Integration over the u-plane in Donaldson theory with surface operators

Abstract

We generalize the analysis by Moore and Witten in [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-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.

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