Integration Over The u-Plane In Donaldson Theory With Surface Operators

We generalize the analysis by Moore and Witten in [arXiv:hep-th/9709193], and consider integration over the u-plane in Donaldson theory with surface operators on a smooth four-manifold X. Several novel aspects will be developed in the process; like a physical interpretation of the "ramified" Donaldson and Seiberg-Witten invariants, and the concept of curved surface operators which are necessarily topological at the outset. Elegant physical proofs -- rooted in R-anomaly cancellations and modular invariance over the u-plane -- of various seminal results in four-dimensional geometric topology obtained by Kronheimer and Mrowka [1,2] -- such as a universal formula relating the "ramified" and ordinary Donaldson invariants, and a generalization of the celebrated Thom conjecture -- will be furnished. Wall-crossing and blow-up formulas of these "ramified" invariants which have not been computed in the mathematical literature before, as well as a generalization and a Seiberg-Witten analog of the universal formula as implied by an electric-magnetic duality of trivially-embedded surface operators in X, will also be presented, among other things.