched upon in this work.

The relevant background theory will be presented and mathematical models of both physical theories will be presented. Subsequently, the full proof of the main theorem will be given. It will rely centrally on the construction of a geometric version of the Knizhnik–Zamolodchikov connection using the Bott– Borel–Weil bundle as a model for the representation space that contains the space of conformal blocks.

Additionally, some elementary topological quantum field theory (TQFT) will be recalled, in order to set up the relevant context. The equivalence that is the main theorem of this thesis has important applications in TQFT, as the conformal field theory has been used to geometrically construct the Reshetikhin– Turaev TQFT, known as the Andersen–Ueno Isomorphism. The Knizhnik–Zamolodchikov connection is crucial to this isomorphism. The projective equivalence between the two connections can then be used to perform the constructions from the TQFT in the gauge-theoretic side. This makes it possible to perform the calculations and analyses with geometric arguments, greatly enriching the theory.

Furthermore, the thesis will treat the moduli space of parabolic Higgs bundles and present two novel results about the moduli space. Higgs bundles are a complexified counterpart to holomorphic vector or principal bundles, equipping them with an additional endormorphism-valued 1-form known as the Higgs field. Higgs fields carry a natural scaling operation from non-zero complex numbers, which defines to a group action on the moduli space of Higgs bundles. The background theory of Higgs bundles is given for context.

Parabolic Higgs bundles are the natural generalisation of Higgs bundles to the parabolic context, and the moduli space of parabolic Higgs bundles also carries a natural scaling action. This action also introduces further stability conditions to the bundles, which provide extra structure to the moduli space. In particular, (parabolic) Higgs bundles can be very stable, whose attracting loci have powerful properties in the field of mirror symmetry. The thesis will provide a full classification theorem for the very stable parabolic Higgs bundles of type (1, … , 1).

Itwill also present a parametrisation theorem for the fullfixed point locus of the moduli space of parabolic Higgs bundles under the scaling action, with explicit formulae. Such parametrisations have important applications in the application of localisation theorems and in understanding the geometry of the space.