Understand
We continue our program initiated in [arXiv:0912.4261] to consider supersymmetric surface operators in a topologically-twisted N=2 pure 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 Ozsvath-Szabo [2,3] and Taubes [4], will be furnished.
- In particular, we will show that Taubes' groundbreaking and difficult result -- that the ordinary Seiberg-Witten 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" Seiberg-Witten invariants of X, which in certain cases, also involve the instanton and monopole Floer homologies of its three-submanifold.