BPS M5-Branes as Defects for the 3d-3d Correspondence

We study supersymmetric probe M5-branes in the AdS$_{4}$ solution that arises from M5-branes wrapped on a hyperbolic 3-manifold M$_{3}$. This amounts to introducing internal defects within the framework of the 3d-3d correspondence. The BPS condition for a probe M5-brane extending along all of AdS$_{4}$ requires it to wrap a surface embedded in an S$^{2}$-fibration over M$_{3}$. We find that the projection of this surface to M$_{3}$ can be either a geodesic or a tubular surface around a geodesic. These configurations preserve an extra U(1) symmetry, in addition to the one corresponding to the R-symmetry of the dual 3d $ \mathcal{N}=2 $ gauge theory. BPS M2-branes can stretch between M5-branes wrapping geodesics. We interpret the addition of probe M5-branes on a closed geodesic in terms of conformal Dehn surgery