World-volume Effective Action of Exotic Five-brane in M-theory

We study the world-volume effective action of an exotic five-brane, known as the M-theory 5${}^3$-brane (M5${}^3$-brane) in eleven dimensions. The supermultiplet of the world-volume theory is the $\mathcal{N}=(2,0)$ tensor multiplet in six dimensions. The world-volume action contains three Killing vectors ${\widehat{k}}_{\widehat{I}}{}^M(\widehat{I}=1,2,3)$ associated with the $U(1{)}^3$ isometry. We find the effective T-duality rule for the eleven-dimensional backgrounds that transforms the M5-brane effective action to that of the M5${}^3$-brane. We also show that our action provides the source term for the M5${}^3$-brane geometry in eleven-dimensional supergravity