World-volume Effective Actions of Exotic Five-branes

We construct world-volume effective actions of exotic $5^2_2$-branes in type IIA and IIB string theories. The effective actions are given in fully space-time covariant forms with two Killing vectors associated with background isometries. The effective theories are governed by the six-dimensional $\mathcal{N}=(2,0)$ tensor multiplet and $\mathcal{N}=(1,1)$ vector multiplet, respectively. Performing the S-duality transformation to the $5^2_2$-brane effective action in type IIB string theory, we also work out the world-volume action of the $5^2_3$-brane. We discuss some additional issues relevant to the exotic five-branes in type I and heterotic string theories.