1006.3313 (Meng-Chwan Tan)

Supersymmetric Surface Operators, Four-Manifold Theory and Invariants in
Various Dimensions    [PDF]

Meng-Chwan Tan

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. 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-Mrowka [5]; and (ii)
monopole Floer homology and Seiberg-Witten theory on symplectic four-manifolds
by Kutluhan-Taubes [4,6]. 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.

View original: http://arxiv.org/abs/1006.3313