  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.
