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We study supersymmetric probe M5-branes in the AdS_4 solution that arisesfrom M5-branes wrapped on a hyperbolic 3-manifold M_3. This amounts tointroducing 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 itto wrap a surface embedded in an S^2-fibration over M_3. We find that theprojection of this surface to M_3 can be either a geodesic or a tubular surfacearound a geodesic. These configurations preserve an extra U(1) symmetry, inaddition to the one corresponding to the R-symmetry of the dual 3d N=2 gaugetheory. BPS M2-branes can stretch between M5-branes wrapping geodesics. Weinterpret the addition of probe M5-branes on a closed geodesic in terms ofconformal Dehn surgery.

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