The Blowup Formula for Higher Rank Donaldson Invariants

In this thesis, I study the relationship between the higher rank Donaldson invariants of a smooth 4-manifold X and the invariants of its blowup X #CP². This relationship can be expressed in terms of a formal power series in several variables, called the blowup function. I compute the restriction of the blowup function to one of its variables, by solving a special system of ordinary differential equations. I also compute the SU(3) blowup function completely, and show that it is a theta function on a family of genus 2 hyperelliptic Jacobians. Finally, I give a formal argument to explain the appearance of Abelian varieties and theta functions in four dimensional topological field theories. (Copies available exclusively from MIT Libraries, libraries.mit.edu/docs - [email protected])