BPS states in the Omega-background and torus knots

We clarify some issues concerning the central charges saturated by the extended objects in the SUSY $U(1)$ $4d$ gauge theory in the $Ω$-background. The configuration involving the monopole localized at the domain wall is considered in some details. At the rational ratio $\frac{ε_1}{ε_2}=\frac{p}{q}$ 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.

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