Discrete gauging in six dimensions

When n M5 branes coincide on an A type singularity, ℂ²/ℤ_k, there is a multitude of tensionless strings which arise in the spectrum. The low energy theory when all M5 branes are separated at the singularity is given by a linear quiver with parameters n and k. The theory has a multitude of phases, as many as partitions of n, each characterized by a different Higgs branch. Each such Higgs branch can be described by a Coulomb branch of a 3d N= 4 quiver. For example, at finite coupling, when all branes are separated, the quiver has a bouquet of n U(1) nodes connected to a single node. There is a natural discrete non Abelian S_n global symmetry which acts in the theory by permuting n identical objects. It acts in particular on the Higgs branch at the above finite coupling phase. It is conjectured that at the coincident point this discrete S_n flavor symmetry is gauged, and at partial coincidence the corresponding subgroup of S_n is gauged. This elegant and simple effect solves several problems which are raised recently on the physics of multiple M5 branes on an A type singularity. Similar results on multitude of phases are concluded for a system of n M5 branes on an A type singularity next to an M9 plane.