Supersymmetric surface operators, four-manifold theory and invariants in various dimensions

Abstract

We continue our program initiated in "Integration over the u-plane in Donaldson theory with surface operators," J. High Energy Phys. 05 (2011), to consider supersymmetric surface operators in a topologically twisted ${\cal N}=2$ pure ${\it SU}(2)$ gauge theory, and apply them to the study of four-manifolds and related invariants. Elegant physical proofs of various seminal theorems in four-manifold theory obtained by Ozsáth and Szabó and Taubes, will be furnished. In particular, we will show that Taubes’ groundbreaking and difficult result — that the ordinary SW invariants are in fact the Gromov invariants which count pseudo-holomorphic curves embedded in a symplectic four-manifold $X$ — nonetheless lends itself to a simple and concrete physical derivation in the presence of “ordinary” surface operators. As an offshoot, we will be led to several interesting and mathematically novel identities among the Gromov and “ramified” SW invariants of $X$, which in certain cases, also involve the instanton and monopole Floer homologies of its three-submanifold. Via these identities, and a physical formulation of the “ramified” Donaldson invariants of four-manifolds with boundaries, we will uncover completely new and economical ways of deriving and understanding various important mathematical results concerning (i) knot homology groups from “ramified” instantons by Kronheimer and Mrowka; and (ii) monopole Floer homology and SW theory on symplectic four-manifolds by Kutluhan–Taubes. Supersymmetry, as well as other physical concepts such as $R$-invariance, electric–magnetic duality, spontaneous gauge symmetry breaking and localization onto supersymmetric configurations in topologically twisted quantum field theories, play a pivotal role in our story.