Exceptional M-brane sigma models and $\eta$-symbols

We develop the M-brane actions proposed in arXiv:1607.04265 by using $\eta$-symbols determined in arXiv:1708.06342. Introducing $\eta$-forms that are defined with the $\eta$-symbols, we present U-duality-covariant M-brane actions which describe the known brane worldvolume theories for M$p$-branes with $p=0,2,5$. We show that the self-duality relation known in the double sigma model is naturally generalized to M-branes. In particular, for an M5-brane, the self-duality relation is nontrivially realized, where the Hodge star operator is defined with the familiar M5-brane metric while the $\eta$-form contains the self-dual 3-form field strength. The action for a Kaluza-Klein monopole is also partially reproduced. Moreover, we explain how to treat type IIB branes in our general formalism. As a demonstration, we reproduce the known action for a $(p,q)$-string.