A Lagrangian for self-dual strings

We propose a Lagrangian for the low-energy theory that resides at the (1+1)-dimensional intersection of N semi-infinite M2-branes ending orthogonally on M M5-branes in ${\mathbb R}^{1,2} \times {\mathbb C}^4/{\mathbb Z}_k$ (for arbitrary positive integers N,M,k). We formulate this theory as a 2d boundary theory with explicit ${\cal N}=(1,1)$ supersymmetry that contains two superfields in the bi-fundamental representation of $U(N)\times U(M)$ interacting with the (2+1)-dimensional $U(N)_k\times U(N)_{-k}$ ABJM Chern-Simons-matter theory in the bulk. We postulate that the boundary theory exhibits in the deep infrared supersymmetry enhancement to ${\cal N}=(4,2)$, or ${\cal N}=(4,4)$ depending on the value of k. Arguments in favor of the proposal follow from the study of the open string theory of a U-dual type IIB Hanany-Witten setup. To formulate the bulk-boundary interactions special care is taken to incorporate all the expected boundary effects on gauge symmetry, supersymmetry, and other global symmetries.