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

TL;DR

This paper explores supersymmetric surface operators in topologically-twisted N=2 gauge theories to provide physical proofs of key four-manifold invariants and establish new identities among them, linking physics and advanced topology.

Contribution

It offers a novel physical derivation of four-manifold invariants and uncovers new identities among Gromov, Seiberg-Witten, and Floer invariants using supersymmetric surface operators.

Findings

Physical proof of Taubes' theorem relating Seiberg-Witten and Gromov invariants

New identities among Gromov and ramified Seiberg-Witten invariants

Connections between knot homology, instantons, and Floer homologies

Abstract

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.