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Notes on the "Ramified" Seiberg-Witten Equations and Invariants

In these notes, we carefully analyze the properties of the "ramified" Seiberg-Witten equations associated with supersymmetric configurations of the Seiberg-Witten abelian gauge theory with surface operators on an oriented closed four-manifold X. We find that in order to have sensible solutions to these equations, only surface operators with certain parameters and embeddings in X, are admissible. In addition, the corresponding "ramified" Seiberg-Witten invariants on X with positive scalar curvature and b^+_2 > 1, vanish, while if X has b^+_2 = 1, there can be wall-crossings whence the invariants will jump. In general, for each of the finite number of basic classes that corresponds to a moduli space of solutions with zero virtual dimension, the perturbed "ramified" Seiberg-Witten invariants on Kahler manifolds will depend - among other parameters associated with the surface operator - on the monopole number "l" and the holonomy parameter "alpha". Nonetheless, the (perturbed) "ramified" and ordinary invariants are found to coincide, albeit up to a sign, in some examples.