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

TL;DR

This paper extends Donaldson theory with surface operators, providing physical insights, novel invariants, and proofs of key topological results using modular invariance and anomaly cancellation.

Contribution

It introduces curved surface operators, offers physical proofs of topological invariants, and generalizes formulas relating ramified and ordinary Donaldson invariants.

Findings

Derived universal formula relating ramified and ordinary Donaldson invariants

Provided physical proofs of the Thom conjecture and other topological results

Computed wall-crossing and blow-up formulas for ramified invariants

Abstract

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.