BPS states in the Ω-background and torus knots

Abstract

We clarify some issues concerning the central charges saturated by the extended objects in the SUSY $U(1)$ $4d$ gauge theory in the $\Omega$-background. The configuration involving the monopole localized at the domain wall is considered in some details. At the rational ratio $\frac{\epsilon_1}{\epsilon_2}=\frac{p}{q}$ the trajectory of the monopole provides the torus $(p,q)$ knot in the squashed three-sphere. Using the relation between the integrable systems of Calogero type at the rational couplings and the torus knots we interpret this configuration in terms of the auxiliary $2d$ quiver theory or $3d$ theory with nontrivial boundary conditions. This realization can be considered as the AGT-like representation of the torus knot invariants.